diff options
| author |  Benjamin Barenblat <bbaren@debian.org> | 2018-12-29 14:31:27 -0500 |
|---|---|---|
| committer | Benjamin Barenblat <bbaren@debian.org> | 2018-12-29 14:31:27 -0500 |
| commit | 9043add656177eeac1491a73d2f3ab92bec0013c (patch) | |
| tree | 2b0092c84bfbf718eca10c81f60b2640dc8cab05 /theories | |
| parent | a4c7f8bd98be2a200489325ff7c5061cf80ab4f3 (diff) | |
Imported Upstream version 8.8.2upstream/8.8.2
Diffstat (limited to 'theories')
460 files changed, 8123 insertions, 19126 deletions
diff --git a/theories/Arith/Arith.v b/theories/Arith/Arith.v index 953f7e4d..1cba8faf 100644 --- a/theories/Arith/Arith.v +++ b/theories/Arith/Arith.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export Arith_base. diff --git a/theories/Arith/Arith_base.v b/theories/Arith/Arith_base.v index b378828e..e3a033a4 100644 --- a/theories/Arith/Arith_base.v +++ b/theories/Arith/Arith_base.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export PeanoNat. diff --git a/theories/Arith/Between.v b/theories/Arith/Between.v index 58d3a2b3..25d84a62 100644 --- a/theories/Arith/Between.v +++ b/theories/Arith/Between.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Le. @@ -16,6 +18,8 @@ Implicit Types k l p q r : nat. Section Between. Variables P Q : nat -> Prop. + (** The [between] type expresses the concept + [forall i: nat, k <= i < l -> P i.]. *) Inductive between k : nat -> Prop := | bet_emp : between k k | bet_S : forall l, between k l -> P l -> between k (S l). @@ -24,7 +28,7 @@ Section Between. Lemma bet_eq : forall k l, l = k -> between k l. Proof. - induction 1; auto with arith. + intros * ->; auto with arith. Qed. Hint Resolve bet_eq: arith. @@ -37,9 +41,7 @@ Section Between. Lemma between_Sk_l : forall k l, between k l -> S k <= l -> between (S k) l. Proof. - intros k l H; induction H as [|l H]. - intros; absurd (S k <= k); auto with arith. - destruct H; auto with arith. + induction 1 as [|* [|]]; auto with arith. Qed. Hint Resolve between_Sk_l: arith. @@ -49,6 +51,8 @@ Section Between. induction 1; auto with arith. Qed. + (** The [exists_between] type expresses the concept + [exists i: nat, k <= i < l /\ Q i]. *) Inductive exists_between k : nat -> Prop := | exists_S : forall l, exists_between k l -> exists_between k (S l) | exists_le : forall l, k <= l -> Q l -> exists_between k (S l). @@ -74,32 +78,32 @@ Section Between. Lemma in_int_intro : forall p q r, p <= r -> r < q -> in_int p q r. Proof. - red; auto with arith. + split; assumption. Qed. Hint Resolve in_int_intro: arith. Lemma in_int_lt : forall p q r, in_int p q r -> p < q. Proof. - induction 1; intros. - apply le_lt_trans with r; auto with arith. + intros * []. + eapply le_lt_trans; eassumption. Qed. Lemma in_int_p_Sq : - forall p q r, in_int p (S q) r -> in_int p q r \/ r = q :>nat. + forall p q r, in_int p (S q) r -> in_int p q r \/ r = q. Proof. - induction 1; intros. - elim (le_lt_or_eq r q); auto with arith. + intros p q r []. + destruct (le_lt_or_eq r q); auto with arith. Qed. Lemma in_int_S : forall p q r, in_int p q r -> in_int p (S q) r. Proof. - induction 1; auto with arith. + intros * []; auto with arith. Qed. Hint Resolve in_int_S: arith. Lemma in_int_Sp_q : forall p q r, in_int (S p) q r -> in_int p q r. Proof. - induction 1; auto with arith. + intros * []; auto with arith. Qed. Hint Immediate in_int_Sp_q: arith. @@ -107,10 +111,9 @@ Section Between. forall k l, between k l -> forall r, in_int k l r -> P r. Proof. induction 1; intros. - absurd (k < k); auto with arith. - apply in_int_lt with r; auto with arith. - elim (in_int_p_Sq k l r); intros; auto with arith. - rewrite H2; trivial with arith. + - absurd (k < k). { auto with arith. } + eapply in_int_lt; eassumption. + - destruct (in_int_p_Sq k l r) as [| ->]; auto with arith. Qed. Lemma in_int_between : @@ -120,17 +123,17 @@ Section Between. Qed. Lemma exists_in_int : - forall k l, exists_between k l -> exists2 m : nat, in_int k l m & Q m. + forall k l, exists_between k l -> exists2 m : nat, in_int k l m & Q m. Proof. - induction 1. - case IHexists_between; intros p inp Qp; exists p; auto with arith. - exists l; auto with arith. + induction 1 as [* ? (p, ?, ?)|]. + - exists p; auto with arith. + - exists l; auto with arith. Qed. Lemma in_int_exists : forall k l r, in_int k l r -> Q r -> exists_between k l. Proof. - destruct 1; intros. - elim H0; auto with arith. + intros * (?, lt_r_l) ?. + induction lt_r_l; auto with arith. Qed. Lemma between_or_exists : @@ -139,9 +142,11 @@ Section Between. (forall n:nat, in_int k l n -> P n \/ Q n) -> between k l \/ exists_between k l. Proof. - induction 1; intros; auto with arith. - elim IHle; intro; auto with arith. - elim (H0 m); auto with arith. + induction 1 as [|m ? IHle]. + - auto with arith. + - intros P_or_Q. + destruct IHle; auto with arith. + destruct (P_or_Q m); auto with arith. Qed. Lemma between_not_exists : @@ -150,14 +155,14 @@ Section Between. (forall n:nat, in_int k l n -> P n -> ~ Q n) -> ~ exists_between k l. Proof. induction 1; red; intros. - absurd (k < k); auto with arith. - absurd (Q l); auto with arith. - elim (exists_in_int k (S l)); auto with arith; intros l' inl' Ql'. - replace l with l'; auto with arith. - elim inl'; intros. - elim (le_lt_or_eq l' l); auto with arith; intros. - absurd (exists_between k l); auto with arith. - apply in_int_exists with l'; auto with arith. + - absurd (k < k); auto with arith. + - absurd (Q l). { auto with arith. } + destruct (exists_in_int k (S l)) as (l',[],?). + + auto with arith. + + replace l with l'. { trivial. } + destruct (le_lt_or_eq l' l); auto with arith. + absurd (exists_between k l). { auto with arith. } + eapply in_int_exists; eauto with arith. Qed. Inductive P_nth (init:nat) : nat -> nat -> Prop := @@ -168,15 +173,16 @@ Section Between. Lemma nth_le : forall (init:nat) l (n:nat), P_nth init l n -> init <= l. Proof. - induction 1; intros; auto with arith. - apply le_trans with (S k); auto with arith. + induction 1. + - auto with arith. + - eapply le_trans; eauto with arith. Qed. Definition eventually (n:nat) := exists2 k : nat, k <= n & Q k. Lemma event_O : eventually 0 -> Q 0. Proof. - induction 1; intros. + intros (x, ?, ?). replace 0 with x; auto with arith. Qed. diff --git a/theories/Arith/Bool_nat.v b/theories/Arith/Bool_nat.v index 602555b6..d892542e 100644 --- a/theories/Arith/Bool_nat.v +++ b/theories/Arith/Bool_nat.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export Compare_dec. diff --git a/theories/Arith/Compare.v b/theories/Arith/Compare.v index 610e9a69..6778d6a0 100644 --- a/theories/Arith/Compare.v +++ b/theories/Arith/Compare.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Equality is decidable on [nat] *) diff --git a/theories/Arith/Compare_dec.v b/theories/Arith/Compare_dec.v index 976507b5..713aef85 100644 --- a/theories/Arith/Compare_dec.v +++ b/theories/Arith/Compare_dec.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Le Lt Gt Decidable PeanoNat. @@ -133,11 +135,11 @@ Qed. See now [Nat.compare] and its properties. In scope [nat_scope], the notation for [Nat.compare] is "?=" *) -Notation nat_compare := Nat.compare (compat "8.4"). +Notation nat_compare := Nat.compare (compat "8.6"). -Notation nat_compare_spec := Nat.compare_spec (compat "8.4"). -Notation nat_compare_eq_iff := Nat.compare_eq_iff (compat "8.4"). -Notation nat_compare_S := Nat.compare_succ (compat "8.4"). +Notation nat_compare_spec := Nat.compare_spec (compat "8.6"). +Notation nat_compare_eq_iff := Nat.compare_eq_iff (compat "8.6"). +Notation nat_compare_S := Nat.compare_succ (only parsing). Lemma nat_compare_lt n m : n<m <-> (n ?= m) = Lt. Proof. @@ -198,9 +200,9 @@ Qed. See now [Nat.leb] and its properties. In scope [nat_scope], the notation for [Nat.leb] is "<=?" *) -Notation leb := Nat.leb (compat "8.4"). +Notation leb := Nat.leb (only parsing). -Notation leb_iff := Nat.leb_le (compat "8.4"). +Notation leb_iff := Nat.leb_le (only parsing). Lemma leb_iff_conv m n : (n <=? m) = false <-> m < n. Proof. diff --git a/theories/Arith/Div2.v b/theories/Arith/Div2.v index 016cb85e..a5e45783 100644 --- a/theories/Arith/Div2.v +++ b/theories/Arith/Div2.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Nota : this file is OBSOLETE, and left only for compatibility. @@ -18,7 +20,7 @@ Implicit Type n : nat. (** Here we define [n/2] and prove some of its properties *) -Notation div2 := Nat.div2 (compat "8.4"). +Notation div2 := Nat.div2 (only parsing). (** Since [div2] is recursively defined on [0], [1] and [(S (S n))], it is useful to prove the corresponding induction principle *) @@ -28,7 +30,7 @@ Lemma ind_0_1_SS : P 0 -> P 1 -> (forall n, P n -> P (S (S n))) -> forall n, P n. Proof. intros P H0 H1 H2. - fix 1. + fix ind_0_1_SS 1. destruct n as [|[|n]]. - exact H0. - exact H1. @@ -84,7 +86,7 @@ Qed. (** Properties related to the double ([2n]) *) -Notation double := Nat.double (compat "8.4"). +Notation double := Nat.double (only parsing). Hint Unfold double Nat.double: arith. @@ -103,7 +105,7 @@ Hint Resolve double_S: arith. Lemma even_odd_double n : (even n <-> n = double (div2 n)) /\ (odd n <-> n = S (double (div2 n))). Proof. - revert n. fix 1. destruct n as [|[|n]]. + revert n. fix even_odd_double 1. destruct n as [|[|n]]. - (* n = 0 *) split; split; auto with arith. inversion 1. - (* n = 1 *) diff --git a/theories/Arith/EqNat.v b/theories/Arith/EqNat.v index f998c19f..4b51dfc0 100644 --- a/theories/Arith/EqNat.v +++ b/theories/Arith/EqNat.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import PeanoNat. @@ -69,10 +71,10 @@ Defined. We reuse the one already defined in module [Nat]. In scope [nat_scope], the notation "=?" can be used. *) -Notation beq_nat := Nat.eqb (compat "8.4"). +Notation beq_nat := Nat.eqb (only parsing). -Notation beq_nat_true_iff := Nat.eqb_eq (compat "8.4"). -Notation beq_nat_false_iff := Nat.eqb_neq (compat "8.4"). +Notation beq_nat_true_iff := Nat.eqb_eq (only parsing). +Notation beq_nat_false_iff := Nat.eqb_neq (only parsing). Lemma beq_nat_refl n : true = (n =? n). Proof. diff --git a/theories/Arith/Euclid.v b/theories/Arith/Euclid.v index fbe98d17..29f4d3e2 100644 --- a/theories/Arith/Euclid.v +++ b/theories/Arith/Euclid.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Mult. diff --git a/theories/Arith/Even.v b/theories/Arith/Even.v index 3c8c250a..a1d0e9fc 100644 --- a/theories/Arith/Even.v +++ b/theories/Arith/Even.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Nota : this file is OBSOLETE, and left only for compatibility. @@ -36,7 +38,7 @@ Hint Constructors odd: arith. Lemma even_equiv : forall n, even n <-> Nat.Even n. Proof. - fix 1. + fix even_equiv 1. destruct n as [|[|n]]; simpl. - split; [now exists 0 | constructor]. - split. @@ -50,7 +52,7 @@ Qed. Lemma odd_equiv : forall n, odd n <-> Nat.Odd n. Proof. - fix 1. + fix odd_equiv 1. destruct n as [|[|n]]; simpl. - split. + inversion_clear 1. diff --git a/theories/Arith/Factorial.v b/theories/Arith/Factorial.v index b119bb00..22f586d7 100644 --- a/theories/Arith/Factorial.v +++ b/theories/Arith/Factorial.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import PeanoNat Plus Mult Lt. diff --git a/theories/Arith/Gt.v b/theories/Arith/Gt.v index 67c94fdf..52ecf131 100644 --- a/theories/Arith/Gt.v +++ b/theories/Arith/Gt.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Theorems about [gt] in [nat]. diff --git a/theories/Arith/Le.v b/theories/Arith/Le.v index 0fbcec57..69626cc1 100644 --- a/theories/Arith/Le.v +++ b/theories/Arith/Le.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Order on natural numbers. @@ -26,17 +28,17 @@ Local Open Scope nat_scope. (** * [le] is an order on [nat] *) -Notation le_refl := Nat.le_refl (compat "8.4"). -Notation le_trans := Nat.le_trans (compat "8.4"). -Notation le_antisym := Nat.le_antisymm (compat "8.4"). +Notation le_refl := Nat.le_refl (only parsing). +Notation le_trans := Nat.le_trans (only parsing). +Notation le_antisym := Nat.le_antisymm (only parsing). Hint Resolve le_trans: arith. Hint Immediate le_antisym: arith. (** * Properties of [le] w.r.t 0 *) -Notation le_0_n := Nat.le_0_l (compat "8.4"). (* 0 <= n *) -Notation le_Sn_0 := Nat.nle_succ_0 (compat "8.4"). (* ~ S n <= 0 *) +Notation le_0_n := Nat.le_0_l (only parsing). (* 0 <= n *) +Notation le_Sn_0 := Nat.nle_succ_0 (only parsing). (* ~ S n <= 0 *) Lemma le_n_0_eq n : n <= 0 -> 0 = n. Proof. @@ -53,8 +55,8 @@ Proof Peano.le_n_S. Theorem le_S_n : forall n m, S n <= S m -> n <= m. Proof Peano.le_S_n. -Notation le_n_Sn := Nat.le_succ_diag_r (compat "8.4"). (* n <= S n *) -Notation le_Sn_n := Nat.nle_succ_diag_l (compat "8.4"). (* ~ S n <= n *) +Notation le_n_Sn := Nat.le_succ_diag_r (only parsing). (* n <= S n *) +Notation le_Sn_n := Nat.nle_succ_diag_l (only parsing). (* ~ S n <= n *) Theorem le_Sn_le : forall n m, S n <= m -> n <= m. Proof Nat.lt_le_incl. @@ -65,8 +67,8 @@ Hint Immediate le_n_0_eq le_Sn_le le_S_n : arith. (** * Properties of [le] w.r.t predecessor *) -Notation le_pred_n := Nat.le_pred_l (compat "8.4"). (* pred n <= n *) -Notation le_pred := Nat.pred_le_mono (compat "8.4"). (* n<=m -> pred n <= pred m *) +Notation le_pred_n := Nat.le_pred_l (only parsing). (* pred n <= n *) +Notation le_pred := Nat.pred_le_mono (only parsing). (* n<=m -> pred n <= pred m *) Hint Resolve le_pred_n: arith. diff --git a/theories/Arith/Lt.v b/theories/Arith/Lt.v index bfc2b91a..0c7515c6 100644 --- a/theories/Arith/Lt.v +++ b/theories/Arith/Lt.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Strict order on natural numbers. @@ -23,7 +25,7 @@ Local Open Scope nat_scope. (** * Irreflexivity *) -Notation lt_irrefl := Nat.lt_irrefl (compat "8.4"). (* ~ x < x *) +Notation lt_irrefl := Nat.lt_irrefl (only parsing). (* ~ x < x *) Hint Resolve lt_irrefl: arith. @@ -62,12 +64,12 @@ Hint Immediate le_not_lt lt_not_le: arith. (** * Asymmetry *) -Notation lt_asym := Nat.lt_asymm (compat "8.4"). (* n<m -> ~m<n *) +Notation lt_asym := Nat.lt_asymm (only parsing). (* n<m -> ~m<n *) (** * Order and 0 *) -Notation lt_0_Sn := Nat.lt_0_succ (compat "8.4"). (* 0 < S n *) -Notation lt_n_0 := Nat.nlt_0_r (compat "8.4"). (* ~ n < 0 *) +Notation lt_0_Sn := Nat.lt_0_succ (only parsing). (* 0 < S n *) +Notation lt_n_0 := Nat.nlt_0_r (only parsing). (* ~ n < 0 *) Theorem neq_0_lt n : 0 <> n -> 0 < n. Proof. @@ -84,8 +86,8 @@ Hint Immediate neq_0_lt lt_0_neq: arith. (** * Order and successor *) -Notation lt_n_Sn := Nat.lt_succ_diag_r (compat "8.4"). (* n < S n *) -Notation lt_S := Nat.lt_lt_succ_r (compat "8.4"). (* n < m -> n < S m *) +Notation lt_n_Sn := Nat.lt_succ_diag_r (only parsing). (* n < S n *) +Notation lt_S := Nat.lt_lt_succ_r (only parsing). (* n < m -> n < S m *) Theorem lt_n_S n m : n < m -> S n < S m. Proof. @@ -107,6 +109,11 @@ Proof. intros. symmetry. now apply Nat.lt_succ_pred with m. Qed. +Lemma S_pred_pos n: O < n -> n = S (pred n). +Proof. + apply S_pred. +Qed. + Lemma lt_pred n m : S n < m -> n < pred m. Proof. apply Nat.lt_succ_lt_pred. @@ -122,28 +129,28 @@ Hint Resolve lt_pred_n_n: arith. (** * Transitivity properties *) -Notation lt_trans := Nat.lt_trans (compat "8.4"). -Notation lt_le_trans := Nat.lt_le_trans (compat "8.4"). -Notation le_lt_trans := Nat.le_lt_trans (compat "8.4"). +Notation lt_trans := Nat.lt_trans (only parsing). +Notation lt_le_trans := Nat.lt_le_trans (only parsing). +Notation le_lt_trans := Nat.le_lt_trans (only parsing). Hint Resolve lt_trans lt_le_trans le_lt_trans: arith. (** * Large = strict or equal *) -Notation le_lt_or_eq_iff := Nat.lt_eq_cases (compat "8.4"). +Notation le_lt_or_eq_iff := Nat.lt_eq_cases (only parsing). Theorem le_lt_or_eq n m : n <= m -> n < m \/ n = m. Proof. apply Nat.lt_eq_cases. Qed. -Notation lt_le_weak := Nat.lt_le_incl (compat "8.4"). +Notation lt_le_weak := Nat.lt_le_incl (only parsing). Hint Immediate lt_le_weak: arith. (** * Dichotomy *) -Notation le_or_lt := Nat.le_gt_cases (compat "8.4"). (* n <= m \/ m < n *) +Notation le_or_lt := Nat.le_gt_cases (only parsing). (* n <= m \/ m < n *) Theorem nat_total_order n m : n <> m -> n < m \/ m < n. Proof. diff --git a/theories/Arith/Max.v b/theories/Arith/Max.v index 49152549..1727fa37 100644 --- a/theories/Arith/Max.v +++ b/theories/Arith/Max.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** THIS FILE IS DEPRECATED. Use [PeanoNat.Nat] instead. *) diff --git a/theories/Arith/Min.v b/theories/Arith/Min.v index 38e59b7b..fcf0b14c 100644 --- a/theories/Arith/Min.v +++ b/theories/Arith/Min.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** THIS FILE IS DEPRECATED. Use [PeanoNat.Nat] instead. *) diff --git a/theories/Arith/Minus.v b/theories/Arith/Minus.v index 1fc8f790..3bf6cd95 100644 --- a/theories/Arith/Minus.v +++ b/theories/Arith/Minus.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Properties of subtraction between natural numbers. @@ -46,7 +48,7 @@ Qed. (** * Diagonal *) -Notation minus_diag := Nat.sub_diag (compat "8.4"). (* n - n = 0 *) +Notation minus_diag := Nat.sub_diag (only parsing). (* n - n = 0 *) Lemma minus_diag_reverse n : 0 = n - n. Proof. @@ -87,13 +89,13 @@ Qed. (** * Relation with order *) Notation minus_le_compat_r := - Nat.sub_le_mono_r (compat "8.4"). (* n <= m -> n - p <= m - p. *) + Nat.sub_le_mono_r (only parsing). (* n <= m -> n - p <= m - p. *) Notation minus_le_compat_l := - Nat.sub_le_mono_l (compat "8.4"). (* n <= m -> p - m <= p - n. *) + Nat.sub_le_mono_l (only parsing). (* n <= m -> p - m <= p - n. *) -Notation le_minus := Nat.le_sub_l (compat "8.4"). (* n - m <= n *) -Notation lt_minus := Nat.sub_lt (compat "8.4"). (* m <= n -> 0 < m -> n-m < n *) +Notation le_minus := Nat.le_sub_l (only parsing). (* n - m <= n *) +Notation lt_minus := Nat.sub_lt (only parsing). (* m <= n -> 0 < m -> n-m < n *) Lemma lt_O_minus_lt n m : 0 < n - m -> m < n. Proof. diff --git a/theories/Arith/Mult.v b/theories/Arith/Mult.v index a173efc1..4f4aa183 100644 --- a/theories/Arith/Mult.v +++ b/theories/Arith/Mult.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Properties of multiplication. @@ -23,35 +25,35 @@ Local Open Scope nat_scope. (** ** Zero property *) -Notation mult_0_l := Nat.mul_0_l (compat "8.4"). (* 0 * n = 0 *) -Notation mult_0_r := Nat.mul_0_r (compat "8.4"). (* n * 0 = 0 *) +Notation mult_0_l := Nat.mul_0_l (only parsing). (* 0 * n = 0 *) +Notation mult_0_r := Nat.mul_0_r (only parsing). (* n * 0 = 0 *) (** ** 1 is neutral *) -Notation mult_1_l := Nat.mul_1_l (compat "8.4"). (* 1 * n = n *) -Notation mult_1_r := Nat.mul_1_r (compat "8.4"). (* n * 1 = n *) +Notation mult_1_l := Nat.mul_1_l (only parsing). (* 1 * n = n *) +Notation mult_1_r := Nat.mul_1_r (only parsing). (* n * 1 = n *) Hint Resolve mult_1_l mult_1_r: arith. (** ** Commutativity *) -Notation mult_comm := Nat.mul_comm (compat "8.4"). (* n * m = m * n *) +Notation mult_comm := Nat.mul_comm (only parsing). (* n * m = m * n *) Hint Resolve mult_comm: arith. (** ** Distributivity *) Notation mult_plus_distr_r := - Nat.mul_add_distr_r (compat "8.4"). (* (n+m)*p = n*p + m*p *) + Nat.mul_add_distr_r (only parsing). (* (n+m)*p = n*p + m*p *) Notation mult_plus_distr_l := - Nat.mul_add_distr_l (compat "8.4"). (* n*(m+p) = n*m + n*p *) + Nat.mul_add_distr_l (only parsing). (* n*(m+p) = n*m + n*p *) Notation mult_minus_distr_r := - Nat.mul_sub_distr_r (compat "8.4"). (* (n-m)*p = n*p - m*p *) + Nat.mul_sub_distr_r (only parsing). (* (n-m)*p = n*p - m*p *) Notation mult_minus_distr_l := - Nat.mul_sub_distr_l (compat "8.4"). (* n*(m-p) = n*m - n*p *) + Nat.mul_sub_distr_l (only parsing). (* n*(m-p) = n*m - n*p *) Hint Resolve mult_plus_distr_r: arith. Hint Resolve mult_minus_distr_r: arith. @@ -59,7 +61,7 @@ Hint Resolve mult_minus_distr_l: arith. (** ** Associativity *) -Notation mult_assoc := Nat.mul_assoc (compat "8.4"). (* n*(m*p)=n*m*p *) +Notation mult_assoc := Nat.mul_assoc (only parsing). (* n*(m*p)=n*m*p *) Lemma mult_assoc_reverse n m p : n * m * p = n * (m * p). Proof. @@ -83,8 +85,8 @@ Qed. (** ** Multiplication and successor *) -Notation mult_succ_l := Nat.mul_succ_l (compat "8.4"). (* S n * m = n * m + m *) -Notation mult_succ_r := Nat.mul_succ_r (compat "8.4"). (* n * S m = n * m + n *) +Notation mult_succ_l := Nat.mul_succ_l (only parsing). (* S n * m = n * m + m *) +Notation mult_succ_r := Nat.mul_succ_r (only parsing). (* n * S m = n * m + n *) (** * Compatibility with orders *) diff --git a/theories/Arith/PeanoNat.v b/theories/Arith/PeanoNat.v index e3240bb7..bc58995f 100644 --- a/theories/Arith/PeanoNat.v +++ b/theories/Arith/PeanoNat.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) @@ -313,7 +315,7 @@ Import Private_Parity. Lemma even_spec : forall n, even n = true <-> Even n. Proof. - fix 1. + fix even_spec 1. destruct n as [|[|n]]; simpl. - split; [ now exists 0 | trivial ]. - split; [ discriminate | intro H; elim (Even_1 H) ]. @@ -323,7 +325,7 @@ Qed. Lemma odd_spec : forall n, odd n = true <-> Odd n. Proof. unfold odd. - fix 1. + fix odd_spec 1. destruct n as [|[|n]]; simpl. - split; [ discriminate | intro H; elim (Odd_0 H) ]. - split; [ now exists 0 | trivial ]. @@ -471,7 +473,7 @@ Notation "( x | y )" := (divide x y) (at level 0) : nat_scope. Lemma gcd_divide : forall a b, (gcd a b | a) /\ (gcd a b | b). Proof. - fix 1. + fix gcd_divide 1. intros [|a] b; simpl. split. now exists 0. @@ -500,7 +502,7 @@ Qed. Lemma gcd_greatest : forall a b c, (c|a) -> (c|b) -> (c|gcd a b). Proof. - fix 1. + fix gcd_greatest 1. intros [|a] b; simpl; auto. fold (b mod (S a)). intros c H H'. apply gcd_greatest; auto. @@ -534,7 +536,7 @@ Qed. Lemma le_div2 n : div2 (S n) <= n. Proof. revert n. - fix 1. + fix le_div2 1. destruct n; simpl; trivial. apply lt_succ_r. destruct n; [simpl|]; trivial. now constructor. Qed. @@ -724,6 +726,26 @@ Definition shiftr_spec a n m (_:0<=m) := shiftr_specif a n m. Include NExtraProp. +(** Properties of tail-recursive addition and multiplication *) + +Lemma tail_add_spec n m : tail_add n m = n + m. +Proof. + revert m. induction n as [|n IH]; simpl; trivial. + intros. now rewrite IH, add_succ_r. +Qed. + +Lemma tail_addmul_spec r n m : tail_addmul r n m = r + n * m. +Proof. + revert m r. induction n as [| n IH]; simpl; trivial. + intros. rewrite IH, tail_add_spec. + rewrite add_assoc. f_equal. apply add_comm. +Qed. + +Lemma tail_mul_spec n m : tail_mul n m = n * m. +Proof. + unfold tail_mul. now rewrite tail_addmul_spec. +Qed. + End Nat. (** Re-export notations that should be available even when diff --git a/theories/Arith/Peano_dec.v b/theories/Arith/Peano_dec.v index f8020a50..9a24c804 100644 --- a/theories/Arith/Peano_dec.v +++ b/theories/Arith/Peano_dec.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Decidable PeanoNat. @@ -19,7 +21,7 @@ Proof. - left; exists n; auto. Defined. -Notation eq_nat_dec := Nat.eq_dec (compat "8.4"). +Notation eq_nat_dec := Nat.eq_dec (only parsing). Hint Resolve O_or_S eq_nat_dec: arith. @@ -57,4 +59,4 @@ now rewrite H. Qed. (** For compatibility *) -Require Import Le Lt.
\ No newline at end of file +Require Import Le Lt. diff --git a/theories/Arith/Plus.v b/theories/Arith/Plus.v index 600e5e51..b8297c2d 100644 --- a/theories/Arith/Plus.v +++ b/theories/Arith/Plus.v @@ -1,9 +1,11 @@ - (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Properties of addition. @@ -27,12 +29,12 @@ Local Open Scope nat_scope. (** * Neutrality of 0, commutativity, associativity *) -Notation plus_0_l := Nat.add_0_l (compat "8.4"). -Notation plus_0_r := Nat.add_0_r (compat "8.4"). -Notation plus_comm := Nat.add_comm (compat "8.4"). -Notation plus_assoc := Nat.add_assoc (compat "8.4"). +Notation plus_0_l := Nat.add_0_l (only parsing). +Notation plus_0_r := Nat.add_0_r (only parsing). +Notation plus_comm := Nat.add_comm (only parsing). +Notation plus_assoc := Nat.add_assoc (only parsing). -Notation plus_permute := Nat.add_shuffle3 (compat "8.4"). +Notation plus_permute := Nat.add_shuffle3 (only parsing). Definition plus_Snm_nSm : forall n m, S n + m = n + S m := Peano.plus_n_Sm. @@ -138,7 +140,7 @@ Defined. (** * Derived properties *) -Notation plus_permute_2_in_4 := Nat.add_shuffle1 (compat "8.4"). +Notation plus_permute_2_in_4 := Nat.add_shuffle1 (only parsing). (** * Tail-recursive plus *) diff --git a/theories/Arith/Wf_nat.v b/theories/Arith/Wf_nat.v index 94bbd50a..b0228898 100644 --- a/theories/Arith/Wf_nat.v +++ b/theories/Arith/Wf_nat.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Well-founded relations and natural numbers *) diff --git a/theories/Arith/vo.itarget b/theories/Arith/vo.itarget deleted file mode 100644 index 0b3d31e9..00000000 --- a/theories/Arith/vo.itarget +++ /dev/null @@ -1,22 +0,0 @@ -PeanoNat.vo -Arith_base.vo -Arith.vo -Between.vo -Bool_nat.vo -Compare_dec.vo -Compare.vo -Div2.vo -EqNat.vo -Euclid.vo -Even.vo -Factorial.vo -Gt.vo -Le.vo -Lt.vo -Max.vo -Minus.vo -Min.vo -Mult.vo -Peano_dec.vo -Plus.vo -Wf_nat.vo diff --git a/theories/Bool/Bool.v b/theories/Bool/Bool.v index 06096c66..edf78ed5 100644 --- a/theories/Bool/Bool.v +++ b/theories/Bool/Bool.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** The type [bool] is defined in the prelude as diff --git a/theories/Bool/BoolEq.v b/theories/Bool/BoolEq.v index aec4f0bb..106a79e8 100644 --- a/theories/Bool/BoolEq.v +++ b/theories/Bool/BoolEq.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Cuihtlauac Alvarado - octobre 2000 *) diff --git a/theories/Bool/Bvector.v b/theories/Bool/Bvector.v index 09f643c8..aecdb59d 100644 --- a/theories/Bool/Bvector.v +++ b/theories/Bool/Bvector.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Bit vectors. Contribution by Jean Duprat (ENS Lyon). *) diff --git a/theories/Bool/DecBool.v b/theories/Bool/DecBool.v index 501366ce..f84aed19 100644 --- a/theories/Bool/DecBool.v +++ b/theories/Bool/DecBool.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Set Implicit Arguments. diff --git a/theories/Bool/IfProp.v b/theories/Bool/IfProp.v index 4257b4bc..6f99ea1d 100644 --- a/theories/Bool/IfProp.v +++ b/theories/Bool/IfProp.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Bool. diff --git a/theories/Bool/Sumbool.v b/theories/Bool/Sumbool.v index fd7f42e2..5333c741 100644 --- a/theories/Bool/Sumbool.v +++ b/theories/Bool/Sumbool.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Here are collected some results about the type sumbool (see INIT/Specif.v) diff --git a/theories/Bool/Zerob.v b/theories/Bool/Zerob.v index 16e47ac5..2420c0fd 100644 --- a/theories/Bool/Zerob.v +++ b/theories/Bool/Zerob.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Arith. diff --git a/theories/Bool/vo.itarget b/theories/Bool/vo.itarget deleted file mode 100644 index 24cbf4ed..00000000 --- a/theories/Bool/vo.itarget +++ /dev/null @@ -1,7 +0,0 @@ -BoolEq.vo -Bool.vo -Bvector.vo -DecBool.vo -IfProp.vo -Sumbool.vo -Zerob.vo diff --git a/theories/Classes/CEquivalence.v b/theories/Classes/CEquivalence.v index bd01de47..03e611f5 100644 --- a/theories/Classes/CEquivalence.v +++ b/theories/Classes/CEquivalence.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Typeclass-based setoids. Definitions on [Equivalence]. diff --git a/theories/Classes/CMorphisms.v b/theories/Classes/CMorphisms.v index 1cfca416..09b35ca7 100644 --- a/theories/Classes/CMorphisms.v +++ b/theories/Classes/CMorphisms.v @@ -1,10 +1,12 @@ -(* -*- coding: utf-8 -*- *) +(* -*- coding: utf-8; coq-prog-args: ("-coqlib" "../.." "-R" ".." "Coq" "-top" "Coq.Classes.CMorphisms") -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Typeclass-based morphism definition and standard, minimal instances diff --git a/theories/Classes/CRelationClasses.v b/theories/Classes/CRelationClasses.v index 3d7ef01f..bc821532 100644 --- a/theories/Classes/CRelationClasses.v +++ b/theories/Classes/CRelationClasses.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Typeclass-based relations, tactics and standard instances @@ -305,9 +307,7 @@ Section Binary. fun x y => sum (R x y) (R' x y). (** Relation equivalence is an equivalence, and subrelation defines a partial order. *) - - Set Automatic Introduction. - + Global Instance relation_equivalence_equivalence : Equivalence relation_equivalence. Proof. split; red; unfold relation_equivalence, iffT. firstorder. diff --git a/theories/Classes/DecidableClass.v b/theories/Classes/DecidableClass.v index a3b7e311..bffad2b3 100644 --- a/theories/Classes/DecidableClass.v +++ b/theories/Classes/DecidableClass.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * A typeclass to ease the handling of decidable properties. *) diff --git a/theories/Classes/EquivDec.v b/theories/Classes/EquivDec.v index 52313735..e9a9d6af 100644 --- a/theories/Classes/EquivDec.v +++ b/theories/Classes/EquivDec.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Decidable equivalences. diff --git a/theories/Classes/Equivalence.v b/theories/Classes/Equivalence.v index 80b0b7e4..5217aedb 100644 --- a/theories/Classes/Equivalence.v +++ b/theories/Classes/Equivalence.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Typeclass-based setoids. Definitions on [Equivalence]. diff --git a/theories/Classes/Init.v b/theories/Classes/Init.v index c13b36fd..8a04206b 100644 --- a/theories/Classes/Init.v +++ b/theories/Classes/Init.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Initialization code for typeclasses, setting up the default tactic diff --git a/theories/Classes/Morphisms.v b/theories/Classes/Morphisms.v index 06511ace..1858ba76 100644 --- a/theories/Classes/Morphisms.v +++ b/theories/Classes/Morphisms.v @@ -1,10 +1,12 @@ -(* -*- coding: utf-8 -*- *) +(* -*- coding: utf-8; coq-prog-args: ("-coqlib" "../.." "-R" ".." "Coq" "-top" "Coq.Classes.Morphisms") -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Typeclass-based morphism definition and standard, minimal instances diff --git a/theories/Classes/Morphisms_Prop.v b/theories/Classes/Morphisms_Prop.v index 15900177..8881fda5 100644 --- a/theories/Classes/Morphisms_Prop.v +++ b/theories/Classes/Morphisms_Prop.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * [Proper] instances for propositional connectives. diff --git a/theories/Classes/Morphisms_Relations.v b/theories/Classes/Morphisms_Relations.v index 6048fe06..a3e57501 100644 --- a/theories/Classes/Morphisms_Relations.v +++ b/theories/Classes/Morphisms_Relations.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Morphism instances for relations. diff --git a/theories/Classes/RelationClasses.v b/theories/Classes/RelationClasses.v index 11c204da..2ab3af20 100644 --- a/theories/Classes/RelationClasses.v +++ b/theories/Classes/RelationClasses.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Typeclass-based relations, tactics and standard instances @@ -433,9 +435,7 @@ Section Binary. @predicate_union (A::A::Tnil) R R'. (** Relation equivalence is an equivalence, and subrelation defines a partial order. *) - - Set Automatic Introduction. - + Global Instance relation_equivalence_equivalence : Equivalence relation_equivalence. Proof. exact (@predicate_equivalence_equivalence (A::A::Tnil)). Qed. diff --git a/theories/Classes/RelationPairs.v b/theories/Classes/RelationPairs.v index 8d1c4982..7af2b0fc 100644 --- a/theories/Classes/RelationPairs.v +++ b/theories/Classes/RelationPairs.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Relations over pairs *) diff --git a/theories/Classes/SetoidClass.v b/theories/Classes/SetoidClass.v index 4b133a4d..2673a119 100644 --- a/theories/Classes/SetoidClass.v +++ b/theories/Classes/SetoidClass.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Typeclass-based setoids, tactics and standard instances. diff --git a/theories/Classes/SetoidDec.v b/theories/Classes/SetoidDec.v index 7201c0b1..f6b240bf 100644 --- a/theories/Classes/SetoidDec.v +++ b/theories/Classes/SetoidDec.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Decidable setoid equality theory. diff --git a/theories/Classes/SetoidTactics.v b/theories/Classes/SetoidTactics.v index 190397ae..3fab3c5a 100644 --- a/theories/Classes/SetoidTactics.v +++ b/theories/Classes/SetoidTactics.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Tactics for typeclass-based setoids. diff --git a/theories/Classes/vo.itarget b/theories/Classes/vo.itarget deleted file mode 100644 index 18147f2a..00000000 --- a/theories/Classes/vo.itarget +++ /dev/null @@ -1,15 +0,0 @@ -DecidableClass.vo -Equivalence.vo -EquivDec.vo -Init.vo -Morphisms_Prop.vo -Morphisms_Relations.vo -Morphisms.vo -RelationClasses.vo -SetoidClass.vo -SetoidDec.vo -SetoidTactics.vo -RelationPairs.vo -CRelationClasses.vo -CMorphisms.vo -CEquivalence.vo diff --git a/theories/Compat/AdmitAxiom.v b/theories/Compat/AdmitAxiom.v index 4d9f55cf..f6b751bd 100644 --- a/theories/Compat/AdmitAxiom.v +++ b/theories/Compat/AdmitAxiom.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Compatibility file for making the admit tactic act similar to Coq v8.4. In diff --git a/theories/Compat/Coq84.v b/theories/Compat/Coq84.v deleted file mode 100644 index a3e23f91..00000000 --- a/theories/Compat/Coq84.v +++ /dev/null @@ -1,79 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -(** Compatibility file for making Coq act similar to Coq v8.4 *) - -(** Any compatibility changes to make future versions of Coq behave like Coq 8.5 are likely needed to make them behave like Coq 8.4. *) -Require Export Coq.Compat.Coq85. - -(** See https://coq.inria.fr/bugs/show_bug.cgi?id=4319 for updates *) -(** This is required in Coq 8.5 to use the [omega] tactic; in Coq 8.4, it's automatically available. But ZArith_base puts infix ~ at level 7, and we don't want that, so we don't [Import] it. *) -Require Coq.omega.Omega. -Ltac omega := Coq.omega.Omega.omega. - -(** The number of arguments given in [match] statements has changed from 8.4 to 8.5. *) -Global Set Asymmetric Patterns. - -(** The automatic elimination schemes for records were dropped by default in 8.5. This restores the default behavior of Coq 8.4. *) -Global Set Nonrecursive Elimination Schemes. - -(** See bug 3545 *) -Global Set Universal Lemma Under Conjunction. - -(** Feature introduced in 8.5, disabled by default and configurable since 8.6. *) -Global Unset Refolding Reduction. - -(** In 8.4, [constructor (tac)] allowed backtracking across the use of [constructor]; it has been subsumed by [constructor; tac]. *) -Ltac constructor_84_n n := constructor n. -Ltac constructor_84_tac tac := once (constructor; tac). - -Tactic Notation "constructor" := Coq.Init.Notations.constructor. -Tactic Notation "constructor" int_or_var(n) := constructor_84_n n. -Tactic Notation "constructor" "(" tactic(tac) ")" := constructor_84_tac tac. - -(** In 8.4, [econstructor (tac)] allowed backtracking across the use of [econstructor]; it has been subsumed by [econstructor; tac]. *) -Ltac econstructor_84_n n := econstructor n. -Ltac econstructor_84_tac tac := once (econstructor; tac). - -Tactic Notation "econstructor" := Coq.Init.Notations.econstructor. -Tactic Notation "econstructor" int_or_var(n) := econstructor_84_n n. -Tactic Notation "econstructor" "(" tactic(tac) ")" := econstructor_84_tac tac. - -(** Some tactic notations do not factor well with tactics; we add global parsing entries for some tactics that would otherwise be overwritten by custom variants. See https://coq.inria.fr/bugs/show_bug.cgi?id=4392. *) -Tactic Notation "reflexivity" := reflexivity. -Tactic Notation "assumption" := assumption. -Tactic Notation "etransitivity" := etransitivity. -Tactic Notation "cut" constr(c) := cut c. -Tactic Notation "exact_no_check" constr(c) := exact_no_check c. -Tactic Notation "vm_cast_no_check" constr(c) := vm_cast_no_check c. -Tactic Notation "casetype" constr(c) := casetype c. -Tactic Notation "elimtype" constr(c) := elimtype c. -Tactic Notation "lapply" constr(c) := lapply c. -Tactic Notation "transitivity" constr(c) := transitivity c. -Tactic Notation "left" := left. -Tactic Notation "eleft" := eleft. -Tactic Notation "right" := right. -Tactic Notation "eright" := eright. -Tactic Notation "symmetry" := symmetry. -Tactic Notation "split" := split. -Tactic Notation "esplit" := esplit. - -(** Many things now import [PeanoNat] rather than [NPeano], so we require it so that the old absolute names in [NPeano.Nat] are available. *) -Require Coq.Numbers.Natural.Peano.NPeano. - -(** The following coercions were declared by default in Specif.v. *) -Coercion sig_of_sig2 : sig2 >-> sig. -Coercion sigT_of_sigT2 : sigT2 >-> sigT. -Coercion sigT_of_sig : sig >-> sigT. -Coercion sig_of_sigT : sigT >-> sig. -Coercion sigT2_of_sig2 : sig2 >-> sigT2. -Coercion sig2_of_sigT2 : sigT2 >-> sig2. - -(** In 8.4, the statement of admitted lemmas did not depend on the section - variables. *) -Unset Keep Admitted Variables. diff --git a/theories/Compat/Coq85.v b/theories/Compat/Coq85.v deleted file mode 100644 index 64ba6b1e..00000000 --- a/theories/Compat/Coq85.v +++ /dev/null @@ -1,36 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -(** Compatibility file for making Coq act similar to Coq v8.5 *) - -(** Any compatibility changes to make future versions of Coq behave like Coq 8.6 - are likely needed to make them behave like Coq 8.5. *) -Require Export Coq.Compat.Coq86. - -(** We use some deprecated options in this file, so we disable the - corresponding warning, to silence the build of this file. *) -Local Set Warnings "-deprecated-option". - -(* In 8.5, "intros [|]", taken e.g. on a goal "A\/B->C", does not - behave as "intros [H|H]" but leave instead hypotheses quantified in - the goal, here producing subgoals A->C and B->C. *) - -Global Unset Bracketing Last Introduction Pattern. -Global Unset Regular Subst Tactic. -Global Unset Structural Injection. -Global Unset Shrink Abstract. -Global Unset Shrink Obligations. -Global Set Refolding Reduction. - -(** The resolution algorithm for type classes has changed. *) -Global Set Typeclasses Legacy Resolution. -Global Set Typeclasses Limit Intros. -Global Unset Typeclasses Filtered Unification. - -(** Allow silently letting unification constraints float after a "." *) -Global Unset Solve Unification Constraints. diff --git a/theories/Compat/Coq86.v b/theories/Compat/Coq86.v index 6952fdf1..666be207 100644 --- a/theories/Compat/Coq86.v +++ b/theories/Compat/Coq86.v @@ -1,9 +1,15 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) -(** Compatibility file for making Coq act similar to Coq v8.6 *)
\ No newline at end of file +(** Compatibility file for making Coq act similar to Coq v8.6 *) +Require Export Coq.Compat.Coq87. + +Require Export Coq.extraction.Extraction. +Require Export Coq.funind.FunInd. diff --git a/theories/Compat/Coq87.v b/theories/Compat/Coq87.v new file mode 100644 index 00000000..dc1397af --- /dev/null +++ b/theories/Compat/Coq87.v @@ -0,0 +1,23 @@ +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) + +(** Compatibility file for making Coq act similar to Coq v8.7 *) +Require Export Coq.Compat.Coq88. + +(* In 8.7, omega wasn't taking advantage of local abbreviations, + see bug 148 and PR#768. For adjusting this flag, we're forced to + first dynlink the omega plugin, but we should avoid doing a full + "Require Omega", since it has some undesired effects (at least on hints) + and breaks at least fiat-crypto. *) +Declare ML Module "omega_plugin". +Unset Omega UseLocalDefs. + + +Set Typeclasses Axioms Are Instances. diff --git a/theories/Compat/Coq88.v b/theories/Compat/Coq88.v new file mode 100644 index 00000000..4142af05 --- /dev/null +++ b/theories/Compat/Coq88.v @@ -0,0 +1,11 @@ +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) + +(** Compatibility file for making Coq act similar to Coq v8.8 *) diff --git a/theories/Compat/vo.itarget b/theories/Compat/vo.itarget deleted file mode 100644 index 7ffb86eb..00000000 --- a/theories/Compat/vo.itarget +++ /dev/null @@ -1,4 +0,0 @@ -AdmitAxiom.vo -Coq84.vo -Coq85.vo -Coq86.vo diff --git a/theories/FSets/FMapAVL.v b/theories/FSets/FMapAVL.v index c9e5b8dd..3485b9c6 100644 --- a/theories/FSets/FMapAVL.v +++ b/theories/FSets/FMapAVL.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (* Finite map library. *) @@ -16,7 +18,7 @@ See the comments at the beginning of FSetAVL for more details. *) -Require Import FMapInterface FMapList ZArith Int. +Require Import FunInd FMapInterface FMapList ZArith Int. Set Implicit Arguments. Unset Strict Implicit. diff --git a/theories/FSets/FMapFacts.v b/theories/FSets/FMapFacts.v index 3c5690a7..99705966 100644 --- a/theories/FSets/FMapFacts.v +++ b/theories/FSets/FMapFacts.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Finite maps library *) @@ -24,7 +26,7 @@ Hint Extern 1 (Equivalence _) => constructor; congruence. Module WFacts_fun (E:DecidableType)(Import M:WSfun E). -Notation option_map := option_map (compat "8.4"). +Notation option_map := option_map (compat "8.6"). Notation eq_dec := E.eq_dec. Definition eqb x y := if eq_dec x y then true else false. @@ -440,7 +442,7 @@ destruct (eq_dec x y); auto. Qed. Lemma map_o : forall m x (f:elt->elt'), - find x (map f m) = option_map f (find x m). + find x (map f m) = Datatypes.option_map f (find x m). Proof. intros. generalize (find_mapsto_iff (map f m) x) (find_mapsto_iff m x) @@ -473,7 +475,7 @@ Qed. Lemma mapi_o : forall m x (f:key->elt->elt'), (forall x y e, E.eq x y -> f x e = f y e) -> - find x (mapi f m) = option_map (f x) (find x m). + find x (mapi f m) = Datatypes.option_map (f x) (find x m). Proof. intros. generalize (find_mapsto_iff (mapi f m) x) (find_mapsto_iff m x) diff --git a/theories/FSets/FMapFullAVL.v b/theories/FSets/FMapFullAVL.v index a7be3232..34529678 100644 --- a/theories/FSets/FMapFullAVL.v +++ b/theories/FSets/FMapFullAVL.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (* Finite map library. *) @@ -25,7 +27,7 @@ *) -Require Import Recdef FMapInterface FMapList ZArith Int FMapAVL ROmega. +Require Import FunInd Recdef FMapInterface FMapList ZArith Int FMapAVL ROmega. Set Implicit Arguments. Unset Strict Implicit. diff --git a/theories/FSets/FMapInterface.v b/theories/FSets/FMapInterface.v index 4d89b562..38a96dc3 100644 --- a/theories/FSets/FMapInterface.v +++ b/theories/FSets/FMapInterface.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Finite map library *) diff --git a/theories/FSets/FMapList.v b/theories/FSets/FMapList.v index 5acdb7eb..3e98d119 100644 --- a/theories/FSets/FMapList.v +++ b/theories/FSets/FMapList.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Finite map library *) @@ -12,7 +14,7 @@ [FMapInterface.S] using lists of pairs ordered (increasing) with respect to left projection. *) -Require Import FMapInterface. +Require Import FunInd FMapInterface. Set Implicit Arguments. Unset Strict Implicit. diff --git a/theories/FSets/FMapPositive.v b/theories/FSets/FMapPositive.v index b1c0fdaa..0fc68b14 100644 --- a/theories/FSets/FMapPositive.v +++ b/theories/FSets/FMapPositive.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * FMapPositive : an implementation of FMapInterface for [positive] keys. *) @@ -654,19 +656,19 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. Lemma xelements_bits_lt_1 : forall p p0 q m v, List.In (p0,v) (xelements m (append p (xO q))) -> E.bits_lt p0 p. - Proof. + Proof using. intros. generalize (xelements_complete _ _ _ _ H); clear H; intros. - revert p0 q m v H. + revert p0 H. induction p; destruct p0; simpl; intros; eauto; try discriminate. Qed. Lemma xelements_bits_lt_2 : forall p p0 q m v, List.In (p0,v) (xelements m (append p (xI q))) -> E.bits_lt p p0. - Proof. + Proof using. intros. generalize (xelements_complete _ _ _ _ H); clear H; intros. - revert p0 q m v H. + revert p0 H. induction p; destruct p0; simpl; intros; eauto; try discriminate. Qed. diff --git a/theories/FSets/FMapWeakList.v b/theories/FSets/FMapWeakList.v index 130cbee8..67360965 100644 --- a/theories/FSets/FMapWeakList.v +++ b/theories/FSets/FMapWeakList.v @@ -1,17 +1,19 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Finite map library *) (** This file proposes an implementation of the non-dependent interface [FMapInterface.WS] using lists of pairs, unordered but without redundancy. *) -Require Import FMapInterface. +Require Import FunInd FMapInterface. Set Implicit Arguments. Unset Strict Implicit. diff --git a/theories/FSets/FMaps.v b/theories/FSets/FMaps.v index 19b25d95..ec507635 100644 --- a/theories/FSets/FMaps.v +++ b/theories/FSets/FMaps.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) Require Export OrderedType OrderedTypeEx OrderedTypeAlt. diff --git a/theories/FSets/FSetAVL.v b/theories/FSets/FSetAVL.v index df627a14..dcaea894 100644 --- a/theories/FSets/FSetAVL.v +++ b/theories/FSets/FSetAVL.v @@ -1,11 +1,13 @@ (* -*- coding: utf-8 -*- *) -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * FSetAVL : Implementation of FSetInterface via AVL trees *) @@ -13,7 +15,7 @@ It follows the implementation from Ocaml's standard library, All operations given here expect and produce well-balanced trees - (in the ocaml sense: heigths of subtrees shouldn't differ by more + (in the ocaml sense: heights of subtrees shouldn't differ by more than 2), and hence has low complexities (e.g. add is logarithmic in the size of the set). But proving these balancing preservations is in fact not necessary for ensuring correct operational behavior diff --git a/theories/FSets/FSetBridge.v b/theories/FSets/FSetBridge.v index 97f140b3..0c4ecb1f 100644 --- a/theories/FSets/FSetBridge.v +++ b/theories/FSets/FSetBridge.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Finite sets library *) diff --git a/theories/FSets/FSetCompat.v b/theories/FSets/FSetCompat.v index b1769da3..736c85da 100644 --- a/theories/FSets/FSetCompat.v +++ b/theories/FSets/FSetCompat.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Compatibility functors between FSetInterface and MSetInterface. *) @@ -165,13 +167,13 @@ End Backport_WSets. (** * From new Sets to new ones *) Module Backport_Sets - (E:OrderedType.OrderedType) - (M:MSetInterface.Sets with Definition E.t := E.t - with Definition E.eq := E.eq - with Definition E.lt := E.lt) - <: FSetInterface.S with Module E:=E. + (O:OrderedType.OrderedType) + (M:MSetInterface.Sets with Definition E.t := O.t + with Definition E.eq := O.eq + with Definition E.lt := O.lt) + <: FSetInterface.S with Module E:=O. - Include Backport_WSets E M. + Include Backport_WSets O M. Implicit Type s : t. Implicit Type x y : elt. @@ -182,21 +184,21 @@ Module Backport_Sets Definition min_elt_1 : forall s x, min_elt s = Some x -> In x s := M.min_elt_spec1. Definition min_elt_2 : forall s x y, - min_elt s = Some x -> In y s -> ~ E.lt y x + min_elt s = Some x -> In y s -> ~ O.lt y x := M.min_elt_spec2. Definition min_elt_3 : forall s, min_elt s = None -> Empty s := M.min_elt_spec3. Definition max_elt_1 : forall s x, max_elt s = Some x -> In x s := M.max_elt_spec1. Definition max_elt_2 : forall s x y, - max_elt s = Some x -> In y s -> ~ E.lt x y + max_elt s = Some x -> In y s -> ~ O.lt x y := M.max_elt_spec2. Definition max_elt_3 : forall s, max_elt s = None -> Empty s := M.max_elt_spec3. - Definition elements_3 : forall s, sort E.lt (elements s) + Definition elements_3 : forall s, sort O.lt (elements s) := M.elements_spec2. Definition choose_3 : forall s s' x y, - choose s = Some x -> choose s' = Some y -> Equal s s' -> E.eq x y + choose s = Some x -> choose s' = Some y -> Equal s s' -> O.eq x y := M.choose_spec3. Definition lt_trans : forall s s' s'', lt s s' -> lt s' s'' -> lt s s'' := @StrictOrder_Transitive _ _ M.lt_strorder. @@ -211,7 +213,7 @@ Module Backport_Sets [ apply EQ | apply LT | apply GT ]; auto. Defined. - Module E := E. + Module E := O. End Backport_Sets. @@ -342,13 +344,13 @@ End Update_WSets. (** * From old Sets to new ones. *) Module Update_Sets - (E:Orders.OrderedType) - (M:FSetInterface.S with Definition E.t := E.t - with Definition E.eq := E.eq - with Definition E.lt := E.lt) - <: MSetInterface.Sets with Module E:=E. + (O:Orders.OrderedType) + (M:FSetInterface.S with Definition E.t := O.t + with Definition E.eq := O.eq + with Definition E.lt := O.lt) + <: MSetInterface.Sets with Module E:=O. - Include Update_WSets E M. + Include Update_WSets O M. Implicit Type s : t. Implicit Type x y : elt. @@ -359,21 +361,21 @@ Module Update_Sets Definition min_elt_spec1 : forall s x, min_elt s = Some x -> In x s := M.min_elt_1. Definition min_elt_spec2 : forall s x y, - min_elt s = Some x -> In y s -> ~ E.lt y x + min_elt s = Some x -> In y s -> ~ O.lt y x := M.min_elt_2. Definition min_elt_spec3 : forall s, min_elt s = None -> Empty s := M.min_elt_3. Definition max_elt_spec1 : forall s x, max_elt s = Some x -> In x s := M.max_elt_1. Definition max_elt_spec2 : forall s x y, - max_elt s = Some x -> In y s -> ~ E.lt x y + max_elt s = Some x -> In y s -> ~ O.lt x y := M.max_elt_2. Definition max_elt_spec3 : forall s, max_elt s = None -> Empty s := M.max_elt_3. - Definition elements_spec2 : forall s, sort E.lt (elements s) + Definition elements_spec2 : forall s, sort O.lt (elements s) := M.elements_3. Definition choose_spec3 : forall s s' x y, - choose s = Some x -> choose s' = Some y -> Equal s s' -> E.eq x y + choose s = Some x -> choose s' = Some y -> Equal s s' -> O.eq x y := M.choose_3. Instance lt_strorder : StrictOrder lt. @@ -407,6 +409,6 @@ Module Update_Sets Lemma compare_spec : forall s s', CompSpec eq lt s s' (compare s s'). Proof. intros; unfold compare; destruct M.compare; auto. Qed. - Module E := E. + Module E := O. End Update_Sets. diff --git a/theories/FSets/FSetDecide.v b/theories/FSets/FSetDecide.v index 1db6a71e..83bb07ff 100644 --- a/theories/FSets/FSetDecide.v +++ b/theories/FSets/FSetDecide.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (**************************************************************) (* FSetDecide.v *) diff --git a/theories/FSets/FSetEqProperties.v b/theories/FSets/FSetEqProperties.v index f2f4cc2c..56844f47 100644 --- a/theories/FSets/FSetEqProperties.v +++ b/theories/FSets/FSetEqProperties.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Finite sets library *) diff --git a/theories/FSets/FSetFacts.v b/theories/FSets/FSetFacts.v index b240ede4..f4d281e9 100644 --- a/theories/FSets/FSetFacts.v +++ b/theories/FSets/FSetFacts.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Finite sets library *) diff --git a/theories/FSets/FSetInterface.v b/theories/FSets/FSetInterface.v index c791f49a..0926d3ae 100644 --- a/theories/FSets/FSetInterface.v +++ b/theories/FSets/FSetInterface.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Finite set library *) diff --git a/theories/FSets/FSetList.v b/theories/FSets/FSetList.v index 9c3ec71a..2036d360 100644 --- a/theories/FSets/FSetList.v +++ b/theories/FSets/FSetList.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Finite sets library *) diff --git a/theories/FSets/FSetPositive.v b/theories/FSets/FSetPositive.v index 507f1cda..8a93f381 100644 --- a/theories/FSets/FSetPositive.v +++ b/theories/FSets/FSetPositive.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** Efficient implementation of [FSetInterface.S] for positive keys, inspired from the [FMapPositive] module. diff --git a/theories/FSets/FSetProperties.v b/theories/FSets/FSetProperties.v index 25b042ca..c9cfb94a 100644 --- a/theories/FSets/FSetProperties.v +++ b/theories/FSets/FSetProperties.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Finite sets library *) @@ -762,7 +764,7 @@ Module WProperties_fun (Import E : DecidableType)(M : WSfun E). rewrite (cardinal_2 (s:=remove x s') (s':=s') (x:=x)); eauto with set. Qed. - Add Morphism cardinal : cardinal_m. + Add Morphism cardinal with signature (Equal ==> Logic.eq) as cardinal_m. Proof. exact Equal_cardinal. Qed. diff --git a/theories/FSets/FSetToFiniteSet.v b/theories/FSets/FSetToFiniteSet.v index 3ac5d9e4..f8d13ed2 100644 --- a/theories/FSets/FSetToFiniteSet.v +++ b/theories/FSets/FSetToFiniteSet.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Finite sets library : conversion to old [Finite_sets] *) diff --git a/theories/FSets/FSetWeakList.v b/theories/FSets/FSetWeakList.v index 9dbea884..1dacd056 100644 --- a/theories/FSets/FSetWeakList.v +++ b/theories/FSets/FSetWeakList.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Finite sets library *) diff --git a/theories/FSets/FSets.v b/theories/FSets/FSets.v index 572f2865..7e9e7aae 100644 --- a/theories/FSets/FSets.v +++ b/theories/FSets/FSets.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) Require Export OrderedType. Require Export OrderedTypeEx. @@ -20,4 +22,4 @@ Require Export FSetEqProperties. Require Export FSetWeakList. Require Export FSetList. Require Export FSetPositive. -Require Export FSetAVL.
\ No newline at end of file +Require Export FSetAVL. diff --git a/theories/FSets/vo.itarget b/theories/FSets/vo.itarget deleted file mode 100644 index 0e7c11fb..00000000 --- a/theories/FSets/vo.itarget +++ /dev/null @@ -1,21 +0,0 @@ -FMapAVL.vo -FMapFacts.vo -FMapFullAVL.vo -FMapInterface.vo -FMapList.vo -FMapPositive.vo -FMaps.vo -FMapWeakList.vo -FSetCompat.vo -FSetAVL.vo -FSetPositive.vo -FSetBridge.vo -FSetDecide.vo -FSetEqProperties.vo -FSetFacts.vo -FSetInterface.vo -FSetList.vo -FSetProperties.vo -FSets.vo -FSetToFiniteSet.vo -FSetWeakList.vo diff --git a/theories/Init/Datatypes.v b/theories/Init/Datatypes.v index ddaf08bf..05b741f0 100644 --- a/theories/Init/Datatypes.v +++ b/theories/Init/Datatypes.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Set Implicit Arguments. @@ -65,7 +67,7 @@ Infix "&&" := andb : bool_scope. Lemma andb_prop : forall a b:bool, andb a b = true -> a = true /\ b = true. Proof. - destruct a; destruct b; intros; split; try (reflexivity || discriminate). + destruct a, b; repeat split; assumption. Qed. Hint Resolve andb_prop: bool. @@ -262,6 +264,11 @@ Inductive comparison : Set := | Lt : comparison | Gt : comparison. +Lemma comparison_eq_stable : forall c c' : comparison, ~~ c = c' -> c = c'. +Proof. + destruct c, c'; intro H; reflexivity || destruct H; discriminate. +Qed. + Definition CompOpp (r:comparison) := match r with | Eq => Eq @@ -326,13 +333,12 @@ Lemma CompSpec2Type : forall A (eq lt:A->A->Prop) x y c, CompSpec eq lt x y c -> CompSpecT eq lt x y c. Proof. intros. apply CompareSpec2Type; assumption. Defined. - (******************************************************************) (** * Misc Other Datatypes *) (** [identity A a] is the family of datatypes on [A] whose sole non-empty member is the singleton datatype [identity A a a] whose - sole inhabitant is denoted [refl_identity A a] *) + sole inhabitant is denoted [identity_refl A a] *) Inductive identity (A:Type) (a:A) : A -> Type := identity_refl : identity a a. @@ -355,14 +361,14 @@ Definition idProp : IDProp := fun A x => x. (* Compatibility *) -Notation prodT := prod (compat "8.2"). -Notation pairT := pair (compat "8.2"). -Notation prodT_rect := prod_rect (compat "8.2"). -Notation prodT_rec := prod_rec (compat "8.2"). -Notation prodT_ind := prod_ind (compat "8.2"). -Notation fstT := fst (compat "8.2"). -Notation sndT := snd (compat "8.2"). -Notation prodT_uncurry := prod_uncurry (compat "8.2"). -Notation prodT_curry := prod_curry (compat "8.2"). +Notation prodT := prod (only parsing). +Notation pairT := pair (only parsing). +Notation prodT_rect := prod_rect (only parsing). +Notation prodT_rec := prod_rec (only parsing). +Notation prodT_ind := prod_ind (only parsing). +Notation fstT := fst (only parsing). +Notation sndT := snd (only parsing). +Notation prodT_uncurry := prod_uncurry (only parsing). +Notation prodT_curry := prod_curry (only parsing). (* end hide *) diff --git a/theories/Init/Decimal.v b/theories/Init/Decimal.v new file mode 100644 index 00000000..57163b1b --- /dev/null +++ b/theories/Init/Decimal.v @@ -0,0 +1,163 @@ +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) + +(** * Decimal numbers *) + +(** These numbers coded in base 10 will be used for parsing and printing + other Coq numeral datatypes in an human-readable way. + See the [Numeral Notation] command. + We represent numbers in base 10 as lists of decimal digits, + in big-endian order (most significant digit comes first). *) + +(** Unsigned integers are just lists of digits. + For instance, ten is (D1 (D0 Nil)) *) + +Inductive uint := + | Nil + | D0 (_:uint) + | D1 (_:uint) + | D2 (_:uint) + | D3 (_:uint) + | D4 (_:uint) + | D5 (_:uint) + | D6 (_:uint) + | D7 (_:uint) + | D8 (_:uint) + | D9 (_:uint). + +(** [Nil] is the number terminator. Taken alone, it behaves as zero, + but rather use [D0 Nil] instead, since this form will be denoted + as [0], while [Nil] will be printed as [Nil]. *) + +Notation zero := (D0 Nil). + +(** For signed integers, we use two constructors [Pos] and [Neg]. *) + +Inductive int := Pos (d:uint) | Neg (d:uint). + +Delimit Scope uint_scope with uint. +Bind Scope uint_scope with uint. +Delimit Scope int_scope with int. +Bind Scope int_scope with int. + +(** This representation favors simplicity over canonicity. + For normalizing numbers, we need to remove head zero digits, + and choose our canonical representation of 0 (here [D0 Nil] + for unsigned numbers and [Pos (D0 Nil)] for signed numbers). *) + +(** [nzhead] removes all head zero digits *) + +Fixpoint nzhead d := + match d with + | D0 d => nzhead d + | _ => d + end. + +(** [unorm] : normalization of unsigned integers *) + +Definition unorm d := + match nzhead d with + | Nil => zero + | d => d + end. + +(** [norm] : normalization of signed integers *) + +Definition norm d := + match d with + | Pos d => Pos (unorm d) + | Neg d => + match nzhead d with + | Nil => Pos zero + | d => Neg d + end + end. + +(** A few easy operations. For more advanced computations, use the conversions + with other Coq numeral datatypes (e.g. Z) and the operations on them. *) + +Definition opp (d:int) := + match d with + | Pos d => Neg d + | Neg d => Pos d + end. + +(** For conversions with binary numbers, it is easier to operate + on little-endian numbers. *) + +Fixpoint revapp (d d' : uint) := + match d with + | Nil => d' + | D0 d => revapp d (D0 d') + | D1 d => revapp d (D1 d') + | D2 d => revapp d (D2 d') + | D3 d => revapp d (D3 d') + | D4 d => revapp d (D4 d') + | D5 d => revapp d (D5 d') + | D6 d => revapp d (D6 d') + | D7 d => revapp d (D7 d') + | D8 d => revapp d (D8 d') + | D9 d => revapp d (D9 d') + end. + +Definition rev d := revapp d Nil. + +Module Little. + +(** Successor of little-endian numbers *) + +Fixpoint succ d := + match d with + | Nil => D1 Nil + | D0 d => D1 d + | D1 d => D2 d + | D2 d => D3 d + | D3 d => D4 d + | D4 d => D5 d + | D5 d => D6 d + | D6 d => D7 d + | D7 d => D8 d + | D8 d => D9 d + | D9 d => D0 (succ d) + end. + +(** Doubling little-endian numbers *) + +Fixpoint double d := + match d with + | Nil => Nil + | D0 d => D0 (double d) + | D1 d => D2 (double d) + | D2 d => D4 (double d) + | D3 d => D6 (double d) + | D4 d => D8 (double d) + | D5 d => D0 (succ_double d) + | D6 d => D2 (succ_double d) + | D7 d => D4 (succ_double d) + | D8 d => D6 (succ_double d) + | D9 d => D8 (succ_double d) + end + +with succ_double d := + match d with + | Nil => D1 Nil + | D0 d => D1 (double d) + | D1 d => D3 (double d) + | D2 d => D5 (double d) + | D3 d => D7 (double d) + | D4 d => D9 (double d) + | D5 d => D1 (succ_double d) + | D6 d => D3 (succ_double d) + | D7 d => D5 (succ_double d) + | D8 d => D7 (succ_double d) + | D9 d => D9 (succ_double d) + end. + +End Little. diff --git a/theories/Init/Logic.v b/theories/Init/Logic.v index 85123cc4..15ca5abc 100644 --- a/theories/Init/Logic.v +++ b/theories/Init/Logic.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Set Implicit Arguments. @@ -125,6 +127,25 @@ Proof. [apply Hl | apply Hr]; assumption. Qed. +Theorem imp_iff_compat_l : forall A B C : Prop, + (B <-> C) -> ((A -> B) <-> (A -> C)). +Proof. + intros ? ? ? [Hl Hr]; split; intros H ?; [apply Hl | apply Hr]; apply H; assumption. +Qed. + +Theorem imp_iff_compat_r : forall A B C : Prop, + (B <-> C) -> ((B -> A) <-> (C -> A)). +Proof. + intros ? ? ? [Hl Hr]; split; intros H ?; [apply H, Hr | apply H, Hl]; assumption. +Qed. + +Theorem not_iff_compat : forall A B : Prop, + (A <-> B) -> (~ A <-> ~B). +Proof. + intros; apply imp_iff_compat_r; assumption. +Qed. + + (** Some equivalences *) Theorem neg_false : forall A : Prop, ~ A <-> (A <-> False). @@ -204,7 +225,7 @@ Proof. Qed. (** [(IF_then_else P Q R)], written [IF P then Q else R] denotes - either [P] and [Q], or [~P] and [Q] *) + either [P] and [Q], or [~P] and [R] *) Definition IF_then_else (P Q R:Prop) := P /\ Q \/ ~ P /\ R. @@ -243,9 +264,16 @@ Notation "'exists' x .. y , p" := (ex (fun x => .. (ex (fun y => p)) ..)) Notation "'exists2' x , p & q" := (ex2 (fun x => p) (fun x => q)) (at level 200, x ident, p at level 200, right associativity) : type_scope. -Notation "'exists2' x : t , p & q" := (ex2 (fun x:t => p) (fun x:t => q)) - (at level 200, x ident, t at level 200, p at level 200, right associativity, - format "'[' 'exists2' '/ ' x : t , '/ ' '[' p & '/' q ']' ']'") +Notation "'exists2' x : A , p & q" := (ex2 (A:=A) (fun x => p) (fun x => q)) + (at level 200, x ident, A at level 200, p at level 200, right associativity, + format "'[' 'exists2' '/ ' x : A , '/ ' '[' p & '/' q ']' ']'") + : type_scope. + +Notation "'exists2' ' x , p & q" := (ex2 (fun x => p) (fun x => q)) + (at level 200, x strict pattern, p at level 200, right associativity) : type_scope. +Notation "'exists2' ' x : A , p & q" := (ex2 (A:=A) (fun x => p) (fun x => q)) + (at level 200, x strict pattern, A at level 200, p at level 200, right associativity, + format "'[' 'exists2' '/ ' ' x : A , '/ ' '[' p & '/' q ']' ']'") : type_scope. (** Derived rules for universal quantification *) @@ -294,8 +322,8 @@ Arguments eq_ind [A] x P _ y _. Arguments eq_rec [A] x P _ y _. Arguments eq_rect [A] x P _ y _. -Hint Resolve I conj or_introl or_intror : core. -Hint Resolve eq_refl: core. +Hint Resolve I conj or_introl or_intror : core. +Hint Resolve eq_refl: core. Hint Resolve ex_intro ex_intro2: core. Section Logic_lemmas. @@ -424,7 +452,7 @@ Proof. destruct e. reflexivity. Defined. -(** The goupoid structure of equality *) +(** The groupoid structure of equality *) Theorem eq_trans_refl_l : forall A (x y:A) (e:x=y), eq_trans eq_refl e = e. Proof. @@ -485,6 +513,11 @@ Proof. reflexivity. Defined. +Lemma eq_refl_map_distr : forall A B x (f:A->B), f_equal f (eq_refl x) = eq_refl (f x). +Proof. + reflexivity. +Qed. + Lemma eq_trans_map_distr : forall A B x y z (f:A->B) (e:x=y) (e':y=z), f_equal f (eq_trans e e') = eq_trans (f_equal f e) (f_equal f e'). Proof. destruct e'. @@ -503,16 +536,29 @@ destruct e, e'. reflexivity. Defined. +Lemma eq_trans_rew_distr : forall A (P:A -> Type) (x y z:A) (e:x=y) (e':y=z) (k:P x), + rew (eq_trans e e') in k = rew e' in rew e in k. +Proof. + destruct e, e'; reflexivity. +Qed. + +Lemma rew_const : forall A P (x y:A) (e:x=y) (k:P), + rew [fun _ => P] e in k = k. +Proof. + destruct e; reflexivity. +Qed. + + (* Aliases *) -Notation sym_eq := eq_sym (compat "8.3"). -Notation trans_eq := eq_trans (compat "8.3"). -Notation sym_not_eq := not_eq_sym (compat "8.3"). +Notation sym_eq := eq_sym (only parsing). +Notation trans_eq := eq_trans (only parsing). +Notation sym_not_eq := not_eq_sym (only parsing). -Notation refl_equal := eq_refl (compat "8.3"). -Notation sym_equal := eq_sym (compat "8.3"). -Notation trans_equal := eq_trans (compat "8.3"). -Notation sym_not_equal := not_eq_sym (compat "8.3"). +Notation refl_equal := eq_refl (only parsing). +Notation sym_equal := eq_sym (only parsing). +Notation trans_equal := eq_trans (only parsing). +Notation sym_not_equal := not_eq_sym (only parsing). Hint Immediate eq_sym not_eq_sym: core. @@ -553,9 +599,10 @@ Proof. intros A P (x & Hp & Huniq); split. - intro; exists x; auto. - intros (x0 & HPx0 & HQx0) x1 HPx1. - replace x1 with x0 by (transitivity x; [symmetry|]; auto). + assert (H : x0 = x1) by (transitivity x; [symmetry|]; auto). + destruct H. assumption. -Qed. +Qed. Lemma forall_exists_coincide_unique_domain : forall A (P:A->Prop), @@ -567,7 +614,7 @@ Proof. exists x. split; [trivial|]. destruct H with (Q:=fun x'=>x=x') as (_,Huniq). apply Huniq. exists x; auto. -Qed. +Qed. (** * Being inhabited *) @@ -589,6 +636,11 @@ Proof. destruct 1; auto. Qed. +Lemma inhabited_covariant (A B : Type) : (A -> B) -> inhabited A -> inhabited B. +Proof. + intros f [x];exact (inhabits (f x)). +Qed. + (** Declaration of stepl and stepr for eq and iff *) Lemma eq_stepl : forall (A : Type) (x y z : A), x = y -> x = z -> z = y. @@ -606,3 +658,97 @@ Qed. Declare Left Step iff_stepl. Declare Right Step iff_trans. + +Local Notation "'rew' 'dependent' H 'in' H'" + := (match H with + | eq_refl => H' + end) + (at level 10, H' at level 10, + format "'[' 'rew' 'dependent' '/ ' H in '/' H' ']'"). + +(** Equality for [ex] *) +Section ex. + Local Unset Implicit Arguments. + Definition eq_ex_uncurried {A : Type} (P : A -> Prop) {u1 v1 : A} {u2 : P u1} {v2 : P v1} + (pq : exists p : u1 = v1, rew p in u2 = v2) + : ex_intro P u1 u2 = ex_intro P v1 v2. + Proof. + destruct pq as [p q]. + destruct q; simpl in *. + destruct p; reflexivity. + Qed. + + Definition eq_ex {A : Type} {P : A -> Prop} (u1 v1 : A) (u2 : P u1) (v2 : P v1) + (p : u1 = v1) (q : rew p in u2 = v2) + : ex_intro P u1 u2 = ex_intro P v1 v2 + := eq_ex_uncurried P (ex_intro _ p q). + + Definition eq_ex_hprop {A} {P : A -> Prop} (P_hprop : forall (x : A) (p q : P x), p = q) + (u1 v1 : A) (u2 : P u1) (v2 : P v1) + (p : u1 = v1) + : ex_intro P u1 u2 = ex_intro P v1 v2 + := eq_ex u1 v1 u2 v2 p (P_hprop _ _ _). + + Lemma rew_ex {A x} {P : A -> Type} (Q : forall a, P a -> Prop) (u : exists p, Q x p) {y} (H : x = y) + : rew [fun a => exists p, Q a p] H in u + = match u with + | ex_intro _ u1 u2 + => ex_intro + (Q y) + (rew H in u1) + (rew dependent H in u2) + end. + Proof. + destruct H, u; reflexivity. + Qed. +End ex. + +(** Equality for [ex2] *) +Section ex2. + Local Unset Implicit Arguments. + + Definition eq_ex2_uncurried {A : Type} (P Q : A -> Prop) {u1 v1 : A} + {u2 : P u1} {v2 : P v1} + {u3 : Q u1} {v3 : Q v1} + (pq : exists2 p : u1 = v1, rew p in u2 = v2 & rew p in u3 = v3) + : ex_intro2 P Q u1 u2 u3 = ex_intro2 P Q v1 v2 v3. + Proof. + destruct pq as [p q r]. + destruct r, q, p; simpl in *. + reflexivity. + Qed. + + Definition eq_ex2 {A : Type} {P Q : A -> Prop} + (u1 v1 : A) + (u2 : P u1) (v2 : P v1) + (u3 : Q u1) (v3 : Q v1) + (p : u1 = v1) (q : rew p in u2 = v2) (r : rew p in u3 = v3) + : ex_intro2 P Q u1 u2 u3 = ex_intro2 P Q v1 v2 v3 + := eq_ex2_uncurried P Q (ex_intro2 _ _ p q r). + + Definition eq_ex2_hprop {A} {P Q : A -> Prop} + (P_hprop : forall (x : A) (p q : P x), p = q) + (Q_hprop : forall (x : A) (p q : Q x), p = q) + (u1 v1 : A) (u2 : P u1) (v2 : P v1) (u3 : Q u1) (v3 : Q v1) + (p : u1 = v1) + : ex_intro2 P Q u1 u2 u3 = ex_intro2 P Q v1 v2 v3 + := eq_ex2 u1 v1 u2 v2 u3 v3 p (P_hprop _ _ _) (Q_hprop _ _ _). + + Lemma rew_ex2 {A x} {P : A -> Type} + (Q : forall a, P a -> Prop) + (R : forall a, P a -> Prop) + (u : exists2 p, Q x p & R x p) {y} (H : x = y) + : rew [fun a => exists2 p, Q a p & R a p] H in u + = match u with + | ex_intro2 _ _ u1 u2 u3 + => ex_intro2 + (Q y) + (R y) + (rew H in u1) + (rew dependent H in u2) + (rew dependent H in u3) + end. + Proof. + destruct H, u; reflexivity. + Qed. +End ex2. diff --git a/theories/Init/Logic_Type.v b/theories/Init/Logic_Type.v index 4536dfc0..6f10a939 100644 --- a/theories/Init/Logic_Type.v +++ b/theories/Init/Logic_Type.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** This module defines type constructors for types in [Type] @@ -66,7 +68,7 @@ Defined. Hint Immediate identity_sym not_identity_sym: core. -Notation refl_id := identity_refl (compat "8.3"). -Notation sym_id := identity_sym (compat "8.3"). -Notation trans_id := identity_trans (compat "8.3"). -Notation sym_not_id := not_identity_sym (compat "8.3"). +Notation refl_id := identity_refl (only parsing). +Notation sym_id := identity_sym (only parsing). +Notation trans_id := identity_trans (only parsing). +Notation sym_not_id := not_identity_sym (only parsing). diff --git a/theories/Init/Nat.v b/theories/Init/Nat.v index b8920586..ad1bc717 100644 --- a/theories/Init/Nat.v +++ b/theories/Init/Nat.v @@ -1,13 +1,15 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Notations Logic Datatypes. - +Require Decimal. Local Open Scope nat_scope. (**********************************************************************) @@ -134,6 +136,62 @@ Fixpoint pow n m := where "n ^ m" := (pow n m) : nat_scope. +(** ** Tail-recursive versions of [add] and [mul] *) + +Fixpoint tail_add n m := + match n with + | O => m + | S n => tail_add n (S m) + end. + +(** [tail_addmul r n m] is [r + n * m]. *) + +Fixpoint tail_addmul r n m := + match n with + | O => r + | S n => tail_addmul (tail_add m r) n m + end. + +Definition tail_mul n m := tail_addmul 0 n m. + +(** ** Conversion with a decimal representation for printing/parsing *) + +Local Notation ten := (S (S (S (S (S (S (S (S (S (S O)))))))))). + +Fixpoint of_uint_acc (d:Decimal.uint)(acc:nat) := + match d with + | Decimal.Nil => acc + | Decimal.D0 d => of_uint_acc d (tail_mul ten acc) + | Decimal.D1 d => of_uint_acc d (S (tail_mul ten acc)) + | Decimal.D2 d => of_uint_acc d (S (S (tail_mul ten acc))) + | Decimal.D3 d => of_uint_acc d (S (S (S (tail_mul ten acc)))) + | Decimal.D4 d => of_uint_acc d (S (S (S (S (tail_mul ten acc))))) + | Decimal.D5 d => of_uint_acc d (S (S (S (S (S (tail_mul ten acc)))))) + | Decimal.D6 d => of_uint_acc d (S (S (S (S (S (S (tail_mul ten acc))))))) + | Decimal.D7 d => of_uint_acc d (S (S (S (S (S (S (S (tail_mul ten acc)))))))) + | Decimal.D8 d => of_uint_acc d (S (S (S (S (S (S (S (S (tail_mul ten acc))))))))) + | Decimal.D9 d => of_uint_acc d (S (S (S (S (S (S (S (S (S (tail_mul ten acc)))))))))) + end. + +Definition of_uint (d:Decimal.uint) := of_uint_acc d O. + +Fixpoint to_little_uint n acc := + match n with + | O => acc + | S n => to_little_uint n (Decimal.Little.succ acc) + end. + +Definition to_uint n := + Decimal.rev (to_little_uint n Decimal.zero). + +Definition of_int (d:Decimal.int) : option nat := + match Decimal.norm d with + | Decimal.Pos u => Some (of_uint u) + | _ => None + end. + +Definition to_int n := Decimal.Pos (to_uint n). + (** ** Euclidean division *) (** This division is linear and tail-recursive. diff --git a/theories/Init/Notations.v b/theories/Init/Notations.v index 48fbe079..72073bb4 100644 --- a/theories/Init/Notations.v +++ b/theories/Init/Notations.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** These are the notations whose level and associativity are imposed by Coq *) @@ -66,15 +68,46 @@ Reserved Notation "{ x }" (at level 0, x at level 99). (** Notations for sigma-types or subsets *) +Reserved Notation "{ A } + { B }" (at level 50, left associativity). +Reserved Notation "A + { B }" (at level 50, left associativity). + Reserved Notation "{ x | P }" (at level 0, x at level 99). Reserved Notation "{ x | P & Q }" (at level 0, x at level 99). Reserved Notation "{ x : A | P }" (at level 0, x at level 99). Reserved Notation "{ x : A | P & Q }" (at level 0, x at level 99). +Reserved Notation "{ x & P }" (at level 0, x at level 99). Reserved Notation "{ x : A & P }" (at level 0, x at level 99). Reserved Notation "{ x : A & P & Q }" (at level 0, x at level 99). +Reserved Notation "{ ' pat | P }" + (at level 0, pat strict pattern, format "{ ' pat | P }"). +Reserved Notation "{ ' pat | P & Q }" + (at level 0, pat strict pattern, format "{ ' pat | P & Q }"). + +Reserved Notation "{ ' pat : A | P }" + (at level 0, pat strict pattern, format "{ ' pat : A | P }"). +Reserved Notation "{ ' pat : A | P & Q }" + (at level 0, pat strict pattern, format "{ ' pat : A | P & Q }"). + +Reserved Notation "{ ' pat : A & P }" + (at level 0, pat strict pattern, format "{ ' pat : A & P }"). +Reserved Notation "{ ' pat : A & P & Q }" + (at level 0, pat strict pattern, format "{ ' pat : A & P & Q }"). + +(** Support for Gonthier-Ssreflect's "if c is pat then u else v" *) + +Module IfNotations. + +Notation "'if' c 'is' p 'then' u 'else' v" := + (match c with p => u | _ => v end) + (at level 200, p pattern at level 100). + +End IfNotations. + +(** Scopes *) + Delimit Scope type_scope with type. Delimit Scope function_scope with function. Delimit Scope core_scope with core. @@ -88,9 +121,6 @@ Open Scope type_scope. (** ML Tactic Notations *) -Declare ML Module "coretactics". -Declare ML Module "extratactics". -Declare ML Module "g_auto". -Declare ML Module "g_class". -Declare ML Module "g_eqdecide". -Declare ML Module "g_rewrite". +Declare ML Module "ltac_plugin". + +Global Set Default Proof Mode "Classic". diff --git a/theories/Init/Peano.v b/theories/Init/Peano.v index 6c4a6350..d5322d09 100644 --- a/theories/Init/Peano.v +++ b/theories/Init/Peano.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** The type [nat] of Peano natural numbers (built from [O] and [S]) @@ -37,7 +39,7 @@ Hint Resolve f_equal_nat: core. (** The predecessor function *) -Notation pred := Nat.pred (compat "8.4"). +Notation pred := Nat.pred (only parsing). Definition f_equal_pred := f_equal pred. @@ -79,7 +81,7 @@ Hint Resolve n_Sn: core. (** Addition *) -Notation plus := Nat.add (compat "8.4"). +Notation plus := Nat.add (only parsing). Infix "+" := Nat.add : nat_scope. Definition f_equal2_plus := f_equal2 plus. @@ -90,7 +92,9 @@ Lemma plus_n_O : forall n:nat, n = n + 0. Proof. induction n; simpl; auto. Qed. -Hint Resolve plus_n_O: core. + +Remove Hints eq_refl : core. +Hint Resolve plus_n_O eq_refl: core. (* We want eq_refl to have higher priority than plus_n_O *) Lemma plus_O_n : forall n:nat, 0 + n = n. Proof. @@ -110,12 +114,12 @@ Qed. (** Standard associated names *) -Notation plus_0_r_reverse := plus_n_O (compat "8.2"). -Notation plus_succ_r_reverse := plus_n_Sm (compat "8.2"). +Notation plus_0_r_reverse := plus_n_O (only parsing). +Notation plus_succ_r_reverse := plus_n_Sm (only parsing). (** Multiplication *) -Notation mult := Nat.mul (compat "8.4"). +Notation mult := Nat.mul (only parsing). Infix "*" := Nat.mul : nat_scope. Definition f_equal2_mult := f_equal2 mult. @@ -137,12 +141,12 @@ Hint Resolve mult_n_Sm: core. (** Standard associated names *) -Notation mult_0_r_reverse := mult_n_O (compat "8.2"). -Notation mult_succ_r_reverse := mult_n_Sm (compat "8.2"). +Notation mult_0_r_reverse := mult_n_O (only parsing). +Notation mult_succ_r_reverse := mult_n_Sm (only parsing). (** Truncated subtraction: [m-n] is [0] if [n>=m] *) -Notation minus := Nat.sub (compat "8.4"). +Notation minus := Nat.sub (only parsing). Infix "-" := Nat.sub : nat_scope. (** Definition of the usual orders, the basic properties of [le] and [lt] @@ -219,8 +223,8 @@ Qed. (** Maximum and minimum : definitions and specifications *) -Notation max := Nat.max (compat "8.4"). -Notation min := Nat.min (compat "8.4"). +Notation max := Nat.max (only parsing). +Notation min := Nat.min (only parsing). Lemma max_l n m : m <= n -> Nat.max n m = n. Proof. diff --git a/theories/Init/Prelude.v b/theories/Init/Prelude.v index 03f2328d..802f18c0 100644 --- a/theories/Init/Prelude.v +++ b/theories/Init/Prelude.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export Notations. @@ -11,6 +13,7 @@ Require Export Logic. Require Export Logic_Type. Require Export Datatypes. Require Export Specif. +Require Coq.Init.Decimal. Require Coq.Init.Nat. Require Export Peano. Require Export Coq.Init.Wf. @@ -18,10 +21,7 @@ Require Export Coq.Init.Tactics. Require Export Coq.Init.Tauto. (* Initially available plugins (+ nat_syntax_plugin loaded in Datatypes) *) -Declare ML Module "extraction_plugin". -Declare ML Module "decl_mode_plugin". Declare ML Module "cc_plugin". Declare ML Module "ground_plugin". -Declare ML Module "recdef_plugin". (* Default substrings not considered by queries like SearchAbout *) -Add Search Blacklist "_subproof" "Private_". +Add Search Blacklist "_subproof" "_subterm" "Private_". diff --git a/theories/Init/Specif.v b/theories/Init/Specif.v index 9fc00e80..b6afba29 100644 --- a/theories/Init/Specif.v +++ b/theories/Init/Specif.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Basic specifications : sets that may contain logical information *) @@ -49,10 +51,20 @@ Notation "{ x | P & Q }" := (sig2 (fun x => P) (fun x => Q)) : type_scope. Notation "{ x : A | P }" := (sig (A:=A) (fun x => P)) : type_scope. Notation "{ x : A | P & Q }" := (sig2 (A:=A) (fun x => P) (fun x => Q)) : type_scope. +Notation "{ x & P }" := (sigT (fun x => P)) : type_scope. Notation "{ x : A & P }" := (sigT (A:=A) (fun x => P)) : type_scope. Notation "{ x : A & P & Q }" := (sigT2 (A:=A) (fun x => P) (fun x => Q)) : type_scope. +Notation "{ ' pat | P }" := (sig (fun pat => P)) : type_scope. +Notation "{ ' pat | P & Q }" := (sig2 (fun pat => P) (fun pat => Q)) : type_scope. +Notation "{ ' pat : A | P }" := (sig (A:=A) (fun pat => P)) : type_scope. +Notation "{ ' pat : A | P & Q }" := (sig2 (A:=A) (fun pat => P) (fun pat => Q)) : + type_scope. +Notation "{ ' pat : A & P }" := (sigT (A:=A) (fun pat => P)) : type_scope. +Notation "{ ' pat : A & P & Q }" := (sigT2 (A:=A) (fun pat => P) (fun pat => Q)) : + type_scope. + Add Printing Let sig. Add Printing Let sig2. Add Printing Let sigT. @@ -103,7 +115,7 @@ Definition sig_of_sig2 (A : Type) (P Q : A -> Prop) (X : sig2 P Q) : sig P of an [a] of type [A], a of a proof [h] that [a] satisfies [P], and a proof [h'] that [a] satisfies [Q]. Then [(proj1_sig (sig_of_sig2 y))] is the witness [a], - [(proj2_sig (sig_of_sig2 y))] is the proof of [(P a)], and + [(proj2_sig (sig_of_sig2 y))] is the proof of [(P a)], and [(proj3_sig y)] is the proof of [(Q a)]. *) Section Subset_projections2. @@ -190,6 +202,435 @@ Definition sig2_of_sigT2 (A : Type) (P Q : A -> Prop) (X : sigT2 P Q) : sig2 P Q Definition sigT2_of_sig2 (A : Type) (P Q : A -> Prop) (X : sig2 P Q) : sigT2 P Q := existT2 P Q (proj1_sig (sig_of_sig2 X)) (proj2_sig (sig_of_sig2 X)) (proj3_sig X). +(** η Principles *) +Definition sigT_eta {A P} (p : { a : A & P a }) + : p = existT _ (projT1 p) (projT2 p). +Proof. destruct p; reflexivity. Defined. + +Definition sig_eta {A P} (p : { a : A | P a }) + : p = exist _ (proj1_sig p) (proj2_sig p). +Proof. destruct p; reflexivity. Defined. + +Definition sigT2_eta {A P Q} (p : { a : A & P a & Q a }) + : p = existT2 _ _ (projT1 (sigT_of_sigT2 p)) (projT2 (sigT_of_sigT2 p)) (projT3 p). +Proof. destruct p; reflexivity. Defined. + +Definition sig2_eta {A P Q} (p : { a : A | P a & Q a }) + : p = exist2 _ _ (proj1_sig (sig_of_sig2 p)) (proj2_sig (sig_of_sig2 p)) (proj3_sig p). +Proof. destruct p; reflexivity. Defined. + +(** [exists x : A, B] is equivalent to [inhabited {x : A | B}] *) +Lemma exists_to_inhabited_sig {A P} : (exists x : A, P x) -> inhabited {x : A | P x}. +Proof. + intros [x y]. exact (inhabits (exist _ x y)). +Qed. + +Lemma inhabited_sig_to_exists {A P} : inhabited {x : A | P x} -> exists x : A, P x. +Proof. + intros [[x y]];exists x;exact y. +Qed. + +(** Equality of sigma types *) +Import EqNotations. +Local Notation "'rew' 'dependent' H 'in' H'" + := (match H with + | eq_refl => H' + end) + (at level 10, H' at level 10, + format "'[' 'rew' 'dependent' '/ ' H in '/' H' ']'"). + +(** Equality for [sigT] *) +Section sigT. + Local Unset Implicit Arguments. + (** Projecting an equality of a pair to equality of the first components *) + Definition projT1_eq {A} {P : A -> Type} {u v : { a : A & P a }} (p : u = v) + : projT1 u = projT1 v + := f_equal (@projT1 _ _) p. + + (** Projecting an equality of a pair to equality of the second components *) + Definition projT2_eq {A} {P : A -> Type} {u v : { a : A & P a }} (p : u = v) + : rew projT1_eq p in projT2 u = projT2 v + := rew dependent p in eq_refl. + + (** Equality of [sigT] is itself a [sigT] (forwards-reasoning version) *) + Definition eq_existT_uncurried {A : Type} {P : A -> Type} {u1 v1 : A} {u2 : P u1} {v2 : P v1} + (pq : { p : u1 = v1 & rew p in u2 = v2 }) + : existT _ u1 u2 = existT _ v1 v2. + Proof. + destruct pq as [p q]. + destruct q; simpl in *. + destruct p; reflexivity. + Defined. + + (** Equality of [sigT] is itself a [sigT] (backwards-reasoning version) *) + Definition eq_sigT_uncurried {A : Type} {P : A -> Type} (u v : { a : A & P a }) + (pq : { p : projT1 u = projT1 v & rew p in projT2 u = projT2 v }) + : u = v. + Proof. + destruct u as [u1 u2], v as [v1 v2]; simpl in *. + apply eq_existT_uncurried; exact pq. + Defined. + + (** Curried version of proving equality of sigma types *) + Definition eq_sigT {A : Type} {P : A -> Type} (u v : { a : A & P a }) + (p : projT1 u = projT1 v) (q : rew p in projT2 u = projT2 v) + : u = v + := eq_sigT_uncurried u v (existT _ p q). + + (** Equality of [sigT] when the property is an hProp *) + Definition eq_sigT_hprop {A P} (P_hprop : forall (x : A) (p q : P x), p = q) + (u v : { a : A & P a }) + (p : projT1 u = projT1 v) + : u = v + := eq_sigT u v p (P_hprop _ _ _). + + (** Equivalence of equality of [sigT] with a [sigT] of equality *) + (** We could actually prove an isomorphism here, and not just [<->], + but for simplicity, we don't. *) + Definition eq_sigT_uncurried_iff {A P} + (u v : { a : A & P a }) + : u = v <-> { p : projT1 u = projT1 v & rew p in projT2 u = projT2 v }. + Proof. + split; [ intro; subst; exists eq_refl; reflexivity | apply eq_sigT_uncurried ]. + Defined. + + (** Induction principle for [@eq (sigT _)] *) + Definition eq_sigT_rect {A P} {u v : { a : A & P a }} (Q : u = v -> Type) + (f : forall p q, Q (eq_sigT u v p q)) + : forall p, Q p. + Proof. intro p; specialize (f (projT1_eq p) (projT2_eq p)); destruct u, p; exact f. Defined. + Definition eq_sigT_rec {A P u v} (Q : u = v :> { a : A & P a } -> Set) := eq_sigT_rect Q. + Definition eq_sigT_ind {A P u v} (Q : u = v :> { a : A & P a } -> Prop) := eq_sigT_rec Q. + + (** Equivalence of equality of [sigT] involving hProps with equality of the first components *) + Definition eq_sigT_hprop_iff {A P} (P_hprop : forall (x : A) (p q : P x), p = q) + (u v : { a : A & P a }) + : u = v <-> (projT1 u = projT1 v) + := conj (fun p => f_equal (@projT1 _ _) p) (eq_sigT_hprop P_hprop u v). + + (** Non-dependent classification of equality of [sigT] *) + Definition eq_sigT_nondep {A B : Type} (u v : { a : A & B }) + (p : projT1 u = projT1 v) (q : projT2 u = projT2 v) + : u = v + := @eq_sigT _ _ u v p (eq_trans (rew_const _ _) q). + + (** Classification of transporting across an equality of [sigT]s *) + Lemma rew_sigT {A x} {P : A -> Type} (Q : forall a, P a -> Prop) (u : { p : P x & Q x p }) {y} (H : x = y) + : rew [fun a => { p : P a & Q a p }] H in u + = existT + (Q y) + (rew H in projT1 u) + (rew dependent H in (projT2 u)). + Proof. + destruct H, u; reflexivity. + Defined. +End sigT. + +(** Equality for [sig] *) +Section sig. + Local Unset Implicit Arguments. + (** Projecting an equality of a pair to equality of the first components *) + Definition proj1_sig_eq {A} {P : A -> Prop} {u v : { a : A | P a }} (p : u = v) + : proj1_sig u = proj1_sig v + := f_equal (@proj1_sig _ _) p. + + (** Projecting an equality of a pair to equality of the second components *) + Definition proj2_sig_eq {A} {P : A -> Prop} {u v : { a : A | P a }} (p : u = v) + : rew proj1_sig_eq p in proj2_sig u = proj2_sig v + := rew dependent p in eq_refl. + + (** Equality of [sig] is itself a [sig] (forwards-reasoning version) *) + Definition eq_exist_uncurried {A : Type} {P : A -> Prop} {u1 v1 : A} {u2 : P u1} {v2 : P v1} + (pq : { p : u1 = v1 | rew p in u2 = v2 }) + : exist _ u1 u2 = exist _ v1 v2. + Proof. + destruct pq as [p q]. + destruct q; simpl in *. + destruct p; reflexivity. + Defined. + + (** Equality of [sig] is itself a [sig] (backwards-reasoning version) *) + Definition eq_sig_uncurried {A : Type} {P : A -> Prop} (u v : { a : A | P a }) + (pq : { p : proj1_sig u = proj1_sig v | rew p in proj2_sig u = proj2_sig v }) + : u = v. + Proof. + destruct u as [u1 u2], v as [v1 v2]; simpl in *. + apply eq_exist_uncurried; exact pq. + Defined. + + (** Curried version of proving equality of sigma types *) + Definition eq_sig {A : Type} {P : A -> Prop} (u v : { a : A | P a }) + (p : proj1_sig u = proj1_sig v) (q : rew p in proj2_sig u = proj2_sig v) + : u = v + := eq_sig_uncurried u v (exist _ p q). + + (** Induction principle for [@eq (sig _)] *) + Definition eq_sig_rect {A P} {u v : { a : A | P a }} (Q : u = v -> Type) + (f : forall p q, Q (eq_sig u v p q)) + : forall p, Q p. + Proof. intro p; specialize (f (proj1_sig_eq p) (proj2_sig_eq p)); destruct u, p; exact f. Defined. + Definition eq_sig_rec {A P u v} (Q : u = v :> { a : A | P a } -> Set) := eq_sig_rect Q. + Definition eq_sig_ind {A P u v} (Q : u = v :> { a : A | P a } -> Prop) := eq_sig_rec Q. + + (** Equality of [sig] when the property is an hProp *) + Definition eq_sig_hprop {A} {P : A -> Prop} (P_hprop : forall (x : A) (p q : P x), p = q) + (u v : { a : A | P a }) + (p : proj1_sig u = proj1_sig v) + : u = v + := eq_sig u v p (P_hprop _ _ _). + + (** Equivalence of equality of [sig] with a [sig] of equality *) + (** We could actually prove an isomorphism here, and not just [<->], + but for simplicity, we don't. *) + Definition eq_sig_uncurried_iff {A} {P : A -> Prop} + (u v : { a : A | P a }) + : u = v <-> { p : proj1_sig u = proj1_sig v | rew p in proj2_sig u = proj2_sig v }. + Proof. + split; [ intro; subst; exists eq_refl; reflexivity | apply eq_sig_uncurried ]. + Defined. + + (** Equivalence of equality of [sig] involving hProps with equality of the first components *) + Definition eq_sig_hprop_iff {A} {P : A -> Prop} (P_hprop : forall (x : A) (p q : P x), p = q) + (u v : { a : A | P a }) + : u = v <-> (proj1_sig u = proj1_sig v) + := conj (fun p => f_equal (@proj1_sig _ _) p) (eq_sig_hprop P_hprop u v). + + Lemma rew_sig {A x} {P : A -> Type} (Q : forall a, P a -> Prop) (u : { p : P x | Q x p }) {y} (H : x = y) + : rew [fun a => { p : P a | Q a p }] H in u + = exist + (Q y) + (rew H in proj1_sig u) + (rew dependent H in proj2_sig u). + Proof. + destruct H, u; reflexivity. + Defined. +End sig. + +(** Equality for [sigT] *) +Section sigT2. + (* We make [sigT_of_sigT2] a coercion so we can use [projT1], [projT2] on [sigT2] *) + Local Coercion sigT_of_sigT2 : sigT2 >-> sigT. + Local Unset Implicit Arguments. + (** Projecting an equality of a pair to equality of the first components *) + Definition sigT_of_sigT2_eq {A} {P Q : A -> Type} {u v : { a : A & P a & Q a }} (p : u = v) + : u = v :> { a : A & P a } + := f_equal _ p. + Definition projT1_of_sigT2_eq {A} {P Q : A -> Type} {u v : { a : A & P a & Q a }} (p : u = v) + : projT1 u = projT1 v + := projT1_eq (sigT_of_sigT2_eq p). + + (** Projecting an equality of a pair to equality of the second components *) + Definition projT2_of_sigT2_eq {A} {P Q : A -> Type} {u v : { a : A & P a & Q a }} (p : u = v) + : rew projT1_of_sigT2_eq p in projT2 u = projT2 v + := rew dependent p in eq_refl. + + (** Projecting an equality of a pair to equality of the third components *) + Definition projT3_eq {A} {P Q : A -> Type} {u v : { a : A & P a & Q a }} (p : u = v) + : rew projT1_of_sigT2_eq p in projT3 u = projT3 v + := rew dependent p in eq_refl. + + (** Equality of [sigT2] is itself a [sigT2] (forwards-reasoning version) *) + Definition eq_existT2_uncurried {A : Type} {P Q : A -> Type} + {u1 v1 : A} {u2 : P u1} {v2 : P v1} {u3 : Q u1} {v3 : Q v1} + (pqr : { p : u1 = v1 + & rew p in u2 = v2 & rew p in u3 = v3 }) + : existT2 _ _ u1 u2 u3 = existT2 _ _ v1 v2 v3. + Proof. + destruct pqr as [p q r]. + destruct r, q, p; simpl. + reflexivity. + Defined. + + (** Equality of [sigT2] is itself a [sigT2] (backwards-reasoning version) *) + Definition eq_sigT2_uncurried {A : Type} {P Q : A -> Type} (u v : { a : A & P a & Q a }) + (pqr : { p : projT1 u = projT1 v + & rew p in projT2 u = projT2 v & rew p in projT3 u = projT3 v }) + : u = v. + Proof. + destruct u as [u1 u2 u3], v as [v1 v2 v3]; simpl in *. + apply eq_existT2_uncurried; exact pqr. + Defined. + + (** Curried version of proving equality of sigma types *) + Definition eq_sigT2 {A : Type} {P Q : A -> Type} (u v : { a : A & P a & Q a }) + (p : projT1 u = projT1 v) + (q : rew p in projT2 u = projT2 v) + (r : rew p in projT3 u = projT3 v) + : u = v + := eq_sigT2_uncurried u v (existT2 _ _ p q r). + + (** Equality of [sigT2] when the second property is an hProp *) + Definition eq_sigT2_hprop {A P Q} (Q_hprop : forall (x : A) (p q : Q x), p = q) + (u v : { a : A & P a & Q a }) + (p : u = v :> { a : A & P a }) + : u = v + := eq_sigT2 u v (projT1_eq p) (projT2_eq p) (Q_hprop _ _ _). + + (** Equivalence of equality of [sigT2] with a [sigT2] of equality *) + (** We could actually prove an isomorphism here, and not just [<->], + but for simplicity, we don't. *) + Definition eq_sigT2_uncurried_iff {A P Q} + (u v : { a : A & P a & Q a }) + : u = v + <-> { p : projT1 u = projT1 v + & rew p in projT2 u = projT2 v & rew p in projT3 u = projT3 v }. + Proof. + split; [ intro; subst; exists eq_refl; reflexivity | apply eq_sigT2_uncurried ]. + Defined. + + (** Induction principle for [@eq (sigT2 _ _)] *) + Definition eq_sigT2_rect {A P Q} {u v : { a : A & P a & Q a }} (R : u = v -> Type) + (f : forall p q r, R (eq_sigT2 u v p q r)) + : forall p, R p. + Proof. + intro p. + specialize (f (projT1_of_sigT2_eq p) (projT2_of_sigT2_eq p) (projT3_eq p)). + destruct u, p; exact f. + Defined. + Definition eq_sigT2_rec {A P Q u v} (R : u = v :> { a : A & P a & Q a } -> Set) := eq_sigT2_rect R. + Definition eq_sigT2_ind {A P Q u v} (R : u = v :> { a : A & P a & Q a } -> Prop) := eq_sigT2_rec R. + + (** Equivalence of equality of [sigT2] involving hProps with equality of the first components *) + Definition eq_sigT2_hprop_iff {A P Q} (Q_hprop : forall (x : A) (p q : Q x), p = q) + (u v : { a : A & P a & Q a }) + : u = v <-> (u = v :> { a : A & P a }) + := conj (fun p => f_equal (@sigT_of_sigT2 _ _ _) p) (eq_sigT2_hprop Q_hprop u v). + + (** Non-dependent classification of equality of [sigT] *) + Definition eq_sigT2_nondep {A B C : Type} (u v : { a : A & B & C }) + (p : projT1 u = projT1 v) (q : projT2 u = projT2 v) (r : projT3 u = projT3 v) + : u = v + := @eq_sigT2 _ _ _ u v p (eq_trans (rew_const _ _) q) (eq_trans (rew_const _ _) r). + + (** Classification of transporting across an equality of [sigT2]s *) + Lemma rew_sigT2 {A x} {P : A -> Type} (Q R : forall a, P a -> Prop) + (u : { p : P x & Q x p & R x p }) + {y} (H : x = y) + : rew [fun a => { p : P a & Q a p & R a p }] H in u + = existT2 + (Q y) + (R y) + (rew H in projT1 u) + (rew dependent H in projT2 u) + (rew dependent H in projT3 u). + Proof. + destruct H, u; reflexivity. + Defined. +End sigT2. + +(** Equality for [sig2] *) +Section sig2. + (* We make [sig_of_sig2] a coercion so we can use [proj1], [proj2] on [sig2] *) + Local Coercion sig_of_sig2 : sig2 >-> sig. + Local Unset Implicit Arguments. + (** Projecting an equality of a pair to equality of the first components *) + Definition sig_of_sig2_eq {A} {P Q : A -> Prop} {u v : { a : A | P a & Q a }} (p : u = v) + : u = v :> { a : A | P a } + := f_equal _ p. + Definition proj1_sig_of_sig2_eq {A} {P Q : A -> Prop} {u v : { a : A | P a & Q a }} (p : u = v) + : proj1_sig u = proj1_sig v + := proj1_sig_eq (sig_of_sig2_eq p). + + (** Projecting an equality of a pair to equality of the second components *) + Definition proj2_sig_of_sig2_eq {A} {P Q : A -> Prop} {u v : { a : A | P a & Q a }} (p : u = v) + : rew proj1_sig_of_sig2_eq p in proj2_sig u = proj2_sig v + := rew dependent p in eq_refl. + + (** Projecting an equality of a pair to equality of the third components *) + Definition proj3_sig_eq {A} {P Q : A -> Prop} {u v : { a : A | P a & Q a }} (p : u = v) + : rew proj1_sig_of_sig2_eq p in proj3_sig u = proj3_sig v + := rew dependent p in eq_refl. + + (** Equality of [sig2] is itself a [sig2] (fowards-reasoning version) *) + Definition eq_exist2_uncurried {A} {P Q : A -> Prop} + {u1 v1 : A} {u2 : P u1} {v2 : P v1} {u3 : Q u1} {v3 : Q v1} + (pqr : { p : u1 = v1 + | rew p in u2 = v2 & rew p in u3 = v3 }) + : exist2 _ _ u1 u2 u3 = exist2 _ _ v1 v2 v3. + Proof. + destruct pqr as [p q r]. + destruct r, q, p; simpl. + reflexivity. + Defined. + + (** Equality of [sig2] is itself a [sig2] (backwards-reasoning version) *) + Definition eq_sig2_uncurried {A} {P Q : A -> Prop} (u v : { a : A | P a & Q a }) + (pqr : { p : proj1_sig u = proj1_sig v + | rew p in proj2_sig u = proj2_sig v & rew p in proj3_sig u = proj3_sig v }) + : u = v. + Proof. + destruct u as [u1 u2 u3], v as [v1 v2 v3]; simpl in *. + apply eq_exist2_uncurried; exact pqr. + Defined. + + (** Curried version of proving equality of sigma types *) + Definition eq_sig2 {A} {P Q : A -> Prop} (u v : { a : A | P a & Q a }) + (p : proj1_sig u = proj1_sig v) + (q : rew p in proj2_sig u = proj2_sig v) + (r : rew p in proj3_sig u = proj3_sig v) + : u = v + := eq_sig2_uncurried u v (exist2 _ _ p q r). + + (** Equality of [sig2] when the second property is an hProp *) + Definition eq_sig2_hprop {A} {P Q : A -> Prop} (Q_hprop : forall (x : A) (p q : Q x), p = q) + (u v : { a : A | P a & Q a }) + (p : u = v :> { a : A | P a }) + : u = v + := eq_sig2 u v (proj1_sig_eq p) (proj2_sig_eq p) (Q_hprop _ _ _). + + (** Equivalence of equality of [sig2] with a [sig2] of equality *) + (** We could actually prove an isomorphism here, and not just [<->], + but for simplicity, we don't. *) + Definition eq_sig2_uncurried_iff {A P Q} + (u v : { a : A | P a & Q a }) + : u = v + <-> { p : proj1_sig u = proj1_sig v + | rew p in proj2_sig u = proj2_sig v & rew p in proj3_sig u = proj3_sig v }. + Proof. + split; [ intro; subst; exists eq_refl; reflexivity | apply eq_sig2_uncurried ]. + Defined. + + (** Induction principle for [@eq (sig2 _ _)] *) + Definition eq_sig2_rect {A P Q} {u v : { a : A | P a & Q a }} (R : u = v -> Type) + (f : forall p q r, R (eq_sig2 u v p q r)) + : forall p, R p. + Proof. + intro p. + specialize (f (proj1_sig_of_sig2_eq p) (proj2_sig_of_sig2_eq p) (proj3_sig_eq p)). + destruct u, p; exact f. + Defined. + Definition eq_sig2_rec {A P Q u v} (R : u = v :> { a : A | P a & Q a } -> Set) := eq_sig2_rect R. + Definition eq_sig2_ind {A P Q u v} (R : u = v :> { a : A | P a & Q a } -> Prop) := eq_sig2_rec R. + + (** Equivalence of equality of [sig2] involving hProps with equality of the first components *) + Definition eq_sig2_hprop_iff {A} {P Q : A -> Prop} (Q_hprop : forall (x : A) (p q : Q x), p = q) + (u v : { a : A | P a & Q a }) + : u = v <-> (u = v :> { a : A | P a }) + := conj (fun p => f_equal (@sig_of_sig2 _ _ _) p) (eq_sig2_hprop Q_hprop u v). + + (** Non-dependent classification of equality of [sig] *) + Definition eq_sig2_nondep {A} {B C : Prop} (u v : @sig2 A (fun _ => B) (fun _ => C)) + (p : proj1_sig u = proj1_sig v) (q : proj2_sig u = proj2_sig v) (r : proj3_sig u = proj3_sig v) + : u = v + := @eq_sig2 _ _ _ u v p (eq_trans (rew_const _ _) q) (eq_trans (rew_const _ _) r). + + (** Classification of transporting across an equality of [sig2]s *) + Lemma rew_sig2 {A x} {P : A -> Type} (Q R : forall a, P a -> Prop) + (u : { p : P x | Q x p & R x p }) + {y} (H : x = y) + : rew [fun a => { p : P a | Q a p & R a p }] H in u + = exist2 + (Q y) + (R y) + (rew H in proj1_sig u) + (rew dependent H in proj2_sig u) + (rew dependent H in proj3_sig u). + Proof. + destruct H, u; reflexivity. + Defined. +End sig2. + + (** [sumbool] is a boolean type equipped with the justification of their value *) @@ -263,10 +704,10 @@ Section Dependent_choice_lemmas. (forall x:X, {y | R x y}) -> forall x0, {f : nat -> X | f O = x0 /\ forall n, R (f n) (f (S n))}. Proof. - intros H x0. + intros H x0. set (f:=fix f n := match n with O => x0 | S n' => proj1_sig (H (f n')) end). exists f. - split. reflexivity. + split. reflexivity. induction n; simpl; apply proj2_sig. Defined. @@ -304,16 +745,16 @@ Hint Resolve exist exist2 existT existT2: core. (* Compatibility *) -Notation sigS := sigT (compat "8.2"). -Notation existS := existT (compat "8.2"). -Notation sigS_rect := sigT_rect (compat "8.2"). -Notation sigS_rec := sigT_rec (compat "8.2"). -Notation sigS_ind := sigT_ind (compat "8.2"). -Notation projS1 := projT1 (compat "8.2"). -Notation projS2 := projT2 (compat "8.2"). - -Notation sigS2 := sigT2 (compat "8.2"). -Notation existS2 := existT2 (compat "8.2"). -Notation sigS2_rect := sigT2_rect (compat "8.2"). -Notation sigS2_rec := sigT2_rec (compat "8.2"). -Notation sigS2_ind := sigT2_ind (compat "8.2"). +Notation sigS := sigT (compat "8.6"). +Notation existS := existT (compat "8.6"). +Notation sigS_rect := sigT_rect (compat "8.6"). +Notation sigS_rec := sigT_rec (compat "8.6"). +Notation sigS_ind := sigT_ind (compat "8.6"). +Notation projS1 := projT1 (compat "8.6"). +Notation projS2 := projT2 (compat "8.6"). + +Notation sigS2 := sigT2 (compat "8.6"). +Notation existS2 := existT2 (compat "8.6"). +Notation sigS2_rect := sigT2_rect (compat "8.6"). +Notation sigS2_rec := sigT2_rec (compat "8.6"). +Notation sigS2_ind := sigT2_ind (compat "8.6"). diff --git a/theories/Init/Tactics.v b/theories/Init/Tactics.v index 5d1e87ae..8df533e7 100644 --- a/theories/Init/Tactics.v +++ b/theories/Init/Tactics.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Notations. @@ -236,3 +238,93 @@ Tactic Notation "clear" "dependent" hyp(h) := Tactic Notation "revert" "dependent" hyp(h) := generalize dependent h. + +(** Provide an error message for dependent induction that reports an import is +required to use it. Importing Coq.Program.Equality will shadow this notation +with the actual [dependent induction] tactic. *) + +Tactic Notation "dependent" "induction" ident(H) := + fail "To use dependent induction, first [Require Import Coq.Program.Equality.]". + +(** *** [inversion_sigma] *) +(** The built-in [inversion] will frequently leave equalities of + dependent pairs. When the first type in the pair is an hProp or + otherwise simplifies, [inversion_sigma] is useful; it will replace + the equality of pairs with a pair of equalities, one involving a + term casted along the other. This might also prove useful for + writing a version of [inversion] / [dependent destruction] which + does not lose information, i.e., does not turn a goal which is + provable into one which requires axiom K / UIP. *) + +Ltac simpl_proj_exist_in H := + repeat match type of H with + | context G[proj1_sig (exist _ ?x ?p)] + => let G' := context G[x] in change G' in H + | context G[proj2_sig (exist _ ?x ?p)] + => let G' := context G[p] in change G' in H + | context G[projT1 (existT _ ?x ?p)] + => let G' := context G[x] in change G' in H + | context G[projT2 (existT _ ?x ?p)] + => let G' := context G[p] in change G' in H + | context G[proj3_sig (exist2 _ _ ?x ?p ?q)] + => let G' := context G[q] in change G' in H + | context G[projT3 (existT2 _ _ ?x ?p ?q)] + => let G' := context G[q] in change G' in H + | context G[sig_of_sig2 (@exist2 ?A ?P ?Q ?x ?p ?q)] + => let G' := context G[@exist A P x p] in change G' in H + | context G[sigT_of_sigT2 (@existT2 ?A ?P ?Q ?x ?p ?q)] + => let G' := context G[@existT A P x p] in change G' in H + end. +Ltac induction_sigma_in_using H rect := + let H0 := fresh H in + let H1 := fresh H in + induction H as [H0 H1] using (rect _ _ _ _); + simpl_proj_exist_in H0; + simpl_proj_exist_in H1. +Ltac induction_sigma2_in_using H rect := + let H0 := fresh H in + let H1 := fresh H in + let H2 := fresh H in + induction H as [H0 H1 H2] using (rect _ _ _ _ _); + simpl_proj_exist_in H0; + simpl_proj_exist_in H1; + simpl_proj_exist_in H2. +Ltac inversion_sigma_step := + match goal with + | [ H : _ = exist _ _ _ |- _ ] + => induction_sigma_in_using H @eq_sig_rect + | [ H : _ = existT _ _ _ |- _ ] + => induction_sigma_in_using H @eq_sigT_rect + | [ H : exist _ _ _ = _ |- _ ] + => induction_sigma_in_using H @eq_sig_rect + | [ H : existT _ _ _ = _ |- _ ] + => induction_sigma_in_using H @eq_sigT_rect + | [ H : _ = exist2 _ _ _ _ _ |- _ ] + => induction_sigma2_in_using H @eq_sig2_rect + | [ H : _ = existT2 _ _ _ _ _ |- _ ] + => induction_sigma2_in_using H @eq_sigT2_rect + | [ H : exist2 _ _ _ _ _ = _ |- _ ] + => induction_sigma_in_using H @eq_sig2_rect + | [ H : existT2 _ _ _ _ _ = _ |- _ ] + => induction_sigma_in_using H @eq_sigT2_rect + end. +Ltac inversion_sigma := repeat inversion_sigma_step. + +(** A version of [time] that works for constrs *) + +Ltac time_constr tac := + let eval_early := match goal with _ => restart_timer end in + let ret := tac () in + let eval_early := match goal with _ => finish_timing ( "Tactic evaluation" ) end in + ret. + +(** Useful combinators *) + +Ltac assert_fails tac := + tryif tac then fail 0 tac "succeeds" else idtac. +Ltac assert_succeeds tac := + tryif (assert_fails tac) then fail 0 tac "fails" else idtac. +Tactic Notation "assert_succeeds" tactic3(tac) := + assert_succeeds tac. +Tactic Notation "assert_fails" tactic3(tac) := + assert_fails tac. diff --git a/theories/Init/Tauto.v b/theories/Init/Tauto.v index 1e409607..87b7a9a3 100644 --- a/theories/Init/Tauto.v +++ b/theories/Init/Tauto.v @@ -2,7 +2,7 @@ Require Import Notations. Require Import Datatypes. Require Import Logic. -Local Declare ML Module "tauto". +Declare ML Module "tauto_plugin". Local Ltac not_dep_intros := repeat match goal with @@ -27,7 +27,7 @@ Local Ltac simplif flags := | id: ?X1 |- _ => is_disj flags X1; elim id; intro; clear id | id0: (forall (_: ?X1), ?X2), id1: ?X1|- _ => (* generalize (id0 id1); intro; clear id0 does not work - (see Marco Maggiesi's bug PR#301) + (see Marco Maggiesi's BZ#301) so we instead use Assert and exact. *) assert X2; [exact (id0 id1) | clear id0] | id: forall (_ : ?X1), ?X2|- _ => diff --git a/theories/Init/Wf.v b/theories/Init/Wf.v index b5b17e5e..c27ffa33 100644 --- a/theories/Init/Wf.v +++ b/theories/Init/Wf.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * This module proves the validity of diff --git a/theories/Init/_CoqProject b/theories/Init/_CoqProject new file mode 100644 index 00000000..bff79d34 --- /dev/null +++ b/theories/Init/_CoqProject @@ -0,0 +1,2 @@ +-R .. Coq +-arg -noinit diff --git a/theories/Init/vo.itarget b/theories/Init/vo.itarget deleted file mode 100644 index 99877065..00000000 --- a/theories/Init/vo.itarget +++ /dev/null @@ -1,11 +0,0 @@ -Datatypes.vo -Logic_Type.vo -Logic.vo -Notations.vo -Peano.vo -Prelude.vo -Specif.vo -Tactics.vo -Wf.vo -Nat.vo -Tauto.vo diff --git a/theories/Lists/List.v b/theories/Lists/List.v index 30f1dec2..ca5f154e 100644 --- a/theories/Lists/List.v +++ b/theories/Lists/List.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Setoid. @@ -27,7 +29,6 @@ Module ListNotations. Notation "[ ]" := nil (format "[ ]") : list_scope. Notation "[ x ]" := (cons x nil) : list_scope. Notation "[ x ; y ; .. ; z ]" := (cons x (cons y .. (cons z nil) ..)) : list_scope. -Notation "[ x ; .. ; y ]" := (cons x .. (cons y nil) ..) (compat "8.4") : list_scope. End ListNotations. Import ListNotations. @@ -419,7 +420,7 @@ Section Elts. Proof. unfold lt; induction n as [| n hn]; simpl. - destruct l; simpl; [ inversion 2 | auto ]. - - destruct l as [| a l hl]; simpl. + - destruct l; simpl. * inversion 2. * intros d ie; right; apply hn; auto with arith. Qed. @@ -1280,7 +1281,7 @@ End Fold_Right_Recursor. partition l = ([], []) <-> l = []. Proof. split. - - destruct l as [|a l' _]. + - destruct l as [|a l']. * intuition. * simpl. destruct (f a), (partition l'); now intros [= -> ->]. - now intros ->. @@ -2110,13 +2111,13 @@ Section Exists_Forall. {Exists l} + {~ Exists l}. Proof. intro Pdec. induction l as [|a l' Hrec]. - - right. now rewrite Exists_nil. + - right. abstract now rewrite Exists_nil. - destruct Hrec as [Hl'|Hl']. * left. now apply Exists_cons_tl. * destruct (Pdec a) as [Ha|Ha]. + left. now apply Exists_cons_hd. - + right. now inversion_clear 1. - Qed. + + right. abstract now inversion 1. + Defined. Inductive Forall : list A -> Prop := | Forall_nil : Forall nil @@ -2152,9 +2153,9 @@ Section Exists_Forall. - destruct Hrec as [Hl'|Hl']. + destruct (Pdec a) as [Ha|Ha]. * left. now apply Forall_cons. - * right. now inversion_clear 1. - + right. now inversion_clear 1. - Qed. + * right. abstract now inversion 1. + + right. abstract now inversion 1. + Defined. End One_predicate. @@ -2179,6 +2180,16 @@ Section Exists_Forall. * now apply Exists_cons_hd. Qed. + Lemma neg_Forall_Exists_neg (P:A->Prop) (l:list A) : + (forall x:A, {P x} + { ~ P x }) -> + ~ Forall P l -> + Exists (fun x => ~ P x) l. + Proof. + intro Dec. + apply Exists_Forall_neg; intros. + destruct (Dec x); auto. + Qed. + Lemma Forall_Exists_dec (P:A->Prop) : (forall x:A, {P x} + { ~ P x }) -> forall l:list A, @@ -2186,9 +2197,8 @@ Section Exists_Forall. Proof. intros Pdec l. destruct (Forall_dec P Pdec l); [left|right]; trivial. - apply Exists_Forall_neg; trivial. - intro x. destruct (Pdec x); [now left|now right]. - Qed. + now apply neg_Forall_Exists_neg. + Defined. Lemma Forall_impl : forall (P Q : A -> Prop), (forall a, P a -> Q a) -> forall l, Forall P l -> Forall Q l. diff --git a/theories/Lists/ListDec.v b/theories/Lists/ListDec.v index 3e2eeac0..e7e2cfc8 100644 --- a/theories/Lists/ListDec.v +++ b/theories/Lists/ListDec.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Decidability results about lists *) diff --git a/theories/Lists/ListSet.v b/theories/Lists/ListSet.v index 655d3940..cc7d6f55 100644 --- a/theories/Lists/ListSet.v +++ b/theories/Lists/ListSet.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** A library for finite sets, implemented as lists *) diff --git a/theories/Lists/ListTactics.v b/theories/Lists/ListTactics.v index 537d5f68..843e3835 100644 --- a/theories/Lists/ListTactics.v +++ b/theories/Lists/ListTactics.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import BinPos. diff --git a/theories/Lists/SetoidList.v b/theories/Lists/SetoidList.v index c95fb4d5..0c5fe55b 100644 --- a/theories/Lists/SetoidList.v +++ b/theories/Lists/SetoidList.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) Require Export List. Require Export Sorted. diff --git a/theories/Lists/SetoidPermutation.v b/theories/Lists/SetoidPermutation.v index cea3e839..24b96514 100644 --- a/theories/Lists/SetoidPermutation.v +++ b/theories/Lists/SetoidPermutation.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) Require Import Permutation SetoidList. (* Set Universe Polymorphism. *) diff --git a/theories/Lists/StreamMemo.v b/theories/Lists/StreamMemo.v index 5a16cc43..57f558de 100644 --- a/theories/Lists/StreamMemo.v +++ b/theories/Lists/StreamMemo.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Eqdep_dec. diff --git a/theories/Lists/Streams.v b/theories/Lists/Streams.v index 1c302b22..8a01b8fb 100644 --- a/theories/Lists/Streams.v +++ b/theories/Lists/Streams.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Set Implicit Arguments. @@ -194,7 +196,7 @@ Lemma ForAll_map : forall (P:Stream B -> Prop) (S:Stream A), ForAll (fun s => P (map s)) S <-> ForAll P (map S). Proof. intros P S. -split; generalize S; clear S; cofix; intros S; constructor; +split; generalize S; clear S; cofix ForAll_map; intros S; constructor; destruct H as [H0 H]; firstorder. Qed. diff --git a/theories/Lists/vo.itarget b/theories/Lists/vo.itarget deleted file mode 100644 index 82dd1be8..00000000 --- a/theories/Lists/vo.itarget +++ /dev/null @@ -1,8 +0,0 @@ -ListSet.vo -ListTactics.vo -List.vo -ListDec.vo -SetoidList.vo -SetoidPermutation.vo -StreamMemo.vo -Streams.vo diff --git a/theories/Logic/Berardi.v b/theories/Logic/Berardi.v index 1e0bd0fe..c6836a1c 100644 --- a/theories/Logic/Berardi.v +++ b/theories/Logic/Berardi.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** This file formalizes Berardi's paradox which says that in diff --git a/theories/Logic/ChoiceFacts.v b/theories/Logic/ChoiceFacts.v index 1420a000..238ac7df 100644 --- a/theories/Logic/ChoiceFacts.v +++ b/theories/Logic/ChoiceFacts.v @@ -1,93 +1,35 @@ -(* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (************************************************************************) (** Some facts and definitions concerning choice and description in - intuitionistic logic. - -We investigate the relations between the following choice and -description principles - -- AC_rel = relational form of the (non extensional) axiom of choice - (a "set-theoretic" axiom of choice) -- AC_fun = functional form of the (non extensional) axiom of choice - (a "type-theoretic" axiom of choice) -- DC_fun = functional form of the dependent axiom of choice -- ACw_fun = functional form of the countable axiom of choice -- AC! = functional relation reification - (known as axiom of unique choice in topos theory, - sometimes called principle of definite description in - the context of constructive type theory) - -- GAC_rel = guarded relational form of the (non extensional) axiom of choice -- GAC_fun = guarded functional form of the (non extensional) axiom of choice -- GAC! = guarded functional relation reification - -- OAC_rel = "omniscient" relational form of the (non extensional) axiom of choice -- OAC_fun = "omniscient" functional form of the (non extensional) axiom of choice - (called AC* in Bell [[Bell]]) -- OAC! - -- ID_iota = intuitionistic definite description -- ID_epsilon = intuitionistic indefinite description - -- D_iota = (weakly classical) definite description principle -- D_epsilon = (weakly classical) indefinite description principle - -- PI = proof irrelevance -- IGP = independence of general premises - (an unconstrained generalisation of the constructive principle of - independence of premises) -- Drinker = drinker's paradox (small form) - (called Ex in Bell [[Bell]]) - -We let also - -- IPL_2 = 2nd-order impredicative minimal predicate logic (with ex. quant.) -- IPL^2 = 2nd-order functional minimal predicate logic (with ex. quant.) -- IPL_2^2 = 2nd-order impredicative, 2nd-order functional minimal pred. logic (with ex. quant.) - -with no prerequisite on the non-emptiness of domains - -Table of contents - -1. Definitions - -2. IPL_2^2 |- AC_rel + AC! = AC_fun - -3.1. typed IPL_2 + Sigma-types + PI |- AC_rel = GAC_rel and IPL_2 |- AC_rel + IGP -> GAC_rel and IPL_2 |- GAC_rel = OAC_rel - -3.2. IPL^2 |- AC_fun + IGP = GAC_fun = OAC_fun = AC_fun + Drinker - -3.3. D_iota -> ID_iota and D_epsilon <-> ID_epsilon + Drinker - -4. Derivability of choice for decidable relations with well-ordered codomain - -5. Equivalence of choices on dependent or non dependent functional types - -6. Non contradiction of constructive descriptions wrt functional choices - -7. Definite description transports classical logic to the computational world - -8. Choice -> Dependent choice -> Countable choice - -References: - + intuitionistic logic. *) +(** * References: *) +(** [[Bell]] John L. Bell, Choice principles in intuitionistic set theory, unpublished. [[Bell93]] John L. Bell, Hilbert's Epsilon Operator in Intuitionistic Type Theories, Mathematical Logic Quarterly, volume 39, 1993. +[[Carlström04]] Jesper Carlström, EM + Ext_ + AC_int is equivalent to +AC_ext, Mathematical Logic Quaterly, vol 50(3), pp 236-240, 2004. + [[Carlström05]] Jesper Carlström, Interpreting descriptions in intentional type theory, Journal of Symbolic Logic 70(2):488-514, 2005. + +[[Werner97]] Benjamin Werner, Sets in Types, Types in Sets, TACS, 1997. *) +Require Import RelationClasses Logic. + Set Implicit Arguments. Local Unset Intuition Negation Unfolding. @@ -108,59 +50,139 @@ Variables A B :Type. Variable P:A->Prop. -Variable R:A->B->Prop. - (** ** Constructive choice and description *) -(** AC_rel *) +(** AC_rel = relational form of the (non extensional) axiom of choice + (a "set-theoretic" axiom of choice) *) Definition RelationalChoice_on := forall R:A->B->Prop, (forall x : A, exists y : B, R x y) -> (exists R' : A->B->Prop, subrelation R' R /\ forall x, exists! y, R' x y). -(** AC_fun *) +(** AC_fun = functional form of the (non extensional) axiom of choice + (a "type-theoretic" axiom of choice) *) + +(* Note: This is called Type-Theoretic Description Axiom (TTDA) in + [[Werner97]] (using a non-standard meaning of "description"). This + is called intensional axiom of choice (AC_int) in [[Carlström04]] *) + +Definition FunctionalChoice_on_rel (R:A->B->Prop) := + (forall x:A, exists y : B, R x y) -> + exists f : A -> B, (forall x:A, R x (f x)). Definition FunctionalChoice_on := forall R:A->B->Prop, (forall x : A, exists y : B, R x y) -> (exists f : A->B, forall x : A, R x (f x)). -(** DC_fun *) +(** AC_fun_dep = functional form of the (non extensional) axiom of + choice, with dependent functions *) +Definition DependentFunctionalChoice_on (A:Type) (B:A -> Type) := + forall R:forall x:A, B x -> Prop, + (forall x:A, exists y : B x, R x y) -> + (exists f : (forall x:A, B x), forall x:A, R x (f x)). + +(** AC_trunc = axiom of choice for propositional truncations + (truncation and quantification commute) *) +Definition InhabitedForallCommute_on (A : Type) (B : A -> Type) := + (forall x, inhabited (B x)) -> inhabited (forall x, B x). + +(** DC_fun = functional form of the dependent axiom of choice *) Definition FunctionalDependentChoice_on := forall (R:A->A->Prop), (forall x, exists y, R x y) -> forall x0, (exists f : nat -> A, f 0 = x0 /\ forall n, R (f n) (f (S n))). -(** ACw_fun *) +(** ACw_fun = functional form of the countable axiom of choice *) Definition FunctionalCountableChoice_on := forall (R:nat->A->Prop), (forall n, exists y, R n y) -> (exists f : nat -> A, forall n, R n (f n)). -(** AC! or Functional Relation Reification (known as Axiom of Unique Choice - in topos theory; also called principle of definite description *) +(** AC! = functional relation reification + (known as axiom of unique choice in topos theory, + sometimes called principle of definite description in + the context of constructive type theory, sometimes + called axiom of no choice) *) Definition FunctionalRelReification_on := forall R:A->B->Prop, (forall x : A, exists! y : B, R x y) -> (exists f : A->B, forall x : A, R x (f x)). -(** ID_epsilon (constructive version of indefinite description; - combined with proof-irrelevance, it may be connected to - Carlström's type theory with a constructive indefinite description - operator) *) +(** AC_dep! = functional relation reification, with dependent functions + see AC! *) +Definition DependentFunctionalRelReification_on (A:Type) (B:A -> Type) := + forall (R:forall x:A, B x -> Prop), + (forall x:A, exists! y : B x, R x y) -> + (exists f : (forall x:A, B x), forall x:A, R x (f x)). + +(** AC_fun_repr = functional choice of a representative in an equivalence class *) + +(* Note: This is called Type-Theoretic Choice Axiom (TTCA) in + [[Werner97]] (by reference to the extensional set-theoretic + formulation of choice); Note also a typo in its intended + formulation in [[Werner97]]. *) + +Definition RepresentativeFunctionalChoice_on := + forall R:A->A->Prop, + (Equivalence R) -> + (exists f : A->A, forall x : A, (R x (f x)) /\ forall x', R x x' -> f x = f x'). + +(** AC_fun_setoid = functional form of the (so-called extensional) axiom of + choice from setoids *) + +Definition SetoidFunctionalChoice_on := + forall R : A -> A -> Prop, + forall T : A -> B -> Prop, + Equivalence R -> + (forall x x' y, R x x' -> T x y -> T x' y) -> + (forall x, exists y, T x y) -> + exists f : A -> B, forall x : A, T x (f x) /\ (forall x' : A, R x x' -> f x = f x'). + +(** AC_fun_setoid_gen = functional form of the general form of the (so-called + extensional) axiom of choice over setoids *) + +(* Note: This is called extensional axiom of choice (AC_ext) in + [[Carlström04]]. *) + +Definition GeneralizedSetoidFunctionalChoice_on := + forall R : A -> A -> Prop, + forall S : B -> B -> Prop, + forall T : A -> B -> Prop, + Equivalence R -> + Equivalence S -> + (forall x x' y y', R x x' -> S y y' -> T x y -> T x' y') -> + (forall x, exists y, T x y) -> + exists f : A -> B, + forall x : A, T x (f x) /\ (forall x' : A, R x x' -> S (f x) (f x')). + +(** AC_fun_setoid_simple = functional form of the (so-called extensional) axiom of + choice from setoids on locally compatible relations *) + +Definition SimpleSetoidFunctionalChoice_on A B := + forall R : A -> A -> Prop, + forall T : A -> B -> Prop, + Equivalence R -> + (forall x, exists y, forall x', R x x' -> T x' y) -> + exists f : A -> B, forall x : A, T x (f x) /\ (forall x' : A, R x x' -> f x = f x'). + +(** ID_epsilon = constructive version of indefinite description; + combined with proof-irrelevance, it may be connected to + Carlström's type theory with a constructive indefinite description + operator *) Definition ConstructiveIndefiniteDescription_on := forall P:A->Prop, (exists x, P x) -> { x:A | P x }. -(** ID_iota (constructive version of definite description; combined - with proof-irrelevance, it may be connected to Carlström's and - Stenlund's type theory with a constructive definite description - operator) *) +(** ID_iota = constructive version of definite description; + combined with proof-irrelevance, it may be connected to + Carlström's and Stenlund's type theory with a + constructive definite description operator) *) Definition ConstructiveDefiniteDescription_on := forall P:A->Prop, @@ -168,7 +190,7 @@ Definition ConstructiveDefiniteDescription_on := (** ** Weakly classical choice and description *) -(** GAC_rel *) +(** GAC_rel = guarded relational form of the (non extensional) axiom of choice *) Definition GuardedRelationalChoice_on := forall P : A->Prop, forall R : A->B->Prop, @@ -176,7 +198,7 @@ Definition GuardedRelationalChoice_on := (exists R' : A->B->Prop, subrelation R' R /\ forall x, P x -> exists! y, R' x y). -(** GAC_fun *) +(** GAC_fun = guarded functional form of the (non extensional) axiom of choice *) Definition GuardedFunctionalChoice_on := forall P : A->Prop, forall R : A->B->Prop, @@ -184,7 +206,7 @@ Definition GuardedFunctionalChoice_on := (forall x : A, P x -> exists y : B, R x y) -> (exists f : A->B, forall x, P x -> R x (f x)). -(** GFR_fun *) +(** GAC! = guarded functional relation reification *) Definition GuardedFunctionalRelReification_on := forall P : A->Prop, forall R : A->B->Prop, @@ -192,27 +214,28 @@ Definition GuardedFunctionalRelReification_on := (forall x : A, P x -> exists! y : B, R x y) -> (exists f : A->B, forall x : A, P x -> R x (f x)). -(** OAC_rel *) +(** OAC_rel = "omniscient" relational form of the (non extensional) axiom of choice *) Definition OmniscientRelationalChoice_on := forall R : A->B->Prop, exists R' : A->B->Prop, subrelation R' R /\ forall x : A, (exists y : B, R x y) -> exists! y, R' x y. -(** OAC_fun *) +(** OAC_fun = "omniscient" functional form of the (non extensional) axiom of choice + (called AC* in Bell [[Bell]]) *) Definition OmniscientFunctionalChoice_on := forall R : A->B->Prop, inhabited B -> exists f : A->B, forall x : A, (exists y : B, R x y) -> R x (f x). -(** D_epsilon *) +(** D_epsilon = (weakly classical) indefinite description principle *) Definition EpsilonStatement_on := forall P:A->Prop, inhabited A -> { x:A | (exists x, P x) -> P x }. -(** D_iota *) +(** D_iota = (weakly classical) definite description principle *) Definition IotaStatement_on := forall P:A->Prop, @@ -226,14 +249,28 @@ Notation RelationalChoice := (forall A B : Type, RelationalChoice_on A B). Notation FunctionalChoice := (forall A B : Type, FunctionalChoice_on A B). -Definition FunctionalDependentChoice := +Notation DependentFunctionalChoice := + (forall A (B:A->Type), DependentFunctionalChoice_on B). +Notation InhabitedForallCommute := + (forall A (B : A -> Type), InhabitedForallCommute_on B). +Notation FunctionalDependentChoice := (forall A : Type, FunctionalDependentChoice_on A). -Definition FunctionalCountableChoice := +Notation FunctionalCountableChoice := (forall A : Type, FunctionalCountableChoice_on A). Notation FunctionalChoiceOnInhabitedSet := (forall A B : Type, inhabited B -> FunctionalChoice_on A B). Notation FunctionalRelReification := (forall A B : Type, FunctionalRelReification_on A B). +Notation DependentFunctionalRelReification := + (forall A (B:A->Type), DependentFunctionalRelReification_on B). +Notation RepresentativeFunctionalChoice := + (forall A : Type, RepresentativeFunctionalChoice_on A). +Notation SetoidFunctionalChoice := + (forall A B: Type, SetoidFunctionalChoice_on A B). +Notation GeneralizedSetoidFunctionalChoice := + (forall A B : Type, GeneralizedSetoidFunctionalChoice_on A B). +Notation SimpleSetoidFunctionalChoice := + (forall A B : Type, SimpleSetoidFunctionalChoice_on A B). Notation GuardedRelationalChoice := (forall A B : Type, GuardedRelationalChoice_on A B). @@ -259,18 +296,87 @@ Notation EpsilonStatement := (** Subclassical schemes *) +(** PI = proof irrelevance *) Definition ProofIrrelevance := forall (A:Prop) (a1 a2:A), a1 = a2. +(** IGP = independence of general premises + (an unconstrained generalisation of the constructive principle of + independence of premises) *) Definition IndependenceOfGeneralPremises := forall (A:Type) (P:A -> Prop) (Q:Prop), inhabited A -> (Q -> exists x, P x) -> exists x, Q -> P x. +(** Drinker = drinker's paradox (small form) + (called Ex in Bell [[Bell]]) *) Definition SmallDrinker'sParadox := forall (A:Type) (P:A -> Prop), inhabited A -> exists x, (exists x, P x) -> P x. +(** EM = excluded-middle *) +Definition ExcludedMiddle := + forall P:Prop, P \/ ~ P. + +(** Extensional schemes *) + +(** Ext_prop_repr = choice of a representative among extensional propositions *) +Local Notation ExtensionalPropositionRepresentative := + (forall (A:Type), + exists h : Prop -> Prop, + forall P : Prop, (P <-> h P) /\ forall Q, (P <-> Q) -> h P = h Q). + +(** Ext_pred_repr = choice of a representative among extensional predicates *) +Local Notation ExtensionalPredicateRepresentative := + (forall (A:Type), + exists h : (A->Prop) -> (A->Prop), + forall (P : A -> Prop), (forall x, P x <-> h P x) /\ forall Q, (forall x, P x <-> Q x) -> h P = h Q). + +(** Ext_fun_repr = choice of a representative among extensional functions *) +Local Notation ExtensionalFunctionRepresentative := + (forall (A B:Type), + exists h : (A->B) -> (A->B), + forall (f : A -> B), (forall x, f x = h f x) /\ forall g, (forall x, f x = g x) -> h f = h g). + +(** We let also + +- IPL_2 = 2nd-order impredicative minimal predicate logic (with ex. quant.) +- IPL^2 = 2nd-order functional minimal predicate logic (with ex. quant.) +- IPL_2^2 = 2nd-order impredicative, 2nd-order functional minimal pred. logic (with ex. quant.) + +with no prerequisite on the non-emptiness of domains +*) + +(**********************************************************************) +(** * Table of contents *) + +(* This is very fragile. *) +(** +1. Definitions + +2. IPL_2^2 |- AC_rel + AC! = AC_fun + +3.1. typed IPL_2 + Sigma-types + PI |- AC_rel = GAC_rel and IPL_2 |- AC_rel + IGP -> GAC_rel and IPL_2 |- GAC_rel = OAC_rel + +3.2. IPL^2 |- AC_fun + IGP = GAC_fun = OAC_fun = AC_fun + Drinker + +3.3. D_iota -> ID_iota and D_epsilon <-> ID_epsilon + Drinker + +4. Derivability of choice for decidable relations with well-ordered codomain + +5. AC_fun = AC_fun_dep = AC_trunc + +6. Non contradiction of constructive descriptions wrt functional choices + +7. Definite description transports classical logic to the computational world + +8. Choice -> Dependent choice -> Countable choice + +9.1. AC_fun_setoid = AC_fun + Ext_fun_repr + EM + +9.2. AC_fun_setoid = AC_fun + Ext_pred_repr + PI + *) + (**********************************************************************) (** * AC_rel + AC! = AC_fun @@ -284,7 +390,7 @@ Definition SmallDrinker'sParadox := relational formulation) without known inconsistency with classical logic, though functional relation reification conflicts with classical logic *) -Lemma description_rel_choice_imp_funct_choice : +Lemma functional_rel_reification_and_rel_choice_imp_fun_choice : forall A B : Type, FunctionalRelReification_on A B -> RelationalChoice_on A B -> FunctionalChoice_on A B. Proof. @@ -298,7 +404,7 @@ Proof. apply HR'R; assumption. Qed. -Lemma funct_choice_imp_rel_choice : +Lemma fun_choice_imp_rel_choice : forall A B : Type, FunctionalChoice_on A B -> RelationalChoice_on A B. Proof. intros A B FunCh R H. @@ -311,7 +417,7 @@ Proof. trivial. Qed. -Lemma funct_choice_imp_description : +Lemma fun_choice_imp_functional_rel_reification : forall A B : Type, FunctionalChoice_on A B -> FunctionalRelReification_on A B. Proof. intros A B FunCh R H. @@ -324,15 +430,15 @@ Proof. exists f; exact H0. Qed. -Corollary FunChoice_Equiv_RelChoice_and_ParamDefinDescr : +Corollary fun_choice_iff_rel_choice_and_functional_rel_reification : forall A B : Type, FunctionalChoice_on A B <-> RelationalChoice_on A B /\ FunctionalRelReification_on A B. Proof. intros A B. split. intro H; split; - [ exact (funct_choice_imp_rel_choice H) - | exact (funct_choice_imp_description H) ]. - intros [H H0]; exact (description_rel_choice_imp_funct_choice H0 H). + [ exact (fun_choice_imp_rel_choice H) + | exact (fun_choice_imp_functional_rel_reification H) ]. + intros [H H0]; exact (functional_rel_reification_and_rel_choice_imp_fun_choice H0 H). Qed. (**********************************************************************) @@ -576,10 +682,6 @@ Qed. Require Import Wf_nat. Require Import Decidable. -Definition FunctionalChoice_on_rel (A B:Type) (R:A->B->Prop) := - (forall x:A, exists y : B, R x y) -> - exists f : A -> B, (forall x:A, R x (f x)). - Lemma classical_denumerable_description_imp_fun_choice : forall A:Type, FunctionalRelReification_on A nat -> @@ -601,18 +703,10 @@ Proof. Qed. (**********************************************************************) -(** * Choice on dependent and non dependent function types are equivalent *) +(** * AC_fun = AC_fun_dep = AC_trunc *) (** ** Choice on dependent and non dependent function types are equivalent *) -Definition DependentFunctionalChoice_on (A:Type) (B:A -> Type) := - forall R:forall x:A, B x -> Prop, - (forall x:A, exists y : B x, R x y) -> - (exists f : (forall x:A, B x), forall x:A, R x (f x)). - -Notation DependentFunctionalChoice := - (forall A (B:A->Type), DependentFunctionalChoice_on B). - (** The easy part *) Theorem dep_non_dep_functional_choice : @@ -649,15 +743,34 @@ Proof. destruct Heq using eq_indd; trivial. Qed. -(** ** Reification of dependent and non dependent functional relation are equivalent *) +(** ** Functional choice and truncation choice are equivalent *) -Definition DependentFunctionalRelReification_on (A:Type) (B:A -> Type) := - forall (R:forall x:A, B x -> Prop), - (forall x:A, exists! y : B x, R x y) -> - (exists f : (forall x:A, B x), forall x:A, R x (f x)). +Theorem functional_choice_to_inhabited_forall_commute : + FunctionalChoice -> InhabitedForallCommute. +Proof. + intros choose0 A B Hinhab. + pose proof (non_dep_dep_functional_choice choose0) as choose;clear choose0. + assert (Hexists : forall x, exists _ : B x, True). + { intros x;apply inhabited_sig_to_exists. + refine (inhabited_covariant _ (Hinhab x)). + intros y;exists y;exact I. } + apply choose in Hexists. + destruct Hexists as [f _]. + exact (inhabits f). +Qed. -Notation DependentFunctionalRelReification := - (forall A (B:A->Type), DependentFunctionalRelReification_on B). +Theorem inhabited_forall_commute_to_functional_choice : + InhabitedForallCommute -> FunctionalChoice. +Proof. + intros choose A B R Hexists. + assert (Hinhab : forall x, inhabited {y : B | R x y}). + { intros x;apply exists_to_inhabited_sig;trivial. } + apply choose in Hinhab. destruct Hinhab as [f]. + exists (fun x => proj1_sig (f x)). + exact (fun x => proj2_sig (f x)). +Qed. + +(** ** Reification of dependent and non dependent functional relation are equivalent *) (** The easy part *) @@ -862,3 +975,346 @@ Proof. rewrite Heq in HR. assumption. Qed. + +(**********************************************************************) +(** * About the axiom of choice over setoids *) + +Require Import ClassicalFacts PropExtensionalityFacts. + +(**********************************************************************) +(** ** Consequences of the choice of a representative in an equivalence class *) + +Theorem repr_fun_choice_imp_ext_prop_repr : + RepresentativeFunctionalChoice -> ExtensionalPropositionRepresentative. +Proof. + intros ReprFunChoice A. + pose (R P Q := P <-> Q). + assert (Hequiv:Equivalence R) by (split; firstorder). + apply (ReprFunChoice _ R Hequiv). +Qed. + +Theorem repr_fun_choice_imp_ext_pred_repr : + RepresentativeFunctionalChoice -> ExtensionalPredicateRepresentative. +Proof. + intros ReprFunChoice A. + pose (R P Q := forall x : A, P x <-> Q x). + assert (Hequiv:Equivalence R) by (split; firstorder). + apply (ReprFunChoice _ R Hequiv). +Qed. + +Theorem repr_fun_choice_imp_ext_function_repr : + RepresentativeFunctionalChoice -> ExtensionalFunctionRepresentative. +Proof. + intros ReprFunChoice A B. + pose (R (f g : A -> B) := forall x : A, f x = g x). + assert (Hequiv:Equivalence R). + { split; try easy. firstorder using eq_trans. } + apply (ReprFunChoice _ R Hequiv). +Qed. + +(** *** This is a variant of Diaconescu and Goodman-Myhill theorems *) + +Theorem repr_fun_choice_imp_excluded_middle : + RepresentativeFunctionalChoice -> ExcludedMiddle. +Proof. + intros ReprFunChoice. + apply representative_boolean_partition_imp_excluded_middle, ReprFunChoice. +Qed. + +Theorem repr_fun_choice_imp_relational_choice : + RepresentativeFunctionalChoice -> RelationalChoice. +Proof. + intros ReprFunChoice A B T Hexists. + pose (D := (A*B)%type). + pose (R (z z':D) := + let x := fst z in + let x' := fst z' in + let y := snd z in + let y' := snd z' in + x = x' /\ (T x y -> y = y' \/ T x y') /\ (T x y' -> y = y' \/ T x y)). + assert (Hequiv : Equivalence R). + { split. + - split. easy. firstorder. + - intros (x,y) (x',y') (H1,(H2,H2')). split. easy. simpl fst in *. simpl snd in *. + subst x'. split; intro H. + + destruct (H2' H); firstorder. + + destruct (H2 H); firstorder. + - intros (x,y) (x',y') (x'',y'') (H1,(H2,H2')) (H3,(H4,H4')). + simpl fst in *. simpl snd in *. subst x'' x'. split. easy. split; intro H. + + simpl fst in *. simpl snd in *. destruct (H2 H) as [<-|H0]. + * destruct (H4 H); firstorder. + * destruct (H2' H0), (H4 H0); try subst y'; try subst y''; try firstorder. + + simpl fst in *. simpl snd in *. destruct (H4' H) as [<-|H0]. + * destruct (H2' H); firstorder. + * destruct (H2' H0), (H4 H0); try subst y'; try subst y''; try firstorder. } + destruct (ReprFunChoice D R Hequiv) as (g,Hg). + set (T' x y := T x y /\ exists y', T x y' /\ g (x,y') = (x,y)). + exists T'. split. + - intros x y (H,_); easy. + - intro x. destruct (Hexists x) as (y,Hy). + exists (snd (g (x,y))). + destruct (Hg (x,y)) as ((Heq1,(H',H'')),Hgxyuniq); clear Hg. + destruct (H' Hy) as [Heq2|Hgy]; clear H'. + + split. split. + * rewrite <- Heq2. assumption. + * exists y. destruct (g (x,y)) as (x',y'). simpl in Heq1, Heq2. subst; easy. + * intros y' (Hy',(y'',(Hy'',Heq))). + rewrite (Hgxyuniq (x,y'')), Heq. easy. split. easy. + split; right; easy. + + split. split. + * assumption. + * exists y. destruct (g (x,y)) as (x',y'). simpl in Heq1. subst x'; easy. + * intros y' (Hy',(y'',(Hy'',Heq))). + rewrite (Hgxyuniq (x,y'')), Heq. easy. split. easy. + split; right; easy. +Qed. + +(**********************************************************************) +(** ** AC_fun_setoid = AC_fun_setoid_gen = AC_fun_setoid_simple *) + +Theorem gen_setoid_fun_choice_imp_setoid_fun_choice : + forall A B, GeneralizedSetoidFunctionalChoice_on A B -> SetoidFunctionalChoice_on A B. +Proof. + intros A B GenSetoidFunChoice R T Hequiv Hcompat Hex. + apply GenSetoidFunChoice; try easy. + apply eq_equivalence. + intros * H <-. firstorder. +Qed. + +Theorem setoid_fun_choice_imp_gen_setoid_fun_choice : + forall A B, SetoidFunctionalChoice_on A B -> GeneralizedSetoidFunctionalChoice_on A B. +Proof. + intros A B SetoidFunChoice R S T HequivR HequivS Hcompat Hex. + destruct SetoidFunChoice with (R:=R) (T:=T) as (f,Hf); try easy. + { intros; apply (Hcompat x x' y y); try easy. } + exists f. intros x; specialize Hf with x as (Hf,Huniq). intuition. now erewrite Huniq. +Qed. + +Corollary setoid_fun_choice_iff_gen_setoid_fun_choice : + forall A B, SetoidFunctionalChoice_on A B <-> GeneralizedSetoidFunctionalChoice_on A B. +Proof. + split; auto using gen_setoid_fun_choice_imp_setoid_fun_choice, setoid_fun_choice_imp_gen_setoid_fun_choice. +Qed. + +Theorem setoid_fun_choice_imp_simple_setoid_fun_choice : + forall A B, SetoidFunctionalChoice_on A B -> SimpleSetoidFunctionalChoice_on A B. +Proof. + intros A B SetoidFunChoice R T Hequiv Hexists. + pose (T' x y := forall x', R x x' -> T x' y). + assert (Hcompat : forall (x x' : A) (y : B), R x x' -> T' x y -> T' x' y) by firstorder. + destruct (SetoidFunChoice R T' Hequiv Hcompat Hexists) as (f,Hf). + exists f. firstorder. +Qed. + +Theorem simple_setoid_fun_choice_imp_setoid_fun_choice : + forall A B, SimpleSetoidFunctionalChoice_on A B -> SetoidFunctionalChoice_on A B. +Proof. + intros A B SimpleSetoidFunChoice R T Hequiv Hcompat Hexists. + destruct (SimpleSetoidFunChoice R T Hequiv) as (f,Hf); firstorder. +Qed. + +Corollary setoid_fun_choice_iff_simple_setoid_fun_choice : + forall A B, SetoidFunctionalChoice_on A B <-> SimpleSetoidFunctionalChoice_on A B. +Proof. + split; auto using simple_setoid_fun_choice_imp_setoid_fun_choice, setoid_fun_choice_imp_simple_setoid_fun_choice. +Qed. + +(**********************************************************************) +(** ** AC_fun_setoid = AC! + AC_fun_repr *) + +Theorem setoid_fun_choice_imp_fun_choice : + forall A B, SetoidFunctionalChoice_on A B -> FunctionalChoice_on A B. +Proof. + intros A B SetoidFunChoice T Hexists. + destruct SetoidFunChoice with (R:=@eq A) (T:=T) as (f,Hf). + - apply eq_equivalence. + - now intros * ->. + - assumption. + - exists f. firstorder. +Qed. + +Corollary setoid_fun_choice_imp_functional_rel_reification : + forall A B, SetoidFunctionalChoice_on A B -> FunctionalRelReification_on A B. +Proof. + intros A B SetoidFunChoice. + apply fun_choice_imp_functional_rel_reification. + now apply setoid_fun_choice_imp_fun_choice. +Qed. + +Theorem setoid_fun_choice_imp_repr_fun_choice : + SetoidFunctionalChoice -> RepresentativeFunctionalChoice . +Proof. + intros SetoidFunChoice A R Hequiv. + apply SetoidFunChoice; firstorder. +Qed. + +Theorem functional_rel_reification_and_repr_fun_choice_imp_setoid_fun_choice : + FunctionalRelReification -> RepresentativeFunctionalChoice -> SetoidFunctionalChoice. +Proof. + intros FunRelReify ReprFunChoice A B R T Hequiv Hcompat Hexists. + assert (FunChoice : FunctionalChoice). + { intros A' B'. apply functional_rel_reification_and_rel_choice_imp_fun_choice. + - apply FunRelReify. + - now apply repr_fun_choice_imp_relational_choice. } + destruct (FunChoice _ _ T Hexists) as (f,Hf). + destruct (ReprFunChoice A R Hequiv) as (g,Hg). + exists (fun a => f (g a)). + intro x. destruct (Hg x) as (Hgx,HRuniq). + split. + - eapply Hcompat. symmetry. apply Hgx. apply Hf. + - intros y Hxy. f_equal. auto. +Qed. + +Theorem functional_rel_reification_and_repr_fun_choice_iff_setoid_fun_choice : + FunctionalRelReification /\ RepresentativeFunctionalChoice <-> SetoidFunctionalChoice. +Proof. + split; intros. + - now apply functional_rel_reification_and_repr_fun_choice_imp_setoid_fun_choice. + - split. + + now intros A B; apply setoid_fun_choice_imp_functional_rel_reification. + + now apply setoid_fun_choice_imp_repr_fun_choice. +Qed. + +(** Note: What characterization to give of +RepresentativeFunctionalChoice? A formulation of it as a functional +relation would certainly be equivalent to the formulation of +SetoidFunctionalChoice as a functional relation, but in their +functional forms, SetoidFunctionalChoice seems strictly stronger *) + +(**********************************************************************) +(** * AC_fun_setoid = AC_fun + Ext_fun_repr + EM *) + +Import EqNotations. + +(** ** This is the main theorem in [[Carlström04]] *) + +(** Note: all ingredients have a computational meaning when taken in + separation. However, to compute with the functional choice, + existential quantification has to be thought as a strong + existential, which is incompatible with the computational content of + excluded-middle *) + +Theorem fun_choice_and_ext_functions_repr_and_excluded_middle_imp_setoid_fun_choice : + FunctionalChoice -> ExtensionalFunctionRepresentative -> ExcludedMiddle -> RepresentativeFunctionalChoice. +Proof. + intros FunChoice SetoidFunRepr EM A R (Hrefl,Hsym,Htrans). + assert (H:forall P:Prop, exists b, b = true <-> P). + { intros P. destruct (EM P). + - exists true; firstorder. + - exists false; easy. } + destruct (FunChoice _ _ _ H) as (c,Hc). + pose (class_of a y := c (R a y)). + pose (isclass f := exists x:A, f x = true). + pose (class := {f:A -> bool | isclass f}). + pose (contains (c:class) (a:A) := proj1_sig c a = true). + destruct (FunChoice class A contains) as (f,Hf). + - intros f. destruct (proj2_sig f) as (x,Hx). + exists x. easy. + - destruct (SetoidFunRepr A bool) as (h,Hh). + assert (Hisclass:forall a, isclass (h (class_of a))). + { intro a. exists a. destruct (Hh (class_of a)) as (Ha,Huniqa). + rewrite <- Ha. apply Hc. apply Hrefl. } + pose (f':= fun a => exist _ (h (class_of a)) (Hisclass a) : class). + exists (fun a => f (f' a)). + intros x. destruct (Hh (class_of x)) as (Hx,Huniqx). split. + + specialize Hf with (f' x). unfold contains in Hf. simpl in Hf. rewrite <- Hx in Hf. apply Hc. assumption. + + intros y Hxy. + f_equal. + assert (Heq1: h (class_of x) = h (class_of y)). + { apply Huniqx. intro z. unfold class_of. + destruct (c (R x z)) eqn:Hxz. + - symmetry. apply Hc. apply -> Hc in Hxz. firstorder. + - destruct (c (R y z)) eqn:Hyz. + + apply -> Hc in Hyz. rewrite <- Hxz. apply Hc. firstorder. + + easy. } + assert (Heq2:rew Heq1 in Hisclass x = Hisclass y). + { apply proof_irrelevance_cci, EM. } + unfold f'. + rewrite <- Heq2. + rewrite <- Heq1. + reflexivity. +Qed. + +Theorem setoid_functional_choice_first_characterization : + FunctionalChoice /\ ExtensionalFunctionRepresentative /\ ExcludedMiddle <-> SetoidFunctionalChoice. +Proof. + split. + - intros (FunChoice & SetoidFunRepr & EM). + apply functional_rel_reification_and_repr_fun_choice_imp_setoid_fun_choice. + + intros A B. apply fun_choice_imp_functional_rel_reification, FunChoice. + + now apply fun_choice_and_ext_functions_repr_and_excluded_middle_imp_setoid_fun_choice. + - intro SetoidFunChoice. repeat split. + + now intros A B; apply setoid_fun_choice_imp_fun_choice. + + apply repr_fun_choice_imp_ext_function_repr. + now apply setoid_fun_choice_imp_repr_fun_choice. + + apply repr_fun_choice_imp_excluded_middle. + now apply setoid_fun_choice_imp_repr_fun_choice. +Qed. + +(**********************************************************************) +(** ** AC_fun_setoid = AC_fun + Ext_pred_repr + PI *) + +(** Note: all ingredients have a computational meaning when taken in + separation. However, to compute with the functional choice, + existential quantification has to be thought as a strong + existential, which is incompatible with proof-irrelevance which + requires existential quantification to be truncated *) + +Theorem fun_choice_and_ext_pred_ext_and_proof_irrel_imp_setoid_fun_choice : + FunctionalChoice -> ExtensionalPredicateRepresentative -> ProofIrrelevance -> RepresentativeFunctionalChoice. +Proof. + intros FunChoice PredExtRepr PI A R (Hrefl,Hsym,Htrans). + pose (isclass P := exists x:A, P x). + pose (class := {P:A -> Prop | isclass P}). + pose (contains (c:class) (a:A) := proj1_sig c a). + pose (class_of a := R a). + destruct (FunChoice class A contains) as (f,Hf). + - intros c. apply proj2_sig. + - destruct (PredExtRepr A) as (h,Hh). + assert (Hisclass:forall a, isclass (h (class_of a))). + { intro a. exists a. destruct (Hh (class_of a)) as (Ha,Huniqa). + rewrite <- Ha; apply Hrefl. } + pose (f':= fun a => exist _ (h (class_of a)) (Hisclass a) : class). + exists (fun a => f (f' a)). + intros x. destruct (Hh (class_of x)) as (Hx,Huniqx). split. + + specialize Hf with (f' x). simpl in Hf. rewrite <- Hx in Hf. assumption. + + intros y Hxy. + f_equal. + assert (Heq1: h (class_of x) = h (class_of y)). + { apply Huniqx. intro z. unfold class_of. firstorder. } + assert (Heq2:rew Heq1 in Hisclass x = Hisclass y). + { apply PI. } + unfold f'. + rewrite <- Heq2. + rewrite <- Heq1. + reflexivity. +Qed. + +Theorem setoid_functional_choice_second_characterization : + FunctionalChoice /\ ExtensionalPredicateRepresentative /\ ProofIrrelevance <-> SetoidFunctionalChoice. +Proof. + split. + - intros (FunChoice & ExtPredRepr & PI). + apply functional_rel_reification_and_repr_fun_choice_imp_setoid_fun_choice. + + intros A B. now apply fun_choice_imp_functional_rel_reification. + + now apply fun_choice_and_ext_pred_ext_and_proof_irrel_imp_setoid_fun_choice. + - intro SetoidFunChoice. repeat split. + + now intros A B; apply setoid_fun_choice_imp_fun_choice. + + apply repr_fun_choice_imp_ext_pred_repr. + now apply setoid_fun_choice_imp_repr_fun_choice. + + red. apply proof_irrelevance_cci. + apply repr_fun_choice_imp_excluded_middle. + now apply setoid_fun_choice_imp_repr_fun_choice. +Qed. + +(**********************************************************************) +(** * Compatibility notations *) +Notation description_rel_choice_imp_funct_choice := + functional_rel_reification_and_rel_choice_imp_fun_choice (only parsing). + +Notation funct_choice_imp_rel_choice := fun_choice_imp_rel_choice (only parsing). + +Notation FunChoice_Equiv_RelChoice_and_ParamDefinDescr := + fun_choice_iff_rel_choice_and_functional_rel_reification (only parsing). + +Notation funct_choice_imp_description := fun_choice_imp_functional_rel_reification (only parsing). diff --git a/theories/Logic/Classical.v b/theories/Logic/Classical.v index 14d83501..72f53a46 100644 --- a/theories/Logic/Classical.v +++ b/theories/Logic/Classical.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* File created for Coq V5.10.14b, Oct 1995 *) diff --git a/theories/Logic/ClassicalChoice.v b/theories/Logic/ClassicalChoice.v index 7041ee40..016fa72f 100644 --- a/theories/Logic/ClassicalChoice.v +++ b/theories/Logic/ClassicalChoice.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** This file provides classical logic and functional choice; this diff --git a/theories/Logic/ClassicalDescription.v b/theories/Logic/ClassicalDescription.v index 9e6d07b2..6867c76e 100644 --- a/theories/Logic/ClassicalDescription.v +++ b/theories/Logic/ClassicalDescription.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** This file provides classical logic and definite description, which is diff --git a/theories/Logic/ClassicalEpsilon.v b/theories/Logic/ClassicalEpsilon.v index 0e91613d..77af9048 100644 --- a/theories/Logic/ClassicalEpsilon.v +++ b/theories/Logic/ClassicalEpsilon.v @@ -1,10 +1,12 @@ -(* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (************************************************************************) (** This file provides classical logic and indefinite description under diff --git a/theories/Logic/ClassicalFacts.v b/theories/Logic/ClassicalFacts.v index afd64efd..b06384e9 100644 --- a/theories/Logic/ClassicalFacts.v +++ b/theories/Logic/ClassicalFacts.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Some facts and definitions about classical logic @@ -34,8 +36,11 @@ Table of contents: 3 3. Independence of general premises and drinker's paradox -4. Classical logic and principle of unrestricted minimization +4. Principles equivalent to classical logic +4.1 Classical logic = principle of unrestricted minimization + +4.2 Classical logic = choice of representatives in a partition of bool *) (************************************************************************) @@ -94,12 +99,14 @@ Qed. (** A weakest form of propositional extensionality: extensionality for provable propositions only *) +Require Import PropExtensionalityFacts. + Definition provable_prop_extensionality := forall A:Prop, A -> A = True. Lemma provable_prop_ext : prop_extensionality -> provable_prop_extensionality. Proof. - intros Ext A Ha; apply Ext; split; trivial. + exact PropExt_imp_ProvPropExt. Qed. (************************************************************************) @@ -516,7 +523,7 @@ End Weak_proof_irrelevance_CCI. (** ** Weak excluded-middle *) (** The weak classical logic based on [~~A \/ ~A] is referred to with - name KC in {[ChagrovZakharyaschev97]] + name KC in [[ChagrovZakharyaschev97]] [[ChagrovZakharyaschev97]] Alexander Chagrov and Michael Zakharyaschev, "Modal Logic", Clarendon Press, 1997. @@ -661,6 +668,8 @@ Proof. exists x0; exact Hnot. Qed. +(** * Axioms equivalent to classical logic *) + (** ** Principle of unrestricted minimization *) Require Import Coq.Arith.PeanoNat. @@ -736,3 +745,56 @@ Section Example_of_undecidable_predicate_with_the_minimization_property. Qed. End Example_of_undecidable_predicate_with_the_minimization_property. + +(** ** Choice of representatives in a partition of bool *) + +(** This is similar to Bell's "weak extensional selection principle" in [[Bell]] + + [[Bell]] John L. Bell, Choice principles in intuitionistic set theory, unpublished. +*) + +Require Import RelationClasses. + +Local Notation representative_boolean_partition := + (forall R:bool->bool->Prop, + Equivalence R -> exists f, forall x, R x (f x) /\ forall y, R x y -> f x = f y). + +Theorem representative_boolean_partition_imp_excluded_middle : + representative_boolean_partition -> excluded_middle. +Proof. + intros ReprFunChoice P. + pose (R (b1 b2 : bool) := b1 = b2 \/ P). + assert (Equivalence R). + { split. + - now left. + - destruct 1. now left. now right. + - destruct 1, 1; try now right. left; now transitivity y. } + destruct (ReprFunChoice R H) as (f,Hf). clear H. + destruct (Bool.bool_dec (f true) (f false)) as [Heq|Hneq]. + + left. + destruct (Hf false) as ([Hfalse|HP],_); try easy. + destruct (Hf true) as ([Htrue|HP],_); try easy. + congruence. + + right. intro HP. + destruct (Hf true) as (_,H). apply Hneq, H. now right. +Qed. + +Theorem excluded_middle_imp_representative_boolean_partition : + excluded_middle -> representative_boolean_partition. +Proof. + intros EM R H. + destruct (EM (R true false)). + - exists (fun _ => true). + intros []; firstorder. + - exists (fun b => b). + intro b. split. + + reflexivity. + + destruct b, y; intros HR; easy || now symmetry in HR. +Qed. + +Theorem excluded_middle_iff_representative_boolean_partition : + excluded_middle <-> representative_boolean_partition. +Proof. + split; auto using excluded_middle_imp_representative_boolean_partition, + representative_boolean_partition_imp_excluded_middle. +Qed. diff --git a/theories/Logic/ClassicalUniqueChoice.v b/theories/Logic/ClassicalUniqueChoice.v index 57f367e5..841bd1be 100644 --- a/theories/Logic/ClassicalUniqueChoice.v +++ b/theories/Logic/ClassicalUniqueChoice.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** This file provides classical logic and unique choice; this is diff --git a/theories/Logic/Classical_Pred_Type.v b/theories/Logic/Classical_Pred_Type.v index 6665798d..18820d3b 100644 --- a/theories/Logic/Classical_Pred_Type.v +++ b/theories/Logic/Classical_Pred_Type.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* This file is a renaming for V5.10.14b, Oct 1995, of file Classical.v diff --git a/theories/Logic/Classical_Prop.v b/theories/Logic/Classical_Prop.v index 2c69d4f0..9f5a2993 100644 --- a/theories/Logic/Classical_Prop.v +++ b/theories/Logic/Classical_Prop.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* File created for Coq V5.10.14b, Oct 1995 *) @@ -97,12 +99,12 @@ Proof proof_irrelevance_cci classic. (* classical_left transforms |- A \/ B into ~B |- A *) (* classical_right transforms |- A \/ B into ~A |- B *) -Ltac classical_right := match goal with - | _:_ |-?X1 \/ _ => (elim (classic X1);intro;[left;trivial|right]) +Ltac classical_right := match goal with +|- ?X \/ _ => (elim (classic X);intro;[left;trivial|right]) end. Ltac classical_left := match goal with -| _:_ |- _ \/?X1 => (elim (classic X1);intro;[right;trivial|left]) +|- _ \/ ?X => (elim (classic X);intro;[right;trivial|left]) end. Require Export EqdepFacts. diff --git a/theories/Logic/ConstructiveEpsilon.v b/theories/Logic/ConstructiveEpsilon.v index a304dd24..6e3da423 100644 --- a/theories/Logic/ConstructiveEpsilon.v +++ b/theories/Logic/ConstructiveEpsilon.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (*i $Id: ConstructiveEpsilon.v 12112 2009-04-28 15:47:34Z herbelin $ i*) diff --git a/theories/Logic/Decidable.v b/theories/Logic/Decidable.v index 8b6054f9..35920d91 100644 --- a/theories/Logic/Decidable.v +++ b/theories/Logic/Decidable.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Properties of decidable propositions *) diff --git a/theories/Logic/Description.v b/theories/Logic/Description.v index 0239222e..5c4f1960 100644 --- a/theories/Logic/Description.v +++ b/theories/Logic/Description.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** This file provides a constructive form of definite description; it diff --git a/theories/Logic/Diaconescu.v b/theories/Logic/Diaconescu.v index 23af5afc..3317766c 100644 --- a/theories/Logic/Diaconescu.v +++ b/theories/Logic/Diaconescu.v @@ -1,16 +1,18 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Diaconescu showed that the Axiom of Choice entails Excluded-Middle - in topoi [Diaconescu75]. Lacas and Werner adapted the proof to show + in topoi [[Diaconescu75]]. Lacas and Werner adapted the proof to show that the axiom of choice in equivalence classes entails - Excluded-Middle in Type Theory [LacasWerner99]. + Excluded-Middle in Type Theory [[LacasWerner99]]. Three variants of Diaconescu's result in type theory are shown below. @@ -23,23 +25,24 @@ Benjamin Werner) C. A proof that extensional Hilbert epsilon's description operator - entails excluded-middle (taken from Bell [Bell93]) + entails excluded-middle (taken from Bell [[Bell93]]) - See also [Carlström] for a discussion of the connection between the + See also [[Carlström04]] for a discussion of the connection between the Extensional Axiom of Choice and Excluded-Middle - [Diaconescu75] Radu Diaconescu, Axiom of Choice and Complementation, + [[Diaconescu75]] Radu Diaconescu, Axiom of Choice and Complementation, in Proceedings of AMS, vol 51, pp 176-178, 1975. - [LacasWerner99] Samuel Lacas, Benjamin Werner, Which Choices imply + [[LacasWerner99]] Samuel Lacas, Benjamin Werner, Which Choices imply the excluded middle?, preprint, 1999. - [Bell93] John L. Bell, Hilbert's epsilon operator and classical + [[Bell93]] John L. Bell, Hilbert's epsilon operator and classical logic, Journal of Philosophical Logic, 22: 1-18, 1993 - [Carlström04] Jesper Carlström, EM + Ext_ + AC_int <-> AC_ext, - Mathematical Logic Quaterly, vol 50(3), pp 236-240, 2004. + [[Carlström04]] Jesper Carlström, EM + Ext_ + AC_int is equivalent + to AC_ext, Mathematical Logic Quaterly, vol 50(3), pp 236-240, 2004. *) +Require ClassicalFacts ChoiceFacts. (**********************************************************************) (** * Pred. Ext. + Rel. Axiom of Choice -> Excluded-Middle *) @@ -54,7 +57,7 @@ Definition PredicateExtensionality := (** From predicate extensionality we get propositional extensionality hence proof-irrelevance *) -Require Import ClassicalFacts. +Import ClassicalFacts. Variable pred_extensionality : PredicateExtensionality. @@ -76,7 +79,7 @@ Qed. (** From proof-irrelevance and relational choice, we get guarded relational choice *) -Require Import ChoiceFacts. +Import ChoiceFacts. Variable rel_choice : RelationalChoice. @@ -89,7 +92,7 @@ Qed. (** The form of choice we need: there is a functional relation which chooses an element in any non empty subset of bool *) -Require Import Bool. +Import Bool. Lemma AC_bool_subset_to_bool : exists R : (bool -> Prop) -> bool -> Prop, @@ -161,6 +164,8 @@ End PredExt_RelChoice_imp_EM. Section ProofIrrel_RelChoice_imp_EqEM. +Import ChoiceFacts. + Variable rel_choice : RelationalChoice. Variable proof_irrelevance : forall P:Prop , forall x y:P, x=y. @@ -263,7 +268,7 @@ End ProofIrrel_RelChoice_imp_EqEM. (**********************************************************************) (** * Extensional Hilbert's epsilon description operator -> Excluded-Middle *) -(** Proof sketch from Bell [Bell93] (with thanks to P. Castéran) *) +(** Proof sketch from Bell [[Bell93]] (with thanks to P. Castéran) *) Local Notation inhabited A := A (only parsing). diff --git a/theories/Logic/Epsilon.v b/theories/Logic/Epsilon.v index ffbb5758..d8c527c6 100644 --- a/theories/Logic/Epsilon.v +++ b/theories/Logic/Epsilon.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** This file provides indefinite description under the form of diff --git a/theories/Logic/Eqdep.v b/theories/Logic/Eqdep.v index 5ef86b8e..35bc4225 100644 --- a/theories/Logic/Eqdep.v +++ b/theories/Logic/Eqdep.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* File Eqdep.v created by Christine Paulin-Mohring in Coq V5.6, May 1992 *) diff --git a/theories/Logic/EqdepFacts.v b/theories/Logic/EqdepFacts.v index bd59159b..d938b315 100644 --- a/theories/Logic/EqdepFacts.v +++ b/theories/Logic/EqdepFacts.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* File Eqdep.v created by Christine Paulin-Mohring in Coq V5.6, May 1992 *) @@ -123,7 +125,7 @@ Proof. apply eq_dep_intro. Qed. -Notation eq_sigS_eq_dep := eq_sigT_eq_dep (compat "8.2"). (* Compatibility *) +Notation eq_sigS_eq_dep := eq_sigT_eq_dep (compat "8.6"). (* Compatibility *) Lemma eq_dep_eq_sigT : forall (U:Type) (P:U -> Type) (p q:U) (x:P p) (y:P q), diff --git a/theories/Logic/Eqdep_dec.v b/theories/Logic/Eqdep_dec.v index b7b4dec2..0560d9ed 100644 --- a/theories/Logic/Eqdep_dec.v +++ b/theories/Logic/Eqdep_dec.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Created by Bruno Barras, Jan 1998 *) diff --git a/theories/Logic/ExtensionalFunctionRepresentative.v b/theories/Logic/ExtensionalFunctionRepresentative.v new file mode 100644 index 00000000..0aac56bb --- /dev/null +++ b/theories/Logic/ExtensionalFunctionRepresentative.v @@ -0,0 +1,26 @@ +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) + +(** This module states a limited form axiom of functional + extensionality which selects a canonical representative in each + class of extensional functions *) + +(** Its main interest is that it is the needed ingredient to provide + axiom of choice on setoids (a.k.a. axiom of extensional choice) + when combined with classical logic and axiom of (intensonal) + choice *) + +(** It provides extensionality of functions while still supporting (a + priori) an intensional interpretation of equality *) + +Axiom extensional_function_representative : + forall A B, exists repr, forall (f : A -> B), + (forall x, f x = repr f x) /\ + (forall g, (forall x, f x = g x) -> repr f = repr g). diff --git a/theories/Logic/ExtensionalityFacts.v b/theories/Logic/ExtensionalityFacts.v index 0e34e7e9..02c8998a 100644 --- a/theories/Logic/ExtensionalityFacts.v +++ b/theories/Logic/ExtensionalityFacts.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Some facts and definitions about extensionality diff --git a/theories/Logic/FinFun.v b/theories/Logic/FinFun.v index 06466801..89f5a82a 100644 --- a/theories/Logic/FinFun.v +++ b/theories/Logic/FinFun.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Functions on finite domains *) diff --git a/theories/Logic/FunctionalExtensionality.v b/theories/Logic/FunctionalExtensionality.v index 04d9a670..95e1af2e 100644 --- a/theories/Logic/FunctionalExtensionality.v +++ b/theories/Logic/FunctionalExtensionality.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** This module states the axiom of (dependent) functional extensionality and (dependent) eta-expansion. @@ -56,6 +58,78 @@ Proof. apply functional_extensionality in H. destruct H. reflexivity. Defined. +(** A version of [functional_extensionality_dep] which is provably + equal to [eq_refl] on [fun _ => eq_refl] *) +Definition functional_extensionality_dep_good + {A} {B : A -> Type} + (f g : forall x : A, B x) + (H : forall x, f x = g x) + : f = g + := eq_trans (eq_sym (functional_extensionality_dep f f (fun _ => eq_refl))) + (functional_extensionality_dep f g H). + +Lemma functional_extensionality_dep_good_refl {A B} f + : @functional_extensionality_dep_good A B f f (fun _ => eq_refl) = eq_refl. +Proof. + unfold functional_extensionality_dep_good; edestruct functional_extensionality_dep; reflexivity. +Defined. + +Opaque functional_extensionality_dep_good. + +Lemma forall_sig_eq_rect + {A B} (f : forall a : A, B a) + (P : { g : _ | (forall a, f a = g a) } -> Type) + (k : P (exist (fun g => forall a, f a = g a) f (fun a => eq_refl))) + g +: P g. +Proof. + destruct g as [g1 g2]. + set (g' := fun x => (exist _ (g1 x) (g2 x))). + change g2 with (fun x => proj2_sig (g' x)). + change g1 with (fun x => proj1_sig (g' x)). + clearbody g'; clear g1 g2. + cut (forall x, (exist _ (f x) eq_refl) = g' x). + { intro H'. + apply functional_extensionality_dep_good in H'. + destruct H'. + exact k. } + { intro x. + destruct (g' x) as [g'x1 g'x2]. + destruct g'x2. + reflexivity. } +Defined. + +Definition forall_eq_rect + {A B} (f : forall a : A, B a) + (P : forall g, (forall a, f a = g a) -> Type) + (k : P f (fun a => eq_refl)) + g H + : P g H + := @forall_sig_eq_rect A B f (fun g => P (proj1_sig g) (proj2_sig g)) k (exist _ g H). + +Definition forall_eq_rect_comp {A B} f P k + : @forall_eq_rect A B f P k f (fun _ => eq_refl) = k. +Proof. + unfold forall_eq_rect, forall_sig_eq_rect; simpl. + rewrite functional_extensionality_dep_good_refl; reflexivity. +Qed. + +Definition f_equal__functional_extensionality_dep_good + {A B f g} H a + : f_equal (fun h => h a) (@functional_extensionality_dep_good A B f g H) = H a. +Proof. + apply forall_eq_rect with (H := H); clear H g. + change (eq_refl (f a)) with (f_equal (fun h => h a) (eq_refl f)). + apply f_equal, functional_extensionality_dep_good_refl. +Defined. + +Definition f_equal__functional_extensionality_dep_good__fun + {A B f g} H + : (fun a => f_equal (fun h => h a) (@functional_extensionality_dep_good A B f g H)) = H. +Proof. + apply functional_extensionality_dep_good; intro a; apply f_equal__functional_extensionality_dep_good. +Defined. + (** Apply [functional_extensionality], introducing variable x. *) Tactic Notation "extensionality" ident(x) := @@ -68,13 +142,93 @@ Tactic Notation "extensionality" ident(x) := apply forall_extensionality) ; intro x end. -(** Eta expansion follows from extensionality. *) +(** Iteratively apply [functional_extensionality] on an hypothesis + until finding an equality statement *) +(* Note that you can write [Ltac extensionality_in_checker tac ::= tac tt.] to get a more informative error message. *) +Ltac extensionality_in_checker tac := + first [ tac tt | fail 1 "Anomaly: Unexpected error in extensionality tactic. Please report." ]. +Tactic Notation "extensionality" "in" hyp(H) := + let rec check_is_extensional_equality H := + lazymatch type of H with + | _ = _ => constr:(Prop) + | forall a : ?A, ?T + => let Ha := fresh in + constr:(forall a : A, match H a with Ha => ltac:(let v := check_is_extensional_equality Ha in exact v) end) + end in + let assert_is_extensional_equality H := + first [ let dummy := check_is_extensional_equality H in idtac + | fail 1 "Not an extensional equality" ] in + let assert_not_intensional_equality H := + lazymatch type of H with + | _ = _ => fail "Already an intensional equality" + | _ => idtac + end in + let enforce_no_body H := + (tryif (let dummy := (eval unfold H in H) in idtac) + then clearbody H + else idtac) in + let rec extensionality_step_make_type H := + lazymatch type of H with + | forall a : ?A, ?f = ?g + => constr:({ H' | (fun a => f_equal (fun h => h a) H') = H }) + | forall a : ?A, _ + => let H' := fresh in + constr:(forall a : A, match H a with H' => ltac:(let ret := extensionality_step_make_type H' in exact ret) end) + end in + let rec eta_contract T := + lazymatch (eval cbv beta in T) with + | context T'[fun a : ?A => ?f a] + => let T'' := context T'[f] in + eta_contract T'' + | ?T => T + end in + let rec lift_sig_extensionality H := + lazymatch type of H with + | sig _ => H + | forall a : ?A, _ + => let Ha := fresh in + let ret := constr:(fun a : A => match H a with Ha => ltac:(let v := lift_sig_extensionality Ha in exact v) end) in + lazymatch type of ret with + | forall a : ?A, sig (fun b : ?B => @?f a b = @?g a b) + => eta_contract (exist (fun b : (forall a : A, B) => (fun a : A => f a (b a)) = (fun a : A => g a (b a))) + (fun a : A => proj1_sig (ret a)) + (@functional_extensionality_dep_good _ _ _ _ (fun a : A => proj2_sig (ret a)))) + end + end in + let extensionality_pre_step H H_out Heq := + let T := extensionality_step_make_type H in + let H' := fresh in + assert (H' : T) by (intros; eexists; apply f_equal__functional_extensionality_dep_good__fun); + let H''b := lift_sig_extensionality H' in + case H''b; clear H'; + intros H_out Heq in + let rec extensionality_rec H H_out Heq := + lazymatch type of H with + | forall a, _ = _ + => extensionality_pre_step H H_out Heq + | _ + => let pre_H_out' := fresh H_out in + let H_out' := fresh pre_H_out' in + extensionality_pre_step H H_out' Heq; + let Heq' := fresh Heq in + extensionality_rec H_out' H_out Heq'; + subst H_out' + end in + first [ assert_is_extensional_equality H | fail 1 "Not an extensional equality" ]; + first [ assert_not_intensional_equality H | fail 1 "Already an intensional equality" ]; + (tryif enforce_no_body H then idtac else clearbody H); + let H_out := fresh in + let Heq := fresh "Heq" in + extensionality_in_checker ltac:(fun tt => extensionality_rec H H_out Heq); + (* If we [subst H], things break if we already have another equation of the form [_ = H] *) + destruct Heq; rename H_out into H. + +(** Eta expansion is built into Coq. *) Lemma eta_expansion_dep {A} {B : A -> Type} (f : forall x : A, B x) : f = fun x => f x. Proof. intros. - extensionality x. reflexivity. Qed. diff --git a/theories/Logic/Hurkens.v b/theories/Logic/Hurkens.v index 841f843c..6c4a8533 100644 --- a/theories/Logic/Hurkens.v +++ b/theories/Logic/Hurkens.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Hurkens.v *) (************************************************************************) @@ -360,13 +362,12 @@ Module NoRetractToModalProposition. Section Paradox. Variable M : Prop -> Prop. -Hypothesis unit : forall A:Prop, A -> M A. -Hypothesis join : forall A:Prop, M (M A) -> M A. Hypothesis incr : forall A B:Prop, (A->B) -> M A -> M B. Lemma strength: forall A (P:A->Prop), M(forall x:A,P x) -> forall x:A,M(P x). Proof. - eauto. + intros A P h x. + eapply incr in h; eauto. Qed. (** ** The universe of modal propositions *) @@ -470,7 +471,7 @@ Hypothesis p2p2 : forall A:NProp, El A -> El (b2p (p2b A)). Theorem paradox : forall B:NProp, El B. Proof. intros B. - unshelve (refine ((fun h => _) (NoRetractToModalProposition.paradox _ _ _ _ _ _ _ _ _ _))). + unshelve (refine ((fun h => _) (NoRetractToModalProposition.paradox _ _ _ _ _ _ _ _))). + exact (fun P => ~~P). + exact bool. + exact p2b. @@ -480,8 +481,6 @@ Proof. + cbn. auto. + cbn. auto. + cbn. auto. - + auto. - + auto. Qed. End Paradox. @@ -516,7 +515,7 @@ Hypothesis p2p2 : forall A:NProp, El A -> El (b2p (p2b A)). Theorem mparadox : forall B:NProp, El B. Proof. intros B. - unshelve (refine ((fun h => _) (NoRetractToModalProposition.paradox _ _ _ _ _ _ _ _ _ _))). + unshelve (refine ((fun h => _) (NoRetractToModalProposition.paradox _ _ _ _ _ _ _ _))). + exact (fun P => P). + exact bool. + exact p2b. @@ -526,8 +525,6 @@ Proof. + cbn. auto. + cbn. auto. + cbn. auto. - + auto. - + auto. Qed. End MParadox. @@ -562,7 +559,7 @@ End Paradox. End NoRetractFromSmallPropositionToProp. -(** * Large universes are no retracts of [Prop]. *) +(** * Large universes are not retracts of [Prop]. *) (** The existence in the Calculus of Constructions with universes of a retract from some [Type] universe into [Prop] is inconsistent. *) diff --git a/theories/Logic/IndefiniteDescription.v b/theories/Logic/IndefiniteDescription.v index 21be5032..86e81529 100644 --- a/theories/Logic/IndefiniteDescription.v +++ b/theories/Logic/IndefiniteDescription.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** This file provides a constructive form of indefinite description that diff --git a/theories/Logic/JMeq.v b/theories/Logic/JMeq.v index 2f95856b..9c56b60a 100644 --- a/theories/Logic/JMeq.v +++ b/theories/Logic/JMeq.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** John Major's Equality as proposed by Conor McBride @@ -130,7 +132,7 @@ Qed. is as strong as [eq_dep U P p x q y] (this uses [JMeq_eq]) *) Lemma JMeq_eq_dep : - forall U (P:U->Prop) p q (x:P p) (y:P q), + forall U (P:U->Type) p q (x:P p) (y:P q), p = q -> JMeq x y -> eq_dep U P p x q y. Proof. intros. diff --git a/theories/Logic/ProofIrrelevance.v b/theories/Logic/ProofIrrelevance.v index 305839cd..134bf649 100644 --- a/theories/Logic/ProofIrrelevance.v +++ b/theories/Logic/ProofIrrelevance.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** This file axiomatizes proof-irrelevance and derives some consequences *) diff --git a/theories/Logic/ProofIrrelevanceFacts.v b/theories/Logic/ProofIrrelevanceFacts.v index 19b3e9e6..10d9dbcd 100644 --- a/theories/Logic/ProofIrrelevanceFacts.v +++ b/theories/Logic/ProofIrrelevanceFacts.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** This defines the functor that build consequences of proof-irrelevance *) diff --git a/theories/Logic/PropExtensionality.v b/theories/Logic/PropExtensionality.v new file mode 100644 index 00000000..80dd4e85 --- /dev/null +++ b/theories/Logic/PropExtensionality.v @@ -0,0 +1,24 @@ +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) + +(** This module states propositional extensionality and draws + consequences of it *) + +Axiom propositional_extensionality : + forall (P Q : Prop), (P <-> Q) -> P = Q. + +Require Import ClassicalFacts. + +Theorem proof_irrelevance : forall (P:Prop) (p1 p2:P), p1 = p2. +Proof. + apply ext_prop_dep_proof_irrel_cic. + exact propositional_extensionality. +Qed. + diff --git a/theories/Logic/PropExtensionalityFacts.v b/theories/Logic/PropExtensionalityFacts.v new file mode 100644 index 00000000..2b303517 --- /dev/null +++ b/theories/Logic/PropExtensionalityFacts.v @@ -0,0 +1,111 @@ +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) + +(** Some facts and definitions about propositional and predicate extensionality + +We investigate the relations between the following extensionality principles + +- Proposition extensionality +- Predicate extensionality +- Propositional functional extensionality +- Provable-proposition extensionality +- Refutable-proposition extensionality +- Extensional proposition representatives +- Extensional predicate representatives +- Extensional propositional function representatives + +Table of contents + +1. Definitions + +2.1 Predicate extensionality <-> Proposition extensionality + Propositional functional extensionality + +2.2 Propositional extensionality -> Provable propositional extensionality + +2.3 Propositional extensionality -> Refutable propositional extensionality + +*) + +Set Implicit Arguments. + +(**********************************************************************) +(** * Definitions *) + +(** Propositional extensionality *) + +Local Notation PropositionalExtensionality := + (forall A B : Prop, (A <-> B) -> A = B). + +(** Provable-proposition extensionality *) + +Local Notation ProvablePropositionExtensionality := + (forall A:Prop, A -> A = True). + +(** Refutable-proposition extensionality *) + +Local Notation RefutablePropositionExtensionality := + (forall A:Prop, ~A -> A = False). + +(** Predicate extensionality *) + +Local Notation PredicateExtensionality := + (forall (A:Type) (P Q : A -> Prop), (forall x, P x <-> Q x) -> P = Q). + +(** Propositional functional extensionality *) + +Local Notation PropositionalFunctionalExtensionality := + (forall (A:Type) (P Q : A -> Prop), (forall x, P x = Q x) -> P = Q). + +(**********************************************************************) +(** * Propositional and predicate extensionality *) + +(**********************************************************************) +(** ** Predicate extensionality <-> Propositional extensionality + Propositional functional extensionality *) + +Lemma PredExt_imp_PropExt : PredicateExtensionality -> PropositionalExtensionality. +Proof. + intros Ext A B Equiv. + change A with ((fun _ => A) I). + now rewrite Ext with (P := fun _ : True =>A) (Q := fun _ => B). +Qed. + +Lemma PredExt_imp_PropFunExt : PredicateExtensionality -> PropositionalFunctionalExtensionality. +Proof. + intros Ext A P Q Eq. apply Ext. intros x. now rewrite (Eq x). +Qed. + +Lemma PropExt_and_PropFunExt_imp_PredExt : + PropositionalExtensionality -> PropositionalFunctionalExtensionality -> PredicateExtensionality. +Proof. + intros Ext FunExt A P Q Equiv. + apply FunExt. intros x. now apply Ext. +Qed. + +Theorem PropExt_and_PropFunExt_iff_PredExt : + PropositionalExtensionality /\ PropositionalFunctionalExtensionality <-> PredicateExtensionality. +Proof. + firstorder using PredExt_imp_PropExt, PredExt_imp_PropFunExt, PropExt_and_PropFunExt_imp_PredExt. +Qed. + +(**********************************************************************) +(** ** Propositional extensionality and provable proposition extensionality *) + +Lemma PropExt_imp_ProvPropExt : PropositionalExtensionality -> ProvablePropositionExtensionality. +Proof. + intros Ext A Ha; apply Ext; split; trivial. +Qed. + +(**********************************************************************) +(** ** Propositional extensionality and refutable proposition extensionality *) + +Lemma PropExt_imp_RefutPropExt : PropositionalExtensionality -> RefutablePropositionExtensionality. +Proof. + intros Ext A Ha; apply Ext; split; easy. +Qed. diff --git a/theories/Logic/PropFacts.v b/theories/Logic/PropFacts.v index 309539e5..06787035 100644 --- a/theories/Logic/PropFacts.v +++ b/theories/Logic/PropFacts.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Basic facts about Prop as a type *) diff --git a/theories/Logic/RelationalChoice.v b/theories/Logic/RelationalChoice.v index d16835f8..994f0785 100644 --- a/theories/Logic/RelationalChoice.v +++ b/theories/Logic/RelationalChoice.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** This file axiomatizes the relational form of the axiom of choice *) diff --git a/theories/Logic/SetIsType.v b/theories/Logic/SetIsType.v index 33fce6cc..afa85514 100644 --- a/theories/Logic/SetIsType.v +++ b/theories/Logic/SetIsType.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * The Set universe seen as a synonym for Type *) diff --git a/theories/Logic/SetoidChoice.v b/theories/Logic/SetoidChoice.v new file mode 100644 index 00000000..21bf7335 --- /dev/null +++ b/theories/Logic/SetoidChoice.v @@ -0,0 +1,62 @@ +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) + +(** This module states the functional form of the axiom of choice over + setoids, commonly called extensional axiom of choice [[Carlström04]], + [[Martin-Löf05]]. This is obtained by a decomposition of the axiom + into the following components: + + - classical logic + - relational axiom of choice + - axiom of unique choice + - a limited form of functional extensionality + + Among other results, it entails: + - proof irrelevance + - choice of a representative in equivalence classes + + [[Carlström04]] Jesper Carlström, EM + Ext_ + AC_int is equivalent to + AC_ext, Mathematical Logic Quaterly, vol 50(3), pp 236-240, 2004. + + [[Martin-Löf05] Per Martin-Löf, 100 years of Zermelo’s axiom of + choice: what was the problem with it?, lecture notes for KTH/SU + colloquium, 2005. + +*) + +Require Export ClassicalChoice. (* classical logic, relational choice, unique choice *) +Require Export ExtensionalFunctionRepresentative. + +Require Import ChoiceFacts. +Require Import ClassicalFacts. +Require Import RelationClasses. + +Theorem setoid_choice : + forall A B, + forall R : A -> A -> Prop, + forall T : A -> B -> Prop, + Equivalence R -> + (forall x x' y, R x x' -> T x y -> T x' y) -> + (forall x, exists y, T x y) -> + exists f : A -> B, forall x : A, T x (f x) /\ (forall x' : A, R x x' -> f x = f x'). +Proof. + apply setoid_functional_choice_first_characterization. split; [|split]. + - exact choice. + - exact extensional_function_representative. + - exact classic. +Qed. + +Theorem representative_choice : + forall A (R:A->A->Prop), (Equivalence R) -> + exists f : A->A, forall x : A, R x (f x) /\ forall x', R x x' -> f x = f x'. +Proof. + apply setoid_fun_choice_imp_repr_fun_choice. + exact setoid_choice. +Qed. diff --git a/theories/Logic/WKL.v b/theories/Logic/WKL.v index 95f3e83f..579800b8 100644 --- a/theories/Logic/WKL.v +++ b/theories/Logic/WKL.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** A constructive proof of a version of Weak König's Lemma over a diff --git a/theories/Logic/WeakFan.v b/theories/Logic/WeakFan.v index 4416d38d..c9822f47 100644 --- a/theories/Logic/WeakFan.v +++ b/theories/Logic/WeakFan.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** A constructive proof of a non-standard version of the weak Fan Theorem diff --git a/theories/Logic/vo.itarget b/theories/Logic/vo.itarget deleted file mode 100644 index 32359739..00000000 --- a/theories/Logic/vo.itarget +++ /dev/null @@ -1,30 +0,0 @@ -Berardi.vo -ChoiceFacts.vo -ClassicalChoice.vo -ClassicalDescription.vo -ClassicalEpsilon.vo -ClassicalFacts.vo -Classical_Pred_Type.vo -Classical_Prop.vo -ClassicalUniqueChoice.vo -Classical.vo -ConstructiveEpsilon.vo -Decidable.vo -Description.vo -Diaconescu.vo -Epsilon.vo -Eqdep_dec.vo -EqdepFacts.vo -Eqdep.vo -WeakFan.vo -WKL.vo -FunctionalExtensionality.vo -ExtensionalityFacts.vo -Hurkens.vo -IndefiniteDescription.vo -JMeq.vo -ProofIrrelevanceFacts.vo -ProofIrrelevance.vo -RelationalChoice.vo -SetIsType.vo -FinFun.vo
\ No newline at end of file diff --git a/theories/MSets/MSetAVL.v b/theories/MSets/MSetAVL.v index a3c265a2..b966f217 100644 --- a/theories/MSets/MSetAVL.v +++ b/theories/MSets/MSetAVL.v @@ -1,11 +1,13 @@ (* -*- coding: utf-8 -*- *) -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * MSetAVL : Implementation of MSetInterface via AVL trees *) @@ -31,7 +33,7 @@ code after extraction. *) -Require Import MSetInterface MSetGenTree BinInt Int. +Require Import FunInd MSetInterface MSetGenTree BinInt Int. Set Implicit Arguments. Unset Strict Implicit. diff --git a/theories/MSets/MSetDecide.v b/theories/MSets/MSetDecide.v index 9c622fd7..f228cbb3 100644 --- a/theories/MSets/MSetDecide.v +++ b/theories/MSets/MSetDecide.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (**************************************************************) (* MSetDecide.v *) diff --git a/theories/MSets/MSetEqProperties.v b/theories/MSets/MSetEqProperties.v index ae20edc8..1ee098cb 100644 --- a/theories/MSets/MSetEqProperties.v +++ b/theories/MSets/MSetEqProperties.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Finite sets library *) diff --git a/theories/MSets/MSetFacts.v b/theories/MSets/MSetFacts.v index 4e17618f..d57a7741 100644 --- a/theories/MSets/MSetFacts.v +++ b/theories/MSets/MSetFacts.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Finite sets library *) diff --git a/theories/MSets/MSetGenTree.v b/theories/MSets/MSetGenTree.v index 154c2384..95868861 100644 --- a/theories/MSets/MSetGenTree.v +++ b/theories/MSets/MSetGenTree.v @@ -1,17 +1,19 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * MSetGenTree : sets via generic trees This module factorizes common parts in implementations of finite sets as AVL trees and as Red-Black trees. The nodes of the trees defined here include an generic information - parameter, that will be the heigth in AVL trees and the color + parameter, that will be the height in AVL trees and the color in Red-Black trees. Without more details here about these information parameters, trees here are not known to be well-balanced, but simply binary-search-trees. @@ -27,7 +29,7 @@ - min_elt max_elt choose *) -Require Import Orders OrdersFacts MSetInterface PeanoNat. +Require Import FunInd Orders OrdersFacts MSetInterface PeanoNat. Local Open Scope list_scope. Local Open Scope lazy_bool_scope. @@ -1144,4 +1146,4 @@ Proof. apply mindepth_cardinal. Qed. -End Props.
\ No newline at end of file +End Props. diff --git a/theories/MSets/MSetInterface.v b/theories/MSets/MSetInterface.v index 74a7f6df..f0e75715 100644 --- a/theories/MSets/MSetInterface.v +++ b/theories/MSets/MSetInterface.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Finite set library *) diff --git a/theories/MSets/MSetList.v b/theories/MSets/MSetList.v index 05c20eb8..35fe4cee 100644 --- a/theories/MSets/MSetList.v +++ b/theories/MSets/MSetList.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Finite sets library *) diff --git a/theories/MSets/MSetPositive.v b/theories/MSets/MSetPositive.v index be95a037..a726eebd 100644 --- a/theories/MSets/MSetPositive.v +++ b/theories/MSets/MSetPositive.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** Efficient implementation of [MSetInterface.S] for positive keys, inspired from the [FMapPositive] module. diff --git a/theories/MSets/MSetProperties.v b/theories/MSets/MSetProperties.v index 396067b5..3c7dea73 100644 --- a/theories/MSets/MSetProperties.v +++ b/theories/MSets/MSetProperties.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Finite sets library *) diff --git a/theories/MSets/MSetRBT.v b/theories/MSets/MSetRBT.v index 83a2343d..eab01a55 100644 --- a/theories/MSets/MSetRBT.v +++ b/theories/MSets/MSetRBT.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * MSetRBT : Implementation of MSetInterface via Red-Black trees *) diff --git a/theories/MSets/MSetToFiniteSet.v b/theories/MSets/MSetToFiniteSet.v index e8087bc5..f456c407 100644 --- a/theories/MSets/MSetToFiniteSet.v +++ b/theories/MSets/MSetToFiniteSet.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Finite sets library : conversion to old [Finite_sets] *) diff --git a/theories/MSets/MSetWeakList.v b/theories/MSets/MSetWeakList.v index 2ac57a93..8df1ff1c 100644 --- a/theories/MSets/MSetWeakList.v +++ b/theories/MSets/MSetWeakList.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * Finite sets library *) diff --git a/theories/MSets/MSets.v b/theories/MSets/MSets.v index f179bcd1..0c9bc20c 100644 --- a/theories/MSets/MSets.v +++ b/theories/MSets/MSets.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) Require Export Orders. Require Export OrdersEx. @@ -18,4 +20,4 @@ Require Export MSetEqProperties. Require Export MSetWeakList. Require Export MSetList. Require Export MSetPositive. -Require Export MSetAVL.
\ No newline at end of file +Require Export MSetAVL. diff --git a/theories/MSets/vo.itarget b/theories/MSets/vo.itarget deleted file mode 100644 index 7c5b6899..00000000 --- a/theories/MSets/vo.itarget +++ /dev/null @@ -1,13 +0,0 @@ -MSetGenTree.vo -MSetAVL.vo -MSetRBT.vo -MSetDecide.vo -MSetEqProperties.vo -MSetFacts.vo -MSetInterface.vo -MSetList.vo -MSetProperties.vo -MSets.vo -MSetToFiniteSet.vo -MSetWeakList.vo -MSetPositive.vo
\ No newline at end of file diff --git a/theories/NArith/BinNat.v b/theories/NArith/BinNat.v index 4c398573..5d3ec5ab 100644 --- a/theories/NArith/BinNat.v +++ b/theories/NArith/BinNat.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export BinNums. @@ -956,95 +958,94 @@ Notation "( p | q )" := (N.divide p q) (at level 0) : N_scope. (** Compatibility notations *) -(*Notation N := N (compat "8.3").*) (*hidden by module N above *) Notation N_rect := N_rect (only parsing). Notation N_rec := N_rec (only parsing). Notation N_ind := N_ind (only parsing). Notation N0 := N0 (only parsing). Notation Npos := N.pos (only parsing). -Notation Ndiscr := N.discr (compat "8.3"). -Notation Ndouble_plus_one := N.succ_double (compat "8.3"). -Notation Ndouble := N.double (compat "8.3"). -Notation Nsucc := N.succ (compat "8.3"). -Notation Npred := N.pred (compat "8.3"). -Notation Nsucc_pos := N.succ_pos (compat "8.3"). -Notation Ppred_N := Pos.pred_N (compat "8.3"). -Notation Nplus := N.add (compat "8.3"). -Notation Nminus := N.sub (compat "8.3"). -Notation Nmult := N.mul (compat "8.3"). -Notation Neqb := N.eqb (compat "8.3"). -Notation Ncompare := N.compare (compat "8.3"). -Notation Nlt := N.lt (compat "8.3"). -Notation Ngt := N.gt (compat "8.3"). -Notation Nle := N.le (compat "8.3"). -Notation Nge := N.ge (compat "8.3"). -Notation Nmin := N.min (compat "8.3"). -Notation Nmax := N.max (compat "8.3"). -Notation Ndiv2 := N.div2 (compat "8.3"). -Notation Neven := N.even (compat "8.3"). -Notation Nodd := N.odd (compat "8.3"). -Notation Npow := N.pow (compat "8.3"). -Notation Nlog2 := N.log2 (compat "8.3"). - -Notation nat_of_N := N.to_nat (compat "8.3"). -Notation N_of_nat := N.of_nat (compat "8.3"). -Notation N_eq_dec := N.eq_dec (compat "8.3"). -Notation Nrect := N.peano_rect (compat "8.3"). -Notation Nrect_base := N.peano_rect_base (compat "8.3"). -Notation Nrect_step := N.peano_rect_succ (compat "8.3"). -Notation Nind := N.peano_ind (compat "8.3"). -Notation Nrec := N.peano_rec (compat "8.3"). -Notation Nrec_base := N.peano_rec_base (compat "8.3"). -Notation Nrec_succ := N.peano_rec_succ (compat "8.3"). - -Notation Npred_succ := N.pred_succ (compat "8.3"). -Notation Npred_minus := N.pred_sub (compat "8.3"). -Notation Nsucc_pred := N.succ_pred (compat "8.3"). -Notation Ppred_N_spec := N.pos_pred_spec (compat "8.3"). -Notation Nsucc_pos_spec := N.succ_pos_spec (compat "8.3"). -Notation Ppred_Nsucc := N.pos_pred_succ (compat "8.3"). -Notation Nplus_0_l := N.add_0_l (compat "8.3"). -Notation Nplus_0_r := N.add_0_r (compat "8.3"). -Notation Nplus_comm := N.add_comm (compat "8.3"). -Notation Nplus_assoc := N.add_assoc (compat "8.3"). -Notation Nplus_succ := N.add_succ_l (compat "8.3"). -Notation Nsucc_0 := N.succ_0_discr (compat "8.3"). -Notation Nsucc_inj := N.succ_inj (compat "8.3"). -Notation Nminus_N0_Nle := N.sub_0_le (compat "8.3"). -Notation Nminus_0_r := N.sub_0_r (compat "8.3"). -Notation Nminus_succ_r:= N.sub_succ_r (compat "8.3"). -Notation Nmult_0_l := N.mul_0_l (compat "8.3"). -Notation Nmult_1_l := N.mul_1_l (compat "8.3"). -Notation Nmult_1_r := N.mul_1_r (compat "8.3"). -Notation Nmult_comm := N.mul_comm (compat "8.3"). -Notation Nmult_assoc := N.mul_assoc (compat "8.3"). -Notation Nmult_plus_distr_r := N.mul_add_distr_r (compat "8.3"). -Notation Neqb_eq := N.eqb_eq (compat "8.3"). -Notation Nle_0 := N.le_0_l (compat "8.3"). -Notation Ncompare_refl := N.compare_refl (compat "8.3"). -Notation Ncompare_Eq_eq := N.compare_eq (compat "8.3"). -Notation Ncompare_eq_correct := N.compare_eq_iff (compat "8.3"). -Notation Nlt_irrefl := N.lt_irrefl (compat "8.3"). -Notation Nlt_trans := N.lt_trans (compat "8.3"). -Notation Nle_lteq := N.lt_eq_cases (compat "8.3"). -Notation Nlt_succ_r := N.lt_succ_r (compat "8.3"). -Notation Nle_trans := N.le_trans (compat "8.3"). -Notation Nle_succ_l := N.le_succ_l (compat "8.3"). -Notation Ncompare_spec := N.compare_spec (compat "8.3"). -Notation Ncompare_0 := N.compare_0_r (compat "8.3"). -Notation Ndouble_div2 := N.div2_double (compat "8.3"). -Notation Ndouble_plus_one_div2 := N.div2_succ_double (compat "8.3"). -Notation Ndouble_inj := N.double_inj (compat "8.3"). -Notation Ndouble_plus_one_inj := N.succ_double_inj (compat "8.3"). -Notation Npow_0_r := N.pow_0_r (compat "8.3"). -Notation Npow_succ_r := N.pow_succ_r (compat "8.3"). -Notation Nlog2_spec := N.log2_spec (compat "8.3"). -Notation Nlog2_nonpos := N.log2_nonpos (compat "8.3"). -Notation Neven_spec := N.even_spec (compat "8.3"). -Notation Nodd_spec := N.odd_spec (compat "8.3"). -Notation Nlt_not_eq := N.lt_neq (compat "8.3"). -Notation Ngt_Nlt := N.gt_lt (compat "8.3"). +Notation Ndiscr := N.discr (compat "8.6"). +Notation Ndouble_plus_one := N.succ_double (only parsing). +Notation Ndouble := N.double (compat "8.6"). +Notation Nsucc := N.succ (compat "8.6"). +Notation Npred := N.pred (compat "8.6"). +Notation Nsucc_pos := N.succ_pos (compat "8.6"). +Notation Ppred_N := Pos.pred_N (compat "8.6"). +Notation Nplus := N.add (only parsing). +Notation Nminus := N.sub (only parsing). +Notation Nmult := N.mul (only parsing). +Notation Neqb := N.eqb (compat "8.6"). +Notation Ncompare := N.compare (compat "8.6"). +Notation Nlt := N.lt (compat "8.6"). +Notation Ngt := N.gt (compat "8.6"). +Notation Nle := N.le (compat "8.6"). +Notation Nge := N.ge (compat "8.6"). +Notation Nmin := N.min (compat "8.6"). +Notation Nmax := N.max (compat "8.6"). +Notation Ndiv2 := N.div2 (compat "8.6"). +Notation Neven := N.even (compat "8.6"). +Notation Nodd := N.odd (compat "8.6"). +Notation Npow := N.pow (compat "8.6"). +Notation Nlog2 := N.log2 (compat "8.6"). + +Notation nat_of_N := N.to_nat (only parsing). +Notation N_of_nat := N.of_nat (only parsing). +Notation N_eq_dec := N.eq_dec (compat "8.6"). +Notation Nrect := N.peano_rect (only parsing). +Notation Nrect_base := N.peano_rect_base (only parsing). +Notation Nrect_step := N.peano_rect_succ (only parsing). +Notation Nind := N.peano_ind (only parsing). +Notation Nrec := N.peano_rec (only parsing). +Notation Nrec_base := N.peano_rec_base (only parsing). +Notation Nrec_succ := N.peano_rec_succ (only parsing). + +Notation Npred_succ := N.pred_succ (compat "8.6"). +Notation Npred_minus := N.pred_sub (only parsing). +Notation Nsucc_pred := N.succ_pred (compat "8.6"). +Notation Ppred_N_spec := N.pos_pred_spec (only parsing). +Notation Nsucc_pos_spec := N.succ_pos_spec (compat "8.6"). +Notation Ppred_Nsucc := N.pos_pred_succ (only parsing). +Notation Nplus_0_l := N.add_0_l (only parsing). +Notation Nplus_0_r := N.add_0_r (only parsing). +Notation Nplus_comm := N.add_comm (only parsing). +Notation Nplus_assoc := N.add_assoc (only parsing). +Notation Nplus_succ := N.add_succ_l (only parsing). +Notation Nsucc_0 := N.succ_0_discr (only parsing). +Notation Nsucc_inj := N.succ_inj (compat "8.6"). +Notation Nminus_N0_Nle := N.sub_0_le (only parsing). +Notation Nminus_0_r := N.sub_0_r (only parsing). +Notation Nminus_succ_r:= N.sub_succ_r (only parsing). +Notation Nmult_0_l := N.mul_0_l (only parsing). +Notation Nmult_1_l := N.mul_1_l (only parsing). +Notation Nmult_1_r := N.mul_1_r (only parsing). +Notation Nmult_comm := N.mul_comm (only parsing). +Notation Nmult_assoc := N.mul_assoc (only parsing). +Notation Nmult_plus_distr_r := N.mul_add_distr_r (only parsing). +Notation Neqb_eq := N.eqb_eq (compat "8.6"). +Notation Nle_0 := N.le_0_l (only parsing). +Notation Ncompare_refl := N.compare_refl (compat "8.6"). +Notation Ncompare_Eq_eq := N.compare_eq (only parsing). +Notation Ncompare_eq_correct := N.compare_eq_iff (only parsing). +Notation Nlt_irrefl := N.lt_irrefl (compat "8.6"). +Notation Nlt_trans := N.lt_trans (compat "8.6"). +Notation Nle_lteq := N.lt_eq_cases (only parsing). +Notation Nlt_succ_r := N.lt_succ_r (compat "8.6"). +Notation Nle_trans := N.le_trans (compat "8.6"). +Notation Nle_succ_l := N.le_succ_l (compat "8.6"). +Notation Ncompare_spec := N.compare_spec (compat "8.6"). +Notation Ncompare_0 := N.compare_0_r (only parsing). +Notation Ndouble_div2 := N.div2_double (only parsing). +Notation Ndouble_plus_one_div2 := N.div2_succ_double (only parsing). +Notation Ndouble_inj := N.double_inj (compat "8.6"). +Notation Ndouble_plus_one_inj := N.succ_double_inj (only parsing). +Notation Npow_0_r := N.pow_0_r (compat "8.6"). +Notation Npow_succ_r := N.pow_succ_r (compat "8.6"). +Notation Nlog2_spec := N.log2_spec (compat "8.6"). +Notation Nlog2_nonpos := N.log2_nonpos (compat "8.6"). +Notation Neven_spec := N.even_spec (compat "8.6"). +Notation Nodd_spec := N.odd_spec (compat "8.6"). +Notation Nlt_not_eq := N.lt_neq (only parsing). +Notation Ngt_Nlt := N.gt_lt (only parsing). (** More complex compatibility facts, expressed as lemmas (to preserve scopes for instance) *) diff --git a/theories/NArith/BinNatDef.v b/theories/NArith/BinNatDef.v index 4df4242e..5de75537 100644 --- a/theories/NArith/BinNatDef.v +++ b/theories/NArith/BinNatDef.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export BinNums. @@ -378,4 +380,22 @@ Definition iter (n:N) {A} (f:A->A) (x:A) : A := | pos p => Pos.iter f x p end. -End N.
\ No newline at end of file +(** Conversion with a decimal representation for printing/parsing *) + +Definition of_uint (d:Decimal.uint) := Pos.of_uint d. + +Definition of_int (d:Decimal.int) := + match Decimal.norm d with + | Decimal.Pos d => Some (Pos.of_uint d) + | Decimal.Neg _ => None + end. + +Definition to_uint n := + match n with + | 0 => Decimal.zero + | pos p => Pos.to_uint p + end. + +Definition to_int n := Decimal.Pos (to_uint n). + +End N. diff --git a/theories/NArith/NArith.v b/theories/NArith/NArith.v index 12d3ad2f..f3007970 100644 --- a/theories/NArith/NArith.v +++ b/theories/NArith/NArith.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Library for binary natural numbers *) diff --git a/theories/NArith/Ndec.v b/theories/NArith/Ndec.v index 37fd9dde..67c30f22 100644 --- a/theories/NArith/Ndec.v +++ b/theories/NArith/Ndec.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Bool. @@ -20,11 +22,11 @@ Local Open Scope N_scope. (** Obsolete results about boolean comparisons over [N], kept for compatibility with IntMap and SMC. *) -Notation Peqb := Pos.eqb (compat "8.3"). -Notation Neqb := N.eqb (compat "8.3"). -Notation Peqb_correct := Pos.eqb_refl (compat "8.3"). -Notation Neqb_correct := N.eqb_refl (compat "8.3"). -Notation Neqb_comm := N.eqb_sym (compat "8.3"). +Notation Peqb := Pos.eqb (compat "8.6"). +Notation Neqb := N.eqb (compat "8.6"). +Notation Peqb_correct := Pos.eqb_refl (only parsing). +Notation Neqb_correct := N.eqb_refl (only parsing). +Notation Neqb_comm := N.eqb_sym (only parsing). Lemma Peqb_complete p p' : Pos.eqb p p' = true -> p = p'. Proof. now apply Pos.eqb_eq. Qed. @@ -274,7 +276,7 @@ Qed. (* Old results about [N.min] *) -Notation Nmin_choice := N.min_dec (compat "8.3"). +Notation Nmin_choice := N.min_dec (only parsing). Lemma Nmin_le_1 a b : Nleb (N.min a b) a = true. Proof. rewrite Nleb_Nle. apply N.le_min_l. Qed. diff --git a/theories/NArith/Ndigits.v b/theories/NArith/Ndigits.v index a1e51c9d..3ccaa721 100644 --- a/theories/NArith/Ndigits.v +++ b/theories/NArith/Ndigits.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Bool Morphisms Setoid Bvector BinPos BinNat PeanoNat Pnat Nnat. @@ -14,17 +16,17 @@ Local Open Scope N_scope. (** Compatibility names for some bitwise operations *) -Notation Pxor := Pos.lxor (compat "8.3"). -Notation Nxor := N.lxor (compat "8.3"). -Notation Pbit := Pos.testbit_nat (compat "8.3"). -Notation Nbit := N.testbit_nat (compat "8.3"). +Notation Pxor := Pos.lxor (only parsing). +Notation Nxor := N.lxor (only parsing). +Notation Pbit := Pos.testbit_nat (only parsing). +Notation Nbit := N.testbit_nat (only parsing). -Notation Nxor_eq := N.lxor_eq (compat "8.3"). -Notation Nxor_comm := N.lxor_comm (compat "8.3"). -Notation Nxor_assoc := N.lxor_assoc (compat "8.3"). -Notation Nxor_neutral_left := N.lxor_0_l (compat "8.3"). -Notation Nxor_neutral_right := N.lxor_0_r (compat "8.3"). -Notation Nxor_nilpotent := N.lxor_nilpotent (compat "8.3"). +Notation Nxor_eq := N.lxor_eq (only parsing). +Notation Nxor_comm := N.lxor_comm (only parsing). +Notation Nxor_assoc := N.lxor_assoc (only parsing). +Notation Nxor_neutral_left := N.lxor_0_l (only parsing). +Notation Nxor_neutral_right := N.lxor_0_r (only parsing). +Notation Nxor_nilpotent := N.lxor_nilpotent (only parsing). (** Equivalence of bit-testing functions, either with index in [N] or in [nat]. *) @@ -249,7 +251,7 @@ Local Close Scope N_scope. (** Checking whether a number is odd, i.e. if its lower bit is set. *) -Notation Nbit0 := N.odd (compat "8.3"). +Notation Nbit0 := N.odd (only parsing). Definition Nodd (n:N) := N.odd n = true. Definition Neven (n:N) := N.odd n = false. @@ -498,7 +500,7 @@ Qed. (** Number of digits in a number *) -Notation Nsize := N.size_nat (compat "8.3"). +Notation Nsize := N.size_nat (only parsing). (** conversions between N and bit vectors. *) diff --git a/theories/NArith/Ndist.v b/theories/NArith/Ndist.v index 0ab8be58..d9a58c05 100644 --- a/theories/NArith/Ndist.v +++ b/theories/NArith/Ndist.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Arith. Require Import Min. diff --git a/theories/NArith/Ndiv_def.v b/theories/NArith/Ndiv_def.v index 4bdf6529..7c9fd869 100644 --- a/theories/NArith/Ndiv_def.v +++ b/theories/NArith/Ndiv_def.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import BinNat. @@ -22,10 +24,10 @@ Lemma Pdiv_eucl_remainder a b : snd (Pdiv_eucl a b) < Npos b. Proof. now apply (N.pos_div_eucl_remainder a (Npos b)). Qed. -Notation Ndiv_eucl := N.div_eucl (compat "8.3"). -Notation Ndiv := N.div (compat "8.3"). -Notation Nmod := N.modulo (compat "8.3"). +Notation Ndiv_eucl := N.div_eucl (compat "8.6"). +Notation Ndiv := N.div (compat "8.6"). +Notation Nmod := N.modulo (only parsing). -Notation Ndiv_eucl_correct := N.div_eucl_spec (compat "8.3"). -Notation Ndiv_mod_eq := N.div_mod' (compat "8.3"). -Notation Nmod_lt := N.mod_lt (compat "8.3"). +Notation Ndiv_eucl_correct := N.div_eucl_spec (only parsing). +Notation Ndiv_mod_eq := N.div_mod' (only parsing). +Notation Nmod_lt := N.mod_lt (compat "8.6"). diff --git a/theories/NArith/Ngcd_def.v b/theories/NArith/Ngcd_def.v index 277fd26f..70784d9c 100644 --- a/theories/NArith/Ngcd_def.v +++ b/theories/NArith/Ngcd_def.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import BinPos BinNat. diff --git a/theories/NArith/Nnat.v b/theories/NArith/Nnat.v index 4e007878..3488c8b4 100644 --- a/theories/NArith/Nnat.v +++ b/theories/NArith/Nnat.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import BinPos BinNat PeanoNat Pnat. @@ -208,30 +210,30 @@ Hint Rewrite Nat2N.id : Nnat. (** Compatibility notations *) -Notation nat_of_N_inj := N2Nat.inj (compat "8.3"). -Notation N_of_nat_of_N := N2Nat.id (compat "8.3"). -Notation nat_of_Ndouble := N2Nat.inj_double (compat "8.3"). -Notation nat_of_Ndouble_plus_one := N2Nat.inj_succ_double (compat "8.3"). -Notation nat_of_Nsucc := N2Nat.inj_succ (compat "8.3"). -Notation nat_of_Nplus := N2Nat.inj_add (compat "8.3"). -Notation nat_of_Nmult := N2Nat.inj_mul (compat "8.3"). -Notation nat_of_Nminus := N2Nat.inj_sub (compat "8.3"). -Notation nat_of_Npred := N2Nat.inj_pred (compat "8.3"). -Notation nat_of_Ndiv2 := N2Nat.inj_div2 (compat "8.3"). -Notation nat_of_Ncompare := N2Nat.inj_compare (compat "8.3"). -Notation nat_of_Nmax := N2Nat.inj_max (compat "8.3"). -Notation nat_of_Nmin := N2Nat.inj_min (compat "8.3"). - -Notation nat_of_N_of_nat := Nat2N.id (compat "8.3"). -Notation N_of_nat_inj := Nat2N.inj (compat "8.3"). -Notation N_of_double := Nat2N.inj_double (compat "8.3"). -Notation N_of_double_plus_one := Nat2N.inj_succ_double (compat "8.3"). -Notation N_of_S := Nat2N.inj_succ (compat "8.3"). -Notation N_of_pred := Nat2N.inj_pred (compat "8.3"). -Notation N_of_plus := Nat2N.inj_add (compat "8.3"). -Notation N_of_minus := Nat2N.inj_sub (compat "8.3"). -Notation N_of_mult := Nat2N.inj_mul (compat "8.3"). -Notation N_of_div2 := Nat2N.inj_div2 (compat "8.3"). -Notation N_of_nat_compare := Nat2N.inj_compare (compat "8.3"). -Notation N_of_min := Nat2N.inj_min (compat "8.3"). -Notation N_of_max := Nat2N.inj_max (compat "8.3"). +Notation nat_of_N_inj := N2Nat.inj (only parsing). +Notation N_of_nat_of_N := N2Nat.id (only parsing). +Notation nat_of_Ndouble := N2Nat.inj_double (only parsing). +Notation nat_of_Ndouble_plus_one := N2Nat.inj_succ_double (only parsing). +Notation nat_of_Nsucc := N2Nat.inj_succ (only parsing). +Notation nat_of_Nplus := N2Nat.inj_add (only parsing). +Notation nat_of_Nmult := N2Nat.inj_mul (only parsing). +Notation nat_of_Nminus := N2Nat.inj_sub (only parsing). +Notation nat_of_Npred := N2Nat.inj_pred (only parsing). +Notation nat_of_Ndiv2 := N2Nat.inj_div2 (only parsing). +Notation nat_of_Ncompare := N2Nat.inj_compare (only parsing). +Notation nat_of_Nmax := N2Nat.inj_max (only parsing). +Notation nat_of_Nmin := N2Nat.inj_min (only parsing). + +Notation nat_of_N_of_nat := Nat2N.id (only parsing). +Notation N_of_nat_inj := Nat2N.inj (only parsing). +Notation N_of_double := Nat2N.inj_double (only parsing). +Notation N_of_double_plus_one := Nat2N.inj_succ_double (only parsing). +Notation N_of_S := Nat2N.inj_succ (only parsing). +Notation N_of_pred := Nat2N.inj_pred (only parsing). +Notation N_of_plus := Nat2N.inj_add (only parsing). +Notation N_of_minus := Nat2N.inj_sub (only parsing). +Notation N_of_mult := Nat2N.inj_mul (only parsing). +Notation N_of_div2 := Nat2N.inj_div2 (only parsing). +Notation N_of_nat_compare := Nat2N.inj_compare (only parsing). +Notation N_of_min := Nat2N.inj_min (only parsing). +Notation N_of_max := Nat2N.inj_max (only parsing). diff --git a/theories/NArith/Nsqrt_def.v b/theories/NArith/Nsqrt_def.v index dd44b6e2..e771fe91 100644 --- a/theories/NArith/Nsqrt_def.v +++ b/theories/NArith/Nsqrt_def.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import BinNat. @@ -11,8 +13,8 @@ Require Import BinNat. (** Obsolete file, see [BinNat] now, only compatibility notations remain here. *) -Notation Nsqrtrem := N.sqrtrem (compat "8.3"). -Notation Nsqrt := N.sqrt (compat "8.3"). -Notation Nsqrtrem_spec := N.sqrtrem_spec (compat "8.3"). -Notation Nsqrt_spec := (fun n => N.sqrt_spec n (N.le_0_l n)) (compat "8.3"). -Notation Nsqrtrem_sqrt := N.sqrtrem_sqrt (compat "8.3"). +Notation Nsqrtrem := N.sqrtrem (compat "8.6"). +Notation Nsqrt := N.sqrt (compat "8.6"). +Notation Nsqrtrem_spec := N.sqrtrem_spec (compat "8.6"). +Notation Nsqrt_spec := (fun n => N.sqrt_spec n (N.le_0_l n)) (only parsing). +Notation Nsqrtrem_sqrt := N.sqrtrem_sqrt (compat "8.6"). diff --git a/theories/NArith/vo.itarget b/theories/NArith/vo.itarget deleted file mode 100644 index e76033f7..00000000 --- a/theories/NArith/vo.itarget +++ /dev/null @@ -1,10 +0,0 @@ -BinNatDef.vo -BinNat.vo -NArith.vo -Ndec.vo -Ndigits.vo -Ndist.vo -Nnat.vo -Ndiv_def.vo -Nsqrt_def.vo -Ngcd_def.vo
\ No newline at end of file diff --git a/theories/Numbers/BigNumPrelude.v b/theories/Numbers/BigNumPrelude.v deleted file mode 100644 index bd893087..00000000 --- a/theories/Numbers/BigNumPrelude.v +++ /dev/null @@ -1,411 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -(** * BigNumPrelude *) - -(** Auxiliary functions & theorems used for arbitrary precision efficient - numbers. *) - - -Require Import ArithRing. -Require Export ZArith. -Require Export Znumtheory. -Require Export Zpow_facts. - -Declare ML Module "numbers_syntax_plugin". - -(* *** Nota Bene *** - All results that were general enough have been moved in ZArith. - Only remain here specialized lemmas and compatibility elements. - (P.L. 5/11/2007). -*) - - -Local Open Scope Z_scope. - -(* For compatibility of scripts, weaker version of some lemmas of Z.div *) - -Lemma Zlt0_not_eq : forall n, 0<n -> n<>0. -Proof. - auto with zarith. -Qed. - -Definition Zdiv_mult_cancel_r a b c H := Zdiv.Zdiv_mult_cancel_r a b c (Zlt0_not_eq _ H). -Definition Zdiv_mult_cancel_l a b c H := Zdiv.Zdiv_mult_cancel_r a b c (Zlt0_not_eq _ H). -Definition Z_div_plus_l a b c H := Zdiv.Z_div_plus_full_l a b c (Zlt0_not_eq _ H). - -(* Automation *) - -Hint Extern 2 (Z.le _ _) => - (match goal with - |- Zpos _ <= Zpos _ => exact (eq_refl _) -| H: _ <= ?p |- _ <= ?p => apply Z.le_trans with (2 := H) -| H: _ < ?p |- _ <= ?p => apply Z.lt_le_incl; apply Z.le_lt_trans with (2 := H) - end). - -Hint Extern 2 (Z.lt _ _) => - (match goal with - |- Zpos _ < Zpos _ => exact (eq_refl _) -| H: _ <= ?p |- _ <= ?p => apply Z.lt_le_trans with (2 := H) -| H: _ < ?p |- _ <= ?p => apply Z.le_lt_trans with (2 := H) - end). - - -Hint Resolve Z.lt_gt Z.le_ge Z_div_pos: zarith. - -(************************************** - Properties of order and product - **************************************) - - Theorem beta_lex: forall a b c d beta, - a * beta + b <= c * beta + d -> - 0 <= b < beta -> 0 <= d < beta -> - a <= c. - Proof. - intros a b c d beta H1 (H3, H4) (H5, H6). - assert (a - c < 1); auto with zarith. - apply Z.mul_lt_mono_pos_r with beta; auto with zarith. - apply Z.le_lt_trans with (d - b); auto with zarith. - rewrite Z.mul_sub_distr_r; auto with zarith. - Qed. - - Theorem beta_lex_inv: forall a b c d beta, - a < c -> 0 <= b < beta -> - 0 <= d < beta -> - a * beta + b < c * beta + d. - Proof. - intros a b c d beta H1 (H3, H4) (H5, H6). - case (Z.le_gt_cases (c * beta + d) (a * beta + b)); auto with zarith. - intros H7. contradict H1. apply Z.le_ngt. apply beta_lex with (1 := H7); auto. - Qed. - - Lemma beta_mult : forall h l beta, - 0 <= h < beta -> 0 <= l < beta -> 0 <= h*beta+l < beta^2. - Proof. - intros h l beta H1 H2;split. auto with zarith. - rewrite <- (Z.add_0_r (beta^2)); rewrite Z.pow_2_r; - apply beta_lex_inv;auto with zarith. - Qed. - - Lemma Zmult_lt_b : - forall b x y, 0 <= x < b -> 0 <= y < b -> 0 <= x * y <= b^2 - 2*b + 1. - Proof. - intros b x y (Hx1,Hx2) (Hy1,Hy2);split;auto with zarith. - apply Z.le_trans with ((b-1)*(b-1)). - apply Z.mul_le_mono_nonneg;auto with zarith. - apply Z.eq_le_incl; ring. - Qed. - - Lemma sum_mul_carry : forall xh xl yh yl wc cc beta, - 1 < beta -> - 0 <= wc < beta -> - 0 <= xh < beta -> - 0 <= xl < beta -> - 0 <= yh < beta -> - 0 <= yl < beta -> - 0 <= cc < beta^2 -> - wc*beta^2 + cc = xh*yl + xl*yh -> - 0 <= wc <= 1. - Proof. - intros xh xl yh yl wc cc beta U H1 H2 H3 H4 H5 H6 H7. - assert (H8 := Zmult_lt_b beta xh yl H2 H5). - assert (H9 := Zmult_lt_b beta xl yh H3 H4). - split;auto with zarith. - apply beta_lex with (cc) (beta^2 - 2) (beta^2); auto with zarith. - Qed. - - Theorem mult_add_ineq: forall x y cross beta, - 0 <= x < beta -> - 0 <= y < beta -> - 0 <= cross < beta -> - 0 <= x * y + cross < beta^2. - Proof. - intros x y cross beta HH HH1 HH2. - split; auto with zarith. - apply Z.le_lt_trans with ((beta-1)*(beta-1)+(beta-1)); auto with zarith. - apply Z.add_le_mono; auto with zarith. - apply Z.mul_le_mono_nonneg; auto with zarith. - rewrite ?Z.mul_sub_distr_l, ?Z.mul_sub_distr_r, Z.pow_2_r; auto with zarith. - Qed. - - Theorem mult_add_ineq2: forall x y c cross beta, - 0 <= x < beta -> - 0 <= y < beta -> - 0 <= c*beta + cross <= 2*beta - 2 -> - 0 <= x * y + (c*beta + cross) < beta^2. - Proof. - intros x y c cross beta HH HH1 HH2. - split; auto with zarith. - apply Z.le_lt_trans with ((beta-1)*(beta-1)+(2*beta-2));auto with zarith. - apply Z.add_le_mono; auto with zarith. - apply Z.mul_le_mono_nonneg; auto with zarith. - rewrite ?Z.mul_sub_distr_l, ?Z.mul_sub_distr_r, Z.pow_2_r; auto with zarith. - Qed. - -Theorem mult_add_ineq3: forall x y c cross beta, - 0 <= x < beta -> - 0 <= y < beta -> - 0 <= cross <= beta - 2 -> - 0 <= c <= 1 -> - 0 <= x * y + (c*beta + cross) < beta^2. - Proof. - intros x y c cross beta HH HH1 HH2 HH3. - apply mult_add_ineq2;auto with zarith. - split;auto with zarith. - apply Z.le_trans with (1*beta+cross);auto with zarith. - Qed. - -Hint Rewrite Z.mul_1_r Z.mul_0_r Z.mul_1_l Z.mul_0_l Z.add_0_l Z.add_0_r Z.sub_0_r: rm10. - - -(************************************** - Properties of Z.div and Z.modulo -**************************************) - -Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. - Proof. - intros a b H H1;case (Z_mod_lt a b);auto with zarith;intros H2 H3;split;auto. - case (Z.le_gt_cases b a); intros H4; auto with zarith. - rewrite Zmod_small; auto with zarith. - Qed. - - - Theorem Zmod_distr: forall a b r t, 0 <= a <= b -> 0 <= r -> 0 <= t < 2 ^a -> - (2 ^a * r + t) mod (2 ^ b) = (2 ^a * r) mod (2 ^ b) + t. - Proof. - intros a b r t (H1, H2) H3 (H4, H5). - assert (t < 2 ^ b). - apply Z.lt_le_trans with (1:= H5); auto with zarith. - apply Zpower_le_monotone; auto with zarith. - rewrite Zplus_mod; auto with zarith. - rewrite Zmod_small with (a := t); auto with zarith. - apply Zmod_small; auto with zarith. - split; auto with zarith. - assert (0 <= 2 ^a * r); auto with zarith. - apply Z.add_nonneg_nonneg; auto with zarith. - match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end; - auto with zarith. - pattern (2 ^ b) at 2; replace (2 ^ b) with ((2 ^ b - 2 ^a) + 2 ^ a); - try ring. - apply Z.add_le_lt_mono; auto with zarith. - replace b with ((b - a) + a); try ring. - rewrite Zpower_exp; auto with zarith. - pattern (2 ^a) at 4; rewrite <- (Z.mul_1_l (2 ^a)); - try rewrite <- Z.mul_sub_distr_r. - rewrite (Z.mul_comm (2 ^(b - a))); rewrite Zmult_mod_distr_l; - auto with zarith. - rewrite (Z.mul_comm (2 ^a)); apply Z.mul_le_mono_nonneg_r; auto with zarith. - match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end; - auto with zarith. - Qed. - - Theorem Zmod_shift_r: - forall a b r t, 0 <= a <= b -> 0 <= r -> 0 <= t < 2 ^a -> - (r * 2 ^a + t) mod (2 ^ b) = (r * 2 ^a) mod (2 ^ b) + t. - Proof. - intros a b r t (H1, H2) H3 (H4, H5). - assert (t < 2 ^ b). - apply Z.lt_le_trans with (1:= H5); auto with zarith. - apply Zpower_le_monotone; auto with zarith. - rewrite Zplus_mod; auto with zarith. - rewrite Zmod_small with (a := t); auto with zarith. - apply Zmod_small; auto with zarith. - split; auto with zarith. - assert (0 <= 2 ^a * r); auto with zarith. - apply Z.add_nonneg_nonneg; auto with zarith. - match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end; - auto with zarith. - pattern (2 ^ b) at 2;replace (2 ^ b) with ((2 ^ b - 2 ^a) + 2 ^ a); try ring. - apply Z.add_le_lt_mono; auto with zarith. - replace b with ((b - a) + a); try ring. - rewrite Zpower_exp; auto with zarith. - pattern (2 ^a) at 4; rewrite <- (Z.mul_1_l (2 ^a)); - try rewrite <- Z.mul_sub_distr_r. - repeat rewrite (fun x => Z.mul_comm x (2 ^ a)); rewrite Zmult_mod_distr_l; - auto with zarith. - apply Z.mul_le_mono_nonneg_l; auto with zarith. - match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end; - auto with zarith. - Qed. - - Theorem Zdiv_shift_r: - forall a b r t, 0 <= a <= b -> 0 <= r -> 0 <= t < 2 ^a -> - (r * 2 ^a + t) / (2 ^ b) = (r * 2 ^a) / (2 ^ b). - Proof. - intros a b r t (H1, H2) H3 (H4, H5). - assert (Eq: t < 2 ^ b); auto with zarith. - apply Z.lt_le_trans with (1 := H5); auto with zarith. - apply Zpower_le_monotone; auto with zarith. - pattern (r * 2 ^ a) at 1; rewrite Z_div_mod_eq with (b := 2 ^ b); - auto with zarith. - rewrite <- Z.add_assoc. - rewrite <- Zmod_shift_r; auto with zarith. - rewrite (Z.mul_comm (2 ^ b)); rewrite Z_div_plus_full_l; auto with zarith. - rewrite (fun x y => @Zdiv_small (x mod y)); auto with zarith. - match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end; - auto with zarith. - Qed. - - - Lemma shift_unshift_mod : forall n p a, - 0 <= a < 2^n -> - 0 <= p <= n -> - a * 2^p = a / 2^(n - p) * 2^n + (a*2^p) mod 2^n. - Proof. - intros n p a H1 H2. - pattern (a*2^p) at 1;replace (a*2^p) with - (a*2^p/2^n * 2^n + a*2^p mod 2^n). - 2:symmetry;rewrite (Z.mul_comm (a*2^p/2^n));apply Z_div_mod_eq. - replace (a * 2 ^ p / 2 ^ n) with (a / 2 ^ (n - p));trivial. - replace (2^n) with (2^(n-p)*2^p). - symmetry;apply Zdiv_mult_cancel_r. - destruct H1;trivial. - cut (0 < 2^p); auto with zarith. - rewrite <- Zpower_exp. - replace (n-p+p) with n;trivial. ring. - omega. omega. - apply Z.lt_gt. apply Z.pow_pos_nonneg;auto with zarith. - Qed. - - - Lemma shift_unshift_mod_2 : forall n p a, 0 <= p <= n -> - ((a * 2 ^ (n - p)) mod (2^n) / 2 ^ (n - p)) mod (2^n) = - a mod 2 ^ p. - Proof. - intros. - rewrite Zmod_small. - rewrite Zmod_eq by (auto with zarith). - unfold Z.sub at 1. - rewrite Z_div_plus_l by (auto with zarith). - assert (2^n = 2^(n-p)*2^p). - rewrite <- Zpower_exp by (auto with zarith). - replace (n-p+p) with n; auto with zarith. - rewrite H0. - rewrite <- Zdiv_Zdiv, Z_div_mult by (auto with zarith). - rewrite (Z.mul_comm (2^(n-p))), Z.mul_assoc. - rewrite <- Z.mul_opp_l. - rewrite Z_div_mult by (auto with zarith). - symmetry; apply Zmod_eq; auto with zarith. - - remember (a * 2 ^ (n - p)) as b. - destruct (Z_mod_lt b (2^n)); auto with zarith. - split. - apply Z_div_pos; auto with zarith. - apply Zdiv_lt_upper_bound; auto with zarith. - apply Z.lt_le_trans with (2^n); auto with zarith. - rewrite <- (Z.mul_1_r (2^n)) at 1. - apply Z.mul_le_mono_nonneg; auto with zarith. - cut (0 < 2 ^ (n-p)); auto with zarith. - Qed. - - Lemma div_le_0 : forall p x, 0 <= x -> 0 <= x / 2 ^ p. - Proof. - intros p x Hle;destruct (Z_le_gt_dec 0 p). - apply Zdiv_le_lower_bound;auto with zarith. - replace (2^p) with 0. - destruct x;compute;intro;discriminate. - destruct p;trivial;discriminate. - Qed. - - Lemma div_lt : forall p x y, 0 <= x < y -> x / 2^p < y. - Proof. - intros p x y H;destruct (Z_le_gt_dec 0 p). - apply Zdiv_lt_upper_bound;auto with zarith. - apply Z.lt_le_trans with y;auto with zarith. - rewrite <- (Z.mul_1_r y);apply Z.mul_le_mono_nonneg;auto with zarith. - assert (0 < 2^p);auto with zarith. - replace (2^p) with 0. - destruct x;change (0<y);auto with zarith. - destruct p;trivial;discriminate. - Qed. - - Theorem Zgcd_div_pos a b: - 0 < b -> 0 < Z.gcd a b -> 0 < b / Z.gcd a b. - Proof. - intros Hb Hg. - assert (H : 0 <= b / Z.gcd a b) by (apply Z.div_pos; auto with zarith). - Z.le_elim H; trivial. - rewrite (Zdivide_Zdiv_eq (Z.gcd a b) b), <- H, Z.mul_0_r in Hb; - auto using Z.gcd_divide_r with zarith. - Qed. - - Theorem Zdiv_neg a b: - a < 0 -> 0 < b -> a / b < 0. - Proof. - intros Ha Hb. - assert (b > 0) by omega. - generalize (Z_mult_div_ge a _ H); intros. - assert (b * (a / b) < 0)%Z. - apply Z.le_lt_trans with a; auto with zarith. - destruct b; try (compute in Hb; discriminate). - destruct (a/Zpos p)%Z. - compute in H1; discriminate. - compute in H1; discriminate. - compute; auto. - Qed. - - Lemma Zdiv_gcd_zero : forall a b, b / Z.gcd a b = 0 -> b <> 0 -> - Z.gcd a b = 0. - Proof. - intros. - generalize (Zgcd_is_gcd a b); destruct 1. - destruct H2 as (k,Hk). - generalize H; rewrite Hk at 1. - destruct (Z.eq_dec (Z.gcd a b) 0) as [H'|H']; auto. - rewrite Z_div_mult_full; auto. - intros; subst k; simpl in *; subst b; elim H0; auto. - Qed. - - Lemma Zgcd_mult_rel_prime : forall a b c, - Z.gcd a c = 1 -> Z.gcd b c = 1 -> Z.gcd (a*b) c = 1. - Proof. - intros. - rewrite Zgcd_1_rel_prime in *. - apply rel_prime_sym; apply rel_prime_mult; apply rel_prime_sym; auto. - Qed. - - Lemma Zcompare_gt : forall (A:Type)(a a':A)(p q:Z), - match (p?=q)%Z with Gt => a | _ => a' end = - if Z_le_gt_dec p q then a' else a. - Proof. - intros. - destruct Z_le_gt_dec as [H|H]. - red in H. - destruct (p?=q)%Z; auto; elim H; auto. - rewrite H; auto. - Qed. - -Theorem Zbounded_induction : - (forall Q : Z -> Prop, forall b : Z, - Q 0 -> - (forall n, 0 <= n -> n < b - 1 -> Q n -> Q (n + 1)) -> - forall n, 0 <= n -> n < b -> Q n)%Z. -Proof. -intros Q b Q0 QS. -set (Q' := fun n => (n < b /\ Q n) \/ (b <= n)). -assert (H : forall n, 0 <= n -> Q' n). -apply natlike_rec2; unfold Q'. -destruct (Z.le_gt_cases b 0) as [H | H]. now right. left; now split. -intros n H IH. destruct IH as [[IH1 IH2] | IH]. -destruct (Z.le_gt_cases (b - 1) n) as [H1 | H1]. -right; auto with zarith. -left. split; [auto with zarith | now apply (QS n)]. -right; auto with zarith. -unfold Q' in *; intros n H1 H2. destruct (H n H1) as [[H3 H4] | H3]. -assumption. now apply Z.le_ngt in H3. -Qed. - -Lemma Zsquare_le x : x <= x*x. -Proof. -destruct (Z.lt_ge_cases 0 x). -- rewrite <- Z.mul_1_l at 1. - rewrite <- Z.mul_le_mono_pos_r; auto with zarith. -- pose proof (Z.square_nonneg x); auto with zarith. -Qed. diff --git a/theories/Numbers/BinNums.v b/theories/Numbers/BinNums.v index e4f3cd6c..f8b3d9e1 100644 --- a/theories/Numbers/BinNums.v +++ b/theories/Numbers/BinNums.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Binary Numerical Datatypes *) diff --git a/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v b/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v index 3312161a..951a4ef2 100644 --- a/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v +++ b/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) (************************************************************************) @@ -17,7 +19,7 @@ Set Implicit Arguments. Require Import ZArith. Require Import Znumtheory. -Require Import BigNumPrelude. +Require Import Zpow_facts. Require Import DoubleType. Local Open Scope Z_scope. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v b/theories/Numbers/Cyclic/Abstract/DoubleType.v index abd567a8..fe0476e4 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v +++ b/theories/Numbers/Cyclic/Abstract/DoubleType.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) (************************************************************************) @@ -67,4 +69,3 @@ Fixpoint word (w:Type) (n:nat) : Type := | O => w | S n => zn2z (word w n) end. - diff --git a/theories/Numbers/Cyclic/Abstract/NZCyclic.v b/theories/Numbers/Cyclic/Abstract/NZCyclic.v index df9b8339..64935ffe 100644 --- a/theories/Numbers/Cyclic/Abstract/NZCyclic.v +++ b/theories/Numbers/Cyclic/Abstract/NZCyclic.v @@ -1,15 +1,18 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) Require Export NZAxioms. -Require Import BigNumPrelude. +Require Import ZArith. +Require Import Zpow_facts. Require Import DoubleType. Require Import CyclicAxioms. @@ -139,6 +142,26 @@ rewrite 2 ZnZ.of_Z_correct; auto with zarith. symmetry; apply Zmod_small; auto with zarith. Qed. +Theorem Zbounded_induction : + (forall Q : Z -> Prop, forall b : Z, + Q 0 -> + (forall n, 0 <= n -> n < b - 1 -> Q n -> Q (n + 1)) -> + forall n, 0 <= n -> n < b -> Q n)%Z. +Proof. +intros Q b Q0 QS. +set (Q' := fun n => (n < b /\ Q n) \/ (b <= n)). +assert (H : forall n, 0 <= n -> Q' n). +apply natlike_rec2; unfold Q'. +destruct (Z.le_gt_cases b 0) as [H | H]. now right. left; now split. +intros n H IH. destruct IH as [[IH1 IH2] | IH]. +destruct (Z.le_gt_cases (b - 1) n) as [H1 | H1]. +right; auto with zarith. +left. split; [auto with zarith | now apply (QS n)]. +right; auto with zarith. +unfold Q' in *; intros n H1 H2. destruct (H n H1) as [[H3 H4] | H3]. +assumption. now apply Z.le_ngt in H3. +Qed. + Lemma B_holds : forall n : Z, 0 <= n < wB -> B n. Proof. intros n [H1 H2]. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v deleted file mode 100644 index 407bcca4..00000000 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v +++ /dev/null @@ -1,317 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Set Implicit Arguments. - -Require Import ZArith. -Require Import BigNumPrelude. -Require Import DoubleType. -Require Import DoubleBase. - -Local Open Scope Z_scope. - -Section DoubleAdd. - Variable w : Type. - Variable w_0 : w. - Variable w_1 : w. - Variable w_WW : w -> w -> zn2z w. - Variable w_W0 : w -> zn2z w. - Variable ww_1 : zn2z w. - Variable w_succ_c : w -> carry w. - Variable w_add_c : w -> w -> carry w. - Variable w_add_carry_c : w -> w -> carry w. - Variable w_succ : w -> w. - Variable w_add : w -> w -> w. - Variable w_add_carry : w -> w -> w. - - Definition ww_succ_c x := - match x with - | W0 => C0 ww_1 - | WW xh xl => - match w_succ_c xl with - | C0 l => C0 (WW xh l) - | C1 l => - match w_succ_c xh with - | C0 h => C0 (WW h w_0) - | C1 h => C1 W0 - end - end - end. - - Definition ww_succ x := - match x with - | W0 => ww_1 - | WW xh xl => - match w_succ_c xl with - | C0 l => WW xh l - | C1 l => w_W0 (w_succ xh) - end - end. - - Definition ww_add_c x y := - match x, y with - | W0, _ => C0 y - | _, W0 => C0 x - | WW xh xl, WW yh yl => - match w_add_c xl yl with - | C0 l => - match w_add_c xh yh with - | C0 h => C0 (WW h l) - | C1 h => C1 (w_WW h l) - end - | C1 l => - match w_add_carry_c xh yh with - | C0 h => C0 (WW h l) - | C1 h => C1 (w_WW h l) - end - end - end. - - Variable R : Type. - Variable f0 f1 : zn2z w -> R. - - Definition ww_add_c_cont x y := - match x, y with - | W0, _ => f0 y - | _, W0 => f0 x - | WW xh xl, WW yh yl => - match w_add_c xl yl with - | C0 l => - match w_add_c xh yh with - | C0 h => f0 (WW h l) - | C1 h => f1 (w_WW h l) - end - | C1 l => - match w_add_carry_c xh yh with - | C0 h => f0 (WW h l) - | C1 h => f1 (w_WW h l) - end - end - end. - - (* ww_add et ww_add_carry conserve la forme normale s'il n'y a pas - de debordement *) - Definition ww_add x y := - match x, y with - | W0, _ => y - | _, W0 => x - | WW xh xl, WW yh yl => - match w_add_c xl yl with - | C0 l => WW (w_add xh yh) l - | C1 l => WW (w_add_carry xh yh) l - end - end. - - Definition ww_add_carry_c x y := - match x, y with - | W0, W0 => C0 ww_1 - | W0, WW yh yl => ww_succ_c (WW yh yl) - | WW xh xl, W0 => ww_succ_c (WW xh xl) - | WW xh xl, WW yh yl => - match w_add_carry_c xl yl with - | C0 l => - match w_add_c xh yh with - | C0 h => C0 (WW h l) - | C1 h => C1 (WW h l) - end - | C1 l => - match w_add_carry_c xh yh with - | C0 h => C0 (WW h l) - | C1 h => C1 (w_WW h l) - end - end - end. - - Definition ww_add_carry x y := - match x, y with - | W0, W0 => ww_1 - | W0, WW yh yl => ww_succ (WW yh yl) - | WW xh xl, W0 => ww_succ (WW xh xl) - | WW xh xl, WW yh yl => - match w_add_carry_c xl yl with - | C0 l => WW (w_add xh yh) l - | C1 l => WW (w_add_carry xh yh) l - end - end. - - (*Section DoubleProof.*) - Variable w_digits : positive. - Variable w_to_Z : w -> Z. - - - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[+| c |]" := - (interp_carry 1 wB w_to_Z c) (at level 0, c at level 99). - Notation "[-| c |]" := - (interp_carry (-1) wB w_to_Z c) (at level 0, c at level 99). - - Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99). - Notation "[+[ c ]]" := - (interp_carry 1 wwB (ww_to_Z w_digits w_to_Z) c) - (at level 0, c at level 99). - Notation "[-[ c ]]" := - (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) - (at level 0, c at level 99). - - Variable spec_w_0 : [|w_0|] = 0. - Variable spec_w_1 : [|w_1|] = 1. - Variable spec_ww_1 : [[ww_1]] = 1. - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. - Variable spec_w_W0 : forall h, [[w_W0 h]] = [|h|] * wB. - Variable spec_w_succ_c : forall x, [+|w_succ_c x|] = [|x|] + 1. - Variable spec_w_add_c : forall x y, [+|w_add_c x y|] = [|x|] + [|y|]. - Variable spec_w_add_carry_c : - forall x y, [+|w_add_carry_c x y|] = [|x|] + [|y|] + 1. - Variable spec_w_succ : forall x, [|w_succ x|] = ([|x|] + 1) mod wB. - Variable spec_w_add : forall x y, [|w_add x y|] = ([|x|] + [|y|]) mod wB. - Variable spec_w_add_carry : - forall x y, [|w_add_carry x y|] = ([|x|] + [|y|] + 1) mod wB. - - Lemma spec_ww_succ_c : forall x, [+[ww_succ_c x]] = [[x]] + 1. - Proof. - destruct x as [ |xh xl];simpl. apply spec_ww_1. - generalize (spec_w_succ_c xl);destruct (w_succ_c xl) as [l|l]; - intro H;unfold interp_carry in H. simpl;rewrite H;ring. - rewrite <- Z.add_assoc;rewrite <- H;rewrite Z.mul_1_l. - assert ([|l|] = 0). generalize (spec_to_Z xl)(spec_to_Z l);omega. - rewrite H0;generalize (spec_w_succ_c xh);destruct (w_succ_c xh) as [h|h]; - intro H1;unfold interp_carry in H1. - simpl;rewrite H1;rewrite spec_w_0;ring. - unfold interp_carry;simpl ww_to_Z;rewrite wwB_wBwB. - assert ([|xh|] = wB - 1). generalize (spec_to_Z xh)(spec_to_Z h);omega. - rewrite H2;ring. - Qed. - - Lemma spec_ww_add_c : forall x y, [+[ww_add_c x y]] = [[x]] + [[y]]. - Proof. - destruct x as [ |xh xl];trivial. - destruct y as [ |yh yl]. rewrite Z.add_0_r;trivial. - simpl. replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|])) - with (([|xh|]+[|yh|])*wB + ([|xl|]+[|yl|])). 2:ring. - generalize (spec_w_add_c xl yl);destruct (w_add_c xl yl) as [l|l]; - intros H;unfold interp_carry in H;rewrite <- H. - generalize (spec_w_add_c xh yh);destruct (w_add_c xh yh) as [h|h]; - intros H1;unfold interp_carry in *;rewrite <- H1. trivial. - repeat rewrite Z.mul_1_l;rewrite spec_w_WW;rewrite wwB_wBwB; ring. - rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - generalize (spec_w_add_carry_c xh yh);destruct (w_add_carry_c xh yh) - as [h|h]; intros H1;unfold interp_carry in *;rewrite <- H1. - simpl;ring. - repeat rewrite Z.mul_1_l;rewrite wwB_wBwB;rewrite spec_w_WW;ring. - Qed. - - Section Cont. - Variable P : zn2z w -> zn2z w -> R -> Prop. - Variable x y : zn2z w. - Variable spec_f0 : forall r, [[r]] = [[x]] + [[y]] -> P x y (f0 r). - Variable spec_f1 : forall r, wwB + [[r]] = [[x]] + [[y]] -> P x y (f1 r). - - Lemma spec_ww_add_c_cont : P x y (ww_add_c_cont x y). - Proof. - destruct x as [ |xh xl];trivial. - apply spec_f0;trivial. - destruct y as [ |yh yl]. - apply spec_f0;rewrite Z.add_0_r;trivial. - simpl. - generalize (spec_w_add_c xl yl);destruct (w_add_c xl yl) as [l|l]; - intros H;unfold interp_carry in H. - generalize (spec_w_add_c xh yh);destruct (w_add_c xh yh) as [h|h]; - intros H1;unfold interp_carry in *. - apply spec_f0. simpl;rewrite H;rewrite H1;ring. - apply spec_f1. simpl;rewrite spec_w_WW;rewrite H. - rewrite Z.add_assoc;rewrite wwB_wBwB. rewrite Z.pow_2_r; rewrite <- Z.mul_add_distr_r. - rewrite Z.mul_1_l in H1;rewrite H1;ring. - generalize (spec_w_add_carry_c xh yh);destruct (w_add_carry_c xh yh) - as [h|h]; intros H1;unfold interp_carry in *. - apply spec_f0;simpl;rewrite H1. rewrite Z.mul_add_distr_r. - rewrite <- Z.add_assoc;rewrite H;ring. - apply spec_f1. rewrite spec_w_WW;rewrite wwB_wBwB. - rewrite Z.add_assoc; rewrite Z.pow_2_r; rewrite <- Z.mul_add_distr_r. - rewrite Z.mul_1_l in H1;rewrite H1. rewrite Z.mul_add_distr_r. - rewrite <- Z.add_assoc;rewrite H; simpl; ring. - Qed. - - End Cont. - - Lemma spec_ww_add_carry_c : - forall x y, [+[ww_add_carry_c x y]] = [[x]] + [[y]] + 1. - Proof. - destruct x as [ |xh xl];intro y. - exact (spec_ww_succ_c y). - destruct y as [ |yh yl]. - rewrite Z.add_0_r;exact (spec_ww_succ_c (WW xh xl)). - simpl; replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|]) + 1) - with (([|xh|]+[|yh|])*wB + ([|xl|]+[|yl|]+1)). 2:ring. - generalize (spec_w_add_carry_c xl yl);destruct (w_add_carry_c xl yl) - as [l|l];intros H;unfold interp_carry in H;rewrite <- H. - generalize (spec_w_add_c xh yh);destruct (w_add_c xh yh) as [h|h]; - intros H1;unfold interp_carry in H1;rewrite <- H1. trivial. - unfold interp_carry;repeat rewrite Z.mul_1_l;simpl;rewrite wwB_wBwB;ring. - rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - generalize (spec_w_add_carry_c xh yh);destruct (w_add_carry_c xh yh) - as [h|h];intros H1;unfold interp_carry in H1;rewrite <- H1. trivial. - unfold interp_carry;rewrite spec_w_WW; - repeat rewrite Z.mul_1_l;simpl;rewrite wwB_wBwB;ring. - Qed. - - Lemma spec_ww_succ : forall x, [[ww_succ x]] = ([[x]] + 1) mod wwB. - Proof. - destruct x as [ |xh xl];simpl. - rewrite spec_ww_1;rewrite Zmod_small;trivial. - split;[intro;discriminate|apply wwB_pos]. - rewrite <- Z.add_assoc;generalize (spec_w_succ_c xl); - destruct (w_succ_c xl) as[l|l];intro H;unfold interp_carry in H;rewrite <-H. - rewrite Zmod_small;trivial. - rewrite wwB_wBwB;apply beta_mult;apply spec_to_Z. - assert ([|l|] = 0). clear spec_ww_1 spec_w_1 spec_w_0. - assert (H1:= spec_to_Z l); assert (H2:= spec_to_Z xl); omega. - rewrite H0;rewrite Z.add_0_r;rewrite <- Z.mul_add_distr_r;rewrite wwB_wBwB. - rewrite Z.pow_2_r; rewrite Zmult_mod_distr_r;try apply lt_0_wB. - rewrite spec_w_W0;rewrite spec_w_succ;trivial. - Qed. - - Lemma spec_ww_add : forall x y, [[ww_add x y]] = ([[x]] + [[y]]) mod wwB. - Proof. - destruct x as [ |xh xl];intros y. - rewrite Zmod_small;trivial. apply spec_ww_to_Z;trivial. - destruct y as [ |yh yl]. - change [[W0]] with 0;rewrite Z.add_0_r. - rewrite Zmod_small;trivial. - exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xh xl)). - simpl. replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|])) - with (([|xh|]+[|yh|])*wB + ([|xl|]+[|yl|])). 2:ring. - generalize (spec_w_add_c xl yl);destruct (w_add_c xl yl) as [l|l]; - unfold interp_carry;intros H;simpl;rewrite <- H. - rewrite (mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_w_add;trivial. - rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - rewrite(mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_w_add_carry;trivial. - Qed. - - Lemma spec_ww_add_carry : - forall x y, [[ww_add_carry x y]] = ([[x]] + [[y]] + 1) mod wwB. - Proof. - destruct x as [ |xh xl];intros y. - exact (spec_ww_succ y). - destruct y as [ |yh yl]. - change [[W0]] with 0;rewrite Z.add_0_r. exact (spec_ww_succ (WW xh xl)). - simpl;replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|]) + 1) - with (([|xh|]+[|yh|])*wB + ([|xl|]+[|yl|]+1)). 2:ring. - generalize (spec_w_add_carry_c xl yl);destruct (w_add_carry_c xl yl) - as [l|l];unfold interp_carry;intros H;rewrite <- H;simpl ww_to_Z. - rewrite(mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_w_add;trivial. - rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - rewrite(mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_w_add_carry;trivial. - Qed. - -(* End DoubleProof. *) -End DoubleAdd. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v deleted file mode 100644 index e94a891d..00000000 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v +++ /dev/null @@ -1,437 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Set Implicit Arguments. - -Require Import ZArith Ndigits. -Require Import BigNumPrelude. -Require Import DoubleType. - -Local Open Scope Z_scope. - -Local Infix "<<" := Pos.shiftl_nat (at level 30). - -Section DoubleBase. - Variable w : Type. - Variable w_0 : w. - Variable w_1 : w. - Variable w_Bm1 : w. - Variable w_WW : w -> w -> zn2z w. - Variable w_0W : w -> zn2z w. - Variable w_digits : positive. - Variable w_zdigits: w. - Variable w_add: w -> w -> zn2z w. - Variable w_to_Z : w -> Z. - Variable w_compare : w -> w -> comparison. - - Definition ww_digits := xO w_digits. - - Definition ww_zdigits := w_add w_zdigits w_zdigits. - - Definition ww_to_Z := zn2z_to_Z (base w_digits) w_to_Z. - - Definition ww_1 := WW w_0 w_1. - - Definition ww_Bm1 := WW w_Bm1 w_Bm1. - - Definition ww_WW xh xl : zn2z (zn2z w) := - match xh, xl with - | W0, W0 => W0 - | _, _ => WW xh xl - end. - - Definition ww_W0 h : zn2z (zn2z w) := - match h with - | W0 => W0 - | _ => WW h W0 - end. - - Definition ww_0W l : zn2z (zn2z w) := - match l with - | W0 => W0 - | _ => WW W0 l - end. - - Definition double_WW (n:nat) := - match n return word w n -> word w n -> word w (S n) with - | O => w_WW - | S n => - fun (h l : zn2z (word w n)) => - match h, l with - | W0, W0 => W0 - | _, _ => WW h l - end - end. - - Definition double_wB n := base (w_digits << n). - - Fixpoint double_to_Z (n:nat) : word w n -> Z := - match n return word w n -> Z with - | O => w_to_Z - | S n => zn2z_to_Z (double_wB n) (double_to_Z n) - end. - - Fixpoint extend_aux (n:nat) (x:zn2z w) {struct n}: word w (S n) := - match n return word w (S n) with - | O => x - | S n1 => WW W0 (extend_aux n1 x) - end. - - Definition extend (n:nat) (x:w) : word w (S n) := - let r := w_0W x in - match r with - | W0 => W0 - | _ => extend_aux n r - end. - - Definition double_0 n : word w n := - match n return word w n with - | O => w_0 - | S _ => W0 - end. - - Definition double_split (n:nat) (x:zn2z (word w n)) := - match x with - | W0 => - match n return word w n * word w n with - | O => (w_0,w_0) - | S _ => (W0, W0) - end - | WW h l => (h,l) - end. - - Definition ww_compare x y := - match x, y with - | W0, W0 => Eq - | W0, WW yh yl => - match w_compare w_0 yh with - | Eq => w_compare w_0 yl - | _ => Lt - end - | WW xh xl, W0 => - match w_compare xh w_0 with - | Eq => w_compare xl w_0 - | _ => Gt - end - | WW xh xl, WW yh yl => - match w_compare xh yh with - | Eq => w_compare xl yl - | Lt => Lt - | Gt => Gt - end - end. - - - (* Return the low part of the composed word*) - Fixpoint get_low (n : nat) {struct n}: - word w n -> w := - match n return (word w n -> w) with - | 0%nat => fun x => x - | S n1 => - fun x => - match x with - | W0 => w_0 - | WW _ x1 => get_low n1 x1 - end - end. - - - Section DoubleProof. - Notation wB := (base w_digits). - Notation wwB := (base ww_digits). - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[[ x ]]" := (ww_to_Z x) (at level 0, x at level 99). - Notation "[+[ c ]]" := - (interp_carry 1 wwB ww_to_Z c) (at level 0, c at level 99). - Notation "[-[ c ]]" := - (interp_carry (-1) wwB ww_to_Z c) (at level 0, c at level 99). - Notation "[! n | x !]" := (double_to_Z n x) (at level 0, x at level 99). - - Variable spec_w_0 : [|w_0|] = 0. - Variable spec_w_1 : [|w_1|] = 1. - Variable spec_w_Bm1 : [|w_Bm1|] = wB - 1. - Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. - Variable spec_w_0W : forall l, [[w_0W l]] = [|l|]. - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - Variable spec_w_compare : forall x y, - w_compare x y = Z.compare [|x|] [|y|]. - - Lemma wwB_wBwB : wwB = wB^2. - Proof. - unfold base, ww_digits;rewrite Z.pow_2_r; rewrite (Pos2Z.inj_xO w_digits). - replace (2 * Zpos w_digits) with (Zpos w_digits + Zpos w_digits). - apply Zpower_exp; unfold Z.ge;simpl;intros;discriminate. - ring. - Qed. - - Lemma spec_ww_1 : [[ww_1]] = 1. - Proof. simpl;rewrite spec_w_0;rewrite spec_w_1;ring. Qed. - - Lemma spec_ww_Bm1 : [[ww_Bm1]] = wwB - 1. - Proof. simpl;rewrite spec_w_Bm1;rewrite wwB_wBwB;ring. Qed. - - Lemma lt_0_wB : 0 < wB. - Proof. - unfold base;apply Z.pow_pos_nonneg. unfold Z.lt;reflexivity. - unfold Z.le;intros H;discriminate H. - Qed. - - Lemma lt_0_wwB : 0 < wwB. - Proof. rewrite wwB_wBwB; rewrite Z.pow_2_r; apply Z.mul_pos_pos;apply lt_0_wB. Qed. - - Lemma wB_pos: 1 < wB. - Proof. - unfold base;apply Z.lt_le_trans with (2^1). unfold Z.lt;reflexivity. - apply Zpower_le_monotone. unfold Z.lt;reflexivity. - split;unfold Z.le;intros H. discriminate H. - clear spec_w_0W w_0W spec_w_Bm1 spec_to_Z spec_w_WW w_WW. - destruct w_digits; discriminate H. - Qed. - - Lemma wwB_pos: 1 < wwB. - Proof. - assert (H:= wB_pos);rewrite wwB_wBwB;rewrite <-(Z.mul_1_r 1). - rewrite Z.pow_2_r. - apply Zmult_lt_compat2;(split;[unfold Z.lt;reflexivity|trivial]). - apply Z.lt_le_incl;trivial. - Qed. - - Theorem wB_div_2: 2 * (wB / 2) = wB. - Proof. - clear spec_w_0 w_0 spec_w_1 w_1 spec_w_Bm1 w_Bm1 spec_w_WW spec_w_0W - spec_to_Z;unfold base. - assert (2 ^ Zpos w_digits = 2 * (2 ^ (Zpos w_digits - 1))). - pattern 2 at 2; rewrite <- Z.pow_1_r. - rewrite <- Zpower_exp; auto with zarith. - f_equal; auto with zarith. - case w_digits; compute; intros; discriminate. - rewrite H; f_equal; auto with zarith. - rewrite Z.mul_comm; apply Z_div_mult; auto with zarith. - Qed. - - Theorem wwB_div_2 : wwB / 2 = wB / 2 * wB. - Proof. - clear spec_w_0 w_0 spec_w_1 w_1 spec_w_Bm1 w_Bm1 spec_w_WW spec_w_0W - spec_to_Z. - rewrite wwB_wBwB; rewrite Z.pow_2_r. - pattern wB at 1; rewrite <- wB_div_2; auto. - rewrite <- Z.mul_assoc. - repeat (rewrite (Z.mul_comm 2); rewrite Z_div_mult); auto with zarith. - Qed. - - Lemma mod_wwB : forall z x, - (z*wB + [|x|]) mod wwB = (z mod wB)*wB + [|x|]. - Proof. - intros z x. - rewrite Zplus_mod. - pattern wwB at 1;rewrite wwB_wBwB; rewrite Z.pow_2_r. - rewrite Zmult_mod_distr_r;try apply lt_0_wB. - rewrite (Zmod_small [|x|]). - apply Zmod_small;rewrite wwB_wBwB;apply beta_mult;try apply spec_to_Z. - apply Z_mod_lt;apply Z.lt_gt;apply lt_0_wB. - destruct (spec_to_Z x);split;trivial. - change [|x|] with (0*wB+[|x|]). rewrite wwB_wBwB. - rewrite Z.pow_2_r;rewrite <- (Z.add_0_r (wB*wB));apply beta_lex_inv. - apply lt_0_wB. apply spec_to_Z. split;[apply Z.le_refl | apply lt_0_wB]. - Qed. - - Lemma wB_div : forall x y, ([|x|] * wB + [|y|]) / wB = [|x|]. - Proof. - clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1. - intros x y;unfold base;rewrite Zdiv_shift_r;auto with zarith. - rewrite Z_div_mult;auto with zarith. - destruct (spec_to_Z x);trivial. - Qed. - - Lemma wB_div_plus : forall x y p, - 0 <= p -> - ([|x|]*wB + [|y|]) / 2^(Zpos w_digits + p) = [|x|] / 2^p. - Proof. - clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1. - intros x y p Hp;rewrite Zpower_exp;auto with zarith. - rewrite <- Zdiv_Zdiv;auto with zarith. - rewrite wB_div;trivial. - Qed. - - Lemma lt_wB_wwB : wB < wwB. - Proof. - clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1. - unfold base;apply Zpower_lt_monotone;auto with zarith. - assert (0 < Zpos w_digits). compute;reflexivity. - unfold ww_digits;rewrite Pos2Z.inj_xO;auto with zarith. - Qed. - - Lemma w_to_Z_wwB : forall x, x < wB -> x < wwB. - Proof. - intros x H;apply Z.lt_trans with wB;trivial;apply lt_wB_wwB. - Qed. - - Lemma spec_ww_to_Z : forall x, 0 <= [[x]] < wwB. - Proof. - clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1. - destruct x as [ |h l];simpl. - split;[apply Z.le_refl|apply lt_0_wwB]. - assert (H:=spec_to_Z h);assert (L:=spec_to_Z l);split. - apply Z.add_nonneg_nonneg;auto with zarith. - rewrite <- (Z.add_0_r wwB);rewrite wwB_wBwB; rewrite Z.pow_2_r; - apply beta_lex_inv;auto with zarith. - Qed. - - Lemma double_wB_wwB : forall n, double_wB n * double_wB n = double_wB (S n). - Proof. - intros n;unfold double_wB;simpl. - unfold base. rewrite (Pos2Z.inj_xO (_ << _)). - replace (2 * Zpos (w_digits << n)) with - (Zpos (w_digits << n) + Zpos (w_digits << n)) by ring. - symmetry; apply Zpower_exp;intro;discriminate. - Qed. - - Lemma double_wB_pos: - forall n, 0 <= double_wB n. - Proof. - intros n; unfold double_wB, base; auto with zarith. - Qed. - - Lemma double_wB_more_digits: - forall n, wB <= double_wB n. - Proof. - clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1. - intros n; elim n; clear n; auto. - unfold double_wB, "<<"; auto with zarith. - intros n H1; rewrite <- double_wB_wwB. - apply Z.le_trans with (wB * 1). - rewrite Z.mul_1_r; apply Z.le_refl. - unfold base; auto with zarith. - apply Z.mul_le_mono_nonneg; auto with zarith. - apply Z.le_trans with wB; auto with zarith. - unfold base. - rewrite <- (Z.pow_0_r 2). - apply Z.pow_le_mono_r; auto with zarith. - Qed. - - Lemma spec_double_to_Z : - forall n (x:word w n), 0 <= [!n | x!] < double_wB n. - Proof. - clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1. - induction n;intros. exact (spec_to_Z x). - unfold double_to_Z;fold double_to_Z. - destruct x;unfold zn2z_to_Z. - unfold double_wB,base;split;auto with zarith. - assert (U0:= IHn w0);assert (U1:= IHn w1). - split;auto with zarith. - apply Z.lt_le_trans with ((double_wB n - 1) * double_wB n + double_wB n). - assert (double_to_Z n w0*double_wB n <= (double_wB n - 1)*double_wB n). - apply Z.mul_le_mono_nonneg_r;auto with zarith. - auto with zarith. - rewrite <- double_wB_wwB. - replace ((double_wB n - 1) * double_wB n + double_wB n) with (double_wB n * double_wB n); - [auto with zarith | ring]. - Qed. - - Lemma spec_get_low: - forall n x, - [!n | x!] < wB -> [|get_low n x|] = [!n | x!]. - Proof. - clear spec_w_1 spec_w_Bm1. - intros n; elim n; auto; clear n. - intros n Hrec x; case x; clear x; auto. - intros xx yy; simpl. - destruct (spec_double_to_Z n xx) as [F1 _]. Z.le_elim F1. - - (* 0 < [!n | xx!] *) - intros; exfalso. - assert (F3 := double_wB_more_digits n). - destruct (spec_double_to_Z n yy) as [F4 _]. - assert (F5: 1 * wB <= [!n | xx!] * double_wB n); - auto with zarith. - apply Z.mul_le_mono_nonneg; auto with zarith. - unfold base; auto with zarith. - - (* 0 = [!n | xx!] *) - rewrite <- F1; rewrite Z.mul_0_l, Z.add_0_l. - intros; apply Hrec; auto. - Qed. - - Lemma spec_double_WW : forall n (h l : word w n), - [!S n|double_WW n h l!] = [!n|h!] * double_wB n + [!n|l!]. - Proof. - induction n;simpl;intros;trivial. - destruct h;auto. - destruct l;auto. - Qed. - - Lemma spec_extend_aux : forall n x, [!S n|extend_aux n x!] = [[x]]. - Proof. induction n;simpl;trivial. Qed. - - Lemma spec_extend : forall n x, [!S n|extend n x!] = [|x|]. - Proof. - intros n x;assert (H:= spec_w_0W x);unfold extend. - destruct (w_0W x);simpl;trivial. - rewrite <- H;exact (spec_extend_aux n (WW w0 w1)). - Qed. - - Lemma spec_double_0 : forall n, [!n|double_0 n!] = 0. - Proof. destruct n;trivial. Qed. - - Lemma spec_double_split : forall n x, - let (h,l) := double_split n x in - [!S n|x!] = [!n|h!] * double_wB n + [!n|l!]. - Proof. - destruct x;simpl;auto. - destruct n;simpl;trivial. - rewrite spec_w_0;trivial. - Qed. - - Lemma wB_lex_inv: forall a b c d, - a < c -> - a * wB + [|b|] < c * wB + [|d|]. - Proof. - intros a b c d H1; apply beta_lex_inv with (1 := H1); auto. - Qed. - - Ltac comp2ord := match goal with - | |- Lt = (?x ?= ?y) => symmetry; change (x < y) - | |- Gt = (?x ?= ?y) => symmetry; change (x > y); apply Z.lt_gt - end. - - Lemma spec_ww_compare : forall x y, - ww_compare x y = Z.compare [[x]] [[y]]. - Proof. - destruct x as [ |xh xl];destruct y as [ |yh yl];simpl;trivial. - (* 1st case *) - rewrite 2 spec_w_compare, spec_w_0. - destruct (Z.compare_spec 0 [|yh|]) as [H|H|H]. - rewrite <- H;simpl. reflexivity. - symmetry. change (0 < [|yh|]*wB+[|yl|]). - change 0 with (0*wB+0). rewrite <- spec_w_0 at 2. - apply wB_lex_inv;trivial. - absurd (0 <= [|yh|]). apply Z.lt_nge; trivial. - destruct (spec_to_Z yh);trivial. - (* 2nd case *) - rewrite 2 spec_w_compare, spec_w_0. - destruct (Z.compare_spec [|xh|] 0) as [H|H|H]. - rewrite H;simpl;reflexivity. - absurd (0 <= [|xh|]). apply Z.lt_nge; trivial. - destruct (spec_to_Z xh);trivial. - comp2ord. - change 0 with (0*wB+0). rewrite <- spec_w_0 at 2. - apply wB_lex_inv;trivial. - (* 3rd case *) - rewrite 2 spec_w_compare. - destruct (Z.compare_spec [|xh|] [|yh|]) as [H|H|H]. - rewrite H. - symmetry. apply Z.add_compare_mono_l. - comp2ord. apply wB_lex_inv;trivial. - comp2ord. apply wB_lex_inv;trivial. - Qed. - - - End DoubleProof. - -End DoubleBase. - diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v deleted file mode 100644 index 4ebe8fac..00000000 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v +++ /dev/null @@ -1,966 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Set Implicit Arguments. - -Require Import ZArith. -Require Import BigNumPrelude. -Require Import DoubleType. -Require Import DoubleBase. -Require Import DoubleAdd. -Require Import DoubleSub. -Require Import DoubleMul. -Require Import DoubleSqrt. -Require Import DoubleLift. -Require Import DoubleDivn1. -Require Import DoubleDiv. -Require Import CyclicAxioms. - -Local Open Scope Z_scope. - - -Section Z_2nZ. - - Context {t : Type}{ops : ZnZ.Ops t}. - - Let w_digits := ZnZ.digits. - Let w_zdigits := ZnZ.zdigits. - - Let w_to_Z := ZnZ.to_Z. - Let w_of_pos := ZnZ.of_pos. - Let w_head0 := ZnZ.head0. - Let w_tail0 := ZnZ.tail0. - - Let w_0 := ZnZ.zero. - Let w_1 := ZnZ.one. - Let w_Bm1 := ZnZ.minus_one. - - Let w_compare := ZnZ.compare. - Let w_eq0 := ZnZ.eq0. - - Let w_opp_c := ZnZ.opp_c. - Let w_opp := ZnZ.opp. - Let w_opp_carry := ZnZ.opp_carry. - - Let w_succ_c := ZnZ.succ_c. - Let w_add_c := ZnZ.add_c. - Let w_add_carry_c := ZnZ.add_carry_c. - Let w_succ := ZnZ.succ. - Let w_add := ZnZ.add. - Let w_add_carry := ZnZ.add_carry. - - Let w_pred_c := ZnZ.pred_c. - Let w_sub_c := ZnZ.sub_c. - Let w_sub_carry_c := ZnZ.sub_carry_c. - Let w_pred := ZnZ.pred. - Let w_sub := ZnZ.sub. - Let w_sub_carry := ZnZ.sub_carry. - - - Let w_mul_c := ZnZ.mul_c. - Let w_mul := ZnZ.mul. - Let w_square_c := ZnZ.square_c. - - Let w_div21 := ZnZ.div21. - Let w_div_gt := ZnZ.div_gt. - Let w_div := ZnZ.div. - - Let w_mod_gt := ZnZ.modulo_gt. - Let w_mod := ZnZ.modulo. - - Let w_gcd_gt := ZnZ.gcd_gt. - Let w_gcd := ZnZ.gcd. - - Let w_add_mul_div := ZnZ.add_mul_div. - - Let w_pos_mod := ZnZ.pos_mod. - - Let w_is_even := ZnZ.is_even. - Let w_sqrt2 := ZnZ.sqrt2. - Let w_sqrt := ZnZ.sqrt. - - Let _zn2z := zn2z t. - - Let wB := base w_digits. - - Let w_Bm2 := w_pred w_Bm1. - - Let ww_1 := ww_1 w_0 w_1. - Let ww_Bm1 := ww_Bm1 w_Bm1. - - Let w_add2 a b := match w_add_c a b with C0 p => WW w_0 p | C1 p => WW w_1 p end. - - Let _ww_digits := xO w_digits. - - Let _ww_zdigits := w_add2 w_zdigits w_zdigits. - - Let to_Z := zn2z_to_Z wB w_to_Z. - - Let w_W0 := ZnZ.WO. - Let w_0W := ZnZ.OW. - Let w_WW := ZnZ.WW. - - Let ww_of_pos p := - match w_of_pos p with - | (N0, l) => (N0, WW w_0 l) - | (Npos ph,l) => - let (n,h) := w_of_pos ph in (n, w_WW h l) - end. - - Let head0 := - Eval lazy beta delta [ww_head0] in - ww_head0 w_0 w_0W w_compare w_head0 w_add2 w_zdigits _ww_zdigits. - - Let tail0 := - Eval lazy beta delta [ww_tail0] in - ww_tail0 w_0 w_0W w_compare w_tail0 w_add2 w_zdigits _ww_zdigits. - - Let ww_WW := Eval lazy beta delta [ww_WW] in (@ww_WW t). - Let ww_0W := Eval lazy beta delta [ww_0W] in (@ww_0W t). - Let ww_W0 := Eval lazy beta delta [ww_W0] in (@ww_W0 t). - - (* ** Comparison ** *) - Let compare := - Eval lazy beta delta[ww_compare] in ww_compare w_0 w_compare. - - Let eq0 (x:zn2z t) := - match x with - | W0 => true - | _ => false - end. - - (* ** Opposites ** *) - Let opp_c := - Eval lazy beta delta [ww_opp_c] in ww_opp_c w_0 w_opp_c w_opp_carry. - - Let opp := - Eval lazy beta delta [ww_opp] in ww_opp w_0 w_opp_c w_opp_carry w_opp. - - Let opp_carry := - Eval lazy beta delta [ww_opp_carry] in ww_opp_carry w_WW ww_Bm1 w_opp_carry. - - (* ** Additions ** *) - - Let succ_c := - Eval lazy beta delta [ww_succ_c] in ww_succ_c w_0 ww_1 w_succ_c. - - Let add_c := - Eval lazy beta delta [ww_add_c] in ww_add_c w_WW w_add_c w_add_carry_c. - - Let add_carry_c := - Eval lazy beta iota delta [ww_add_carry_c ww_succ_c] in - ww_add_carry_c w_0 w_WW ww_1 w_succ_c w_add_c w_add_carry_c. - - Let succ := - Eval lazy beta delta [ww_succ] in ww_succ w_W0 ww_1 w_succ_c w_succ. - - Let add := - Eval lazy beta delta [ww_add] in ww_add w_add_c w_add w_add_carry. - - Let add_carry := - Eval lazy beta iota delta [ww_add_carry ww_succ] in - ww_add_carry w_W0 ww_1 w_succ_c w_add_carry_c w_succ w_add w_add_carry. - - (* ** Subtractions ** *) - - Let pred_c := - Eval lazy beta delta [ww_pred_c] in ww_pred_c w_Bm1 w_WW ww_Bm1 w_pred_c. - - Let sub_c := - Eval lazy beta iota delta [ww_sub_c ww_opp_c] in - ww_sub_c w_0 w_WW w_opp_c w_opp_carry w_sub_c w_sub_carry_c. - - Let sub_carry_c := - Eval lazy beta iota delta [ww_sub_carry_c ww_pred_c ww_opp_carry] in - ww_sub_carry_c w_Bm1 w_WW ww_Bm1 w_opp_carry w_pred_c w_sub_c w_sub_carry_c. - - Let pred := - Eval lazy beta delta [ww_pred] in ww_pred w_Bm1 w_WW ww_Bm1 w_pred_c w_pred. - - Let sub := - Eval lazy beta iota delta [ww_sub ww_opp] in - ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c w_opp w_sub w_sub_carry. - - Let sub_carry := - Eval lazy beta iota delta [ww_sub_carry ww_pred ww_opp_carry] in - ww_sub_carry w_Bm1 w_WW ww_Bm1 w_opp_carry w_pred_c w_sub_carry_c w_pred - w_sub w_sub_carry. - - - (* ** Multiplication ** *) - - Let mul_c := - Eval lazy beta iota delta [ww_mul_c double_mul_c] in - ww_mul_c w_0 w_1 w_WW w_W0 w_mul_c add_c add add_carry. - - Let karatsuba_c := - Eval lazy beta iota delta [ww_karatsuba_c double_mul_c kara_prod] in - ww_karatsuba_c w_0 w_1 w_WW w_W0 w_compare w_add w_sub w_mul_c - add_c add add_carry sub_c sub. - - Let mul := - Eval lazy beta delta [ww_mul] in - ww_mul w_W0 w_add w_mul_c w_mul add. - - Let square_c := - Eval lazy beta delta [ww_square_c] in - ww_square_c w_0 w_1 w_WW w_W0 w_mul_c w_square_c add_c add add_carry. - - (* Division operation *) - - Let div32 := - Eval lazy beta iota delta [w_div32] in - w_div32 w_0 w_Bm1 w_Bm2 w_WW w_compare w_add_c w_add_carry_c - w_add w_add_carry w_pred w_sub w_mul_c w_div21 sub_c. - - Let div21 := - Eval lazy beta iota delta [ww_div21] in - ww_div21 w_0 w_0W div32 ww_1 compare sub. - - Let low (p: zn2z t) := match p with WW _ p1 => p1 | _ => w_0 end. - - Let add_mul_div := - Eval lazy beta delta [ww_add_mul_div] in - ww_add_mul_div w_0 w_WW w_W0 w_0W compare w_add_mul_div sub w_zdigits low. - - Let div_gt := - Eval lazy beta delta [ww_div_gt] in - ww_div_gt w_0 w_WW w_0W w_compare w_eq0 w_opp_c w_opp - w_opp_carry w_sub_c w_sub w_sub_carry - w_div_gt w_add_mul_div w_head0 w_div21 div32 _ww_zdigits ww_1 add_mul_div w_zdigits. - - Let div := - Eval lazy beta delta [ww_div] in ww_div ww_1 compare div_gt. - - Let mod_gt := - Eval lazy beta delta [ww_mod_gt] in - ww_mod_gt w_0 w_WW w_0W w_compare w_eq0 w_opp_c w_opp w_opp_carry w_sub_c w_sub w_sub_carry - w_mod_gt w_add_mul_div w_head0 w_div21 div32 _ww_zdigits add_mul_div w_zdigits. - - Let mod_ := - Eval lazy beta delta [ww_mod] in ww_mod compare mod_gt. - - Let pos_mod := - Eval lazy beta delta [ww_pos_mod] in - ww_pos_mod w_0 w_zdigits w_WW w_pos_mod compare w_0W low sub _ww_zdigits. - - Let is_even := - Eval lazy beta delta [ww_is_even] in ww_is_even w_is_even. - - Let sqrt2 := - Eval lazy beta delta [ww_sqrt2] in - ww_sqrt2 w_is_even w_compare w_0 w_1 w_Bm1 w_0W w_sub w_square_c - w_div21 w_add_mul_div w_zdigits w_add_c w_sqrt2 w_pred pred_c - pred add_c add sub_c add_mul_div. - - Let sqrt := - Eval lazy beta delta [ww_sqrt] in - ww_sqrt w_is_even w_0 w_sub w_add_mul_div w_zdigits - _ww_zdigits w_sqrt2 pred add_mul_div head0 compare low. - - Let gcd_gt_fix := - Eval cbv beta delta [ww_gcd_gt_aux ww_gcd_gt_body] in - ww_gcd_gt_aux w_0 w_WW w_0W w_compare w_opp_c w_opp w_opp_carry - w_sub_c w_sub w_sub_carry w_gcd_gt - w_add_mul_div w_head0 w_div21 div32 _ww_zdigits add_mul_div - w_zdigits. - - Let gcd_cont := - Eval lazy beta delta [gcd_cont] in gcd_cont ww_1 w_1 w_compare. - - Let gcd_gt := - Eval lazy beta delta [ww_gcd_gt] in - ww_gcd_gt w_0 w_eq0 w_gcd_gt _ww_digits gcd_gt_fix gcd_cont. - - Let gcd := - Eval lazy beta delta [ww_gcd] in - ww_gcd compare w_0 w_eq0 w_gcd_gt _ww_digits gcd_gt_fix gcd_cont. - - Definition lor (x y : zn2z t) := - match x, y with - | W0, _ => y - | _, W0 => x - | WW hx lx, WW hy ly => WW (ZnZ.lor hx hy) (ZnZ.lor lx ly) - end. - - Definition land (x y : zn2z t) := - match x, y with - | W0, _ => W0 - | _, W0 => W0 - | WW hx lx, WW hy ly => WW (ZnZ.land hx hy) (ZnZ.land lx ly) - end. - - Definition lxor (x y : zn2z t) := - match x, y with - | W0, _ => y - | _, W0 => x - | WW hx lx, WW hy ly => WW (ZnZ.lxor hx hy) (ZnZ.lxor lx ly) - end. - - (* ** Record of operators on 2 words *) - - Global Instance mk_zn2z_ops : ZnZ.Ops (zn2z t) | 1 := - ZnZ.MkOps _ww_digits _ww_zdigits - to_Z ww_of_pos head0 tail0 - W0 ww_1 ww_Bm1 - compare eq0 - opp_c opp opp_carry - succ_c add_c add_carry_c - succ add add_carry - pred_c sub_c sub_carry_c - pred sub sub_carry - mul_c mul square_c - div21 div_gt div - mod_gt mod_ - gcd_gt gcd - add_mul_div - pos_mod - is_even - sqrt2 - sqrt - lor - land - lxor. - - Global Instance mk_zn2z_ops_karatsuba : ZnZ.Ops (zn2z t) | 2 := - ZnZ.MkOps _ww_digits _ww_zdigits - to_Z ww_of_pos head0 tail0 - W0 ww_1 ww_Bm1 - compare eq0 - opp_c opp opp_carry - succ_c add_c add_carry_c - succ add add_carry - pred_c sub_c sub_carry_c - pred sub sub_carry - karatsuba_c mul square_c - div21 div_gt div - mod_gt mod_ - gcd_gt gcd - add_mul_div - pos_mod - is_even - sqrt2 - sqrt - lor - land - lxor. - - (* Proof *) - Context {specs : ZnZ.Specs ops}. - - Create HintDb ZnZ. - - Hint Resolve - ZnZ.spec_to_Z - ZnZ.spec_of_pos - ZnZ.spec_0 - ZnZ.spec_1 - ZnZ.spec_m1 - ZnZ.spec_compare - ZnZ.spec_eq0 - ZnZ.spec_opp_c - ZnZ.spec_opp - ZnZ.spec_opp_carry - ZnZ.spec_succ_c - ZnZ.spec_add_c - ZnZ.spec_add_carry_c - ZnZ.spec_succ - ZnZ.spec_add - ZnZ.spec_add_carry - ZnZ.spec_pred_c - ZnZ.spec_sub_c - ZnZ.spec_sub_carry_c - ZnZ.spec_pred - ZnZ.spec_sub - ZnZ.spec_sub_carry - ZnZ.spec_mul_c - ZnZ.spec_mul - ZnZ.spec_square_c - ZnZ.spec_div21 - ZnZ.spec_div_gt - ZnZ.spec_div - ZnZ.spec_modulo_gt - ZnZ.spec_modulo - ZnZ.spec_gcd_gt - ZnZ.spec_gcd - ZnZ.spec_head0 - ZnZ.spec_tail0 - ZnZ.spec_add_mul_div - ZnZ.spec_pos_mod - ZnZ.spec_is_even - ZnZ.spec_sqrt2 - ZnZ.spec_sqrt - ZnZ.spec_WO - ZnZ.spec_OW - ZnZ.spec_WW : ZnZ. - - Ltac wwauto := unfold ww_to_Z; eauto with ZnZ. - - Let wwB := base _ww_digits. - - Notation "[| x |]" := (to_Z x) (at level 0, x at level 99). - - Notation "[+| c |]" := - (interp_carry 1 wwB to_Z c) (at level 0, c at level 99). - - Notation "[-| c |]" := - (interp_carry (-1) wwB to_Z c) (at level 0, c at level 99). - - Notation "[[ x ]]" := (zn2z_to_Z wwB to_Z x) (at level 0, x at level 99). - - Let spec_ww_to_Z : forall x, 0 <= [| x |] < wwB. - Proof. refine (spec_ww_to_Z w_digits w_to_Z _); wwauto. Qed. - - Let spec_ww_of_pos : forall p, - Zpos p = (Z.of_N (fst (ww_of_pos p)))*wwB + [|(snd (ww_of_pos p))|]. - Proof. - unfold ww_of_pos;intros. - rewrite (ZnZ.spec_of_pos p). unfold w_of_pos. - case (ZnZ.of_pos p); intros. simpl. - destruct n; simpl ZnZ.to_Z. - simpl;unfold w_to_Z,w_0; rewrite ZnZ.spec_0;trivial. - unfold Z.of_N. - rewrite (ZnZ.spec_of_pos p0). - case (ZnZ.of_pos p0); intros. simpl. - unfold fst, snd,Z.of_N, to_Z, wB, w_digits, w_to_Z, w_WW. - rewrite ZnZ.spec_WW. - replace wwB with (wB*wB). - unfold wB,w_to_Z,w_digits;destruct n;ring. - symmetry. rewrite <- Z.pow_2_r; exact (wwB_wBwB w_digits). - Qed. - - Let spec_ww_0 : [|W0|] = 0. - Proof. reflexivity. Qed. - - Let spec_ww_1 : [|ww_1|] = 1. - Proof. refine (spec_ww_1 w_0 w_1 w_digits w_to_Z _ _);wwauto. Qed. - - Let spec_ww_Bm1 : [|ww_Bm1|] = wwB - 1. - Proof. refine (spec_ww_Bm1 w_Bm1 w_digits w_to_Z _);wwauto. Qed. - - Let spec_ww_compare : - forall x y, compare x y = Z.compare [|x|] [|y|]. - Proof. - refine (spec_ww_compare w_0 w_digits w_to_Z w_compare _ _ _);wwauto. - Qed. - - Let spec_ww_eq0 : forall x, eq0 x = true -> [|x|] = 0. - Proof. destruct x;simpl;intros;trivial;discriminate. Qed. - - Let spec_ww_opp_c : forall x, [-|opp_c x|] = -[|x|]. - Proof. - refine(spec_ww_opp_c w_0 w_0 W0 w_opp_c w_opp_carry w_digits w_to_Z _ _ _ _); - wwauto. - Qed. - - Let spec_ww_opp : forall x, [|opp x|] = (-[|x|]) mod wwB. - Proof. - refine(spec_ww_opp w_0 w_0 W0 w_opp_c w_opp_carry w_opp - w_digits w_to_Z _ _ _ _ _); - wwauto. - Qed. - - Let spec_ww_opp_carry : forall x, [|opp_carry x|] = wwB - [|x|] - 1. - Proof. - refine (spec_ww_opp_carry w_WW ww_Bm1 w_opp_carry w_digits w_to_Z _ _ _); - wwauto. - Qed. - - Let spec_ww_succ_c : forall x, [+|succ_c x|] = [|x|] + 1. - Proof. - refine (spec_ww_succ_c w_0 w_0 ww_1 w_succ_c w_digits w_to_Z _ _ _ _);wwauto. - Qed. - - Let spec_ww_add_c : forall x y, [+|add_c x y|] = [|x|] + [|y|]. - Proof. - refine (spec_ww_add_c w_WW w_add_c w_add_carry_c w_digits w_to_Z _ _ _);wwauto. - Qed. - - Let spec_ww_add_carry_c : forall x y, [+|add_carry_c x y|] = [|x|]+[|y|]+1. - Proof. - refine (spec_ww_add_carry_c w_0 w_0 w_WW ww_1 w_succ_c w_add_c w_add_carry_c - w_digits w_to_Z _ _ _ _ _ _ _);wwauto. - Qed. - - Let spec_ww_succ : forall x, [|succ x|] = ([|x|] + 1) mod wwB. - Proof. - refine (spec_ww_succ w_W0 ww_1 w_succ_c w_succ w_digits w_to_Z _ _ _ _ _); - wwauto. - Qed. - - Let spec_ww_add : forall x y, [|add x y|] = ([|x|] + [|y|]) mod wwB. - Proof. - refine (spec_ww_add w_add_c w_add w_add_carry w_digits w_to_Z _ _ _ _);wwauto. - Qed. - - Let spec_ww_add_carry : forall x y, [|add_carry x y|]=([|x|]+[|y|]+1)mod wwB. - Proof. - refine (spec_ww_add_carry w_W0 ww_1 w_succ_c w_add_carry_c w_succ - w_add w_add_carry w_digits w_to_Z _ _ _ _ _ _ _ _);wwauto. - Qed. - - Let spec_ww_pred_c : forall x, [-|pred_c x|] = [|x|] - 1. - Proof. - refine (spec_ww_pred_c w_0 w_Bm1 w_WW ww_Bm1 w_pred_c w_digits w_to_Z - _ _ _ _ _);wwauto. - Qed. - - Let spec_ww_sub_c : forall x y, [-|sub_c x y|] = [|x|] - [|y|]. - Proof. - refine (spec_ww_sub_c w_0 w_0 w_WW W0 w_opp_c w_opp_carry w_sub_c - w_sub_carry_c w_digits w_to_Z _ _ _ _ _ _ _);wwauto. - Qed. - - Let spec_ww_sub_carry_c : forall x y, [-|sub_carry_c x y|] = [|x|]-[|y|]-1. - Proof. - refine (spec_ww_sub_carry_c w_0 w_Bm1 w_WW ww_Bm1 w_opp_carry w_pred_c - w_sub_c w_sub_carry_c w_digits w_to_Z _ _ _ _ _ _ _ _);wwauto. - Qed. - - Let spec_ww_pred : forall x, [|pred x|] = ([|x|] - 1) mod wwB. - Proof. - refine (spec_ww_pred w_0 w_Bm1 w_WW ww_Bm1 w_pred_c w_pred w_digits w_to_Z - _ _ _ _ _ _);wwauto. - Qed. - - Let spec_ww_sub : forall x y, [|sub x y|] = ([|x|] - [|y|]) mod wwB. - Proof. - refine (spec_ww_sub w_0 w_0 w_WW W0 w_opp_c w_opp_carry w_sub_c w_opp - w_sub w_sub_carry w_digits w_to_Z _ _ _ _ _ _ _ _ _);wwauto. - Qed. - - Let spec_ww_sub_carry : forall x y, [|sub_carry x y|]=([|x|]-[|y|]-1) mod wwB. - Proof. - refine (spec_ww_sub_carry w_0 w_Bm1 w_WW ww_Bm1 w_opp_carry w_pred_c - w_sub_carry_c w_pred w_sub w_sub_carry w_digits w_to_Z _ _ _ _ _ _ _ _ _ _); - wwauto. - Qed. - - Let spec_ww_mul_c : forall x y, [[mul_c x y ]] = [|x|] * [|y|]. - Proof. - refine (spec_ww_mul_c w_0 w_1 w_WW w_W0 w_mul_c add_c add add_carry w_digits - w_to_Z _ _ _ _ _ _ _ _ _);wwauto. - Qed. - - Let spec_ww_karatsuba_c : forall x y, [[karatsuba_c x y ]] = [|x|] * [|y|]. - Proof. - refine (spec_ww_karatsuba_c _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _); wwauto. - unfold w_digits; apply ZnZ.spec_more_than_1_digit; auto. - Qed. - - Let spec_ww_mul : forall x y, [|mul x y|] = ([|x|] * [|y|]) mod wwB. - Proof. - refine (spec_ww_mul w_W0 w_add w_mul_c w_mul add w_digits w_to_Z _ _ _ _ _); - wwauto. - Qed. - - Let spec_ww_square_c : forall x, [[square_c x]] = [|x|] * [|x|]. - Proof. - refine (spec_ww_square_c w_0 w_1 w_WW w_W0 w_mul_c w_square_c add_c add - add_carry w_digits w_to_Z _ _ _ _ _ _ _ _ _ _);wwauto. - Qed. - - Let spec_w_div32 : forall a1 a2 a3 b1 b2, - wB / 2 <= (w_to_Z b1) -> - [|WW a1 a2|] < [|WW b1 b2|] -> - let (q, r) := div32 a1 a2 a3 b1 b2 in - (w_to_Z a1) * wwB + (w_to_Z a2) * wB + (w_to_Z a3) = - (w_to_Z q) * ((w_to_Z b1)*wB + (w_to_Z b2)) + [|r|] /\ - 0 <= [|r|] < (w_to_Z b1)*wB + w_to_Z b2. - Proof. - refine (spec_w_div32 w_0 w_Bm1 w_Bm2 w_WW w_compare w_add_c w_add_carry_c - w_add w_add_carry w_pred w_sub w_mul_c w_div21 sub_c w_digits w_to_Z - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _);wwauto. - unfold w_Bm2, w_to_Z, w_pred, w_Bm1. - rewrite ZnZ.spec_pred, ZnZ.spec_m1. - unfold w_digits;rewrite Zmod_small. ring. - assert (H:= wB_pos(ZnZ.digits)). omega. - exact ZnZ.spec_div21. - Qed. - - Let spec_ww_div21 : forall a1 a2 b, - wwB/2 <= [|b|] -> - [|a1|] < [|b|] -> - let (q,r) := div21 a1 a2 b in - [|a1|] *wwB+ [|a2|] = [|q|] * [|b|] + [|r|] /\ - 0 <= [|r|] < [|b|]. - Proof. - refine (spec_ww_div21 w_0 w_0W div32 ww_1 compare sub w_digits w_to_Z - _ _ _ _ _ _ _);wwauto. - Qed. - - Let spec_add2: forall x y, - [|w_add2 x y|] = w_to_Z x + w_to_Z y. - unfold w_add2. - intros xh xl; generalize (ZnZ.spec_add_c xh xl). - unfold w_add_c; case ZnZ.add_c; unfold interp_carry; simpl ww_to_Z. - intros w0 Hw0; simpl; unfold w_to_Z; rewrite Hw0. - unfold w_0; rewrite ZnZ.spec_0; simpl; auto with zarith. - intros w0; rewrite Z.mul_1_l; simpl. - unfold w_to_Z, w_1; rewrite ZnZ.spec_1; auto with zarith. - rewrite Z.mul_1_l; auto. - Qed. - - Let spec_low: forall x, - w_to_Z (low x) = [|x|] mod wB. - intros x; case x; simpl low. - unfold ww_to_Z, w_to_Z, w_0; rewrite ZnZ.spec_0; simpl; wwauto. - intros xh xl; simpl. - rewrite Z.add_comm; rewrite Z_mod_plus; auto with zarith. - rewrite Zmod_small; auto with zarith. - unfold wB, base; eauto with ZnZ zarith. - unfold wB, base; eauto with ZnZ zarith. - Qed. - - Let spec_ww_digits: - [|_ww_zdigits|] = Zpos (xO w_digits). - Proof. - unfold w_to_Z, _ww_zdigits. - rewrite spec_add2. - unfold w_to_Z, w_zdigits, w_digits. - rewrite ZnZ.spec_zdigits; auto. - rewrite Pos2Z.inj_xO; auto with zarith. - Qed. - - - Let spec_ww_head00 : forall x, [|x|] = 0 -> [|head0 x|] = Zpos _ww_digits. - Proof. - refine (spec_ww_head00 w_0 w_0W - w_compare w_head0 w_add2 w_zdigits _ww_zdigits - w_to_Z _ _ _ (eq_refl _ww_digits) _ _ _ _); wwauto. - exact ZnZ.spec_head00. - exact ZnZ.spec_zdigits. - Qed. - - Let spec_ww_head0 : forall x, 0 < [|x|] -> - wwB/ 2 <= 2 ^ [|head0 x|] * [|x|] < wwB. - Proof. - refine (spec_ww_head0 w_0 w_0W w_compare w_head0 - w_add2 w_zdigits _ww_zdigits - w_to_Z _ _ _ _ _ _ _);wwauto. - exact ZnZ.spec_zdigits. - Qed. - - Let spec_ww_tail00 : forall x, [|x|] = 0 -> [|tail0 x|] = Zpos _ww_digits. - Proof. - refine (spec_ww_tail00 w_0 w_0W - w_compare w_tail0 w_add2 w_zdigits _ww_zdigits - w_to_Z _ _ _ (eq_refl _ww_digits) _ _ _ _); wwauto. - exact ZnZ.spec_tail00. - exact ZnZ.spec_zdigits. - Qed. - - - Let spec_ww_tail0 : forall x, 0 < [|x|] -> - exists y, 0 <= y /\ [|x|] = (2 * y + 1) * 2 ^ [|tail0 x|]. - Proof. - refine (spec_ww_tail0 (w_digits := w_digits) w_0 w_0W w_compare w_tail0 - w_add2 w_zdigits _ww_zdigits w_to_Z _ _ _ _ _ _ _);wwauto. - exact ZnZ.spec_zdigits. - Qed. - - Lemma spec_ww_add_mul_div : forall x y p, - [|p|] <= Zpos _ww_digits -> - [| add_mul_div p x y |] = - ([|x|] * (2 ^ [|p|]) + - [|y|] / (2 ^ ((Zpos _ww_digits) - [|p|]))) mod wwB. - Proof. - refine (@spec_ww_add_mul_div t w_0 w_WW w_W0 w_0W compare w_add_mul_div - sub w_digits w_zdigits low w_to_Z - _ _ _ _ _ _ _ _ _ _ _);wwauto. - exact ZnZ.spec_zdigits. - Qed. - - Let spec_ww_div_gt : forall a b, - [|a|] > [|b|] -> 0 < [|b|] -> - let (q,r) := div_gt a b in - [|a|] = [|q|] * [|b|] + [|r|] /\ 0 <= [|r|] < [|b|]. - Proof. -refine -(@spec_ww_div_gt t w_digits w_0 w_WW w_0W w_compare w_eq0 - w_opp_c w_opp w_opp_carry w_sub_c w_sub w_sub_carry w_div_gt - w_add_mul_div w_head0 w_div21 div32 _ww_zdigits ww_1 add_mul_div w_zdigits w_to_Z - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ -). - exact ZnZ.spec_0. - exact ZnZ.spec_to_Z. - wwauto. - wwauto. - exact ZnZ.spec_compare. - exact ZnZ.spec_eq0. - exact ZnZ.spec_opp_c. - exact ZnZ.spec_opp. - exact ZnZ.spec_opp_carry. - exact ZnZ.spec_sub_c. - exact ZnZ.spec_sub. - exact ZnZ.spec_sub_carry. - exact ZnZ.spec_div_gt. - exact ZnZ.spec_add_mul_div. - exact ZnZ.spec_head0. - exact ZnZ.spec_div21. - exact spec_w_div32. - exact ZnZ.spec_zdigits. - exact spec_ww_digits. - exact spec_ww_1. - exact spec_ww_add_mul_div. - Qed. - - Let spec_ww_div : forall a b, 0 < [|b|] -> - let (q,r) := div a b in - [|a|] = [|q|] * [|b|] + [|r|] /\ - 0 <= [|r|] < [|b|]. - Proof. - refine (spec_ww_div w_digits ww_1 compare div_gt w_to_Z _ _ _ _);wwauto. - Qed. - - Let spec_ww_mod_gt : forall a b, - [|a|] > [|b|] -> 0 < [|b|] -> - [|mod_gt a b|] = [|a|] mod [|b|]. - Proof. - refine (@spec_ww_mod_gt t w_digits w_0 w_WW w_0W w_compare w_eq0 - w_opp_c w_opp w_opp_carry w_sub_c w_sub w_sub_carry w_div_gt w_mod_gt - w_add_mul_div w_head0 w_div21 div32 _ww_zdigits ww_1 add_mul_div - w_zdigits w_to_Z - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _);wwauto. - exact ZnZ.spec_div_gt. - exact ZnZ.spec_div21. - exact ZnZ.spec_zdigits. - exact spec_ww_add_mul_div. - Qed. - - Let spec_ww_mod : forall a b, 0 < [|b|] -> [|mod_ a b|] = [|a|] mod [|b|]. - Proof. - refine (spec_ww_mod w_digits W0 compare mod_gt w_to_Z _ _ _);wwauto. - Qed. - - Let spec_ww_gcd_gt : forall a b, [|a|] > [|b|] -> - Zis_gcd [|a|] [|b|] [|gcd_gt a b|]. - Proof. - refine (@spec_ww_gcd_gt t w_digits W0 w_to_Z _ - w_0 w_0 w_eq0 w_gcd_gt _ww_digits - _ gcd_gt_fix _ _ _ _ gcd_cont _);wwauto. - refine (@spec_ww_gcd_gt_aux t w_digits w_0 w_WW w_0W w_compare w_opp_c w_opp - w_opp_carry w_sub_c w_sub w_sub_carry w_gcd_gt w_add_mul_div w_head0 - w_div21 div32 _ww_zdigits ww_1 add_mul_div w_zdigits w_to_Z - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _);wwauto. - exact ZnZ.spec_div21. - exact ZnZ.spec_zdigits. - exact spec_ww_add_mul_div. - refine (@spec_gcd_cont t w_digits ww_1 w_to_Z _ _ w_0 w_1 w_compare - _ _);wwauto. - Qed. - - Let spec_ww_gcd : forall a b, Zis_gcd [|a|] [|b|] [|gcd a b|]. - Proof. - refine (@spec_ww_gcd t w_digits W0 compare w_to_Z _ _ w_0 w_0 w_eq0 w_gcd_gt - _ww_digits _ gcd_gt_fix _ _ _ _ gcd_cont _);wwauto. - refine (@spec_ww_gcd_gt_aux t w_digits w_0 w_WW w_0W w_compare w_opp_c w_opp - w_opp_carry w_sub_c w_sub w_sub_carry w_gcd_gt w_add_mul_div w_head0 - w_div21 div32 _ww_zdigits ww_1 add_mul_div w_zdigits w_to_Z - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _);wwauto. - exact ZnZ.spec_div21. - exact ZnZ.spec_zdigits. - exact spec_ww_add_mul_div. - refine (@spec_gcd_cont t w_digits ww_1 w_to_Z _ _ w_0 w_1 w_compare - _ _);wwauto. - Qed. - - Let spec_ww_is_even : forall x, - match is_even x with - true => [|x|] mod 2 = 0 - | false => [|x|] mod 2 = 1 - end. - Proof. - refine (@spec_ww_is_even t w_is_even w_digits _ _ ). - exact ZnZ.spec_is_even. - Qed. - - Let spec_ww_sqrt2 : forall x y, - wwB/ 4 <= [|x|] -> - let (s,r) := sqrt2 x y in - [[WW x y]] = [|s|] ^ 2 + [+|r|] /\ - [+|r|] <= 2 * [|s|]. - Proof. - intros x y H. - refine (@spec_ww_sqrt2 t w_is_even w_compare w_0 w_1 w_Bm1 - w_0W w_sub w_square_c w_div21 w_add_mul_div w_digits w_zdigits - _ww_zdigits - w_add_c w_sqrt2 w_pred pred_c pred add_c add sub_c add_mul_div - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _); wwauto. - exact ZnZ.spec_zdigits. - exact ZnZ.spec_more_than_1_digit. - exact ZnZ.spec_is_even. - exact ZnZ.spec_div21. - exact spec_ww_add_mul_div. - exact ZnZ.spec_sqrt2. - Qed. - - Let spec_ww_sqrt : forall x, - [|sqrt x|] ^ 2 <= [|x|] < ([|sqrt x|] + 1) ^ 2. - Proof. - refine (@spec_ww_sqrt t w_is_even w_0 w_1 w_Bm1 - w_sub w_add_mul_div w_digits w_zdigits _ww_zdigits - w_sqrt2 pred add_mul_div head0 compare - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _); wwauto. - exact ZnZ.spec_zdigits. - exact ZnZ.spec_more_than_1_digit. - exact ZnZ.spec_is_even. - exact spec_ww_add_mul_div. - exact ZnZ.spec_sqrt2. - Qed. - - Let wB_pos : 0 < wB. - Proof. - unfold wB, base; apply Z.pow_pos_nonneg; auto with zarith. - Qed. - - Hint Transparent ww_to_Z. - - Let ww_testbit_high n x y : Z.pos w_digits <= n -> - Z.testbit [|WW x y|] n = - Z.testbit (ZnZ.to_Z x) (n - Z.pos w_digits). - Proof. - intros Hn. - assert (E : ZnZ.to_Z x = [|WW x y|] / wB). - { simpl. - rewrite Z.div_add_l; eauto with ZnZ zarith. - now rewrite Z.div_small, Z.add_0_r; wwauto. } - rewrite E. - unfold wB, base. rewrite Z.div_pow2_bits. - - f_equal; auto with zarith. - - easy. - - auto with zarith. - Qed. - - Let ww_testbit_low n x y : 0 <= n < Z.pos w_digits -> - Z.testbit [|WW x y|] n = Z.testbit (ZnZ.to_Z y) n. - Proof. - intros (Hn,Hn'). - assert (E : ZnZ.to_Z y = [|WW x y|] mod wB). - { simpl; symmetry. - rewrite Z.add_comm, Z.mod_add; auto with zarith nocore. - apply Z.mod_small; eauto with ZnZ zarith. } - rewrite E. - unfold wB, base. symmetry. apply Z.mod_pow2_bits_low; auto. - Qed. - - Let spec_lor x y : [|lor x y|] = Z.lor [|x|] [|y|]. - Proof. - destruct x as [ |hx lx]. trivial. - destruct y as [ |hy ly]. now rewrite Z.lor_comm. - change ([|WW (ZnZ.lor hx hy) (ZnZ.lor lx ly)|] = - Z.lor [|WW hx lx|] [|WW hy ly|]). - apply Z.bits_inj'; intros n Hn. - rewrite Z.lor_spec. - destruct (Z.le_gt_cases (Z.pos w_digits) n) as [LE|GT]. - - now rewrite !ww_testbit_high, ZnZ.spec_lor, Z.lor_spec. - - rewrite !ww_testbit_low; auto. - now rewrite ZnZ.spec_lor, Z.lor_spec. - Qed. - - Let spec_land x y : [|land x y|] = Z.land [|x|] [|y|]. - Proof. - destruct x as [ |hx lx]. trivial. - destruct y as [ |hy ly]. now rewrite Z.land_comm. - change ([|WW (ZnZ.land hx hy) (ZnZ.land lx ly)|] = - Z.land [|WW hx lx|] [|WW hy ly|]). - apply Z.bits_inj'; intros n Hn. - rewrite Z.land_spec. - destruct (Z.le_gt_cases (Z.pos w_digits) n) as [LE|GT]. - - now rewrite !ww_testbit_high, ZnZ.spec_land, Z.land_spec. - - rewrite !ww_testbit_low; auto. - now rewrite ZnZ.spec_land, Z.land_spec. - Qed. - - Let spec_lxor x y : [|lxor x y|] = Z.lxor [|x|] [|y|]. - Proof. - destruct x as [ |hx lx]. trivial. - destruct y as [ |hy ly]. now rewrite Z.lxor_comm. - change ([|WW (ZnZ.lxor hx hy) (ZnZ.lxor lx ly)|] = - Z.lxor [|WW hx lx|] [|WW hy ly|]). - apply Z.bits_inj'; intros n Hn. - rewrite Z.lxor_spec. - destruct (Z.le_gt_cases (Z.pos w_digits) n) as [LE|GT]. - - now rewrite !ww_testbit_high, ZnZ.spec_lxor, Z.lxor_spec. - - rewrite !ww_testbit_low; auto. - now rewrite ZnZ.spec_lxor, Z.lxor_spec. - Qed. - - Global Instance mk_zn2z_specs : ZnZ.Specs mk_zn2z_ops. - Proof. - apply ZnZ.MkSpecs; auto. - exact spec_ww_add_mul_div. - - refine (@spec_ww_pos_mod t w_0 w_digits w_zdigits w_WW - w_pos_mod compare w_0W low sub _ww_zdigits w_to_Z - _ _ _ _ _ _ _ _ _ _ _ _);wwauto. - exact ZnZ.spec_zdigits. - unfold w_to_Z, w_zdigits. - rewrite ZnZ.spec_zdigits. - rewrite <- Pos2Z.inj_xO; exact spec_ww_digits. - Qed. - - Global Instance mk_zn2z_specs_karatsuba : ZnZ.Specs mk_zn2z_ops_karatsuba. - Proof. - apply ZnZ.MkSpecs; auto. - exact spec_ww_add_mul_div. - refine (@spec_ww_pos_mod t w_0 w_digits w_zdigits w_WW - w_pos_mod compare w_0W low sub _ww_zdigits w_to_Z - _ _ _ _ _ _ _ _ _ _ _ _);wwauto. - exact ZnZ.spec_zdigits. - unfold w_to_Z, w_zdigits. - rewrite ZnZ.spec_zdigits. - rewrite <- Pos2Z.inj_xO; exact spec_ww_digits. - Qed. - -End Z_2nZ. - - -Section MulAdd. - - Context {t : Type}{ops : ZnZ.Ops t}{specs : ZnZ.Specs ops}. - - Definition mul_add:= w_mul_add ZnZ.zero ZnZ.succ ZnZ.add_c ZnZ.mul_c. - - Notation "[| x |]" := (ZnZ.to_Z x) (at level 0, x at level 99). - - Notation "[|| x ||]" := - (zn2z_to_Z (base ZnZ.digits) ZnZ.to_Z x) (at level 0, x at level 99). - - Lemma spec_mul_add: forall x y z, - let (zh, zl) := mul_add x y z in - [||WW zh zl||] = [|x|] * [|y|] + [|z|]. - Proof. - intros x y z. - refine (spec_w_mul_add _ _ _ _ _ _ _ _ _ _ _ _ x y z); auto. - exact ZnZ.spec_0. - exact ZnZ.spec_to_Z. - exact ZnZ.spec_succ. - exact ZnZ.spec_add_c. - exact ZnZ.spec_mul_c. - Qed. - -End MulAdd. - - -(** Modular versions of DoubleCyclic *) - -Module DoubleCyclic (C:CyclicType) <: CyclicType. - Definition t := zn2z C.t. - Instance ops : ZnZ.Ops t := mk_zn2z_ops. - Instance specs : ZnZ.Specs ops := mk_zn2z_specs. -End DoubleCyclic. - -Module DoubleCyclicKaratsuba (C:CyclicType) <: CyclicType. - Definition t := zn2z C.t. - Definition ops : ZnZ.Ops t := mk_zn2z_ops_karatsuba. - Definition specs : ZnZ.Specs ops := mk_zn2z_specs_karatsuba. -End DoubleCyclicKaratsuba. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v deleted file mode 100644 index 09d7329b..00000000 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v +++ /dev/null @@ -1,1494 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Set Implicit Arguments. - -Require Import ZArith. -Require Import BigNumPrelude. -Require Import DoubleType. -Require Import DoubleBase. -Require Import DoubleDivn1. -Require Import DoubleAdd. -Require Import DoubleSub. - -Local Open Scope Z_scope. - -Ltac zarith := auto with zarith. - - -Section POS_MOD. - - Variable w:Type. - Variable w_0 : w. - Variable w_digits : positive. - Variable w_zdigits : w. - Variable w_WW : w -> w -> zn2z w. - Variable w_pos_mod : w -> w -> w. - Variable w_compare : w -> w -> comparison. - Variable ww_compare : zn2z w -> zn2z w -> comparison. - Variable w_0W : w -> zn2z w. - Variable low: zn2z w -> w. - Variable ww_sub: zn2z w -> zn2z w -> zn2z w. - Variable ww_zdigits : zn2z w. - - - Definition ww_pos_mod p x := - let zdigits := w_0W w_zdigits in - match x with - | W0 => W0 - | WW xh xl => - match ww_compare p zdigits with - | Eq => w_WW w_0 xl - | Lt => w_WW w_0 (w_pos_mod (low p) xl) - | Gt => - match ww_compare p ww_zdigits with - | Lt => - let n := low (ww_sub p zdigits) in - w_WW (w_pos_mod n xh) xl - | _ => x - end - end - end. - - - Variable w_to_Z : w -> Z. - - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - - Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99). - - - Variable spec_w_0 : [|w_0|] = 0. - - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - - Variable spec_to_w_Z : forall x, 0 <= [[x]] < wwB. - - Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. - - Variable spec_pos_mod : forall w p, - [|w_pos_mod p w|] = [|w|] mod (2 ^ [|p|]). - - Variable spec_w_0W : forall l, [[w_0W l]] = [|l|]. - Variable spec_ww_compare : forall x y, - ww_compare x y = Z.compare [[x]] [[y]]. - Variable spec_ww_sub: forall x y, - [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB. - - Variable spec_zdigits : [| w_zdigits |] = Zpos w_digits. - Variable spec_low: forall x, [| low x|] = [[x]] mod wB. - Variable spec_ww_zdigits : [[ww_zdigits]] = 2 * [|w_zdigits|]. - Variable spec_ww_digits : ww_digits w_digits = xO w_digits. - - - Hint Rewrite spec_w_0 spec_w_WW : w_rewrite. - - Lemma spec_ww_pos_mod : forall w p, - [[ww_pos_mod p w]] = [[w]] mod (2 ^ [[p]]). - assert (HHHHH:= lt_0_wB w_digits). - assert (F0: forall x y, x - y + y = x); auto with zarith. - intros w1 p; case (spec_to_w_Z p); intros HH1 HH2. - unfold ww_pos_mod; case w1. reflexivity. - intros xh xl; rewrite spec_ww_compare. - case Z.compare_spec; - rewrite spec_w_0W; rewrite spec_zdigits; fold wB; - intros H1. - rewrite H1; simpl ww_to_Z. - autorewrite with w_rewrite rm10. - rewrite Zplus_mod; auto with zarith. - rewrite Z_mod_mult; auto with zarith. - autorewrite with rm10. - rewrite Zmod_mod; auto with zarith. - rewrite Zmod_small; auto with zarith. - autorewrite with w_rewrite rm10. - simpl ww_to_Z. - rewrite spec_pos_mod. - assert (HH0: [|low p|] = [[p]]). - rewrite spec_low. - apply Zmod_small; auto with zarith. - case (spec_to_w_Z p); intros HHH1 HHH2; split; auto with zarith. - apply Z.lt_le_trans with (1 := H1). - unfold base; apply Zpower2_le_lin; auto with zarith. - rewrite HH0. - rewrite Zplus_mod; auto with zarith. - unfold base. - rewrite <- (F0 (Zpos w_digits) [[p]]). - rewrite Zpower_exp; auto with zarith. - rewrite Z.mul_assoc. - rewrite Z_mod_mult; auto with zarith. - autorewrite with w_rewrite rm10. - rewrite Zmod_mod; auto with zarith. - rewrite spec_ww_compare. - case Z.compare_spec; rewrite spec_ww_zdigits; - rewrite spec_zdigits; intros H2. - replace (2^[[p]]) with wwB. - rewrite Zmod_small; auto with zarith. - unfold base; rewrite H2. - rewrite spec_ww_digits; auto. - assert (HH0: [|low (ww_sub p (w_0W w_zdigits))|] = - [[p]] - Zpos w_digits). - rewrite spec_low. - rewrite spec_ww_sub. - rewrite spec_w_0W; rewrite spec_zdigits. - rewrite <- Zmod_div_mod; auto with zarith. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - apply Z.lt_le_trans with (Zpos w_digits); auto with zarith. - unfold base; apply Zpower2_le_lin; auto with zarith. - exists wB; unfold base; rewrite <- Zpower_exp; auto with zarith. - rewrite spec_ww_digits; - apply f_equal with (f := Z.pow 2); rewrite Pos2Z.inj_xO; auto with zarith. - simpl ww_to_Z; autorewrite with w_rewrite. - rewrite spec_pos_mod; rewrite HH0. - pattern [|xh|] at 2; - rewrite Z_div_mod_eq with (b := 2 ^ ([[p]] - Zpos w_digits)); - auto with zarith. - rewrite (fun x => (Z.mul_comm (2 ^ x))); rewrite Z.mul_add_distr_r. - unfold base; rewrite <- Z.mul_assoc; rewrite <- Zpower_exp; - auto with zarith. - rewrite F0; auto with zarith. - rewrite <- Z.add_assoc; rewrite Zplus_mod; auto with zarith. - rewrite Z_mod_mult; auto with zarith. - autorewrite with rm10. - rewrite Zmod_mod; auto with zarith. - symmetry; apply Zmod_small; auto with zarith. - case (spec_to_Z xh); intros U1 U2. - case (spec_to_Z xl); intros U3 U4. - split; auto with zarith. - apply Z.add_nonneg_nonneg; auto with zarith. - apply Z.mul_nonneg_nonneg; auto with zarith. - match goal with |- 0 <= ?X mod ?Y => - case (Z_mod_lt X Y); auto with zarith - end. - match goal with |- ?X mod ?Y * ?U + ?Z < ?T => - apply Z.le_lt_trans with ((Y - 1) * U + Z ); - [case (Z_mod_lt X Y); auto with zarith | idtac] - end. - match goal with |- ?X * ?U + ?Y < ?Z => - apply Z.le_lt_trans with (X * U + (U - 1)) - end. - apply Z.add_le_mono_l; auto with zarith. - case (spec_to_Z xl); unfold base; auto with zarith. - rewrite Z.mul_sub_distr_r; rewrite <- Zpower_exp; auto with zarith. - rewrite F0; auto with zarith. - rewrite Zmod_small; auto with zarith. - case (spec_to_w_Z (WW xh xl)); intros U1 U2. - split; auto with zarith. - apply Z.lt_le_trans with (1:= U2). - unfold base; rewrite spec_ww_digits. - apply Zpower_le_monotone; auto with zarith. - split; auto with zarith. - rewrite Pos2Z.inj_xO; auto with zarith. - Qed. - -End POS_MOD. - -Section DoubleDiv32. - - Variable w : Type. - Variable w_0 : w. - Variable w_Bm1 : w. - Variable w_Bm2 : w. - Variable w_WW : w -> w -> zn2z w. - Variable w_compare : w -> w -> comparison. - Variable w_add_c : w -> w -> carry w. - Variable w_add_carry_c : w -> w -> carry w. - Variable w_add : w -> w -> w. - Variable w_add_carry : w -> w -> w. - Variable w_pred : w -> w. - Variable w_sub : w -> w -> w. - Variable w_mul_c : w -> w -> zn2z w. - Variable w_div21 : w -> w -> w -> w*w. - Variable ww_sub_c : zn2z w -> zn2z w -> carry (zn2z w). - - Definition w_div32_body a1 a2 a3 b1 b2 := - match w_compare a1 b1 with - | Lt => - let (q,r) := w_div21 a1 a2 b1 in - match ww_sub_c (w_WW r a3) (w_mul_c q b2) with - | C0 r1 => (q,r1) - | C1 r1 => - let q := w_pred q in - ww_add_c_cont w_WW w_add_c w_add_carry_c - (fun r2=>(w_pred q, ww_add w_add_c w_add w_add_carry r2 (WW b1 b2))) - (fun r2 => (q,r2)) - r1 (WW b1 b2) - end - | Eq => - ww_add_c_cont w_WW w_add_c w_add_carry_c - (fun r => (w_Bm2, ww_add w_add_c w_add w_add_carry r (WW b1 b2))) - (fun r => (w_Bm1,r)) - (WW (w_sub a2 b2) a3) (WW b1 b2) - | Gt => (w_0, W0) (* cas absurde *) - end. - - Definition w_div32 a1 a2 a3 b1 b2 := - Eval lazy beta iota delta [ww_add_c_cont ww_add w_div32_body] in - w_div32_body a1 a2 a3 b1 b2. - - (* Proof *) - - Variable w_digits : positive. - Variable w_to_Z : w -> Z. - - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[+| c |]" := - (interp_carry 1 wB w_to_Z c) (at level 0, c at level 99). - Notation "[-| c |]" := - (interp_carry (-1) wB w_to_Z c) (at level 0, c at level 99). - - Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99). - Notation "[-[ c ]]" := - (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) - (at level 0, c at level 99). - - - Variable spec_w_0 : [|w_0|] = 0. - Variable spec_w_Bm1 : [|w_Bm1|] = wB - 1. - Variable spec_w_Bm2 : [|w_Bm2|] = wB - 2. - - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - - Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. - Variable spec_compare : - forall x y, w_compare x y = Z.compare [|x|] [|y|]. - Variable spec_w_add_c : forall x y, [+|w_add_c x y|] = [|x|] + [|y|]. - Variable spec_w_add_carry_c : - forall x y, [+|w_add_carry_c x y|] = [|x|] + [|y|] + 1. - - Variable spec_w_add : forall x y, [|w_add x y|] = ([|x|] + [|y|]) mod wB. - Variable spec_w_add_carry : - forall x y, [|w_add_carry x y|] = ([|x|] + [|y|] + 1) mod wB. - - Variable spec_pred : forall x, [|w_pred x|] = ([|x|] - 1) mod wB. - Variable spec_sub : forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB. - - Variable spec_mul_c : forall x y, [[ w_mul_c x y ]] = [|x|] * [|y|]. - Variable spec_div21 : forall a1 a2 b, - wB/2 <= [|b|] -> - [|a1|] < [|b|] -> - let (q,r) := w_div21 a1 a2 b in - [|a1|] *wB+ [|a2|] = [|q|] * [|b|] + [|r|] /\ - 0 <= [|r|] < [|b|]. - - Variable spec_ww_sub_c : forall x y, [-[ww_sub_c x y]] = [[x]] - [[y]]. - - Ltac Spec_w_to_Z x := - let H:= fresh "HH" in - assert (H:= spec_to_Z x). - Ltac Spec_ww_to_Z x := - let H:= fresh "HH" in - assert (H:= spec_ww_to_Z w_digits w_to_Z spec_to_Z x). - - Theorem wB_div2: forall x, wB/2 <= x -> wB <= 2 * x. - intros x H; rewrite <- wB_div_2; apply Z.mul_le_mono_nonneg_l; auto with zarith. - Qed. - - Lemma Zmult_lt_0_reg_r_2 : forall n m : Z, 0 <= n -> 0 < m * n -> 0 < m. - Proof. - intros n m H1 H2;apply Z.mul_pos_cancel_r with n;trivial. - Z.le_elim H1; trivial. - subst;rewrite Z.mul_0_r in H2;discriminate H2. - Qed. - - Theorem spec_w_div32 : forall a1 a2 a3 b1 b2, - wB/2 <= [|b1|] -> - [[WW a1 a2]] < [[WW b1 b2]] -> - let (q,r) := w_div32 a1 a2 a3 b1 b2 in - [|a1|] * wwB + [|a2|] * wB + [|a3|] = - [|q|] * ([|b1|] * wB + [|b2|]) + [[r]] /\ - 0 <= [[r]] < [|b1|] * wB + [|b2|]. - Proof. - intros a1 a2 a3 b1 b2 Hle Hlt. - assert (U:= lt_0_wB w_digits); assert (U1:= lt_0_wwB w_digits). - Spec_w_to_Z a1;Spec_w_to_Z a2;Spec_w_to_Z a3;Spec_w_to_Z b1;Spec_w_to_Z b2. - rewrite wwB_wBwB; rewrite Z.pow_2_r; rewrite Z.mul_assoc;rewrite <- Z.mul_add_distr_r. - change (w_div32 a1 a2 a3 b1 b2) with (w_div32_body a1 a2 a3 b1 b2). - unfold w_div32_body. - rewrite spec_compare. case Z.compare_spec; intro Hcmp. - simpl in Hlt. - rewrite Hcmp in Hlt;assert ([|a2|] < [|b2|]). omega. - assert ([[WW (w_sub a2 b2) a3]] = ([|a2|]-[|b2|])*wB + [|a3|] + wwB). - simpl;rewrite spec_sub. - assert ([|a2|] - [|b2|] = wB*(-1) + ([|a2|] - [|b2|] + wB)). ring. - assert (0 <= [|a2|] - [|b2|] + wB < wB). omega. - rewrite <-(Zmod_unique ([|a2|]-[|b2|]) wB (-1) ([|a2|]-[|b2|]+wB) H1 H0). - rewrite wwB_wBwB;ring. - assert (U2 := wB_pos w_digits). - eapply spec_ww_add_c_cont with (P := - fun (x y:zn2z w) (res:w*zn2z w) => - let (q, r) := res in - ([|a1|] * wB + [|a2|]) * wB + [|a3|] = - [|q|] * ([|b1|] * wB + [|b2|]) + [[r]] /\ - 0 <= [[r]] < [|b1|] * wB + [|b2|]);eauto. - rewrite H0;intros r. - repeat - (rewrite spec_ww_add;eauto || rewrite spec_w_Bm1 || rewrite spec_w_Bm2); - simpl ww_to_Z;try rewrite Z.mul_1_l;intros H1. - assert (0<= ([[r]] + ([|b1|] * wB + [|b2|])) - wwB < [|b1|] * wB + [|b2|]). - Spec_ww_to_Z r;split;zarith. - rewrite H1. - assert (H12:= wB_div2 Hle). assert (wwB <= 2 * [|b1|] * wB). - rewrite wwB_wBwB; rewrite Z.pow_2_r; zarith. - assert (-wwB < ([|a2|] - [|b2|]) * wB + [|a3|] < 0). - split. apply Z.lt_le_trans with (([|a2|] - [|b2|]) * wB);zarith. - rewrite wwB_wBwB;replace (-(wB^2)) with (-wB*wB);[zarith | ring]. - apply Z.mul_lt_mono_pos_r;zarith. - apply Z.le_lt_trans with (([|a2|] - [|b2|]) * wB + (wB -1));zarith. - replace ( ([|a2|] - [|b2|]) * wB + (wB - 1)) with - (([|a2|] - [|b2|] + 1) * wB + - 1);[zarith | ring]. - assert (([|a2|] - [|b2|] + 1) * wB <= 0);zarith. - replace 0 with (0*wB);zarith. - replace (([|a2|] - [|b2|]) * wB + [|a3|] + wwB + ([|b1|] * wB + [|b2|]) + - ([|b1|] * wB + [|b2|]) - wwB) with - (([|a2|] - [|b2|]) * wB + [|a3|] + 2*[|b1|] * wB + 2*[|b2|]); - [zarith | ring]. - rewrite <- (Zmod_unique ([[r]] + ([|b1|] * wB + [|b2|])) wwB - 1 ([[r]] + ([|b1|] * wB + [|b2|]) - wwB));zarith;try (ring;fail). - split. rewrite H1;rewrite Hcmp;ring. trivial. - Spec_ww_to_Z (WW b1 b2). simpl in HH4;zarith. - rewrite H0;intros r;repeat - (rewrite spec_w_Bm1 || rewrite spec_w_Bm2); - simpl ww_to_Z;try rewrite Z.mul_1_l;intros H1. - assert ([[r]]=([|a2|]-[|b2|])*wB+[|a3|]+([|b1|]*wB+[|b2|])). zarith. - split. rewrite H2;rewrite Hcmp;ring. - split. Spec_ww_to_Z r;zarith. - rewrite H2. - assert (([|a2|] - [|b2|]) * wB + [|a3|] < 0);zarith. - apply Z.le_lt_trans with (([|a2|] - [|b2|]) * wB + (wB -1));zarith. - replace ( ([|a2|] - [|b2|]) * wB + (wB - 1)) with - (([|a2|] - [|b2|] + 1) * wB + - 1);[zarith|ring]. - assert (([|a2|] - [|b2|] + 1) * wB <= 0);zarith. - replace 0 with (0*wB);zarith. - (* Cas Lt *) - assert (Hdiv21 := spec_div21 a2 Hle Hcmp); - destruct (w_div21 a1 a2 b1) as (q, r);destruct Hdiv21. - rewrite H. - assert (Hq := spec_to_Z q). - generalize - (spec_ww_sub_c (w_WW r a3) (w_mul_c q b2)); - destruct (ww_sub_c (w_WW r a3) (w_mul_c q b2)) - as [r1|r1];repeat (rewrite spec_w_WW || rewrite spec_mul_c); - unfold interp_carry;intros H1. - rewrite H1. - split. ring. split. - rewrite <- H1;destruct (spec_ww_to_Z w_digits w_to_Z spec_to_Z r1);trivial. - apply Z.le_lt_trans with ([|r|] * wB + [|a3|]). - assert ( 0 <= [|q|] * [|b2|]);zarith. - apply beta_lex_inv;zarith. - assert ([[r1]] = [|r|] * wB + [|a3|] - [|q|] * [|b2|] + wwB). - rewrite <- H1;ring. - Spec_ww_to_Z r1; assert (0 <= [|r|]*wB). zarith. - assert (0 < [|q|] * [|b2|]). zarith. - assert (0 < [|q|]). - apply Zmult_lt_0_reg_r_2 with [|b2|];zarith. - eapply spec_ww_add_c_cont with (P := - fun (x y:zn2z w) (res:w*zn2z w) => - let (q0, r0) := res in - ([|q|] * [|b1|] + [|r|]) * wB + [|a3|] = - [|q0|] * ([|b1|] * wB + [|b2|]) + [[r0]] /\ - 0 <= [[r0]] < [|b1|] * wB + [|b2|]);eauto. - intros r2;repeat (rewrite spec_pred || rewrite spec_ww_add;eauto); - simpl ww_to_Z;intros H7. - assert (0 < [|q|] - 1). - assert (H6 : 1 <= [|q|]) by zarith. - Z.le_elim H6;zarith. - rewrite <- H6 in H2;rewrite H2 in H7. - assert (0 < [|b1|]*wB). apply Z.mul_pos_pos;zarith. - Spec_ww_to_Z r2. zarith. - rewrite (Zmod_small ([|q|] -1));zarith. - rewrite (Zmod_small ([|q|] -1 -1));zarith. - assert ([[r2]] + ([|b1|] * wB + [|b2|]) = - wwB * 1 + - ([|r|] * wB + [|a3|] - [|q|] * [|b2|] + 2 * ([|b1|] * wB + [|b2|]))). - rewrite H7;rewrite H2;ring. - assert - ([|r|]*wB + [|a3|] - [|q|]*[|b2|] + 2 * ([|b1|]*wB + [|b2|]) - < [|b1|]*wB + [|b2|]). - Spec_ww_to_Z r2;omega. - Spec_ww_to_Z (WW b1 b2). simpl in HH5. - assert - (0 <= [|r|]*wB + [|a3|] - [|q|]*[|b2|] + 2 * ([|b1|]*wB + [|b2|]) - < wwB). split;try omega. - replace (2*([|b1|]*wB+[|b2|])) with ((2*[|b1|])*wB+2*[|b2|]). 2:ring. - assert (H12:= wB_div2 Hle). assert (wwB <= 2 * [|b1|] * wB). - rewrite wwB_wBwB; rewrite Z.pow_2_r; zarith. omega. - rewrite <- (Zmod_unique - ([[r2]] + ([|b1|] * wB + [|b2|])) - wwB - 1 - ([|r|] * wB + [|a3|] - [|q|] * [|b2|] + 2*([|b1|] * wB + [|b2|])) - H10 H8). - split. ring. zarith. - intros r2;repeat (rewrite spec_pred);simpl ww_to_Z;intros H7. - rewrite (Zmod_small ([|q|] -1));zarith. - split. - replace [[r2]] with ([[r1]] + ([|b1|] * wB + [|b2|]) -wwB). - rewrite H2; ring. rewrite <- H7; ring. - Spec_ww_to_Z r2;Spec_ww_to_Z r1. omega. - simpl in Hlt. - assert ([|a1|] * wB + [|a2|] <= [|b1|] * wB + [|b2|]). zarith. - assert (H1 := beta_lex _ _ _ _ _ H HH0 HH3). rewrite spec_w_0;simpl;zarith. - Qed. - - -End DoubleDiv32. - -Section DoubleDiv21. - Variable w : Type. - Variable w_0 : w. - - Variable w_0W : w -> zn2z w. - Variable w_div32 : w -> w -> w -> w -> w -> w * zn2z w. - - Variable ww_1 : zn2z w. - Variable ww_compare : zn2z w -> zn2z w -> comparison. - Variable ww_sub : zn2z w -> zn2z w -> zn2z w. - - - Definition ww_div21 a1 a2 b := - match a1 with - | W0 => - match ww_compare a2 b with - | Gt => (ww_1, ww_sub a2 b) - | Eq => (ww_1, W0) - | Lt => (W0, a2) - end - | WW a1h a1l => - match a2 with - | W0 => - match b with - | W0 => (W0,W0) (* cas absurde *) - | WW b1 b2 => - let (q1, r) := w_div32 a1h a1l w_0 b1 b2 in - match r with - | W0 => (WW q1 w_0, W0) - | WW r1 r2 => - let (q2, s) := w_div32 r1 r2 w_0 b1 b2 in - (WW q1 q2, s) - end - end - | WW a2h a2l => - match b with - | W0 => (W0,W0) (* cas absurde *) - | WW b1 b2 => - let (q1, r) := w_div32 a1h a1l a2h b1 b2 in - match r with - | W0 => (WW q1 w_0, w_0W a2l) - | WW r1 r2 => - let (q2, s) := w_div32 r1 r2 a2l b1 b2 in - (WW q1 q2, s) - end - end - end - end. - - (* Proof *) - - Variable w_digits : positive. - Variable w_to_Z : w -> Z. - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99). - Notation "[-[ c ]]" := - (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) - (at level 0, c at level 99). - - Variable spec_w_0 : [|w_0|] = 0. - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - Variable spec_w_0W : forall l, [[w_0W l]] = [|l|]. - Variable spec_w_div32 : forall a1 a2 a3 b1 b2, - wB/2 <= [|b1|] -> - [[WW a1 a2]] < [[WW b1 b2]] -> - let (q,r) := w_div32 a1 a2 a3 b1 b2 in - [|a1|] * wwB + [|a2|] * wB + [|a3|] = - [|q|] * ([|b1|] * wB + [|b2|]) + [[r]] /\ - 0 <= [[r]] < [|b1|] * wB + [|b2|]. - Variable spec_ww_1 : [[ww_1]] = 1. - Variable spec_ww_compare : forall x y, - ww_compare x y = Z.compare [[x]] [[y]]. - Variable spec_ww_sub : forall x y, [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB. - - Theorem wwB_div: wwB = 2 * (wwB / 2). - Proof. - rewrite wwB_div_2; rewrite Z.mul_assoc; rewrite wB_div_2; auto. - rewrite <- Z.pow_2_r; apply wwB_wBwB. - Qed. - - Ltac Spec_w_to_Z x := - let H:= fresh "HH" in - assert (H:= spec_to_Z x). - Ltac Spec_ww_to_Z x := - let H:= fresh "HH" in - assert (H:= spec_ww_to_Z w_digits w_to_Z spec_to_Z x). - - Theorem spec_ww_div21 : forall a1 a2 b, - wwB/2 <= [[b]] -> - [[a1]] < [[b]] -> - let (q,r) := ww_div21 a1 a2 b in - [[a1]] *wwB+[[a2]] = [[q]] * [[b]] + [[r]] /\ 0 <= [[r]] < [[b]]. - Proof. - assert (U:= lt_0_wB w_digits). - assert (U1:= lt_0_wwB w_digits). - intros a1 a2 b H Hlt; unfold ww_div21. - Spec_ww_to_Z b; assert (Eq: 0 < [[b]]). Spec_ww_to_Z a1;omega. - generalize Hlt H ;clear Hlt H;case a1. - intros H1 H2;simpl in H1;Spec_ww_to_Z a2. - rewrite spec_ww_compare. case Z.compare_spec; - simpl;try rewrite spec_ww_1;autorewrite with rm10; intros;zarith. - rewrite spec_ww_sub;simpl. rewrite Zmod_small;zarith. - split. ring. - assert (wwB <= 2*[[b]]);zarith. - rewrite wwB_div;zarith. - intros a1h a1l. Spec_w_to_Z a1h;Spec_w_to_Z a1l. Spec_ww_to_Z a2. - destruct a2 as [ |a3 a4]; - (destruct b as [ |b1 b2];[unfold Z.le in Eq;discriminate Eq|idtac]); - try (Spec_w_to_Z a3; Spec_w_to_Z a4); Spec_w_to_Z b1; Spec_w_to_Z b2; - intros Hlt H; match goal with |-context [w_div32 ?X ?Y ?Z ?T ?U] => - generalize (@spec_w_div32 X Y Z T U); case (w_div32 X Y Z T U); - intros q1 r H0 - end; (assert (Eq1: wB / 2 <= [|b1|]);[ - apply (@beta_lex (wB / 2) 0 [|b1|] [|b2|] wB); auto with zarith; - autorewrite with rm10;repeat rewrite (Z.mul_comm wB); - rewrite <- wwB_div_2; trivial - | generalize (H0 Eq1 Hlt);clear H0;destruct r as [ |r1 r2];simpl; - try rewrite spec_w_0; try rewrite spec_w_0W;repeat rewrite Z.add_0_r; - intros (H1,H2) ]). - split;[rewrite wwB_wBwB; rewrite Z.pow_2_r | trivial]. - rewrite Z.mul_assoc;rewrite Z.mul_add_distr_r;rewrite <- Z.mul_assoc; - rewrite <- Z.pow_2_r; rewrite <- wwB_wBwB;rewrite H1;ring. - destruct H2 as (H2,H3);match goal with |-context [w_div32 ?X ?Y ?Z ?T ?U] => - generalize (@spec_w_div32 X Y Z T U); case (w_div32 X Y Z T U); - intros q r H0;generalize (H0 Eq1 H3);clear H0;intros (H4,H5) end. - split;[rewrite wwB_wBwB | trivial]. - rewrite Z.pow_2_r. - rewrite Z.mul_assoc;rewrite Z.mul_add_distr_r;rewrite <- Z.mul_assoc; - rewrite <- Z.pow_2_r. - rewrite <- wwB_wBwB;rewrite H1. - rewrite spec_w_0 in H4;rewrite Z.add_0_r in H4. - repeat rewrite Z.mul_add_distr_r. rewrite <- (Z.mul_assoc [|r1|]). - rewrite <- Z.pow_2_r; rewrite <- wwB_wBwB;rewrite H4;simpl;ring. - split;[rewrite wwB_wBwB | split;zarith]. - replace (([|a1h|] * wB + [|a1l|]) * wB^2 + ([|a3|] * wB + [|a4|])) - with (([|a1h|] * wwB + [|a1l|] * wB + [|a3|])*wB+ [|a4|]). - rewrite H1;ring. rewrite wwB_wBwB;ring. - change [|a4|] with (0*wB+[|a4|]);apply beta_lex_inv;zarith. - assert (1 <= wB/2);zarith. - assert (H_:= wB_pos w_digits);apply Zdiv_le_lower_bound;zarith. - destruct H2 as (H2,H3);match goal with |-context [w_div32 ?X ?Y ?Z ?T ?U] => - generalize (@spec_w_div32 X Y Z T U); case (w_div32 X Y Z T U); - intros q r H0;generalize (H0 Eq1 H3);clear H0;intros (H4,H5) end. - split;trivial. - replace (([|a1h|] * wB + [|a1l|]) * wwB + ([|a3|] * wB + [|a4|])) with - (([|a1h|] * wwB + [|a1l|] * wB + [|a3|])*wB + [|a4|]); - [rewrite H1 | rewrite wwB_wBwB;ring]. - replace (([|q1|]*([|b1|]*wB+[|b2|])+([|r1|]*wB+[|r2|]))*wB+[|a4|]) with - (([|q1|]*([|b1|]*wB+[|b2|]))*wB+([|r1|]*wwB+[|r2|]*wB+[|a4|])); - [rewrite H4;simpl|rewrite wwB_wBwB];ring. - Qed. - -End DoubleDiv21. - -Section DoubleDivGt. - Variable w : Type. - Variable w_digits : positive. - Variable w_0 : w. - - Variable w_WW : w -> w -> zn2z w. - Variable w_0W : w -> zn2z w. - Variable w_compare : w -> w -> comparison. - Variable w_eq0 : w -> bool. - Variable w_opp_c : w -> carry w. - Variable w_opp w_opp_carry : w -> w. - Variable w_sub_c : w -> w -> carry w. - Variable w_sub w_sub_carry : w -> w -> w. - - Variable w_div_gt : w -> w -> w*w. - Variable w_mod_gt : w -> w -> w. - Variable w_gcd_gt : w -> w -> w. - Variable w_add_mul_div : w -> w -> w -> w. - Variable w_head0 : w -> w. - Variable w_div21 : w -> w -> w -> w * w. - Variable w_div32 : w -> w -> w -> w -> w -> w * zn2z w. - - - Variable _ww_zdigits : zn2z w. - Variable ww_1 : zn2z w. - Variable ww_add_mul_div : zn2z w -> zn2z w -> zn2z w -> zn2z w. - - Variable w_zdigits : w. - - Definition ww_div_gt_aux ah al bh bl := - Eval lazy beta iota delta [ww_sub ww_opp] in - let p := w_head0 bh in - match w_compare p w_0 with - | Gt => - let b1 := w_add_mul_div p bh bl in - let b2 := w_add_mul_div p bl w_0 in - let a1 := w_add_mul_div p w_0 ah in - let a2 := w_add_mul_div p ah al in - let a3 := w_add_mul_div p al w_0 in - let (q,r) := w_div32 a1 a2 a3 b1 b2 in - (WW w_0 q, ww_add_mul_div - (ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c - w_opp w_sub w_sub_carry _ww_zdigits (w_0W p)) W0 r) - | _ => (ww_1, ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c - w_opp w_sub w_sub_carry (WW ah al) (WW bh bl)) - end. - - Definition ww_div_gt a b := - Eval lazy beta iota delta [ww_div_gt_aux double_divn1 - double_divn1_p double_divn1_p_aux double_divn1_0 double_divn1_0_aux - double_split double_0 double_WW] in - match a, b with - | W0, _ => (W0,W0) - | _, W0 => (W0,W0) - | WW ah al, WW bh bl => - if w_eq0 ah then - let (q,r) := w_div_gt al bl in - (WW w_0 q, w_0W r) - else - match w_compare w_0 bh with - | Eq => - let(q,r):= - double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 - w_compare w_sub 1 a bl in - (q, w_0W r) - | Lt => ww_div_gt_aux ah al bh bl - | Gt => (W0,W0) (* cas absurde *) - end - end. - - Definition ww_mod_gt_aux ah al bh bl := - Eval lazy beta iota delta [ww_sub ww_opp] in - let p := w_head0 bh in - match w_compare p w_0 with - | Gt => - let b1 := w_add_mul_div p bh bl in - let b2 := w_add_mul_div p bl w_0 in - let a1 := w_add_mul_div p w_0 ah in - let a2 := w_add_mul_div p ah al in - let a3 := w_add_mul_div p al w_0 in - let (q,r) := w_div32 a1 a2 a3 b1 b2 in - ww_add_mul_div (ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c - w_opp w_sub w_sub_carry _ww_zdigits (w_0W p)) W0 r - | _ => - ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c - w_opp w_sub w_sub_carry (WW ah al) (WW bh bl) - end. - - Definition ww_mod_gt a b := - Eval lazy beta iota delta [ww_mod_gt_aux double_modn1 - double_modn1_p double_modn1_p_aux double_modn1_0 double_modn1_0_aux - double_split double_0 double_WW snd] in - match a, b with - | W0, _ => W0 - | _, W0 => W0 - | WW ah al, WW bh bl => - if w_eq0 ah then w_0W (w_mod_gt al bl) - else - match w_compare w_0 bh with - | Eq => - w_0W (double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 - w_compare w_sub 1 a bl) - | Lt => ww_mod_gt_aux ah al bh bl - | Gt => W0 (* cas absurde *) - end - end. - - Definition ww_gcd_gt_body (cont: w->w->w->w->zn2z w) (ah al bh bl: w) := - Eval lazy beta iota delta [ww_mod_gt_aux double_modn1 - double_modn1_p double_modn1_p_aux double_modn1_0 double_modn1_0_aux - double_split double_0 double_WW snd] in - match w_compare w_0 bh with - | Eq => - match w_compare w_0 bl with - | Eq => WW ah al (* normalement n'arrive pas si forme normale *) - | Lt => - let m := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 - w_compare w_sub 1 (WW ah al) bl in - WW w_0 (w_gcd_gt bl m) - | Gt => W0 (* absurde *) - end - | Lt => - let m := ww_mod_gt_aux ah al bh bl in - match m with - | W0 => WW bh bl - | WW mh ml => - match w_compare w_0 mh with - | Eq => - match w_compare w_0 ml with - | Eq => WW bh bl - | _ => - let r := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 - w_compare w_sub 1 (WW bh bl) ml in - WW w_0 (w_gcd_gt ml r) - end - | Lt => - let r := ww_mod_gt_aux bh bl mh ml in - match r with - | W0 => m - | WW rh rl => cont mh ml rh rl - end - | Gt => W0 (* absurde *) - end - end - | Gt => W0 (* absurde *) - end. - - Fixpoint ww_gcd_gt_aux - (p:positive) (cont: w -> w -> w -> w -> zn2z w) (ah al bh bl : w) - {struct p} : zn2z w := - ww_gcd_gt_body - (fun mh ml rh rl => match p with - | xH => cont mh ml rh rl - | xO p => ww_gcd_gt_aux p (ww_gcd_gt_aux p cont) mh ml rh rl - | xI p => ww_gcd_gt_aux p (ww_gcd_gt_aux p cont) mh ml rh rl - end) ah al bh bl. - - - (* Proof *) - - Variable w_to_Z : w -> Z. - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[-| c |]" := - (interp_carry (-1) wB w_to_Z c) (at level 0, c at level 99). - - Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99). - - Variable spec_w_0 : [|w_0|] = 0. - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - Variable spec_to_w_Z : forall x, 0 <= [[x]] < wwB. - - Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. - Variable spec_w_0W : forall l, [[w_0W l]] = [|l|]. - Variable spec_compare : - forall x y, w_compare x y = Z.compare [|x|] [|y|]. - Variable spec_eq0 : forall x, w_eq0 x = true -> [|x|] = 0. - - Variable spec_opp_c : forall x, [-|w_opp_c x|] = -[|x|]. - Variable spec_opp : forall x, [|w_opp x|] = (-[|x|]) mod wB. - Variable spec_opp_carry : forall x, [|w_opp_carry x|] = wB - [|x|] - 1. - - Variable spec_sub_c : forall x y, [-|w_sub_c x y|] = [|x|] - [|y|]. - Variable spec_sub : forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB. - Variable spec_sub_carry : - forall x y, [|w_sub_carry x y|] = ([|x|] - [|y|] - 1) mod wB. - - Variable spec_div_gt : forall a b, [|a|] > [|b|] -> 0 < [|b|] -> - let (q,r) := w_div_gt a b in - [|a|] = [|q|] * [|b|] + [|r|] /\ - 0 <= [|r|] < [|b|]. - Variable spec_mod_gt : forall a b, [|a|] > [|b|] -> 0 < [|b|] -> - [|w_mod_gt a b|] = [|a|] mod [|b|]. - Variable spec_gcd_gt : forall a b, [|a|] > [|b|] -> - Zis_gcd [|a|] [|b|] [|w_gcd_gt a b|]. - - Variable spec_add_mul_div : forall x y p, - [|p|] <= Zpos w_digits -> - [| w_add_mul_div p x y |] = - ([|x|] * (2 ^ ([|p|])) + - [|y|] / (2 ^ ((Zpos w_digits) - [|p|]))) mod wB. - Variable spec_head0 : forall x, 0 < [|x|] -> - wB/ 2 <= 2 ^ [|w_head0 x|] * [|x|] < wB. - - Variable spec_div21 : forall a1 a2 b, - wB/2 <= [|b|] -> - [|a1|] < [|b|] -> - let (q,r) := w_div21 a1 a2 b in - [|a1|] *wB+ [|a2|] = [|q|] * [|b|] + [|r|] /\ - 0 <= [|r|] < [|b|]. - - Variable spec_w_div32 : forall a1 a2 a3 b1 b2, - wB/2 <= [|b1|] -> - [[WW a1 a2]] < [[WW b1 b2]] -> - let (q,r) := w_div32 a1 a2 a3 b1 b2 in - [|a1|] * wwB + [|a2|] * wB + [|a3|] = - [|q|] * ([|b1|] * wB + [|b2|]) + [[r]] /\ - 0 <= [[r]] < [|b1|] * wB + [|b2|]. - - Variable spec_w_zdigits: [|w_zdigits|] = Zpos w_digits. - - Variable spec_ww_digits_ : [[_ww_zdigits]] = Zpos (xO w_digits). - Variable spec_ww_1 : [[ww_1]] = 1. - Variable spec_ww_add_mul_div : forall x y p, - [[p]] <= Zpos (xO w_digits) -> - [[ ww_add_mul_div p x y ]] = - ([[x]] * (2^[[p]]) + - [[y]] / (2^(Zpos (xO w_digits) - [[p]]))) mod wwB. - - Ltac Spec_w_to_Z x := - let H:= fresh "HH" in - assert (H:= spec_to_Z x). - - Ltac Spec_ww_to_Z x := - let H:= fresh "HH" in - assert (H:= spec_ww_to_Z w_digits w_to_Z spec_to_Z x). - - Lemma to_Z_div_minus_p : forall x p, - 0 < [|p|] < Zpos w_digits -> - 0 <= [|x|] / 2 ^ (Zpos w_digits - [|p|]) < 2 ^ [|p|]. - Proof. - intros x p H;Spec_w_to_Z x. - split. apply Zdiv_le_lower_bound;zarith. - apply Zdiv_lt_upper_bound;zarith. - rewrite <- Zpower_exp;zarith. - ring_simplify ([|p|] + (Zpos w_digits - [|p|])); unfold base in HH;zarith. - Qed. - Hint Resolve to_Z_div_minus_p : zarith. - - Lemma spec_ww_div_gt_aux : forall ah al bh bl, - [[WW ah al]] > [[WW bh bl]] -> - 0 < [|bh|] -> - let (q,r) := ww_div_gt_aux ah al bh bl in - [[WW ah al]] = [[q]] * [[WW bh bl]] + [[r]] /\ - 0 <= [[r]] < [[WW bh bl]]. - Proof. - intros ah al bh bl Hgt Hpos;unfold ww_div_gt_aux. - change - (let (q, r) := let p := w_head0 bh in - match w_compare p w_0 with - | Gt => - let b1 := w_add_mul_div p bh bl in - let b2 := w_add_mul_div p bl w_0 in - let a1 := w_add_mul_div p w_0 ah in - let a2 := w_add_mul_div p ah al in - let a3 := w_add_mul_div p al w_0 in - let (q,r) := w_div32 a1 a2 a3 b1 b2 in - (WW w_0 q, ww_add_mul_div - (ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c - w_opp w_sub w_sub_carry _ww_zdigits (w_0W p)) W0 r) - | _ => (ww_1, ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c - w_opp w_sub w_sub_carry (WW ah al) (WW bh bl)) - end in [[WW ah al]]=[[q]]*[[WW bh bl]]+[[r]] /\ 0 <=[[r]]< [[WW bh bl]]). - assert (Hh := spec_head0 Hpos). - lazy zeta. - rewrite spec_compare; case Z.compare_spec; - rewrite spec_w_0; intros HH. - generalize Hh; rewrite HH; simpl Z.pow; - rewrite Z.mul_1_l; intros (HH1, HH2); clear HH. - assert (wwB <= 2*[[WW bh bl]]). - apply Z.le_trans with (2*[|bh|]*wB). - rewrite wwB_wBwB; rewrite Z.pow_2_r; apply Z.mul_le_mono_nonneg_r; zarith. - rewrite <- wB_div_2; apply Z.mul_le_mono_nonneg_l; zarith. - simpl ww_to_Z;rewrite Z.mul_add_distr_l;rewrite Z.mul_assoc. - Spec_w_to_Z bl;zarith. - Spec_ww_to_Z (WW ah al). - rewrite spec_ww_sub;eauto. - simpl;rewrite spec_ww_1;rewrite Z.mul_1_l;simpl. - simpl ww_to_Z in Hgt, H, HH;rewrite Zmod_small;split;zarith. - case (spec_to_Z (w_head0 bh)); auto with zarith. - assert ([|w_head0 bh|] < Zpos w_digits). - destruct (Z_lt_ge_dec [|w_head0 bh|] (Zpos w_digits));trivial. - exfalso. - assert (2 ^ [|w_head0 bh|] * [|bh|] >= wB);auto with zarith. - apply Z.le_ge; replace wB with (wB * 1);try ring. - Spec_w_to_Z bh;apply Z.mul_le_mono_nonneg;zarith. - unfold base;apply Zpower_le_monotone;zarith. - assert (HHHH : 0 < [|w_head0 bh|] < Zpos w_digits); auto with zarith. - assert (Hb:= Z.lt_le_incl _ _ H). - generalize (spec_add_mul_div w_0 ah Hb) - (spec_add_mul_div ah al Hb) - (spec_add_mul_div al w_0 Hb) - (spec_add_mul_div bh bl Hb) - (spec_add_mul_div bl w_0 Hb); - rewrite spec_w_0; repeat rewrite Z.mul_0_l;repeat rewrite Z.add_0_l; - rewrite Zdiv_0_l;repeat rewrite Z.add_0_r. - Spec_w_to_Z ah;Spec_w_to_Z bh. - unfold base;repeat rewrite Zmod_shift_r;zarith. - assert (H3:=to_Z_div_minus_p ah HHHH);assert(H4:=to_Z_div_minus_p al HHHH); - assert (H5:=to_Z_div_minus_p bl HHHH). - rewrite Z.mul_comm in Hh. - assert (2^[|w_head0 bh|] < wB). unfold base;apply Zpower_lt_monotone;zarith. - unfold base in H0;rewrite Zmod_small;zarith. - fold wB; rewrite (Zmod_small ([|bh|] * 2 ^ [|w_head0 bh|]));zarith. - intros U1 U2 U3 V1 V2. - generalize (@spec_w_div32 (w_add_mul_div (w_head0 bh) w_0 ah) - (w_add_mul_div (w_head0 bh) ah al) - (w_add_mul_div (w_head0 bh) al w_0) - (w_add_mul_div (w_head0 bh) bh bl) - (w_add_mul_div (w_head0 bh) bl w_0)). - destruct (w_div32 (w_add_mul_div (w_head0 bh) w_0 ah) - (w_add_mul_div (w_head0 bh) ah al) - (w_add_mul_div (w_head0 bh) al w_0) - (w_add_mul_div (w_head0 bh) bh bl) - (w_add_mul_div (w_head0 bh) bl w_0)) as (q,r). - rewrite V1;rewrite V2. rewrite Z.mul_add_distr_r. - rewrite <- (Z.add_assoc ([|bh|] * 2 ^ [|w_head0 bh|] * wB)). - unfold base;rewrite <- shift_unshift_mod;zarith. fold wB. - replace ([|bh|] * 2 ^ [|w_head0 bh|] * wB + [|bl|] * 2 ^ [|w_head0 bh|]) with - ([[WW bh bl]] * 2^[|w_head0 bh|]). 2:simpl;ring. - fold wwB. rewrite wwB_wBwB. rewrite Z.pow_2_r. rewrite U1;rewrite U2;rewrite U3. - rewrite Z.mul_assoc. rewrite Z.mul_add_distr_r. - rewrite (Z.add_assoc ([|ah|] / 2^(Zpos(w_digits) - [|w_head0 bh|])*wB * wB)). - rewrite <- Z.mul_add_distr_r. rewrite <- Z.add_assoc. - unfold base;repeat rewrite <- shift_unshift_mod;zarith. fold wB. - replace ([|ah|] * 2 ^ [|w_head0 bh|] * wB + [|al|] * 2 ^ [|w_head0 bh|]) with - ([[WW ah al]] * 2^[|w_head0 bh|]). 2:simpl;ring. - intros Hd;destruct Hd;zarith. - simpl. apply beta_lex_inv;zarith. rewrite U1;rewrite V1. - assert ([|ah|] / 2 ^ (Zpos (w_digits) - [|w_head0 bh|]) < wB/2);zarith. - apply Zdiv_lt_upper_bound;zarith. - unfold base. - replace (2^Zpos (w_digits)) with (2^(Zpos (w_digits) - 1)*2). - rewrite Z_div_mult;zarith. rewrite <- Zpower_exp;zarith. - apply Z.lt_le_trans with wB;zarith. - unfold base;apply Zpower_le_monotone;zarith. - pattern 2 at 2;replace 2 with (2^1);trivial. - rewrite <- Zpower_exp;zarith. ring_simplify (Zpos (w_digits) - 1 + 1);trivial. - change [[WW w_0 q]] with ([|w_0|]*wB+[|q|]);rewrite spec_w_0;rewrite - Z.mul_0_l;rewrite Z.add_0_l. - replace [[ww_add_mul_div (ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c w_opp w_sub w_sub_carry - _ww_zdigits (w_0W (w_head0 bh))) W0 r]] with ([[r]]/2^[|w_head0 bh|]). - assert (0 < 2^[|w_head0 bh|]). apply Z.pow_pos_nonneg;zarith. - split. - rewrite <- (Z_div_mult [[WW ah al]] (2^[|w_head0 bh|]));zarith. - rewrite H1;rewrite Z.mul_assoc;apply Z_div_plus_l;trivial. - split;[apply Zdiv_le_lower_bound| apply Zdiv_lt_upper_bound];zarith. - rewrite spec_ww_add_mul_div. - rewrite spec_ww_sub; auto with zarith. - rewrite spec_ww_digits_. - change (Zpos (xO (w_digits))) with (2*Zpos (w_digits));zarith. - simpl ww_to_Z;rewrite Z.mul_0_l;rewrite Z.add_0_l. - rewrite spec_w_0W. - rewrite (fun x y => Zmod_small (x-y)); auto with zarith. - ring_simplify (2 * Zpos w_digits - (2 * Zpos w_digits - [|w_head0 bh|])). - rewrite Zmod_small;zarith. - split;[apply Zdiv_le_lower_bound| apply Zdiv_lt_upper_bound];zarith. - Spec_ww_to_Z r. - apply Z.lt_le_trans with wwB;zarith. - rewrite <- (Z.mul_1_r wwB);apply Z.mul_le_mono_nonneg;zarith. - split; auto with zarith. - apply Z.le_lt_trans with (2 * Zpos w_digits); auto with zarith. - unfold base, ww_digits; rewrite (Pos2Z.inj_xO w_digits). - apply Zpower2_lt_lin; auto with zarith. - rewrite spec_ww_sub; auto with zarith. - rewrite spec_ww_digits_; rewrite spec_w_0W. - rewrite Zmod_small;zarith. - rewrite Pos2Z.inj_xO; split; auto with zarith. - apply Z.le_lt_trans with (2 * Zpos w_digits); auto with zarith. - unfold base, ww_digits; rewrite (Pos2Z.inj_xO w_digits). - apply Zpower2_lt_lin; auto with zarith. - Qed. - - Lemma spec_ww_div_gt : forall a b, [[a]] > [[b]] -> 0 < [[b]] -> - let (q,r) := ww_div_gt a b in - [[a]] = [[q]] * [[b]] + [[r]] /\ - 0 <= [[r]] < [[b]]. - Proof. - intros a b Hgt Hpos;unfold ww_div_gt. - change (let (q,r) := match a, b with - | W0, _ => (W0,W0) - | _, W0 => (W0,W0) - | WW ah al, WW bh bl => - if w_eq0 ah then - let (q,r) := w_div_gt al bl in - (WW w_0 q, w_0W r) - else - match w_compare w_0 bh with - | Eq => - let(q,r):= - double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 - w_compare w_sub 1 a bl in - (q, w_0W r) - | Lt => ww_div_gt_aux ah al bh bl - | Gt => (W0,W0) (* cas absurde *) - end - end in [[a]] = [[q]] * [[b]] + [[r]] /\ 0 <= [[r]] < [[b]]). - destruct a as [ |ah al]. simpl in Hgt;omega. - destruct b as [ |bh bl]. simpl in Hpos;omega. - Spec_w_to_Z ah; Spec_w_to_Z al; Spec_w_to_Z bh; Spec_w_to_Z bl. - assert (H:=@spec_eq0 ah);destruct (w_eq0 ah). - simpl ww_to_Z;rewrite H;trivial. simpl in Hgt;rewrite H in Hgt;trivial. - assert ([|bh|] <= 0). - apply beta_lex with (d:=[|al|])(b:=[|bl|]) (beta := wB);zarith. - assert ([|bh|] = 0);zarith. rewrite H1 in Hgt;rewrite H1;simpl in Hgt. - simpl. simpl in Hpos;rewrite H1 in Hpos;simpl in Hpos. - assert (H2:=spec_div_gt Hgt Hpos);destruct (w_div_gt al bl). - repeat rewrite spec_w_0W;simpl;rewrite spec_w_0;simpl;trivial. - clear H. - rewrite spec_compare; case Z.compare_spec; intros Hcmp. - rewrite spec_w_0 in Hcmp. change [[WW bh bl]] with ([|bh|]*wB+[|bl|]). - rewrite <- Hcmp;rewrite Z.mul_0_l;rewrite Z.add_0_l. - simpl in Hpos;rewrite <- Hcmp in Hpos;simpl in Hpos. - assert (H2:= @spec_double_divn1 w w_digits w_zdigits w_0 w_WW w_head0 w_add_mul_div - w_div21 w_compare w_sub w_to_Z spec_to_Z spec_w_zdigits spec_w_0 spec_w_WW spec_head0 - spec_add_mul_div spec_div21 spec_compare spec_sub 1 (WW ah al) bl Hpos). - destruct (double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 - w_compare w_sub 1 - (WW ah al) bl). - rewrite spec_w_0W;unfold ww_to_Z;trivial. - apply spec_ww_div_gt_aux;trivial. rewrite spec_w_0 in Hcmp;trivial. - rewrite spec_w_0 in Hcmp;exfalso;omega. - Qed. - - Lemma spec_ww_mod_gt_aux_eq : forall ah al bh bl, - ww_mod_gt_aux ah al bh bl = snd (ww_div_gt_aux ah al bh bl). - Proof. - intros ah al bh bl. unfold ww_mod_gt_aux, ww_div_gt_aux. - case w_compare; auto. - case w_div32; auto. - Qed. - - Lemma spec_ww_mod_gt_aux : forall ah al bh bl, - [[WW ah al]] > [[WW bh bl]] -> - 0 < [|bh|] -> - [[ww_mod_gt_aux ah al bh bl]] = [[WW ah al]] mod [[WW bh bl]]. - Proof. - intros. rewrite spec_ww_mod_gt_aux_eq;trivial. - assert (H3 := spec_ww_div_gt_aux ah al bl H H0). - destruct (ww_div_gt_aux ah al bh bl) as (q,r);simpl. simpl in H,H3. - destruct H3;apply Zmod_unique with [[q]];zarith. - rewrite H1;ring. - Qed. - - Lemma spec_w_mod_gt_eq : forall a b, [|a|] > [|b|] -> 0 <[|b|] -> - [|w_mod_gt a b|] = [|snd (w_div_gt a b)|]. - Proof. - intros a b Hgt Hpos. - rewrite spec_mod_gt;trivial. - assert (H:=spec_div_gt Hgt Hpos). - destruct (w_div_gt a b) as (q,r);simpl. - rewrite Z.mul_comm in H;destruct H. - symmetry;apply Zmod_unique with [|q|];trivial. - Qed. - - Lemma spec_ww_mod_gt_eq : forall a b, [[a]] > [[b]] -> 0 < [[b]] -> - [[ww_mod_gt a b]] = [[snd (ww_div_gt a b)]]. - Proof. - intros a b Hgt Hpos. - change (ww_mod_gt a b) with - (match a, b with - | W0, _ => W0 - | _, W0 => W0 - | WW ah al, WW bh bl => - if w_eq0 ah then w_0W (w_mod_gt al bl) - else - match w_compare w_0 bh with - | Eq => - w_0W (double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 - w_compare w_sub 1 a bl) - | Lt => ww_mod_gt_aux ah al bh bl - | Gt => W0 (* cas absurde *) - end end). - change (ww_div_gt a b) with - (match a, b with - | W0, _ => (W0,W0) - | _, W0 => (W0,W0) - | WW ah al, WW bh bl => - if w_eq0 ah then - let (q,r) := w_div_gt al bl in - (WW w_0 q, w_0W r) - else - match w_compare w_0 bh with - | Eq => - let(q,r):= - double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 - w_compare w_sub 1 a bl in - (q, w_0W r) - | Lt => ww_div_gt_aux ah al bh bl - | Gt => (W0,W0) (* cas absurde *) - end - end). - destruct a as [ |ah al];trivial. - destruct b as [ |bh bl];trivial. - Spec_w_to_Z ah; Spec_w_to_Z al; Spec_w_to_Z bh; Spec_w_to_Z bl. - assert (H:=@spec_eq0 ah);destruct (w_eq0 ah). - simpl in Hgt;rewrite H in Hgt;trivial. - assert ([|bh|] <= 0). - apply beta_lex with (d:=[|al|])(b:=[|bl|]) (beta := wB);zarith. - assert ([|bh|] = 0);zarith. rewrite H1 in Hgt;simpl in Hgt. - simpl in Hpos;rewrite H1 in Hpos;simpl in Hpos. - rewrite spec_w_0W;rewrite spec_w_mod_gt_eq;trivial. - destruct (w_div_gt al bl);simpl;rewrite spec_w_0W;trivial. - clear H. - rewrite spec_compare; case Z.compare_spec; intros H2. - rewrite (@spec_double_modn1_aux w w_zdigits w_0 w_WW w_head0 w_add_mul_div - w_div21 w_compare w_sub w_to_Z spec_w_0 spec_compare 1 (WW ah al) bl). - destruct (double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 w_compare w_sub 1 - (WW ah al) bl);simpl;trivial. - rewrite spec_ww_mod_gt_aux_eq;trivial;symmetry;trivial. - trivial. - Qed. - - Lemma spec_ww_mod_gt : forall a b, [[a]] > [[b]] -> 0 < [[b]] -> - [[ww_mod_gt a b]] = [[a]] mod [[b]]. - Proof. - intros a b Hgt Hpos. - assert (H:= spec_ww_div_gt a b Hgt Hpos). - rewrite (spec_ww_mod_gt_eq a b Hgt Hpos). - destruct (ww_div_gt a b)as(q,r);destruct H. - apply Zmod_unique with[[q]];simpl;trivial. - rewrite Z.mul_comm;trivial. - Qed. - - Lemma Zis_gcd_mod : forall a b d, - 0 < b -> Zis_gcd b (a mod b) d -> Zis_gcd a b d. - Proof. - intros a b d H H1; apply Zis_gcd_for_euclid with (a/b). - pattern a at 1;rewrite (Z_div_mod_eq a b). - ring_simplify (b * (a / b) + a mod b - a / b * b);trivial. zarith. - Qed. - - Lemma spec_ww_gcd_gt_aux_body : - forall ah al bh bl n cont, - [[WW bh bl]] <= 2^n -> - [[WW ah al]] > [[WW bh bl]] -> - (forall xh xl yh yl, - [[WW xh xl]] > [[WW yh yl]] -> [[WW yh yl]] <= 2^(n-1) -> - Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]]) -> - Zis_gcd [[WW ah al]] [[WW bh bl]] [[ww_gcd_gt_body cont ah al bh bl]]. - Proof. - intros ah al bh bl n cont Hlog Hgt Hcont. - change (ww_gcd_gt_body cont ah al bh bl) with (match w_compare w_0 bh with - | Eq => - match w_compare w_0 bl with - | Eq => WW ah al (* normalement n'arrive pas si forme normale *) - | Lt => - let m := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 - w_compare w_sub 1 (WW ah al) bl in - WW w_0 (w_gcd_gt bl m) - | Gt => W0 (* absurde *) - end - | Lt => - let m := ww_mod_gt_aux ah al bh bl in - match m with - | W0 => WW bh bl - | WW mh ml => - match w_compare w_0 mh with - | Eq => - match w_compare w_0 ml with - | Eq => WW bh bl - | _ => - let r := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 - w_compare w_sub 1 (WW bh bl) ml in - WW w_0 (w_gcd_gt ml r) - end - | Lt => - let r := ww_mod_gt_aux bh bl mh ml in - match r with - | W0 => m - | WW rh rl => cont mh ml rh rl - end - | Gt => W0 (* absurde *) - end - end - | Gt => W0 (* absurde *) - end). - rewrite spec_compare, spec_w_0. - case Z.compare_spec; intros Hbh. - simpl ww_to_Z in *. rewrite <- Hbh. - rewrite Z.mul_0_l;rewrite Z.add_0_l. - rewrite spec_compare, spec_w_0. - case Z.compare_spec; intros Hbl. - rewrite <- Hbl;apply Zis_gcd_0. - simpl;rewrite spec_w_0;rewrite Z.mul_0_l;rewrite Z.add_0_l. - apply Zis_gcd_mod;zarith. - change ([|ah|] * wB + [|al|]) with (double_to_Z w_digits w_to_Z 1 (WW ah al)). - rewrite <- (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW w_head0 w_add_mul_div - w_div21 w_compare w_sub w_to_Z spec_to_Z spec_w_zdigits spec_w_0 spec_w_WW spec_head0 spec_add_mul_div - spec_div21 spec_compare spec_sub 1 (WW ah al) bl Hbl). - apply spec_gcd_gt. - rewrite (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW); trivial. - apply Z.lt_gt;match goal with | |- ?x mod ?y < ?y => - destruct (Z_mod_lt x y);zarith end. - Spec_w_to_Z bl;exfalso;omega. - assert (H:= spec_ww_mod_gt_aux _ _ _ Hgt Hbh). - assert (H2 : 0 < [[WW bh bl]]). - simpl;Spec_w_to_Z bl. apply Z.lt_le_trans with ([|bh|]*wB);zarith. - apply Z.mul_pos_pos;zarith. - apply Zis_gcd_mod;trivial. rewrite <- H. - simpl in *;destruct (ww_mod_gt_aux ah al bh bl) as [ |mh ml]. - simpl;apply Zis_gcd_0;zarith. - rewrite spec_compare, spec_w_0; case Z.compare_spec; intros Hmh. - simpl;rewrite <- Hmh;simpl. - rewrite spec_compare, spec_w_0; case Z.compare_spec; intros Hml. - rewrite <- Hml;simpl;apply Zis_gcd_0. - simpl; rewrite spec_w_0; simpl. - apply Zis_gcd_mod;zarith. - change ([|bh|] * wB + [|bl|]) with (double_to_Z w_digits w_to_Z 1 (WW bh bl)). - rewrite <- (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW w_head0 w_add_mul_div - w_div21 w_compare w_sub w_to_Z spec_to_Z spec_w_zdigits spec_w_0 spec_w_WW spec_head0 spec_add_mul_div - spec_div21 spec_compare spec_sub 1 (WW bh bl) ml Hml). - apply spec_gcd_gt. - rewrite (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW); trivial. - apply Z.lt_gt;match goal with | |- ?x mod ?y < ?y => - destruct (Z_mod_lt x y);zarith end. - Spec_w_to_Z ml;exfalso;omega. - assert ([[WW bh bl]] > [[WW mh ml]]). - rewrite H;simpl; apply Z.lt_gt;match goal with | |- ?x mod ?y < ?y => - destruct (Z_mod_lt x y);zarith end. - assert (H1:= spec_ww_mod_gt_aux _ _ _ H0 Hmh). - assert (H3 : 0 < [[WW mh ml]]). - simpl;Spec_w_to_Z ml. apply Z.lt_le_trans with ([|mh|]*wB);zarith. - apply Z.mul_pos_pos;zarith. - apply Zis_gcd_mod;zarith. simpl in *;rewrite <- H1. - destruct (ww_mod_gt_aux bh bl mh ml) as [ |rh rl]. simpl; apply Zis_gcd_0. - simpl;apply Hcont. simpl in H1;rewrite H1. - apply Z.lt_gt;match goal with | |- ?x mod ?y < ?y => - destruct (Z_mod_lt x y);zarith end. - apply Z.le_trans with (2^n/2). - apply Zdiv_le_lower_bound;zarith. - apply Z.le_trans with ([|bh|] * wB + [|bl|]);zarith. - assert (H3' := Z_div_mod_eq [[WW bh bl]] [[WW mh ml]] (Z.lt_gt _ _ H3)). - assert (H4 : 0 <= [[WW bh bl]]/[[WW mh ml]]). - apply Z.ge_le;apply Z_div_ge0;zarith. simpl in *;rewrite H1. - pattern ([|bh|] * wB + [|bl|]) at 2;rewrite H3'. - Z.le_elim H4. - assert (H6' : [[WW bh bl]] mod [[WW mh ml]] = - [[WW bh bl]] - [[WW mh ml]] * ([[WW bh bl]]/[[WW mh ml]])). - simpl;pattern ([|bh|] * wB + [|bl|]) at 2;rewrite H3';ring. simpl in H6'. - assert ([[WW mh ml]] <= [[WW mh ml]] * ([[WW bh bl]]/[[WW mh ml]])). - simpl;pattern ([|mh|]*wB+[|ml|]) at 1;rewrite <- Z.mul_1_r;zarith. - simpl in *;assert (H8 := Z_mod_lt [[WW bh bl]] [[WW mh ml]]);simpl in H8; - zarith. - assert (H8 := Z_mod_lt [[WW bh bl]] [[WW mh ml]]);simpl in *;zarith. - rewrite <- H4 in H3';rewrite Z.mul_0_r in H3';simpl in H3';zarith. - pattern n at 1;replace n with (n-1+1);try ring. - rewrite Zpower_exp;zarith. change (2^1) with 2. - rewrite Z_div_mult;zarith. - assert (2^1 <= 2^n). change (2^1) with 2;zarith. - assert (H7 := @Zpower_le_monotone_inv 2 1 n);zarith. - Spec_w_to_Z mh;exfalso;zarith. - Spec_w_to_Z bh;exfalso;zarith. - Qed. - - Lemma spec_ww_gcd_gt_aux : - forall p cont n, - (forall xh xl yh yl, - [[WW xh xl]] > [[WW yh yl]] -> - [[WW yh yl]] <= 2^n -> - Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]]) -> - forall ah al bh bl , [[WW ah al]] > [[WW bh bl]] -> - [[WW bh bl]] <= 2^(Zpos p + n) -> - Zis_gcd [[WW ah al]] [[WW bh bl]] - [[ww_gcd_gt_aux p cont ah al bh bl]]. - Proof. - induction p;intros cont n Hcont ah al bh bl Hgt Hs;simpl ww_gcd_gt_aux. - assert (0 < Zpos p). unfold Z.lt;reflexivity. - apply spec_ww_gcd_gt_aux_body with (n := Zpos (xI p) + n); - trivial;rewrite Pos2Z.inj_xI. - intros. apply IHp with (n := Zpos p + n);zarith. - intros. apply IHp with (n := n );zarith. - apply Z.le_trans with (2 ^ (2* Zpos p + 1+ n -1));zarith. - apply Z.pow_le_mono_r;zarith. - assert (0 < Zpos p). unfold Z.lt;reflexivity. - apply spec_ww_gcd_gt_aux_body with (n := Zpos (xO p) + n );trivial. - rewrite (Pos2Z.inj_xO p). - intros. apply IHp with (n := Zpos p + n - 1);zarith. - intros. apply IHp with (n := n -1 );zarith. - intros;apply Hcont;zarith. - apply Z.le_trans with (2^(n-1));zarith. - apply Z.pow_le_mono_r;zarith. - apply Z.le_trans with (2 ^ (Zpos p + n -1));zarith. - apply Z.pow_le_mono_r;zarith. - apply Z.le_trans with (2 ^ (2*Zpos p + n -1));zarith. - apply Z.pow_le_mono_r;zarith. - apply spec_ww_gcd_gt_aux_body with (n := n+1);trivial. - rewrite Z.add_comm;trivial. - ring_simplify (n + 1 - 1);trivial. - Qed. - -End DoubleDivGt. - -Section DoubleDiv. - - Variable w : Type. - Variable w_digits : positive. - Variable ww_1 : zn2z w. - Variable ww_compare : zn2z w -> zn2z w -> comparison. - - Variable ww_div_gt : zn2z w -> zn2z w -> zn2z w * zn2z w. - Variable ww_mod_gt : zn2z w -> zn2z w -> zn2z w. - - Definition ww_div a b := - match ww_compare a b with - | Gt => ww_div_gt a b - | Eq => (ww_1, W0) - | Lt => (W0, a) - end. - - Definition ww_mod a b := - match ww_compare a b with - | Gt => ww_mod_gt a b - | Eq => W0 - | Lt => a - end. - - Variable w_to_Z : w -> Z. - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99). - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - Variable spec_ww_1 : [[ww_1]] = 1. - Variable spec_ww_compare : forall x y, - ww_compare x y = Z.compare [[x]] [[y]]. - Variable spec_ww_div_gt : forall a b, [[a]] > [[b]] -> 0 < [[b]] -> - let (q,r) := ww_div_gt a b in - [[a]] = [[q]] * [[b]] + [[r]] /\ - 0 <= [[r]] < [[b]]. - Variable spec_ww_mod_gt : forall a b, [[a]] > [[b]] -> 0 < [[b]] -> - [[ww_mod_gt a b]] = [[a]] mod [[b]]. - - Ltac Spec_w_to_Z x := - let H:= fresh "HH" in - assert (H:= spec_to_Z x). - - Ltac Spec_ww_to_Z x := - let H:= fresh "HH" in - assert (H:= spec_ww_to_Z w_digits w_to_Z spec_to_Z x). - - Lemma spec_ww_div : forall a b, 0 < [[b]] -> - let (q,r) := ww_div a b in - [[a]] = [[q]] * [[b]] + [[r]] /\ - 0 <= [[r]] < [[b]]. - Proof. - intros a b Hpos;unfold ww_div. - rewrite spec_ww_compare; case Z.compare_spec; intros. - simpl;rewrite spec_ww_1;split;zarith. - simpl;split;[ring|Spec_ww_to_Z a;zarith]. - apply spec_ww_div_gt;auto with zarith. - Qed. - - Lemma spec_ww_mod : forall a b, 0 < [[b]] -> - [[ww_mod a b]] = [[a]] mod [[b]]. - Proof. - intros a b Hpos;unfold ww_mod. - rewrite spec_ww_compare; case Z.compare_spec; intros. - simpl;apply Zmod_unique with 1;try rewrite H;zarith. - Spec_ww_to_Z a;symmetry;apply Zmod_small;zarith. - apply spec_ww_mod_gt;auto with zarith. - Qed. - - - Variable w_0 : w. - Variable w_1 : w. - Variable w_compare : w -> w -> comparison. - Variable w_eq0 : w -> bool. - Variable w_gcd_gt : w -> w -> w. - Variable _ww_digits : positive. - Variable spec_ww_digits_ : _ww_digits = xO w_digits. - Variable ww_gcd_gt_fix : - positive -> (w -> w -> w -> w -> zn2z w) -> - w -> w -> w -> w -> zn2z w. - - Variable spec_w_0 : [|w_0|] = 0. - Variable spec_w_1 : [|w_1|] = 1. - Variable spec_compare : - forall x y, w_compare x y = Z.compare [|x|] [|y|]. - Variable spec_eq0 : forall x, w_eq0 x = true -> [|x|] = 0. - Variable spec_gcd_gt : forall a b, [|a|] > [|b|] -> - Zis_gcd [|a|] [|b|] [|w_gcd_gt a b|]. - Variable spec_gcd_gt_fix : - forall p cont n, - (forall xh xl yh yl, - [[WW xh xl]] > [[WW yh yl]] -> - [[WW yh yl]] <= 2^n -> - Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]]) -> - forall ah al bh bl , [[WW ah al]] > [[WW bh bl]] -> - [[WW bh bl]] <= 2^(Zpos p + n) -> - Zis_gcd [[WW ah al]] [[WW bh bl]] - [[ww_gcd_gt_fix p cont ah al bh bl]]. - - Definition gcd_cont (xh xl yh yl:w) := - match w_compare w_1 yl with - | Eq => ww_1 - | _ => WW xh xl - end. - - Lemma spec_gcd_cont : forall xh xl yh yl, - [[WW xh xl]] > [[WW yh yl]] -> - [[WW yh yl]] <= 1 -> - Zis_gcd [[WW xh xl]] [[WW yh yl]] [[gcd_cont xh xl yh yl]]. - Proof. - intros xh xl yh yl Hgt' Hle. simpl in Hle. - assert ([|yh|] = 0). - change 1 with (0*wB+1) in Hle. - assert (0 <= 1 < wB). split;zarith. apply wB_pos. - assert (H1:= beta_lex _ _ _ _ _ Hle (spec_to_Z yl) H). - Spec_w_to_Z yh;zarith. - unfold gcd_cont; rewrite spec_compare, spec_w_1. - case Z.compare_spec; intros Hcmpy. - simpl;rewrite H;simpl; - rewrite spec_ww_1;rewrite <- Hcmpy;apply Zis_gcd_mod;zarith. - rewrite <- (Zmod_unique ([|xh|]*wB+[|xl|]) 1 ([|xh|]*wB+[|xl|]) 0);zarith. - rewrite H in Hle; exfalso;zarith. - assert (H0 : [|yl|] = 0) by (Spec_w_to_Z yl;zarith). - simpl. rewrite H0, H;simpl;apply Zis_gcd_0;trivial. - Qed. - - - Variable cont : w -> w -> w -> w -> zn2z w. - Variable spec_cont : forall xh xl yh yl, - [[WW xh xl]] > [[WW yh yl]] -> - [[WW yh yl]] <= 1 -> - Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]]. - - Definition ww_gcd_gt a b := - match a, b with - | W0, _ => b - | _, W0 => a - | WW ah al, WW bh bl => - if w_eq0 ah then (WW w_0 (w_gcd_gt al bl)) - else ww_gcd_gt_fix _ww_digits cont ah al bh bl - end. - - Definition ww_gcd a b := - Eval lazy beta delta [ww_gcd_gt] in - match ww_compare a b with - | Gt => ww_gcd_gt a b - | Eq => a - | Lt => ww_gcd_gt b a - end. - - Lemma spec_ww_gcd_gt : forall a b, [[a]] > [[b]] -> - Zis_gcd [[a]] [[b]] [[ww_gcd_gt a b]]. - Proof. - intros a b Hgt;unfold ww_gcd_gt. - destruct a as [ |ah al]. simpl;apply Zis_gcd_sym;apply Zis_gcd_0. - destruct b as [ |bh bl]. simpl;apply Zis_gcd_0. - simpl in Hgt. generalize (@spec_eq0 ah);destruct (w_eq0 ah);intros. - simpl;rewrite H in Hgt;trivial;rewrite H;trivial;rewrite spec_w_0;simpl. - assert ([|bh|] <= 0). - apply beta_lex with (d:=[|al|])(b:=[|bl|]) (beta := wB);zarith. - Spec_w_to_Z bh;assert ([|bh|] = 0);zarith. rewrite H1 in Hgt;simpl in Hgt. - rewrite H1;simpl;auto. clear H. - apply spec_gcd_gt_fix with (n:= 0);trivial. - rewrite Z.add_0_r;rewrite spec_ww_digits_. - change (2 ^ Zpos (xO w_digits)) with wwB. Spec_ww_to_Z (WW bh bl);zarith. - Qed. - - Lemma spec_ww_gcd : forall a b, Zis_gcd [[a]] [[b]] [[ww_gcd a b]]. - Proof. - intros a b. - change (ww_gcd a b) with - (match ww_compare a b with - | Gt => ww_gcd_gt a b - | Eq => a - | Lt => ww_gcd_gt b a - end). - rewrite spec_ww_compare; case Z.compare_spec; intros Hcmp. - Spec_ww_to_Z b;rewrite Hcmp. - apply Zis_gcd_for_euclid with 1;zarith. - ring_simplify ([[b]] - 1 * [[b]]). apply Zis_gcd_0;zarith. - apply Zis_gcd_sym;apply spec_ww_gcd_gt;zarith. - apply spec_ww_gcd_gt;zarith. - Qed. - -End DoubleDiv. - diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v deleted file mode 100644 index 195527dd..00000000 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v +++ /dev/null @@ -1,519 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Set Implicit Arguments. - -Require Import ZArith Ndigits. -Require Import BigNumPrelude. -Require Import DoubleType. -Require Import DoubleBase. - -Local Open Scope Z_scope. - -Local Infix "<<" := Pos.shiftl_nat (at level 30). - -Section GENDIVN1. - - Variable w : Type. - Variable w_digits : positive. - Variable w_zdigits : w. - Variable w_0 : w. - Variable w_WW : w -> w -> zn2z w. - Variable w_head0 : w -> w. - Variable w_add_mul_div : w -> w -> w -> w. - Variable w_div21 : w -> w -> w -> w * w. - Variable w_compare : w -> w -> comparison. - Variable w_sub : w -> w -> w. - - - - (* ** For proofs ** *) - Variable w_to_Z : w -> Z. - - Notation wB := (base w_digits). - - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[! n | x !]" := (double_to_Z w_digits w_to_Z n x) - (at level 0, x at level 99). - Notation "[[ x ]]" := (zn2z_to_Z wB w_to_Z x) (at level 0, x at level 99). - - Variable spec_to_Z : forall x, 0 <= [| x |] < wB. - Variable spec_w_zdigits: [|w_zdigits|] = Zpos w_digits. - Variable spec_0 : [|w_0|] = 0. - Variable spec_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. - Variable spec_head0 : forall x, 0 < [|x|] -> - wB/ 2 <= 2 ^ [|w_head0 x|] * [|x|] < wB. - Variable spec_add_mul_div : forall x y p, - [|p|] <= Zpos w_digits -> - [| w_add_mul_div p x y |] = - ([|x|] * (2 ^ [|p|]) + - [|y|] / (2 ^ ((Zpos w_digits) - [|p|]))) mod wB. - Variable spec_div21 : forall a1 a2 b, - wB/2 <= [|b|] -> - [|a1|] < [|b|] -> - let (q,r) := w_div21 a1 a2 b in - [|a1|] *wB+ [|a2|] = [|q|] * [|b|] + [|r|] /\ - 0 <= [|r|] < [|b|]. - Variable spec_compare : - forall x y, w_compare x y = Z.compare [|x|] [|y|]. - Variable spec_sub: forall x y, - [|w_sub x y|] = ([|x|] - [|y|]) mod wB. - - - - Section DIVAUX. - Variable b2p : w. - Variable b2p_le : wB/2 <= [|b2p|]. - - Definition double_divn1_0_aux n (divn1: w -> word w n -> word w n * w) r h := - let (hh,hl) := double_split w_0 n h in - let (qh,rh) := divn1 r hh in - let (ql,rl) := divn1 rh hl in - (double_WW w_WW n qh ql, rl). - - Fixpoint double_divn1_0 (n:nat) : w -> word w n -> word w n * w := - match n return w -> word w n -> word w n * w with - | O => fun r x => w_div21 r x b2p - | S n => double_divn1_0_aux n (double_divn1_0 n) - end. - - Lemma spec_split : forall (n : nat) (x : zn2z (word w n)), - let (h, l) := double_split w_0 n x in - [!S n | x!] = [!n | h!] * double_wB w_digits n + [!n | l!]. - Proof (spec_double_split w_0 w_digits w_to_Z spec_0). - - Lemma spec_double_divn1_0 : forall n r a, - [|r|] < [|b2p|] -> - let (q,r') := double_divn1_0 n r a in - [|r|] * double_wB w_digits n + [!n|a!] = [!n|q!] * [|b2p|] + [|r'|] /\ - 0 <= [|r'|] < [|b2p|]. - Proof. - induction n;intros. - exact (spec_div21 a b2p_le H). - simpl (double_divn1_0 (S n) r a); unfold double_divn1_0_aux. - assert (H1 := spec_split n a);destruct (double_split w_0 n a) as (hh,hl). - rewrite H1. - assert (H2 := IHn r hh H);destruct (double_divn1_0 n r hh) as (qh,rh). - destruct H2. - assert ([|rh|] < [|b2p|]). omega. - assert (H4 := IHn rh hl H3);destruct (double_divn1_0 n rh hl) as (ql,rl). - destruct H4;split;trivial. - rewrite spec_double_WW;trivial. - rewrite <- double_wB_wwB. - rewrite Z.mul_assoc;rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - rewrite H0;rewrite Z.mul_add_distr_r;rewrite <- Z.add_assoc. - rewrite H4;ring. - Qed. - - Definition double_modn1_0_aux n (modn1:w -> word w n -> w) r h := - let (hh,hl) := double_split w_0 n h in modn1 (modn1 r hh) hl. - - Fixpoint double_modn1_0 (n:nat) : w -> word w n -> w := - match n return w -> word w n -> w with - | O => fun r x => snd (w_div21 r x b2p) - | S n => double_modn1_0_aux n (double_modn1_0 n) - end. - - Lemma spec_double_modn1_0 : forall n r x, - double_modn1_0 n r x = snd (double_divn1_0 n r x). - Proof. - induction n;simpl;intros;trivial. - unfold double_modn1_0_aux, double_divn1_0_aux. - destruct (double_split w_0 n x) as (hh,hl). - rewrite (IHn r hh). - destruct (double_divn1_0 n r hh) as (qh,rh);simpl. - rewrite IHn. destruct (double_divn1_0 n rh hl);trivial. - Qed. - - Variable p : w. - Variable p_bounded : [|p|] <= Zpos w_digits. - - Lemma spec_add_mul_divp : forall x y, - [| w_add_mul_div p x y |] = - ([|x|] * (2 ^ [|p|]) + - [|y|] / (2 ^ ((Zpos w_digits) - [|p|]))) mod wB. - Proof. - intros;apply spec_add_mul_div;auto. - Qed. - - Definition double_divn1_p_aux n - (divn1 : w -> word w n -> word w n -> word w n * w) r h l := - let (hh,hl) := double_split w_0 n h in - let (lh,ll) := double_split w_0 n l in - let (qh,rh) := divn1 r hh hl in - let (ql,rl) := divn1 rh hl lh in - (double_WW w_WW n qh ql, rl). - - Fixpoint double_divn1_p (n:nat) : w -> word w n -> word w n -> word w n * w := - match n return w -> word w n -> word w n -> word w n * w with - | O => fun r h l => w_div21 r (w_add_mul_div p h l) b2p - | S n => double_divn1_p_aux n (double_divn1_p n) - end. - - Lemma p_lt_double_digits : forall n, [|p|] <= Zpos (w_digits << n). - Proof. - induction n;simpl. trivial. - case (spec_to_Z p); rewrite Pos2Z.inj_xO;auto with zarith. - Qed. - - Lemma spec_double_divn1_p : forall n r h l, - [|r|] < [|b2p|] -> - let (q,r') := double_divn1_p n r h l in - [|r|] * double_wB w_digits n + - ([!n|h!]*2^[|p|] + - [!n|l!] / (2^(Zpos(w_digits << n) - [|p|]))) - mod double_wB w_digits n = [!n|q!] * [|b2p|] + [|r'|] /\ - 0 <= [|r'|] < [|b2p|]. - Proof. - case (spec_to_Z p); intros HH0 HH1. - induction n;intros. - simpl (double_divn1_p 0 r h l). - unfold double_to_Z, double_wB, "<<". - rewrite <- spec_add_mul_divp. - exact (spec_div21 (w_add_mul_div p h l) b2p_le H). - simpl (double_divn1_p (S n) r h l). - unfold double_divn1_p_aux. - assert (H1 := spec_split n h);destruct (double_split w_0 n h) as (hh,hl). - rewrite H1. rewrite <- double_wB_wwB. - assert (H2 := spec_split n l);destruct (double_split w_0 n l) as (lh,ll). - rewrite H2. - replace ([|r|] * (double_wB w_digits n * double_wB w_digits n) + - (([!n|hh!] * double_wB w_digits n + [!n|hl!]) * 2 ^ [|p|] + - ([!n|lh!] * double_wB w_digits n + [!n|ll!]) / - 2^(Zpos (w_digits << (S n)) - [|p|])) mod - (double_wB w_digits n * double_wB w_digits n)) with - (([|r|] * double_wB w_digits n + ([!n|hh!] * 2^[|p|] + - [!n|hl!] / 2^(Zpos (w_digits << n) - [|p|])) mod - double_wB w_digits n) * double_wB w_digits n + - ([!n|hl!] * 2^[|p|] + - [!n|lh!] / 2^(Zpos (w_digits << n) - [|p|])) mod - double_wB w_digits n). - generalize (IHn r hh hl H);destruct (double_divn1_p n r hh hl) as (qh,rh); - intros (H3,H4);rewrite H3. - assert ([|rh|] < [|b2p|]). omega. - replace (([!n|qh!] * [|b2p|] + [|rh|]) * double_wB w_digits n + - ([!n|hl!] * 2 ^ [|p|] + - [!n|lh!] / 2 ^ (Zpos (w_digits << n) - [|p|])) mod - double_wB w_digits n) with - ([!n|qh!] * [|b2p|] *double_wB w_digits n + ([|rh|]*double_wB w_digits n + - ([!n|hl!] * 2 ^ [|p|] + - [!n|lh!] / 2 ^ (Zpos (w_digits << n) - [|p|])) mod - double_wB w_digits n)). 2:ring. - generalize (IHn rh hl lh H0);destruct (double_divn1_p n rh hl lh) as (ql,rl); - intros (H5,H6);rewrite H5. - split;[rewrite spec_double_WW;trivial;ring|trivial]. - assert (Uhh := spec_double_to_Z w_digits w_to_Z spec_to_Z n hh); - unfold double_wB,base in Uhh. - assert (Uhl := spec_double_to_Z w_digits w_to_Z spec_to_Z n hl); - unfold double_wB,base in Uhl. - assert (Ulh := spec_double_to_Z w_digits w_to_Z spec_to_Z n lh); - unfold double_wB,base in Ulh. - assert (Ull := spec_double_to_Z w_digits w_to_Z spec_to_Z n ll); - unfold double_wB,base in Ull. - unfold double_wB,base. - assert (UU:=p_lt_double_digits n). - rewrite Zdiv_shift_r;auto with zarith. - 2:change (Zpos (w_digits << (S n))) - with (2*Zpos (w_digits << n));auto with zarith. - replace (2 ^ (Zpos (w_digits << (S n)) - [|p|])) with - (2^(Zpos (w_digits << n) - [|p|])*2^Zpos (w_digits << n)). - rewrite Zdiv_mult_cancel_r;auto with zarith. - rewrite Z.mul_add_distr_r with (p:= 2^[|p|]). - pattern ([!n|hl!] * 2^[|p|]) at 2; - rewrite (shift_unshift_mod (Zpos(w_digits << n))([|p|])([!n|hl!])); - auto with zarith. - rewrite Z.add_assoc. - replace - ([!n|hh!] * 2^Zpos (w_digits << n)* 2^[|p|] + - ([!n|hl!] / 2^(Zpos (w_digits << n)-[|p|])* - 2^Zpos(w_digits << n))) - with - (([!n|hh!] *2^[|p|] + double_to_Z w_digits w_to_Z n hl / - 2^(Zpos (w_digits << n)-[|p|])) - * 2^Zpos(w_digits << n));try (ring;fail). - rewrite <- Z.add_assoc. - rewrite <- (Zmod_shift_r ([|p|]));auto with zarith. - replace - (2 ^ Zpos (w_digits << n) * 2 ^ Zpos (w_digits << n)) with - (2 ^ (Zpos (w_digits << n) + Zpos (w_digits << n))). - rewrite (Zmod_shift_r (Zpos (w_digits << n)));auto with zarith. - replace (2 ^ (Zpos (w_digits << n) + Zpos (w_digits << n))) - with (2^Zpos(w_digits << n) *2^Zpos(w_digits << n)). - rewrite (Z.mul_comm (([!n|hh!] * 2 ^ [|p|] + - [!n|hl!] / 2 ^ (Zpos (w_digits << n) - [|p|])))). - rewrite Zmult_mod_distr_l;auto with zarith. - ring. - rewrite Zpower_exp;auto with zarith. - assert (0 < Zpos (w_digits << n)). unfold Z.lt;reflexivity. - auto with zarith. - apply Z_mod_lt;auto with zarith. - rewrite Zpower_exp;auto with zarith. - split;auto with zarith. - apply Zdiv_lt_upper_bound;auto with zarith. - rewrite <- Zpower_exp;auto with zarith. - replace ([|p|] + (Zpos (w_digits << n) - [|p|])) with - (Zpos(w_digits << n));auto with zarith. - rewrite <- Zpower_exp;auto with zarith. - replace (Zpos (w_digits << (S n)) - [|p|]) with - (Zpos (w_digits << n) - [|p|] + - Zpos (w_digits << n));trivial. - change (Zpos (w_digits << (S n))) with - (2*Zpos (w_digits << n)). ring. - Qed. - - Definition double_modn1_p_aux n (modn1 : w -> word w n -> word w n -> w) r h l:= - let (hh,hl) := double_split w_0 n h in - let (lh,ll) := double_split w_0 n l in - modn1 (modn1 r hh hl) hl lh. - - Fixpoint double_modn1_p (n:nat) : w -> word w n -> word w n -> w := - match n return w -> word w n -> word w n -> w with - | O => fun r h l => snd (w_div21 r (w_add_mul_div p h l) b2p) - | S n => double_modn1_p_aux n (double_modn1_p n) - end. - - Lemma spec_double_modn1_p : forall n r h l , - double_modn1_p n r h l = snd (double_divn1_p n r h l). - Proof. - induction n;simpl;intros;trivial. - unfold double_modn1_p_aux, double_divn1_p_aux. - destruct(double_split w_0 n h)as(hh,hl);destruct(double_split w_0 n l) as (lh,ll). - rewrite (IHn r hh hl);destruct (double_divn1_p n r hh hl) as (qh,rh). - rewrite IHn;simpl;destruct (double_divn1_p n rh hl lh);trivial. - Qed. - - End DIVAUX. - - Fixpoint high (n:nat) : word w n -> w := - match n return word w n -> w with - | O => fun a => a - | S n => - fun (a:zn2z (word w n)) => - match a with - | W0 => w_0 - | WW h l => high n h - end - end. - - Lemma spec_double_digits:forall n, Zpos w_digits <= Zpos (w_digits << n). - Proof. - induction n;simpl;auto with zarith. - change (Zpos (xO (w_digits << n))) with - (2*Zpos (w_digits << n)). - assert (0 < Zpos w_digits) by reflexivity. - auto with zarith. - Qed. - - Lemma spec_high : forall n (x:word w n), - [|high n x|] = [!n|x!] / 2^(Zpos (w_digits << n) - Zpos w_digits). - Proof. - induction n;intros. - unfold high,double_to_Z. rewrite Pshiftl_nat_0. - replace (Zpos w_digits - Zpos w_digits) with 0;try ring. - simpl. rewrite <- (Zdiv_unique [|x|] 1 [|x|] 0);auto with zarith. - assert (U2 := spec_double_digits n). - assert (U3 : 0 < Zpos w_digits). exact (eq_refl Lt). - destruct x;unfold high;fold high. - unfold double_to_Z,zn2z_to_Z;rewrite spec_0. - rewrite Zdiv_0_l;trivial. - assert (U0 := spec_double_to_Z w_digits w_to_Z spec_to_Z n w0); - assert (U1 := spec_double_to_Z w_digits w_to_Z spec_to_Z n w1). - simpl [!S n|WW w0 w1!]. - unfold double_wB,base;rewrite Zdiv_shift_r;auto with zarith. - replace (2 ^ (Zpos (w_digits << (S n)) - Zpos w_digits)) with - (2^(Zpos (w_digits << n) - Zpos w_digits) * - 2^Zpos (w_digits << n)). - rewrite Zdiv_mult_cancel_r;auto with zarith. - rewrite <- Zpower_exp;auto with zarith. - replace (Zpos (w_digits << n) - Zpos w_digits + - Zpos (w_digits << n)) with - (Zpos (w_digits << (S n)) - Zpos w_digits);trivial. - change (Zpos (w_digits << (S n))) with - (2*Zpos (w_digits << n));ring. - change (Zpos (w_digits << (S n))) with - (2*Zpos (w_digits << n)); auto with zarith. - Qed. - - Definition double_divn1 (n:nat) (a:word w n) (b:w) := - let p := w_head0 b in - match w_compare p w_0 with - | Gt => - let b2p := w_add_mul_div p b w_0 in - let ha := high n a in - let k := w_sub w_zdigits p in - let lsr_n := w_add_mul_div k w_0 in - let r0 := w_add_mul_div p w_0 ha in - let (q,r) := double_divn1_p b2p p n r0 a (double_0 w_0 n) in - (q, lsr_n r) - | _ => double_divn1_0 b n w_0 a - end. - - Lemma spec_double_divn1 : forall n a b, - 0 < [|b|] -> - let (q,r) := double_divn1 n a b in - [!n|a!] = [!n|q!] * [|b|] + [|r|] /\ - 0 <= [|r|] < [|b|]. - Proof. - intros n a b H. unfold double_divn1. - case (spec_head0 H); intros H0 H1. - case (spec_to_Z (w_head0 b)); intros HH1 HH2. - rewrite spec_compare; case Z.compare_spec; - rewrite spec_0; intros H2; auto with zarith. - assert (Hv1: wB/2 <= [|b|]). - generalize H0; rewrite H2; rewrite Z.pow_0_r; - rewrite Z.mul_1_l; auto. - assert (Hv2: [|w_0|] < [|b|]). - rewrite spec_0; auto. - generalize (spec_double_divn1_0 Hv1 n a Hv2). - rewrite spec_0;rewrite Z.mul_0_l; rewrite Z.add_0_l; auto. - contradict H2; auto with zarith. - assert (HHHH : 0 < [|w_head0 b|]); auto with zarith. - assert ([|w_head0 b|] < Zpos w_digits). - case (Z.le_gt_cases (Zpos w_digits) [|w_head0 b|]); auto; intros HH. - assert (2 ^ [|w_head0 b|] < wB). - apply Z.le_lt_trans with (2 ^ [|w_head0 b|] * [|b|]);auto with zarith. - replace (2 ^ [|w_head0 b|]) with (2^[|w_head0 b|] * 1);try (ring;fail). - apply Z.mul_le_mono_nonneg;auto with zarith. - assert (wB <= 2^[|w_head0 b|]). - unfold base;apply Zpower_le_monotone;auto with zarith. omega. - assert ([|w_add_mul_div (w_head0 b) b w_0|] = - 2 ^ [|w_head0 b|] * [|b|]). - rewrite (spec_add_mul_div b w_0); auto with zarith. - rewrite spec_0;rewrite Zdiv_0_l; try omega. - rewrite Z.add_0_r; rewrite Z.mul_comm. - rewrite Zmod_small; auto with zarith. - assert (H5 := spec_to_Z (high n a)). - assert - ([|w_add_mul_div (w_head0 b) w_0 (high n a)|] - <[|w_add_mul_div (w_head0 b) b w_0|]). - rewrite H4. - rewrite spec_add_mul_div;auto with zarith. - rewrite spec_0;rewrite Z.mul_0_l;rewrite Z.add_0_l. - assert (([|high n a|]/2^(Zpos w_digits - [|w_head0 b|])) < wB). - apply Zdiv_lt_upper_bound;auto with zarith. - apply Z.lt_le_trans with wB;auto with zarith. - pattern wB at 1;replace wB with (wB*1);try ring. - apply Z.mul_le_mono_nonneg;auto with zarith. - assert (H6 := Z.pow_pos_nonneg 2 (Zpos w_digits - [|w_head0 b|])); - auto with zarith. - rewrite Zmod_small;auto with zarith. - apply Zdiv_lt_upper_bound;auto with zarith. - apply Z.lt_le_trans with wB;auto with zarith. - apply Z.le_trans with (2 ^ [|w_head0 b|] * [|b|] * 2). - rewrite <- wB_div_2; try omega. - apply Z.mul_le_mono_nonneg;auto with zarith. - pattern 2 at 1;rewrite <- Z.pow_1_r. - apply Zpower_le_monotone;split;auto with zarith. - rewrite <- H4 in H0. - assert (Hb3: [|w_head0 b|] <= Zpos w_digits); auto with zarith. - assert (H7:= spec_double_divn1_p H0 Hb3 n a (double_0 w_0 n) H6). - destruct (double_divn1_p (w_add_mul_div (w_head0 b) b w_0) (w_head0 b) n - (w_add_mul_div (w_head0 b) w_0 (high n a)) a - (double_0 w_0 n)) as (q,r). - assert (U:= spec_double_digits n). - rewrite spec_double_0 in H7;trivial;rewrite Zdiv_0_l in H7. - rewrite Z.add_0_r in H7. - rewrite spec_add_mul_div in H7;auto with zarith. - rewrite spec_0 in H7;rewrite Z.mul_0_l in H7;rewrite Z.add_0_l in H7. - assert (([|high n a|] / 2 ^ (Zpos w_digits - [|w_head0 b|])) mod wB - = [!n|a!] / 2^(Zpos (w_digits << n) - [|w_head0 b|])). - rewrite Zmod_small;auto with zarith. - rewrite spec_high. rewrite Zdiv_Zdiv;auto with zarith. - rewrite <- Zpower_exp;auto with zarith. - replace (Zpos (w_digits << n) - Zpos w_digits + - (Zpos w_digits - [|w_head0 b|])) - with (Zpos (w_digits << n) - [|w_head0 b|]);trivial;ring. - assert (H8 := Z.pow_pos_nonneg 2 (Zpos w_digits - [|w_head0 b|]));auto with zarith. - split;auto with zarith. - apply Z.le_lt_trans with ([|high n a|]);auto with zarith. - apply Zdiv_le_upper_bound;auto with zarith. - pattern ([|high n a|]) at 1;rewrite <- Z.mul_1_r. - apply Z.mul_le_mono_nonneg;auto with zarith. - rewrite H8 in H7;unfold double_wB,base in H7. - rewrite <- shift_unshift_mod in H7;auto with zarith. - rewrite H4 in H7. - assert ([|w_add_mul_div (w_sub w_zdigits (w_head0 b)) w_0 r|] - = [|r|]/2^[|w_head0 b|]). - rewrite spec_add_mul_div. - rewrite spec_0;rewrite Z.mul_0_l;rewrite Z.add_0_l. - replace (Zpos w_digits - [|w_sub w_zdigits (w_head0 b)|]) - with ([|w_head0 b|]). - rewrite Zmod_small;auto with zarith. - assert (H9 := spec_to_Z r). - split;auto with zarith. - apply Z.le_lt_trans with ([|r|]);auto with zarith. - apply Zdiv_le_upper_bound;auto with zarith. - pattern ([|r|]) at 1;rewrite <- Z.mul_1_r. - apply Z.mul_le_mono_nonneg;auto with zarith. - assert (H10 := Z.pow_pos_nonneg 2 ([|w_head0 b|]));auto with zarith. - rewrite spec_sub. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - case (spec_to_Z w_zdigits); auto with zarith. - rewrite spec_sub. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - case (spec_to_Z w_zdigits); auto with zarith. - case H7; intros H71 H72. - split. - rewrite <- (Z_div_mult [!n|a!] (2^[|w_head0 b|]));auto with zarith. - rewrite H71;rewrite H9. - replace ([!n|q!] * (2 ^ [|w_head0 b|] * [|b|])) - with ([!n|q!] *[|b|] * 2^[|w_head0 b|]); - try (ring;fail). - rewrite Z_div_plus_l;auto with zarith. - assert (H10 := spec_to_Z - (w_add_mul_div (w_sub w_zdigits (w_head0 b)) w_0 r));split; - auto with zarith. - rewrite H9. - apply Zdiv_lt_upper_bound;auto with zarith. - rewrite Z.mul_comm;auto with zarith. - exact (spec_double_to_Z w_digits w_to_Z spec_to_Z n a). - Qed. - - - Definition double_modn1 (n:nat) (a:word w n) (b:w) := - let p := w_head0 b in - match w_compare p w_0 with - | Gt => - let b2p := w_add_mul_div p b w_0 in - let ha := high n a in - let k := w_sub w_zdigits p in - let lsr_n := w_add_mul_div k w_0 in - let r0 := w_add_mul_div p w_0 ha in - let r := double_modn1_p b2p p n r0 a (double_0 w_0 n) in - lsr_n r - | _ => double_modn1_0 b n w_0 a - end. - - Lemma spec_double_modn1_aux : forall n a b, - double_modn1 n a b = snd (double_divn1 n a b). - Proof. - intros n a b;unfold double_divn1,double_modn1. - rewrite spec_compare; case Z.compare_spec; - rewrite spec_0; intros H2; auto with zarith. - apply spec_double_modn1_0. - apply spec_double_modn1_0. - rewrite spec_double_modn1_p. - destruct (double_divn1_p (w_add_mul_div (w_head0 b) b w_0) (w_head0 b) n - (w_add_mul_div (w_head0 b) w_0 (high n a)) a (double_0 w_0 n));simpl;trivial. - Qed. - - Lemma spec_double_modn1 : forall n a b, 0 < [|b|] -> - [|double_modn1 n a b|] = [!n|a!] mod [|b|]. - Proof. - intros n a b H;assert (H1 := spec_double_divn1 n a H). - assert (H2 := spec_double_modn1_aux n a b). - rewrite H2;destruct (double_divn1 n a b) as (q,r). - simpl;apply Zmod_unique with (double_to_Z w_digits w_to_Z n q);auto with zarith. - destruct H1 as (h1,h2);rewrite h1;ring. - Qed. - -End GENDIVN1. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v deleted file mode 100644 index f65b47c8..00000000 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v +++ /dev/null @@ -1,475 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Set Implicit Arguments. - -Require Import ZArith. -Require Import BigNumPrelude. -Require Import DoubleType. -Require Import DoubleBase. - -Local Open Scope Z_scope. - -Section DoubleLift. - Variable w : Type. - Variable w_0 : w. - Variable w_WW : w -> w -> zn2z w. - Variable w_W0 : w -> zn2z w. - Variable w_0W : w -> zn2z w. - Variable w_compare : w -> w -> comparison. - Variable ww_compare : zn2z w -> zn2z w -> comparison. - Variable w_head0 : w -> w. - Variable w_tail0 : w -> w. - Variable w_add: w -> w -> zn2z w. - Variable w_add_mul_div : w -> w -> w -> w. - Variable ww_sub: zn2z w -> zn2z w -> zn2z w. - Variable w_digits : positive. - Variable ww_Digits : positive. - Variable w_zdigits : w. - Variable ww_zdigits : zn2z w. - Variable low: zn2z w -> w. - - Definition ww_head0 x := - match x with - | W0 => ww_zdigits - | WW xh xl => - match w_compare w_0 xh with - | Eq => w_add w_zdigits (w_head0 xl) - | _ => w_0W (w_head0 xh) - end - end. - - - Definition ww_tail0 x := - match x with - | W0 => ww_zdigits - | WW xh xl => - match w_compare w_0 xl with - | Eq => w_add w_zdigits (w_tail0 xh) - | _ => w_0W (w_tail0 xl) - end - end. - - - (* 0 < p < ww_digits *) - Definition ww_add_mul_div p x y := - let zdigits := w_0W w_zdigits in - match x, y with - | W0, W0 => W0 - | W0, WW yh yl => - match ww_compare p zdigits with - | Eq => w_0W yh - | Lt => w_0W (w_add_mul_div (low p) w_0 yh) - | Gt => - let n := low (ww_sub p zdigits) in - w_WW (w_add_mul_div n w_0 yh) (w_add_mul_div n yh yl) - end - | WW xh xl, W0 => - match ww_compare p zdigits with - | Eq => w_W0 xl - | Lt => w_WW (w_add_mul_div (low p) xh xl) (w_add_mul_div (low p) xl w_0) - | Gt => - let n := low (ww_sub p zdigits) in - w_W0 (w_add_mul_div n xl w_0) - end - | WW xh xl, WW yh yl => - match ww_compare p zdigits with - | Eq => w_WW xl yh - | Lt => w_WW (w_add_mul_div (low p) xh xl) (w_add_mul_div (low p) xl yh) - | Gt => - let n := low (ww_sub p zdigits) in - w_WW (w_add_mul_div n xl yh) (w_add_mul_div n yh yl) - end - end. - - Section DoubleProof. - Variable w_to_Z : w -> Z. - - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99). - - Variable spec_w_0 : [|w_0|] = 0. - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - Variable spec_to_w_Z : forall x, 0 <= [[x]] < wwB. - Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. - Variable spec_w_W0 : forall h, [[w_W0 h]] = [|h|] * wB. - Variable spec_w_0W : forall l, [[w_0W l]] = [|l|]. - Variable spec_compare : forall x y, - w_compare x y = Z.compare [|x|] [|y|]. - Variable spec_ww_compare : forall x y, - ww_compare x y = Z.compare [[x]] [[y]]. - Variable spec_ww_digits : ww_Digits = xO w_digits. - Variable spec_w_head00 : forall x, [|x|] = 0 -> [|w_head0 x|] = Zpos w_digits. - Variable spec_w_head0 : forall x, 0 < [|x|] -> - wB/ 2 <= 2 ^ ([|w_head0 x|]) * [|x|] < wB. - Variable spec_w_tail00 : forall x, [|x|] = 0 -> [|w_tail0 x|] = Zpos w_digits. - Variable spec_w_tail0 : forall x, 0 < [|x|] -> - exists y, 0 <= y /\ [|x|] = (2* y + 1) * (2 ^ [|w_tail0 x|]). - Variable spec_w_add_mul_div : forall x y p, - [|p|] <= Zpos w_digits -> - [| w_add_mul_div p x y |] = - ([|x|] * (2 ^ [|p|]) + - [|y|] / (2 ^ ((Zpos w_digits) - [|p|]))) mod wB. - Variable spec_w_add: forall x y, - [[w_add x y]] = [|x|] + [|y|]. - Variable spec_ww_sub: forall x y, - [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB. - - Variable spec_zdigits : [| w_zdigits |] = Zpos w_digits. - Variable spec_low: forall x, [| low x|] = [[x]] mod wB. - - Variable spec_ww_zdigits : [[ww_zdigits]] = Zpos ww_Digits. - - Hint Resolve div_le_0 div_lt w_to_Z_wwB: lift. - Ltac zarith := auto with zarith lift. - - Lemma spec_ww_head00 : forall x, [[x]] = 0 -> [[ww_head0 x]] = Zpos ww_Digits. - Proof. - intros x; case x; unfold ww_head0. - intros HH; rewrite spec_ww_zdigits; auto. - intros xh xl; simpl; intros Hx. - case (spec_to_Z xh); intros Hx1 Hx2. - case (spec_to_Z xl); intros Hy1 Hy2. - assert (F1: [|xh|] = 0). - { Z.le_elim Hy1; auto. - - absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. - apply Z.lt_le_trans with (1 := Hy1); auto with zarith. - pattern [|xl|] at 1; rewrite <- (Z.add_0_l [|xl|]). - apply Z.add_le_mono_r; auto with zarith. - - Z.le_elim Hx1; auto. - absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. - rewrite <- Hy1; rewrite Z.add_0_r; auto with zarith. - apply Z.mul_pos_pos; auto with zarith. } - rewrite spec_compare. case Z.compare_spec. - intros H; simpl. - rewrite spec_w_add; rewrite spec_w_head00. - rewrite spec_zdigits; rewrite spec_ww_digits. - rewrite Pos2Z.inj_xO; auto with zarith. - rewrite F1 in Hx; auto with zarith. - rewrite spec_w_0; auto with zarith. - rewrite spec_w_0; auto with zarith. - Qed. - - Lemma spec_ww_head0 : forall x, 0 < [[x]] -> - wwB/ 2 <= 2 ^ [[ww_head0 x]] * [[x]] < wwB. - Proof. - clear spec_ww_zdigits. - rewrite wwB_div_2;rewrite Z.mul_comm;rewrite wwB_wBwB. - assert (U:= lt_0_wB w_digits); destruct x as [ |xh xl];simpl ww_to_Z;intros H. - unfold Z.lt in H;discriminate H. - rewrite spec_compare, spec_w_0. case Z.compare_spec; intros H0. - rewrite <- H0 in *. simpl Z.add. simpl in H. - case (spec_to_Z w_zdigits); - case (spec_to_Z (w_head0 xl)); intros HH1 HH2 HH3 HH4. - rewrite spec_w_add. - rewrite spec_zdigits; rewrite Zpower_exp; auto with zarith. - case (spec_w_head0 H); intros H1 H2. - rewrite Z.pow_2_r; fold wB; rewrite <- Z.mul_assoc; split. - apply Z.mul_le_mono_nonneg_l; auto with zarith. - apply Z.mul_lt_mono_pos_l; auto with zarith. - assert (H1 := spec_w_head0 H0). - rewrite spec_w_0W. - split. - rewrite Z.mul_add_distr_l;rewrite Z.mul_assoc. - apply Z.le_trans with (2 ^ [|w_head0 xh|] * [|xh|] * wB). - rewrite Z.mul_comm; zarith. - assert (0 <= 2 ^ [|w_head0 xh|] * [|xl|]);zarith. - assert (H2:=spec_to_Z xl);apply Z.mul_nonneg_nonneg;zarith. - case (spec_to_Z (w_head0 xh)); intros H2 _. - generalize ([|w_head0 xh|]) H1 H2;clear H1 H2; - intros p H1 H2. - assert (Eq1 : 2^p < wB). - rewrite <- (Z.mul_1_r (2^p));apply Z.le_lt_trans with (2^p*[|xh|]);zarith. - assert (Eq2: p < Zpos w_digits). - destruct (Z.le_gt_cases (Zpos w_digits) p);trivial;contradict Eq1. - apply Z.le_ngt;unfold base;apply Zpower_le_monotone;zarith. - assert (Zpos w_digits = p + (Zpos w_digits - p)). ring. - rewrite Z.pow_2_r. - unfold base at 2;rewrite H3;rewrite Zpower_exp;zarith. - rewrite <- Z.mul_assoc; apply Z.mul_lt_mono_pos_l; zarith. - rewrite <- (Z.add_0_r (2^(Zpos w_digits - p)*wB));apply beta_lex_inv;zarith. - apply Z.mul_lt_mono_pos_r with (2 ^ p); zarith. - rewrite <- Zpower_exp;zarith. - rewrite Z.mul_comm;ring_simplify (Zpos w_digits - p + p);fold wB;zarith. - assert (H1 := spec_to_Z xh);zarith. - Qed. - - Lemma spec_ww_tail00 : forall x, [[x]] = 0 -> [[ww_tail0 x]] = Zpos ww_Digits. - Proof. - intros x; case x; unfold ww_tail0. - intros HH; rewrite spec_ww_zdigits; auto. - intros xh xl; simpl; intros Hx. - case (spec_to_Z xh); intros Hx1 Hx2. - case (spec_to_Z xl); intros Hy1 Hy2. - assert (F1: [|xh|] = 0). - { Z.le_elim Hy1; auto. - - absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. - apply Z.lt_le_trans with (1 := Hy1); auto with zarith. - pattern [|xl|] at 1; rewrite <- (Z.add_0_l [|xl|]). - apply Z.add_le_mono_r; auto with zarith. - - Z.le_elim Hx1; auto. - absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. - rewrite <- Hy1; rewrite Z.add_0_r; auto with zarith. - apply Z.mul_pos_pos; auto with zarith. } - assert (F2: [|xl|] = 0). - rewrite F1 in Hx; auto with zarith. - rewrite spec_compare; case Z.compare_spec. - intros H; simpl. - rewrite spec_w_add; rewrite spec_w_tail00; auto. - rewrite spec_zdigits; rewrite spec_ww_digits. - rewrite Pos2Z.inj_xO; auto with zarith. - rewrite spec_w_0; auto with zarith. - rewrite spec_w_0; auto with zarith. - Qed. - - Lemma spec_ww_tail0 : forall x, 0 < [[x]] -> - exists y, 0 <= y /\ [[x]] = (2 * y + 1) * 2 ^ [[ww_tail0 x]]. - Proof. - clear spec_ww_zdigits. - destruct x as [ |xh xl];simpl ww_to_Z;intros H. - unfold Z.lt in H;discriminate H. - rewrite spec_compare, spec_w_0. case Z.compare_spec; intros H0. - rewrite <- H0; rewrite Z.add_0_r. - case (spec_to_Z (w_tail0 xh)); intros HH1 HH2. - generalize H; rewrite <- H0; rewrite Z.add_0_r; clear H; intros H. - case (@spec_w_tail0 xh). - apply Z.mul_lt_mono_pos_r with wB; auto with zarith. - unfold base; auto with zarith. - intros z (Hz1, Hz2); exists z; split; auto. - rewrite spec_w_add; rewrite (fun x => Z.add_comm [|x|]). - rewrite spec_zdigits; rewrite Zpower_exp; auto with zarith. - rewrite Z.mul_assoc; rewrite <- Hz2; auto. - - case (spec_to_Z (w_tail0 xh)); intros HH1 HH2. - case (spec_w_tail0 H0); intros z (Hz1, Hz2). - assert (Hp: [|w_tail0 xl|] < Zpos w_digits). - case (Z.le_gt_cases (Zpos w_digits) [|w_tail0 xl|]); auto; intros H1. - absurd (2 ^ (Zpos w_digits) <= 2 ^ [|w_tail0 xl|]). - apply Z.lt_nge. - case (spec_to_Z xl); intros HH3 HH4. - apply Z.le_lt_trans with (2 := HH4). - apply Z.le_trans with (1 * 2 ^ [|w_tail0 xl|]); auto with zarith. - rewrite Hz2. - apply Z.mul_le_mono_nonneg_r; auto with zarith. - apply Zpower_le_monotone; auto with zarith. - exists ([|xh|] * (2 ^ ((Zpos w_digits - [|w_tail0 xl|]) - 1)) + z); split. - apply Z.add_nonneg_nonneg; auto. - apply Z.mul_nonneg_nonneg; auto with zarith. - case (spec_to_Z xh); auto. - rewrite spec_w_0W. - rewrite (Z.mul_add_distr_l 2); rewrite <- Z.add_assoc. - rewrite Z.mul_add_distr_r; rewrite <- Hz2. - apply f_equal2 with (f := Z.add); auto. - rewrite (Z.mul_comm 2). - repeat rewrite <- Z.mul_assoc. - apply f_equal2 with (f := Z.mul); auto. - case (spec_to_Z (w_tail0 xl)); intros HH3 HH4. - pattern 2 at 2; rewrite <- Z.pow_1_r. - lazy beta; repeat rewrite <- Zpower_exp; auto with zarith. - unfold base; apply f_equal with (f := Z.pow 2); auto with zarith. - - contradict H0; case (spec_to_Z xl); auto with zarith. - Qed. - - Hint Rewrite Zdiv_0_l Z.mul_0_l Z.add_0_l Z.mul_0_r Z.add_0_r - spec_w_W0 spec_w_0W spec_w_WW spec_w_0 - (wB_div w_digits w_to_Z spec_to_Z) - (wB_div_plus w_digits w_to_Z spec_to_Z) : w_rewrite. - Ltac w_rewrite := autorewrite with w_rewrite;trivial. - - Lemma spec_ww_add_mul_div_aux : forall xh xl yh yl p, - let zdigits := w_0W w_zdigits in - [[p]] <= Zpos (xO w_digits) -> - [[match ww_compare p zdigits with - | Eq => w_WW xl yh - | Lt => w_WW (w_add_mul_div (low p) xh xl) - (w_add_mul_div (low p) xl yh) - | Gt => - let n := low (ww_sub p zdigits) in - w_WW (w_add_mul_div n xl yh) (w_add_mul_div n yh yl) - end]] = - ([[WW xh xl]] * (2^[[p]]) + - [[WW yh yl]] / (2^(Zpos (xO w_digits) - [[p]]))) mod wwB. - Proof. - clear spec_ww_zdigits. - intros xh xl yh yl p zdigits;assert (HwwB := wwB_pos w_digits). - case (spec_to_w_Z p); intros Hv1 Hv2. - replace (Zpos (xO w_digits)) with (Zpos w_digits + Zpos w_digits). - 2 : rewrite Pos2Z.inj_xO;ring. - replace (Zpos w_digits + Zpos w_digits - [[p]]) with - (Zpos w_digits + (Zpos w_digits - [[p]])). 2:ring. - intros Hp; assert (Hxh := spec_to_Z xh);assert (Hxl:=spec_to_Z xl); - assert (Hx := spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xh xl)); - simpl in Hx;assert (Hyh := spec_to_Z yh);assert (Hyl:=spec_to_Z yl); - assert (Hy:=spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW yh yl));simpl in Hy. - rewrite spec_ww_compare; case Z.compare_spec; intros H1. - rewrite H1; unfold zdigits; rewrite spec_w_0W. - rewrite spec_zdigits; rewrite Z.sub_diag; rewrite Z.add_0_r. - simpl ww_to_Z; w_rewrite;zarith. - fold wB. - rewrite Z.mul_add_distr_r;rewrite <- Z.mul_assoc;rewrite <- Z.add_assoc. - rewrite <- Z.pow_2_r. - rewrite <- wwB_wBwB;apply Zmod_unique with [|xh|]. - exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xl yh)). ring. - simpl ww_to_Z; w_rewrite;zarith. - assert (HH0: [|low p|] = [[p]]). - rewrite spec_low. - apply Zmod_small. - case (spec_to_w_Z p); intros HH1 HH2; split; auto. - generalize H1; unfold zdigits; rewrite spec_w_0W; - rewrite spec_zdigits; intros tmp. - apply Z.lt_le_trans with (1 := tmp). - unfold base. - apply Zpower2_le_lin; auto with zarith. - 2: generalize H1; unfold zdigits; rewrite spec_w_0W; - rewrite spec_zdigits; auto with zarith. - generalize H1; unfold zdigits; rewrite spec_w_0W; - rewrite spec_zdigits; auto; clear H1; intros H1. - assert (HH: [|low p|] <= Zpos w_digits). - rewrite HH0; auto with zarith. - repeat rewrite spec_w_add_mul_div with (1 := HH). - rewrite HH0. - rewrite Z.mul_add_distr_r. - pattern ([|xl|] * 2 ^ [[p]]) at 2; - rewrite shift_unshift_mod with (n:= Zpos w_digits);fold wB;zarith. - replace ([|xh|] * wB * 2^[[p]]) with ([|xh|] * 2^[[p]] * wB). 2:ring. - rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. rewrite <- Z.add_assoc. - unfold base at 5;rewrite <- Zmod_shift_r;zarith. - unfold base;rewrite Zmod_shift_r with (b:= Zpos (ww_digits w_digits)); - fold wB;fold wwB;zarith. - rewrite wwB_wBwB;rewrite Z.pow_2_r; rewrite Zmult_mod_distr_r;zarith. - unfold ww_digits;rewrite Pos2Z.inj_xO;zarith. apply Z_mod_lt;zarith. - split;zarith. apply Zdiv_lt_upper_bound;zarith. - rewrite <- Zpower_exp;zarith. - ring_simplify ([[p]] + (Zpos w_digits - [[p]]));fold wB;zarith. - assert (Hv: [[p]] > Zpos w_digits). - generalize H1; clear H1. - unfold zdigits; rewrite spec_w_0W; rewrite spec_zdigits; auto with zarith. - clear H1. - assert (HH0: [|low (ww_sub p zdigits)|] = [[p]] - Zpos w_digits). - rewrite spec_low. - rewrite spec_ww_sub. - unfold zdigits; rewrite spec_w_0W; rewrite spec_zdigits. - rewrite <- Zmod_div_mod; auto with zarith. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - apply Z.le_lt_trans with (Zpos w_digits); auto with zarith. - unfold base; apply Zpower2_lt_lin; auto with zarith. - exists wB; unfold base. - unfold ww_digits; rewrite (Pos2Z.inj_xO w_digits). - rewrite <- Zpower_exp; auto with zarith. - apply f_equal with (f := fun x => 2 ^ x); auto with zarith. - assert (HH: [|low (ww_sub p zdigits)|] <= Zpos w_digits). - rewrite HH0; auto with zarith. - replace (Zpos w_digits + (Zpos w_digits - [[p]])) with - (Zpos w_digits - ([[p]] - Zpos w_digits)); zarith. - lazy zeta; simpl ww_to_Z; w_rewrite;zarith. - repeat rewrite spec_w_add_mul_div;zarith. - rewrite HH0. - pattern wB at 5;replace wB with - (2^(([[p]] - Zpos w_digits) - + (Zpos w_digits - ([[p]] - Zpos w_digits)))). - rewrite Zpower_exp;zarith. rewrite Z.mul_assoc. - rewrite Z_div_plus_l;zarith. - rewrite shift_unshift_mod with (a:= [|yh|]) (p:= [[p]] - Zpos w_digits) - (n := Zpos w_digits);zarith. fold wB. - set (u := [[p]] - Zpos w_digits). - replace [[p]] with (u + Zpos w_digits);zarith. - rewrite Zpower_exp;zarith. rewrite Z.mul_assoc. fold wB. - repeat rewrite Z.add_assoc. rewrite <- Z.mul_add_distr_r. - repeat rewrite <- Z.add_assoc. - unfold base;rewrite Zmod_shift_r with (b:= Zpos (ww_digits w_digits)); - fold wB;fold wwB;zarith. - unfold base;rewrite Zmod_shift_r with (a:= Zpos w_digits) - (b:= Zpos w_digits);fold wB;fold wwB;zarith. - rewrite wwB_wBwB; rewrite Z.pow_2_r; rewrite Zmult_mod_distr_r;zarith. - rewrite Z.mul_add_distr_r. - replace ([|xh|] * wB * 2 ^ u) with - ([|xh|]*2^u*wB). 2:ring. - repeat rewrite <- Z.add_assoc. - rewrite (Z.add_comm ([|xh|] * 2 ^ u * wB)). - rewrite Z_mod_plus;zarith. rewrite Z_mod_mult;zarith. - unfold base;rewrite <- Zmod_shift_r;zarith. fold base;apply Z_mod_lt;zarith. - unfold u; split;zarith. - split;zarith. unfold u; apply Zdiv_lt_upper_bound;zarith. - rewrite <- Zpower_exp;zarith. - fold u. - ring_simplify (u + (Zpos w_digits - u)); fold - wB;zarith. unfold ww_digits;rewrite Pos2Z.inj_xO;zarith. - unfold base;rewrite <- Zmod_shift_r;zarith. fold base;apply Z_mod_lt;zarith. - unfold u; split;zarith. - unfold u; split;zarith. - apply Zdiv_lt_upper_bound;zarith. - rewrite <- Zpower_exp;zarith. - fold u. - ring_simplify (u + (Zpos w_digits - u)); fold wB; auto with zarith. - unfold u;zarith. - unfold u;zarith. - set (u := [[p]] - Zpos w_digits). - ring_simplify (u + (Zpos w_digits - u)); fold wB; auto with zarith. - Qed. - - Lemma spec_ww_add_mul_div : forall x y p, - [[p]] <= Zpos (xO w_digits) -> - [[ ww_add_mul_div p x y ]] = - ([[x]] * (2^[[p]]) + - [[y]] / (2^(Zpos (xO w_digits) - [[p]]))) mod wwB. - Proof. - clear spec_ww_zdigits. - intros x y p H. - destruct x as [ |xh xl]; - [assert (H1 := @spec_ww_add_mul_div_aux w_0 w_0) - |assert (H1 := @spec_ww_add_mul_div_aux xh xl)]; - (destruct y as [ |yh yl]; - [generalize (H1 w_0 w_0 p H) | generalize (H1 yh yl p H)]; - clear H1;w_rewrite);simpl ww_add_mul_div. - replace [[WW w_0 w_0]] with 0;[w_rewrite|simpl;w_rewrite;trivial]. - intros Heq;rewrite <- Heq;clear Heq; auto. - rewrite spec_ww_compare. case Z.compare_spec; intros H1; w_rewrite. - rewrite (spec_w_add_mul_div w_0 w_0);w_rewrite;zarith. - generalize H1; w_rewrite; rewrite spec_zdigits; clear H1; intros H1. - assert (HH0: [|low p|] = [[p]]). - rewrite spec_low. - apply Zmod_small. - case (spec_to_w_Z p); intros HH1 HH2; split; auto. - apply Z.lt_le_trans with (1 := H1). - unfold base; apply Zpower2_le_lin; auto with zarith. - rewrite HH0; auto with zarith. - replace [[WW w_0 w_0]] with 0;[w_rewrite|simpl;w_rewrite;trivial]. - intros Heq;rewrite <- Heq;clear Heq. - generalize (spec_ww_compare p (w_0W w_zdigits)); - case ww_compare; intros H1; w_rewrite. - rewrite (spec_w_add_mul_div w_0 w_0);w_rewrite;zarith. - rewrite Pos2Z.inj_xO in H;zarith. - assert (HH: [|low (ww_sub p (w_0W w_zdigits)) |] = [[p]] - Zpos w_digits). - symmetry in H1; change ([[p]] > [[w_0W w_zdigits]]) in H1. - revert H1. - rewrite spec_low. - rewrite spec_ww_sub; w_rewrite; intros H1. - rewrite <- Zmod_div_mod; auto with zarith. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - apply Z.le_lt_trans with (Zpos w_digits); auto with zarith. - unfold base; apply Zpower2_lt_lin; auto with zarith. - unfold base; auto with zarith. - unfold base; auto with zarith. - exists wB; unfold base. - unfold ww_digits; rewrite (Pos2Z.inj_xO w_digits). - rewrite <- Zpower_exp; auto with zarith. - apply f_equal with (f := fun x => 2 ^ x); auto with zarith. - case (spec_to_Z xh); auto with zarith. - Qed. - - End DoubleProof. - -End DoubleLift. - diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v deleted file mode 100644 index b9901390..00000000 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v +++ /dev/null @@ -1,621 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Set Implicit Arguments. - -Require Import ZArith. -Require Import BigNumPrelude. -Require Import DoubleType. -Require Import DoubleBase. - -Local Open Scope Z_scope. - -Section DoubleMul. - Variable w : Type. - Variable w_0 : w. - Variable w_1 : w. - Variable w_WW : w -> w -> zn2z w. - Variable w_W0 : w -> zn2z w. - Variable w_0W : w -> zn2z w. - Variable w_compare : w -> w -> comparison. - Variable w_succ : w -> w. - Variable w_add_c : w -> w -> carry w. - Variable w_add : w -> w -> w. - Variable w_sub: w -> w -> w. - Variable w_mul_c : w -> w -> zn2z w. - Variable w_mul : w -> w -> w. - Variable w_square_c : w -> zn2z w. - Variable ww_add_c : zn2z w -> zn2z w -> carry (zn2z w). - Variable ww_add : zn2z w -> zn2z w -> zn2z w. - Variable ww_add_carry : zn2z w -> zn2z w -> zn2z w. - Variable ww_sub_c : zn2z w -> zn2z w -> carry (zn2z w). - Variable ww_sub : zn2z w -> zn2z w -> zn2z w. - - (* ** Multiplication ** *) - - (* (xh*B+xl) (yh*B + yl) - xh*yh = hh = |hhh|hhl|B2 - xh*yl +xl*yh = cc = |cch|ccl|B - xl*yl = ll = |llh|lll - *) - - Definition double_mul_c (cross:w->w->w->w->zn2z w -> zn2z w -> w*zn2z w) x y := - match x, y with - | W0, _ => W0 - | _, W0 => W0 - | WW xh xl, WW yh yl => - let hh := w_mul_c xh yh in - let ll := w_mul_c xl yl in - let (wc,cc) := cross xh xl yh yl hh ll in - match cc with - | W0 => WW (ww_add hh (w_W0 wc)) ll - | WW cch ccl => - match ww_add_c (w_W0 ccl) ll with - | C0 l => WW (ww_add hh (w_WW wc cch)) l - | C1 l => WW (ww_add_carry hh (w_WW wc cch)) l - end - end - end. - - Definition ww_mul_c := - double_mul_c - (fun xh xl yh yl hh ll=> - match ww_add_c (w_mul_c xh yl) (w_mul_c xl yh) with - | C0 cc => (w_0, cc) - | C1 cc => (w_1, cc) - end). - - Definition w_2 := w_add w_1 w_1. - - Definition kara_prod xh xl yh yl hh ll := - match ww_add_c hh ll with - C0 m => - match w_compare xl xh with - Eq => (w_0, m) - | Lt => - match w_compare yl yh with - Eq => (w_0, m) - | Lt => (w_0, ww_sub m (w_mul_c (w_sub xh xl) (w_sub yh yl))) - | Gt => match ww_add_c m (w_mul_c (w_sub xh xl) (w_sub yl yh)) with - C1 m1 => (w_1, m1) | C0 m1 => (w_0, m1) - end - end - | Gt => - match w_compare yl yh with - Eq => (w_0, m) - | Lt => match ww_add_c m (w_mul_c (w_sub xl xh) (w_sub yh yl)) with - C1 m1 => (w_1, m1) | C0 m1 => (w_0, m1) - end - | Gt => (w_0, ww_sub m (w_mul_c (w_sub xl xh) (w_sub yl yh))) - end - end - | C1 m => - match w_compare xl xh with - Eq => (w_1, m) - | Lt => - match w_compare yl yh with - Eq => (w_1, m) - | Lt => match ww_sub_c m (w_mul_c (w_sub xh xl) (w_sub yh yl)) with - C0 m1 => (w_1, m1) | C1 m1 => (w_0, m1) - end - | Gt => match ww_add_c m (w_mul_c (w_sub xh xl) (w_sub yl yh)) with - C1 m1 => (w_2, m1) | C0 m1 => (w_1, m1) - end - end - | Gt => - match w_compare yl yh with - Eq => (w_1, m) - | Lt => match ww_add_c m (w_mul_c (w_sub xl xh) (w_sub yh yl)) with - C1 m1 => (w_2, m1) | C0 m1 => (w_1, m1) - end - | Gt => match ww_sub_c m (w_mul_c (w_sub xl xh) (w_sub yl yh)) with - C1 m1 => (w_0, m1) | C0 m1 => (w_1, m1) - end - end - end - end. - - Definition ww_karatsuba_c := double_mul_c kara_prod. - - Definition ww_mul x y := - match x, y with - | W0, _ => W0 - | _, W0 => W0 - | WW xh xl, WW yh yl => - let ccl := w_add (w_mul xh yl) (w_mul xl yh) in - ww_add (w_W0 ccl) (w_mul_c xl yl) - end. - - Definition ww_square_c x := - match x with - | W0 => W0 - | WW xh xl => - let hh := w_square_c xh in - let ll := w_square_c xl in - let xhxl := w_mul_c xh xl in - let (wc,cc) := - match ww_add_c xhxl xhxl with - | C0 cc => (w_0, cc) - | C1 cc => (w_1, cc) - end in - match cc with - | W0 => WW (ww_add hh (w_W0 wc)) ll - | WW cch ccl => - match ww_add_c (w_W0 ccl) ll with - | C0 l => WW (ww_add hh (w_WW wc cch)) l - | C1 l => WW (ww_add_carry hh (w_WW wc cch)) l - end - end - end. - - Section DoubleMulAddn1. - Variable w_mul_add : w -> w -> w -> w * w. - - Fixpoint double_mul_add_n1 (n:nat) : word w n -> w -> w -> w * word w n := - match n return word w n -> w -> w -> w * word w n with - | O => w_mul_add - | S n1 => - let mul_add := double_mul_add_n1 n1 in - fun x y r => - match x with - | W0 => (w_0,extend w_0W n1 r) - | WW xh xl => - let (rl,l) := mul_add xl y r in - let (rh,h) := mul_add xh y rl in - (rh, double_WW w_WW n1 h l) - end - end. - - End DoubleMulAddn1. - - Section DoubleMulAddmn1. - Variable wn: Type. - Variable extend_n : w -> wn. - Variable wn_0W : wn -> zn2z wn. - Variable wn_WW : wn -> wn -> zn2z wn. - Variable w_mul_add_n1 : wn -> w -> w -> w*wn. - Fixpoint double_mul_add_mn1 (m:nat) : - word wn m -> w -> w -> w*word wn m := - match m return word wn m -> w -> w -> w*word wn m with - | O => w_mul_add_n1 - | S m1 => - let mul_add := double_mul_add_mn1 m1 in - fun x y r => - match x with - | W0 => (w_0,extend wn_0W m1 (extend_n r)) - | WW xh xl => - let (rl,l) := mul_add xl y r in - let (rh,h) := mul_add xh y rl in - (rh, double_WW wn_WW m1 h l) - end - end. - - End DoubleMulAddmn1. - - Definition w_mul_add x y r := - match w_mul_c x y with - | W0 => (w_0, r) - | WW h l => - match w_add_c l r with - | C0 lr => (h,lr) - | C1 lr => (w_succ h, lr) - end - end. - - - (*Section DoubleProof. *) - Variable w_digits : positive. - Variable w_to_Z : w -> Z. - - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[+| c |]" := - (interp_carry 1 wB w_to_Z c) (at level 0, c at level 99). - Notation "[-| c |]" := - (interp_carry (-1) wB w_to_Z c) (at level 0, c at level 99). - - Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99). - Notation "[+[ c ]]" := - (interp_carry 1 wwB (ww_to_Z w_digits w_to_Z) c) - (at level 0, c at level 99). - Notation "[-[ c ]]" := - (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) - (at level 0, c at level 99). - - Notation "[|| x ||]" := - (zn2z_to_Z wwB (ww_to_Z w_digits w_to_Z) x) (at level 0, x at level 99). - - Notation "[! n | x !]" := (double_to_Z w_digits w_to_Z n x) - (at level 0, x at level 99). - - Variable spec_more_than_1_digit: 1 < Zpos w_digits. - Variable spec_w_0 : [|w_0|] = 0. - Variable spec_w_1 : [|w_1|] = 1. - - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - - Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. - Variable spec_w_W0 : forall h, [[w_W0 h]] = [|h|] * wB. - Variable spec_w_0W : forall l, [[w_0W l]] = [|l|]. - Variable spec_w_compare : - forall x y, w_compare x y = Z.compare [|x|] [|y|]. - Variable spec_w_succ : forall x, [|w_succ x|] = ([|x|] + 1) mod wB. - Variable spec_w_add_c : forall x y, [+|w_add_c x y|] = [|x|] + [|y|]. - Variable spec_w_add : forall x y, [|w_add x y|] = ([|x|] + [|y|]) mod wB. - Variable spec_w_sub : forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB. - - Variable spec_w_mul_c : forall x y, [[ w_mul_c x y ]] = [|x|] * [|y|]. - Variable spec_w_mul : forall x y, [|w_mul x y|] = ([|x|] * [|y|]) mod wB. - Variable spec_w_square_c : forall x, [[ w_square_c x]] = [|x|] * [|x|]. - - Variable spec_ww_add_c : forall x y, [+[ww_add_c x y]] = [[x]] + [[y]]. - Variable spec_ww_add : forall x y, [[ww_add x y]] = ([[x]] + [[y]]) mod wwB. - Variable spec_ww_add_carry : - forall x y, [[ww_add_carry x y]] = ([[x]] + [[y]] + 1) mod wwB. - Variable spec_ww_sub : forall x y, [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB. - Variable spec_ww_sub_c : forall x y, [-[ww_sub_c x y]] = [[x]] - [[y]]. - - - Lemma spec_ww_to_Z : forall x, 0 <= [[x]] < wwB. - Proof. intros x;apply spec_ww_to_Z;auto. Qed. - - Lemma spec_ww_to_Z_wBwB : forall x, 0 <= [[x]] < wB^2. - Proof. rewrite <- wwB_wBwB;apply spec_ww_to_Z. Qed. - - Hint Resolve spec_ww_to_Z spec_ww_to_Z_wBwB : mult. - Ltac zarith := auto with zarith mult. - - Lemma wBwB_lex: forall a b c d, - a * wB^2 + [[b]] <= c * wB^2 + [[d]] -> - a <= c. - Proof. - intros a b c d H; apply beta_lex with [[b]] [[d]] (wB^2);zarith. - Qed. - - Lemma wBwB_lex_inv: forall a b c d, - a < c -> - a * wB^2 + [[b]] < c * wB^2 + [[d]]. - Proof. - intros a b c d H; apply beta_lex_inv; zarith. - Qed. - - Lemma sum_mul_carry : forall xh xl yh yl wc cc, - [|wc|]*wB^2 + [[cc]] = [|xh|] * [|yl|] + [|xl|] * [|yh|] -> - 0 <= [|wc|] <= 1. - Proof. - intros. - apply (sum_mul_carry [|xh|] [|xl|] [|yh|] [|yl|] [|wc|][[cc]] wB);zarith. - apply wB_pos. - Qed. - - Theorem mult_add_ineq: forall xH yH crossH, - 0 <= [|xH|] * [|yH|] + [|crossH|] < wwB. - Proof. - intros;rewrite wwB_wBwB;apply mult_add_ineq;zarith. - Qed. - - Hint Resolve mult_add_ineq : mult. - - Lemma spec_mul_aux : forall xh xl yh yl wc (cc:zn2z w) hh ll, - [[hh]] = [|xh|] * [|yh|] -> - [[ll]] = [|xl|] * [|yl|] -> - [|wc|]*wB^2 + [[cc]] = [|xh|] * [|yl|] + [|xl|] * [|yh|] -> - [||match cc with - | W0 => WW (ww_add hh (w_W0 wc)) ll - | WW cch ccl => - match ww_add_c (w_W0 ccl) ll with - | C0 l => WW (ww_add hh (w_WW wc cch)) l - | C1 l => WW (ww_add_carry hh (w_WW wc cch)) l - end - end||] = ([|xh|] * wB + [|xl|]) * ([|yh|] * wB + [|yl|]). - Proof. - intros;assert (U1 := wB_pos w_digits). - replace (([|xh|] * wB + [|xl|]) * ([|yh|] * wB + [|yl|])) with - ([|xh|]*[|yh|]*wB^2 + ([|xh|]*[|yl|] + [|xl|]*[|yh|])*wB + [|xl|]*[|yl|]). - 2:ring. rewrite <- H1;rewrite <- H;rewrite <- H0. - assert (H2 := sum_mul_carry _ _ _ _ _ _ H1). - destruct cc as [ | cch ccl]; simpl zn2z_to_Z; simpl ww_to_Z. - rewrite spec_ww_add;rewrite spec_w_W0;rewrite Zmod_small; - rewrite wwB_wBwB. ring. - rewrite <- (Z.add_0_r ([|wc|]*wB));rewrite H;apply mult_add_ineq3;zarith. - simpl ww_to_Z in H1. assert (U:=spec_to_Z cch). - assert ([|wc|]*wB + [|cch|] <= 2*wB - 3). - destruct (Z_le_gt_dec ([|wc|]*wB + [|cch|]) (2*wB - 3)) as [Hle|Hgt];trivial. - assert ([|xh|] * [|yl|] + [|xl|] * [|yh|] <= (2*wB - 4)*wB + 2). - ring_simplify ((2*wB - 4)*wB + 2). - assert (H4 := Zmult_lt_b _ _ _ (spec_to_Z xh) (spec_to_Z yl)). - assert (H5 := Zmult_lt_b _ _ _ (spec_to_Z xl) (spec_to_Z yh)). - omega. - generalize H3;clear H3;rewrite <- H1. - rewrite Z.add_assoc; rewrite Z.pow_2_r; rewrite Z.mul_assoc; - rewrite <- Z.mul_add_distr_r. - assert (((2 * wB - 4) + 2)*wB <= ([|wc|] * wB + [|cch|])*wB). - apply Z.mul_le_mono_nonneg;zarith. - rewrite Z.mul_add_distr_r in H3. - intros. assert (U2 := spec_to_Z ccl);omega. - generalize (spec_ww_add_c (w_W0 ccl) ll);destruct (ww_add_c (w_W0 ccl) ll) - as [l|l];unfold interp_carry;rewrite spec_w_W0;try rewrite Z.mul_1_l; - simpl zn2z_to_Z; - try rewrite spec_ww_add;try rewrite spec_ww_add_carry;rewrite spec_w_WW; - rewrite Zmod_small;rewrite wwB_wBwB;intros. - rewrite H4;ring. rewrite H;apply mult_add_ineq2;zarith. - rewrite Z.add_assoc;rewrite Z.mul_add_distr_r. - rewrite Z.mul_1_l;rewrite <- Z.add_assoc;rewrite H4;ring. - repeat rewrite <- Z.add_assoc;rewrite H;apply mult_add_ineq2;zarith. - Qed. - - Lemma spec_double_mul_c : forall cross:w->w->w->w->zn2z w -> zn2z w -> w*zn2z w, - (forall xh xl yh yl hh ll, - [[hh]] = [|xh|]*[|yh|] -> - [[ll]] = [|xl|]*[|yl|] -> - let (wc,cc) := cross xh xl yh yl hh ll in - [|wc|]*wwB + [[cc]] = [|xh|]*[|yl|] + [|xl|]*[|yh|]) -> - forall x y, [||double_mul_c cross x y||] = [[x]] * [[y]]. - Proof. - intros cross Hcross x y;destruct x as [ |xh xl];simpl;trivial. - destruct y as [ |yh yl];simpl. rewrite Z.mul_0_r;trivial. - assert (H1:= spec_w_mul_c xh yh);assert (H2:= spec_w_mul_c xl yl). - generalize (Hcross _ _ _ _ _ _ H1 H2). - destruct (cross xh xl yh yl (w_mul_c xh yh) (w_mul_c xl yl)) as (wc,cc). - intros;apply spec_mul_aux;trivial. - rewrite <- wwB_wBwB;trivial. - Qed. - - Lemma spec_ww_mul_c : forall x y, [||ww_mul_c x y||] = [[x]] * [[y]]. - Proof. - intros x y;unfold ww_mul_c;apply spec_double_mul_c. - intros xh xl yh yl hh ll H1 H2. - generalize (spec_ww_add_c (w_mul_c xh yl) (w_mul_c xl yh)); - destruct (ww_add_c (w_mul_c xh yl) (w_mul_c xl yh)) as [c|c]; - unfold interp_carry;repeat rewrite spec_w_mul_c;intros H; - (rewrite spec_w_0 || rewrite spec_w_1);rewrite H;ring. - Qed. - - Lemma spec_w_2: [|w_2|] = 2. - unfold w_2; rewrite spec_w_add; rewrite spec_w_1; simpl. - apply Zmod_small; split; auto with zarith. - rewrite <- (Z.pow_1_r 2); unfold base; apply Zpower_lt_monotone; auto with zarith. - Qed. - - Lemma kara_prod_aux : forall xh xl yh yl, - xh*yh + xl*yl - (xh-xl)*(yh-yl) = xh*yl + xl*yh. - Proof. intros;ring. Qed. - - Lemma spec_kara_prod : forall xh xl yh yl hh ll, - [[hh]] = [|xh|]*[|yh|] -> - [[ll]] = [|xl|]*[|yl|] -> - let (wc,cc) := kara_prod xh xl yh yl hh ll in - [|wc|]*wwB + [[cc]] = [|xh|]*[|yl|] + [|xl|]*[|yh|]. - Proof. - intros xh xl yh yl hh ll H H0; rewrite <- kara_prod_aux; - rewrite <- H; rewrite <- H0; unfold kara_prod. - assert (Hxh := (spec_to_Z xh)); assert (Hxl := (spec_to_Z xl)); - assert (Hyh := (spec_to_Z yh)); assert (Hyl := (spec_to_Z yl)). - generalize (spec_ww_add_c hh ll); case (ww_add_c hh ll); - intros z Hz; rewrite <- Hz; unfold interp_carry; assert (Hz1 := (spec_ww_to_Z z)). - rewrite spec_w_compare; case Z.compare_spec; intros Hxlh; - try rewrite Hxlh; try rewrite spec_w_0; try (ring; fail). - rewrite spec_w_compare; case Z.compare_spec; intros Hylh. - rewrite Hylh; rewrite spec_w_0; try (ring; fail). - rewrite spec_w_0; try (ring; fail). - repeat (rewrite spec_ww_sub || rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - split; auto with zarith. - simpl in Hz; rewrite Hz; rewrite H; rewrite H0. - rewrite kara_prod_aux; apply Z.add_nonneg_nonneg; apply Z.mul_nonneg_nonneg; auto with zarith. - apply Z.le_lt_trans with ([[z]]-0); auto with zarith. - unfold Z.sub; apply Z.add_le_mono_l; apply Z.le_0_sub; simpl; rewrite Z.opp_involutive. - apply Z.mul_nonneg_nonneg; auto with zarith. - match goal with |- context[ww_add_c ?x ?y] => - generalize (spec_ww_add_c x y); case (ww_add_c x y); try rewrite spec_w_0; - intros z1 Hz2 - end. - simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_1; unfold interp_carry in Hz2; rewrite Hz2; - repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_compare; case Z.compare_spec; intros Hylh. - rewrite Hylh; rewrite spec_w_0; try (ring; fail). - match goal with |- context[ww_add_c ?x ?y] => - generalize (spec_ww_add_c x y); case (ww_add_c x y); try rewrite spec_w_0; - intros z1 Hz2 - end. - simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_1; unfold interp_carry in Hz2; rewrite Hz2; - repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_0; try (ring; fail). - repeat (rewrite spec_ww_sub || rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - split. - match goal with |- context[(?x - ?y) * (?z - ?t)] => - replace ((x - y) * (z - t)) with ((y - x) * (t - z)); [idtac | ring] - end. - simpl in Hz; rewrite Hz; rewrite H; rewrite H0. - rewrite kara_prod_aux; apply Z.add_nonneg_nonneg; apply Z.mul_nonneg_nonneg; auto with zarith. - apply Z.le_lt_trans with ([[z]]-0); auto with zarith. - unfold Z.sub; apply Z.add_le_mono_l; apply Z.le_0_sub; simpl; rewrite Z.opp_involutive. - apply Z.mul_nonneg_nonneg; auto with zarith. - (** there is a carry in hh + ll **) - rewrite Z.mul_1_l. - rewrite spec_w_compare; case Z.compare_spec; intros Hxlh; - try rewrite Hxlh; try rewrite spec_w_1; try (ring; fail). - rewrite spec_w_compare; case Z.compare_spec; intros Hylh; - try rewrite Hylh; try rewrite spec_w_1; try (ring; fail). - match goal with |- context[ww_sub_c ?x ?y] => - generalize (spec_ww_sub_c x y); case (ww_sub_c x y); try rewrite spec_w_1; - intros z1 Hz2 - end. - simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_0; rewrite Z.mul_0_l; rewrite Z.add_0_l. - generalize Hz2; clear Hz2; unfold interp_carry. - repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - match goal with |- context[ww_add_c ?x ?y] => - generalize (spec_ww_add_c x y); case (ww_add_c x y); try rewrite spec_w_1; - intros z1 Hz2 - end. - simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_2; unfold interp_carry in Hz2. - transitivity (wwB + (1 * wwB + [[z1]])). - ring. - rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_compare; case Z.compare_spec; intros Hylh; - try rewrite Hylh; try rewrite spec_w_1; try (ring; fail). - match goal with |- context[ww_add_c ?x ?y] => - generalize (spec_ww_add_c x y); case (ww_add_c x y); try rewrite spec_w_1; - intros z1 Hz2 - end. - simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_2; unfold interp_carry in Hz2. - transitivity (wwB + (1 * wwB + [[z1]])). - ring. - rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - match goal with |- context[ww_sub_c ?x ?y] => - generalize (spec_ww_sub_c x y); case (ww_sub_c x y); try rewrite spec_w_1; - intros z1 Hz2 - end. - simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_0; rewrite Z.mul_0_l; rewrite Z.add_0_l. - match goal with |- context[(?x - ?y) * (?z - ?t)] => - replace ((x - y) * (z - t)) with ((y - x) * (t - z)); [idtac | ring] - end. - generalize Hz2; clear Hz2; unfold interp_carry. - repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - Qed. - - Lemma sub_carry : forall xh xl yh yl z, - 0 <= z -> - [|xh|]*[|yl|] + [|xl|]*[|yh|] = wwB + z -> - z < wwB. - Proof. - intros xh xl yh yl z Hle Heq. - destruct (Z_le_gt_dec wwB z);auto with zarith. - generalize (Zmult_lt_b _ _ _ (spec_to_Z xh) (spec_to_Z yl)). - generalize (Zmult_lt_b _ _ _ (spec_to_Z xl) (spec_to_Z yh)). - rewrite <- wwB_wBwB;intros H1 H2. - assert (H3 := wB_pos w_digits). - assert (2*wB <= wwB). - rewrite wwB_wBwB; rewrite Z.pow_2_r; apply Z.mul_le_mono_nonneg;zarith. - omega. - Qed. - - Ltac Spec_ww_to_Z x := - let H:= fresh "H" in - assert (H:= spec_ww_to_Z x). - - Ltac Zmult_lt_b x y := - let H := fresh "H" in - assert (H := Zmult_lt_b _ _ _ (spec_to_Z x) (spec_to_Z y)). - - Lemma spec_ww_karatsuba_c : forall x y, [||ww_karatsuba_c x y||]=[[x]]*[[y]]. - Proof. - intros x y; unfold ww_karatsuba_c;apply spec_double_mul_c. - intros; apply spec_kara_prod; auto. - Qed. - - Lemma spec_ww_mul : forall x y, [[ww_mul x y]] = [[x]]*[[y]] mod wwB. - Proof. - assert (U:= lt_0_wB w_digits). - assert (U1:= lt_0_wwB w_digits). - intros x y; case x; auto; intros xh xl. - case y; auto. - simpl; rewrite Z.mul_0_r; rewrite Zmod_small; auto with zarith. - intros yh yl;simpl. - repeat (rewrite spec_ww_add || rewrite spec_w_W0 || rewrite spec_w_mul_c - || rewrite spec_w_add || rewrite spec_w_mul). - rewrite <- Zplus_mod; auto with zarith. - repeat (rewrite Z.mul_add_distr_r || rewrite Z.mul_add_distr_l). - rewrite <- Zmult_mod_distr_r; auto with zarith. - rewrite <- Z.pow_2_r; rewrite <- wwB_wBwB; auto with zarith. - rewrite Zplus_mod; auto with zarith. - rewrite Zmod_mod; auto with zarith. - rewrite <- Zplus_mod; auto with zarith. - match goal with |- ?X mod _ = _ => - rewrite <- Z_mod_plus with (a := X) (b := [|xh|] * [|yh|]) - end; auto with zarith. - f_equal; auto; rewrite wwB_wBwB; ring. - Qed. - - Lemma spec_ww_square_c : forall x, [||ww_square_c x||] = [[x]]*[[x]]. - Proof. - destruct x as [ |xh xl];simpl;trivial. - case_eq match ww_add_c (w_mul_c xh xl) (w_mul_c xh xl) with - | C0 cc => (w_0, cc) - | C1 cc => (w_1, cc) - end;intros wc cc Heq. - apply (spec_mul_aux xh xl xh xl wc cc);trivial. - generalize Heq (spec_ww_add_c (w_mul_c xh xl) (w_mul_c xh xl));clear Heq. - rewrite spec_w_mul_c;destruct (ww_add_c (w_mul_c xh xl) (w_mul_c xh xl)); - unfold interp_carry;try rewrite Z.mul_1_l;intros Heq Heq';inversion Heq; - rewrite (Z.mul_comm [|xl|]);subst. - rewrite spec_w_0;rewrite Z.mul_0_l;rewrite Z.add_0_l;trivial. - rewrite spec_w_1;rewrite Z.mul_1_l;rewrite <- wwB_wBwB;trivial. - Qed. - - Section DoubleMulAddn1Proof. - - Variable w_mul_add : w -> w -> w -> w * w. - Variable spec_w_mul_add : forall x y r, - let (h,l):= w_mul_add x y r in - [|h|]*wB+[|l|] = [|x|]*[|y|] + [|r|]. - - Lemma spec_double_mul_add_n1 : forall n x y r, - let (h,l) := double_mul_add_n1 w_mul_add n x y r in - [|h|]*double_wB w_digits n + [!n|l!] = [!n|x!]*[|y|]+[|r|]. - Proof. - induction n;intros x y r;trivial. - exact (spec_w_mul_add x y r). - unfold double_mul_add_n1;destruct x as[ |xh xl]; - fold(double_mul_add_n1 w_mul_add). - rewrite spec_w_0;rewrite spec_extend;simpl;trivial. - assert(H:=IHn xl y r);destruct (double_mul_add_n1 w_mul_add n xl y r)as(rl,l). - assert(U:=IHn xh y rl);destruct(double_mul_add_n1 w_mul_add n xh y rl)as(rh,h). - rewrite <- double_wB_wwB. rewrite spec_double_WW;simpl;trivial. - rewrite Z.mul_add_distr_r;rewrite <- Z.add_assoc;rewrite <- H. - rewrite Z.mul_assoc;rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - rewrite U;ring. - Qed. - - End DoubleMulAddn1Proof. - - Lemma spec_w_mul_add : forall x y r, - let (h,l):= w_mul_add x y r in - [|h|]*wB+[|l|] = [|x|]*[|y|] + [|r|]. - Proof. - intros x y r;unfold w_mul_add;assert (H:=spec_w_mul_c x y); - destruct (w_mul_c x y) as [ |h l];simpl;rewrite <- H. - rewrite spec_w_0;trivial. - assert (U:=spec_w_add_c l r);destruct (w_add_c l r) as [lr|lr];unfold - interp_carry in U;try rewrite Z.mul_1_l in H;simpl. - rewrite U;ring. rewrite spec_w_succ. rewrite Zmod_small. - rewrite <- Z.add_assoc;rewrite <- U;ring. - simpl in H;assert (H1:= Zmult_lt_b _ _ _ (spec_to_Z x) (spec_to_Z y)). - rewrite <- H in H1. - assert (H2:=spec_to_Z h);split;zarith. - case H1;clear H1;intro H1;clear H1. - replace (wB ^ 2 - 2 * wB) with ((wB - 2)*wB). 2:ring. - intros H0;assert (U1:= wB_pos w_digits). - assert (H1 := beta_lex _ _ _ _ _ H0 (spec_to_Z l));zarith. - Qed. - -(* End DoubleProof. *) - -End DoubleMul. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v deleted file mode 100644 index d07ce301..00000000 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v +++ /dev/null @@ -1,1369 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Set Implicit Arguments. - -Require Import ZArith. -Require Import BigNumPrelude. -Require Import DoubleType. -Require Import DoubleBase. - -Local Open Scope Z_scope. - -Section DoubleSqrt. - Variable w : Type. - Variable w_is_even : w -> bool. - Variable w_compare : w -> w -> comparison. - Variable w_0 : w. - Variable w_1 : w. - Variable w_Bm1 : w. - Variable w_WW : w -> w -> zn2z w. - Variable w_W0 : w -> zn2z w. - Variable w_0W : w -> zn2z w. - Variable w_sub : w -> w -> w. - Variable w_sub_c : w -> w -> carry w. - Variable w_square_c : w -> zn2z w. - Variable w_div21 : w -> w -> w -> w * w. - Variable w_add_mul_div : w -> w -> w -> w. - Variable w_digits : positive. - Variable w_zdigits : w. - Variable ww_zdigits : zn2z w. - Variable w_add_c : w -> w -> carry w. - Variable w_sqrt2 : w -> w -> w * carry w. - Variable w_pred : w -> w. - Variable ww_pred_c : zn2z w -> carry (zn2z w). - Variable ww_pred : zn2z w -> zn2z w. - Variable ww_add_c : zn2z w -> zn2z w -> carry (zn2z w). - Variable ww_add : zn2z w -> zn2z w -> zn2z w. - Variable ww_sub_c : zn2z w -> zn2z w -> carry (zn2z w). - Variable ww_add_mul_div : zn2z w -> zn2z w -> zn2z w -> zn2z w. - Variable ww_head0 : zn2z w -> zn2z w. - Variable ww_compare : zn2z w -> zn2z w -> comparison. - Variable low : zn2z w -> w. - - Let wwBm1 := ww_Bm1 w_Bm1. - - Definition ww_is_even x := - match x with - | W0 => true - | WW xh xl => w_is_even xl - end. - - Let w_div21c x y z := - match w_compare x z with - | Eq => - match w_compare y z with - Eq => (C1 w_1, w_0) - | Gt => (C1 w_1, w_sub y z) - | Lt => (C1 w_0, y) - end - | Gt => - let x1 := w_sub x z in - let (q, r) := w_div21 x1 y z in - (C1 q, r) - | Lt => - let (q, r) := w_div21 x y z in - (C0 q, r) - end. - - Let w_div2s x y s := - match x with - C1 x1 => - let x2 := w_sub x1 s in - let (q, r) := w_div21c x2 y s in - match q with - C0 q1 => - if w_is_even q1 then - (C0 (w_add_mul_div (w_pred w_zdigits) w_1 q1), C0 r) - else - (C0 (w_add_mul_div (w_pred w_zdigits) w_1 q1), w_add_c r s) - | C1 q1 => - if w_is_even q1 then - (C1 (w_add_mul_div (w_pred w_zdigits) w_0 q1), C0 r) - else - (C1 (w_add_mul_div (w_pred w_zdigits) w_0 q1), w_add_c r s) - end - | C0 x1 => - let (q, r) := w_div21c x1 y s in - match q with - C0 q1 => - if w_is_even q1 then - (C0 (w_add_mul_div (w_pred w_zdigits) w_0 q1), C0 r) - else - (C0 (w_add_mul_div (w_pred w_zdigits) w_0 q1), w_add_c r s) - | C1 q1 => - if w_is_even q1 then - (C0 (w_add_mul_div (w_pred w_zdigits) w_1 q1), C0 r) - else - (C0 (w_add_mul_div (w_pred w_zdigits) w_1 q1), w_add_c r s) - end - end. - - Definition split x := - match x with - | W0 => (w_0,w_0) - | WW h l => (h,l) - end. - - Definition ww_sqrt2 x y := - let (x1, x2) := split x in - let (y1, y2) := split y in - let ( q, r) := w_sqrt2 x1 x2 in - let (q1, r1) := w_div2s r y1 q in - match q1 with - C0 q1 => - let q2 := w_square_c q1 in - let a := WW q q1 in - match r1 with - C1 r2 => - match ww_sub_c (WW r2 y2) q2 with - C0 r3 => (a, C1 r3) - | C1 r3 => (a, C0 r3) - end - | C0 r2 => - match ww_sub_c (WW r2 y2) q2 with - C0 r3 => (a, C0 r3) - | C1 r3 => - let a2 := ww_add_mul_div (w_0W w_1) a W0 in - match ww_pred_c a2 with - C0 a3 => - (ww_pred a, ww_add_c a3 r3) - | C1 a3 => - (ww_pred a, C0 (ww_add a3 r3)) - end - end - end - | C1 q1 => - let a1 := WW q w_Bm1 in - let a2 := ww_add_mul_div (w_0W w_1) a1 wwBm1 in - (a1, ww_add_c a2 y) - end. - - Definition ww_is_zero x := - match ww_compare W0 x with - Eq => true - | _ => false - end. - - Definition ww_head1 x := - let p := ww_head0 x in - if (ww_is_even p) then p else ww_pred p. - - Definition ww_sqrt x := - if (ww_is_zero x) then W0 - else - let p := ww_head1 x in - match ww_compare p W0 with - | Gt => - match ww_add_mul_div p x W0 with - W0 => W0 - | WW x1 x2 => - let (r, _) := w_sqrt2 x1 x2 in - WW w_0 (w_add_mul_div - (w_sub w_zdigits - (low (ww_add_mul_div (ww_pred ww_zdigits) - W0 p))) w_0 r) - end - | _ => - match x with - W0 => W0 - | WW x1 x2 => WW w_0 (fst (w_sqrt2 x1 x2)) - end - end. - - - Variable w_to_Z : w -> Z. - - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[+| c |]" := - (interp_carry 1 wB w_to_Z c) (at level 0, c at level 99). - Notation "[-| c |]" := - (interp_carry (-1) wB w_to_Z c) (at level 0, c at level 99). - - Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99). - Notation "[+[ c ]]" := - (interp_carry 1 wwB (ww_to_Z w_digits w_to_Z) c) - (at level 0, c at level 99). - Notation "[-[ c ]]" := - (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) - (at level 0, c at level 99). - - Notation "[|| x ||]" := - (zn2z_to_Z wwB (ww_to_Z w_digits w_to_Z) x) (at level 0, x at level 99). - - Notation "[! n | x !]" := (double_to_Z w_digits w_to_Z n x) - (at level 0, x at level 99). - - Variable spec_w_0 : [|w_0|] = 0. - Variable spec_w_1 : [|w_1|] = 1. - Variable spec_w_Bm1 : [|w_Bm1|] = wB - 1. - Variable spec_w_zdigits : [|w_zdigits|] = Zpos w_digits. - Variable spec_more_than_1_digit: 1 < Zpos w_digits. - - Variable spec_ww_zdigits : [[ww_zdigits]] = Zpos (xO w_digits). - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - Variable spec_to_w_Z : forall x, 0 <= [[x]] < wwB. - - Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. - Variable spec_w_W0 : forall h, [[w_W0 h]] = [|h|] * wB. - Variable spec_w_0W : forall l, [[w_0W l]] = [|l|]. - Variable spec_w_is_even : forall x, - if w_is_even x then [|x|] mod 2 = 0 else [|x|] mod 2 = 1. - Variable spec_w_compare : forall x y, - w_compare x y = Z.compare [|x|] [|y|]. - Variable spec_w_sub : forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB. - Variable spec_w_square_c : forall x, [[ w_square_c x]] = [|x|] * [|x|]. - Variable spec_w_div21 : forall a1 a2 b, - wB/2 <= [|b|] -> - [|a1|] < [|b|] -> - let (q,r) := w_div21 a1 a2 b in - [|a1|] *wB+ [|a2|] = [|q|] * [|b|] + [|r|] /\ - 0 <= [|r|] < [|b|]. - Variable spec_w_add_mul_div : forall x y p, - [|p|] <= Zpos w_digits -> - [| w_add_mul_div p x y |] = - ([|x|] * (2 ^ [|p|]) + - [|y|] / (Z.pow 2 ((Zpos w_digits) - [|p|]))) mod wB. - Variable spec_ww_add_mul_div : forall x y p, - [[p]] <= Zpos (xO w_digits) -> - [[ ww_add_mul_div p x y ]] = - ([[x]] * (2^ [[p]]) + - [[y]] / (2^ (Zpos (xO w_digits) - [[p]]))) mod wwB. - Variable spec_w_add_c : forall x y, [+|w_add_c x y|] = [|x|] + [|y|]. - Variable spec_ww_add : forall x y, [[ww_add x y]] = ([[x]] + [[y]]) mod wwB. - Variable spec_w_sqrt2 : forall x y, - wB/ 4 <= [|x|] -> - let (s,r) := w_sqrt2 x y in - [[WW x y]] = [|s|] ^ 2 + [+|r|] /\ - [+|r|] <= 2 * [|s|]. - Variable spec_ww_sub_c : forall x y, [-[ww_sub_c x y]] = [[x]] - [[y]]. - Variable spec_ww_pred_c : forall x, [-[ww_pred_c x]] = [[x]] - 1. - Variable spec_pred : forall x, [|w_pred x|] = ([|x|] - 1) mod wB. - Variable spec_ww_pred : forall x, [[ww_pred x]] = ([[x]] - 1) mod wwB. - Variable spec_ww_add_c : forall x y, [+[ww_add_c x y]] = [[x]] + [[y]]. - Variable spec_ww_compare : forall x y, - ww_compare x y = Z.compare [[x]] [[y]]. - Variable spec_ww_head0 : forall x, 0 < [[x]] -> - wwB/ 2 <= 2 ^ [[ww_head0 x]] * [[x]] < wwB. - Variable spec_low: forall x, [|low x|] = [[x]] mod wB. - - Let spec_ww_Bm1 : [[wwBm1]] = wwB - 1. - Proof. refine (spec_ww_Bm1 w_Bm1 w_digits w_to_Z _);auto. Qed. - - Hint Rewrite spec_w_0 spec_w_1 spec_w_WW spec_w_sub - spec_w_add_mul_div spec_ww_Bm1 spec_w_add_c : w_rewrite. - - Lemma spec_ww_is_even : forall x, - if ww_is_even x then [[x]] mod 2 = 0 else [[x]] mod 2 = 1. -clear spec_more_than_1_digit. -intros x; case x; simpl ww_is_even. - reflexivity. - simpl. - intros w1 w2; simpl. - unfold base. - rewrite Zplus_mod; auto with zarith. - rewrite (fun x y => (Zdivide_mod (x * y))); auto with zarith. - rewrite Z.add_0_l; rewrite Zmod_mod; auto with zarith. - apply spec_w_is_even; auto with zarith. - apply Z.divide_mul_r; apply Zpower_divide; auto with zarith. - Qed. - - - Theorem spec_w_div21c : forall a1 a2 b, - wB/2 <= [|b|] -> - let (q,r) := w_div21c a1 a2 b in - [|a1|] * wB + [|a2|] = [+|q|] * [|b|] + [|r|] /\ 0 <= [|r|] < [|b|]. - intros a1 a2 b Hb; unfold w_div21c. - assert (H: 0 < [|b|]); auto with zarith. - assert (U := wB_pos w_digits). - apply Z.lt_le_trans with (2 := Hb); auto with zarith. - apply Z.lt_le_trans with 1; auto with zarith. - apply Zdiv_le_lower_bound; auto with zarith. - rewrite !spec_w_compare. repeat case Z.compare_spec. - intros H1 H2; split. - unfold interp_carry; autorewrite with w_rewrite rm10; auto with zarith. - rewrite H1; rewrite H2; ring. - autorewrite with w_rewrite; auto with zarith. - intros H1 H2; split. - unfold interp_carry; autorewrite with w_rewrite rm10; auto with zarith. - rewrite H2; ring. - destruct (spec_to_Z a2);auto with zarith. - intros H1 H2; split. - unfold interp_carry; autorewrite with w_rewrite rm10; auto with zarith. - rewrite H2; rewrite Zmod_small; auto with zarith. - ring. - destruct (spec_to_Z a2);auto with zarith. - rewrite spec_w_sub; auto with zarith. - destruct (spec_to_Z a2) as [H3 H4];auto with zarith. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - assert ([|a2|] < 2 * [|b|]); auto with zarith. - apply Z.lt_le_trans with (2 * (wB / 2)); auto with zarith. - rewrite wB_div_2; auto. - intros H1. - match goal with |- context[w_div21 ?y ?z ?t] => - generalize (@spec_w_div21 y z t Hb H1); - case (w_div21 y z t); simpl; autorewrite with w_rewrite; - auto - end. - intros H1. - assert (H2: [|w_sub a1 b|] < [|b|]). - rewrite spec_w_sub; auto with zarith. - rewrite Zmod_small; auto with zarith. - assert ([|a1|] < 2 * [|b|]); auto with zarith. - apply Z.lt_le_trans with (2 * (wB / 2)); auto with zarith. - rewrite wB_div_2; auto. - destruct (spec_to_Z a1);auto with zarith. - destruct (spec_to_Z a1);auto with zarith. - match goal with |- context[w_div21 ?y ?z ?t] => - generalize (@spec_w_div21 y z t Hb H2); - case (w_div21 y z t); autorewrite with w_rewrite; - auto - end. - intros w0 w1; replace [+|C1 w0|] with (wB + [|w0|]). - rewrite Zmod_small; auto with zarith. - intros (H3, H4); split; auto. - rewrite Z.mul_add_distr_r. - rewrite <- Z.add_assoc; rewrite <- H3; ring. - split; auto with zarith. - assert ([|a1|] < 2 * [|b|]); auto with zarith. - apply Z.lt_le_trans with (2 * (wB / 2)); auto with zarith. - rewrite wB_div_2; auto. - destruct (spec_to_Z a1);auto with zarith. - destruct (spec_to_Z a1);auto with zarith. - simpl; case wB; auto. - Qed. - - Theorem C0_id: forall p, [+|C0 p|] = [|p|]. - intros p; simpl; auto. - Qed. - - Theorem add_mult_div_2: forall w, - [|w_add_mul_div (w_pred w_zdigits) w_0 w|] = [|w|] / 2. - intros w1. - assert (Hp: [|w_pred w_zdigits|] = Zpos w_digits - 1). - rewrite spec_pred; rewrite spec_w_zdigits. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - apply Z.lt_le_trans with (Zpos w_digits); auto with zarith. - unfold base; apply Zpower2_le_lin; auto with zarith. - rewrite spec_w_add_mul_div; auto with zarith. - autorewrite with w_rewrite rm10. - match goal with |- context[?X - ?Y] => - replace (X - Y) with 1 - end. - rewrite Z.pow_1_r; rewrite Zmod_small; auto with zarith. - destruct (spec_to_Z w1) as [H1 H2];auto with zarith. - split; auto with zarith. - apply Zdiv_lt_upper_bound; auto with zarith. - rewrite Hp; ring. - Qed. - - Theorem add_mult_div_2_plus_1: forall w, - [|w_add_mul_div (w_pred w_zdigits) w_1 w|] = - [|w|] / 2 + 2 ^ Zpos (w_digits - 1). - intros w1. - assert (Hp: [|w_pred w_zdigits|] = Zpos w_digits - 1). - rewrite spec_pred; rewrite spec_w_zdigits. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - apply Z.lt_le_trans with (Zpos w_digits); auto with zarith. - unfold base; apply Zpower2_le_lin; auto with zarith. - autorewrite with w_rewrite rm10; auto with zarith. - match goal with |- context[?X - ?Y] => - replace (X - Y) with 1 - end; rewrite Hp; try ring. - rewrite Pos2Z.inj_sub_max; auto with zarith. - rewrite Z.max_r; auto with zarith. - rewrite Z.pow_1_r; rewrite Zmod_small; auto with zarith. - destruct (spec_to_Z w1) as [H1 H2];auto with zarith. - split; auto with zarith. - unfold base. - match goal with |- _ < _ ^ ?X => - assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; - rewrite <- (tmp X); clear tmp - end. - rewrite Zpower_exp; try rewrite Z.pow_1_r; auto with zarith. - assert (tmp: forall p, 1 + (p -1) - 1 = p - 1); auto with zarith; - rewrite tmp; clear tmp; auto with zarith. - match goal with |- ?X + ?Y < _ => - assert (Y < X); auto with zarith - end. - apply Zdiv_lt_upper_bound; auto with zarith. - pattern 2 at 2; rewrite <- Z.pow_1_r; rewrite <- Zpower_exp; - auto with zarith. - assert (tmp: forall p, (p - 1) + 1 = p); auto with zarith; - rewrite tmp; clear tmp; auto with zarith. - Qed. - - Theorem add_mult_mult_2: forall w, - [|w_add_mul_div w_1 w w_0|] = 2 * [|w|] mod wB. - intros w1. - autorewrite with w_rewrite rm10; auto with zarith. - rewrite Z.pow_1_r; auto with zarith. - rewrite Z.mul_comm; auto. - Qed. - - Theorem ww_add_mult_mult_2: forall w, - [[ww_add_mul_div (w_0W w_1) w W0]] = 2 * [[w]] mod wwB. - intros w1. - rewrite spec_ww_add_mul_div; auto with zarith. - autorewrite with w_rewrite rm10. - rewrite spec_w_0W; rewrite spec_w_1. - rewrite Z.pow_1_r; auto with zarith. - rewrite Z.mul_comm; auto. - rewrite spec_w_0W; rewrite spec_w_1; auto with zarith. - red; simpl; intros; discriminate. - Qed. - - Theorem ww_add_mult_mult_2_plus_1: forall w, - [[ww_add_mul_div (w_0W w_1) w wwBm1]] = - (2 * [[w]] + 1) mod wwB. - intros w1. - rewrite spec_ww_add_mul_div; auto with zarith. - rewrite spec_w_0W; rewrite spec_w_1; auto with zarith. - rewrite Z.pow_1_r; auto with zarith. - f_equal; auto. - rewrite Z.mul_comm; f_equal; auto. - autorewrite with w_rewrite rm10. - unfold ww_digits, base. - symmetry; apply Zdiv_unique with (r := 2 ^ (Zpos (ww_digits w_digits) - 1) -1); - auto with zarith. - unfold ww_digits; split; auto with zarith. - match goal with |- 0 <= ?X - 1 => - assert (0 < X); auto with zarith - end. - apply Z.pow_pos_nonneg; auto with zarith. - match goal with |- 0 <= ?X - 1 => - assert (0 < X); auto with zarith; red; reflexivity - end. - unfold ww_digits; autorewrite with rm10. - assert (tmp: forall p q r, p + (q - r) = p + q - r); auto with zarith; - rewrite tmp; clear tmp. - assert (tmp: forall p, p + p = 2 * p); auto with zarith; - rewrite tmp; clear tmp. - f_equal; auto. - pattern 2 at 2; rewrite <- Z.pow_1_r; rewrite <- Zpower_exp; - auto with zarith. - assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; - rewrite tmp; clear tmp; auto. - match goal with |- ?X - 1 >= 0 => - assert (0 < X); auto with zarith; red; reflexivity - end. - rewrite spec_w_0W; rewrite spec_w_1; auto with zarith. - red; simpl; intros; discriminate. - Qed. - - Theorem Zplus_mod_one: forall a1 b1, 0 < b1 -> (a1 + b1) mod b1 = a1 mod b1. - intros a1 b1 H; rewrite Zplus_mod; auto with zarith. - rewrite Z_mod_same; try rewrite Z.add_0_r; auto with zarith. - apply Zmod_mod; auto. - Qed. - - Lemma C1_plus_wB: forall x, [+|C1 x|] = wB + [|x|]. - unfold interp_carry; auto with zarith. - Qed. - - Theorem spec_w_div2s : forall a1 a2 b, - wB/2 <= [|b|] -> [+|a1|] <= 2 * [|b|] -> - let (q,r) := w_div2s a1 a2 b in - [+|a1|] * wB + [|a2|] = [+|q|] * (2 * [|b|]) + [+|r|] /\ 0 <= [+|r|] < 2 * [|b|]. - intros a1 a2 b H. - assert (HH: 0 < [|b|]); auto with zarith. - assert (U := wB_pos w_digits). - apply Z.lt_le_trans with (2 := H); auto with zarith. - apply Z.lt_le_trans with 1; auto with zarith. - apply Zdiv_le_lower_bound; auto with zarith. - unfold w_div2s; case a1; intros w0 H0. - match goal with |- context[w_div21c ?y ?z ?t] => - generalize (@spec_w_div21c y z t H); - case (w_div21c y z t); autorewrite with w_rewrite; - auto - end. - intros c w1; case c. - simpl interp_carry; intros w2 (Hw1, Hw2). - match goal with |- context[w_is_even ?y] => - generalize (spec_w_is_even y); - case (w_is_even y) - end. - repeat rewrite C0_id. - rewrite add_mult_div_2. - intros H1; split; auto with zarith. - rewrite Hw1. - pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); - auto with zarith. - rewrite H1; ring. - repeat rewrite C0_id. - rewrite add_mult_div_2. - rewrite spec_w_add_c; auto with zarith. - intros H1; split; auto with zarith. - rewrite Hw1. - pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); - auto with zarith. - rewrite H1; ring. - intros w2; rewrite C1_plus_wB. - intros (Hw1, Hw2). - match goal with |- context[w_is_even ?y] => - generalize (spec_w_is_even y); - case (w_is_even y) - end. - repeat rewrite C0_id. - intros H1; split; auto with zarith. - rewrite Hw1. - pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); - auto with zarith. - rewrite H1. - repeat rewrite C0_id. - rewrite add_mult_div_2_plus_1; unfold base. - match goal with |- context[_ ^ ?X] => - assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; - rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; - try rewrite Z.pow_1_r; auto with zarith - end. - rewrite Pos2Z.inj_sub_max; auto with zarith. - rewrite Z.max_r; auto with zarith. - ring. - repeat rewrite C0_id. - rewrite spec_w_add_c; auto with zarith. - intros H1; split; auto with zarith. - rewrite add_mult_div_2_plus_1. - rewrite Hw1. - pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); - auto with zarith. - rewrite H1. - unfold base. - match goal with |- context[_ ^ ?X] => - assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; - rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; - try rewrite Z.pow_1_r; auto with zarith - end. - rewrite Pos2Z.inj_sub_max; auto with zarith. - rewrite Z.max_r; auto with zarith. - ring. - repeat rewrite C1_plus_wB in H0. - rewrite C1_plus_wB. - match goal with |- context[w_div21c ?y ?z ?t] => - generalize (@spec_w_div21c y z t H); - case (w_div21c y z t); autorewrite with w_rewrite; - auto - end. - intros c w1; case c. - intros w2 (Hw1, Hw2); rewrite C0_id in Hw1. - rewrite <- Zplus_mod_one in Hw1; auto with zarith. - rewrite Zmod_small in Hw1; auto with zarith. - match goal with |- context[w_is_even ?y] => - generalize (spec_w_is_even y); - case (w_is_even y) - end. - repeat rewrite C0_id. - intros H1; split; auto with zarith. - rewrite add_mult_div_2_plus_1. - replace (wB + [|w0|]) with ([|b|] + ([|w0|] - [|b|] + wB)); - auto with zarith. - rewrite Z.mul_add_distr_r; rewrite <- Z.add_assoc. - rewrite Hw1. - pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); - auto with zarith. - rewrite H1; unfold base. - match goal with |- context[_ ^ ?X] => - assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; - rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; - try rewrite Z.pow_1_r; auto with zarith - end. - rewrite Pos2Z.inj_sub_max; auto with zarith. - rewrite Z.max_r; auto with zarith. - ring. - repeat rewrite C0_id. - rewrite add_mult_div_2_plus_1. - rewrite spec_w_add_c; auto with zarith. - intros H1; split; auto with zarith. - replace (wB + [|w0|]) with ([|b|] + ([|w0|] - [|b|] + wB)); - auto with zarith. - rewrite Z.mul_add_distr_r; rewrite <- Z.add_assoc. - rewrite Hw1. - pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); - auto with zarith. - rewrite H1; unfold base. - match goal with |- context[_ ^ ?X] => - assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; - rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; - try rewrite Z.pow_1_r; auto with zarith - end. - rewrite Pos2Z.inj_sub_max; auto with zarith. - rewrite Z.max_r; auto with zarith. - ring. - split; auto with zarith. - destruct (spec_to_Z b);auto with zarith. - destruct (spec_to_Z w0);auto with zarith. - destruct (spec_to_Z b);auto with zarith. - destruct (spec_to_Z b);auto with zarith. - intros w2; rewrite C1_plus_wB. - rewrite <- Zplus_mod_one; auto with zarith. - rewrite Zmod_small; auto with zarith. - intros (Hw1, Hw2). - match goal with |- context[w_is_even ?y] => - generalize (spec_w_is_even y); - case (w_is_even y) - end. - repeat (rewrite C0_id || rewrite C1_plus_wB). - intros H1; split; auto with zarith. - rewrite add_mult_div_2. - replace (wB + [|w0|]) with ([|b|] + ([|w0|] - [|b|] + wB)); - auto with zarith. - rewrite Z.mul_add_distr_r; rewrite <- Z.add_assoc. - rewrite Hw1. - pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); - auto with zarith. - rewrite H1; ring. - repeat (rewrite C0_id || rewrite C1_plus_wB). - rewrite spec_w_add_c; auto with zarith. - intros H1; split; auto with zarith. - rewrite add_mult_div_2. - replace (wB + [|w0|]) with ([|b|] + ([|w0|] - [|b|] + wB)); - auto with zarith. - rewrite Z.mul_add_distr_r; rewrite <- Z.add_assoc. - rewrite Hw1. - pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); - auto with zarith. - rewrite H1; ring. - split; auto with zarith. - destruct (spec_to_Z b);auto with zarith. - destruct (spec_to_Z w0);auto with zarith. - destruct (spec_to_Z b);auto with zarith. - destruct (spec_to_Z b);auto with zarith. - Qed. - - Theorem wB_div_4: 4 * (wB / 4) = wB. - Proof. - unfold base. - assert (2 ^ Zpos w_digits = - 4 * (2 ^ (Zpos w_digits - 2))). - change 4 with (2 ^ 2). - rewrite <- Zpower_exp; auto with zarith. - f_equal; auto with zarith. - rewrite H. - rewrite (fun x => (Z.mul_comm 4 (2 ^x))). - rewrite Z_div_mult; auto with zarith. - Qed. - - Theorem Zsquare_mult: forall p, p ^ 2 = p * p. - intros p; change 2 with (1 + 1); rewrite Zpower_exp; - try rewrite Z.pow_1_r; auto with zarith. - Qed. - - Theorem Zsquare_pos: forall p, 0 <= p ^ 2. - intros p; case (Z.le_gt_cases 0 p); intros H1. - rewrite Zsquare_mult; apply Z.mul_nonneg_nonneg; auto with zarith. - rewrite Zsquare_mult; replace (p * p) with ((- p) * (- p)); try ring. - apply Z.mul_nonneg_nonneg; auto with zarith. - Qed. - - Lemma spec_split: forall x, - [|fst (split x)|] * wB + [|snd (split x)|] = [[x]]. - intros x; case x; simpl; autorewrite with w_rewrite; - auto with zarith. - Qed. - - Theorem mult_wwB: forall x y, [|x|] * [|y|] < wwB. - Proof. - intros x y; rewrite wwB_wBwB; rewrite Z.pow_2_r. - generalize (spec_to_Z x); intros U. - generalize (spec_to_Z y); intros U1. - apply Z.le_lt_trans with ((wB -1 ) * (wB - 1)); auto with zarith. - apply Z.mul_le_mono_nonneg; auto with zarith. - rewrite ?Z.mul_sub_distr_l, ?Z.mul_sub_distr_r; auto with zarith. - Qed. - Hint Resolve mult_wwB. - - Lemma spec_ww_sqrt2 : forall x y, - wwB/ 4 <= [[x]] -> - let (s,r) := ww_sqrt2 x y in - [||WW x y||] = [[s]] ^ 2 + [+[r]] /\ - [+[r]] <= 2 * [[s]]. - intros x y H; unfold ww_sqrt2. - repeat match goal with |- context[split ?x] => - generalize (spec_split x); case (split x) - end; simpl @fst; simpl @snd. - intros w0 w1 Hw0 w2 w3 Hw1. - assert (U: wB/4 <= [|w2|]). - case (Z.le_gt_cases (wB / 4) [|w2|]); auto; intros H1. - contradict H; apply Z.lt_nge. - rewrite wwB_wBwB; rewrite Z.pow_2_r. - pattern wB at 1; rewrite <- wB_div_4; rewrite <- Z.mul_assoc; - rewrite Z.mul_comm. - rewrite Z_div_mult; auto with zarith. - rewrite <- Hw1. - match goal with |- _ < ?X => - pattern X; rewrite <- Z.add_0_r; apply beta_lex_inv; - auto with zarith - end. - destruct (spec_to_Z w3);auto with zarith. - generalize (@spec_w_sqrt2 w2 w3 U); case (w_sqrt2 w2 w3). - intros w4 c (H1, H2). - assert (U1: wB/2 <= [|w4|]). - case (Z.le_gt_cases (wB/2) [|w4|]); auto with zarith. - intros U1. - assert (U2 : [|w4|] <= wB/2 -1); auto with zarith. - assert (U3 : [|w4|] ^ 2 <= wB/4 * wB - wB + 1); auto with zarith. - match goal with |- ?X ^ 2 <= ?Y => - rewrite Zsquare_mult; - replace Y with ((wB/2 - 1) * (wB/2 -1)) - end. - apply Z.mul_le_mono_nonneg; auto with zarith. - destruct (spec_to_Z w4);auto with zarith. - destruct (spec_to_Z w4);auto with zarith. - pattern wB at 4 5; rewrite <- wB_div_2. - rewrite Z.mul_assoc. - replace ((wB / 4) * 2) with (wB / 2). - ring. - pattern wB at 1; rewrite <- wB_div_4. - change 4 with (2 * 2). - rewrite <- Z.mul_assoc; rewrite (Z.mul_comm 2). - rewrite Z_div_mult; try ring; auto with zarith. - assert (U4 : [+|c|] <= wB -2); auto with zarith. - apply Z.le_trans with (1 := H2). - match goal with |- ?X <= ?Y => - replace Y with (2 * (wB/ 2 - 1)); auto with zarith - end. - pattern wB at 2; rewrite <- wB_div_2; auto with zarith. - match type of H1 with ?X = _ => - assert (U5: X < wB / 4 * wB) - end. - rewrite H1; auto with zarith. - contradict U; apply Z.lt_nge. - apply Z.mul_lt_mono_pos_r with wB; auto with zarith. - destruct (spec_to_Z w4);auto with zarith. - apply Z.le_lt_trans with (2 := U5). - unfold ww_to_Z, zn2z_to_Z. - destruct (spec_to_Z w3);auto with zarith. - generalize (@spec_w_div2s c w0 w4 U1 H2). - case (w_div2s c w0 w4). - intros c0; case c0; intros w5; - repeat (rewrite C0_id || rewrite C1_plus_wB). - intros c1; case c1; intros w6; - repeat (rewrite C0_id || rewrite C1_plus_wB). - intros (H3, H4). - match goal with |- context [ww_sub_c ?y ?z] => - generalize (spec_ww_sub_c y z); case (ww_sub_c y z) - end. - intros z; change [-[C0 z]] with ([[z]]). - change [+[C0 z]] with ([[z]]). - intros H5; rewrite spec_w_square_c in H5; - auto. - split. - unfold zn2z_to_Z; rewrite <- Hw1. - unfold ww_to_Z, zn2z_to_Z in H1. rewrite H1. - rewrite <- Hw0. - match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U => - transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) - end. - repeat rewrite Zsquare_mult. - rewrite wwB_wBwB; ring. - rewrite H3. - rewrite H5. - unfold ww_to_Z, zn2z_to_Z. - repeat rewrite Zsquare_mult; ring. - rewrite H5. - unfold ww_to_Z, zn2z_to_Z. - match goal with |- ?X - ?Y * ?Y <= _ => - assert (V := Zsquare_pos Y); - rewrite Zsquare_mult in V; - apply Z.le_trans with X; auto with zarith; - clear V - end. - match goal with |- ?X * wB + ?Y <= 2 * (?Z * wB + ?T) => - apply Z.le_trans with ((2 * Z - 1) * wB + wB); auto with zarith - end. - destruct (spec_to_Z w1);auto with zarith. - match goal with |- ?X <= _ => - replace X with (2 * [|w4|] * wB); auto with zarith - end. - rewrite Z.mul_add_distr_l; rewrite Z.mul_assoc. - destruct (spec_to_Z w5); auto with zarith. - ring. - intros z; replace [-[C1 z]] with (- wwB + [[z]]). - 2: simpl; case wwB; auto with zarith. - intros H5; rewrite spec_w_square_c in H5; - auto. - match goal with |- context [ww_pred_c ?y] => - generalize (spec_ww_pred_c y); case (ww_pred_c y) - end. - intros z1; change [-[C0 z1]] with ([[z1]]). - rewrite ww_add_mult_mult_2. - rewrite spec_ww_add_c. - rewrite spec_ww_pred. - rewrite <- Zmod_unique with (q := 1) (r := -wwB + 2 * [[WW w4 w5]]); - auto with zarith. - intros Hz1; rewrite Zmod_small; auto with zarith. - match type of H5 with -?X + ?Y = ?Z => - assert (V: Y = Z + X); - try (rewrite <- H5; ring) - end. - split. - unfold zn2z_to_Z; rewrite <- Hw1. - unfold ww_to_Z, zn2z_to_Z in H1; rewrite H1. - rewrite <- Hw0. - match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U => - transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) - end. - repeat rewrite Zsquare_mult. - rewrite wwB_wBwB; ring. - rewrite H3. - rewrite V. - rewrite Hz1. - unfold ww_to_Z; simpl zn2z_to_Z. - repeat rewrite Zsquare_mult; ring. - rewrite Hz1. - destruct (spec_ww_to_Z w_digits w_to_Z spec_to_Z z);auto with zarith. - assert (V1 := spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW w4 w5)). - assert (0 < [[WW w4 w5]]); auto with zarith. - apply Z.lt_le_trans with (wB/ 2 * wB + 0); auto with zarith. - autorewrite with rm10; apply Z.mul_pos_pos; auto with zarith. - apply Z.mul_lt_mono_pos_r with 2; auto with zarith. - autorewrite with rm10. - rewrite Z.mul_comm; rewrite wB_div_2; auto with zarith. - case (spec_to_Z w5);auto with zarith. - case (spec_to_Z w5);auto with zarith. - simpl. - assert (V2 := spec_to_Z w5);auto with zarith. - assert (V1 := spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW w4 w5)); auto with zarith. - split; auto with zarith. - assert (wwB <= 2 * [[WW w4 w5]]); auto with zarith. - apply Z.le_trans with (2 * ([|w4|] * wB)). - rewrite wwB_wBwB; rewrite Z.pow_2_r. - rewrite Z.mul_assoc; apply Z.mul_le_mono_nonneg_r; auto with zarith. - assert (V2 := spec_to_Z w5);auto with zarith. - rewrite <- wB_div_2; auto with zarith. - simpl ww_to_Z; assert (V2 := spec_to_Z w5);auto with zarith. - assert (V1 := spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW w4 w5)); auto with zarith. - intros z1; change [-[C1 z1]] with (-wwB + [[z1]]). - match goal with |- context[([+[C0 ?z]])] => - change [+[C0 z]] with ([[z]]) - end. - rewrite spec_ww_add; auto with zarith. - rewrite spec_ww_pred; auto with zarith. - rewrite ww_add_mult_mult_2. - rename V1 into VV1. - assert (VV2: 0 < [[WW w4 w5]]); auto with zarith. - apply Z.lt_le_trans with (wB/ 2 * wB + 0); auto with zarith. - autorewrite with rm10; apply Z.mul_pos_pos; auto with zarith. - apply Z.mul_lt_mono_pos_r with 2; auto with zarith. - autorewrite with rm10. - rewrite Z.mul_comm; rewrite wB_div_2; auto with zarith. - assert (VV3 := spec_to_Z w5);auto with zarith. - assert (VV3 := spec_to_Z w5);auto with zarith. - simpl. - assert (VV3 := spec_to_Z w5);auto with zarith. - assert (VV3: wwB <= 2 * [[WW w4 w5]]); auto with zarith. - apply Z.le_trans with (2 * ([|w4|] * wB)). - rewrite wwB_wBwB; rewrite Z.pow_2_r. - rewrite Z.mul_assoc; apply Z.mul_le_mono_nonneg_r; auto with zarith. - case (spec_to_Z w5);auto with zarith. - rewrite <- wB_div_2; auto with zarith. - simpl ww_to_Z; assert (V4 := spec_to_Z w5);auto with zarith. - rewrite <- Zmod_unique with (q := 1) (r := -wwB + 2 * [[WW w4 w5]]); - auto with zarith. - intros Hz1; rewrite Zmod_small; auto with zarith. - match type of H5 with -?X + ?Y = ?Z => - assert (V: Y = Z + X); - try (rewrite <- H5; ring) - end. - match type of Hz1 with -?X + ?Y = -?X + ?Z - 1 => - assert (V1: Y = Z - 1); - [replace (Z - 1) with (X + (-X + Z -1)); - [rewrite <- Hz1 | idtac]; ring - | idtac] - end. - rewrite <- Zmod_unique with (q := 1) (r := -wwB + [[z1]] + [[z]]); - auto with zarith. - unfold zn2z_to_Z; rewrite <- Hw1. - unfold ww_to_Z, zn2z_to_Z in H1; rewrite H1. - rewrite <- Hw0. - split. - match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U => - transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) - end. - repeat rewrite Zsquare_mult. - rewrite wwB_wBwB; ring. - rewrite H3. - rewrite V. - rewrite Hz1. - unfold ww_to_Z; simpl zn2z_to_Z. - repeat rewrite Zsquare_mult; ring. - assert (V2 := spec_ww_to_Z w_digits w_to_Z spec_to_Z z);auto with zarith. - assert (V2 := spec_ww_to_Z w_digits w_to_Z spec_to_Z z);auto with zarith. - assert (V3 := spec_ww_to_Z w_digits w_to_Z spec_to_Z z1);auto with zarith. - split; auto with zarith. - rewrite (Z.add_comm (-wwB)); rewrite <- Z.add_assoc. - rewrite H5. - match goal with |- 0 <= ?X + (?Y - ?Z) => - apply Z.le_trans with (X - Z); auto with zarith - end. - 2: generalize (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW w6 w1)); unfold ww_to_Z; auto with zarith. - rewrite V1. - match goal with |- 0 <= ?X - 1 - ?Y => - assert (Y < X); auto with zarith - end. - apply Z.lt_le_trans with wwB; auto with zarith. - intros (H3, H4). - match goal with |- context [ww_sub_c ?y ?z] => - generalize (spec_ww_sub_c y z); case (ww_sub_c y z) - end. - intros z; change [-[C0 z]] with ([[z]]). - match goal with |- context[([+[C1 ?z]])] => - replace [+[C1 z]] with (wwB + [[z]]) - end. - 2: simpl; case wwB; auto. - intros H5; rewrite spec_w_square_c in H5; - auto. - split. - change ([||WW x y||]) with ([[x]] * wwB + [[y]]). - rewrite <- Hw1. - unfold ww_to_Z, zn2z_to_Z in H1; rewrite H1. - rewrite <- Hw0. - match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U => - transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) - end. - repeat rewrite Zsquare_mult. - rewrite wwB_wBwB; ring. - rewrite H3. - rewrite H5. - unfold ww_to_Z; simpl zn2z_to_Z. - rewrite wwB_wBwB. - repeat rewrite Zsquare_mult; ring. - simpl ww_to_Z. - rewrite H5. - simpl ww_to_Z. - rewrite wwB_wBwB; rewrite Z.pow_2_r. - match goal with |- ?X * ?Y + (?Z * ?Y + ?T - ?U) <= _ => - apply Z.le_trans with (X * Y + (Z * Y + T - 0)); - auto with zarith - end. - assert (V := Zsquare_pos [|w5|]); - rewrite Zsquare_mult in V; auto with zarith. - autorewrite with rm10. - match goal with |- _ <= 2 * (?U * ?V + ?W) => - apply Z.le_trans with (2 * U * V + 0); - auto with zarith - end. - match goal with |- ?X * ?Y + (?Z * ?Y + ?T) <= _ => - replace (X * Y + (Z * Y + T)) with ((X + Z) * Y + T); - try ring - end. - apply Z.lt_le_incl; apply beta_lex_inv; auto with zarith. - destruct (spec_to_Z w1);auto with zarith. - destruct (spec_to_Z w5);auto with zarith. - rewrite Z.mul_add_distr_l; auto with zarith. - rewrite Z.mul_assoc; auto with zarith. - intros z; replace [-[C1 z]] with (- wwB + [[z]]). - 2: simpl; case wwB; auto with zarith. - intros H5; rewrite spec_w_square_c in H5; - auto. - match goal with |- context[([+[C0 ?z]])] => - change [+[C0 z]] with ([[z]]) - end. - match type of H5 with -?X + ?Y = ?Z => - assert (V: Y = Z + X); - try (rewrite <- H5; ring) - end. - change ([||WW x y||]) with ([[x]] * wwB + [[y]]). - simpl ww_to_Z. - rewrite <- Hw1. - simpl ww_to_Z in H1; rewrite H1. - rewrite <- Hw0. - split. - match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U => - transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) - end. - repeat rewrite Zsquare_mult. - rewrite wwB_wBwB; ring. - rewrite H3. - rewrite V. - simpl ww_to_Z. - rewrite wwB_wBwB. - repeat rewrite Zsquare_mult; ring. - rewrite V. - simpl ww_to_Z. - rewrite wwB_wBwB; rewrite Z.pow_2_r. - match goal with |- (?Z * ?Y + ?T - ?U) + ?X * ?Y <= _ => - apply Z.le_trans with ((Z * Y + T - 0) + X * Y); - auto with zarith - end. - assert (V1 := Zsquare_pos [|w5|]); - rewrite Zsquare_mult in V1; auto with zarith. - autorewrite with rm10. - match goal with |- _ <= 2 * (?U * ?V + ?W) => - apply Z.le_trans with (2 * U * V + 0); - auto with zarith - end. - match goal with |- (?Z * ?Y + ?T) + ?X * ?Y <= _ => - replace ((Z * Y + T) + X * Y) with ((X + Z) * Y + T); - try ring - end. - apply Z.lt_le_incl; apply beta_lex_inv; auto with zarith. - destruct (spec_to_Z w1);auto with zarith. - destruct (spec_to_Z w5);auto with zarith. - rewrite Z.mul_add_distr_l; auto with zarith. - rewrite Z.mul_assoc; auto with zarith. - Z.le_elim H2. - intros c1 (H3, H4). - match type of H3 with ?X = ?Y => absurd (X < Y) end. - apply Z.le_ngt; rewrite <- H3; auto with zarith. - rewrite Z.mul_add_distr_r. - apply Z.lt_le_trans with ((2 * [|w4|]) * wB + 0); - auto with zarith. - apply beta_lex_inv; auto with zarith. - destruct (spec_to_Z w0);auto with zarith. - assert (V1 := spec_to_Z w5);auto with zarith. - rewrite (Z.mul_comm wB); auto with zarith. - assert (0 <= [|w5|] * (2 * [|w4|])); auto with zarith. - intros c1 (H3, H4); rewrite H2 in H3. - match type of H3 with ?X + ?Y = (?Z + ?T) * ?U + ?V => - assert (VV: (Y = (T * U) + V)); - [replace Y with ((X + Y) - X); - [rewrite H3; ring | ring] | idtac] - end. - assert (V1 := spec_to_Z w0);auto with zarith. - assert (V2 := spec_to_Z w5);auto with zarith. - case V2; intros V3 _. - Z.le_elim V3; auto with zarith. - match type of VV with ?X = ?Y => absurd (X < Y) end. - apply Z.le_ngt; rewrite <- VV; auto with zarith. - apply Z.lt_le_trans with wB; auto with zarith. - match goal with |- _ <= ?X + _ => - apply Z.le_trans with X; auto with zarith - end. - match goal with |- _ <= _ * ?X => - apply Z.le_trans with (1 * X); auto with zarith - end. - autorewrite with rm10. - rewrite <- wB_div_2; apply Z.mul_le_mono_nonneg_l; auto with zarith. - rewrite <- V3 in VV; generalize VV; autorewrite with rm10; - clear VV; intros VV. - rewrite spec_ww_add_c; auto with zarith. - rewrite ww_add_mult_mult_2_plus_1. - match goal with |- context[?X mod wwB] => - rewrite <- Zmod_unique with (q := 1) (r := -wwB + X) - end; auto with zarith. - simpl ww_to_Z. - rewrite spec_w_Bm1; auto with zarith. - split. - change ([||WW x y||]) with ([[x]] * wwB + [[y]]). - rewrite <- Hw1. - simpl ww_to_Z in H1; rewrite H1. - rewrite <- Hw0. - match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U => - transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) - end. - repeat rewrite Zsquare_mult. - rewrite wwB_wBwB; ring. - rewrite H2. - rewrite wwB_wBwB. - repeat rewrite Zsquare_mult; ring. - assert (V4 := spec_ww_to_Z w_digits w_to_Z spec_to_Z y);auto with zarith. - assert (V4 := spec_ww_to_Z w_digits w_to_Z spec_to_Z y);auto with zarith. - simpl ww_to_Z; unfold ww_to_Z. - rewrite spec_w_Bm1; auto with zarith. - split. - rewrite wwB_wBwB; rewrite Z.pow_2_r. - match goal with |- _ <= -?X + (2 * (?Z * ?T + ?U) + ?V) => - assert (X <= 2 * Z * T); auto with zarith - end. - apply Z.mul_le_mono_nonneg_r; auto with zarith. - rewrite <- wB_div_2; apply Z.mul_le_mono_nonneg_l; auto with zarith. - rewrite Z.mul_add_distr_l; auto with zarith. - rewrite Z.mul_assoc; auto with zarith. - match goal with |- _ + ?X < _ => - replace X with ((2 * (([|w4|]) + 1) * wB) - 1); try ring - end. - assert (2 * ([|w4|] + 1) * wB <= 2 * wwB); auto with zarith. - rewrite <- Z.mul_assoc; apply Z.mul_le_mono_nonneg_l; auto with zarith. - rewrite wwB_wBwB; rewrite Z.pow_2_r. - apply Z.mul_le_mono_nonneg_r; auto with zarith. - case (spec_to_Z w4);auto with zarith. -Qed. - - Lemma spec_ww_is_zero: forall x, - if ww_is_zero x then [[x]] = 0 else 0 < [[x]]. - intro x; unfold ww_is_zero. - rewrite spec_ww_compare. case Z.compare_spec; - auto with zarith. - simpl ww_to_Z. - assert (V4 := spec_ww_to_Z w_digits w_to_Z spec_to_Z x);auto with zarith. - Qed. - - Lemma wwB_4_2: 2 * (wwB / 4) = wwB/ 2. - pattern wwB at 1; rewrite wwB_wBwB; rewrite Z.pow_2_r. - rewrite <- wB_div_2. - match goal with |- context[(2 * ?X) * (2 * ?Z)] => - replace ((2 * X) * (2 * Z)) with ((X * Z) * 4); try ring - end. - rewrite Z_div_mult; auto with zarith. - rewrite Z.mul_assoc; rewrite wB_div_2. - rewrite wwB_div_2; ring. - Qed. - - - Lemma spec_ww_head1 - : forall x : zn2z w, - (ww_is_even (ww_head1 x) = true) /\ - (0 < [[x]] -> wwB / 4 <= 2 ^ [[ww_head1 x]] * [[x]] < wwB). - assert (U := wB_pos w_digits). - intros x; unfold ww_head1. - generalize (spec_ww_is_even (ww_head0 x)); case_eq (ww_is_even (ww_head0 x)). - intros HH H1; rewrite HH; split; auto. - intros H2. - generalize (spec_ww_head0 x H2); case (ww_head0 x); autorewrite with rm10. - intros (H3, H4); split; auto with zarith. - apply Z.le_trans with (2 := H3). - apply Zdiv_le_compat_l; auto with zarith. - intros xh xl (H3, H4); split; auto with zarith. - apply Z.le_trans with (2 := H3). - apply Zdiv_le_compat_l; auto with zarith. - intros H1. - case (spec_to_w_Z (ww_head0 x)); intros Hv1 Hv2. - assert (Hp0: 0 < [[ww_head0 x]]). - generalize (spec_ww_is_even (ww_head0 x)); rewrite H1. - generalize Hv1; case [[ww_head0 x]]. - rewrite Zmod_small; auto with zarith. - intros; assert (0 < Zpos p); auto with zarith. - red; simpl; auto. - intros p H2; case H2; auto. - assert (Hp: [[ww_pred (ww_head0 x)]] = [[ww_head0 x]] - 1). - rewrite spec_ww_pred. - rewrite Zmod_small; auto with zarith. - intros H2; split. - generalize (spec_ww_is_even (ww_pred (ww_head0 x))); - case ww_is_even; auto. - rewrite Hp. - rewrite Zminus_mod; auto with zarith. - rewrite H2; repeat rewrite Zmod_small; auto with zarith. - intros H3; rewrite Hp. - case (spec_ww_head0 x); auto; intros Hv3 Hv4. - assert (Hu: forall u, 0 < u -> 2 * 2 ^ (u - 1) = 2 ^u). - intros u Hu. - pattern 2 at 1; rewrite <- Z.pow_1_r. - rewrite <- Zpower_exp; auto with zarith. - ring_simplify (1 + (u - 1)); auto with zarith. - split; auto with zarith. - apply Z.mul_le_mono_pos_r with 2; auto with zarith. - repeat rewrite (fun x => Z.mul_comm x 2). - rewrite wwB_4_2. - rewrite Z.mul_assoc; rewrite Hu; auto with zarith. - apply Z.le_lt_trans with (2 * 2 ^ ([[ww_head0 x]] - 1) * [[x]]); auto with zarith; - rewrite Hu; auto with zarith. - apply Z.mul_le_mono_nonneg_r; auto with zarith. - apply Zpower_le_monotone; auto with zarith. - Qed. - - Theorem wwB_4_wB_4: wwB / 4 = wB / 4 * wB. - Proof. - symmetry; apply Zdiv_unique with 0; auto with zarith. - rewrite Z.mul_assoc; rewrite wB_div_4; auto with zarith. - rewrite wwB_wBwB; ring. - Qed. - - Lemma spec_ww_sqrt : forall x, - [[ww_sqrt x]] ^ 2 <= [[x]] < ([[ww_sqrt x]] + 1) ^ 2. - assert (U := wB_pos w_digits). - intro x; unfold ww_sqrt. - generalize (spec_ww_is_zero x); case (ww_is_zero x). - simpl ww_to_Z; simpl Z.pow; unfold Z.pow_pos; simpl; - auto with zarith. - intros H1. - rewrite spec_ww_compare. case Z.compare_spec; - simpl ww_to_Z; autorewrite with rm10. - generalize H1; case x. - intros HH; contradict HH; simpl ww_to_Z; auto with zarith. - intros w0 w1; simpl ww_to_Z; autorewrite with w_rewrite rm10. - intros H2; case (spec_ww_head1 (WW w0 w1)); intros H3 H4 H5. - generalize (H4 H2); clear H4; rewrite H5; clear H5; autorewrite with rm10. - intros (H4, H5). - assert (V: wB/4 <= [|w0|]). - apply beta_lex with 0 [|w1|] wB; auto with zarith; autorewrite with rm10. - rewrite <- wwB_4_wB_4; auto. - generalize (@spec_w_sqrt2 w0 w1 V);auto with zarith. - case (w_sqrt2 w0 w1); intros w2 c. - simpl ww_to_Z; simpl @fst. - case c; unfold interp_carry; autorewrite with rm10. - intros w3 (H6, H7); rewrite H6. - assert (V1 := spec_to_Z w3);auto with zarith. - split; auto with zarith. - apply Z.le_lt_trans with ([|w2|] ^2 + 2 * [|w2|]); auto with zarith. - match goal with |- ?X < ?Z => - replace Z with (X + 1); auto with zarith - end. - repeat rewrite Zsquare_mult; ring. - intros w3 (H6, H7); rewrite H6. - assert (V1 := spec_to_Z w3);auto with zarith. - split; auto with zarith. - apply Z.le_lt_trans with ([|w2|] ^2 + 2 * [|w2|]); auto with zarith. - match goal with |- ?X < ?Z => - replace Z with (X + 1); auto with zarith - end. - repeat rewrite Zsquare_mult; ring. - intros HH; case (spec_to_w_Z (ww_head1 x)); auto with zarith. - intros Hv1. - case (spec_ww_head1 x); intros Hp1 Hp2. - generalize (Hp2 H1); clear Hp2; intros Hp2. - assert (Hv2: [[ww_head1 x]] <= Zpos (xO w_digits)). - case (Z.le_gt_cases (Zpos (xO w_digits)) [[ww_head1 x]]); auto with zarith; intros HH1. - case Hp2; intros _ HH2; contradict HH2. - apply Z.le_ngt; unfold base. - apply Z.le_trans with (2 ^ [[ww_head1 x]]). - apply Zpower_le_monotone; auto with zarith. - pattern (2 ^ [[ww_head1 x]]) at 1; - rewrite <- (Z.mul_1_r (2 ^ [[ww_head1 x]])). - apply Z.mul_le_mono_nonneg_l; auto with zarith. - generalize (spec_ww_add_mul_div x W0 (ww_head1 x) Hv2); - case ww_add_mul_div. - simpl ww_to_Z; autorewrite with w_rewrite rm10. - rewrite Zmod_small; auto with zarith. - intros H2. symmetry in H2. rewrite Z.mul_eq_0 in H2. destruct H2 as [H2|H2]. - rewrite H2; unfold Z.pow, Z.pow_pos; simpl; auto with zarith. - match type of H2 with ?X = ?Y => - absurd (Y < X); try (rewrite H2; auto with zarith; fail) - end. - apply Z.pow_pos_nonneg; auto with zarith. - split; auto with zarith. - case Hp2; intros _ tmp; apply Z.le_lt_trans with (2 := tmp); - clear tmp. - rewrite Z.mul_comm; apply Z.mul_le_mono_nonneg_r; auto with zarith. - assert (Hv0: [[ww_head1 x]] = 2 * ([[ww_head1 x]]/2)). - pattern [[ww_head1 x]] at 1; rewrite (Z_div_mod_eq [[ww_head1 x]] 2); - auto with zarith. - generalize (spec_ww_is_even (ww_head1 x)); rewrite Hp1; - intros tmp; rewrite tmp; rewrite Z.add_0_r; auto. - intros w0 w1; autorewrite with w_rewrite rm10. - rewrite Zmod_small; auto with zarith. - 2: rewrite Z.mul_comm; auto with zarith. - intros H2. - assert (V: wB/4 <= [|w0|]). - apply beta_lex with 0 [|w1|] wB; auto with zarith; autorewrite with rm10. - simpl ww_to_Z in H2; rewrite H2. - rewrite <- wwB_4_wB_4; auto with zarith. - rewrite Z.mul_comm; auto with zarith. - assert (V1 := spec_to_Z w1);auto with zarith. - generalize (@spec_w_sqrt2 w0 w1 V);auto with zarith. - case (w_sqrt2 w0 w1); intros w2 c. - case (spec_to_Z w2); intros HH1 HH2. - simpl ww_to_Z; simpl @fst. - assert (Hv3: [[ww_pred ww_zdigits]] - = Zpos (xO w_digits) - 1). - rewrite spec_ww_pred; rewrite spec_ww_zdigits. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - apply Z.lt_le_trans with (Zpos (xO w_digits)); auto with zarith. - unfold base; apply Zpower2_le_lin; auto with zarith. - assert (Hv4: [[ww_head1 x]]/2 < wB). - apply Z.le_lt_trans with (Zpos w_digits). - apply Z.mul_le_mono_pos_r with 2; auto with zarith. - repeat rewrite (fun x => Z.mul_comm x 2). - rewrite <- Hv0; rewrite <- Pos2Z.inj_xO; auto. - unfold base; apply Zpower2_lt_lin; auto with zarith. - assert (Hv5: [[(ww_add_mul_div (ww_pred ww_zdigits) W0 (ww_head1 x))]] - = [[ww_head1 x]]/2). - rewrite spec_ww_add_mul_div. - simpl ww_to_Z; autorewrite with rm10. - rewrite Hv3. - ring_simplify (Zpos (xO w_digits) - (Zpos (xO w_digits) - 1)). - rewrite Z.pow_1_r. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - apply Z.lt_le_trans with (1 := Hv4); auto with zarith. - unfold base; apply Zpower_le_monotone; auto with zarith. - split; unfold ww_digits; try rewrite Pos2Z.inj_xO; auto with zarith. - rewrite Hv3; auto with zarith. - assert (Hv6: [|low(ww_add_mul_div (ww_pred ww_zdigits) W0 (ww_head1 x))|] - = [[ww_head1 x]]/2). - rewrite spec_low. - rewrite Hv5; rewrite Zmod_small; auto with zarith. - rewrite spec_w_add_mul_div; auto with zarith. - rewrite spec_w_sub; auto with zarith. - rewrite spec_w_0. - simpl ww_to_Z; autorewrite with rm10. - rewrite Hv6; rewrite spec_w_zdigits. - rewrite (fun x y => Zmod_small (x - y)). - ring_simplify (Zpos w_digits - (Zpos w_digits - [[ww_head1 x]] / 2)). - rewrite Zmod_small. - simpl ww_to_Z in H2; rewrite H2; auto with zarith. - intros (H4, H5); split. - apply Z.mul_le_mono_pos_r with (2 ^ [[ww_head1 x]]); auto with zarith. - rewrite H4. - apply Z.le_trans with ([|w2|] ^ 2); auto with zarith. - rewrite Z.mul_comm. - pattern [[ww_head1 x]] at 1; - rewrite Hv0; auto with zarith. - rewrite (Z.mul_comm 2); rewrite Z.pow_mul_r; - auto with zarith. - assert (tmp: forall p q, p ^ 2 * q ^ 2 = (p * q) ^2); - try (intros; repeat rewrite Zsquare_mult; ring); - rewrite tmp; clear tmp. - apply Zpower_le_monotone3; auto with zarith. - split; auto with zarith. - pattern [|w2|] at 2; - rewrite (Z_div_mod_eq [|w2|] (2 ^ ([[ww_head1 x]] / 2))); - auto with zarith. - match goal with |- ?X <= ?X + ?Y => - assert (0 <= Y); auto with zarith - end. - case (Z_mod_lt [|w2|] (2 ^ ([[ww_head1 x]] / 2))); auto with zarith. - case c; unfold interp_carry; autorewrite with rm10; - intros w3; assert (V3 := spec_to_Z w3);auto with zarith. - apply Z.mul_lt_mono_pos_r with (2 ^ [[ww_head1 x]]); auto with zarith. - rewrite H4. - apply Z.le_lt_trans with ([|w2|] ^ 2 + 2 * [|w2|]); auto with zarith. - apply Z.lt_le_trans with (([|w2|] + 1) ^ 2); auto with zarith. - match goal with |- ?X < ?Y => - replace Y with (X + 1); auto with zarith - end. - repeat rewrite (Zsquare_mult); ring. - rewrite Z.mul_comm. - pattern [[ww_head1 x]] at 1; rewrite Hv0. - rewrite (Z.mul_comm 2); rewrite Z.pow_mul_r; - auto with zarith. - assert (tmp: forall p q, p ^ 2 * q ^ 2 = (p * q) ^2); - try (intros; repeat rewrite Zsquare_mult; ring); - rewrite tmp; clear tmp. - apply Zpower_le_monotone3; auto with zarith. - split; auto with zarith. - pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] (2 ^ ([[ww_head1 x]]/2))); - auto with zarith. - rewrite <- Z.add_assoc; rewrite Z.mul_add_distr_l. - autorewrite with rm10; apply Z.add_le_mono_l; auto with zarith. - case (Z_mod_lt [|w2|] (2 ^ ([[ww_head1 x]]/2))); auto with zarith. - split; auto with zarith. - apply Z.le_lt_trans with ([|w2|]); auto with zarith. - apply Zdiv_le_upper_bound; auto with zarith. - pattern [|w2|] at 1; replace [|w2|] with ([|w2|] * 2 ^0); - auto with zarith. - apply Z.mul_le_mono_nonneg_l; auto with zarith. - apply Zpower_le_monotone; auto with zarith. - rewrite Z.pow_0_r; autorewrite with rm10; auto. - split; auto with zarith. - rewrite Hv0 in Hv2; rewrite (Pos2Z.inj_xO w_digits) in Hv2; auto with zarith. - apply Z.le_lt_trans with (Zpos w_digits); auto with zarith. - unfold base; apply Zpower2_lt_lin; auto with zarith. - rewrite spec_w_sub; auto with zarith. - rewrite Hv6; rewrite spec_w_zdigits; auto with zarith. - assert (Hv7: 0 < [[ww_head1 x]]/2); auto with zarith. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - assert ([[ww_head1 x]]/2 <= Zpos w_digits); auto with zarith. - apply Z.mul_le_mono_pos_r with 2; auto with zarith. - repeat rewrite (fun x => Z.mul_comm x 2). - rewrite <- Hv0; rewrite <- Pos2Z.inj_xO; auto with zarith. - apply Z.le_lt_trans with (Zpos w_digits); auto with zarith. - unfold base; apply Zpower2_lt_lin; auto with zarith. - Qed. - -End DoubleSqrt. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v deleted file mode 100644 index a2df2600..00000000 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v +++ /dev/null @@ -1,356 +0,0 @@ - -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Set Implicit Arguments. - -Require Import ZArith. -Require Import BigNumPrelude. -Require Import DoubleType. -Require Import DoubleBase. - -Local Open Scope Z_scope. - -Section DoubleSub. - Variable w : Type. - Variable w_0 : w. - Variable w_Bm1 : w. - Variable w_WW : w -> w -> zn2z w. - Variable ww_Bm1 : zn2z w. - Variable w_opp_c : w -> carry w. - Variable w_opp_carry : w -> w. - Variable w_pred_c : w -> carry w. - Variable w_sub_c : w -> w -> carry w. - Variable w_sub_carry_c : w -> w -> carry w. - Variable w_opp : w -> w. - Variable w_pred : w -> w. - Variable w_sub : w -> w -> w. - Variable w_sub_carry : w -> w -> w. - - (* ** Opposites ** *) - Definition ww_opp_c x := - match x with - | W0 => C0 W0 - | WW xh xl => - match w_opp_c xl with - | C0 _ => - match w_opp_c xh with - | C0 h => C0 W0 - | C1 h => C1 (WW h w_0) - end - | C1 l => C1 (WW (w_opp_carry xh) l) - end - end. - - Definition ww_opp x := - match x with - | W0 => W0 - | WW xh xl => - match w_opp_c xl with - | C0 _ => WW (w_opp xh) w_0 - | C1 l => WW (w_opp_carry xh) l - end - end. - - Definition ww_opp_carry x := - match x with - | W0 => ww_Bm1 - | WW xh xl => w_WW (w_opp_carry xh) (w_opp_carry xl) - end. - - Definition ww_pred_c x := - match x with - | W0 => C1 ww_Bm1 - | WW xh xl => - match w_pred_c xl with - | C0 l => C0 (w_WW xh l) - | C1 _ => - match w_pred_c xh with - | C0 h => C0 (WW h w_Bm1) - | C1 _ => C1 ww_Bm1 - end - end - end. - - Definition ww_pred x := - match x with - | W0 => ww_Bm1 - | WW xh xl => - match w_pred_c xl with - | C0 l => w_WW xh l - | C1 l => WW (w_pred xh) w_Bm1 - end - end. - - Definition ww_sub_c x y := - match y, x with - | W0, _ => C0 x - | WW yh yl, W0 => ww_opp_c (WW yh yl) - | WW yh yl, WW xh xl => - match w_sub_c xl yl with - | C0 l => - match w_sub_c xh yh with - | C0 h => C0 (w_WW h l) - | C1 h => C1 (WW h l) - end - | C1 l => - match w_sub_carry_c xh yh with - | C0 h => C0 (WW h l) - | C1 h => C1 (WW h l) - end - end - end. - - Definition ww_sub x y := - match y, x with - | W0, _ => x - | WW yh yl, W0 => ww_opp (WW yh yl) - | WW yh yl, WW xh xl => - match w_sub_c xl yl with - | C0 l => w_WW (w_sub xh yh) l - | C1 l => WW (w_sub_carry xh yh) l - end - end. - - Definition ww_sub_carry_c x y := - match y, x with - | W0, W0 => C1 ww_Bm1 - | W0, WW xh xl => ww_pred_c (WW xh xl) - | WW yh yl, W0 => C1 (ww_opp_carry (WW yh yl)) - | WW yh yl, WW xh xl => - match w_sub_carry_c xl yl with - | C0 l => - match w_sub_c xh yh with - | C0 h => C0 (w_WW h l) - | C1 h => C1 (WW h l) - end - | C1 l => - match w_sub_carry_c xh yh with - | C0 h => C0 (w_WW h l) - | C1 h => C1 (w_WW h l) - end - end - end. - - Definition ww_sub_carry x y := - match y, x with - | W0, W0 => ww_Bm1 - | W0, WW xh xl => ww_pred (WW xh xl) - | WW yh yl, W0 => ww_opp_carry (WW yh yl) - | WW yh yl, WW xh xl => - match w_sub_carry_c xl yl with - | C0 l => w_WW (w_sub xh yh) l - | C1 l => w_WW (w_sub_carry xh yh) l - end - end. - - (*Section DoubleProof.*) - Variable w_digits : positive. - Variable w_to_Z : w -> Z. - - - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[+| c |]" := - (interp_carry 1 wB w_to_Z c) (at level 0, c at level 99). - Notation "[-| c |]" := - (interp_carry (-1) wB w_to_Z c) (at level 0, c at level 99). - - Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99). - Notation "[+[ c ]]" := - (interp_carry 1 wwB (ww_to_Z w_digits w_to_Z) c) - (at level 0, c at level 99). - Notation "[-[ c ]]" := - (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) - (at level 0, c at level 99). - - Variable spec_w_0 : [|w_0|] = 0. - Variable spec_w_Bm1 : [|w_Bm1|] = wB - 1. - Variable spec_ww_Bm1 : [[ww_Bm1]] = wwB - 1. - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. - - Variable spec_opp_c : forall x, [-|w_opp_c x|] = -[|x|]. - Variable spec_opp : forall x, [|w_opp x|] = (-[|x|]) mod wB. - Variable spec_opp_carry : forall x, [|w_opp_carry x|] = wB - [|x|] - 1. - - Variable spec_pred_c : forall x, [-|w_pred_c x|] = [|x|] - 1. - Variable spec_sub_c : forall x y, [-|w_sub_c x y|] = [|x|] - [|y|]. - Variable spec_sub_carry_c : - forall x y, [-|w_sub_carry_c x y|] = [|x|] - [|y|] - 1. - - Variable spec_pred : forall x, [|w_pred x|] = ([|x|] - 1) mod wB. - Variable spec_sub : forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB. - Variable spec_sub_carry : - forall x y, [|w_sub_carry x y|] = ([|x|] - [|y|] - 1) mod wB. - - - Lemma spec_ww_opp_c : forall x, [-[ww_opp_c x]] = -[[x]]. - Proof. - destruct x as [ |xh xl];simpl. reflexivity. - rewrite Z.opp_add_distr;generalize (spec_opp_c xl);destruct (w_opp_c xl) - as [l|l];intros H;unfold interp_carry in H;rewrite <- H; - rewrite <- Z.mul_opp_l. - assert ([|l|] = 0). - assert (H1:= spec_to_Z l);assert (H2 := spec_to_Z xl);omega. - rewrite H0;generalize (spec_opp_c xh);destruct (w_opp_c xh) - as [h|h];intros H1;unfold interp_carry in *;rewrite <- H1. - assert ([|h|] = 0). - assert (H3:= spec_to_Z h);assert (H2 := spec_to_Z xh);omega. - rewrite H2;reflexivity. - simpl ww_to_Z;rewrite wwB_wBwB;rewrite spec_w_0;ring. - unfold interp_carry;simpl ww_to_Z;rewrite wwB_wBwB;rewrite spec_opp_carry; - ring. - Qed. - - Lemma spec_ww_opp : forall x, [[ww_opp x]] = (-[[x]]) mod wwB. - Proof. - destruct x as [ |xh xl];simpl. reflexivity. - rewrite Z.opp_add_distr, <- Z.mul_opp_l. - generalize (spec_opp_c xl);destruct (w_opp_c xl) - as [l|l];intros H;unfold interp_carry in H;rewrite <- H;simpl ww_to_Z. - rewrite spec_w_0;rewrite Z.add_0_r;rewrite wwB_wBwB. - assert ([|l|] = 0). - assert (H1:= spec_to_Z l);assert (H2 := spec_to_Z xl);omega. - rewrite H0;rewrite Z.add_0_r; rewrite Z.pow_2_r; - rewrite Zmult_mod_distr_r;try apply lt_0_wB. - rewrite spec_opp;trivial. - apply Zmod_unique with (q:= -1). - exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW (w_opp_carry xh) l)). - rewrite spec_opp_carry;rewrite wwB_wBwB;ring. - Qed. - - Lemma spec_ww_opp_carry : forall x, [[ww_opp_carry x]] = wwB - [[x]] - 1. - Proof. - destruct x as [ |xh xl];simpl. rewrite spec_ww_Bm1;ring. - rewrite spec_w_WW;simpl;repeat rewrite spec_opp_carry;rewrite wwB_wBwB;ring. - Qed. - - Lemma spec_ww_pred_c : forall x, [-[ww_pred_c x]] = [[x]] - 1. - Proof. - destruct x as [ |xh xl];unfold ww_pred_c. - unfold interp_carry;rewrite spec_ww_Bm1;simpl ww_to_Z;ring. - simpl ww_to_Z;replace (([|xh|]*wB+[|xl|])-1) with ([|xh|]*wB+([|xl|]-1)). - 2:ring. generalize (spec_pred_c xl);destruct (w_pred_c xl) as [l|l]; - intros H;unfold interp_carry in H;rewrite <- H. simpl;apply spec_w_WW. - rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - assert ([|l|] = wB - 1). - assert (H1:= spec_to_Z l);assert (H2 := spec_to_Z xl);omega. - rewrite H0;change ([|xh|] + -1) with ([|xh|] - 1). - generalize (spec_pred_c xh);destruct (w_pred_c xh) as [h|h]; - intros H1;unfold interp_carry in H1;rewrite <- H1. - simpl;rewrite spec_w_Bm1;ring. - assert ([|h|] = wB - 1). - assert (H3:= spec_to_Z h);assert (H2 := spec_to_Z xh);omega. - rewrite H2;unfold interp_carry;rewrite spec_ww_Bm1;rewrite wwB_wBwB;ring. - Qed. - - Lemma spec_ww_sub_c : forall x y, [-[ww_sub_c x y]] = [[x]] - [[y]]. - Proof. - destruct y as [ |yh yl];simpl. ring. - destruct x as [ |xh xl];simpl. exact (spec_ww_opp_c (WW yh yl)). - replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|])) - with (([|xh|]-[|yh|])*wB + ([|xl|]-[|yl|])). 2:ring. - generalize (spec_sub_c xl yl);destruct (w_sub_c xl yl) as [l|l];intros H; - unfold interp_carry in H;rewrite <- H. - generalize (spec_sub_c xh yh);destruct (w_sub_c xh yh) as [h|h];intros H1; - unfold interp_carry in H1;rewrite <- H1;unfold interp_carry; - try rewrite spec_w_WW;simpl ww_to_Z;try rewrite wwB_wBwB;ring. - rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - change ([|xh|] - [|yh|] + -1) with ([|xh|] - [|yh|] - 1). - generalize (spec_sub_carry_c xh yh);destruct (w_sub_carry_c xh yh) as [h|h]; - intros H1;unfold interp_carry in *;rewrite <- H1;simpl ww_to_Z; - try rewrite wwB_wBwB;ring. - Qed. - - Lemma spec_ww_sub_carry_c : - forall x y, [-[ww_sub_carry_c x y]] = [[x]] - [[y]] - 1. - Proof. - destruct y as [ |yh yl];simpl. - unfold Z.sub;simpl;rewrite Z.add_0_r;exact (spec_ww_pred_c x). - destruct x as [ |xh xl]. - unfold interp_carry;rewrite spec_w_WW;simpl ww_to_Z;rewrite wwB_wBwB; - repeat rewrite spec_opp_carry;ring. - simpl ww_to_Z. - replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|]) - 1) - with (([|xh|]-[|yh|])*wB + ([|xl|]-[|yl|]-1)). 2:ring. - generalize (spec_sub_carry_c xl yl);destruct (w_sub_carry_c xl yl) - as [l|l];intros H;unfold interp_carry in H;rewrite <- H. - generalize (spec_sub_c xh yh);destruct (w_sub_c xh yh) as [h|h];intros H1; - unfold interp_carry in H1;rewrite <- H1;unfold interp_carry; - try rewrite spec_w_WW;simpl ww_to_Z;try rewrite wwB_wBwB;ring. - rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - change ([|xh|] - [|yh|] + -1) with ([|xh|] - [|yh|] - 1). - generalize (spec_sub_carry_c xh yh);destruct (w_sub_carry_c xh yh) as [h|h]; - intros H1;unfold interp_carry in *;rewrite <- H1;try rewrite spec_w_WW; - simpl ww_to_Z; try rewrite wwB_wBwB;ring. - Qed. - - Lemma spec_ww_pred : forall x, [[ww_pred x]] = ([[x]] - 1) mod wwB. - Proof. - destruct x as [ |xh xl];simpl. - apply Zmod_unique with (-1). apply spec_ww_to_Z;trivial. - rewrite spec_ww_Bm1;ring. - replace ([|xh|]*wB + [|xl|] - 1) with ([|xh|]*wB + ([|xl|] - 1)). 2:ring. - generalize (spec_pred_c xl);destruct (w_pred_c xl) as [l|l];intro H; - unfold interp_carry in H;rewrite <- H;simpl ww_to_Z. - rewrite Zmod_small. apply spec_w_WW. - exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xh l)). - rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - change ([|xh|] + -1) with ([|xh|] - 1). - assert ([|l|] = wB - 1). - assert (H1:= spec_to_Z l);assert (H2:= spec_to_Z xl);omega. - rewrite (mod_wwB w_digits w_to_Z);trivial. - rewrite spec_pred;rewrite spec_w_Bm1;rewrite <- H0;trivial. - Qed. - - Lemma spec_ww_sub : forall x y, [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB. - Proof. - destruct y as [ |yh yl];simpl. - ring_simplify ([[x]] - 0);rewrite Zmod_small;trivial. apply spec_ww_to_Z;trivial. - destruct x as [ |xh xl];simpl. exact (spec_ww_opp (WW yh yl)). - replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|])) - with (([|xh|] - [|yh|]) * wB + ([|xl|] - [|yl|])). 2:ring. - generalize (spec_sub_c xl yl);destruct (w_sub_c xl yl)as[l|l];intros H; - unfold interp_carry in H;rewrite <- H. - rewrite spec_w_WW;rewrite (mod_wwB w_digits w_to_Z spec_to_Z). - rewrite spec_sub;trivial. - simpl ww_to_Z;rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - rewrite (mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_sub_carry;trivial. - Qed. - - Lemma spec_ww_sub_carry : - forall x y, [[ww_sub_carry x y]] = ([[x]] - [[y]] - 1) mod wwB. - Proof. - destruct y as [ |yh yl];simpl. - ring_simplify ([[x]] - 0);exact (spec_ww_pred x). - destruct x as [ |xh xl];simpl. - apply Zmod_unique with (-1). - apply spec_ww_to_Z;trivial. - fold (ww_opp_carry (WW yh yl)). - rewrite (spec_ww_opp_carry (WW yh yl));simpl ww_to_Z;ring. - replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|]) - 1) - with (([|xh|] - [|yh|]) * wB + ([|xl|] - [|yl|] - 1)). 2:ring. - generalize (spec_sub_carry_c xl yl);destruct (w_sub_carry_c xl yl)as[l|l]; - intros H;unfold interp_carry in H;rewrite <- H;rewrite spec_w_WW. - rewrite (mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_sub;trivial. - rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - rewrite (mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_sub_carry;trivial. - Qed. - -(* End DoubleProof. *) - -End DoubleSub. - - - - - diff --git a/theories/Numbers/Cyclic/Int31/Cyclic31.v b/theories/Numbers/Cyclic/Int31/Cyclic31.v index 0e58b815..bd4f0279 100644 --- a/theories/Numbers/Cyclic/Int31/Cyclic31.v +++ b/theories/Numbers/Cyclic/Int31/Cyclic31.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Int31 numbers defines indeed a cyclic structure : Z/(2^31)Z *) @@ -18,13 +20,16 @@ Require Export Int31. Require Import Znumtheory. Require Import Zgcd_alt. Require Import Zpow_facts. -Require Import BigNumPrelude. Require Import CyclicAxioms. Require Import ROmega. +Declare ML Module "int31_syntax_plugin". + Local Open Scope nat_scope. Local Open Scope int31_scope. +Local Hint Resolve Z.lt_gt Z.div_pos : zarith. + Section Basics. (** * Basic results about [iszero], [shiftl], [shiftr] *) @@ -455,12 +460,19 @@ Section Basics. rewrite Z.succ_double_spec; auto with zarith. Qed. - Lemma phi_bounded : forall x, (0 <= phi x < 2 ^ (Z.of_nat size))%Z. + Lemma phi_nonneg : forall x, (0 <= phi x)%Z. Proof. intros. rewrite <- phibis_aux_equiv. - split. apply phibis_aux_pos. + Qed. + + Hint Resolve phi_nonneg : zarith. + + Lemma phi_bounded : forall x, (0 <= phi x < 2 ^ (Z.of_nat size))%Z. + Proof. + intros. split; [auto with zarith|]. + rewrite <- phibis_aux_equiv. change x with (nshiftr x (size-size)). apply phibis_aux_bounded; auto. Qed. @@ -1624,6 +1636,37 @@ Section Int31_Specs. rewrite Z.mul_comm, Z_div_mult; auto with zarith. Qed. + Lemma shift_unshift_mod_2 : forall n p a, 0 <= p <= n -> + ((a * 2 ^ (n - p)) mod (2^n) / 2 ^ (n - p)) mod (2^n) = + a mod 2 ^ p. + Proof. + intros. + rewrite Zmod_small. + rewrite Zmod_eq by (auto with zarith). + unfold Z.sub at 1. + rewrite Z_div_plus_full_l + by (cut (0 < 2^(n-p)); auto with zarith). + assert (2^n = 2^(n-p)*2^p). + rewrite <- Zpower_exp by (auto with zarith). + replace (n-p+p) with n; auto with zarith. + rewrite H0. + rewrite <- Zdiv_Zdiv, Z_div_mult by (auto with zarith). + rewrite (Z.mul_comm (2^(n-p))), Z.mul_assoc. + rewrite <- Z.mul_opp_l. + rewrite Z_div_mult by (auto with zarith). + symmetry; apply Zmod_eq; auto with zarith. + + remember (a * 2 ^ (n - p)) as b. + destruct (Z_mod_lt b (2^n)); auto with zarith. + split. + apply Z_div_pos; auto with zarith. + apply Zdiv_lt_upper_bound; auto with zarith. + apply Z.lt_le_trans with (2^n); auto with zarith. + rewrite <- (Z.mul_1_r (2^n)) at 1. + apply Z.mul_le_mono_nonneg; auto with zarith. + cut (0 < 2 ^ (n-p)); auto with zarith. + Qed. + Lemma spec_pos_mod : forall w p, [|ZnZ.pos_mod p w|] = [|w|] mod (2 ^ [|p|]). Proof. @@ -1654,7 +1697,7 @@ Section Int31_Specs. rewrite spec_add_mul_div by (rewrite H4; auto with zarith). change [|0|] with 0%Z; rewrite Zdiv_0_l, Z.add_0_r. rewrite H4. - apply shift_unshift_mod_2; auto with zarith. + apply shift_unshift_mod_2; simpl; auto with zarith. Qed. @@ -1973,32 +2016,24 @@ Section Int31_Specs. assert (Hp2: 0 < [|2|]) by exact (eq_refl Lt). intros Hi Hj Hij H31 Hrec; rewrite sqrt31_step_def. rewrite spec_compare, div31_phi; auto. - case Z.compare_spec; auto; intros Hc; + case Z.compare_spec; auto; intros Hc; try (split; auto; apply sqrt_test_true; auto with zarith; fail). - apply Hrec; repeat rewrite div31_phi; auto with zarith. - replace [|(j + fst (i / j)%int31)|] with ([|j|] + [|i|] / [|j|]). - split. + assert (E : [|(j + fst (i / j)%int31)|] = [|j|] + [|i|] / [|j|]). + { rewrite spec_add, div31_phi; auto using Z.mod_small with zarith. } + apply Hrec; rewrite !div31_phi, E; auto using sqrt_main with zarith. + split; try apply sqrt_test_false; auto with zarith. apply Z.le_succ_l in Hj. change (1 <= [|j|]) in Hj. Z.le_elim Hj. - replace ([|j|] + [|i|]/[|j|]) with - (1 * 2 + (([|j|] - 2) + [|i|] / [|j|])); try ring. - rewrite Z_div_plus_full_l; auto with zarith. - assert (0 <= [|i|]/ [|j|]) by (apply Z_div_pos; auto with zarith). - assert (0 <= ([|j|] - 2 + [|i|] / [|j|]) / [|2|]) ; auto with zarith. - rewrite <- Hj, Zdiv_1_r. - replace (1 + [|i|])%Z with (1 * 2 + ([|i|] - 1))%Z; try ring. - rewrite Z_div_plus_full_l; auto with zarith. - assert (0 <= ([|i|] - 1) /2)%Z by (apply Z_div_pos; auto with zarith). - change ([|2|]) with 2%Z; auto with zarith. - apply sqrt_test_false; auto with zarith. - rewrite spec_add, div31_phi; auto. - symmetry; apply Zmod_small. - split; auto with zarith. - replace [|j + fst (i / j)%int31|] with ([|j|] + [|i|] / [|j|]). - apply sqrt_main; auto with zarith. - rewrite spec_add, div31_phi; auto. - symmetry; apply Zmod_small. - split; auto with zarith. + - replace ([|j|] + [|i|]/[|j|]) with + (1 * 2 + (([|j|] - 2) + [|i|] / [|j|])) by ring. + rewrite Z_div_plus_full_l; auto with zarith. + assert (0 <= [|i|]/ [|j|]) by auto with zarith. + assert (0 <= ([|j|] - 2 + [|i|] / [|j|]) / [|2|]); auto with zarith. + - rewrite <- Hj, Zdiv_1_r. + replace (1 + [|i|]) with (1 * 2 + ([|i|] - 1)) by ring. + rewrite Z_div_plus_full_l; auto with zarith. + assert (0 <= ([|i|] - 1) /2) by auto with zarith. + change ([|2|]) with 2; auto with zarith. Qed. Lemma iter31_sqrt_correct n rec i j: 0 < [|i|] -> 0 < [|j|] -> @@ -2078,11 +2113,12 @@ Section Int31_Specs. case (phi_bounded j); intros Hbj _. case (phi_bounded il); intros Hbil _. case (phi_bounded ih); intros Hbih Hbih1. - assert (([|ih|] < [|j|] + 1)%Z); auto with zarith. + assert ([|ih|] < [|j|] + 1); auto with zarith. apply Z.square_lt_simpl_nonneg; auto with zarith. - repeat rewrite <-Z.pow_2_r; apply Z.le_lt_trans with (2 := H1). - apply Z.le_trans with ([|ih|] * base)%Z; unfold phi2, base; - try rewrite Z.pow_2_r; auto with zarith. + rewrite <- ?Z.pow_2_r; apply Z.le_lt_trans with (2 := H1). + apply Z.le_trans with ([|ih|] * wB). + - rewrite ? Z.pow_2_r; auto with zarith. + - unfold phi2. change base with wB; auto with zarith. Qed. Lemma div312_phi ih il j: (2^30 <= [|j|] -> [|ih|] < [|j|] -> @@ -2104,90 +2140,89 @@ Section Int31_Specs. Proof. assert (Hp2: (0 < [|2|])%Z) by exact (eq_refl Lt). intros Hih Hj Hij Hrec; rewrite sqrt312_step_def. - assert (H1: ([|ih|] <= [|j|])%Z) by (apply sqrt312_lower_bound with il; auto). + assert (H1: ([|ih|] <= [|j|])) by (apply sqrt312_lower_bound with il; auto). case (phi_bounded ih); intros Hih1 _. case (phi_bounded il); intros Hil1 _. case (phi_bounded j); intros _ Hj1. assert (Hp3: (0 < phi2 ih il)). - unfold phi2; apply Z.lt_le_trans with ([|ih|] * base)%Z; auto with zarith. - apply Z.mul_pos_pos; auto with zarith. - apply Z.lt_le_trans with (2:= Hih); auto with zarith. + { unfold phi2; apply Z.lt_le_trans with ([|ih|] * base); auto with zarith. + apply Z.mul_pos_pos; auto with zarith. + apply Z.lt_le_trans with (2:= Hih); auto with zarith. } rewrite spec_compare. case Z.compare_spec; intros Hc1. - split; auto. - apply sqrt_test_true; auto. - unfold phi2, base; auto with zarith. - unfold phi2; rewrite Hc1. - assert (0 <= [|il|]/[|j|]) by (apply Z_div_pos; auto with zarith). - rewrite Z.mul_comm, Z_div_plus_full_l; unfold base; auto with zarith. - simpl wB in Hj1. unfold Z.pow_pos in Hj1. simpl in Hj1. auto with zarith. - case (Z.le_gt_cases (2 ^ 30) [|j|]); intros Hjj. - rewrite spec_compare; case Z.compare_spec; - rewrite div312_phi; auto; intros Hc; - try (split; auto; apply sqrt_test_true; auto with zarith; fail). - apply Hrec. - assert (Hf1: 0 <= phi2 ih il/ [|j|]) by (apply Z_div_pos; auto with zarith). - apply Z.le_succ_l in Hj. change (1 <= [|j|]) in Hj. - Z.le_elim Hj. - 2: contradict Hc; apply Z.le_ngt; rewrite <- Hj, Zdiv_1_r; auto with zarith. - assert (Hf3: 0 < ([|j|] + phi2 ih il / [|j|]) / 2). - replace ([|j|] + phi2 ih il/ [|j|])%Z with - (1 * 2 + (([|j|] - 2) + phi2 ih il / [|j|])); try ring. - rewrite Z_div_plus_full_l; auto with zarith. - assert (0 <= ([|j|] - 2 + phi2 ih il / [|j|]) / 2) ; auto with zarith. - assert (Hf4: ([|j|] + phi2 ih il / [|j|]) / 2 < [|j|]). - apply sqrt_test_false; auto with zarith. - generalize (spec_add_c j (fst (div3121 ih il j))). - unfold interp_carry; case add31c; intros r; - rewrite div312_phi; auto with zarith. - rewrite div31_phi; change [|2|] with 2%Z; auto with zarith. - intros HH; rewrite HH; clear HH; auto with zarith. - rewrite spec_add, div31_phi; change [|2|] with 2%Z; auto. - rewrite Z.mul_1_l; intros HH. - rewrite Z.add_comm, <- Z_div_plus_full_l; auto with zarith. - change (phi v30 * 2) with (2 ^ Z.of_nat size). - rewrite HH, Zmod_small; auto with zarith. - replace (phi - match j +c fst (div3121 ih il j) with - | C0 m1 => fst (m1 / 2)%int31 - | C1 m1 => fst (m1 / 2)%int31 + v30 - end) with ((([|j|] + (phi2 ih il)/([|j|]))/2)). - apply sqrt_main; auto with zarith. - generalize (spec_add_c j (fst (div3121 ih il j))). - unfold interp_carry; case add31c; intros r; - rewrite div312_phi; auto with zarith. - rewrite div31_phi; auto with zarith. - intros HH; rewrite HH; auto with zarith. - intros HH; rewrite <- HH. - change (1 * 2 ^ Z.of_nat size) with (phi (v30) * 2). - rewrite Z_div_plus_full_l; auto with zarith. - rewrite Z.add_comm. - rewrite spec_add, Zmod_small. - rewrite div31_phi; auto. - split; auto with zarith. - case (phi_bounded (fst (r/2)%int31)); - case (phi_bounded v30); auto with zarith. - rewrite div31_phi; change (phi 2) with 2%Z; auto. - change (2 ^Z.of_nat size) with (base/2 + phi v30). - assert (phi r / 2 < base/2); auto with zarith. - apply Z.mul_lt_mono_pos_r with 2; auto with zarith. - change (base/2 * 2) with base. - apply Z.le_lt_trans with (phi r). - rewrite Z.mul_comm; apply Z_mult_div_ge; auto with zarith. - case (phi_bounded r); auto with zarith. - contradict Hij; apply Z.le_ngt. - assert ((1 + [|j|]) <= 2 ^ 30); auto with zarith. - apply Z.le_trans with ((2 ^ 30) * (2 ^ 30)); auto with zarith. - assert (0 <= 1 + [|j|]); auto with zarith. - apply Z.mul_le_mono_nonneg; auto with zarith. - change ((2 ^ 30) * (2 ^ 30)) with ((2 ^ 29) * base). - apply Z.le_trans with ([|ih|] * base); auto with zarith. - unfold phi2, base; auto with zarith. - split; auto. - apply sqrt_test_true; auto. - unfold phi2, base; auto with zarith. - apply Z.le_ge; apply Z.le_trans with (([|j|] * base)/[|j|]). - rewrite Z.mul_comm, Z_div_mult; auto with zarith. - apply Z.ge_le; apply Z_div_ge; auto with zarith. + - split; auto. + apply sqrt_test_true; auto. + + unfold phi2, base; auto with zarith. + + unfold phi2; rewrite Hc1. + assert (0 <= [|il|]/[|j|]) by (apply Z_div_pos; auto with zarith). + rewrite Z.mul_comm, Z_div_plus_full_l; auto with zarith. + change base with wB. auto with zarith. + - case (Z.le_gt_cases (2 ^ 30) [|j|]); intros Hjj. + + rewrite spec_compare; case Z.compare_spec; + rewrite div312_phi; auto; intros Hc; + try (split; auto; apply sqrt_test_true; auto with zarith; fail). + apply Hrec. + * assert (Hf1: 0 <= phi2 ih il/ [|j|]) by auto with zarith. + apply Z.le_succ_l in Hj. change (1 <= [|j|]) in Hj. + Z.le_elim Hj; + [ | contradict Hc; apply Z.le_ngt; + rewrite <- Hj, Zdiv_1_r; auto with zarith ]. + assert (Hf3: 0 < ([|j|] + phi2 ih il / [|j|]) / 2). + { replace ([|j|] + phi2 ih il/ [|j|]) with + (1 * 2 + (([|j|] - 2) + phi2 ih il / [|j|])); try ring. + rewrite Z_div_plus_full_l; auto with zarith. + assert (0 <= ([|j|] - 2 + phi2 ih il / [|j|]) / 2) ; + auto with zarith. } + assert (Hf4: ([|j|] + phi2 ih il / [|j|]) / 2 < [|j|]). + { apply sqrt_test_false; auto with zarith. } + generalize (spec_add_c j (fst (div3121 ih il j))). + unfold interp_carry; case add31c; intros r; + rewrite div312_phi; auto with zarith. + { rewrite div31_phi; change [|2|] with 2; auto with zarith. + intros HH; rewrite HH; clear HH; auto with zarith. } + { rewrite spec_add, div31_phi; change [|2|] with 2; auto. + rewrite Z.mul_1_l; intros HH. + rewrite Z.add_comm, <- Z_div_plus_full_l; auto with zarith. + change (phi v30 * 2) with (2 ^ Z.of_nat size). + rewrite HH, Zmod_small; auto with zarith. } + * replace (phi _) with (([|j|] + (phi2 ih il)/([|j|]))/2); + [ apply sqrt_main; auto with zarith | ]. + generalize (spec_add_c j (fst (div3121 ih il j))). + unfold interp_carry; case add31c; intros r; + rewrite div312_phi; auto with zarith. + { rewrite div31_phi; auto with zarith. + intros HH; rewrite HH; auto with zarith. } + { intros HH; rewrite <- HH. + change (1 * 2 ^ Z.of_nat size) with (phi (v30) * 2). + rewrite Z_div_plus_full_l; auto with zarith. + rewrite Z.add_comm. + rewrite spec_add, Zmod_small. + - rewrite div31_phi; auto. + - split; auto with zarith. + + case (phi_bounded (fst (r/2)%int31)); + case (phi_bounded v30); auto with zarith. + + rewrite div31_phi; change (phi 2) with 2; auto. + change (2 ^Z.of_nat size) with (base/2 + phi v30). + assert (phi r / 2 < base/2); auto with zarith. + apply Z.mul_lt_mono_pos_r with 2; auto with zarith. + change (base/2 * 2) with base. + apply Z.le_lt_trans with (phi r). + * rewrite Z.mul_comm; apply Z_mult_div_ge; auto with zarith. + * case (phi_bounded r); auto with zarith. } + + contradict Hij; apply Z.le_ngt. + assert ((1 + [|j|]) <= 2 ^ 30); auto with zarith. + apply Z.le_trans with ((2 ^ 30) * (2 ^ 30)); auto with zarith. + * assert (0 <= 1 + [|j|]); auto with zarith. + apply Z.mul_le_mono_nonneg; auto with zarith. + * change ((2 ^ 30) * (2 ^ 30)) with ((2 ^ 29) * base). + apply Z.le_trans with ([|ih|] * base); + change wB with base in *; auto with zarith. + unfold phi2, base; auto with zarith. + - split; auto. + apply sqrt_test_true; auto. + + unfold phi2, base; auto with zarith. + + apply Z.le_ge; apply Z.le_trans with (([|j|] * base)/[|j|]). + * rewrite Z.mul_comm, Z_div_mult; auto with zarith. + * apply Z.ge_le; apply Z_div_ge; auto with zarith. Qed. Lemma iter312_sqrt_correct n rec ih il j: @@ -2209,7 +2244,7 @@ Section Int31_Specs. intros j3 Hj3 Hpj3. apply HHrec; auto. rewrite Nat2Z.inj_succ, Z.pow_succ_r. - apply Z.le_trans with (2 ^Z.of_nat n + [|j2|])%Z; auto with zarith. + apply Z.le_trans with (2 ^Z.of_nat n + [|j2|]); auto with zarith. apply Nat2Z.is_nonneg. Qed. diff --git a/theories/Numbers/Cyclic/Int31/Int31.v b/theories/Numbers/Cyclic/Int31/Int31.v index ff4b998e..9f8da831 100644 --- a/theories/Numbers/Cyclic/Int31/Int31.v +++ b/theories/Numbers/Cyclic/Int31/Int31.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Cyclic/Int31/Ring31.v b/theories/Numbers/Cyclic/Int31/Ring31.v index d160f5f1..b6935294 100644 --- a/theories/Numbers/Cyclic/Int31/Ring31.v +++ b/theories/Numbers/Cyclic/Int31/Ring31.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Int31 numbers defines Z/(2^31)Z, and can hence be equipped diff --git a/theories/Numbers/Cyclic/ZModulo/ZModulo.v b/theories/Numbers/Cyclic/ZModulo/ZModulo.v index 04fc5a8d..784e8175 100644 --- a/theories/Numbers/Cyclic/ZModulo/ZModulo.v +++ b/theories/Numbers/Cyclic/ZModulo/ZModulo.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Type [Z] viewed modulo a particular constant corresponds to [Z/nZ] @@ -18,7 +20,7 @@ Set Implicit Arguments. Require Import Bool. Require Import ZArith. Require Import Znumtheory. -Require Import BigNumPrelude. +Require Import Zpow_facts. Require Import DoubleType. Require Import CyclicAxioms. @@ -48,13 +50,14 @@ Section ZModulo. Lemma spec_more_than_1_digit: 1 < Zpos digits. Proof. - generalize digits_ne_1; destruct digits; auto. + generalize digits_ne_1; destruct digits; red; auto. destruct 1; auto. Qed. Let digits_gt_1 := spec_more_than_1_digit. Lemma wB_pos : wB > 0. Proof. + apply Z.lt_gt. unfold wB, base; auto with zarith. Qed. Hint Resolve wB_pos. @@ -558,7 +561,7 @@ Section ZModulo. apply Zmod_small. generalize (Z_mod_lt [|w|] (2 ^ [|p|])); intros. split. - destruct H; auto with zarith. + destruct H; auto using Z.lt_gt with zarith. apply Z.le_lt_trans with [|w|]; auto with zarith. apply Zmod_le; auto with zarith. Qed. diff --git a/theories/Numbers/DecimalFacts.v b/theories/Numbers/DecimalFacts.v new file mode 100644 index 00000000..0f490527 --- /dev/null +++ b/theories/Numbers/DecimalFacts.v @@ -0,0 +1,143 @@ +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) + +(** * DecimalFacts : some facts about Decimal numbers *) + +Require Import Decimal. + +Lemma uint_dec (d d' : uint) : { d = d' } + { d <> d' }. +Proof. + decide equality. +Defined. + +Lemma rev_revapp d d' : + rev (revapp d d') = revapp d' d. +Proof. + revert d'. induction d; simpl; intros; now rewrite ?IHd. +Qed. + +Lemma rev_rev d : rev (rev d) = d. +Proof. + apply rev_revapp. +Qed. + +(** Normalization on little-endian numbers *) + +Fixpoint nztail d := + match d with + | Nil => Nil + | D0 d => match nztail d with Nil => Nil | d' => D0 d' end + | D1 d => D1 (nztail d) + | D2 d => D2 (nztail d) + | D3 d => D3 (nztail d) + | D4 d => D4 (nztail d) + | D5 d => D5 (nztail d) + | D6 d => D6 (nztail d) + | D7 d => D7 (nztail d) + | D8 d => D8 (nztail d) + | D9 d => D9 (nztail d) + end. + +Definition lnorm d := + match nztail d with + | Nil => zero + | d => d + end. + +Lemma nzhead_revapp_0 d d' : nztail d = Nil -> + nzhead (revapp d d') = nzhead d'. +Proof. + revert d'. induction d; intros d' [=]; simpl; trivial. + destruct (nztail d); now rewrite IHd. +Qed. + +Lemma nzhead_revapp d d' : nztail d <> Nil -> + nzhead (revapp d d') = revapp (nztail d) d'. +Proof. + revert d'. + induction d; intros d' H; simpl in *; + try destruct (nztail d) eqn:E; + (now rewrite ?nzhead_revapp_0) || (now rewrite IHd). +Qed. + +Lemma nzhead_rev d : nztail d <> Nil -> + nzhead (rev d) = rev (nztail d). +Proof. + apply nzhead_revapp. +Qed. + +Lemma rev_nztail_rev d : + rev (nztail (rev d)) = nzhead d. +Proof. + destruct (uint_dec (nztail (rev d)) Nil) as [H|H]. + - rewrite H. unfold rev; simpl. + rewrite <- (rev_rev d). symmetry. + now apply nzhead_revapp_0. + - now rewrite <- nzhead_rev, rev_rev. +Qed. + +Lemma revapp_nil_inv d d' : revapp d d' = Nil -> d = Nil /\ d' = Nil. +Proof. + revert d'. + induction d; simpl; intros d' H; auto; now apply IHd in H. +Qed. + +Lemma rev_nil_inv d : rev d = Nil -> d = Nil. +Proof. + apply revapp_nil_inv. +Qed. + +Lemma rev_lnorm_rev d : + rev (lnorm (rev d)) = unorm d. +Proof. + unfold unorm, lnorm. + rewrite <- rev_nztail_rev. + destruct nztail; simpl; trivial; + destruct rev eqn:E; trivial; now apply rev_nil_inv in E. +Qed. + +Lemma nzhead_nonzero d d' : nzhead d <> D0 d'. +Proof. + induction d; easy. +Qed. + +Lemma unorm_0 d : unorm d = zero <-> nzhead d = Nil. +Proof. + unfold unorm. split. + - generalize (nzhead_nonzero d). + destruct nzhead; intros H [=]; trivial. now destruct (H u). + - now intros ->. +Qed. + +Lemma unorm_nonnil d : unorm d <> Nil. +Proof. + unfold unorm. now destruct nzhead. +Qed. + +Lemma nzhead_invol d : nzhead (nzhead d) = nzhead d. +Proof. + now induction d. +Qed. + +Lemma unorm_invol d : unorm (unorm d) = unorm d. +Proof. + unfold unorm. + destruct (nzhead d) eqn:E; trivial. + destruct (nzhead_nonzero _ _ E). +Qed. + +Lemma norm_invol d : norm (norm d) = norm d. +Proof. + unfold norm. + destruct d. + - f_equal. apply unorm_invol. + - destruct (nzhead d) eqn:E; auto. + destruct (nzhead_nonzero _ _ E). +Qed. diff --git a/theories/Numbers/DecimalN.v b/theories/Numbers/DecimalN.v new file mode 100644 index 00000000..ef00e280 --- /dev/null +++ b/theories/Numbers/DecimalN.v @@ -0,0 +1,107 @@ +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) + +(** * DecimalN + + Proofs that conversions between decimal numbers and [N] + are bijections *) + +Require Import Decimal DecimalFacts DecimalPos PArith NArith. + +Module Unsigned. + +Lemma of_to (n:N) : N.of_uint (N.to_uint n) = n. +Proof. + destruct n. + - reflexivity. + - apply DecimalPos.Unsigned.of_to. +Qed. + +Lemma to_of (d:uint) : N.to_uint (N.of_uint d) = unorm d. +Proof. + exact (DecimalPos.Unsigned.to_of d). +Qed. + +Lemma to_uint_inj n n' : N.to_uint n = N.to_uint n' -> n = n'. +Proof. + intros E. now rewrite <- (of_to n), <- (of_to n'), E. +Qed. + +Lemma to_uint_surj d : exists p, N.to_uint p = unorm d. +Proof. + exists (N.of_uint d). apply to_of. +Qed. + +Lemma of_uint_norm d : N.of_uint (unorm d) = N.of_uint d. +Proof. + now induction d. +Qed. + +Lemma of_inj d d' : + N.of_uint d = N.of_uint d' -> unorm d = unorm d'. +Proof. + intros. rewrite <- !to_of. now f_equal. +Qed. + +Lemma of_iff d d' : N.of_uint d = N.of_uint d' <-> unorm d = unorm d'. +Proof. + split. apply of_inj. intros E. rewrite <- of_uint_norm, E. + apply of_uint_norm. +Qed. + +End Unsigned. + +(** Conversion from/to signed decimal numbers *) + +Module Signed. + +Lemma of_to (n:N) : N.of_int (N.to_int n) = Some n. +Proof. + unfold N.to_int, N.of_int, norm. f_equal. + rewrite Unsigned.of_uint_norm. apply Unsigned.of_to. +Qed. + +Lemma to_of (d:int)(n:N) : N.of_int d = Some n -> N.to_int n = norm d. +Proof. + unfold N.of_int. + destruct (norm d) eqn:Hd; intros [= <-]. + unfold N.to_int. rewrite Unsigned.to_of. f_equal. + revert Hd; destruct d; simpl. + - intros [= <-]. apply unorm_invol. + - destruct (nzhead d); now intros [= <-]. +Qed. + +Lemma to_int_inj n n' : N.to_int n = N.to_int n' -> n = n'. +Proof. + intro E. + assert (E' : Some n = Some n'). + { now rewrite <- (of_to n), <- (of_to n'), E. } + now injection E'. +Qed. + +Lemma to_int_pos_surj d : exists n, N.to_int n = norm (Pos d). +Proof. + exists (N.of_uint d). unfold N.to_int. now rewrite Unsigned.to_of. +Qed. + +Lemma of_int_norm d : N.of_int (norm d) = N.of_int d. +Proof. + unfold N.of_int. now rewrite norm_invol. +Qed. + +Lemma of_inj_pos d d' : + N.of_int (Pos d) = N.of_int (Pos d') -> unorm d = unorm d'. +Proof. + unfold N.of_int. simpl. intros [= H]. apply Unsigned.of_inj. + change Pos.of_uint with N.of_uint in H. + now rewrite <- Unsigned.of_uint_norm, H, Unsigned.of_uint_norm. +Qed. + +End Signed. diff --git a/theories/Numbers/DecimalNat.v b/theories/Numbers/DecimalNat.v new file mode 100644 index 00000000..5ffe1688 --- /dev/null +++ b/theories/Numbers/DecimalNat.v @@ -0,0 +1,302 @@ +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) + +(** * DecimalNat + + Proofs that conversions between decimal numbers and [nat] + are bijections. *) + +Require Import Decimal DecimalFacts Arith. + +Module Unsigned. + +(** A few helper functions used during proofs *) + +Definition hd d := + match d with + | Nil => 0 + | D0 _ => 0 + | D1 _ => 1 + | D2 _ => 2 + | D3 _ => 3 + | D4 _ => 4 + | D5 _ => 5 + | D6 _ => 6 + | D7 _ => 7 + | D8 _ => 8 + | D9 _ => 9 +end. + +Definition tl d := + match d with + | Nil => d + | D0 d | D1 d | D2 d | D3 d | D4 d | D5 d | D6 d | D7 d | D8 d | D9 d => d +end. + +Fixpoint usize (d:uint) : nat := + match d with + | Nil => 0 + | D0 d => S (usize d) + | D1 d => S (usize d) + | D2 d => S (usize d) + | D3 d => S (usize d) + | D4 d => S (usize d) + | D5 d => S (usize d) + | D6 d => S (usize d) + | D7 d => S (usize d) + | D8 d => S (usize d) + | D9 d => S (usize d) + end. + +(** A direct version of [to_little_uint], not tail-recursive *) +Fixpoint to_lu n := + match n with + | 0 => Decimal.zero + | S n => Little.succ (to_lu n) + end. + +(** A direct version of [of_little_uint] *) +Fixpoint of_lu (d:uint) : nat := + match d with + | Nil => 0 + | D0 d => 10 * of_lu d + | D1 d => 1 + 10 * of_lu d + | D2 d => 2 + 10 * of_lu d + | D3 d => 3 + 10 * of_lu d + | D4 d => 4 + 10 * of_lu d + | D5 d => 5 + 10 * of_lu d + | D6 d => 6 + 10 * of_lu d + | D7 d => 7 + 10 * of_lu d + | D8 d => 8 + 10 * of_lu d + | D9 d => 9 + 10 * of_lu d + end. + +(** Properties of [to_lu] *) + +Lemma to_lu_succ n : to_lu (S n) = Little.succ (to_lu n). +Proof. + reflexivity. +Qed. + +Lemma to_little_uint_succ n d : + Nat.to_little_uint n (Little.succ d) = + Little.succ (Nat.to_little_uint n d). +Proof. + revert d; induction n; simpl; trivial. +Qed. + +Lemma to_lu_equiv n : + to_lu n = Nat.to_little_uint n zero. +Proof. + induction n; simpl; trivial. + now rewrite IHn, <- to_little_uint_succ. +Qed. + +Lemma to_uint_alt n : + Nat.to_uint n = rev (to_lu n). +Proof. + unfold Nat.to_uint. f_equal. symmetry. apply to_lu_equiv. +Qed. + +(** Properties of [of_lu] *) + +Lemma of_lu_eqn d : + of_lu d = hd d + 10 * of_lu (tl d). +Proof. + induction d; simpl; trivial. +Qed. + +Ltac simpl_of_lu := + match goal with + | |- context [ of_lu (?f ?x) ] => + rewrite (of_lu_eqn (f x)); simpl hd; simpl tl + end. + +Lemma of_lu_succ d : + of_lu (Little.succ d) = S (of_lu d). +Proof. + induction d; trivial. + simpl_of_lu. rewrite IHd. simpl_of_lu. + now rewrite Nat.mul_succ_r, <- (Nat.add_comm 10). +Qed. + +Lemma of_to_lu n : + of_lu (to_lu n) = n. +Proof. + induction n; simpl; trivial. rewrite of_lu_succ. now f_equal. +Qed. + +Lemma of_lu_revapp d d' : +of_lu (revapp d d') = + of_lu (rev d) + of_lu d' * 10^usize d. +Proof. + revert d'. + induction d; intro d'; simpl usize; + [ simpl; now rewrite Nat.mul_1_r | .. ]; + unfold rev; simpl revapp; rewrite 2 IHd; + rewrite <- Nat.add_assoc; f_equal; simpl_of_lu; simpl of_lu; + rewrite Nat.pow_succ_r'; ring. +Qed. + +Lemma of_uint_acc_spec n d : + Nat.of_uint_acc d n = of_lu (rev d) + n * 10^usize d. +Proof. + revert n. induction d; intros; + simpl Nat.of_uint_acc; rewrite ?Nat.tail_mul_spec, ?IHd; + simpl rev; simpl usize; rewrite ?Nat.pow_succ_r'; + [ simpl; now rewrite Nat.mul_1_r | .. ]; + unfold rev at 2; simpl revapp; rewrite of_lu_revapp; + simpl of_lu; ring. +Qed. + +Lemma of_uint_alt d : Nat.of_uint d = of_lu (rev d). +Proof. + unfold Nat.of_uint. now rewrite of_uint_acc_spec. +Qed. + +(** First main bijection result *) + +Lemma of_to (n:nat) : Nat.of_uint (Nat.to_uint n) = n. +Proof. + rewrite to_uint_alt, of_uint_alt, rev_rev. apply of_to_lu. +Qed. + +(** The other direction *) + +Lemma to_lu_tenfold n : n<>0 -> + to_lu (10 * n) = D0 (to_lu n). +Proof. + induction n. + - simpl. now destruct 1. + - intros _. + destruct (Nat.eq_dec n 0) as [->|H]; simpl; trivial. + rewrite !Nat.add_succ_r. + simpl in *. rewrite (IHn H). now destruct (to_lu n). +Qed. + +Lemma of_lu_0 d : of_lu d = 0 <-> nztail d = Nil. +Proof. + induction d; try simpl_of_lu; try easy. + rewrite Nat.add_0_l. + split; intros H. + - apply Nat.eq_mul_0_r in H; auto. + rewrite IHd in H. simpl. now rewrite H. + - simpl in H. destruct (nztail d); try discriminate. + now destruct IHd as [_ ->]. +Qed. + +Lemma to_of_lu_tenfold d : + to_lu (of_lu d) = lnorm d -> + to_lu (10 * of_lu d) = lnorm (D0 d). +Proof. + intro IH. + destruct (Nat.eq_dec (of_lu d) 0) as [H|H]. + - rewrite H. simpl. rewrite of_lu_0 in H. + unfold lnorm. simpl. now rewrite H. + - rewrite (to_lu_tenfold _ H), IH. + rewrite of_lu_0 in H. + unfold lnorm. simpl. now destruct (nztail d). +Qed. + +Lemma to_of_lu d : to_lu (of_lu d) = lnorm d. +Proof. + induction d; [ reflexivity | .. ]; + simpl_of_lu; + rewrite ?Nat.add_succ_l, Nat.add_0_l, ?to_lu_succ, to_of_lu_tenfold + by assumption; + unfold lnorm; simpl; now destruct nztail. +Qed. + +(** Second bijection result *) + +Lemma to_of (d:uint) : Nat.to_uint (Nat.of_uint d) = unorm d. +Proof. + rewrite to_uint_alt, of_uint_alt, to_of_lu. + apply rev_lnorm_rev. +Qed. + +(** Some consequences *) + +Lemma to_uint_inj n n' : Nat.to_uint n = Nat.to_uint n' -> n = n'. +Proof. + intro EQ. + now rewrite <- (of_to n), <- (of_to n'), EQ. +Qed. + +Lemma to_uint_surj d : exists n, Nat.to_uint n = unorm d. +Proof. + exists (Nat.of_uint d). apply to_of. +Qed. + +Lemma of_uint_norm d : Nat.of_uint (unorm d) = Nat.of_uint d. +Proof. + unfold Nat.of_uint. now induction d. +Qed. + +Lemma of_inj d d' : + Nat.of_uint d = Nat.of_uint d' -> unorm d = unorm d'. +Proof. + intros. rewrite <- !to_of. now f_equal. +Qed. + +Lemma of_iff d d' : Nat.of_uint d = Nat.of_uint d' <-> unorm d = unorm d'. +Proof. + split. apply of_inj. intros E. rewrite <- of_uint_norm, E. + apply of_uint_norm. +Qed. + +End Unsigned. + +(** Conversion from/to signed decimal numbers *) + +Module Signed. + +Lemma of_to (n:nat) : Nat.of_int (Nat.to_int n) = Some n. +Proof. + unfold Nat.to_int, Nat.of_int, norm. f_equal. + rewrite Unsigned.of_uint_norm. apply Unsigned.of_to. +Qed. + +Lemma to_of (d:int)(n:nat) : Nat.of_int d = Some n -> Nat.to_int n = norm d. +Proof. + unfold Nat.of_int. + destruct (norm d) eqn:Hd; intros [= <-]. + unfold Nat.to_int. rewrite Unsigned.to_of. f_equal. + revert Hd; destruct d; simpl. + - intros [= <-]. apply unorm_invol. + - destruct (nzhead d); now intros [= <-]. +Qed. + +Lemma to_int_inj n n' : Nat.to_int n = Nat.to_int n' -> n = n'. +Proof. + intro E. + assert (E' : Some n = Some n'). + { now rewrite <- (of_to n), <- (of_to n'), E. } + now injection E'. +Qed. + +Lemma to_int_pos_surj d : exists n, Nat.to_int n = norm (Pos d). +Proof. + exists (Nat.of_uint d). unfold Nat.to_int. now rewrite Unsigned.to_of. +Qed. + +Lemma of_int_norm d : Nat.of_int (norm d) = Nat.of_int d. +Proof. + unfold Nat.of_int. now rewrite norm_invol. +Qed. + +Lemma of_inj_pos d d' : + Nat.of_int (Pos d) = Nat.of_int (Pos d') -> unorm d = unorm d'. +Proof. + unfold Nat.of_int. simpl. intros [= H]. apply Unsigned.of_inj. + now rewrite <- Unsigned.of_uint_norm, H, Unsigned.of_uint_norm. +Qed. + +End Signed. diff --git a/theories/Numbers/DecimalPos.v b/theories/Numbers/DecimalPos.v new file mode 100644 index 00000000..722e73d9 --- /dev/null +++ b/theories/Numbers/DecimalPos.v @@ -0,0 +1,383 @@ +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) + +(** * DecimalPos + + Proofs that conversions between decimal numbers and [positive] + are bijections. *) + +Require Import Decimal DecimalFacts PArith NArith. + +Module Unsigned. + +Local Open Scope N. + +(** A direct version of [of_little_uint] *) +Fixpoint of_lu (d:uint) : N := + match d with + | Nil => 0 + | D0 d => 10 * of_lu d + | D1 d => 1 + 10 * of_lu d + | D2 d => 2 + 10 * of_lu d + | D3 d => 3 + 10 * of_lu d + | D4 d => 4 + 10 * of_lu d + | D5 d => 5 + 10 * of_lu d + | D6 d => 6 + 10 * of_lu d + | D7 d => 7 + 10 * of_lu d + | D8 d => 8 + 10 * of_lu d + | D9 d => 9 + 10 * of_lu d + end. + +Definition hd d := +match d with + | Nil => 0 + | D0 _ => 0 + | D1 _ => 1 + | D2 _ => 2 + | D3 _ => 3 + | D4 _ => 4 + | D5 _ => 5 + | D6 _ => 6 + | D7 _ => 7 + | D8 _ => 8 + | D9 _ => 9 +end. + +Definition tl d := + match d with + | Nil => d + | D0 d | D1 d | D2 d | D3 d | D4 d | D5 d | D6 d | D7 d | D8 d | D9 d => d +end. + +Lemma of_lu_eqn d : + of_lu d = hd d + 10 * (of_lu (tl d)). +Proof. + induction d; simpl; trivial. +Qed. + +Ltac simpl_of_lu := + match goal with + | |- context [ of_lu (?f ?x) ] => + rewrite (of_lu_eqn (f x)); simpl hd; simpl tl + end. + +Fixpoint usize (d:uint) : N := + match d with + | Nil => 0 + | D0 d => N.succ (usize d) + | D1 d => N.succ (usize d) + | D2 d => N.succ (usize d) + | D3 d => N.succ (usize d) + | D4 d => N.succ (usize d) + | D5 d => N.succ (usize d) + | D6 d => N.succ (usize d) + | D7 d => N.succ (usize d) + | D8 d => N.succ (usize d) + | D9 d => N.succ (usize d) + end. + +Lemma of_lu_revapp d d' : + of_lu (revapp d d') = + of_lu (rev d) + of_lu d' * 10^usize d. +Proof. + revert d'. + induction d; simpl; intro d'; [ now rewrite N.mul_1_r | .. ]; + unfold rev; simpl revapp; rewrite 2 IHd; + rewrite <- N.add_assoc; f_equal; simpl_of_lu; simpl of_lu; + rewrite N.pow_succ_r'; ring. +Qed. + +Definition Nadd n p := + match n with + | N0 => p + | Npos p0 => (p0+p)%positive + end. + +Lemma Nadd_simpl n p q : Npos (Nadd n (p * q)) = n + Npos p * Npos q. +Proof. + now destruct n. +Qed. + +Lemma of_uint_acc_eqn d acc : d<>Nil -> + Pos.of_uint_acc d acc = Pos.of_uint_acc (tl d) (Nadd (hd d) (10*acc)). +Proof. + destruct d; simpl; trivial. now destruct 1. +Qed. + +Lemma of_uint_acc_rev d acc : + Npos (Pos.of_uint_acc d acc) = + of_lu (rev d) + (Npos acc) * 10^usize d. +Proof. + revert acc. + induction d; intros; simpl usize; + [ simpl; now rewrite Pos.mul_1_r | .. ]; + rewrite N.pow_succ_r'; + unfold rev; simpl revapp; try rewrite of_lu_revapp; simpl of_lu; + rewrite of_uint_acc_eqn by easy; simpl tl; simpl hd; + rewrite IHd, Nadd_simpl; ring. +Qed. + +Lemma of_uint_alt d : Pos.of_uint d = of_lu (rev d). +Proof. + induction d; simpl; trivial; unfold rev; simpl revapp; + rewrite of_lu_revapp; simpl of_lu; try apply of_uint_acc_rev. + rewrite IHd. ring. +Qed. + +Lemma of_lu_rev d : Pos.of_uint (rev d) = of_lu d. +Proof. + rewrite of_uint_alt. now rewrite rev_rev. +Qed. + +Lemma of_lu_double_gen d : + of_lu (Little.double d) = N.double (of_lu d) /\ + of_lu (Little.succ_double d) = N.succ_double (of_lu d). +Proof. + rewrite N.double_spec, N.succ_double_spec. + induction d; try destruct IHd as (IH1,IH2); + simpl Little.double; simpl Little.succ_double; + repeat (simpl_of_lu; rewrite ?IH1, ?IH2); split; reflexivity || ring. +Qed. + +Lemma of_lu_double d : + of_lu (Little.double d) = N.double (of_lu d). +Proof. + apply of_lu_double_gen. +Qed. + +Lemma of_lu_succ_double d : + of_lu (Little.succ_double d) = N.succ_double (of_lu d). +Proof. + apply of_lu_double_gen. +Qed. + +(** First bijection result *) + +Lemma of_to (p:positive) : Pos.of_uint (Pos.to_uint p) = Npos p. +Proof. + unfold Pos.to_uint. + rewrite of_lu_rev. + induction p; simpl; trivial. + - now rewrite of_lu_succ_double, IHp. + - now rewrite of_lu_double, IHp. +Qed. + +(** The other direction *) + +Definition to_lu n := + match n with + | N0 => Decimal.zero + | Npos p => Pos.to_little_uint p + end. + +Lemma succ_double_alt d : + Little.succ_double d = Little.succ (Little.double d). +Proof. + now induction d. +Qed. + +Lemma double_succ d : + Little.double (Little.succ d) = + Little.succ (Little.succ_double d). +Proof. + induction d; simpl; f_equal; auto using succ_double_alt. +Qed. + +Lemma to_lu_succ n : + to_lu (N.succ n) = Little.succ (to_lu n). +Proof. + destruct n; simpl; trivial. + induction p; simpl; rewrite ?IHp; + auto using succ_double_alt, double_succ. +Qed. + +Lemma nat_iter_S n {A} (f:A->A) i : + Nat.iter (S n) f i = f (Nat.iter n f i). +Proof. + reflexivity. +Qed. + +Lemma nat_iter_0 {A} (f:A->A) i : Nat.iter 0 f i = i. +Proof. + reflexivity. +Qed. + +Lemma to_ldec_tenfold p : + to_lu (10 * Npos p) = D0 (to_lu (Npos p)). +Proof. + induction p using Pos.peano_rect. + - trivial. + - change (N.pos (Pos.succ p)) with (N.succ (N.pos p)). + rewrite N.mul_succ_r. + change 10 at 2 with (Nat.iter 10%nat N.succ 0). + rewrite ?nat_iter_S, nat_iter_0. + rewrite !N.add_succ_r, N.add_0_r, !to_lu_succ, IHp. + destruct (to_lu (N.pos p)); simpl; auto. +Qed. + +Lemma of_lu_0 d : of_lu d = 0 <-> nztail d = Nil. +Proof. + induction d; try simpl_of_lu; split; trivial; try discriminate; + try (intros H; now apply N.eq_add_0 in H). + - rewrite N.add_0_l. intros H. + apply N.eq_mul_0_r in H; [|easy]. rewrite IHd in H. + simpl. now rewrite H. + - simpl. destruct (nztail d); try discriminate. + now destruct IHd as [_ ->]. +Qed. + +Lemma to_of_lu_tenfold d : + to_lu (of_lu d) = lnorm d -> + to_lu (10 * of_lu d) = lnorm (D0 d). +Proof. + intro IH. + destruct (N.eq_dec (of_lu d) 0) as [H|H]. + - rewrite H. simpl. rewrite of_lu_0 in H. + unfold lnorm. simpl. now rewrite H. + - destruct (of_lu d) eqn:Eq; [easy| ]. + rewrite to_ldec_tenfold; auto. rewrite IH. + rewrite <- Eq in H. rewrite of_lu_0 in H. + unfold lnorm. simpl. now destruct (nztail d). +Qed. + +Lemma Nadd_alt n m : n + m = Nat.iter (N.to_nat n) N.succ m. +Proof. + destruct n. trivial. + induction p using Pos.peano_rect. + - now rewrite N.add_1_l. + - change (N.pos (Pos.succ p)) with (N.succ (N.pos p)). + now rewrite N.add_succ_l, IHp, N2Nat.inj_succ. +Qed. + +Ltac simpl_to_nat := simpl N.to_nat; unfold Pos.to_nat; simpl Pos.iter_op. + +Lemma to_of_lu d : to_lu (of_lu d) = lnorm d. +Proof. + induction d; [reflexivity|..]; + simpl_of_lu; rewrite Nadd_alt; simpl_to_nat; + rewrite ?nat_iter_S, nat_iter_0, ?to_lu_succ, to_of_lu_tenfold by assumption; + unfold lnorm; simpl; destruct nztail; auto. +Qed. + +(** Second bijection result *) + +Lemma to_of (d:uint) : N.to_uint (Pos.of_uint d) = unorm d. +Proof. + rewrite of_uint_alt. + unfold N.to_uint, Pos.to_uint. + destruct (of_lu (rev d)) eqn:H. + - rewrite of_lu_0 in H. rewrite <- rev_lnorm_rev. + unfold lnorm. now rewrite H. + - change (Pos.to_little_uint p) with (to_lu (N.pos p)). + rewrite <- H. rewrite to_of_lu. apply rev_lnorm_rev. +Qed. + +(** Some consequences *) + +Lemma to_uint_nonzero p : Pos.to_uint p <> zero. +Proof. + intro E. generalize (of_to p). now rewrite E. +Qed. + +Lemma to_uint_nonnil p : Pos.to_uint p <> Nil. +Proof. + intros E. generalize (of_to p). now rewrite E. +Qed. + +Lemma to_uint_inj p p' : Pos.to_uint p = Pos.to_uint p' -> p = p'. +Proof. + intro E. + assert (E' : N.pos p = N.pos p'). + { now rewrite <- (of_to p), <- (of_to p'), E. } + now injection E'. +Qed. + +Lemma to_uint_pos_surj d : + unorm d<>zero -> exists p, Pos.to_uint p = unorm d. +Proof. + intros. + destruct (Pos.of_uint d) eqn:E. + - destruct H. generalize (to_of d). now rewrite E. + - exists p. generalize (to_of d). now rewrite E. +Qed. + +Lemma of_uint_norm d : Pos.of_uint (unorm d) = Pos.of_uint d. +Proof. + now induction d. +Qed. + +Lemma of_inj d d' : + Pos.of_uint d = Pos.of_uint d' -> unorm d = unorm d'. +Proof. + intros. rewrite <- !to_of. now f_equal. +Qed. + +Lemma of_iff d d' : Pos.of_uint d = Pos.of_uint d' <-> unorm d = unorm d'. +Proof. + split. apply of_inj. intros E. rewrite <- of_uint_norm, E. + apply of_uint_norm. +Qed. + +End Unsigned. + +(** Conversion from/to signed decimal numbers *) + +Module Signed. + +Lemma of_to (p:positive) : Pos.of_int (Pos.to_int p) = Some p. +Proof. + unfold Pos.to_int, Pos.of_int, norm. + now rewrite Unsigned.of_to. +Qed. + +Lemma to_of (d:int)(p:positive) : + Pos.of_int d = Some p -> Pos.to_int p = norm d. +Proof. + unfold Pos.of_int. + destruct d; [ | intros [=]]. + simpl norm. rewrite <- Unsigned.to_of. + destruct (Pos.of_uint d); now intros [= <-]. +Qed. + +Lemma to_int_inj p p' : Pos.to_int p = Pos.to_int p' -> p = p'. +Proof. + intro E. + assert (E' : Some p = Some p'). + { now rewrite <- (of_to p), <- (of_to p'), E. } + now injection E'. +Qed. + +Lemma to_int_pos_surj d : + unorm d <> zero -> exists p, Pos.to_int p = norm (Pos d). +Proof. + simpl. unfold Pos.to_int. intros H. + destruct (Unsigned.to_uint_pos_surj d H) as (p,Hp). + exists p. now f_equal. +Qed. + +Lemma of_int_norm d : Pos.of_int (norm d) = Pos.of_int d. +Proof. + unfold Pos.of_int. + destruct d. + - simpl. now rewrite Unsigned.of_uint_norm. + - simpl. now destruct (nzhead d) eqn:H. +Qed. + +Lemma of_inj_pos d d' : + Pos.of_int (Pos d) = Pos.of_int (Pos d') -> unorm d = unorm d'. +Proof. + unfold Pos.of_int. + destruct (Pos.of_uint d) eqn:Hd, (Pos.of_uint d') eqn:Hd'; + intros [=]. + - apply Unsigned.of_inj; now rewrite Hd, Hd'. + - apply Unsigned.of_inj; rewrite Hd, Hd'; now f_equal. +Qed. + +End Signed. diff --git a/theories/Numbers/DecimalString.v b/theories/Numbers/DecimalString.v new file mode 100644 index 00000000..1a3220f6 --- /dev/null +++ b/theories/Numbers/DecimalString.v @@ -0,0 +1,265 @@ +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) + +Require Import Decimal Ascii String. + +(** * Conversion between decimal numbers and Coq strings *) + +(** Pretty straightforward, which is precisely the point of the + [Decimal.int] datatype. The only catch is [Decimal.Nil] : we could + choose to convert it as [""] or as ["0"]. In the first case, it is + awkward to consider "" (or "-") as a number, while in the second case + we don't have a perfect bijection. Since the second variant is implemented + thanks to the first one, we provide both. *) + +Local Open Scope string_scope. + +(** Parsing one char *) + +Definition uint_of_char (a:ascii)(d:option uint) := + match d with + | None => None + | Some d => + match a with + | "0" => Some (D0 d) + | "1" => Some (D1 d) + | "2" => Some (D2 d) + | "3" => Some (D3 d) + | "4" => Some (D4 d) + | "5" => Some (D5 d) + | "6" => Some (D6 d) + | "7" => Some (D7 d) + | "8" => Some (D8 d) + | "9" => Some (D9 d) + | _ => None + end + end%char. + +Lemma uint_of_char_spec c d d' : + uint_of_char c (Some d) = Some d' -> + (c = "0" /\ d' = D0 d \/ + c = "1" /\ d' = D1 d \/ + c = "2" /\ d' = D2 d \/ + c = "3" /\ d' = D3 d \/ + c = "4" /\ d' = D4 d \/ + c = "5" /\ d' = D5 d \/ + c = "6" /\ d' = D6 d \/ + c = "7" /\ d' = D7 d \/ + c = "8" /\ d' = D8 d \/ + c = "9" /\ d' = D9 d)%char. +Proof. + destruct c as [[|] [|] [|] [|] [|] [|] [|] [|]]; + intros [= <-]; intuition. +Qed. + +(** Decimal/String conversion where [Nil] is [""] *) + +Module NilEmpty. + +Fixpoint string_of_uint (d:uint) := + match d with + | Nil => EmptyString + | D0 d => String "0" (string_of_uint d) + | D1 d => String "1" (string_of_uint d) + | D2 d => String "2" (string_of_uint d) + | D3 d => String "3" (string_of_uint d) + | D4 d => String "4" (string_of_uint d) + | D5 d => String "5" (string_of_uint d) + | D6 d => String "6" (string_of_uint d) + | D7 d => String "7" (string_of_uint d) + | D8 d => String "8" (string_of_uint d) + | D9 d => String "9" (string_of_uint d) + end. + +Fixpoint uint_of_string s := + match s with + | EmptyString => Some Nil + | String a s => uint_of_char a (uint_of_string s) + end. + +Definition string_of_int (d:int) := + match d with + | Pos d => string_of_uint d + | Neg d => String "-" (string_of_uint d) + end. + +Definition int_of_string s := + match s with + | EmptyString => Some (Pos Nil) + | String a s' => + if ascii_dec a "-" then option_map Neg (uint_of_string s') + else option_map Pos (uint_of_string s) + end. + +(* NB: For the moment whitespace between - and digits are not accepted. + And in this variant [int_of_string "-" = Some (Neg Nil)]. + +Compute int_of_string "-123456890123456890123456890123456890". +Compute string_of_int (-123456890123456890123456890123456890). +*) + +(** Corresponding proofs *) + +Lemma usu d : + uint_of_string (string_of_uint d) = Some d. +Proof. + induction d; simpl; rewrite ?IHd; simpl; auto. +Qed. + +Lemma sus s d : + uint_of_string s = Some d -> string_of_uint d = s. +Proof. + revert d. + induction s; simpl. + - now intros d [= <-]. + - intros d. + destruct (uint_of_string s); [intros H | intros [=]]. + apply uint_of_char_spec in H. + intuition subst; simpl; f_equal; auto. +Qed. + +Lemma isi d : int_of_string (string_of_int d) = Some d. +Proof. + destruct d; simpl. + - unfold int_of_string. + destruct (string_of_uint d) eqn:Hd. + + now destruct d. + + destruct ascii_dec; subst. + * now destruct d. + * rewrite <- Hd, usu; auto. + - rewrite usu; auto. +Qed. + +Lemma sis s d : + int_of_string s = Some d -> string_of_int d = s. +Proof. + destruct s; [intros [= <-]| ]; simpl; trivial. + destruct ascii_dec; subst; simpl. + - destruct (uint_of_string s) eqn:Hs; simpl; intros [= <-]. + simpl; f_equal. now apply sus. + - destruct d; [ | now destruct uint_of_char]. + simpl string_of_int. + intros. apply sus; simpl. + destruct uint_of_char; simpl in *; congruence. +Qed. + +End NilEmpty. + +(** Decimal/String conversions where [Nil] is ["0"] *) + +Module NilZero. + +Definition string_of_uint (d:uint) := + match d with + | Nil => "0" + | _ => NilEmpty.string_of_uint d + end. + +Definition uint_of_string s := + match s with + | EmptyString => None + | _ => NilEmpty.uint_of_string s + end. + +Definition string_of_int (d:int) := + match d with + | Pos d => string_of_uint d + | Neg d => String "-" (string_of_uint d) + end. + +Definition int_of_string s := + match s with + | EmptyString => None + | String a s' => + if ascii_dec a "-" then option_map Neg (uint_of_string s') + else option_map Pos (uint_of_string s) + end. + +(** Corresponding proofs *) + +Lemma uint_of_string_nonnil s : uint_of_string s <> Some Nil. +Proof. + destruct s; simpl. + - easy. + - destruct (NilEmpty.uint_of_string s); [intros H | intros [=]]. + apply uint_of_char_spec in H. + now intuition subst. +Qed. + +Lemma sus s d : + uint_of_string s = Some d -> string_of_uint d = s. +Proof. + destruct s; [intros [=] | intros H]. + apply NilEmpty.sus in H. now destruct d. +Qed. + +Lemma usu d : + d<>Nil -> uint_of_string (string_of_uint d) = Some d. +Proof. + destruct d; (now destruct 1) || (intros _; apply NilEmpty.usu). +Qed. + +Lemma usu_nil : + uint_of_string (string_of_uint Nil) = Some Decimal.zero. +Proof. + reflexivity. +Qed. + +Lemma usu_gen d : + uint_of_string (string_of_uint d) = Some d \/ + uint_of_string (string_of_uint d) = Some Decimal.zero. +Proof. + destruct d; (now right) || (left; now apply usu). +Qed. + +Lemma isi d : + d<>Pos Nil -> d<>Neg Nil -> + int_of_string (string_of_int d) = Some d. +Proof. + destruct d; simpl. + - intros H _. + unfold int_of_string. + destruct (string_of_uint d) eqn:Hd. + + now destruct d. + + destruct ascii_dec; subst. + * now destruct d. + * rewrite <- Hd, usu; auto. now intros ->. + - intros _ H. + rewrite usu; auto. now intros ->. +Qed. + +Lemma isi_posnil : + int_of_string (string_of_int (Pos Nil)) = Some (Pos Decimal.zero). +Proof. + reflexivity. +Qed. + +(** Warning! (-0) won't parse (compatibility with the behavior of Z). *) + +Lemma isi_negnil : + int_of_string (string_of_int (Neg Nil)) = Some (Neg (D0 Nil)). +Proof. + reflexivity. +Qed. + +Lemma sis s d : + int_of_string s = Some d -> string_of_int d = s. +Proof. + destruct s; [intros [=]| ]; simpl. + destruct ascii_dec; subst; simpl. + - destruct (uint_of_string s) eqn:Hs; simpl; intros [= <-]. + simpl; f_equal. now apply sus. + - destruct d; [ | now destruct uint_of_char]. + simpl string_of_int. + intros. apply sus; simpl. + destruct uint_of_char; simpl in *; congruence. +Qed. + +End NilZero. diff --git a/theories/Numbers/DecimalZ.v b/theories/Numbers/DecimalZ.v new file mode 100644 index 00000000..3a083796 --- /dev/null +++ b/theories/Numbers/DecimalZ.v @@ -0,0 +1,75 @@ +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) + +(** * DecimalZ + + Proofs that conversions between decimal numbers and [Z] + are bijections. *) + +Require Import Decimal DecimalFacts DecimalPos DecimalN ZArith. + +Lemma of_to (z:Z) : Z.of_int (Z.to_int z) = z. +Proof. + destruct z; simpl. + - trivial. + - unfold Z.of_uint. rewrite DecimalPos.Unsigned.of_to. now destruct p. + - unfold Z.of_uint. rewrite DecimalPos.Unsigned.of_to. destruct p; auto. +Qed. + +Lemma to_of (d:int) : Z.to_int (Z.of_int d) = norm d. +Proof. + destruct d; simpl; unfold Z.to_int, Z.of_uint. + - rewrite <- (DecimalN.Unsigned.to_of d). unfold N.of_uint. + now destruct (Pos.of_uint d). + - destruct (Pos.of_uint d) eqn:Hd; simpl; f_equal. + + generalize (DecimalPos.Unsigned.to_of d). rewrite Hd. simpl. + intros H. symmetry in H. apply unorm_0 in H. now rewrite H. + + assert (Hp := DecimalPos.Unsigned.to_of d). rewrite Hd in Hp. simpl in *. + rewrite Hp. unfold unorm in *. + destruct (nzhead d); trivial. + generalize (DecimalPos.Unsigned.of_to p). now rewrite Hp. +Qed. + +(** Some consequences *) + +Lemma to_int_inj n n' : Z.to_int n = Z.to_int n' -> n = n'. +Proof. + intro EQ. + now rewrite <- (of_to n), <- (of_to n'), EQ. +Qed. + +Lemma to_int_surj d : exists n, Z.to_int n = norm d. +Proof. + exists (Z.of_int d). apply to_of. +Qed. + +Lemma of_int_norm d : Z.of_int (norm d) = Z.of_int d. +Proof. + unfold Z.of_int, Z.of_uint. + destruct d. + - simpl. now rewrite DecimalPos.Unsigned.of_uint_norm. + - simpl. destruct (nzhead d) eqn:H; + [ induction d; simpl; auto; discriminate | + destruct (nzhead_nonzero _ _ H) | .. ]; + f_equal; f_equal; apply DecimalPos.Unsigned.of_iff; + unfold unorm; now rewrite H. +Qed. + +Lemma of_inj d d' : + Z.of_int d = Z.of_int d' -> norm d = norm d'. +Proof. + intros. rewrite <- !to_of. now f_equal. +Qed. + +Lemma of_iff d d' : Z.of_int d = Z.of_int d' <-> norm d = norm d'. +Proof. + split. apply of_inj. intros E. rewrite <- of_int_norm, E. + apply of_int_norm. +Qed. diff --git a/theories/Numbers/Integer/Abstract/ZAdd.v b/theories/Numbers/Integer/Abstract/ZAdd.v index f7fdc179..c4c5174d 100644 --- a/theories/Numbers/Integer/Abstract/ZAdd.v +++ b/theories/Numbers/Integer/Abstract/ZAdd.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Integer/Abstract/ZAddOrder.v b/theories/Numbers/Integer/Abstract/ZAddOrder.v index 6bf5e9d4..7f5b0df6 100644 --- a/theories/Numbers/Integer/Abstract/ZAddOrder.v +++ b/theories/Numbers/Integer/Abstract/ZAddOrder.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Integer/Abstract/ZAxioms.v b/theories/Numbers/Integer/Abstract/ZAxioms.v index ad10e65f..4f1ab775 100644 --- a/theories/Numbers/Integer/Abstract/ZAxioms.v +++ b/theories/Numbers/Integer/Abstract/ZAxioms.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Integer/Abstract/ZBase.v b/theories/Numbers/Integer/Abstract/ZBase.v index 9b1b30f8..7fdd018d 100644 --- a/theories/Numbers/Integer/Abstract/ZBase.v +++ b/theories/Numbers/Integer/Abstract/ZBase.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Integer/Abstract/ZBits.v b/theories/Numbers/Integer/Abstract/ZBits.v index c919e121..2da44528 100644 --- a/theories/Numbers/Integer/Abstract/ZBits.v +++ b/theories/Numbers/Integer/Abstract/ZBits.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import diff --git a/theories/Numbers/Integer/Abstract/ZDivEucl.v b/theories/Numbers/Integer/Abstract/ZDivEucl.v index c2fa69e5..d7f25a66 100644 --- a/theories/Numbers/Integer/Abstract/ZDivEucl.v +++ b/theories/Numbers/Integer/Abstract/ZDivEucl.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import ZAxioms ZMulOrder ZSgnAbs NZDiv. @@ -602,6 +604,14 @@ Proof. apply div_mod; order. Qed. +Lemma mod_div: forall a b, b~=0 -> + a mod b / b == 0. +Proof. + intros a b Hb. + rewrite div_small_iff by assumption. + auto using mod_always_pos. +Qed. + (** A last inequality: *) Theorem div_mul_le: diff --git a/theories/Numbers/Integer/Abstract/ZDivFloor.v b/theories/Numbers/Integer/Abstract/ZDivFloor.v index 310748dd..a0d1821b 100644 --- a/theories/Numbers/Integer/Abstract/ZDivFloor.v +++ b/theories/Numbers/Integer/Abstract/ZDivFloor.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import ZAxioms ZMulOrder ZSgnAbs NZDiv. @@ -650,6 +652,14 @@ Proof. apply div_mod; order. Qed. +Lemma mod_div: forall a b, b~=0 -> + a mod b / b == 0. +Proof. + intros a b Hb. + rewrite div_small_iff by assumption. + auto using mod_bound_or. +Qed. + (** A last inequality: *) Theorem div_mul_le: diff --git a/theories/Numbers/Integer/Abstract/ZDivTrunc.v b/theories/Numbers/Integer/Abstract/ZDivTrunc.v index 04301077..31e42738 100644 --- a/theories/Numbers/Integer/Abstract/ZDivTrunc.v +++ b/theories/Numbers/Integer/Abstract/ZDivTrunc.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import ZAxioms ZMulOrder ZSgnAbs NZDiv. @@ -622,6 +624,14 @@ Proof. apply quot_rem; order. Qed. +Lemma rem_quot: forall a b, b~=0 -> + a rem b ÷ b == 0. +Proof. + intros a b Hb. + rewrite quot_small_iff by assumption. + auto using rem_bound_abs. +Qed. + (** A last inequality: *) Theorem quot_mul_le: diff --git a/theories/Numbers/Integer/Abstract/ZGcd.v b/theories/Numbers/Integer/Abstract/ZGcd.v index 30adaeb4..f0b7bf9d 100644 --- a/theories/Numbers/Integer/Abstract/ZGcd.v +++ b/theories/Numbers/Integer/Abstract/ZGcd.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Properties of the greatest common divisor *) diff --git a/theories/Numbers/Integer/Abstract/ZLcm.v b/theories/Numbers/Integer/Abstract/ZLcm.v index ef33308c..0ab528de 100644 --- a/theories/Numbers/Integer/Abstract/ZLcm.v +++ b/theories/Numbers/Integer/Abstract/ZLcm.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import ZAxioms ZMulOrder ZSgnAbs ZGcd ZDivTrunc ZDivFloor. diff --git a/theories/Numbers/Integer/Abstract/ZLt.v b/theories/Numbers/Integer/Abstract/ZLt.v index 0c92918d..726b041c 100644 --- a/theories/Numbers/Integer/Abstract/ZLt.v +++ b/theories/Numbers/Integer/Abstract/ZLt.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Integer/Abstract/ZMaxMin.v b/theories/Numbers/Integer/Abstract/ZMaxMin.v index 24a47f00..f3f3a861 100644 --- a/theories/Numbers/Integer/Abstract/ZMaxMin.v +++ b/theories/Numbers/Integer/Abstract/ZMaxMin.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import ZAxioms ZMulOrder GenericMinMax. diff --git a/theories/Numbers/Integer/Abstract/ZMul.v b/theories/Numbers/Integer/Abstract/ZMul.v index 830c0a7d..120647dc 100644 --- a/theories/Numbers/Integer/Abstract/ZMul.v +++ b/theories/Numbers/Integer/Abstract/ZMul.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Integer/Abstract/ZMulOrder.v b/theories/Numbers/Integer/Abstract/ZMulOrder.v index 320c8f35..cd9523d3 100644 --- a/theories/Numbers/Integer/Abstract/ZMulOrder.v +++ b/theories/Numbers/Integer/Abstract/ZMulOrder.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Integer/Abstract/ZParity.v b/theories/Numbers/Integer/Abstract/ZParity.v index 952d8e9c..a5e53b36 100644 --- a/theories/Numbers/Integer/Abstract/ZParity.v +++ b/theories/Numbers/Integer/Abstract/ZParity.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Bool ZMulOrder NZParity. diff --git a/theories/Numbers/Integer/Abstract/ZPow.v b/theories/Numbers/Integer/Abstract/ZPow.v index 02b8501c..a4b964e5 100644 --- a/theories/Numbers/Integer/Abstract/ZPow.v +++ b/theories/Numbers/Integer/Abstract/ZPow.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Properties of the power function *) diff --git a/theories/Numbers/Integer/Abstract/ZProperties.v b/theories/Numbers/Integer/Abstract/ZProperties.v index 1dec3c58..e4b997cf 100644 --- a/theories/Numbers/Integer/Abstract/ZProperties.v +++ b/theories/Numbers/Integer/Abstract/ZProperties.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export ZAxioms ZMaxMin ZSgnAbs ZParity ZPow ZDivTrunc ZDivFloor diff --git a/theories/Numbers/Integer/Abstract/ZSgnAbs.v b/theories/Numbers/Integer/Abstract/ZSgnAbs.v index a10552ab..dda12872 100644 --- a/theories/Numbers/Integer/Abstract/ZSgnAbs.v +++ b/theories/Numbers/Integer/Abstract/ZSgnAbs.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Properties of [abs] and [sgn] *) diff --git a/theories/Numbers/Integer/BigZ/BigZ.v b/theories/Numbers/Integer/BigZ/BigZ.v deleted file mode 100644 index 7c76011f..00000000 --- a/theories/Numbers/Integer/BigZ/BigZ.v +++ /dev/null @@ -1,208 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Require Export BigN. -Require Import ZProperties ZDivFloor ZSig ZSigZAxioms ZMake. - -(** * [BigZ] : arbitrary large efficient integers. - - The following [BigZ] module regroups both the operations and - all the abstract properties: - - - [ZMake.Make BigN] provides the operations and basic specs w.r.t. ZArith - - [ZTypeIsZAxioms] shows (mainly) that these operations implement - the interface [ZAxioms] - - [ZProp] adds all generic properties derived from [ZAxioms] - - [MinMax*Properties] provides properties of [min] and [max] - -*) - -Delimit Scope bigZ_scope with bigZ. - -Module BigZ <: ZType <: OrderedTypeFull <: TotalOrder := - ZMake.Make BigN - <+ ZTypeIsZAxioms - <+ ZBasicProp [no inline] <+ ZExtraProp [no inline] - <+ HasEqBool2Dec [no inline] - <+ MinMaxLogicalProperties [no inline] - <+ MinMaxDecProperties [no inline]. - -(** For precision concerning the above scope handling, see comment in BigN *) - -(** Notations about [BigZ] *) - -Local Open Scope bigZ_scope. - -Notation bigZ := BigZ.t. -Bind Scope bigZ_scope with bigZ BigZ.t BigZ.t_. -Arguments BigZ.Pos _%bigN. -Arguments BigZ.Neg _%bigN. -Local Notation "0" := BigZ.zero : bigZ_scope. -Local Notation "1" := BigZ.one : bigZ_scope. -Local Notation "2" := BigZ.two : bigZ_scope. -Infix "+" := BigZ.add : bigZ_scope. -Infix "-" := BigZ.sub : bigZ_scope. -Notation "- x" := (BigZ.opp x) : bigZ_scope. -Infix "*" := BigZ.mul : bigZ_scope. -Infix "/" := BigZ.div : bigZ_scope. -Infix "^" := BigZ.pow : bigZ_scope. -Infix "?=" := BigZ.compare : bigZ_scope. -Infix "=?" := BigZ.eqb (at level 70, no associativity) : bigZ_scope. -Infix "<=?" := BigZ.leb (at level 70, no associativity) : bigZ_scope. -Infix "<?" := BigZ.ltb (at level 70, no associativity) : bigZ_scope. -Infix "==" := BigZ.eq (at level 70, no associativity) : bigZ_scope. -Notation "x != y" := (~x==y) (at level 70, no associativity) : bigZ_scope. -Infix "<" := BigZ.lt : bigZ_scope. -Infix "<=" := BigZ.le : bigZ_scope. -Notation "x > y" := (y < x) (only parsing) : bigZ_scope. -Notation "x >= y" := (y <= x) (only parsing) : bigZ_scope. -Notation "x < y < z" := (x<y /\ y<z) : bigZ_scope. -Notation "x < y <= z" := (x<y /\ y<=z) : bigZ_scope. -Notation "x <= y < z" := (x<=y /\ y<z) : bigZ_scope. -Notation "x <= y <= z" := (x<=y /\ y<=z) : bigZ_scope. -Notation "[ i ]" := (BigZ.to_Z i) : bigZ_scope. -Infix "mod" := BigZ.modulo (at level 40, no associativity) : bigZ_scope. -Infix "÷" := BigZ.quot (at level 40, left associativity) : bigZ_scope. - -(** Some additional results about [BigZ] *) - -Theorem spec_to_Z: forall n : bigZ, - BigN.to_Z (BigZ.to_N n) = ((Z.sgn [n]) * [n])%Z. -Proof. -intros n; case n; simpl; intros p; - generalize (BigN.spec_pos p); case (BigN.to_Z p); auto. -intros p1 H1; case H1; auto. -intros p1 H1; case H1; auto. -Qed. - -Theorem spec_to_N n: - ([n] = Z.sgn [n] * (BigN.to_Z (BigZ.to_N n)))%Z. -Proof. -case n; simpl; intros p; - generalize (BigN.spec_pos p); case (BigN.to_Z p); auto. -intros p1 H1; case H1; auto. -intros p1 H1; case H1; auto. -Qed. - -Theorem spec_to_Z_pos: forall n, (0 <= [n])%Z -> - BigN.to_Z (BigZ.to_N n) = [n]. -Proof. -intros n; case n; simpl; intros p; - generalize (BigN.spec_pos p); case (BigN.to_Z p); auto. -intros p1 _ H1; case H1; auto. -intros p1 H1; case H1; auto. -Qed. - -(** [BigZ] is a ring *) - -Lemma BigZring : - ring_theory 0 1 BigZ.add BigZ.mul BigZ.sub BigZ.opp BigZ.eq. -Proof. -constructor. -exact BigZ.add_0_l. exact BigZ.add_comm. exact BigZ.add_assoc. -exact BigZ.mul_1_l. exact BigZ.mul_comm. exact BigZ.mul_assoc. -exact BigZ.mul_add_distr_r. -symmetry. apply BigZ.add_opp_r. -exact BigZ.add_opp_diag_r. -Qed. - -Lemma BigZeqb_correct : forall x y, (x =? y) = true -> x==y. -Proof. now apply BigZ.eqb_eq. Qed. - -Definition BigZ_of_N n := BigZ.of_Z (Z.of_N n). - -Lemma BigZpower : power_theory 1 BigZ.mul BigZ.eq BigZ_of_N BigZ.pow. -Proof. -constructor. -intros. unfold BigZ.eq, BigZ_of_N. rewrite BigZ.spec_pow, BigZ.spec_of_Z. -rewrite Zpower_theory.(rpow_pow_N). -destruct n; simpl. reflexivity. -induction p; simpl; intros; BigZ.zify; rewrite ?IHp; auto. -Qed. - -Lemma BigZdiv : div_theory BigZ.eq BigZ.add BigZ.mul (@id _) - (fun a b => if b =? 0 then (0,a) else BigZ.div_eucl a b). -Proof. -constructor. unfold id. intros a b. -BigZ.zify. -case Z.eqb_spec. -BigZ.zify. auto with zarith. -intros NEQ. -generalize (BigZ.spec_div_eucl a b). -generalize (Z_div_mod_full [a] [b] NEQ). -destruct BigZ.div_eucl as (q,r), Z.div_eucl as (q',r'). -intros (EQ,_). injection 1 as EQr EQq. -BigZ.zify. rewrite EQr, EQq; auto. -Qed. - -(** Detection of constants *) - -Ltac isBigZcst t := - match t with - | BigZ.Pos ?t => isBigNcst t - | BigZ.Neg ?t => isBigNcst t - | BigZ.zero => constr:(true) - | BigZ.one => constr:(true) - | BigZ.two => constr:(true) - | BigZ.minus_one => constr:(true) - | _ => constr:(false) - end. - -Ltac BigZcst t := - match isBigZcst t with - | true => constr:(t) - | false => constr:(NotConstant) - end. - -Ltac BigZ_to_N t := - match t with - | BigZ.Pos ?t => BigN_to_N t - | BigZ.zero => constr:(0%N) - | BigZ.one => constr:(1%N) - | BigZ.two => constr:(2%N) - | _ => constr:(NotConstant) - end. - -(** Registration for the "ring" tactic *) - -Add Ring BigZr : BigZring - (decidable BigZeqb_correct, - constants [BigZcst], - power_tac BigZpower [BigZ_to_N], - div BigZdiv). - -Section TestRing. -Let test : forall x y, 1 + x*y + x^2 + 1 == 1*1 + 1 + (y + 1*x)*x. -Proof. -intros. ring_simplify. reflexivity. -Qed. -Let test' : forall x y, 1 + x*y + x^2 - 1*1 - y*x + 1*(-x)*x == 0. -Proof. -intros. ring_simplify. reflexivity. -Qed. -End TestRing. - -(** [BigZ] also benefits from an "order" tactic *) - -Ltac bigZ_order := BigZ.order. - -Section TestOrder. -Let test : forall x y : bigZ, x<=y -> y<=x -> x==y. -Proof. bigZ_order. Qed. -End TestOrder. - -(** We can use at least a bit of (r)omega by translating to [Z]. *) - -Section TestOmega. -Let test : forall x y : bigZ, x<=y -> y<=x -> x==y. -Proof. intros x y. BigZ.zify. omega. Qed. -End TestOmega. - -(** Todo: micromega *) diff --git a/theories/Numbers/Integer/BigZ/ZMake.v b/theories/Numbers/Integer/BigZ/ZMake.v deleted file mode 100644 index fec6e068..00000000 --- a/theories/Numbers/Integer/BigZ/ZMake.v +++ /dev/null @@ -1,759 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Require Import ZArith. -Require Import BigNumPrelude. -Require Import NSig. -Require Import ZSig. - -Open Scope Z_scope. - -(** * ZMake - - A generic transformation from a structure of natural numbers - [NSig.NType] to a structure of integers [ZSig.ZType]. -*) - -Module Make (NN:NType) <: ZType. - - Inductive t_ := - | Pos : NN.t -> t_ - | Neg : NN.t -> t_. - - Definition t := t_. - - Definition zero := Pos NN.zero. - Definition one := Pos NN.one. - Definition two := Pos NN.two. - Definition minus_one := Neg NN.one. - - Definition of_Z x := - match x with - | Zpos x => Pos (NN.of_N (Npos x)) - | Z0 => zero - | Zneg x => Neg (NN.of_N (Npos x)) - end. - - Definition to_Z x := - match x with - | Pos nx => NN.to_Z nx - | Neg nx => Z.opp (NN.to_Z nx) - end. - - Theorem spec_of_Z: forall x, to_Z (of_Z x) = x. - Proof. - intros x; case x; unfold to_Z, of_Z, zero. - exact NN.spec_0. - intros; rewrite NN.spec_of_N; auto. - intros; rewrite NN.spec_of_N; auto. - Qed. - - Definition eq x y := (to_Z x = to_Z y). - - Theorem spec_0: to_Z zero = 0. - exact NN.spec_0. - Qed. - - Theorem spec_1: to_Z one = 1. - exact NN.spec_1. - Qed. - - Theorem spec_2: to_Z two = 2. - exact NN.spec_2. - Qed. - - Theorem spec_m1: to_Z minus_one = -1. - simpl; rewrite NN.spec_1; auto. - Qed. - - Definition compare x y := - match x, y with - | Pos nx, Pos ny => NN.compare nx ny - | Pos nx, Neg ny => - match NN.compare nx NN.zero with - | Gt => Gt - | _ => NN.compare ny NN.zero - end - | Neg nx, Pos ny => - match NN.compare NN.zero nx with - | Lt => Lt - | _ => NN.compare NN.zero ny - end - | Neg nx, Neg ny => NN.compare ny nx - end. - - Theorem spec_compare : - forall x y, compare x y = Z.compare (to_Z x) (to_Z y). - Proof. - unfold compare, to_Z. - destruct x as [x|x], y as [y|y]; - rewrite ?NN.spec_compare, ?NN.spec_0, ?Z.compare_opp; auto; - assert (Hx:=NN.spec_pos x); assert (Hy:=NN.spec_pos y); - set (X:=NN.to_Z x) in *; set (Y:=NN.to_Z y) in *; clearbody X Y. - - destruct (Z.compare_spec X 0) as [EQ|LT|GT]. - + rewrite <- Z.opp_0 in EQ. now rewrite EQ, Z.compare_opp. - + exfalso. omega. - + symmetry. change (X > -Y). omega. - - destruct (Z.compare_spec 0 X) as [EQ|LT|GT]. - + rewrite <- EQ, Z.opp_0; auto. - + symmetry. change (-X < Y). omega. - + exfalso. omega. - Qed. - - Definition eqb x y := - match compare x y with - | Eq => true - | _ => false - end. - - Theorem spec_eqb x y : eqb x y = Z.eqb (to_Z x) (to_Z y). - Proof. - apply Bool.eq_iff_eq_true. - unfold eqb. rewrite Z.eqb_eq, <- Z.compare_eq_iff, spec_compare. - split; [now destruct Z.compare | now intros ->]. - Qed. - - Definition lt n m := to_Z n < to_Z m. - Definition le n m := to_Z n <= to_Z m. - - - Definition ltb (x y : t) : bool := - match compare x y with - | Lt => true - | _ => false - end. - - Theorem spec_ltb x y : ltb x y = Z.ltb (to_Z x) (to_Z y). - Proof. - apply Bool.eq_iff_eq_true. - rewrite Z.ltb_lt. unfold Z.lt, ltb. rewrite spec_compare. - split; [now destruct Z.compare | now intros ->]. - Qed. - - Definition leb (x y : t) : bool := - match compare x y with - | Gt => false - | _ => true - end. - - Theorem spec_leb x y : leb x y = Z.leb (to_Z x) (to_Z y). - Proof. - apply Bool.eq_iff_eq_true. - rewrite Z.leb_le. unfold Z.le, leb. rewrite spec_compare. - now destruct Z.compare; split. - Qed. - - Definition min n m := match compare n m with Gt => m | _ => n end. - Definition max n m := match compare n m with Lt => m | _ => n end. - - Theorem spec_min : forall n m, to_Z (min n m) = Z.min (to_Z n) (to_Z m). - Proof. - unfold min, Z.min. intros. rewrite spec_compare. destruct Z.compare; auto. - Qed. - - Theorem spec_max : forall n m, to_Z (max n m) = Z.max (to_Z n) (to_Z m). - Proof. - unfold max, Z.max. intros. rewrite spec_compare. destruct Z.compare; auto. - Qed. - - Definition to_N x := - match x with - | Pos nx => nx - | Neg nx => nx - end. - - Definition abs x := Pos (to_N x). - - Theorem spec_abs: forall x, to_Z (abs x) = Z.abs (to_Z x). - Proof. - intros x; case x; clear x; intros x; assert (F:=NN.spec_pos x). - simpl; rewrite Z.abs_eq; auto. - simpl; rewrite Z.abs_neq; simpl; auto with zarith. - Qed. - - Definition opp x := - match x with - | Pos nx => Neg nx - | Neg nx => Pos nx - end. - - Theorem spec_opp: forall x, to_Z (opp x) = - to_Z x. - Proof. - intros x; case x; simpl; auto with zarith. - Qed. - - Definition succ x := - match x with - | Pos n => Pos (NN.succ n) - | Neg n => - match NN.compare NN.zero n with - | Lt => Neg (NN.pred n) - | _ => one - end - end. - - Theorem spec_succ: forall n, to_Z (succ n) = to_Z n + 1. - Proof. - intros x; case x; clear x; intros x. - exact (NN.spec_succ x). - simpl. rewrite NN.spec_compare. case Z.compare_spec; rewrite ?NN.spec_0; simpl. - intros HH; rewrite <- HH; rewrite NN.spec_1; ring. - intros HH; rewrite NN.spec_pred, Z.max_r; auto with zarith. - generalize (NN.spec_pos x); auto with zarith. - Qed. - - Definition add x y := - match x, y with - | Pos nx, Pos ny => Pos (NN.add nx ny) - | Pos nx, Neg ny => - match NN.compare nx ny with - | Gt => Pos (NN.sub nx ny) - | Eq => zero - | Lt => Neg (NN.sub ny nx) - end - | Neg nx, Pos ny => - match NN.compare nx ny with - | Gt => Neg (NN.sub nx ny) - | Eq => zero - | Lt => Pos (NN.sub ny nx) - end - | Neg nx, Neg ny => Neg (NN.add nx ny) - end. - - Theorem spec_add: forall x y, to_Z (add x y) = to_Z x + to_Z y. - Proof. - unfold add, to_Z; intros [x | x] [y | y]; - try (rewrite NN.spec_add; auto with zarith); - rewrite NN.spec_compare; case Z.compare_spec; - unfold zero; rewrite ?NN.spec_0, ?NN.spec_sub; omega with *. - Qed. - - Definition pred x := - match x with - | Pos nx => - match NN.compare NN.zero nx with - | Lt => Pos (NN.pred nx) - | _ => minus_one - end - | Neg nx => Neg (NN.succ nx) - end. - - Theorem spec_pred: forall x, to_Z (pred x) = to_Z x - 1. - Proof. - unfold pred, to_Z, minus_one; intros [x | x]; - try (rewrite NN.spec_succ; ring). - rewrite NN.spec_compare; case Z.compare_spec; - rewrite ?NN.spec_0, ?NN.spec_1, ?NN.spec_pred; - generalize (NN.spec_pos x); omega with *. - Qed. - - Definition sub x y := - match x, y with - | Pos nx, Pos ny => - match NN.compare nx ny with - | Gt => Pos (NN.sub nx ny) - | Eq => zero - | Lt => Neg (NN.sub ny nx) - end - | Pos nx, Neg ny => Pos (NN.add nx ny) - | Neg nx, Pos ny => Neg (NN.add nx ny) - | Neg nx, Neg ny => - match NN.compare nx ny with - | Gt => Neg (NN.sub nx ny) - | Eq => zero - | Lt => Pos (NN.sub ny nx) - end - end. - - Theorem spec_sub: forall x y, to_Z (sub x y) = to_Z x - to_Z y. - Proof. - unfold sub, to_Z; intros [x | x] [y | y]; - try (rewrite NN.spec_add; auto with zarith); - rewrite NN.spec_compare; case Z.compare_spec; - unfold zero; rewrite ?NN.spec_0, ?NN.spec_sub; omega with *. - Qed. - - Definition mul x y := - match x, y with - | Pos nx, Pos ny => Pos (NN.mul nx ny) - | Pos nx, Neg ny => Neg (NN.mul nx ny) - | Neg nx, Pos ny => Neg (NN.mul nx ny) - | Neg nx, Neg ny => Pos (NN.mul nx ny) - end. - - Theorem spec_mul: forall x y, to_Z (mul x y) = to_Z x * to_Z y. - Proof. - unfold mul, to_Z; intros [x | x] [y | y]; rewrite NN.spec_mul; ring. - Qed. - - Definition square x := - match x with - | Pos nx => Pos (NN.square nx) - | Neg nx => Pos (NN.square nx) - end. - - Theorem spec_square: forall x, to_Z (square x) = to_Z x * to_Z x. - Proof. - unfold square, to_Z; intros [x | x]; rewrite NN.spec_square; ring. - Qed. - - Definition pow_pos x p := - match x with - | Pos nx => Pos (NN.pow_pos nx p) - | Neg nx => - match p with - | xH => x - | xO _ => Pos (NN.pow_pos nx p) - | xI _ => Neg (NN.pow_pos nx p) - end - end. - - Theorem spec_pow_pos: forall x n, to_Z (pow_pos x n) = to_Z x ^ Zpos n. - Proof. - assert (F0: forall x, (-x)^2 = x^2). - intros x; rewrite Z.pow_2_r; ring. - unfold pow_pos, to_Z; intros [x | x] [p | p |]; - try rewrite NN.spec_pow_pos; try ring. - assert (F: 0 <= 2 * Zpos p). - assert (0 <= Zpos p); auto with zarith. - rewrite Pos2Z.inj_xI; repeat rewrite Zpower_exp; auto with zarith. - repeat rewrite Z.pow_mul_r; auto with zarith. - rewrite F0; ring. - assert (F: 0 <= 2 * Zpos p). - assert (0 <= Zpos p); auto with zarith. - rewrite Pos2Z.inj_xO; repeat rewrite Zpower_exp; auto with zarith. - repeat rewrite Z.pow_mul_r; auto with zarith. - rewrite F0; ring. - Qed. - - Definition pow_N x n := - match n with - | N0 => one - | Npos p => pow_pos x p - end. - - Theorem spec_pow_N: forall x n, to_Z (pow_N x n) = to_Z x ^ Z.of_N n. - Proof. - destruct n; simpl. apply NN.spec_1. - apply spec_pow_pos. - Qed. - - Definition pow x y := - match to_Z y with - | Z0 => one - | Zpos p => pow_pos x p - | Zneg p => zero - end. - - Theorem spec_pow: forall x y, to_Z (pow x y) = to_Z x ^ to_Z y. - Proof. - intros. unfold pow. destruct (to_Z y); simpl. - apply NN.spec_1. - apply spec_pow_pos. - apply NN.spec_0. - Qed. - - Definition log2 x := - match x with - | Pos nx => Pos (NN.log2 nx) - | Neg nx => zero - end. - - Theorem spec_log2: forall x, to_Z (log2 x) = Z.log2 (to_Z x). - Proof. - intros. destruct x as [p|p]; simpl. apply NN.spec_log2. - rewrite NN.spec_0. - destruct (Z_le_lt_eq_dec _ _ (NN.spec_pos p)) as [LT|EQ]. - rewrite Z.log2_nonpos; auto with zarith. - now rewrite <- EQ. - Qed. - - Definition sqrt x := - match x with - | Pos nx => Pos (NN.sqrt nx) - | Neg nx => Neg NN.zero - end. - - Theorem spec_sqrt: forall x, to_Z (sqrt x) = Z.sqrt (to_Z x). - Proof. - destruct x as [p|p]; simpl. - apply NN.spec_sqrt. - rewrite NN.spec_0. - destruct (Z_le_lt_eq_dec _ _ (NN.spec_pos p)) as [LT|EQ]. - rewrite Z.sqrt_neg; auto with zarith. - now rewrite <- EQ. - Qed. - - Definition div_eucl x y := - match x, y with - | Pos nx, Pos ny => - let (q, r) := NN.div_eucl nx ny in - (Pos q, Pos r) - | Pos nx, Neg ny => - let (q, r) := NN.div_eucl nx ny in - if NN.eqb NN.zero r - then (Neg q, zero) - else (Neg (NN.succ q), Neg (NN.sub ny r)) - | Neg nx, Pos ny => - let (q, r) := NN.div_eucl nx ny in - if NN.eqb NN.zero r - then (Neg q, zero) - else (Neg (NN.succ q), Pos (NN.sub ny r)) - | Neg nx, Neg ny => - let (q, r) := NN.div_eucl nx ny in - (Pos q, Neg r) - end. - - Ltac break_nonneg x px EQx := - let H := fresh "H" in - assert (H:=NN.spec_pos x); - destruct (NN.to_Z x) as [|px|px] eqn:EQx; - [clear H|clear H|elim H; reflexivity]. - - Theorem spec_div_eucl: forall x y, - let (q,r) := div_eucl x y in - (to_Z q, to_Z r) = Z.div_eucl (to_Z x) (to_Z y). - Proof. - unfold div_eucl, to_Z. intros [x | x] [y | y]. - (* Pos Pos *) - generalize (NN.spec_div_eucl x y); destruct (NN.div_eucl x y); auto. - (* Pos Neg *) - generalize (NN.spec_div_eucl x y); destruct (NN.div_eucl x y) as (q,r). - break_nonneg x px EQx; break_nonneg y py EQy; - try (injection 1 as Hq Hr; rewrite NN.spec_eqb, NN.spec_0, Hr; - simpl; rewrite Hq, NN.spec_0; auto). - change (- Zpos py) with (Zneg py). - assert (GT : Zpos py > 0) by (compute; auto). - generalize (Z_div_mod (Zpos px) (Zpos py) GT). - unfold Z.div_eucl. destruct (Z.pos_div_eucl px (Zpos py)) as (q',r'). - intros (EQ,MOD). injection 1 as Hq' Hr'. - rewrite NN.spec_eqb, NN.spec_0, Hr'. - break_nonneg r pr EQr. - subst; simpl. rewrite NN.spec_0; auto. - subst. lazy iota beta delta [Z.eqb]. - rewrite NN.spec_sub, NN.spec_succ, EQy, EQr. f_equal. omega with *. - (* Neg Pos *) - generalize (NN.spec_div_eucl x y); destruct (NN.div_eucl x y) as (q,r). - break_nonneg x px EQx; break_nonneg y py EQy; - try (injection 1 as Hq Hr; rewrite NN.spec_eqb, NN.spec_0, Hr; - simpl; rewrite Hq, NN.spec_0; auto). - change (- Zpos px) with (Zneg px). - assert (GT : Zpos py > 0) by (compute; auto). - generalize (Z_div_mod (Zpos px) (Zpos py) GT). - unfold Z.div_eucl. destruct (Z.pos_div_eucl px (Zpos py)) as (q',r'). - intros (EQ,MOD). injection 1 as Hq' Hr'. - rewrite NN.spec_eqb, NN.spec_0, Hr'. - break_nonneg r pr EQr. - subst; simpl. rewrite NN.spec_0; auto. - subst. lazy iota beta delta [Z.eqb]. - rewrite NN.spec_sub, NN.spec_succ, EQy, EQr. f_equal. omega with *. - (* Neg Neg *) - generalize (NN.spec_div_eucl x y); destruct (NN.div_eucl x y) as (q,r). - break_nonneg x px EQx; break_nonneg y py EQy; - try (injection 1 as -> ->; auto). - simpl. intros <-; auto. - Qed. - - Definition div x y := fst (div_eucl x y). - - Definition spec_div: forall x y, - to_Z (div x y) = to_Z x / to_Z y. - Proof. - intros x y; generalize (spec_div_eucl x y); unfold div, Z.div. - case div_eucl; case Z.div_eucl; simpl; auto. - intros q r q11 r1 H; injection H; auto. - Qed. - - Definition modulo x y := snd (div_eucl x y). - - Theorem spec_modulo: - forall x y, to_Z (modulo x y) = to_Z x mod to_Z y. - Proof. - intros x y; generalize (spec_div_eucl x y); unfold modulo, Z.modulo. - case div_eucl; case Z.div_eucl; simpl; auto. - intros q r q11 r1 H; injection H; auto. - Qed. - - Definition quot x y := - match x, y with - | Pos nx, Pos ny => Pos (NN.div nx ny) - | Pos nx, Neg ny => Neg (NN.div nx ny) - | Neg nx, Pos ny => Neg (NN.div nx ny) - | Neg nx, Neg ny => Pos (NN.div nx ny) - end. - - Definition rem x y := - if eqb y zero then x - else - match x, y with - | Pos nx, Pos ny => Pos (NN.modulo nx ny) - | Pos nx, Neg ny => Pos (NN.modulo nx ny) - | Neg nx, Pos ny => Neg (NN.modulo nx ny) - | Neg nx, Neg ny => Neg (NN.modulo nx ny) - end. - - Lemma spec_quot : forall x y, to_Z (quot x y) = (to_Z x) ÷ (to_Z y). - Proof. - intros [x|x] [y|y]; simpl; symmetry; rewrite NN.spec_div; - (* Nota: we rely here on [forall a b, a ÷ 0 = b / 0] *) - destruct (Z.eq_dec (NN.to_Z y) 0) as [EQ|NEQ]; - try (rewrite EQ; now destruct (NN.to_Z x)); - rewrite ?Z.quot_opp_r, ?Z.quot_opp_l, ?Z.opp_involutive, ?Z.opp_inj_wd; - trivial; apply Z.quot_div_nonneg; - generalize (NN.spec_pos x) (NN.spec_pos y); Z.order. - Qed. - - Lemma spec_rem : forall x y, - to_Z (rem x y) = Z.rem (to_Z x) (to_Z y). - Proof. - intros x y. unfold rem. rewrite spec_eqb, spec_0. - case Z.eqb_spec; intros Hy. - (* Nota: we rely here on [Z.rem a 0 = a] *) - rewrite Hy. now destruct (to_Z x). - destruct x as [x|x], y as [y|y]; simpl in *; symmetry; - rewrite ?Z.eq_opp_l, ?Z.opp_0 in Hy; - rewrite NN.spec_modulo, ?Z.rem_opp_r, ?Z.rem_opp_l, ?Z.opp_involutive, - ?Z.opp_inj_wd; - trivial; apply Z.rem_mod_nonneg; - generalize (NN.spec_pos x) (NN.spec_pos y); Z.order. - Qed. - - Definition gcd x y := - match x, y with - | Pos nx, Pos ny => Pos (NN.gcd nx ny) - | Pos nx, Neg ny => Pos (NN.gcd nx ny) - | Neg nx, Pos ny => Pos (NN.gcd nx ny) - | Neg nx, Neg ny => Pos (NN.gcd nx ny) - end. - - Theorem spec_gcd: forall a b, to_Z (gcd a b) = Z.gcd (to_Z a) (to_Z b). - Proof. - unfold gcd, Z.gcd, to_Z; intros [x | x] [y | y]; rewrite NN.spec_gcd; unfold Z.gcd; - auto; case NN.to_Z; simpl; auto with zarith; - try rewrite Z.abs_opp; auto; - case NN.to_Z; simpl; auto with zarith. - Qed. - - Definition sgn x := - match compare zero x with - | Lt => one - | Eq => zero - | Gt => minus_one - end. - - Lemma spec_sgn : forall x, to_Z (sgn x) = Z.sgn (to_Z x). - Proof. - intros. unfold sgn. rewrite spec_compare. case Z.compare_spec. - rewrite spec_0. intros <-; auto. - rewrite spec_0, spec_1. symmetry. rewrite Z.sgn_pos_iff; auto. - rewrite spec_0, spec_m1. symmetry. rewrite Z.sgn_neg_iff; auto with zarith. - Qed. - - Definition even z := - match z with - | Pos n => NN.even n - | Neg n => NN.even n - end. - - Definition odd z := - match z with - | Pos n => NN.odd n - | Neg n => NN.odd n - end. - - Lemma spec_even : forall z, even z = Z.even (to_Z z). - Proof. - intros [n|n]; simpl; rewrite NN.spec_even; trivial. - destruct (NN.to_Z n) as [|p|p]; now try destruct p. - Qed. - - Lemma spec_odd : forall z, odd z = Z.odd (to_Z z). - Proof. - intros [n|n]; simpl; rewrite NN.spec_odd; trivial. - destruct (NN.to_Z n) as [|p|p]; now try destruct p. - Qed. - - Definition norm_pos z := - match z with - | Pos _ => z - | Neg n => if NN.eqb n NN.zero then Pos n else z - end. - - Definition testbit a n := - match norm_pos n, norm_pos a with - | Pos p, Pos a => NN.testbit a p - | Pos p, Neg a => negb (NN.testbit (NN.pred a) p) - | Neg p, _ => false - end. - - Definition shiftl a n := - match norm_pos a, n with - | Pos a, Pos n => Pos (NN.shiftl a n) - | Pos a, Neg n => Pos (NN.shiftr a n) - | Neg a, Pos n => Neg (NN.shiftl a n) - | Neg a, Neg n => Neg (NN.succ (NN.shiftr (NN.pred a) n)) - end. - - Definition shiftr a n := shiftl a (opp n). - - Definition lor a b := - match norm_pos a, norm_pos b with - | Pos a, Pos b => Pos (NN.lor a b) - | Neg a, Pos b => Neg (NN.succ (NN.ldiff (NN.pred a) b)) - | Pos a, Neg b => Neg (NN.succ (NN.ldiff (NN.pred b) a)) - | Neg a, Neg b => Neg (NN.succ (NN.land (NN.pred a) (NN.pred b))) - end. - - Definition land a b := - match norm_pos a, norm_pos b with - | Pos a, Pos b => Pos (NN.land a b) - | Neg a, Pos b => Pos (NN.ldiff b (NN.pred a)) - | Pos a, Neg b => Pos (NN.ldiff a (NN.pred b)) - | Neg a, Neg b => Neg (NN.succ (NN.lor (NN.pred a) (NN.pred b))) - end. - - Definition ldiff a b := - match norm_pos a, norm_pos b with - | Pos a, Pos b => Pos (NN.ldiff a b) - | Neg a, Pos b => Neg (NN.succ (NN.lor (NN.pred a) b)) - | Pos a, Neg b => Pos (NN.land a (NN.pred b)) - | Neg a, Neg b => Pos (NN.ldiff (NN.pred b) (NN.pred a)) - end. - - Definition lxor a b := - match norm_pos a, norm_pos b with - | Pos a, Pos b => Pos (NN.lxor a b) - | Neg a, Pos b => Neg (NN.succ (NN.lxor (NN.pred a) b)) - | Pos a, Neg b => Neg (NN.succ (NN.lxor a (NN.pred b))) - | Neg a, Neg b => Pos (NN.lxor (NN.pred a) (NN.pred b)) - end. - - Definition div2 x := shiftr x one. - - Lemma Zlnot_alt1 : forall x, -(x+1) = Z.lnot x. - Proof. - unfold Z.lnot, Z.pred; auto with zarith. - Qed. - - Lemma Zlnot_alt2 : forall x, Z.lnot (x-1) = -x. - Proof. - unfold Z.lnot, Z.pred; auto with zarith. - Qed. - - Lemma Zlnot_alt3 : forall x, Z.lnot (-x) = x-1. - Proof. - unfold Z.lnot, Z.pred; auto with zarith. - Qed. - - Lemma spec_norm_pos : forall x, to_Z (norm_pos x) = to_Z x. - Proof. - intros [x|x]; simpl; trivial. - rewrite NN.spec_eqb, NN.spec_0. - case Z.eqb_spec; simpl; auto with zarith. - Qed. - - Lemma spec_norm_pos_pos : forall x y, norm_pos x = Neg y -> - 0 < NN.to_Z y. - Proof. - intros [x|x] y; simpl; try easy. - rewrite NN.spec_eqb, NN.spec_0. - case Z.eqb_spec; simpl; try easy. - inversion 2. subst. generalize (NN.spec_pos y); auto with zarith. - Qed. - - Ltac destr_norm_pos x := - rewrite <- (spec_norm_pos x); - let H := fresh in - let x' := fresh x in - assert (H := spec_norm_pos_pos x); - destruct (norm_pos x) as [x'|x']; - specialize (H x' (eq_refl _)) || clear H. - - Lemma spec_testbit: forall x p, testbit x p = Z.testbit (to_Z x) (to_Z p). - Proof. - intros x p. unfold testbit. - destr_norm_pos p; simpl. destr_norm_pos x; simpl. - apply NN.spec_testbit. - rewrite NN.spec_testbit, NN.spec_pred, Z.max_r by auto with zarith. - symmetry. apply Z.bits_opp. apply NN.spec_pos. - symmetry. apply Z.testbit_neg_r; auto with zarith. - Qed. - - Lemma spec_shiftl: forall x p, to_Z (shiftl x p) = Z.shiftl (to_Z x) (to_Z p). - Proof. - intros x p. unfold shiftl. - destr_norm_pos x; destruct p as [p|p]; simpl; - assert (Hp := NN.spec_pos p). - apply NN.spec_shiftl. - rewrite Z.shiftl_opp_r. apply NN.spec_shiftr. - rewrite !NN.spec_shiftl. - rewrite !Z.shiftl_mul_pow2 by apply NN.spec_pos. - symmetry. apply Z.mul_opp_l. - rewrite Z.shiftl_opp_r, NN.spec_succ, NN.spec_shiftr, NN.spec_pred, Z.max_r - by auto with zarith. - now rewrite Zlnot_alt1, Z.lnot_shiftr, Zlnot_alt2. - Qed. - - Lemma spec_shiftr: forall x p, to_Z (shiftr x p) = Z.shiftr (to_Z x) (to_Z p). - Proof. - intros. unfold shiftr. rewrite spec_shiftl, spec_opp. - apply Z.shiftl_opp_r. - Qed. - - Lemma spec_land: forall x y, to_Z (land x y) = Z.land (to_Z x) (to_Z y). - Proof. - intros x y. unfold land. - destr_norm_pos x; destr_norm_pos y; simpl; - rewrite ?NN.spec_succ, ?NN.spec_land, ?NN.spec_ldiff, ?NN.spec_lor, - ?NN.spec_pred, ?Z.max_r, ?Zlnot_alt1; auto with zarith. - now rewrite Z.ldiff_land, Zlnot_alt2. - now rewrite Z.ldiff_land, Z.land_comm, Zlnot_alt2. - now rewrite Z.lnot_lor, !Zlnot_alt2. - Qed. - - Lemma spec_lor: forall x y, to_Z (lor x y) = Z.lor (to_Z x) (to_Z y). - Proof. - intros x y. unfold lor. - destr_norm_pos x; destr_norm_pos y; simpl; - rewrite ?NN.spec_succ, ?NN.spec_land, ?NN.spec_ldiff, ?NN.spec_lor, - ?NN.spec_pred, ?Z.max_r, ?Zlnot_alt1; auto with zarith. - now rewrite Z.lnot_ldiff, Z.lor_comm, Zlnot_alt2. - now rewrite Z.lnot_ldiff, Zlnot_alt2. - now rewrite Z.lnot_land, !Zlnot_alt2. - Qed. - - Lemma spec_ldiff: forall x y, to_Z (ldiff x y) = Z.ldiff (to_Z x) (to_Z y). - Proof. - intros x y. unfold ldiff. - destr_norm_pos x; destr_norm_pos y; simpl; - rewrite ?NN.spec_succ, ?NN.spec_land, ?NN.spec_ldiff, ?NN.spec_lor, - ?NN.spec_pred, ?Z.max_r, ?Zlnot_alt1; auto with zarith. - now rewrite Z.ldiff_land, Zlnot_alt3. - now rewrite Z.lnot_lor, Z.ldiff_land, <- Zlnot_alt2. - now rewrite 2 Z.ldiff_land, Zlnot_alt2, Z.land_comm, Zlnot_alt3. - Qed. - - Lemma spec_lxor: forall x y, to_Z (lxor x y) = Z.lxor (to_Z x) (to_Z y). - Proof. - intros x y. unfold lxor. - destr_norm_pos x; destr_norm_pos y; simpl; - rewrite ?NN.spec_succ, ?NN.spec_lxor, ?NN.spec_pred, ?Z.max_r, ?Zlnot_alt1; - auto with zarith. - now rewrite !Z.lnot_lxor_r, Zlnot_alt2. - now rewrite !Z.lnot_lxor_l, Zlnot_alt2. - now rewrite <- Z.lxor_lnot_lnot, !Zlnot_alt2. - Qed. - - Lemma spec_div2: forall x, to_Z (div2 x) = Z.div2 (to_Z x). - Proof. - intros x. unfold div2. now rewrite spec_shiftr, Z.div2_spec, spec_1. - Qed. - -End Make. diff --git a/theories/Numbers/Integer/Binary/ZBinary.v b/theories/Numbers/Integer/Binary/ZBinary.v index eae8204d..bed827fd 100644 --- a/theories/Numbers/Integer/Binary/ZBinary.v +++ b/theories/Numbers/Integer/Binary/ZBinary.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Integer/NatPairs/ZNatPairs.v b/theories/Numbers/Integer/NatPairs/ZNatPairs.v index 0aaf3365..4b2d5c13 100644 --- a/theories/Numbers/Integer/NatPairs/ZNatPairs.v +++ b/theories/Numbers/Integer/NatPairs/ZNatPairs.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Integer/SpecViaZ/ZSig.v b/theories/Numbers/Integer/SpecViaZ/ZSig.v deleted file mode 100644 index a360327a..00000000 --- a/theories/Numbers/Integer/SpecViaZ/ZSig.v +++ /dev/null @@ -1,135 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Require Import BinInt. - -Open Scope Z_scope. - -(** * ZSig *) - -(** Interface of a rich structure about integers. - Specifications are written via translation to Z. -*) - -Module Type ZType. - - Parameter t : Type. - - Parameter to_Z : t -> Z. - Local Notation "[ x ]" := (to_Z x). - - Definition eq x y := [x] = [y]. - Definition lt x y := [x] < [y]. - Definition le x y := [x] <= [y]. - - Parameter of_Z : Z -> t. - Parameter spec_of_Z: forall x, to_Z (of_Z x) = x. - - Parameter compare : t -> t -> comparison. - Parameter eqb : t -> t -> bool. - Parameter ltb : t -> t -> bool. - Parameter leb : t -> t -> bool. - Parameter min : t -> t -> t. - Parameter max : t -> t -> t. - Parameter zero : t. - Parameter one : t. - Parameter two : t. - Parameter minus_one : t. - Parameter succ : t -> t. - Parameter add : t -> t -> t. - Parameter pred : t -> t. - Parameter sub : t -> t -> t. - Parameter opp : t -> t. - Parameter mul : t -> t -> t. - Parameter square : t -> t. - Parameter pow_pos : t -> positive -> t. - Parameter pow_N : t -> N -> t. - Parameter pow : t -> t -> t. - Parameter sqrt : t -> t. - Parameter log2 : t -> t. - Parameter div_eucl : t -> t -> t * t. - Parameter div : t -> t -> t. - Parameter modulo : t -> t -> t. - Parameter quot : t -> t -> t. - Parameter rem : t -> t -> t. - Parameter gcd : t -> t -> t. - Parameter sgn : t -> t. - Parameter abs : t -> t. - Parameter even : t -> bool. - Parameter odd : t -> bool. - Parameter testbit : t -> t -> bool. - Parameter shiftr : t -> t -> t. - Parameter shiftl : t -> t -> t. - Parameter land : t -> t -> t. - Parameter lor : t -> t -> t. - Parameter ldiff : t -> t -> t. - Parameter lxor : t -> t -> t. - Parameter div2 : t -> t. - - Parameter spec_compare: forall x y, compare x y = ([x] ?= [y]). - Parameter spec_eqb : forall x y, eqb x y = ([x] =? [y]). - Parameter spec_ltb : forall x y, ltb x y = ([x] <? [y]). - Parameter spec_leb : forall x y, leb x y = ([x] <=? [y]). - Parameter spec_min : forall x y, [min x y] = Z.min [x] [y]. - Parameter spec_max : forall x y, [max x y] = Z.max [x] [y]. - Parameter spec_0: [zero] = 0. - Parameter spec_1: [one] = 1. - Parameter spec_2: [two] = 2. - Parameter spec_m1: [minus_one] = -1. - Parameter spec_succ: forall n, [succ n] = [n] + 1. - Parameter spec_add: forall x y, [add x y] = [x] + [y]. - Parameter spec_pred: forall x, [pred x] = [x] - 1. - Parameter spec_sub: forall x y, [sub x y] = [x] - [y]. - Parameter spec_opp: forall x, [opp x] = - [x]. - Parameter spec_mul: forall x y, [mul x y] = [x] * [y]. - Parameter spec_square: forall x, [square x] = [x] * [x]. - Parameter spec_pow_pos: forall x n, [pow_pos x n] = [x] ^ Zpos n. - Parameter spec_pow_N: forall x n, [pow_N x n] = [x] ^ Z.of_N n. - Parameter spec_pow: forall x n, [pow x n] = [x] ^ [n]. - Parameter spec_sqrt: forall x, [sqrt x] = Z.sqrt [x]. - Parameter spec_log2: forall x, [log2 x] = Z.log2 [x]. - Parameter spec_div_eucl: forall x y, - let (q,r) := div_eucl x y in ([q], [r]) = Z.div_eucl [x] [y]. - Parameter spec_div: forall x y, [div x y] = [x] / [y]. - Parameter spec_modulo: forall x y, [modulo x y] = [x] mod [y]. - Parameter spec_quot: forall x y, [quot x y] = [x] ÷ [y]. - Parameter spec_rem: forall x y, [rem x y] = Z.rem [x] [y]. - Parameter spec_gcd: forall a b, [gcd a b] = Z.gcd [a] [b]. - Parameter spec_sgn : forall x, [sgn x] = Z.sgn [x]. - Parameter spec_abs : forall x, [abs x] = Z.abs [x]. - Parameter spec_even : forall x, even x = Z.even [x]. - Parameter spec_odd : forall x, odd x = Z.odd [x]. - Parameter spec_testbit: forall x p, testbit x p = Z.testbit [x] [p]. - Parameter spec_shiftr: forall x p, [shiftr x p] = Z.shiftr [x] [p]. - Parameter spec_shiftl: forall x p, [shiftl x p] = Z.shiftl [x] [p]. - Parameter spec_land: forall x y, [land x y] = Z.land [x] [y]. - Parameter spec_lor: forall x y, [lor x y] = Z.lor [x] [y]. - Parameter spec_ldiff: forall x y, [ldiff x y] = Z.ldiff [x] [y]. - Parameter spec_lxor: forall x y, [lxor x y] = Z.lxor [x] [y]. - Parameter spec_div2: forall x, [div2 x] = Z.div2 [x]. - -End ZType. - -Module Type ZType_Notation (Import Z:ZType). - Notation "[ x ]" := (to_Z x). - Infix "==" := eq (at level 70). - Notation "0" := zero. - Notation "1" := one. - Notation "2" := two. - Infix "+" := add. - Infix "-" := sub. - Infix "*" := mul. - Infix "^" := pow. - Notation "- x" := (opp x). - Infix "<=" := le. - Infix "<" := lt. -End ZType_Notation. - -Module Type ZType' := ZType <+ ZType_Notation. diff --git a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v deleted file mode 100644 index 32410d1d..00000000 --- a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v +++ /dev/null @@ -1,527 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -Require Import Bool ZArith OrdersFacts Nnat ZAxioms ZSig. - -(** * The interface [ZSig.ZType] implies the interface [ZAxiomsSig] *) - -Module ZTypeIsZAxioms (Import ZZ : ZType'). - -Hint Rewrite - spec_0 spec_1 spec_2 spec_add spec_sub spec_pred spec_succ - spec_mul spec_opp spec_of_Z spec_div spec_modulo spec_square spec_sqrt - spec_compare spec_eqb spec_ltb spec_leb spec_max spec_min - spec_abs spec_sgn spec_pow spec_log2 spec_even spec_odd spec_gcd - spec_quot spec_rem spec_testbit spec_shiftl spec_shiftr - spec_land spec_lor spec_ldiff spec_lxor spec_div2 - : zsimpl. - -Ltac zsimpl := autorewrite with zsimpl. -Ltac zcongruence := repeat red; intros; zsimpl; congruence. -Ltac zify := unfold eq, lt, le in *; zsimpl. - -Instance eq_equiv : Equivalence eq. -Proof. unfold eq. firstorder. Qed. - -Local Obligation Tactic := zcongruence. - -Program Instance succ_wd : Proper (eq ==> eq) succ. -Program Instance pred_wd : Proper (eq ==> eq) pred. -Program Instance add_wd : Proper (eq ==> eq ==> eq) add. -Program Instance sub_wd : Proper (eq ==> eq ==> eq) sub. -Program Instance mul_wd : Proper (eq ==> eq ==> eq) mul. - -Theorem pred_succ : forall n, pred (succ n) == n. -Proof. -intros. zify. auto with zarith. -Qed. - -Theorem one_succ : 1 == succ 0. -Proof. -now zify. -Qed. - -Theorem two_succ : 2 == succ 1. -Proof. -now zify. -Qed. - -Section Induction. - -Variable A : ZZ.t -> Prop. -Hypothesis A_wd : Proper (eq==>iff) A. -Hypothesis A0 : A 0. -Hypothesis AS : forall n, A n <-> A (succ n). - -Let B (z : Z) := A (of_Z z). - -Lemma B0 : B 0. -Proof. -unfold B; simpl. -rewrite <- (A_wd 0); auto. -zify. auto. -Qed. - -Lemma BS : forall z : Z, B z -> B (z + 1). -Proof. -intros z H. -unfold B in *. apply -> AS in H. -setoid_replace (of_Z (z + 1)) with (succ (of_Z z)); auto. -zify. auto. -Qed. - -Lemma BP : forall z : Z, B z -> B (z - 1). -Proof. -intros z H. -unfold B in *. rewrite AS. -setoid_replace (succ (of_Z (z - 1))) with (of_Z z); auto. -zify. auto with zarith. -Qed. - -Lemma B_holds : forall z : Z, B z. -Proof. -intros; destruct (Z_lt_le_dec 0 z). -apply natlike_ind; auto with zarith. -apply B0. -intros; apply BS; auto. -replace z with (-(-z))%Z in * by (auto with zarith). -remember (-z)%Z as z'. -pattern z'; apply natlike_ind. -apply B0. -intros; rewrite Z.opp_succ; unfold Z.pred; apply BP; auto. -subst z'; auto with zarith. -Qed. - -Theorem bi_induction : forall n, A n. -Proof. -intro n. setoid_replace n with (of_Z (to_Z n)). -apply B_holds. -zify. auto. -Qed. - -End Induction. - -Theorem add_0_l : forall n, 0 + n == n. -Proof. -intros. zify. auto with zarith. -Qed. - -Theorem add_succ_l : forall n m, (succ n) + m == succ (n + m). -Proof. -intros. zify. auto with zarith. -Qed. - -Theorem sub_0_r : forall n, n - 0 == n. -Proof. -intros. zify. auto with zarith. -Qed. - -Theorem sub_succ_r : forall n m, n - (succ m) == pred (n - m). -Proof. -intros. zify. auto with zarith. -Qed. - -Theorem mul_0_l : forall n, 0 * n == 0. -Proof. -intros. zify. auto with zarith. -Qed. - -Theorem mul_succ_l : forall n m, (succ n) * m == n * m + m. -Proof. -intros. zify. ring. -Qed. - -(** Order *) - -Lemma eqb_eq x y : eqb x y = true <-> x == y. -Proof. - zify. apply Z.eqb_eq. -Qed. - -Lemma leb_le x y : leb x y = true <-> x <= y. -Proof. - zify. apply Z.leb_le. -Qed. - -Lemma ltb_lt x y : ltb x y = true <-> x < y. -Proof. - zify. apply Z.ltb_lt. -Qed. - -Lemma compare_eq_iff n m : compare n m = Eq <-> n == m. -Proof. - intros. zify. apply Z.compare_eq_iff. -Qed. - -Lemma compare_lt_iff n m : compare n m = Lt <-> n < m. -Proof. - intros. zify. reflexivity. -Qed. - -Lemma compare_le_iff n m : compare n m <> Gt <-> n <= m. -Proof. - intros. zify. reflexivity. -Qed. - -Lemma compare_antisym n m : compare m n = CompOpp (compare n m). -Proof. - intros. zify. apply Z.compare_antisym. -Qed. - -Include BoolOrderFacts ZZ ZZ ZZ [no inline]. - -Instance compare_wd : Proper (eq ==> eq ==> Logic.eq) compare. -Proof. -intros x x' Hx y y' Hy. zify. now rewrite Hx, Hy. -Qed. - -Instance eqb_wd : Proper (eq ==> eq ==> Logic.eq) eqb. -Proof. -intros x x' Hx y y' Hy. zify. now rewrite Hx, Hy. -Qed. - -Instance ltb_wd : Proper (eq ==> eq ==> Logic.eq) ltb. -Proof. -intros x x' Hx y y' Hy. zify. now rewrite Hx, Hy. -Qed. - -Instance leb_wd : Proper (eq ==> eq ==> Logic.eq) leb. -Proof. -intros x x' Hx y y' Hy. zify. now rewrite Hx, Hy. -Qed. - -Instance lt_wd : Proper (eq ==> eq ==> iff) lt. -Proof. -intros x x' Hx y y' Hy; unfold lt; rewrite Hx, Hy; intuition. -Qed. - -Theorem lt_succ_r : forall n m, n < (succ m) <-> n <= m. -Proof. -intros. zify. omega. -Qed. - -Theorem min_l : forall n m, n <= m -> min n m == n. -Proof. -intros n m. zify. omega with *. -Qed. - -Theorem min_r : forall n m, m <= n -> min n m == m. -Proof. -intros n m. zify. omega with *. -Qed. - -Theorem max_l : forall n m, m <= n -> max n m == n. -Proof. -intros n m. zify. omega with *. -Qed. - -Theorem max_r : forall n m, n <= m -> max n m == m. -Proof. -intros n m. zify. omega with *. -Qed. - -(** Part specific to integers, not natural numbers *) - -Theorem succ_pred : forall n, succ (pred n) == n. -Proof. -intros. zify. auto with zarith. -Qed. - -(** Opp *) - -Program Instance opp_wd : Proper (eq ==> eq) opp. - -Theorem opp_0 : - 0 == 0. -Proof. -intros. zify. auto with zarith. -Qed. - -Theorem opp_succ : forall n, - (succ n) == pred (- n). -Proof. -intros. zify. auto with zarith. -Qed. - -(** Abs / Sgn *) - -Theorem abs_eq : forall n, 0 <= n -> abs n == n. -Proof. -intros n. zify. omega with *. -Qed. - -Theorem abs_neq : forall n, n <= 0 -> abs n == -n. -Proof. -intros n. zify. omega with *. -Qed. - -Theorem sgn_null : forall n, n==0 -> sgn n == 0. -Proof. -intros n. zify. omega with *. -Qed. - -Theorem sgn_pos : forall n, 0<n -> sgn n == 1. -Proof. -intros n. zify. omega with *. -Qed. - -Theorem sgn_neg : forall n, n<0 -> sgn n == opp 1. -Proof. -intros n. zify. omega with *. -Qed. - -(** Power *) - -Program Instance pow_wd : Proper (eq==>eq==>eq) pow. - -Lemma pow_0_r : forall a, a^0 == 1. -Proof. - intros. now zify. -Qed. - -Lemma pow_succ_r : forall a b, 0<=b -> a^(succ b) == a * a^b. -Proof. - intros a b. zify. intros. now rewrite Z.add_1_r, Z.pow_succ_r. -Qed. - -Lemma pow_neg_r : forall a b, b<0 -> a^b == 0. -Proof. - intros a b. zify. intros Hb. - destruct [b]; reflexivity || discriminate. -Qed. - -Lemma pow_pow_N : forall a b, 0<=b -> a^b == pow_N a (Z.to_N (to_Z b)). -Proof. - intros a b. zify. intros Hb. now rewrite spec_pow_N, Z2N.id. -Qed. - -Lemma pow_pos_N : forall a p, pow_pos a p == pow_N a (Npos p). -Proof. - intros a b. red. now rewrite spec_pow_N, spec_pow_pos. -Qed. - -(** Square *) - -Lemma square_spec n : square n == n * n. -Proof. - now zify. -Qed. - -(** Sqrt *) - -Lemma sqrt_spec : forall n, 0<=n -> - (sqrt n)*(sqrt n) <= n /\ n < (succ (sqrt n))*(succ (sqrt n)). -Proof. - intros n. zify. apply Z.sqrt_spec. -Qed. - -Lemma sqrt_neg : forall n, n<0 -> sqrt n == 0. -Proof. - intros n. zify. apply Z.sqrt_neg. -Qed. - -(** Log2 *) - -Lemma log2_spec : forall n, 0<n -> - 2^(log2 n) <= n /\ n < 2^(succ (log2 n)). -Proof. - intros n. zify. apply Z.log2_spec. -Qed. - -Lemma log2_nonpos : forall n, n<=0 -> log2 n == 0. -Proof. - intros n. zify. apply Z.log2_nonpos. -Qed. - -(** Even / Odd *) - -Definition Even n := exists m, n == 2*m. -Definition Odd n := exists m, n == 2*m+1. - -Lemma even_spec n : even n = true <-> Even n. -Proof. - unfold Even. zify. rewrite Z.even_spec. - split; intros (m,Hm). - - exists (of_Z m). now zify. - - exists [m]. revert Hm. now zify. -Qed. - -Lemma odd_spec n : odd n = true <-> Odd n. -Proof. - unfold Odd. zify. rewrite Z.odd_spec. - split; intros (m,Hm). - - exists (of_Z m). now zify. - - exists [m]. revert Hm. now zify. -Qed. - -(** Div / Mod *) - -Program Instance div_wd : Proper (eq==>eq==>eq) div. -Program Instance mod_wd : Proper (eq==>eq==>eq) modulo. - -Theorem div_mod : forall a b, ~b==0 -> a == b*(div a b) + (modulo a b). -Proof. -intros a b. zify. intros. apply Z.div_mod; auto. -Qed. - -Theorem mod_pos_bound : - forall a b, 0 < b -> 0 <= modulo a b /\ modulo a b < b. -Proof. -intros a b. zify. intros. apply Z_mod_lt; auto with zarith. -Qed. - -Theorem mod_neg_bound : - forall a b, b < 0 -> b < modulo a b /\ modulo a b <= 0. -Proof. -intros a b. zify. intros. apply Z_mod_neg; auto with zarith. -Qed. - -Definition mod_bound_pos : - forall a b, 0<=a -> 0<b -> 0 <= modulo a b /\ modulo a b < b := - fun a b _ H => mod_pos_bound a b H. - -(** Quot / Rem *) - -Program Instance quot_wd : Proper (eq==>eq==>eq) quot. -Program Instance rem_wd : Proper (eq==>eq==>eq) rem. - -Theorem quot_rem : forall a b, ~b==0 -> a == b*(quot a b) + rem a b. -Proof. -intros a b. zify. apply Z.quot_rem. -Qed. - -Theorem rem_bound_pos : - forall a b, 0<=a -> 0<b -> 0 <= rem a b /\ rem a b < b. -Proof. -intros a b. zify. apply Z.rem_bound_pos. -Qed. - -Theorem rem_opp_l : forall a b, ~b==0 -> rem (-a) b == -(rem a b). -Proof. -intros a b. zify. apply Z.rem_opp_l. -Qed. - -Theorem rem_opp_r : forall a b, ~b==0 -> rem a (-b) == rem a b. -Proof. -intros a b. zify. apply Z.rem_opp_r. -Qed. - -(** Gcd *) - -Definition divide n m := exists p, m == p*n. -Local Notation "( x | y )" := (divide x y) (at level 0). - -Lemma spec_divide : forall n m, (n|m) <-> Z.divide [n] [m]. -Proof. - intros n m. split. - - intros (p,H). exists [p]. revert H; now zify. - - intros (z,H). exists (of_Z z). now zify. -Qed. - -Lemma gcd_divide_l : forall n m, (gcd n m | n). -Proof. - intros n m. apply spec_divide. zify. apply Z.gcd_divide_l. -Qed. - -Lemma gcd_divide_r : forall n m, (gcd n m | m). -Proof. - intros n m. apply spec_divide. zify. apply Z.gcd_divide_r. -Qed. - -Lemma gcd_greatest : forall n m p, (p|n) -> (p|m) -> (p|gcd n m). -Proof. - intros n m p. rewrite !spec_divide. zify. apply Z.gcd_greatest. -Qed. - -Lemma gcd_nonneg : forall n m, 0 <= gcd n m. -Proof. - intros. zify. apply Z.gcd_nonneg. -Qed. - -(** Bitwise operations *) - -Program Instance testbit_wd : Proper (eq==>eq==>Logic.eq) testbit. - -Lemma testbit_odd_0 : forall a, testbit (2*a+1) 0 = true. -Proof. - intros. zify. apply Z.testbit_odd_0. -Qed. - -Lemma testbit_even_0 : forall a, testbit (2*a) 0 = false. -Proof. - intros. zify. apply Z.testbit_even_0. -Qed. - -Lemma testbit_odd_succ : forall a n, 0<=n -> - testbit (2*a+1) (succ n) = testbit a n. -Proof. - intros a n. zify. apply Z.testbit_odd_succ. -Qed. - -Lemma testbit_even_succ : forall a n, 0<=n -> - testbit (2*a) (succ n) = testbit a n. -Proof. - intros a n. zify. apply Z.testbit_even_succ. -Qed. - -Lemma testbit_neg_r : forall a n, n<0 -> testbit a n = false. -Proof. - intros a n. zify. apply Z.testbit_neg_r. -Qed. - -Lemma shiftr_spec : forall a n m, 0<=m -> - testbit (shiftr a n) m = testbit a (m+n). -Proof. - intros a n m. zify. apply Z.shiftr_spec. -Qed. - -Lemma shiftl_spec_high : forall a n m, 0<=m -> n<=m -> - testbit (shiftl a n) m = testbit a (m-n). -Proof. - intros a n m. zify. intros Hn H. - now apply Z.shiftl_spec_high. -Qed. - -Lemma shiftl_spec_low : forall a n m, m<n -> - testbit (shiftl a n) m = false. -Proof. - intros a n m. zify. intros H. now apply Z.shiftl_spec_low. -Qed. - -Lemma land_spec : forall a b n, - testbit (land a b) n = testbit a n && testbit b n. -Proof. - intros a n m. zify. now apply Z.land_spec. -Qed. - -Lemma lor_spec : forall a b n, - testbit (lor a b) n = testbit a n || testbit b n. -Proof. - intros a n m. zify. now apply Z.lor_spec. -Qed. - -Lemma ldiff_spec : forall a b n, - testbit (ldiff a b) n = testbit a n && negb (testbit b n). -Proof. - intros a n m. zify. now apply Z.ldiff_spec. -Qed. - -Lemma lxor_spec : forall a b n, - testbit (lxor a b) n = xorb (testbit a n) (testbit b n). -Proof. - intros a n m. zify. now apply Z.lxor_spec. -Qed. - -Lemma div2_spec : forall a, div2 a == shiftr a 1. -Proof. - intros a. zify. now apply Z.div2_spec. -Qed. - -End ZTypeIsZAxioms. - -Module ZType_ZAxioms (ZZ : ZType) - <: ZAxiomsSig <: OrderFunctions ZZ <: HasMinMax ZZ - := ZZ <+ ZTypeIsZAxioms. diff --git a/theories/Numbers/NaryFunctions.v b/theories/Numbers/NaryFunctions.v index eaac7d69..ee28628e 100644 --- a/theories/Numbers/NaryFunctions.v +++ b/theories/Numbers/NaryFunctions.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Pierre Letouzey, Jerome Vouillon, PPS, Paris 7, 2008 *) (************************************************************************) diff --git a/theories/Numbers/NatInt/NZAdd.v b/theories/Numbers/NatInt/NZAdd.v index 8cc52940..bc366c50 100644 --- a/theories/Numbers/NatInt/NZAdd.v +++ b/theories/Numbers/NatInt/NZAdd.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/NatInt/NZAddOrder.v b/theories/Numbers/NatInt/NZAddOrder.v index 705f2910..99812ee3 100644 --- a/theories/Numbers/NatInt/NZAddOrder.v +++ b/theories/Numbers/NatInt/NZAddOrder.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/NatInt/NZAxioms.v b/theories/Numbers/NatInt/NZAxioms.v index 5a7f2072..8c364cde 100644 --- a/theories/Numbers/NatInt/NZAxioms.v +++ b/theories/Numbers/NatInt/NZAxioms.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Initial Author : Evgeny Makarov, INRIA, 2007 *) diff --git a/theories/Numbers/NatInt/NZBase.v b/theories/Numbers/NatInt/NZBase.v index cf4ad354..595b2182 100644 --- a/theories/Numbers/NatInt/NZBase.v +++ b/theories/Numbers/NatInt/NZBase.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/NatInt/NZBits.v b/theories/Numbers/NatInt/NZBits.v index 209bd947..eefa5157 100644 --- a/theories/Numbers/NatInt/NZBits.v +++ b/theories/Numbers/NatInt/NZBits.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Bool NZAxioms NZMulOrder NZParity NZPow NZDiv NZLog. diff --git a/theories/Numbers/NatInt/NZDiv.v b/theories/Numbers/NatInt/NZDiv.v index b2c0be6f..550aa226 100644 --- a/theories/Numbers/NatInt/NZDiv.v +++ b/theories/Numbers/NatInt/NZDiv.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Euclidean Division *) diff --git a/theories/Numbers/NatInt/NZDomain.v b/theories/Numbers/NatInt/NZDomain.v index 3881a27f..3d0c005f 100644 --- a/theories/Numbers/NatInt/NZDomain.v +++ b/theories/Numbers/NatInt/NZDomain.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export NumPrelude NZAxioms. diff --git a/theories/Numbers/NatInt/NZGcd.v b/theories/Numbers/NatInt/NZGcd.v index 44088f8c..c38d1aac 100644 --- a/theories/Numbers/NatInt/NZGcd.v +++ b/theories/Numbers/NatInt/NZGcd.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Greatest Common Divisor *) diff --git a/theories/Numbers/NatInt/NZLog.v b/theories/Numbers/NatInt/NZLog.v index 3dfa2eef..794851a9 100644 --- a/theories/Numbers/NatInt/NZLog.v +++ b/theories/Numbers/NatInt/NZLog.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Base-2 Logarithm *) diff --git a/theories/Numbers/NatInt/NZMul.v b/theories/Numbers/NatInt/NZMul.v index 36cd0827..44cbc517 100644 --- a/theories/Numbers/NatInt/NZMul.v +++ b/theories/Numbers/NatInt/NZMul.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/NatInt/NZMulOrder.v b/theories/Numbers/NatInt/NZMulOrder.v index 8569fc56..292f0837 100644 --- a/theories/Numbers/NatInt/NZMulOrder.v +++ b/theories/Numbers/NatInt/NZMulOrder.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/NatInt/NZOrder.v b/theories/Numbers/NatInt/NZOrder.v index 0b9d6598..60e1123b 100644 --- a/theories/Numbers/NatInt/NZOrder.v +++ b/theories/Numbers/NatInt/NZOrder.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/NatInt/NZParity.v b/theories/Numbers/NatInt/NZParity.v index 4a1352cc..93d99f08 100644 --- a/theories/Numbers/NatInt/NZParity.v +++ b/theories/Numbers/NatInt/NZParity.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Bool NZAxioms NZMulOrder. @@ -260,4 +262,4 @@ Proof. intros. apply odd_add_mul_even. apply even_spec, even_2. Qed. -End NZParityProp.
\ No newline at end of file +End NZParityProp. diff --git a/theories/Numbers/NatInt/NZPow.v b/theories/Numbers/NatInt/NZPow.v index 261febbc..a1310667 100644 --- a/theories/Numbers/NatInt/NZPow.v +++ b/theories/Numbers/NatInt/NZPow.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Power Function *) diff --git a/theories/Numbers/NatInt/NZProperties.v b/theories/Numbers/NatInt/NZProperties.v index 79a9d733..fbcf43e8 100644 --- a/theories/Numbers/NatInt/NZProperties.v +++ b/theories/Numbers/NatInt/NZProperties.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/NatInt/NZSqrt.v b/theories/Numbers/NatInt/NZSqrt.v index 7f9bb9c2..c2d2c4ae 100644 --- a/theories/Numbers/NatInt/NZSqrt.v +++ b/theories/Numbers/NatInt/NZSqrt.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Square Root Function *) diff --git a/theories/Numbers/Natural/Abstract/NAdd.v b/theories/Numbers/Natural/Abstract/NAdd.v index eecec3ac..dc5f8e53 100644 --- a/theories/Numbers/Natural/Abstract/NAdd.v +++ b/theories/Numbers/Natural/Abstract/NAdd.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Natural/Abstract/NAddOrder.v b/theories/Numbers/Natural/Abstract/NAddOrder.v index 9d68a006..2da3f0bf 100644 --- a/theories/Numbers/Natural/Abstract/NAddOrder.v +++ b/theories/Numbers/Natural/Abstract/NAddOrder.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Natural/Abstract/NAxioms.v b/theories/Numbers/Natural/Abstract/NAxioms.v index 25e285d5..dd09ac5f 100644 --- a/theories/Numbers/Natural/Abstract/NAxioms.v +++ b/theories/Numbers/Natural/Abstract/NAxioms.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Natural/Abstract/NBase.v b/theories/Numbers/Natural/Abstract/NBase.v index f949a0f6..ad0b3d3d 100644 --- a/theories/Numbers/Natural/Abstract/NBase.v +++ b/theories/Numbers/Natural/Abstract/NBase.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Natural/Abstract/NBits.v b/theories/Numbers/Natural/Abstract/NBits.v index 3dd603e2..e1391f59 100644 --- a/theories/Numbers/Natural/Abstract/NBits.v +++ b/theories/Numbers/Natural/Abstract/NBits.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Bool NAxioms NSub NPow NDiv NParity NLog. diff --git a/theories/Numbers/Natural/Abstract/NDefOps.v b/theories/Numbers/Natural/Abstract/NDefOps.v index b3a53617..8e1be0d7 100644 --- a/theories/Numbers/Natural/Abstract/NDefOps.v +++ b/theories/Numbers/Natural/Abstract/NDefOps.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Natural/Abstract/NDiv.v b/theories/Numbers/Natural/Abstract/NDiv.v index 84e1219e..4c26a071 100644 --- a/theories/Numbers/Natural/Abstract/NDiv.v +++ b/theories/Numbers/Natural/Abstract/NDiv.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import NAxioms NSub NZDiv. diff --git a/theories/Numbers/Natural/Abstract/NGcd.v b/theories/Numbers/Natural/Abstract/NGcd.v index 1eac134d..96fb4247 100644 --- a/theories/Numbers/Natural/Abstract/NGcd.v +++ b/theories/Numbers/Natural/Abstract/NGcd.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Properties of the greatest common divisor *) diff --git a/theories/Numbers/Natural/Abstract/NIso.v b/theories/Numbers/Natural/Abstract/NIso.v index e6cac0ba..d41d0aff 100644 --- a/theories/Numbers/Natural/Abstract/NIso.v +++ b/theories/Numbers/Natural/Abstract/NIso.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Natural/Abstract/NLcm.v b/theories/Numbers/Natural/Abstract/NLcm.v index 3d749799..47b74193 100644 --- a/theories/Numbers/Natural/Abstract/NLcm.v +++ b/theories/Numbers/Natural/Abstract/NLcm.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import NAxioms NSub NDiv NGcd. diff --git a/theories/Numbers/Natural/Abstract/NLog.v b/theories/Numbers/Natural/Abstract/NLog.v index 0c9da29b..fe6fcee5 100644 --- a/theories/Numbers/Natural/Abstract/NLog.v +++ b/theories/Numbers/Natural/Abstract/NLog.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Base-2 Logarithm Properties *) diff --git a/theories/Numbers/Natural/Abstract/NMaxMin.v b/theories/Numbers/Natural/Abstract/NMaxMin.v index b4c9db91..3cf4d3f9 100644 --- a/theories/Numbers/Natural/Abstract/NMaxMin.v +++ b/theories/Numbers/Natural/Abstract/NMaxMin.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import NAxioms NSub GenericMinMax. diff --git a/theories/Numbers/Natural/Abstract/NMulOrder.v b/theories/Numbers/Natural/Abstract/NMulOrder.v index a4b52396..b7f1c8e4 100644 --- a/theories/Numbers/Natural/Abstract/NMulOrder.v +++ b/theories/Numbers/Natural/Abstract/NMulOrder.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Natural/Abstract/NOrder.v b/theories/Numbers/Natural/Abstract/NOrder.v index 60e955f5..acaecad9 100644 --- a/theories/Numbers/Natural/Abstract/NOrder.v +++ b/theories/Numbers/Natural/Abstract/NOrder.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Natural/Abstract/NParity.v b/theories/Numbers/Natural/Abstract/NParity.v index e1f573f6..cb89e1d7 100644 --- a/theories/Numbers/Natural/Abstract/NParity.v +++ b/theories/Numbers/Natural/Abstract/NParity.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Bool NSub NZParity. diff --git a/theories/Numbers/Natural/Abstract/NPow.v b/theories/Numbers/Natural/Abstract/NPow.v index c9b2177f..fc1cc93b 100644 --- a/theories/Numbers/Natural/Abstract/NPow.v +++ b/theories/Numbers/Natural/Abstract/NPow.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Properties of the power function *) diff --git a/theories/Numbers/Natural/Abstract/NProperties.v b/theories/Numbers/Natural/Abstract/NProperties.v index 819a5be9..bcf906cf 100644 --- a/theories/Numbers/Natural/Abstract/NProperties.v +++ b/theories/Numbers/Natural/Abstract/NProperties.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export NAxioms. diff --git a/theories/Numbers/Natural/Abstract/NSqrt.v b/theories/Numbers/Natural/Abstract/NSqrt.v index 4ae41407..6bffe693 100644 --- a/theories/Numbers/Natural/Abstract/NSqrt.v +++ b/theories/Numbers/Natural/Abstract/NSqrt.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Properties of Square Root Function *) diff --git a/theories/Numbers/Natural/Abstract/NStrongRec.v b/theories/Numbers/Natural/Abstract/NStrongRec.v index fa3a5351..f76d8ae8 100644 --- a/theories/Numbers/Natural/Abstract/NStrongRec.v +++ b/theories/Numbers/Natural/Abstract/NStrongRec.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Natural/Abstract/NSub.v b/theories/Numbers/Natural/Abstract/NSub.v index 21f12037..453b0c0d 100644 --- a/theories/Numbers/Natural/Abstract/NSub.v +++ b/theories/Numbers/Natural/Abstract/NSub.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Natural/BigN/BigN.v b/theories/Numbers/Natural/BigN/BigN.v deleted file mode 100644 index e8ff516f..00000000 --- a/theories/Numbers/Natural/BigN/BigN.v +++ /dev/null @@ -1,198 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -(** * Efficient arbitrary large natural numbers in base 2^31 *) - -(** Initial Author: Arnaud Spiwack *) - -Require Export Int31. -Require Import CyclicAxioms Cyclic31 Ring31 NSig NSigNAxioms NMake - NProperties GenericMinMax. - -(** The following [BigN] module regroups both the operations and - all the abstract properties: - - - [NMake.Make Int31Cyclic] provides the operations and basic specs - w.r.t. ZArith - - [NTypeIsNAxioms] shows (mainly) that these operations implement - the interface [NAxioms] - - [NProp] adds all generic properties derived from [NAxioms] - - [MinMax*Properties] provides properties of [min] and [max]. - -*) - -Delimit Scope bigN_scope with bigN. - -Module BigN <: NType <: OrderedTypeFull <: TotalOrder := - NMake.Make Int31Cyclic - <+ NTypeIsNAxioms - <+ NBasicProp [no inline] <+ NExtraProp [no inline] - <+ HasEqBool2Dec [no inline] - <+ MinMaxLogicalProperties [no inline] - <+ MinMaxDecProperties [no inline]. - -(** Notations about [BigN] *) - -Local Open Scope bigN_scope. - -Notation bigN := BigN.t. -Bind Scope bigN_scope with bigN BigN.t BigN.t'. -Arguments BigN.N0 _%int31. -Local Notation "0" := BigN.zero : bigN_scope. (* temporary notation *) -Local Notation "1" := BigN.one : bigN_scope. (* temporary notation *) -Local Notation "2" := BigN.two : bigN_scope. (* temporary notation *) -Infix "+" := BigN.add : bigN_scope. -Infix "-" := BigN.sub : bigN_scope. -Infix "*" := BigN.mul : bigN_scope. -Infix "/" := BigN.div : bigN_scope. -Infix "^" := BigN.pow : bigN_scope. -Infix "?=" := BigN.compare : bigN_scope. -Infix "=?" := BigN.eqb (at level 70, no associativity) : bigN_scope. -Infix "<=?" := BigN.leb (at level 70, no associativity) : bigN_scope. -Infix "<?" := BigN.ltb (at level 70, no associativity) : bigN_scope. -Infix "==" := BigN.eq (at level 70, no associativity) : bigN_scope. -Notation "x != y" := (~x==y) (at level 70, no associativity) : bigN_scope. -Infix "<" := BigN.lt : bigN_scope. -Infix "<=" := BigN.le : bigN_scope. -Notation "x > y" := (y < x) (only parsing) : bigN_scope. -Notation "x >= y" := (y <= x) (only parsing) : bigN_scope. -Notation "x < y < z" := (x<y /\ y<z) : bigN_scope. -Notation "x < y <= z" := (x<y /\ y<=z) : bigN_scope. -Notation "x <= y < z" := (x<=y /\ y<z) : bigN_scope. -Notation "x <= y <= z" := (x<=y /\ y<=z) : bigN_scope. -Notation "[ i ]" := (BigN.to_Z i) : bigN_scope. -Infix "mod" := BigN.modulo (at level 40, no associativity) : bigN_scope. - -(** Example of reasoning about [BigN] *) - -Theorem succ_pred: forall q : bigN, - 0 < q -> BigN.succ (BigN.pred q) == q. -Proof. -intros; apply BigN.succ_pred. -intro H'; rewrite H' in H; discriminate. -Qed. - -(** [BigN] is a semi-ring *) - -Lemma BigNring : semi_ring_theory 0 1 BigN.add BigN.mul BigN.eq. -Proof. -constructor. -exact BigN.add_0_l. exact BigN.add_comm. exact BigN.add_assoc. -exact BigN.mul_1_l. exact BigN.mul_0_l. exact BigN.mul_comm. -exact BigN.mul_assoc. exact BigN.mul_add_distr_r. -Qed. - -Lemma BigNeqb_correct : forall x y, (x =? y) = true -> x==y. -Proof. now apply BigN.eqb_eq. Qed. - -Lemma BigNpower : power_theory 1 BigN.mul BigN.eq BigN.of_N BigN.pow. -Proof. -constructor. -intros. red. rewrite BigN.spec_pow, BigN.spec_of_N. -rewrite Zpower_theory.(rpow_pow_N). -destruct n; simpl. reflexivity. -induction p; simpl; intros; BigN.zify; rewrite ?IHp; auto. -Qed. - -Lemma BigNdiv : div_theory BigN.eq BigN.add BigN.mul (@id _) - (fun a b => if b =? 0 then (0,a) else BigN.div_eucl a b). -Proof. -constructor. unfold id. intros a b. -BigN.zify. -case Z.eqb_spec. -BigN.zify. auto with zarith. -intros NEQ. -generalize (BigN.spec_div_eucl a b). -generalize (Z_div_mod_full [a] [b] NEQ). -destruct BigN.div_eucl as (q,r), Z.div_eucl as (q',r'). -intros (EQ,_). injection 1 as EQr EQq. -BigN.zify. rewrite EQr, EQq; auto. -Qed. - - -(** Detection of constants *) - -Ltac isStaticWordCst t := - match t with - | W0 => constr:(true) - | WW ?t1 ?t2 => - match isStaticWordCst t1 with - | false => constr:(false) - | true => isStaticWordCst t2 - end - | _ => isInt31cst t - end. - -Ltac isBigNcst t := - match t with - | BigN.N0 ?t => isStaticWordCst t - | BigN.N1 ?t => isStaticWordCst t - | BigN.N2 ?t => isStaticWordCst t - | BigN.N3 ?t => isStaticWordCst t - | BigN.N4 ?t => isStaticWordCst t - | BigN.N5 ?t => isStaticWordCst t - | BigN.N6 ?t => isStaticWordCst t - | BigN.Nn ?n ?t => match isnatcst n with - | true => isStaticWordCst t - | false => constr:(false) - end - | BigN.zero => constr:(true) - | BigN.one => constr:(true) - | BigN.two => constr:(true) - | _ => constr:(false) - end. - -Ltac BigNcst t := - match isBigNcst t with - | true => constr:(t) - | false => constr:(NotConstant) - end. - -Ltac BigN_to_N t := - match isBigNcst t with - | true => eval vm_compute in (BigN.to_N t) - | false => constr:(NotConstant) - end. - -Ltac Ncst t := - match isNcst t with - | true => constr:(t) - | false => constr:(NotConstant) - end. - -(** Registration for the "ring" tactic *) - -Add Ring BigNr : BigNring - (decidable BigNeqb_correct, - constants [BigNcst], - power_tac BigNpower [BigN_to_N], - div BigNdiv). - -Section TestRing. -Let test : forall x y, 1 + x*y^1 + x^2 + 1 == 1*1 + 1 + y*x + 1*x*x. -intros. ring_simplify. reflexivity. -Qed. -End TestRing. - -(** We benefit also from an "order" tactic *) - -Ltac bigN_order := BigN.order. - -Section TestOrder. -Let test : forall x y : bigN, x<=y -> y<=x -> x==y. -Proof. bigN_order. Qed. -End TestOrder. - -(** We can use at least a bit of (r)omega by translating to [Z]. *) - -Section TestOmega. -Let test : forall x y : bigN, x<=y -> y<=x -> x==y. -Proof. intros x y. BigN.zify. omega. Qed. -End TestOmega. - -(** Todo: micromega *) diff --git a/theories/Numbers/Natural/BigN/NMake.v b/theories/Numbers/Natural/BigN/NMake.v deleted file mode 100644 index 1425041a..00000000 --- a/theories/Numbers/Natural/BigN/NMake.v +++ /dev/null @@ -1,1706 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -(** * NMake *) - -(** From a cyclic Z/nZ representation to arbitrary precision natural numbers.*) - -(** NB: This file contain the part which is independent from the underlying - representation. The representation-dependent (and macro-generated) part - is now in [NMake_gen]. *) - -Require Import Bool BigNumPrelude ZArith Nnat Ndigits CyclicAxioms DoubleType - Nbasic Wf_nat StreamMemo NSig NMake_gen. - -Module Make (W0:CyclicType) <: NType. - - (** Let's include the macro-generated part. Even if we can't functorize - things (due to Eval red_t below), the rest of the module only uses - elements mentionned in interface [NAbstract]. *) - - Include NMake_gen.Make W0. - - Open Scope Z_scope. - - Local Notation "[ x ]" := (to_Z x). - - Definition eq (x y : t) := [x] = [y]. - - Declare Reduction red_t := - lazy beta iota delta - [iter_t reduce same_level mk_t mk_t_S succ_t dom_t dom_op]. - - Ltac red_t := - match goal with |- ?u => let v := (eval red_t in u) in change v end. - - (** * Generic results *) - - Tactic Notation "destr_t" constr(x) "as" simple_intropattern(pat) := - destruct (destr_t x) as pat; cbv zeta; - rewrite ?iter_mk_t, ?spec_mk_t, ?spec_reduce. - - Lemma spec_same_level : forall A (P:Z->Z->A->Prop) - (f : forall n, dom_t n -> dom_t n -> A), - (forall n x y, P (ZnZ.to_Z x) (ZnZ.to_Z y) (f n x y)) -> - forall x y, P [x] [y] (same_level f x y). - Proof. - intros. apply spec_same_level_dep with (P:=fun _ => P); auto. - Qed. - - Theorem spec_pos: forall x, 0 <= [x]. - Proof. - intros x. destr_t x as (n,x). now case (ZnZ.spec_to_Z x). - Qed. - - Lemma digits_dom_op_incr : forall n m, (n<=m)%nat -> - (ZnZ.digits (dom_op n) <= ZnZ.digits (dom_op m))%positive. - Proof. - intros. - change (Zpos (ZnZ.digits (dom_op n)) <= Zpos (ZnZ.digits (dom_op m))). - rewrite !digits_dom_op, !Pshiftl_nat_Zpower. - apply Z.mul_le_mono_nonneg_l; auto with zarith. - apply Z.pow_le_mono_r; auto with zarith. - Qed. - - Definition to_N (x : t) := Z.to_N (to_Z x). - - (** * Zero, One *) - - Definition zero := mk_t O ZnZ.zero. - Definition one := mk_t O ZnZ.one. - - Theorem spec_0: [zero] = 0. - Proof. - unfold zero. rewrite spec_mk_t. exact ZnZ.spec_0. - Qed. - - Theorem spec_1: [one] = 1. - Proof. - unfold one. rewrite spec_mk_t. exact ZnZ.spec_1. - Qed. - - (** * Successor *) - - (** NB: it is crucial here and for the rest of this file to preserve - the let-in's. They allow to pre-compute once and for all the - field access to Z/nZ initial structures (when n=0..6). *) - - Local Notation succn := (fun n => - let op := dom_op n in - let succ_c := ZnZ.succ_c in - let one := ZnZ.one in - fun x => match succ_c x with - | C0 r => mk_t n r - | C1 r => mk_t_S n (WW one r) - end). - - Definition succ : t -> t := Eval red_t in iter_t succn. - - Lemma succ_fold : succ = iter_t succn. - Proof. red_t; reflexivity. Qed. - - Theorem spec_succ: forall n, [succ n] = [n] + 1. - Proof. - intros x. rewrite succ_fold. destr_t x as (n,x). - generalize (ZnZ.spec_succ_c x); case ZnZ.succ_c. - intros. rewrite spec_mk_t. assumption. - intros. unfold interp_carry in *. - rewrite spec_mk_t_S. simpl. rewrite ZnZ.spec_1. assumption. - Qed. - - (** Two *) - - (** Not really pretty, but since W0 might be Z/2Z, we're not sure - there's a proper 2 there. *) - - Definition two := succ one. - - Lemma spec_2 : [two] = 2. - Proof. - unfold two. now rewrite spec_succ, spec_1. - Qed. - - (** * Addition *) - - Local Notation addn := (fun n => - let op := dom_op n in - let add_c := ZnZ.add_c in - let one := ZnZ.one in - fun x y =>match add_c x y with - | C0 r => mk_t n r - | C1 r => mk_t_S n (WW one r) - end). - - Definition add : t -> t -> t := Eval red_t in same_level addn. - - Lemma add_fold : add = same_level addn. - Proof. red_t; reflexivity. Qed. - - Theorem spec_add: forall x y, [add x y] = [x] + [y]. - Proof. - intros x y. rewrite add_fold. apply spec_same_level; clear x y. - intros n x y. cbv beta iota zeta. - generalize (ZnZ.spec_add_c x y); case ZnZ.add_c; intros z H. - rewrite spec_mk_t. assumption. - rewrite spec_mk_t_S. unfold interp_carry in H. - simpl. rewrite ZnZ.spec_1. assumption. - Qed. - - (** * Predecessor *) - - Local Notation predn := (fun n => - let pred_c := ZnZ.pred_c in - fun x => match pred_c x with - | C0 r => reduce n r - | C1 _ => zero - end). - - Definition pred : t -> t := Eval red_t in iter_t predn. - - Lemma pred_fold : pred = iter_t predn. - Proof. red_t; reflexivity. Qed. - - Theorem spec_pred_pos : forall x, 0 < [x] -> [pred x] = [x] - 1. - Proof. - intros x. rewrite pred_fold. destr_t x as (n,x). intros H. - generalize (ZnZ.spec_pred_c x); case ZnZ.pred_c; intros y H'. - rewrite spec_reduce. assumption. - exfalso. unfold interp_carry in *. - generalize (ZnZ.spec_to_Z x) (ZnZ.spec_to_Z y); auto with zarith. - Qed. - - Theorem spec_pred0 : forall x, [x] = 0 -> [pred x] = 0. - Proof. - intros x. rewrite pred_fold. destr_t x as (n,x). intros H. - generalize (ZnZ.spec_pred_c x); case ZnZ.pred_c; intros y H'. - rewrite spec_reduce. - unfold interp_carry in H'. - generalize (ZnZ.spec_to_Z y); auto with zarith. - exact spec_0. - Qed. - - Lemma spec_pred x : [pred x] = Z.max 0 ([x]-1). - Proof. - rewrite Z.max_comm. - destruct (Z.max_spec ([x]-1) 0) as [(H,->)|(H,->)]. - - apply spec_pred0; generalize (spec_pos x); auto with zarith. - - apply spec_pred_pos; auto with zarith. - Qed. - - (** * Subtraction *) - - Local Notation subn := (fun n => - let sub_c := ZnZ.sub_c in - fun x y => match sub_c x y with - | C0 r => reduce n r - | C1 r => zero - end). - - Definition sub : t -> t -> t := Eval red_t in same_level subn. - - Lemma sub_fold : sub = same_level subn. - Proof. red_t; reflexivity. Qed. - - Theorem spec_sub_pos : forall x y, [y] <= [x] -> [sub x y] = [x] - [y]. - Proof. - intros x y. rewrite sub_fold. apply spec_same_level. clear x y. - intros n x y. simpl. - generalize (ZnZ.spec_sub_c x y); case ZnZ.sub_c; intros z H LE. - rewrite spec_reduce. assumption. - unfold interp_carry in H. - exfalso. - generalize (ZnZ.spec_to_Z z); auto with zarith. - Qed. - - Theorem spec_sub0 : forall x y, [x] < [y] -> [sub x y] = 0. - Proof. - intros x y. rewrite sub_fold. apply spec_same_level. clear x y. - intros n x y. simpl. - generalize (ZnZ.spec_sub_c x y); case ZnZ.sub_c; intros z H LE. - rewrite spec_reduce. - unfold interp_carry in H. - generalize (ZnZ.spec_to_Z z); auto with zarith. - exact spec_0. - Qed. - - Lemma spec_sub : forall x y, [sub x y] = Z.max 0 ([x]-[y]). - Proof. - intros. destruct (Z.le_gt_cases [y] [x]). - rewrite Z.max_r; auto with zarith. apply spec_sub_pos; auto. - rewrite Z.max_l; auto with zarith. apply spec_sub0; auto. - Qed. - - (** * Comparison *) - - Definition comparen_m n : - forall m, word (dom_t n) (S m) -> dom_t n -> comparison := - let op := dom_op n in - let zero := ZnZ.zero (Ops:=op) in - let compare := ZnZ.compare (Ops:=op) in - let compare0 := compare zero in - fun m => compare_mn_1 (dom_t n) (dom_t n) zero compare compare0 compare (S m). - - Let spec_comparen_m: - forall n m (x : word (dom_t n) (S m)) (y : dom_t n), - comparen_m n m x y = Z.compare (eval n (S m) x) (ZnZ.to_Z y). - Proof. - intros n m x y. - unfold comparen_m, eval. - rewrite nmake_double. - apply spec_compare_mn_1. - exact ZnZ.spec_0. - intros. apply ZnZ.spec_compare. - exact ZnZ.spec_to_Z. - exact ZnZ.spec_compare. - exact ZnZ.spec_compare. - exact ZnZ.spec_to_Z. - Qed. - - Definition comparenm n m wx wy := - let mn := Max.max n m in - let d := diff n m in - let op := make_op mn in - ZnZ.compare - (castm (diff_r n m) (extend_tr wx (snd d))) - (castm (diff_l n m) (extend_tr wy (fst d))). - - Local Notation compare_folded := - (iter_sym _ - (fun n => ZnZ.compare (Ops:=dom_op n)) - comparen_m - comparenm - CompOpp). - - Definition compare : t -> t -> comparison := - Eval lazy beta iota delta [iter_sym dom_op dom_t comparen_m] in - compare_folded. - - Lemma compare_fold : compare = compare_folded. - Proof. - lazy beta iota delta [iter_sym dom_op dom_t comparen_m]. reflexivity. - Qed. - - Theorem spec_compare : forall x y, - compare x y = Z.compare [x] [y]. - Proof. - intros x y. rewrite compare_fold. apply spec_iter_sym; clear x y. - intros. apply ZnZ.spec_compare. - intros. cbv beta zeta. apply spec_comparen_m. - intros n m x y; unfold comparenm. - rewrite (spec_cast_l n m x), (spec_cast_r n m y). - unfold to_Z; apply ZnZ.spec_compare. - intros. subst. now rewrite <- Z.compare_antisym. - Qed. - - Definition eqb (x y : t) : bool := - match compare x y with - | Eq => true - | _ => false - end. - - Theorem spec_eqb x y : eqb x y = Z.eqb [x] [y]. - Proof. - apply eq_iff_eq_true. - unfold eqb. rewrite Z.eqb_eq, <- Z.compare_eq_iff, spec_compare. - split; [now destruct Z.compare | now intros ->]. - Qed. - - Definition lt (n m : t) := [n] < [m]. - Definition le (n m : t) := [n] <= [m]. - - Definition ltb (x y : t) : bool := - match compare x y with - | Lt => true - | _ => false - end. - - Theorem spec_ltb x y : ltb x y = Z.ltb [x] [y]. - Proof. - apply eq_iff_eq_true. - rewrite Z.ltb_lt. unfold Z.lt, ltb. rewrite spec_compare. - split; [now destruct Z.compare | now intros ->]. - Qed. - - Definition leb (x y : t) : bool := - match compare x y with - | Gt => false - | _ => true - end. - - Theorem spec_leb x y : leb x y = Z.leb [x] [y]. - Proof. - apply eq_iff_eq_true. - rewrite Z.leb_le. unfold Z.le, leb. rewrite spec_compare. - now destruct Z.compare; split. - Qed. - - Definition min (n m : t) : t := match compare n m with Gt => m | _ => n end. - Definition max (n m : t) : t := match compare n m with Lt => m | _ => n end. - - Theorem spec_max : forall n m, [max n m] = Z.max [n] [m]. - Proof. - intros. unfold max, Z.max. rewrite spec_compare; destruct Z.compare; reflexivity. - Qed. - - Theorem spec_min : forall n m, [min n m] = Z.min [n] [m]. - Proof. - intros. unfold min, Z.min. rewrite spec_compare; destruct Z.compare; reflexivity. - Qed. - - (** * Multiplication *) - - Definition wn_mul n : forall m, word (dom_t n) (S m) -> dom_t n -> t := - let op := dom_op n in - let zero := ZnZ.zero in - let succ := ZnZ.succ (Ops:=op) in - let add_c := ZnZ.add_c (Ops:=op) in - let mul_c := ZnZ.mul_c (Ops:=op) in - let ww := @ZnZ.WW _ op in - let ow := @ZnZ.OW _ op in - let eq0 := ZnZ.eq0 in - let mul_add := @DoubleMul.w_mul_add _ zero succ add_c mul_c in - let mul_add_n1 := @DoubleMul.double_mul_add_n1 _ zero ww ow mul_add in - fun m x y => - let (w,r) := mul_add_n1 (S m) x y zero in - if eq0 w then mk_t_w' n m r - else mk_t_w' n (S m) (WW (extend n m w) r). - - Definition mulnm n m x y := - let mn := Max.max n m in - let d := diff n m in - let op := make_op mn in - reduce_n (S mn) (ZnZ.mul_c - (castm (diff_r n m) (extend_tr x (snd d))) - (castm (diff_l n m) (extend_tr y (fst d)))). - - Local Notation mul_folded := - (iter_sym _ - (fun n => let mul_c := ZnZ.mul_c in - fun x y => reduce (S n) (succ_t _ (mul_c x y))) - wn_mul - mulnm - (fun x => x)). - - Definition mul : t -> t -> t := - Eval lazy beta iota delta - [iter_sym dom_op dom_t reduce succ_t extend zeron - wn_mul DoubleMul.w_mul_add mk_t_w'] in - mul_folded. - - Lemma mul_fold : mul = mul_folded. - Proof. - lazy beta iota delta - [iter_sym dom_op dom_t reduce succ_t extend zeron - wn_mul DoubleMul.w_mul_add mk_t_w']. reflexivity. - Qed. - - Lemma spec_muln: - forall n (x: word _ (S n)) y, - [Nn (S n) (ZnZ.mul_c (Ops:=make_op n) x y)] = [Nn n x] * [Nn n y]. - Proof. - intros n x y; unfold to_Z. - rewrite <- ZnZ.spec_mul_c. - rewrite make_op_S. - case ZnZ.mul_c; auto. - Qed. - - Lemma spec_mul_add_n1: forall n m x y z, - let (q,r) := DoubleMul.double_mul_add_n1 ZnZ.zero ZnZ.WW ZnZ.OW - (DoubleMul.w_mul_add ZnZ.zero ZnZ.succ ZnZ.add_c ZnZ.mul_c) - (S m) x y z in - ZnZ.to_Z q * (base (ZnZ.digits (nmake_op _ (dom_op n) (S m)))) - + eval n (S m) r = - eval n (S m) x * ZnZ.to_Z y + ZnZ.to_Z z. - Proof. - intros n m x y z. - rewrite digits_nmake. - unfold eval. rewrite nmake_double. - apply DoubleMul.spec_double_mul_add_n1. - apply ZnZ.spec_0. - exact ZnZ.spec_WW. - exact ZnZ.spec_OW. - apply DoubleCyclic.spec_mul_add. - Qed. - - Lemma spec_wn_mul : forall n m x y, - [wn_mul n m x y] = (eval n (S m) x) * ZnZ.to_Z y. - Proof. - intros; unfold wn_mul. - generalize (spec_mul_add_n1 n m x y ZnZ.zero). - case DoubleMul.double_mul_add_n1; intros q r Hqr. - rewrite ZnZ.spec_0, Z.add_0_r in Hqr. rewrite <- Hqr. - generalize (ZnZ.spec_eq0 q); case ZnZ.eq0; intros HH. - rewrite HH; auto. simpl. apply spec_mk_t_w'. - clear. - rewrite spec_mk_t_w'. - set (m' := S m) in *. - unfold eval. - rewrite nmake_WW. f_equal. f_equal. - rewrite <- spec_mk_t. - symmetry. apply spec_extend. - Qed. - - Theorem spec_mul : forall x y, [mul x y] = [x] * [y]. - Proof. - intros x y. rewrite mul_fold. apply spec_iter_sym; clear x y. - intros n x y. cbv zeta beta. - rewrite spec_reduce, spec_succ_t, <- ZnZ.spec_mul_c; auto. - apply spec_wn_mul. - intros n m x y; unfold mulnm. rewrite spec_reduce_n. - rewrite (spec_cast_l n m x), (spec_cast_r n m y). - apply spec_muln. - intros. rewrite Z.mul_comm; auto. - Qed. - - (** * Division by a smaller number *) - - Definition wn_divn1 n := - let op := dom_op n in - let zd := ZnZ.zdigits op in - let zero := ZnZ.zero in - let ww := ZnZ.WW in - let head0 := ZnZ.head0 in - let add_mul_div := ZnZ.add_mul_div in - let div21 := ZnZ.div21 in - let compare := ZnZ.compare in - let sub := ZnZ.sub in - let ddivn1 := - DoubleDivn1.double_divn1 zd zero ww head0 add_mul_div div21 compare sub in - fun m x y => let (u,v) := ddivn1 (S m) x y in (mk_t_w' n m u, mk_t n v). - - Definition div_gtnm n m wx wy := - let mn := Max.max n m in - let d := diff n m in - let op := make_op mn in - let (q, r):= ZnZ.div_gt - (castm (diff_r n m) (extend_tr wx (snd d))) - (castm (diff_l n m) (extend_tr wy (fst d))) in - (reduce_n mn q, reduce_n mn r). - - Local Notation div_gt_folded := - (iter _ - (fun n => let div_gt := ZnZ.div_gt in - fun x y => let (u,v) := div_gt x y in (reduce n u, reduce n v)) - (fun n => - let div_gt := ZnZ.div_gt in - fun m x y => - let y' := DoubleBase.get_low (zeron n) (S m) y in - let (u,v) := div_gt x y' in (reduce n u, reduce n v)) - wn_divn1 - div_gtnm). - - Definition div_gt := - Eval lazy beta iota delta - [iter dom_op dom_t reduce zeron wn_divn1 mk_t_w' mk_t] in - div_gt_folded. - - Lemma div_gt_fold : div_gt = div_gt_folded. - Proof. - lazy beta iota delta [iter dom_op dom_t reduce zeron wn_divn1 mk_t_w' mk_t]. - reflexivity. - Qed. - - Lemma spec_get_endn: forall n m x y, - eval n m x <= [mk_t n y] -> - [mk_t n (DoubleBase.get_low (zeron n) m x)] = eval n m x. - Proof. - intros n m x y H. - unfold eval. rewrite nmake_double. - rewrite spec_mk_t in *. - apply DoubleBase.spec_get_low. - apply spec_zeron. - exact ZnZ.spec_to_Z. - apply Z.le_lt_trans with (ZnZ.to_Z y); auto. - rewrite <- nmake_double; auto. - case (ZnZ.spec_to_Z y); auto. - Qed. - - Definition spec_divn1 n := - DoubleDivn1.spec_double_divn1 - (ZnZ.zdigits (dom_op n)) (ZnZ.zero:dom_t n) - ZnZ.WW ZnZ.head0 - ZnZ.add_mul_div ZnZ.div21 - ZnZ.compare ZnZ.sub ZnZ.to_Z - ZnZ.spec_to_Z - ZnZ.spec_zdigits - ZnZ.spec_0 ZnZ.spec_WW ZnZ.spec_head0 - ZnZ.spec_add_mul_div ZnZ.spec_div21 - ZnZ.spec_compare ZnZ.spec_sub. - - Lemma spec_div_gt_aux : forall x y, [x] > [y] -> 0 < [y] -> - let (q,r) := div_gt x y in - [x] = [q] * [y] + [r] /\ 0 <= [r] < [y]. - Proof. - intros x y. rewrite div_gt_fold. apply spec_iter; clear x y. - intros n x y H1 H2. simpl. - generalize (ZnZ.spec_div_gt x y H1 H2); case ZnZ.div_gt. - intros u v. rewrite 2 spec_reduce. auto. - intros n m x y H1 H2. cbv zeta beta. - generalize (ZnZ.spec_div_gt x - (DoubleBase.get_low (zeron n) (S m) y)). - case ZnZ.div_gt. - intros u v H3; repeat rewrite spec_reduce. - generalize (spec_get_endn n (S m) y x). rewrite !spec_mk_t. intros H4. - rewrite H4 in H3; auto with zarith. - intros n m x y H1 H2. - generalize (spec_divn1 n (S m) x y H2). - unfold wn_divn1; case DoubleDivn1.double_divn1. - intros u v H3. - rewrite spec_mk_t_w', spec_mk_t. - rewrite <- !nmake_double in H3; auto. - intros n m x y H1 H2; unfold div_gtnm. - generalize (ZnZ.spec_div_gt - (castm (diff_r n m) - (extend_tr x (snd (diff n m)))) - (castm (diff_l n m) - (extend_tr y (fst (diff n m))))). - case ZnZ.div_gt. - intros xx yy HH. - repeat rewrite spec_reduce_n. - rewrite (spec_cast_l n m x), (spec_cast_r n m y). - unfold to_Z; apply HH. - rewrite (spec_cast_l n m x) in H1; auto. - rewrite (spec_cast_r n m y) in H1; auto. - rewrite (spec_cast_r n m y) in H2; auto. - Qed. - - Theorem spec_div_gt: forall x y, [x] > [y] -> 0 < [y] -> - let (q,r) := div_gt x y in - [q] = [x] / [y] /\ [r] = [x] mod [y]. - Proof. - intros x y H1 H2; generalize (spec_div_gt_aux x y H1 H2); case div_gt. - intros q r (H3, H4); split. - apply (Zdiv_unique [x] [y] [q] [r]); auto. - rewrite Z.mul_comm; auto. - apply (Zmod_unique [x] [y] [q] [r]); auto. - rewrite Z.mul_comm; auto. - Qed. - - (** * General Division *) - - Definition div_eucl (x y : t) : t * t := - if eqb y zero then (zero,zero) else - match compare x y with - | Eq => (one, zero) - | Lt => (zero, x) - | Gt => div_gt x y - end. - - Theorem spec_div_eucl: forall x y, - let (q,r) := div_eucl x y in - ([q], [r]) = Z.div_eucl [x] [y]. - Proof. - intros x y. unfold div_eucl. - rewrite spec_eqb, spec_compare, spec_0. - case Z.eqb_spec. - intros ->. rewrite spec_0. destruct [x]; auto. - intros H'. - assert (H : 0 < [y]) by (generalize (spec_pos y); auto with zarith). - clear H'. - case Z.compare_spec; intros Cmp; - rewrite ?spec_0, ?spec_1; intros; auto with zarith. - rewrite Cmp; generalize (Z_div_same [y] (Z.lt_gt _ _ H)) - (Z_mod_same [y] (Z.lt_gt _ _ H)); - unfold Z.div, Z.modulo; case Z.div_eucl; intros; subst; auto. - assert (LeLt: 0 <= [x] < [y]) by (generalize (spec_pos x); auto). - generalize (Zdiv_small _ _ LeLt) (Zmod_small _ _ LeLt); - unfold Z.div, Z.modulo; case Z.div_eucl; intros; subst; auto. - generalize (spec_div_gt _ _ (Z.lt_gt _ _ Cmp) H); auto. - unfold Z.div, Z.modulo; case Z.div_eucl; case div_gt. - intros a b c d (H1, H2); subst; auto. - Qed. - - Definition div (x y : t) : t := fst (div_eucl x y). - - Theorem spec_div: - forall x y, [div x y] = [x] / [y]. - Proof. - intros x y; unfold div; generalize (spec_div_eucl x y); - case div_eucl; simpl fst. - intros xx yy; unfold Z.div; case Z.div_eucl; intros qq rr H; - injection H; auto. - Qed. - - (** * Modulo by a smaller number *) - - Definition wn_modn1 n := - let op := dom_op n in - let zd := ZnZ.zdigits op in - let zero := ZnZ.zero in - let head0 := ZnZ.head0 in - let add_mul_div := ZnZ.add_mul_div in - let div21 := ZnZ.div21 in - let compare := ZnZ.compare in - let sub := ZnZ.sub in - let dmodn1 := - DoubleDivn1.double_modn1 zd zero head0 add_mul_div div21 compare sub in - fun m x y => reduce n (dmodn1 (S m) x y). - - Definition mod_gtnm n m wx wy := - let mn := Max.max n m in - let d := diff n m in - let op := make_op mn in - reduce_n mn (ZnZ.modulo_gt - (castm (diff_r n m) (extend_tr wx (snd d))) - (castm (diff_l n m) (extend_tr wy (fst d)))). - - Local Notation mod_gt_folded := - (iter _ - (fun n => let modulo_gt := ZnZ.modulo_gt in - fun x y => reduce n (modulo_gt x y)) - (fun n => let modulo_gt := ZnZ.modulo_gt in - fun m x y => - reduce n (modulo_gt x (DoubleBase.get_low (zeron n) (S m) y))) - wn_modn1 - mod_gtnm). - - Definition mod_gt := - Eval lazy beta iota delta [iter dom_op dom_t reduce wn_modn1 zeron] in - mod_gt_folded. - - Lemma mod_gt_fold : mod_gt = mod_gt_folded. - Proof. - lazy beta iota delta [iter dom_op dom_t reduce wn_modn1 zeron]. - reflexivity. - Qed. - - Definition spec_modn1 n := - DoubleDivn1.spec_double_modn1 - (ZnZ.zdigits (dom_op n)) (ZnZ.zero:dom_t n) - ZnZ.WW ZnZ.head0 - ZnZ.add_mul_div ZnZ.div21 - ZnZ.compare ZnZ.sub ZnZ.to_Z - ZnZ.spec_to_Z - ZnZ.spec_zdigits - ZnZ.spec_0 ZnZ.spec_WW ZnZ.spec_head0 - ZnZ.spec_add_mul_div ZnZ.spec_div21 - ZnZ.spec_compare ZnZ.spec_sub. - - Theorem spec_mod_gt: - forall x y, [x] > [y] -> 0 < [y] -> [mod_gt x y] = [x] mod [y]. - Proof. - intros x y. rewrite mod_gt_fold. apply spec_iter; clear x y. - intros n x y H1 H2. simpl. rewrite spec_reduce. - exact (ZnZ.spec_modulo_gt x y H1 H2). - intros n m x y H1 H2. cbv zeta beta. rewrite spec_reduce. - rewrite <- spec_mk_t in H1. - rewrite <- (spec_get_endn n (S m) y x); auto with zarith. - rewrite spec_mk_t. - apply ZnZ.spec_modulo_gt; auto. - rewrite <- (spec_get_endn n (S m) y x), !spec_mk_t in H1; auto with zarith. - rewrite <- (spec_get_endn n (S m) y x), !spec_mk_t in H2; auto with zarith. - intros n m x y H1 H2. unfold wn_modn1. rewrite spec_reduce. - unfold eval; rewrite nmake_double. - apply (spec_modn1 n); auto. - intros n m x y H1 H2; unfold mod_gtnm. - repeat rewrite spec_reduce_n. - rewrite (spec_cast_l n m x), (spec_cast_r n m y). - unfold to_Z; apply ZnZ.spec_modulo_gt. - rewrite (spec_cast_l n m x) in H1; auto. - rewrite (spec_cast_r n m y) in H1; auto. - rewrite (spec_cast_r n m y) in H2; auto. - Qed. - - (** * General Modulo *) - - Definition modulo (x y : t) : t := - if eqb y zero then zero else - match compare x y with - | Eq => zero - | Lt => x - | Gt => mod_gt x y - end. - - Theorem spec_modulo: - forall x y, [modulo x y] = [x] mod [y]. - Proof. - intros x y. unfold modulo. - rewrite spec_eqb, spec_compare, spec_0. - case Z.eqb_spec. - intros ->; rewrite spec_0. destruct [x]; auto. - intro H'. - assert (H : 0 < [y]) by (generalize (spec_pos y); auto with zarith). - clear H'. - case Z.compare_spec; - rewrite ?spec_0, ?spec_1; intros; try split; auto with zarith. - rewrite H0; symmetry; apply Z_mod_same; auto with zarith. - symmetry; apply Zmod_small; auto with zarith. - generalize (spec_pos x); auto with zarith. - apply spec_mod_gt; auto with zarith. - Qed. - - (** * Square *) - - Local Notation squaren := (fun n => - let square_c := ZnZ.square_c in - fun x => reduce (S n) (succ_t _ (square_c x))). - - Definition square : t -> t := Eval red_t in iter_t squaren. - - Lemma square_fold : square = iter_t squaren. - Proof. red_t; reflexivity. Qed. - - Theorem spec_square: forall x, [square x] = [x] * [x]. - Proof. - intros x. rewrite square_fold. destr_t x as (n,x). - rewrite spec_succ_t. exact (ZnZ.spec_square_c x). - Qed. - - (** * Square Root *) - - Local Notation sqrtn := (fun n => - let sqrt := ZnZ.sqrt in - fun x => reduce n (sqrt x)). - - Definition sqrt : t -> t := Eval red_t in iter_t sqrtn. - - Lemma sqrt_fold : sqrt = iter_t sqrtn. - Proof. red_t; reflexivity. Qed. - - Theorem spec_sqrt_aux: forall x, [sqrt x] ^ 2 <= [x] < ([sqrt x] + 1) ^ 2. - Proof. - intros x. rewrite sqrt_fold. destr_t x as (n,x). exact (ZnZ.spec_sqrt x). - Qed. - - Theorem spec_sqrt: forall x, [sqrt x] = Z.sqrt [x]. - Proof. - intros x. - symmetry. apply Z.sqrt_unique. - rewrite <- ! Z.pow_2_r. apply spec_sqrt_aux. - Qed. - - (** * Power *) - - Fixpoint pow_pos (x:t)(p:positive) : t := - match p with - | xH => x - | xO p => square (pow_pos x p) - | xI p => mul (square (pow_pos x p)) x - end. - - Theorem spec_pow_pos: forall x n, [pow_pos x n] = [x] ^ Zpos n. - Proof. - intros x n; generalize x; elim n; clear n x; simpl pow_pos. - intros; rewrite spec_mul; rewrite spec_square; rewrite H. - rewrite Pos2Z.inj_xI; rewrite Zpower_exp; auto with zarith. - rewrite (Z.mul_comm 2); rewrite Z.pow_mul_r; auto with zarith. - rewrite Z.pow_2_r; rewrite Z.pow_1_r; auto. - intros; rewrite spec_square; rewrite H. - rewrite Pos2Z.inj_xO; auto with zarith. - rewrite (Z.mul_comm 2); rewrite Z.pow_mul_r; auto with zarith. - rewrite Z.pow_2_r; auto. - intros; rewrite Z.pow_1_r; auto. - Qed. - - Definition pow_N (x:t)(n:N) : t := match n with - | BinNat.N0 => one - | BinNat.Npos p => pow_pos x p - end. - - Theorem spec_pow_N: forall x n, [pow_N x n] = [x] ^ Z.of_N n. - Proof. - destruct n; simpl. apply spec_1. - apply spec_pow_pos. - Qed. - - Definition pow (x y:t) : t := pow_N x (to_N y). - - Theorem spec_pow : forall x y, [pow x y] = [x] ^ [y]. - Proof. - intros. unfold pow, to_N. - now rewrite spec_pow_N, Z2N.id by apply spec_pos. - Qed. - - - (** * digits - - Number of digits in the representation of a numbers - (including head zero's). - NB: This function isn't a morphism for setoid [eq]. - *) - - Local Notation digitsn := (fun n => - let digits := ZnZ.digits (dom_op n) in - fun _ => digits). - - Definition digits : t -> positive := Eval red_t in iter_t digitsn. - - Lemma digits_fold : digits = iter_t digitsn. - Proof. red_t; reflexivity. Qed. - - Theorem spec_digits: forall x, 0 <= [x] < 2 ^ Zpos (digits x). - Proof. - intros x. rewrite digits_fold. destr_t x as (n,x). exact (ZnZ.spec_to_Z x). - Qed. - - Lemma digits_level : forall x, digits x = ZnZ.digits (dom_op (level x)). - Proof. - intros x. rewrite digits_fold. unfold level. destr_t x as (n,x). reflexivity. - Qed. - - (** * Gcd *) - - Definition gcd_gt_body a b cont := - match compare b zero with - | Gt => - let r := mod_gt a b in - match compare r zero with - | Gt => cont r (mod_gt b r) - | _ => b - end - | _ => a - end. - - Theorem Zspec_gcd_gt_body: forall a b cont p, - [a] > [b] -> [a] < 2 ^ p -> - (forall a1 b1, [a1] < 2 ^ (p - 1) -> [a1] > [b1] -> - Zis_gcd [a1] [b1] [cont a1 b1]) -> - Zis_gcd [a] [b] [gcd_gt_body a b cont]. - Proof. - intros a b cont p H2 H3 H4; unfold gcd_gt_body. - rewrite ! spec_compare, spec_0. case Z.compare_spec. - intros ->; apply Zis_gcd_0. - intros HH; absurd (0 <= [b]); auto with zarith. - case (spec_digits b); auto with zarith. - intros H5; case Z.compare_spec. - intros H6; rewrite <- (Z.mul_1_r [b]). - rewrite (Z_div_mod_eq [a] [b]); auto with zarith. - rewrite <- spec_mod_gt; auto with zarith. - rewrite H6; rewrite Z.add_0_r. - apply Zis_gcd_mult; apply Zis_gcd_1. - intros; apply False_ind. - case (spec_digits (mod_gt a b)); auto with zarith. - intros H6; apply DoubleDiv.Zis_gcd_mod; auto with zarith. - apply DoubleDiv.Zis_gcd_mod; auto with zarith. - rewrite <- spec_mod_gt; auto with zarith. - assert (F2: [b] > [mod_gt a b]). - case (Z_mod_lt [a] [b]); auto with zarith. - repeat rewrite <- spec_mod_gt; auto with zarith. - assert (F3: [mod_gt a b] > [mod_gt b (mod_gt a b)]). - case (Z_mod_lt [b] [mod_gt a b]); auto with zarith. - rewrite <- spec_mod_gt; auto with zarith. - repeat rewrite <- spec_mod_gt; auto with zarith. - apply H4; auto with zarith. - apply Z.mul_lt_mono_pos_r with 2; auto with zarith. - apply Z.le_lt_trans with ([b] + [mod_gt a b]); auto with zarith. - apply Z.le_lt_trans with (([a]/[b]) * [b] + [mod_gt a b]); auto with zarith. - - apply Z.add_le_mono_r. - rewrite <- (Z.mul_1_l [b]) at 1. - apply Z.mul_le_mono_nonneg_r; auto with zarith. - change 1 with (Z.succ 0). apply Z.le_succ_l. - apply Z.div_str_pos; auto with zarith. - - rewrite Z.mul_comm; rewrite spec_mod_gt; auto with zarith. - rewrite <- Z_div_mod_eq; auto with zarith. - rewrite Z.mul_comm, <- Z.pow_succ_r, Z.sub_1_r, Z.succ_pred; auto. - apply Z.le_0_sub. change 1 with (Z.succ 0). apply Z.le_succ_l. - destruct p; simpl in H3; auto with zarith. - Qed. - - Fixpoint gcd_gt_aux (p:positive) (cont:t->t->t) (a b:t) : t := - gcd_gt_body a b - (fun a b => - match p with - | xH => cont a b - | xO p => gcd_gt_aux p (gcd_gt_aux p cont) a b - | xI p => gcd_gt_aux p (gcd_gt_aux p cont) a b - end). - - Theorem Zspec_gcd_gt_aux: forall p n a b cont, - [a] > [b] -> [a] < 2 ^ (Zpos p + n) -> - (forall a1 b1, [a1] < 2 ^ n -> [a1] > [b1] -> - Zis_gcd [a1] [b1] [cont a1 b1]) -> - Zis_gcd [a] [b] [gcd_gt_aux p cont a b]. - intros p; elim p; clear p. - intros p Hrec n a b cont H2 H3 H4. - unfold gcd_gt_aux; apply Zspec_gcd_gt_body with (Zpos (xI p) + n); auto. - intros a1 b1 H6 H7. - apply Hrec with (Zpos p + n); auto. - replace (Zpos p + (Zpos p + n)) with - (Zpos (xI p) + n - 1); auto. - rewrite Pos2Z.inj_xI; ring. - intros a2 b2 H9 H10. - apply Hrec with n; auto. - intros p Hrec n a b cont H2 H3 H4. - unfold gcd_gt_aux; apply Zspec_gcd_gt_body with (Zpos (xO p) + n); auto. - intros a1 b1 H6 H7. - apply Hrec with (Zpos p + n - 1); auto. - replace (Zpos p + (Zpos p + n - 1)) with - (Zpos (xO p) + n - 1); auto. - rewrite Pos2Z.inj_xO; ring. - intros a2 b2 H9 H10. - apply Hrec with (n - 1); auto. - replace (Zpos p + (n - 1)) with - (Zpos p + n - 1); auto with zarith. - intros a3 b3 H12 H13; apply H4; auto with zarith. - apply Z.lt_le_trans with (1 := H12). - apply Z.pow_le_mono_r; auto with zarith. - intros n a b cont H H2 H3. - simpl gcd_gt_aux. - apply Zspec_gcd_gt_body with (n + 1); auto with zarith. - rewrite Z.add_comm; auto. - intros a1 b1 H5 H6; apply H3; auto. - replace n with (n + 1 - 1); auto; try ring. - Qed. - - Definition gcd_cont a b := - match compare one b with - | Eq => one - | _ => a - end. - - Definition gcd_gt a b := gcd_gt_aux (digits a) gcd_cont a b. - - Theorem spec_gcd_gt: forall a b, - [a] > [b] -> [gcd_gt a b] = Z.gcd [a] [b]. - Proof. - intros a b H2. - case (spec_digits (gcd_gt a b)); intros H3 H4. - case (spec_digits a); intros H5 H6. - symmetry; apply Zis_gcd_gcd; auto with zarith. - unfold gcd_gt; apply Zspec_gcd_gt_aux with 0; auto with zarith. - intros a1 a2; rewrite Z.pow_0_r. - case (spec_digits a2); intros H7 H8; - intros; apply False_ind; auto with zarith. - Qed. - - Definition gcd (a b : t) : t := - match compare a b with - | Eq => a - | Lt => gcd_gt b a - | Gt => gcd_gt a b - end. - - Theorem spec_gcd: forall a b, [gcd a b] = Z.gcd [a] [b]. - Proof. - intros a b. - case (spec_digits a); intros H1 H2. - case (spec_digits b); intros H3 H4. - unfold gcd. rewrite spec_compare. case Z.compare_spec. - intros HH; rewrite HH; symmetry; apply Zis_gcd_gcd; auto. - apply Zis_gcd_refl. - intros; transitivity (Z.gcd [b] [a]). - apply spec_gcd_gt; auto with zarith. - apply Zis_gcd_gcd; auto with zarith. - apply Z.gcd_nonneg. - apply Zis_gcd_sym; apply Zgcd_is_gcd. - intros; apply spec_gcd_gt; auto with zarith. - Qed. - - (** * Parity test *) - - Definition even : t -> bool := Eval red_t in - iter_t (fun n x => ZnZ.is_even x). - - Definition odd x := negb (even x). - - Lemma even_fold : even = iter_t (fun n x => ZnZ.is_even x). - Proof. red_t; reflexivity. Qed. - - Theorem spec_even_aux: forall x, - if even x then [x] mod 2 = 0 else [x] mod 2 = 1. - Proof. - intros x. rewrite even_fold. destr_t x as (n,x). - exact (ZnZ.spec_is_even x). - Qed. - - Theorem spec_even: forall x, even x = Z.even [x]. - Proof. - intros x. assert (H := spec_even_aux x). symmetry. - rewrite (Z.div_mod [x] 2); auto with zarith. - destruct (even x); rewrite H, ?Z.add_0_r. - rewrite Zeven_bool_iff. apply Zeven_2p. - apply not_true_is_false. rewrite Zeven_bool_iff. - apply Zodd_not_Zeven. apply Zodd_2p_plus_1. - Qed. - - Theorem spec_odd: forall x, odd x = Z.odd [x]. - Proof. - intros x. unfold odd. - assert (H := spec_even_aux x). symmetry. - rewrite (Z.div_mod [x] 2); auto with zarith. - destruct (even x); rewrite H, ?Z.add_0_r; simpl negb. - apply not_true_is_false. rewrite Zodd_bool_iff. - apply Zeven_not_Zodd. apply Zeven_2p. - apply Zodd_bool_iff. apply Zodd_2p_plus_1. - Qed. - - (** * Conversion *) - - Definition pheight p := - Peano.pred (Pos.to_nat (get_height (ZnZ.digits (dom_op 0)) (plength p))). - - Theorem pheight_correct: forall p, - Zpos p < 2 ^ (Zpos (ZnZ.digits (dom_op 0)) * 2 ^ (Z.of_nat (pheight p))). - Proof. - intros p; unfold pheight. - rewrite Nat2Z.inj_pred by apply Pos2Nat.is_pos. - rewrite positive_nat_Z. - rewrite <- Z.sub_1_r. - assert (F2:= (get_height_correct (ZnZ.digits (dom_op 0)) (plength p))). - apply Z.lt_le_trans with (Zpos (Pos.succ p)). - rewrite Pos2Z.inj_succ; auto with zarith. - apply Z.le_trans with (1 := plength_pred_correct (Pos.succ p)). - rewrite Pos.pred_succ. - apply Z.pow_le_mono_r; auto with zarith. - Qed. - - Definition of_pos (x:positive) : t := - let n := pheight x in - reduce n (snd (ZnZ.of_pos x)). - - Theorem spec_of_pos: forall x, - [of_pos x] = Zpos x. - Proof. - intros x; unfold of_pos. - rewrite spec_reduce. - simpl. - apply ZnZ.of_pos_correct. - unfold base. - apply Z.lt_le_trans with (1 := pheight_correct x). - apply Z.pow_le_mono_r; auto with zarith. - rewrite (digits_dom_op (_ _)), Pshiftl_nat_Zpower. auto with zarith. - Qed. - - Definition of_N (x:N) : t := - match x with - | BinNat.N0 => zero - | Npos p => of_pos p - end. - - Theorem spec_of_N: forall x, - [of_N x] = Z.of_N x. - Proof. - intros x; case x. - simpl of_N. exact spec_0. - intros p; exact (spec_of_pos p). - Qed. - - (** * [head0] and [tail0] - - Number of zero at the beginning and at the end of - the representation of the number. - NB: these functions are not morphism for setoid [eq]. - *) - - Local Notation head0n := (fun n => - let head0 := ZnZ.head0 in - fun x => reduce n (head0 x)). - - Definition head0 : t -> t := Eval red_t in iter_t head0n. - - Lemma head0_fold : head0 = iter_t head0n. - Proof. red_t; reflexivity. Qed. - - Theorem spec_head00: forall x, [x] = 0 -> [head0 x] = Zpos (digits x). - Proof. - intros x. rewrite head0_fold, digits_fold. destr_t x as (n,x). - exact (ZnZ.spec_head00 x). - Qed. - - Lemma pow2_pos_minus_1 : forall z, 0<z -> 2^(z-1) = 2^z / 2. - Proof. - intros. apply Zdiv_unique with 0; auto with zarith. - change 2 with (2^1) at 2. - rewrite <- Zpower_exp; auto with zarith. - rewrite Z.add_0_r. f_equal. auto with zarith. - Qed. - - Theorem spec_head0: forall x, 0 < [x] -> - 2 ^ (Zpos (digits x) - 1) <= 2 ^ [head0 x] * [x] < 2 ^ Zpos (digits x). - Proof. - intros x. rewrite pow2_pos_minus_1 by (red; auto). - rewrite head0_fold, digits_fold. destr_t x as (n,x). exact (ZnZ.spec_head0 x). - Qed. - - Local Notation tail0n := (fun n => - let tail0 := ZnZ.tail0 in - fun x => reduce n (tail0 x)). - - Definition tail0 : t -> t := Eval red_t in iter_t tail0n. - - Lemma tail0_fold : tail0 = iter_t tail0n. - Proof. red_t; reflexivity. Qed. - - Theorem spec_tail00: forall x, [x] = 0 -> [tail0 x] = Zpos (digits x). - Proof. - intros x. rewrite tail0_fold, digits_fold. destr_t x as (n,x). - exact (ZnZ.spec_tail00 x). - Qed. - - Theorem spec_tail0: forall x, - 0 < [x] -> exists y, 0 <= y /\ [x] = (2 * y + 1) * 2 ^ [tail0 x]. - Proof. - intros x. rewrite tail0_fold. destr_t x as (n,x). exact (ZnZ.spec_tail0 x). - Qed. - - (** * [Ndigits] - - Same as [digits] but encoded using large integers - NB: this function is not a morphism for setoid [eq]. - *) - - Local Notation Ndigitsn := (fun n => - let d := reduce n (ZnZ.zdigits (dom_op n)) in - fun _ => d). - - Definition Ndigits : t -> t := Eval red_t in iter_t Ndigitsn. - - Lemma Ndigits_fold : Ndigits = iter_t Ndigitsn. - Proof. red_t; reflexivity. Qed. - - Theorem spec_Ndigits: forall x, [Ndigits x] = Zpos (digits x). - Proof. - intros x. rewrite Ndigits_fold, digits_fold. destr_t x as (n,x). - apply ZnZ.spec_zdigits. - Qed. - - (** * Binary logarithm *) - - Local Notation log2n := (fun n => - let op := dom_op n in - let zdigits := ZnZ.zdigits op in - let head0 := ZnZ.head0 in - let sub_carry := ZnZ.sub_carry in - fun x => reduce n (sub_carry zdigits (head0 x))). - - Definition log2 : t -> t := Eval red_t in - let log2 := iter_t log2n in - fun x => if eqb x zero then zero else log2 x. - - Lemma log2_fold : - log2 = fun x => if eqb x zero then zero else iter_t log2n x. - Proof. red_t; reflexivity. Qed. - - Lemma spec_log2_0 : forall x, [x] = 0 -> [log2 x] = 0. - Proof. - intros x H. rewrite log2_fold. - rewrite spec_eqb, H. rewrite spec_0. simpl. exact spec_0. - Qed. - - Lemma head0_zdigits : forall n (x : dom_t n), - 0 < ZnZ.to_Z x -> - ZnZ.to_Z (ZnZ.head0 x) < ZnZ.to_Z (ZnZ.zdigits (dom_op n)). - Proof. - intros n x H. - destruct (ZnZ.spec_head0 x H) as (_,H0). - intros. - assert (H1 := ZnZ.spec_to_Z (ZnZ.head0 x)). - assert (H2 := ZnZ.spec_to_Z (ZnZ.zdigits (dom_op n))). - unfold base in *. - rewrite ZnZ.spec_zdigits in H2 |- *. - set (h := ZnZ.to_Z (ZnZ.head0 x)) in *; clearbody h. - set (d := ZnZ.digits (dom_op n)) in *; clearbody d. - destruct (Z_lt_le_dec h (Zpos d)); auto. exfalso. - assert (1 * 2^Zpos d <= ZnZ.to_Z x * 2^h). - apply Z.mul_le_mono_nonneg; auto with zarith. - apply Z.pow_le_mono_r; auto with zarith. - rewrite Z.mul_comm in H0. auto with zarith. - Qed. - - Lemma spec_log2_pos : forall x, [x]<>0 -> - 2^[log2 x] <= [x] < 2^([log2 x]+1). - Proof. - intros x H. rewrite log2_fold. - rewrite spec_eqb. rewrite spec_0. - case Z.eqb_spec. - auto with zarith. - clear H. - destr_t x as (n,x). intros H. - rewrite ZnZ.spec_sub_carry. - assert (H0 := ZnZ.spec_to_Z x). - assert (H1 := ZnZ.spec_to_Z (ZnZ.head0 x)). - assert (H2 := ZnZ.spec_to_Z (ZnZ.zdigits (dom_op n))). - assert (H3 := head0_zdigits n x). - rewrite Zmod_small by auto with zarith. - rewrite Z.sub_simpl_r. - rewrite (Z.mul_lt_mono_pos_l (2^(ZnZ.to_Z (ZnZ.head0 x)))); - auto with zarith. - rewrite (Z.mul_le_mono_pos_l _ _ (2^(ZnZ.to_Z (ZnZ.head0 x)))); - auto with zarith. - rewrite <- 2 Zpower_exp; auto with zarith. - rewrite !Z.add_sub_assoc, !Z.add_simpl_l. - rewrite ZnZ.spec_zdigits. - rewrite pow2_pos_minus_1 by (red; auto). - apply ZnZ.spec_head0; auto with zarith. - Qed. - - Lemma spec_log2 : forall x, [log2 x] = Z.log2 [x]. - Proof. - intros. destruct (Z_lt_ge_dec 0 [x]). - symmetry. apply Z.log2_unique. apply spec_pos. - apply spec_log2_pos. intro EQ; rewrite EQ in *; auto with zarith. - rewrite spec_log2_0. rewrite Z.log2_nonpos; auto with zarith. - generalize (spec_pos x); auto with zarith. - Qed. - - Lemma log2_digits_head0 : forall x, 0 < [x] -> - [log2 x] = Zpos (digits x) - [head0 x] - 1. - Proof. - intros. rewrite log2_fold. - rewrite spec_eqb. rewrite spec_0. - case Z.eqb_spec. - auto with zarith. - intros _. revert H. rewrite digits_fold, head0_fold. destr_t x as (n,x). - rewrite ZnZ.spec_sub_carry. - intros. - generalize (head0_zdigits n x H). - generalize (ZnZ.spec_to_Z (ZnZ.head0 x)). - generalize (ZnZ.spec_to_Z (ZnZ.zdigits (dom_op n))). - rewrite ZnZ.spec_zdigits. intros. apply Zmod_small. - auto with zarith. - Qed. - - (** * Right shift *) - - Local Notation shiftrn := (fun n => - let op := dom_op n in - let zdigits := ZnZ.zdigits op in - let sub_c := ZnZ.sub_c in - let add_mul_div := ZnZ.add_mul_div in - let zzero := ZnZ.zero in - fun x p => match sub_c zdigits p with - | C0 d => reduce n (add_mul_div d zzero x) - | C1 _ => zero - end). - - Definition shiftr : t -> t -> t := Eval red_t in - same_level shiftrn. - - Lemma shiftr_fold : shiftr = same_level shiftrn. - Proof. red_t; reflexivity. Qed. - - Lemma div_pow2_bound :forall x y z, - 0 <= x -> 0 <= y -> x < z -> 0 <= x / 2 ^ y < z. - Proof. - intros x y z HH HH1 HH2. - split; auto with zarith. - apply Z.le_lt_trans with (2 := HH2); auto with zarith. - apply Zdiv_le_upper_bound; auto with zarith. - pattern x at 1; replace x with (x * 2 ^ 0); auto with zarith. - apply Z.mul_le_mono_nonneg_l; auto. - apply Z.pow_le_mono_r; auto with zarith. - rewrite Z.pow_0_r; ring. - Qed. - - Theorem spec_shiftr_pow2 : forall x n, - [shiftr x n] = [x] / 2 ^ [n]. - Proof. - intros x y. rewrite shiftr_fold. apply spec_same_level. clear x y. - intros n x p. simpl. - assert (Hx := ZnZ.spec_to_Z x). - assert (Hy := ZnZ.spec_to_Z p). - generalize (ZnZ.spec_sub_c (ZnZ.zdigits (dom_op n)) p). - case ZnZ.sub_c; intros d H; unfold interp_carry in *; simpl. - (** Subtraction without underflow : [ p <= digits ] *) - rewrite spec_reduce. - rewrite ZnZ.spec_zdigits in H. - rewrite ZnZ.spec_add_mul_div by auto with zarith. - rewrite ZnZ.spec_0, Z.mul_0_l, Z.add_0_l. - rewrite Zmod_small. - f_equal. f_equal. auto with zarith. - split. auto with zarith. - apply div_pow2_bound; auto with zarith. - (** Subtraction with underflow : [ digits < p ] *) - rewrite ZnZ.spec_0. symmetry. - apply Zdiv_small. - split; auto with zarith. - apply Z.lt_le_trans with (base (ZnZ.digits (dom_op n))); auto with zarith. - unfold base. apply Z.pow_le_mono_r; auto with zarith. - rewrite ZnZ.spec_zdigits in H. - generalize (ZnZ.spec_to_Z d); auto with zarith. - Qed. - - Lemma spec_shiftr: forall x p, [shiftr x p] = Z.shiftr [x] [p]. - Proof. - intros. - now rewrite spec_shiftr_pow2, Z.shiftr_div_pow2 by apply spec_pos. - Qed. - - (** * Left shift *) - - (** First an unsafe version, working correctly only if - the representation is large enough *) - - Local Notation unsafe_shiftln := (fun n => - let op := dom_op n in - let add_mul_div := ZnZ.add_mul_div in - let zero := ZnZ.zero in - fun x p => reduce n (add_mul_div p x zero)). - - Definition unsafe_shiftl : t -> t -> t := Eval red_t in - same_level unsafe_shiftln. - - Lemma unsafe_shiftl_fold : unsafe_shiftl = same_level unsafe_shiftln. - Proof. red_t; reflexivity. Qed. - - Theorem spec_unsafe_shiftl_aux : forall x p K, - 0 <= K -> - [x] < 2^K -> - [p] + K <= Zpos (digits x) -> - [unsafe_shiftl x p] = [x] * 2 ^ [p]. - Proof. - intros x p. - rewrite unsafe_shiftl_fold. rewrite digits_level. - apply spec_same_level_dep. - intros n m z z' r LE H K HK H1 H2. apply (H K); auto. - transitivity (Zpos (ZnZ.digits (dom_op n))); auto. - apply digits_dom_op_incr; auto. - clear x p. - intros n x p K HK Hx Hp. simpl. rewrite spec_reduce. - destruct (ZnZ.spec_to_Z x). - destruct (ZnZ.spec_to_Z p). - rewrite ZnZ.spec_add_mul_div by (omega with *). - rewrite ZnZ.spec_0, Zdiv_0_l, Z.add_0_r. - apply Zmod_small. unfold base. - split; auto with zarith. - rewrite Z.mul_comm. - apply Z.lt_le_trans with (2^(ZnZ.to_Z p + K)). - rewrite Zpower_exp; auto with zarith. - apply Z.mul_lt_mono_pos_l; auto with zarith. - apply Z.pow_le_mono_r; auto with zarith. - Qed. - - Theorem spec_unsafe_shiftl: forall x p, - [p] <= [head0 x] -> [unsafe_shiftl x p] = [x] * 2 ^ [p]. - Proof. - intros. - destruct (Z.eq_dec [x] 0) as [EQ|NEQ]. - (* [x] = 0 *) - apply spec_unsafe_shiftl_aux with 0; auto with zarith. - now rewrite EQ. - rewrite spec_head00 in *; auto with zarith. - (* [x] <> 0 *) - apply spec_unsafe_shiftl_aux with ([log2 x] + 1); auto with zarith. - generalize (spec_pos (log2 x)); auto with zarith. - destruct (spec_log2_pos x); auto with zarith. - rewrite log2_digits_head0; auto with zarith. - generalize (spec_pos x); auto with zarith. - Qed. - - (** Then we define a function doubling the size of the representation - but without changing the value of the number. *) - - Local Notation double_size_n := (fun n => - let zero := ZnZ.zero in - fun x => mk_t_S n (WW zero x)). - - Definition double_size : t -> t := Eval red_t in - iter_t double_size_n. - - Lemma double_size_fold : double_size = iter_t double_size_n. - Proof. red_t; reflexivity. Qed. - - Lemma double_size_level : forall x, level (double_size x) = S (level x). - Proof. - intros x. rewrite double_size_fold; unfold level at 2. destr_t x as (n,x). - apply mk_t_S_level. - Qed. - - Theorem spec_double_size_digits: - forall x, Zpos (digits (double_size x)) = 2 * (Zpos (digits x)). - Proof. - intros x. rewrite ! digits_level, double_size_level. - rewrite 2 digits_dom_op, 2 Pshiftl_nat_Zpower, - Nat2Z.inj_succ, Z.pow_succ_r; auto with zarith. - ring. - Qed. - - Theorem spec_double_size: forall x, [double_size x] = [x]. - Proof. - intros x. rewrite double_size_fold. destr_t x as (n,x). - rewrite spec_mk_t_S. simpl. rewrite ZnZ.spec_0. auto with zarith. - Qed. - - Theorem spec_double_size_head0: - forall x, 2 * [head0 x] <= [head0 (double_size x)]. - Proof. - intros x. - assert (F1:= spec_pos (head0 x)). - assert (F2: 0 < Zpos (digits x)). - red; auto. - assert (HH := spec_pos x). Z.le_elim HH. - generalize HH; rewrite <- (spec_double_size x); intros HH1. - case (spec_head0 x HH); intros _ HH2. - case (spec_head0 _ HH1). - rewrite (spec_double_size x); rewrite (spec_double_size_digits x). - intros HH3 _. - case (Z.le_gt_cases ([head0 (double_size x)]) (2 * [head0 x])); auto; intros HH4. - absurd (2 ^ (2 * [head0 x] )* [x] < 2 ^ [head0 (double_size x)] * [x]); auto. - apply Z.le_ngt. - apply Z.mul_le_mono_nonneg_r; auto with zarith. - apply Z.pow_le_mono_r; auto; auto with zarith. - assert (HH5: 2 ^[head0 x] <= 2 ^(Zpos (digits x) - 1)). - { apply Z.le_succ_l in HH. change (1 <= [x]) in HH. - Z.le_elim HH. - - apply Z.mul_le_mono_pos_r with (2 ^ 1); auto with zarith. - rewrite <- (fun x y z => Z.pow_add_r x (y - z)); auto with zarith. - rewrite Z.sub_add. - apply Z.le_trans with (2 := Z.lt_le_incl _ _ HH2). - apply Z.mul_le_mono_nonneg_l; auto with zarith. - rewrite Z.pow_1_r; auto with zarith. - - apply Z.pow_le_mono_r; auto with zarith. - case (Z.le_gt_cases (Zpos (digits x)) [head0 x]); auto with zarith; intros HH6. - absurd (2 ^ Zpos (digits x) <= 2 ^ [head0 x] * [x]); auto with zarith. - rewrite <- HH; rewrite Z.mul_1_r. - apply Z.pow_le_mono_r; auto with zarith. } - rewrite (Z.mul_comm 2). - rewrite Z.pow_mul_r; auto with zarith. - rewrite Z.pow_2_r. - apply Z.lt_le_trans with (2 := HH3). - rewrite <- Z.mul_assoc. - replace (2 * Zpos (digits x) - 1) with - ((Zpos (digits x) - 1) + (Zpos (digits x))). - rewrite Zpower_exp; auto with zarith. - apply Zmult_lt_compat2; auto with zarith. - split; auto with zarith. - apply Z.mul_pos_pos; auto with zarith. - rewrite Pos2Z.inj_xO; ring. - apply Z.lt_le_incl; auto. - repeat rewrite spec_head00; auto. - rewrite spec_double_size_digits. - rewrite Pos2Z.inj_xO; auto with zarith. - rewrite spec_double_size; auto. - Qed. - - Theorem spec_double_size_head0_pos: - forall x, 0 < [head0 (double_size x)]. - Proof. - intros x. - assert (F := Pos2Z.is_pos (digits x)). - assert (F0 := spec_pos (head0 (double_size x))). - Z.le_elim F0; auto. - assert (F1 := spec_pos (head0 x)). - Z.le_elim F1. - apply Z.lt_le_trans with (2 := (spec_double_size_head0 x)); auto with zarith. - assert (F3 := spec_pos x). - Z.le_elim F3. - generalize F3; rewrite <- (spec_double_size x); intros F4. - absurd (2 ^ (Zpos (xO (digits x)) - 1) < 2 ^ (Zpos (digits x))). - { apply Z.le_ngt. - apply Z.pow_le_mono_r; auto with zarith. - rewrite Pos2Z.inj_xO; auto with zarith. } - case (spec_head0 x F3). - rewrite <- F1; rewrite Z.pow_0_r; rewrite Z.mul_1_l; intros _ HH. - apply Z.le_lt_trans with (2 := HH). - case (spec_head0 _ F4). - rewrite (spec_double_size x); rewrite (spec_double_size_digits x). - rewrite <- F0; rewrite Z.pow_0_r; rewrite Z.mul_1_l; auto. - generalize F1; rewrite (spec_head00 _ (eq_sym F3)); auto with zarith. - Qed. - - (** Finally we iterate [double_size] enough before [unsafe_shiftl] - in order to get a fully correct [shiftl]. *) - - Definition shiftl_aux_body cont x n := - match compare n (head0 x) with - Gt => cont (double_size x) n - | _ => unsafe_shiftl x n - end. - - Theorem spec_shiftl_aux_body: forall n x p cont, - 2^ Zpos p <= [head0 x] -> - (forall x, 2 ^ (Zpos p + 1) <= [head0 x]-> - [cont x n] = [x] * 2 ^ [n]) -> - [shiftl_aux_body cont x n] = [x] * 2 ^ [n]. - Proof. - intros n x p cont H1 H2; unfold shiftl_aux_body. - rewrite spec_compare; case Z.compare_spec; intros H. - apply spec_unsafe_shiftl; auto with zarith. - apply spec_unsafe_shiftl; auto with zarith. - rewrite H2. - rewrite spec_double_size; auto. - rewrite Z.add_comm; rewrite Zpower_exp; auto with zarith. - apply Z.le_trans with (2 := spec_double_size_head0 x). - rewrite Z.pow_1_r; apply Z.mul_le_mono_nonneg_l; auto with zarith. - Qed. - - Fixpoint shiftl_aux p cont x n := - shiftl_aux_body - (fun x n => match p with - | xH => cont x n - | xO p => shiftl_aux p (shiftl_aux p cont) x n - | xI p => shiftl_aux p (shiftl_aux p cont) x n - end) x n. - - Theorem spec_shiftl_aux: forall p q x n cont, - 2 ^ (Zpos q) <= [head0 x] -> - (forall x, 2 ^ (Zpos p + Zpos q) <= [head0 x] -> - [cont x n] = [x] * 2 ^ [n]) -> - [shiftl_aux p cont x n] = [x] * 2 ^ [n]. - Proof. - intros p; elim p; unfold shiftl_aux; fold shiftl_aux; clear p. - intros p Hrec q x n cont H1 H2. - apply spec_shiftl_aux_body with (q); auto. - intros x1 H3; apply Hrec with (q + 1)%positive; auto. - intros x2 H4; apply Hrec with (p + q + 1)%positive; auto. - rewrite <- Pos.add_assoc. - rewrite Pos2Z.inj_add; auto. - intros x3 H5; apply H2. - rewrite Pos2Z.inj_xI. - replace (2 * Zpos p + 1 + Zpos q) with (Zpos p + Zpos (p + q + 1)); - auto. - rewrite !Pos2Z.inj_add; ring. - intros p Hrec q n x cont H1 H2. - apply spec_shiftl_aux_body with (q); auto. - intros x1 H3; apply Hrec with (q); auto. - apply Z.le_trans with (2 := H3); auto with zarith. - apply Z.pow_le_mono_r; auto with zarith. - intros x2 H4; apply Hrec with (p + q)%positive; auto. - intros x3 H5; apply H2. - rewrite (Pos2Z.inj_xO p). - replace (2 * Zpos p + Zpos q) with (Zpos p + Zpos (p + q)); - auto. - rewrite Pos2Z.inj_add; ring. - intros q n x cont H1 H2. - apply spec_shiftl_aux_body with (q); auto. - rewrite Z.add_comm; auto. - Qed. - - Definition shiftl x n := - shiftl_aux_body - (shiftl_aux_body - (shiftl_aux (digits n) unsafe_shiftl)) x n. - - Theorem spec_shiftl_pow2 : forall x n, - [shiftl x n] = [x] * 2 ^ [n]. - Proof. - intros x n; unfold shiftl, shiftl_aux_body. - rewrite spec_compare; case Z.compare_spec; intros H. - apply spec_unsafe_shiftl; auto with zarith. - apply spec_unsafe_shiftl; auto with zarith. - rewrite <- (spec_double_size x). - rewrite spec_compare; case Z.compare_spec; intros H1. - apply spec_unsafe_shiftl; auto with zarith. - apply spec_unsafe_shiftl; auto with zarith. - rewrite <- (spec_double_size (double_size x)). - apply spec_shiftl_aux with 1%positive. - apply Z.le_trans with (2 := spec_double_size_head0 (double_size x)). - replace (2 ^ 1) with (2 * 1). - apply Z.mul_le_mono_nonneg_l; auto with zarith. - generalize (spec_double_size_head0_pos x); auto with zarith. - rewrite Z.pow_1_r; ring. - intros x1 H2; apply spec_unsafe_shiftl. - apply Z.le_trans with (2 := H2). - apply Z.le_trans with (2 ^ Zpos (digits n)); auto with zarith. - case (spec_digits n); auto with zarith. - apply Z.pow_le_mono_r; auto with zarith. - Qed. - - Lemma spec_shiftl: forall x p, [shiftl x p] = Z.shiftl [x] [p]. - Proof. - intros. - now rewrite spec_shiftl_pow2, Z.shiftl_mul_pow2 by apply spec_pos. - Qed. - - (** Other bitwise operations *) - - Definition testbit x n := odd (shiftr x n). - - Lemma spec_testbit: forall x p, testbit x p = Z.testbit [x] [p]. - Proof. - intros. unfold testbit. symmetry. - rewrite spec_odd, spec_shiftr. apply Z.testbit_odd. - Qed. - - Definition div2 x := shiftr x one. - - Lemma spec_div2: forall x, [div2 x] = Z.div2 [x]. - Proof. - intros. unfold div2. symmetry. - rewrite spec_shiftr, spec_1. apply Z.div2_spec. - Qed. - - Local Notation lorn := (fun n => - let op := dom_op n in - let lor := ZnZ.lor in - fun x y => reduce n (lor x y)). - - Definition lor : t -> t -> t := Eval red_t in same_level lorn. - - Lemma lor_fold : lor = same_level lorn. - Proof. red_t; reflexivity. Qed. - - Theorem spec_lor x y : [lor x y] = Z.lor [x] [y]. - Proof. - rewrite lor_fold. apply spec_same_level; clear x y. - intros n x y. simpl. rewrite spec_reduce. apply ZnZ.spec_lor. - Qed. - - Local Notation landn := (fun n => - let op := dom_op n in - let land := ZnZ.land in - fun x y => reduce n (land x y)). - - Definition land : t -> t -> t := Eval red_t in same_level landn. - - Lemma land_fold : land = same_level landn. - Proof. red_t; reflexivity. Qed. - - Theorem spec_land x y : [land x y] = Z.land [x] [y]. - Proof. - rewrite land_fold. apply spec_same_level; clear x y. - intros n x y. simpl. rewrite spec_reduce. apply ZnZ.spec_land. - Qed. - - Local Notation lxorn := (fun n => - let op := dom_op n in - let lxor := ZnZ.lxor in - fun x y => reduce n (lxor x y)). - - Definition lxor : t -> t -> t := Eval red_t in same_level lxorn. - - Lemma lxor_fold : lxor = same_level lxorn. - Proof. red_t; reflexivity. Qed. - - Theorem spec_lxor x y : [lxor x y] = Z.lxor [x] [y]. - Proof. - rewrite lxor_fold. apply spec_same_level; clear x y. - intros n x y. simpl. rewrite spec_reduce. apply ZnZ.spec_lxor. - Qed. - - Local Notation ldiffn := (fun n => - let op := dom_op n in - let lxor := ZnZ.lxor in - let land := ZnZ.land in - let m1 := ZnZ.minus_one in - fun x y => reduce n (land x (lxor y m1))). - - Definition ldiff : t -> t -> t := Eval red_t in same_level ldiffn. - - Lemma ldiff_fold : ldiff = same_level ldiffn. - Proof. red_t; reflexivity. Qed. - - Lemma ldiff_alt x y p : - 0 <= x < 2^p -> 0 <= y < 2^p -> - Z.ldiff x y = Z.land x (Z.lxor y (2^p - 1)). - Proof. - intros (Hx,Hx') (Hy,Hy'). - destruct p as [|p|p]. - - simpl in *; replace x with 0; replace y with 0; auto with zarith. - - rewrite <- Z.shiftl_1_l. change (_ - 1) with (Z.ones (Z.pos p)). - rewrite <- Z.ldiff_ones_l_low; trivial. - rewrite !Z.ldiff_land, Z.land_assoc. f_equal. - rewrite Z.land_ones; try easy. - symmetry. apply Z.mod_small; now split. - Z.le_elim Hy. - + now apply Z.log2_lt_pow2. - + now subst. - - simpl in *; omega. - Qed. - - Theorem spec_ldiff x y : [ldiff x y] = Z.ldiff [x] [y]. - Proof. - rewrite ldiff_fold. apply spec_same_level; clear x y. - intros n x y. simpl. rewrite spec_reduce. - rewrite ZnZ.spec_land, ZnZ.spec_lxor, ZnZ.spec_m1. - symmetry. apply ldiff_alt; apply ZnZ.spec_to_Z. - Qed. - -End Make. diff --git a/theories/Numbers/Natural/BigN/NMake_gen.ml b/theories/Numbers/Natural/BigN/NMake_gen.ml deleted file mode 100644 index 5177fae6..00000000 --- a/theories/Numbers/Natural/BigN/NMake_gen.ml +++ /dev/null @@ -1,1017 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -(*S NMake_gen.ml : this file generates NMake_gen.v *) - - -(*s The parameter that control the generation: *) - -let size = 6 (* how many times should we repeat the Z/nZ --> Z/2nZ - process before relying on a generic construct *) - -(*s Some utilities *) - -let rec iter_str n s = if n = 0 then "" else (iter_str (n-1) s) ^ s - -let rec iter_str_gen n f = if n < 0 then "" else (iter_str_gen (n-1) f) ^ (f n) - -let rec iter_name i j base sep = - if i >= j then base^(string_of_int i) - else (iter_name i (j-1) base sep)^sep^" "^base^(string_of_int j) - -let pr s = Printf.printf (s^^"\n") - -(*s The actual printing *) - -let _ = - -pr -"(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) -(* \\VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -(** * NMake_gen *) - -(** From a cyclic Z/nZ representation to arbitrary precision natural numbers.*) - -(** Remark: File automatically generated by NMake_gen.ml, DO NOT EDIT ! *) - -Require Import BigNumPrelude ZArith Ndigits CyclicAxioms - DoubleType DoubleMul DoubleDivn1 DoubleCyclic Nbasic - Wf_nat StreamMemo. - -Module Make (W0:CyclicType) <: NAbstract. - - (** * The word types *) -"; - -pr " Local Notation w0 := W0.t."; -for i = 1 to size do - pr " Definition w%i := zn2z w%i." i (i-1) -done; -pr ""; - -pr " (** * The operation type classes for the word types *) -"; - -pr " Local Notation w0_op := W0.ops."; -for i = 1 to min 3 size do - pr " Instance w%i_op : ZnZ.Ops w%i := mk_zn2z_ops w%i_op." i i (i-1) -done; -for i = 4 to size do - pr " Instance w%i_op : ZnZ.Ops w%i := mk_zn2z_ops_karatsuba w%i_op." i i (i-1) -done; -for i = size+1 to size+3 do - pr " Instance w%i_op : ZnZ.Ops (word w%i %i) := mk_zn2z_ops_karatsuba w%i_op." i size (i-size) (i-1) -done; -pr ""; - - pr " Section Make_op."; - pr " Variable mk : forall w', ZnZ.Ops w' -> ZnZ.Ops (zn2z w')."; - pr ""; - pr " Fixpoint make_op_aux (n:nat) : ZnZ.Ops (word w%i (S n)):=" size; - pr " match n return ZnZ.Ops (word w%i (S n)) with" size; - pr " | O => w%i_op" (size+1); - pr " | S n1 =>"; - pr " match n1 return ZnZ.Ops (word w%i (S (S n1))) with" size; - pr " | O => w%i_op" (size+2); - pr " | S n2 =>"; - pr " match n2 return ZnZ.Ops (word w%i (S (S (S n2)))) with" size; - pr " | O => w%i_op" (size+3); - pr " | S n3 => mk _ (mk _ (mk _ (make_op_aux n3)))"; - pr " end"; - pr " end"; - pr " end."; - pr ""; - pr " End Make_op."; - pr ""; - pr " Definition omake_op := make_op_aux mk_zn2z_ops_karatsuba."; - pr ""; - pr ""; - pr " Definition make_op_list := dmemo_list _ omake_op."; - pr ""; - pr " Instance make_op n : ZnZ.Ops (word w%i (S n))" size; - pr " := dmemo_get _ omake_op n make_op_list."; - pr ""; - -pr " Ltac unfold_ops := unfold omake_op, make_op_aux, w%i_op, w%i_op." (size+3) (size+2); - -pr -" - Lemma make_op_omake: forall n, make_op n = omake_op n. - Proof. - intros n; unfold make_op, make_op_list. - refine (dmemo_get_correct _ _ _). - Qed. - - Theorem make_op_S: forall n, - make_op (S n) = mk_zn2z_ops_karatsuba (make_op n). - Proof. - intros n. do 2 rewrite make_op_omake. - revert n. fix IHn 1. - do 3 (destruct n; [unfold_ops; reflexivity|]). - simpl mk_zn2z_ops_karatsuba. simpl word in *. - rewrite <- (IHn n). auto. - Qed. - - (** * The main type [t], isomorphic with [exists n, word w0 n] *) -"; - - pr " Inductive t' :="; - for i = 0 to size do - pr " | N%i : w%i -> t'" i i - done; - pr " | Nn : forall n, word w%i (S n) -> t'." size; - pr ""; - pr " Definition t := t'."; - pr ""; - - pr " (** * A generic toolbox for building and deconstructing [t] *)"; - pr ""; - - pr " Local Notation SizePlus n := %sn%s." - (iter_str size "(S ") (iter_str size ")"); - pr " Local Notation Size := (SizePlus O)."; - pr ""; - - pr " Tactic Notation (at level 3) \"do_size\" tactic3(t) := do %i t." (size+1); - pr ""; - - pr " Definition dom_t n := match n with"; - for i = 0 to size do - pr " | %i => w%i" i i; - done; - pr " | %sn => word w%i n" (if size=0 then "" else "SizePlus ") size; - pr " end."; - pr ""; - -pr -" Instance dom_op n : ZnZ.Ops (dom_t n) | 10. - Proof. - do_size (destruct n; [simpl;auto with *|]). - unfold dom_t. auto with *. - Defined. -"; - - pr " Definition iter_t {A:Type}(f : forall n, dom_t n -> A) : t -> A :="; - for i = 0 to size do - pr " let f%i := f %i in" i i; - done; - pr " let fn n := f (SizePlus (S n)) in"; - pr " fun x => match x with"; - for i = 0 to size do - pr " | N%i wx => f%i wx" i i; - done; - pr " | Nn n wx => fn n wx"; - pr " end."; - pr ""; - - pr " Definition mk_t (n:nat) : dom_t n -> t :="; - pr " match n as n' return dom_t n' -> t with"; - for i = 0 to size do - pr " | %i => N%i" i i; - done; - pr " | %s(S n) => Nn n" (if size=0 then "" else "SizePlus "); - pr " end."; - pr ""; - -pr -" Definition level := iter_t (fun n _ => n). - - Inductive View_t : t -> Prop := - Mk_t : forall n (x : dom_t n), View_t (mk_t n x). - - Lemma destr_t : forall x, View_t x. - Proof. - intros x. generalize (Mk_t (level x)). destruct x; simpl; auto. - Defined. - - Lemma iter_mk_t : forall A (f:forall n, dom_t n -> A), - forall n x, iter_t f (mk_t n x) = f n x. - Proof. - do_size (destruct n; try reflexivity). - Qed. - - (** * Projection to ZArith *) - - Definition to_Z : t -> Z := - Eval lazy beta iota delta [iter_t dom_t dom_op] in - iter_t (fun _ x => ZnZ.to_Z x). - - Notation \"[ x ]\" := (to_Z x). - - Theorem spec_mk_t : forall n (x:dom_t n), [mk_t n x] = ZnZ.to_Z x. - Proof. - intros. change to_Z with (iter_t (fun _ x => ZnZ.to_Z x)). - rewrite iter_mk_t; auto. - Qed. - - (** * Regular make op, without memoization or karatsuba - - This will normally never be used for actual computations, - but only for specification purpose when using - [word (dom_t n) m] intermediate values. *) - - Fixpoint nmake_op (ww:Type) (ww_op: ZnZ.Ops ww) (n: nat) : - ZnZ.Ops (word ww n) := - match n return ZnZ.Ops (word ww n) with - O => ww_op - | S n1 => mk_zn2z_ops (nmake_op ww ww_op n1) - end. - - Definition eval n m := ZnZ.to_Z (Ops:=nmake_op _ (dom_op n) m). - - Theorem nmake_op_S: forall ww (w_op: ZnZ.Ops ww) x, - nmake_op _ w_op (S x) = mk_zn2z_ops (nmake_op _ w_op x). - Proof. - auto. - Qed. - - Theorem digits_nmake_S :forall n ww (w_op: ZnZ.Ops ww), - ZnZ.digits (nmake_op _ w_op (S n)) = - xO (ZnZ.digits (nmake_op _ w_op n)). - Proof. - auto. - Qed. - - Theorem digits_nmake : forall n ww (w_op: ZnZ.Ops ww), - ZnZ.digits (nmake_op _ w_op n) = Pos.shiftl_nat (ZnZ.digits w_op) n. - Proof. - induction n. auto. - intros ww ww_op. rewrite Pshiftl_nat_S, <- IHn; auto. - Qed. - - Theorem nmake_double: forall n ww (w_op: ZnZ.Ops ww), - ZnZ.to_Z (Ops:=nmake_op _ w_op n) = - @DoubleBase.double_to_Z _ (ZnZ.digits w_op) (ZnZ.to_Z (Ops:=w_op)) n. - Proof. - intros n; elim n; auto; clear n. - intros n Hrec ww ww_op; simpl DoubleBase.double_to_Z; unfold zn2z_to_Z. - rewrite <- Hrec; auto. - unfold DoubleBase.double_wB; rewrite <- digits_nmake; auto. - Qed. - - Theorem nmake_WW: forall ww ww_op n xh xl, - (ZnZ.to_Z (Ops:=nmake_op ww ww_op (S n)) (WW xh xl) = - ZnZ.to_Z (Ops:=nmake_op ww ww_op n) xh * - base (ZnZ.digits (nmake_op ww ww_op n)) + - ZnZ.to_Z (Ops:=nmake_op ww ww_op n) xl)%%Z. - Proof. - auto. - Qed. - - (** * The specification proofs for the word operators *) -"; - - if size <> 0 then - pr " Typeclasses Opaque %s." (iter_name 1 size "w" ""); - pr ""; - - pr " Instance w0_spec: ZnZ.Specs w0_op := W0.specs."; - for i = 1 to min 3 size do - pr " Instance w%i_spec: ZnZ.Specs w%i_op := mk_zn2z_specs w%i_spec." i i (i-1) - done; - for i = 4 to size do - pr " Instance w%i_spec: ZnZ.Specs w%i_op := mk_zn2z_specs_karatsuba w%i_spec." i i (i-1) - done; - pr " Instance w%i_spec: ZnZ.Specs w%i_op := mk_zn2z_specs_karatsuba w%i_spec." (size+1) (size+1) size; - - -pr " - Instance wn_spec (n:nat) : ZnZ.Specs (make_op n). - Proof. - induction n. - rewrite make_op_omake; simpl; auto with *. - rewrite make_op_S. exact (mk_zn2z_specs_karatsuba IHn). - Qed. - - Instance dom_spec n : ZnZ.Specs (dom_op n) | 10. - Proof. - do_size (destruct n; auto with *). apply wn_spec. - Qed. - - Let make_op_WW : forall n x y, - (ZnZ.to_Z (Ops:=make_op (S n)) (WW x y) = - ZnZ.to_Z (Ops:=make_op n) x * base (ZnZ.digits (make_op n)) - + ZnZ.to_Z (Ops:=make_op n) y)%%Z. - Proof. - intros n x y; rewrite make_op_S; auto. - Qed. - - (** * Zero *) - - Definition zero0 : w0 := ZnZ.zero. - - Definition zeron n : dom_t n := - match n with - | O => zero0 - | SizePlus (S n) => W0 - | _ => W0 - end. - - Lemma spec_zeron : forall n, ZnZ.to_Z (zeron n) = 0%%Z. - Proof. - do_size (destruct n; - [match goal with - |- @eq Z (_ (zeron ?n)) _ => - apply (ZnZ.spec_0 (Specs:=dom_spec n)) - end|]). - destruct n; auto. simpl. rewrite make_op_S. fold word. - apply (ZnZ.spec_0 (Specs:=wn_spec (SizePlus 0))). - Qed. - - (** * Digits *) - - Lemma digits_make_op_0 : forall n, - ZnZ.digits (make_op n) = Pos.shiftl_nat (ZnZ.digits (dom_op Size)) (S n). - Proof. - induction n. - auto. - replace (ZnZ.digits (make_op (S n))) with (xO (ZnZ.digits (make_op n))). - rewrite IHn; auto. - rewrite make_op_S; auto. - Qed. - - Lemma digits_make_op : forall n, - ZnZ.digits (make_op n) = Pos.shiftl_nat (ZnZ.digits w0_op) (SizePlus (S n)). - Proof. - intros. rewrite digits_make_op_0. - replace (SizePlus (S n)) with (S n + Size) by (rewrite <- plus_comm; auto). - rewrite Pshiftl_nat_plus. auto. - Qed. - - Lemma digits_dom_op : forall n, - ZnZ.digits (dom_op n) = Pos.shiftl_nat (ZnZ.digits w0_op) n. - Proof. - do_size (destruct n; try reflexivity). - exact (digits_make_op n). - Qed. - - Lemma digits_dom_op_nmake : forall n m, - ZnZ.digits (dom_op (m+n)) = ZnZ.digits (nmake_op _ (dom_op n) m). - Proof. - intros. rewrite digits_nmake, 2 digits_dom_op. apply Pshiftl_nat_plus. - Qed. - - (** * Conversion between [zn2z (dom_t n)] and [dom_t (S n)]. - - These two types are provably equal, but not convertible, - hence we need some work. We now avoid using generic casts - (i.e. rewrite via proof of equalities in types), since - proving things with them is a mess. - *) - - Definition succ_t n : zn2z (dom_t n) -> dom_t (S n) := - match n with - | SizePlus (S _) => fun x => x - | _ => fun x => x - end. - - Lemma spec_succ_t : forall n x, - ZnZ.to_Z (succ_t n x) = - zn2z_to_Z (base (ZnZ.digits (dom_op n))) ZnZ.to_Z x. - Proof. - do_size (destruct n ; [reflexivity|]). - intros. simpl. rewrite make_op_S. simpl. auto. - Qed. - - Definition pred_t n : dom_t (S n) -> zn2z (dom_t n) := - match n with - | SizePlus (S _) => fun x => x - | _ => fun x => x - end. - - Lemma succ_pred_t : forall n x, succ_t n (pred_t n x) = x. - Proof. - do_size (destruct n ; [reflexivity|]). reflexivity. - Qed. - - (** We can hence project from [zn2z (dom_t n)] to [t] : *) - - Definition mk_t_S n (x : zn2z (dom_t n)) : t := - mk_t (S n) (succ_t n x). - - Lemma spec_mk_t_S : forall n x, - [mk_t_S n x] = zn2z_to_Z (base (ZnZ.digits (dom_op n))) ZnZ.to_Z x. - Proof. - intros. unfold mk_t_S. rewrite spec_mk_t. apply spec_succ_t. - Qed. - - Lemma mk_t_S_level : forall n x, level (mk_t_S n x) = S n. - Proof. - intros. unfold mk_t_S, level. rewrite iter_mk_t; auto. - Qed. - - (** * Conversion from [word (dom_t n) m] to [dom_t (m+n)]. - - Things are more complex here. We start with a naive version - that breaks zn2z-trees and reconstruct them. Doing this is - quite unfortunate, but I don't know how to fully avoid that. - (cast someday ?). Then we build an optimized version where - all basic cases (n<=6 or m<=7) are nicely handled. - *) - - Definition zn2z_map {A} {B} (f:A->B) (x:zn2z A) : zn2z B := - match x with - | W0 => W0 - | WW h l => WW (f h) (f l) - end. - - Lemma zn2z_map_id : forall A f (x:zn2z A), (forall u, f u = u) -> - zn2z_map f x = x. - Proof. - destruct x; auto; intros. - simpl; f_equal; auto. - Qed. - - (** The naive version *) - - Fixpoint plus_t n m : word (dom_t n) m -> dom_t (m+n) := - match m as m' return word (dom_t n) m' -> dom_t (m'+n) with - | O => fun x => x - | S m => fun x => succ_t _ (zn2z_map (plus_t n m) x) - end. - - Theorem spec_plus_t : forall n m (x:word (dom_t n) m), - ZnZ.to_Z (plus_t n m x) = eval n m x. - Proof. - unfold eval. - induction m. - simpl; auto. - intros. - simpl plus_t; simpl plus. rewrite spec_succ_t. - destruct x. - simpl; auto. - fold word in w, w0. - simpl. rewrite 2 IHm. f_equal. f_equal. f_equal. - apply digits_dom_op_nmake. - Qed. - - Definition mk_t_w n m (x:word (dom_t n) m) : t := - mk_t (m+n) (plus_t n m x). - - Theorem spec_mk_t_w : forall n m (x:word (dom_t n) m), - [mk_t_w n m x] = eval n m x. - Proof. - intros. unfold mk_t_w. rewrite spec_mk_t. apply spec_plus_t. - Qed. - - (** The optimized version. - - NB: the last particular case for m could depend on n, - but it's simplier to just expand everywhere up to m=7 - (cf [mk_t_w'] later). - *) - - Definition plus_t' n : forall m, word (dom_t n) m -> dom_t (m+n) := - match n return (forall m, word (dom_t n) m -> dom_t (m+n)) with - | SizePlus (S n') as n => plus_t n - | _ as n => - fun m => match m return (word (dom_t n) m -> dom_t (m+n)) with - | SizePlus (S (S m')) as m => plus_t n m - | _ => fun x => x - end - end. - - Lemma plus_t_equiv : forall n m x, - plus_t' n m x = plus_t n m x. - Proof. - (do_size try destruct n); try reflexivity; - (do_size try destruct m); try destruct m; try reflexivity; - simpl; symmetry; repeat (intros; apply zn2z_map_id; trivial). - Qed. - - Lemma spec_plus_t' : forall n m x, - ZnZ.to_Z (plus_t' n m x) = eval n m x. - Proof. - intros; rewrite plus_t_equiv. apply spec_plus_t. - Qed. - - (** Particular cases [Nk x] = eval i j x with specific k,i,j - can be solved by the following tactic *) - - Ltac solve_eval := - intros; rewrite <- spec_plus_t'; unfold to_Z; simpl dom_op; reflexivity. - - (** The last particular case that remains useful *) - - Lemma spec_eval_size : forall n x, [Nn n x] = eval Size (S n) x. - Proof. - induction n. - solve_eval. - destruct x as [ | xh xl ]. - simpl. unfold eval. rewrite make_op_S. rewrite nmake_op_S. auto. - simpl word in xh, xl |- *. - unfold to_Z in *. rewrite make_op_WW. - unfold eval in *. rewrite nmake_WW. - f_equal; auto. - f_equal; auto. - f_equal. - rewrite <- digits_dom_op_nmake. rewrite plus_comm; auto. - Qed. - - (** An optimized [mk_t_w]. - - We could say mk_t_w' := mk_t _ (plus_t' n m x) - (TODO: WHY NOT, BTW ??). - Instead we directly define functions for all intersting [n], - reverting to naive [mk_t_w] at places that should normally - never be used (see [mul] and [div_gt]). - *) -"; - -for i = 0 to size-1 do -let pattern = (iter_str (size+1-i) "(S ") ^ "_" ^ (iter_str (size+1-i) ")") in -pr -" Definition mk_t_%iw m := Eval cbv beta zeta iota delta [ mk_t plus ] in - match m return word w%i (S m) -> t with - | %s as p => mk_t_w %i (S p) - | p => mk_t (%i+p) - end. -" i i pattern i (i+1) -done; - -pr -" Definition mk_t_w' n : forall m, word (dom_t n) (S m) -> t := - match n return (forall m, word (dom_t n) (S m) -> t) with"; -for i = 0 to size-1 do pr " | %i => mk_t_%iw" i i done; -pr -" | Size => Nn - | _ as n' => fun m => mk_t_w n' (S m) - end. -"; - -pr -" Ltac solve_spec_mk_t_w' := - rewrite <- spec_plus_t'; - match goal with _ : word (dom_t ?n) ?m |- _ => apply (spec_mk_t (n+m)) end. - - Theorem spec_mk_t_w' : - forall n m x, [mk_t_w' n m x] = eval n (S m) x. - Proof. - intros. - repeat (apply spec_mk_t_w || (destruct n; - [repeat (apply spec_mk_t_w || (destruct m; [solve_spec_mk_t_w'|]))|])). - apply spec_eval_size. - Qed. - - (** * Extend : injecting [dom_t n] into [word (dom_t n) (S m)] *) - - Definition extend n m (x:dom_t n) : word (dom_t n) (S m) := - DoubleBase.extend_aux m (WW (zeron n) x). - - Lemma spec_extend : forall n m x, - [mk_t n x] = eval n (S m) (extend n m x). - Proof. - intros. unfold eval, extend. - rewrite spec_mk_t. - assert (H : forall (x:dom_t n), - (ZnZ.to_Z (zeron n) * base (ZnZ.digits (dom_op n)) + ZnZ.to_Z x = - ZnZ.to_Z x)%%Z). - clear; intros; rewrite spec_zeron; auto. - rewrite <- (@DoubleBase.spec_extend _ - (WW (zeron n)) (ZnZ.digits (dom_op n)) ZnZ.to_Z H m x). - simpl. rewrite digits_nmake, <- nmake_double. auto. - Qed. - - (** A particular case of extend, used in [same_level]: - [extend_size] is [extend Size] *) - - Definition extend_size := DoubleBase.extend (WW (W0:dom_t Size)). - - Lemma spec_extend_size : forall n x, [mk_t Size x] = [Nn n (extend_size n x)]. - Proof. - intros. rewrite spec_eval_size. apply (spec_extend Size n). - Qed. - - (** Misc results about extensions *) - - Let spec_extend_WW : forall n x, - [Nn (S n) (WW W0 x)] = [Nn n x]. - Proof. - intros n x. - set (N:=SizePlus (S n)). - change ([Nn (S n) (extend N 0 x)]=[mk_t N x]). - rewrite (spec_extend N 0). - solve_eval. - Qed. - - Let spec_extend_tr: forall m n w, - [Nn (m + n) (extend_tr w m)] = [Nn n w]. - Proof. - induction m; auto. - intros n x; simpl extend_tr. - simpl plus; rewrite spec_extend_WW; auto. - Qed. - - Let spec_cast_l: forall n m x1, - [Nn n x1] = - [Nn (Max.max n m) (castm (diff_r n m) (extend_tr x1 (snd (diff n m))))]. - Proof. - intros n m x1; case (diff_r n m); simpl castm. - rewrite spec_extend_tr; auto. - Qed. - - Let spec_cast_r: forall n m x1, - [Nn m x1] = - [Nn (Max.max n m) (castm (diff_l n m) (extend_tr x1 (fst (diff n m))))]. - Proof. - intros n m x1; case (diff_l n m); simpl castm. - rewrite spec_extend_tr; auto. - Qed. - - Ltac unfold_lets := - match goal with - | h : _ |- _ => unfold h; clear h; unfold_lets - | _ => idtac - end. - - (** * [same_level] - - Generic binary operator construction, by extending the smaller - argument to the level of the other. - *) - - Section SameLevel. - - Variable res: Type. - Variable P : Z -> Z -> res -> Prop. - Variable f : forall n, dom_t n -> dom_t n -> res. - Variable Pf : forall n x y, P (ZnZ.to_Z x) (ZnZ.to_Z y) (f n x y). -"; - -for i = 0 to size do -pr " Let f%i : w%i -> w%i -> res := f %i." i i i i -done; -pr -" Let fn n := f (SizePlus (S n)). - - Let Pf' : - forall n x y u v, u = [mk_t n x] -> v = [mk_t n y] -> P u v (f n x y). - Proof. - intros. subst. rewrite 2 spec_mk_t. apply Pf. - Qed. -"; - -let ext i j s = - if j <= i then s else Printf.sprintf "(extend %i %i %s)" i (j-i-1) s -in - -pr " Notation same_level_folded := (fun x y => match x, y with"; -for i = 0 to size do - for j = 0 to size do - pr " | N%i wx, N%i wy => f%i %s %s" i j (max i j) (ext i j "wx") (ext j i "wy") - done; - pr " | N%i wx, Nn m wy => fn m (extend_size m %s) wy" i (ext i size "wx") -done; -for i = 0 to size do - pr " | Nn n wx, N%i wy => fn n wx (extend_size n %s)" i (ext i size "wy") -done; -pr -" | Nn n wx, Nn m wy => - let mn := Max.max n m in - let d := diff n m in - fn mn - (castm (diff_r n m) (extend_tr wx (snd d))) - (castm (diff_l n m) (extend_tr wy (fst d))) - end). -"; - -pr -" Definition same_level := Eval lazy beta iota delta - [ DoubleBase.extend DoubleBase.extend_aux extend zeron ] - in same_level_folded. - - Lemma spec_same_level_0: forall x y, P [x] [y] (same_level x y). - Proof. - change same_level with same_level_folded. unfold_lets. - destruct x, y; apply Pf'; simpl mk_t; rewrite <- ?spec_extend_size; - match goal with - | |- context [ extend ?n ?m _ ] => apply (spec_extend n m) - | |- context [ castm _ _ ] => apply spec_cast_l || apply spec_cast_r - | _ => reflexivity - end. - Qed. - - End SameLevel. - - Arguments same_level [res] f x y. - - Theorem spec_same_level_dep : - forall res - (P : nat -> Z -> Z -> res -> Prop) - (Pantimon : forall n m z z' r, n <= m -> P m z z' r -> P n z z' r) - (f : forall n, dom_t n -> dom_t n -> res) - (Pf: forall n x y, P n (ZnZ.to_Z x) (ZnZ.to_Z y) (f n x y)), - forall x y, P (level x) [x] [y] (same_level f x y). - Proof. - intros res P Pantimon f Pf. - set (f' := fun n x y => (n, f n x y)). - set (P' := fun z z' r => P (fst r) z z' (snd r)). - assert (FST : forall x y, level x <= fst (same_level f' x y)) - by (destruct x, y; simpl; omega with * ). - assert (SND : forall x y, same_level f x y = snd (same_level f' x y)) - by (destruct x, y; reflexivity). - intros. eapply Pantimon; [eapply FST|]. - rewrite SND. eapply (@spec_same_level_0 _ P' f'); eauto. - Qed. - - (** * [iter] - - Generic binary operator construction, by splitting the larger - argument in blocks and applying the smaller argument to them. - *) - - Section Iter. - - Variable res: Type. - Variable P: Z -> Z -> res -> Prop. - - Variable f : forall n, dom_t n -> dom_t n -> res. - Variable Pf : forall n x y, P (ZnZ.to_Z x) (ZnZ.to_Z y) (f n x y). - - Variable fd : forall n m, dom_t n -> word (dom_t n) (S m) -> res. - Variable fg : forall n m, word (dom_t n) (S m) -> dom_t n -> res. - Variable Pfd : forall n m x y, P (ZnZ.to_Z x) (eval n (S m) y) (fd n m x y). - Variable Pfg : forall n m x y, P (eval n (S m) x) (ZnZ.to_Z y) (fg n m x y). - - Variable fnm: forall n m, word (dom_t Size) (S n) -> word (dom_t Size) (S m) -> res. - Variable Pfnm: forall n m x y, P [Nn n x] [Nn m y] (fnm n m x y). - - Let Pf' : - forall n x y u v, u = [mk_t n x] -> v = [mk_t n y] -> P u v (f n x y). - Proof. - intros. subst. rewrite 2 spec_mk_t. apply Pf. - Qed. - - Let Pfd' : forall n m x y u v, u = [mk_t n x] -> v = eval n (S m) y -> - P u v (fd n m x y). - Proof. - intros. subst. rewrite spec_mk_t. apply Pfd. - Qed. - - Let Pfg' : forall n m x y u v, u = eval n (S m) x -> v = [mk_t n y] -> - P u v (fg n m x y). - Proof. - intros. subst. rewrite spec_mk_t. apply Pfg. - Qed. -"; - -for i = 0 to size do -pr " Let f%i := f %i." i i -done; - -for i = 0 to size do -pr " Let f%in := fd %i." i i; -pr " Let fn%i := fg %i." i i; -done; - -pr " Notation iter_folded := (fun x y => match x, y with"; -for i = 0 to size do - for j = 0 to size do - pr " | N%i wx, N%i wy => f%s wx wy" i j - (if i = j then string_of_int i - else if i < j then string_of_int i ^ "n " ^ string_of_int (j-i-1) - else "n" ^ string_of_int j ^ " " ^ string_of_int (i-j-1)) - done; - pr " | N%i wx, Nn m wy => f%in m %s wy" i size (ext i size "wx") -done; -for i = 0 to size do - pr " | Nn n wx, N%i wy => fn%i n wx %s" i size (ext i size "wy") -done; -pr -" | Nn n wx, Nn m wy => fnm n m wx wy - end). -"; - -pr -" Definition iter := Eval lazy beta iota delta - [extend DoubleBase.extend DoubleBase.extend_aux zeron] - in iter_folded. - - Lemma spec_iter: forall x y, P [x] [y] (iter x y). - Proof. - change iter with iter_folded; unfold_lets. - destruct x; destruct y; apply Pf' || apply Pfd' || apply Pfg' || apply Pfnm; - simpl mk_t; - match goal with - | |- ?x = ?x => reflexivity - | |- [Nn _ _] = _ => apply spec_eval_size - | |- context [extend ?n ?m _] => apply (spec_extend n m) - | _ => idtac - end; - unfold to_Z; rewrite <- spec_plus_t'; simpl dom_op; reflexivity. - Qed. - - End Iter. -"; - -pr -" Definition switch - (P:nat->Type)%s - (fn:forall n, P n) n := - match n return P n with" - (iter_str_gen size (fun i -> Printf.sprintf "(f%i:P %i)" i i)); -for i = 0 to size do pr " | %i => f%i" i i done; -pr -" | n => fn n - end. -"; - -pr -" Lemma spec_switch : forall P (f:forall n, P n) n, - switch P %sf n = f n. - Proof. - repeat (destruct n; try reflexivity). - Qed. -" (iter_str_gen size (fun i -> Printf.sprintf "(f %i) " i)); - -pr -" (** * [iter_sym] - - A variant of [iter] for symmetric functions, or pseudo-symmetric - functions (when f y x can be deduced from f x y). - *) - - Section IterSym. - - Variable res: Type. - Variable P: Z -> Z -> res -> Prop. - - Variable f : forall n, dom_t n -> dom_t n -> res. - Variable Pf : forall n x y, P (ZnZ.to_Z x) (ZnZ.to_Z y) (f n x y). - - Variable fg : forall n m, word (dom_t n) (S m) -> dom_t n -> res. - Variable Pfg : forall n m x y, P (eval n (S m) x) (ZnZ.to_Z y) (fg n m x y). - - Variable fnm: forall n m, word (dom_t Size) (S n) -> word (dom_t Size) (S m) -> res. - Variable Pfnm: forall n m x y, P [Nn n x] [Nn m y] (fnm n m x y). - - Variable opp: res -> res. - Variable Popp : forall u v r, P u v r -> P v u (opp r). -"; - -for i = 0 to size do -pr " Let f%i := f %i." i i -done; - -for i = 0 to size do -pr " Let fn%i := fg %i." i i; -done; - -pr " Let f' := switch _ %s f." (iter_name 0 size "f" ""); -pr " Let fg' := switch _ %s fg." (iter_name 0 size "fn" ""); - -pr -" Local Notation iter_sym_folded := - (iter res f' (fun n m x y => opp (fg' n m y x)) fg' fnm). - - Definition iter_sym := - Eval lazy beta zeta iota delta [iter f' fg' switch] in iter_sym_folded. - - Lemma spec_iter_sym: forall x y, P [x] [y] (iter_sym x y). - Proof. - intros. change iter_sym with iter_sym_folded. apply spec_iter; clear x y. - unfold_lets. - intros. rewrite spec_switch. auto. - intros. apply Popp. unfold_lets. rewrite spec_switch; auto. - intros. unfold_lets. rewrite spec_switch; auto. - auto. - Qed. - - End IterSym. - - (** * Reduction - - [reduce] can be used instead of [mk_t], it will choose the - lowest possible level. NB: We only search and remove leftmost - W0's via ZnZ.eq0, any non-W0 block ends the process, even - if its value is 0. - *) - - (** First, a direct version ... *) - - Fixpoint red_t n : dom_t n -> t := - match n return dom_t n -> t with - | O => N0 - | S n => fun x => - let x' := pred_t n x in - reduce_n1 _ _ (N0 zero0) ZnZ.eq0 (red_t n) (mk_t_S n) x' - end. - - Lemma spec_red_t : forall n x, [red_t n x] = [mk_t n x]. - Proof. - induction n. - reflexivity. - intros. - simpl red_t. unfold reduce_n1. - rewrite <- (succ_pred_t n x) at 2. - remember (pred_t n x) as x'. - rewrite spec_mk_t, spec_succ_t. - destruct x' as [ | xh xl]. simpl. apply ZnZ.spec_0. - generalize (ZnZ.spec_eq0 xh); case ZnZ.eq0; intros H. - rewrite IHn, spec_mk_t. simpl. rewrite H; auto. - apply spec_mk_t_S. - Qed. - - (** ... then a specialized one *) -"; - -for i = 0 to size do -pr " Definition eq0%i := @ZnZ.eq0 _ w%i_op." i i; -done; - -pr " - Definition reduce_0 := N0."; -for i = 1 to size do - pr " Definition reduce_%i :=" i; - pr " Eval lazy beta iota delta [reduce_n1] in"; - pr " reduce_n1 _ _ (N0 zero0) eq0%i reduce_%i N%i." (i-1) (i-1) i -done; - - pr " Definition reduce_%i :=" (size+1); - pr " Eval lazy beta iota delta [reduce_n1] in"; - pr " reduce_n1 _ _ (N0 zero0) eq0%i reduce_%i (Nn 0)." size size; - - pr " Definition reduce_n n :="; - pr " Eval lazy beta iota delta [reduce_n] in"; - pr " reduce_n _ _ (N0 zero0) reduce_%i Nn n." (size + 1); - pr ""; - -pr " Definition reduce n : dom_t n -> t :="; -pr " match n with"; -for i = 0 to size do -pr " | %i => reduce_%i" i i; -done; -pr " | %s(S n) => reduce_n n" (if size=0 then "" else "SizePlus "); -pr " end."; -pr ""; - -pr " Ltac unfold_red := unfold reduce, %s." (iter_name 1 size "reduce_" ","); -pr ""; -for i = 0 to size do -pr " Declare Equivalent Keys reduce reduce_%i." i; -done; -pr " Declare Equivalent Keys reduce_n reduce_%i." (size + 1); - -pr " - Ltac solve_red := - let H := fresh in let G := fresh in - match goal with - | |- ?P (S ?n) => assert (H:P n) by solve_red - | _ => idtac - end; - intros n G x; destruct (le_lt_eq_dec _ _ G) as [LT|EQ]; - solve [ - apply (H _ (lt_n_Sm_le _ _ LT)) | - inversion LT | - subst; change (reduce 0 x = red_t 0 x); reflexivity | - specialize (H (pred n)); subst; destruct x; - [|unfold_red; rewrite H; auto]; reflexivity - ]. - - Lemma reduce_equiv : forall n x, n <= Size -> reduce n x = red_t n x. - Proof. - set (P N := forall n, n <= N -> forall x, reduce n x = red_t n x). - intros n x H. revert n H x. change (P Size). solve_red. - Qed. - - Lemma spec_reduce_n : forall n x, [reduce_n n x] = [Nn n x]. - Proof. - assert (H : forall x, reduce_%i x = red_t (SizePlus 1) x). - destruct x; [|unfold reduce_%i; rewrite (reduce_equiv Size)]; auto. - induction n. - intros. rewrite H. apply spec_red_t. - destruct x as [|xh xl]. - simpl. rewrite make_op_S. exact ZnZ.spec_0. - fold word in *. - destruct xh; auto. - simpl reduce_n. - rewrite IHn. - rewrite spec_extend_WW; auto. - Qed. -" (size+1) (size+1); - -pr -" Lemma spec_reduce : forall n x, [reduce n x] = ZnZ.to_Z x. - Proof. - do_size (destruct n; - [intros; rewrite reduce_equiv;[apply spec_red_t|auto with arith]|]). - apply spec_reduce_n. - Qed. - -End Make. -"; diff --git a/theories/Numbers/Natural/BigN/Nbasic.v b/theories/Numbers/Natural/BigN/Nbasic.v deleted file mode 100644 index 18d0262c..00000000 --- a/theories/Numbers/Natural/BigN/Nbasic.v +++ /dev/null @@ -1,569 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Require Import ZArith Ndigits. -Require Import BigNumPrelude. -Require Import Max. -Require Import DoubleType. -Require Import DoubleBase. -Require Import CyclicAxioms. -Require Import DoubleCyclic. - -Arguments mk_zn2z_ops [t] ops. -Arguments mk_zn2z_ops_karatsuba [t] ops. -Arguments mk_zn2z_specs [t ops] specs. -Arguments mk_zn2z_specs_karatsuba [t ops] specs. -Arguments ZnZ.digits [t] Ops. -Arguments ZnZ.zdigits [t] Ops. - -Lemma Pshiftl_nat_Zpower : forall n p, - Zpos (Pos.shiftl_nat p n) = Zpos p * 2 ^ Z.of_nat n. -Proof. - intros. - rewrite Z.mul_comm. - induction n. simpl; auto. - transitivity (2 * (2 ^ Z.of_nat n * Zpos p)). - rewrite <- IHn. auto. - rewrite Z.mul_assoc. - rewrite Nat2Z.inj_succ. - rewrite <- Z.pow_succ_r; auto with zarith. -Qed. - -(* To compute the necessary height *) - -Fixpoint plength (p: positive) : positive := - match p with - xH => xH - | xO p1 => Pos.succ (plength p1) - | xI p1 => Pos.succ (plength p1) - end. - -Theorem plength_correct: forall p, (Zpos p < 2 ^ Zpos (plength p))%Z. -assert (F: (forall p, 2 ^ (Zpos (Pos.succ p)) = 2 * 2 ^ Zpos p)%Z). -intros p; replace (Zpos (Pos.succ p)) with (1 + Zpos p)%Z. -rewrite Zpower_exp; auto with zarith. -rewrite Pos2Z.inj_succ; unfold Z.succ; auto with zarith. -intros p; elim p; simpl plength; auto. -intros p1 Hp1; rewrite F; repeat rewrite Pos2Z.inj_xI. -assert (tmp: (forall p, 2 * p = p + p)%Z); - try repeat rewrite tmp; auto with zarith. -intros p1 Hp1; rewrite F; rewrite (Pos2Z.inj_xO p1). -assert (tmp: (forall p, 2 * p = p + p)%Z); - try repeat rewrite tmp; auto with zarith. -rewrite Z.pow_1_r; auto with zarith. -Qed. - -Theorem plength_pred_correct: forall p, (Zpos p <= 2 ^ Zpos (plength (Pos.pred p)))%Z. -intros p; case (Pos.succ_pred_or p); intros H1. -subst; simpl plength. -rewrite Z.pow_1_r; auto with zarith. -pattern p at 1; rewrite <- H1. -rewrite Pos2Z.inj_succ; unfold Z.succ; auto with zarith. -generalize (plength_correct (Pos.pred p)); auto with zarith. -Qed. - -Definition Pdiv p q := - match Z.div (Zpos p) (Zpos q) with - Zpos q1 => match (Zpos p) - (Zpos q) * (Zpos q1) with - Z0 => q1 - | _ => (Pos.succ q1) - end - | _ => xH - end. - -Theorem Pdiv_le: forall p q, - Zpos p <= Zpos q * Zpos (Pdiv p q). -intros p q. -unfold Pdiv. -assert (H1: Zpos q > 0); auto with zarith. -assert (H1b: Zpos p >= 0); auto with zarith. -generalize (Z_div_ge0 (Zpos p) (Zpos q) H1 H1b). -generalize (Z_div_mod_eq (Zpos p) (Zpos q) H1); case Z.div. - intros HH _; rewrite HH; rewrite Z.mul_0_r; rewrite Z.mul_1_r; simpl. -case (Z_mod_lt (Zpos p) (Zpos q) H1); auto with zarith. -intros q1 H2. -replace (Zpos p - Zpos q * Zpos q1) with (Zpos p mod Zpos q). - 2: pattern (Zpos p) at 2; rewrite H2; auto with zarith. -generalize H2 (Z_mod_lt (Zpos p) (Zpos q) H1); clear H2; - case Z.modulo. - intros HH _; rewrite HH; auto with zarith. - intros r1 HH (_,HH1); rewrite HH; rewrite Pos2Z.inj_succ. - unfold Z.succ; rewrite Z.mul_add_distr_l; auto with zarith. - intros r1 _ (HH,_); case HH; auto. -intros q1 HH; rewrite HH. -unfold Z.ge; simpl Z.compare; intros HH1; case HH1; auto. -Qed. - -Definition is_one p := match p with xH => true | _ => false end. - -Theorem is_one_one: forall p, is_one p = true -> p = xH. -intros p; case p; auto; intros p1 H1; discriminate H1. -Qed. - -Definition get_height digits p := - let r := Pdiv p digits in - if is_one r then xH else Pos.succ (plength (Pos.pred r)). - -Theorem get_height_correct: - forall digits N, - Zpos N <= Zpos digits * (2 ^ (Zpos (get_height digits N) -1)). -intros digits N. -unfold get_height. -assert (H1 := Pdiv_le N digits). -case_eq (is_one (Pdiv N digits)); intros H2. -rewrite (is_one_one _ H2) in H1. -rewrite Z.mul_1_r in H1. -change (2^(1-1))%Z with 1; rewrite Z.mul_1_r; auto. -clear H2. -apply Z.le_trans with (1 := H1). -apply Z.mul_le_mono_nonneg_l; auto with zarith. -rewrite Pos2Z.inj_succ; unfold Z.succ. -rewrite Z.add_comm; rewrite Z.add_simpl_l. -apply plength_pred_correct. -Qed. - -Definition zn2z_word_comm : forall w n, zn2z (word w n) = word (zn2z w) n. - fix zn2z_word_comm 2. - intros w n; case n. - reflexivity. - intros n0;simpl. - case (zn2z_word_comm w n0). - reflexivity. -Defined. - -Fixpoint extend (n:nat) {struct n} : forall w:Type, zn2z w -> word w (S n) := - match n return forall w:Type, zn2z w -> word w (S n) with - | O => fun w x => x - | S m => - let aux := extend m in - fun w x => WW W0 (aux w x) - end. - -Section ExtendMax. - -Open Scope nat_scope. - -Fixpoint plusnS (n m: nat) {struct n} : (n + S m = S (n + m))%nat := - match n return (n + S m = S (n + m))%nat with - | 0 => eq_refl (S m) - | S n1 => - let v := S (S n1 + m) in - eq_ind_r (fun n => S n = v) (eq_refl v) (plusnS n1 m) - end. - -Fixpoint plusn0 n : n + 0 = n := - match n return (n + 0 = n) with - | 0 => eq_refl 0 - | S n1 => - let v := S n1 in - eq_ind_r (fun n : nat => S n = v) (eq_refl v) (plusn0 n1) - end. - - Fixpoint diff (m n: nat) {struct m}: nat * nat := - match m, n with - O, n => (O, n) - | m, O => (m, O) - | S m1, S n1 => diff m1 n1 - end. - -Fixpoint diff_l (m n : nat) {struct m} : fst (diff m n) + n = max m n := - match m return fst (diff m n) + n = max m n with - | 0 => - match n return (n = max 0 n) with - | 0 => eq_refl _ - | S n0 => eq_refl _ - end - | S m1 => - match n return (fst (diff (S m1) n) + n = max (S m1) n) - with - | 0 => plusn0 _ - | S n1 => - let v := fst (diff m1 n1) + n1 in - let v1 := fst (diff m1 n1) + S n1 in - eq_ind v (fun n => v1 = S n) - (eq_ind v1 (fun n => v1 = n) (eq_refl v1) (S v) (plusnS _ _)) - _ (diff_l _ _) - end - end. - -Fixpoint diff_r (m n: nat) {struct m}: snd (diff m n) + m = max m n := - match m return (snd (diff m n) + m = max m n) with - | 0 => - match n return (snd (diff 0 n) + 0 = max 0 n) with - | 0 => eq_refl _ - | S _ => plusn0 _ - end - | S m => - match n return (snd (diff (S m) n) + S m = max (S m) n) with - | 0 => eq_refl (snd (diff (S m) 0) + S m) - | S n1 => - let v := S (max m n1) in - eq_ind_r (fun n => n = v) - (eq_ind_r (fun n => S n = v) - (eq_refl v) (diff_r _ _)) (plusnS _ _) - end - end. - - Variable w: Type. - - Definition castm (m n: nat) (H: m = n) (x: word w (S m)): - (word w (S n)) := - match H in (_ = y) return (word w (S y)) with - | eq_refl => x - end. - -Variable m: nat. -Variable v: (word w (S m)). - -Fixpoint extend_tr (n : nat) {struct n}: (word w (S (n + m))) := - match n return (word w (S (n + m))) with - | O => v - | S n1 => WW W0 (extend_tr n1) - end. - -End ExtendMax. - -Arguments extend_tr [w m] v n. -Arguments castm [w m n] H x. - - - -Section Reduce. - - Variable w : Type. - Variable nT : Type. - Variable N0 : nT. - Variable eq0 : w -> bool. - Variable reduce_n : w -> nT. - Variable zn2z_to_Nt : zn2z w -> nT. - - Definition reduce_n1 (x:zn2z w) := - match x with - | W0 => N0 - | WW xh xl => - if eq0 xh then reduce_n xl - else zn2z_to_Nt x - end. - -End Reduce. - -Section ReduceRec. - - Variable w : Type. - Variable nT : Type. - Variable N0 : nT. - Variable reduce_1n : zn2z w -> nT. - Variable c : forall n, word w (S n) -> nT. - - Fixpoint reduce_n (n:nat) : word w (S n) -> nT := - match n return word w (S n) -> nT with - | O => reduce_1n - | S m => fun x => - match x with - | W0 => N0 - | WW xh xl => - match xh with - | W0 => @reduce_n m xl - | _ => @c (S m) x - end - end - end. - -End ReduceRec. - -Section CompareRec. - - Variable wm w : Type. - Variable w_0 : w. - Variable compare : w -> w -> comparison. - Variable compare0_m : wm -> comparison. - Variable compare_m : wm -> w -> comparison. - - Fixpoint compare0_mn (n:nat) : word wm n -> comparison := - match n return word wm n -> comparison with - | O => compare0_m - | S m => fun x => - match x with - | W0 => Eq - | WW xh xl => - match compare0_mn m xh with - | Eq => compare0_mn m xl - | r => Lt - end - end - end. - - Variable wm_base: positive. - Variable wm_to_Z: wm -> Z. - Variable w_to_Z: w -> Z. - Variable w_to_Z_0: w_to_Z w_0 = 0. - Variable spec_compare0_m: forall x, - compare0_m x = (w_to_Z w_0 ?= wm_to_Z x). - Variable wm_to_Z_pos: forall x, 0 <= wm_to_Z x < base wm_base. - - Let double_to_Z := double_to_Z wm_base wm_to_Z. - Let double_wB := double_wB wm_base. - - Lemma base_xO: forall n, base (xO n) = (base n)^2. - Proof. - intros n1; unfold base. - rewrite (Pos2Z.inj_xO n1); rewrite Z.mul_comm; rewrite Z.pow_mul_r; auto with zarith. - Qed. - - Let double_to_Z_pos: forall n x, 0 <= double_to_Z n x < double_wB n := - (spec_double_to_Z wm_base wm_to_Z wm_to_Z_pos). - - Declare Equivalent Keys compare0_mn compare0_m. - - Lemma spec_compare0_mn: forall n x, - compare0_mn n x = (0 ?= double_to_Z n x). - Proof. - intros n; elim n; clear n; auto. - intros x; rewrite spec_compare0_m; rewrite w_to_Z_0; auto. - intros n Hrec x; case x; unfold compare0_mn; fold compare0_mn; auto. - fold word in *. - intros xh xl. - rewrite 2 Hrec. - simpl double_to_Z. - set (wB := DoubleBase.double_wB wm_base n). - case Z.compare_spec; intros Cmp. - rewrite <- Cmp. reflexivity. - symmetry. apply Z.gt_lt, Z.lt_gt. (* ;-) *) - assert (0 < wB). - unfold wB, DoubleBase.double_wB, base; auto with zarith. - change 0 with (0 + 0); apply Z.add_lt_le_mono; auto with zarith. - apply Z.mul_pos_pos; auto with zarith. - case (double_to_Z_pos n xl); auto with zarith. - case (double_to_Z_pos n xh); intros; exfalso; omega. - Qed. - - Fixpoint compare_mn_1 (n:nat) : word wm n -> w -> comparison := - match n return word wm n -> w -> comparison with - | O => compare_m - | S m => fun x y => - match x with - | W0 => compare w_0 y - | WW xh xl => - match compare0_mn m xh with - | Eq => compare_mn_1 m xl y - | r => Gt - end - end - end. - - Variable spec_compare: forall x y, - compare x y = Z.compare (w_to_Z x) (w_to_Z y). - Variable spec_compare_m: forall x y, - compare_m x y = Z.compare (wm_to_Z x) (w_to_Z y). - Variable wm_base_lt: forall x, - 0 <= w_to_Z x < base (wm_base). - - Let double_wB_lt: forall n x, - 0 <= w_to_Z x < (double_wB n). - Proof. - intros n x; elim n; simpl; auto; clear n. - intros n (H0, H); split; auto. - apply Z.lt_le_trans with (1:= H). - unfold double_wB, DoubleBase.double_wB; simpl. - rewrite base_xO. - set (u := base (Pos.shiftl_nat wm_base n)). - assert (0 < u). - unfold u, base; auto with zarith. - replace (u^2) with (u * u); simpl; auto with zarith. - apply Z.le_trans with (1 * u); auto with zarith. - unfold Z.pow_pos; simpl; ring. - Qed. - - - Lemma spec_compare_mn_1: forall n x y, - compare_mn_1 n x y = Z.compare (double_to_Z n x) (w_to_Z y). - Proof. - intros n; elim n; simpl; auto; clear n. - intros n Hrec x; case x; clear x; auto. - intros y; rewrite spec_compare; rewrite w_to_Z_0. reflexivity. - intros xh xl y; simpl; - rewrite spec_compare0_mn, Hrec. case Z.compare_spec. - intros H1b. - rewrite <- H1b; rewrite Z.mul_0_l; rewrite Z.add_0_l; auto. - symmetry. apply Z.lt_gt. - case (double_wB_lt n y); intros _ H0. - apply Z.lt_le_trans with (1:= H0). - fold double_wB. - case (double_to_Z_pos n xl); intros H1 H2. - apply Z.le_trans with (double_to_Z n xh * double_wB n); auto with zarith. - apply Z.le_trans with (1 * double_wB n); auto with zarith. - case (double_to_Z_pos n xh); intros; exfalso; omega. - Qed. - -End CompareRec. - - -Section AddS. - - Variable w wm : Type. - Variable incr : wm -> carry wm. - Variable addr : w -> wm -> carry wm. - Variable injr : w -> zn2z wm. - - Variable w_0 u: w. - Fixpoint injs (n:nat): word w (S n) := - match n return (word w (S n)) with - O => WW w_0 u - | S n1 => (WW W0 (injs n1)) - end. - - Definition adds x y := - match y with - W0 => C0 (injr x) - | WW hy ly => match addr x ly with - C0 z => C0 (WW hy z) - | C1 z => match incr hy with - C0 z1 => C0 (WW z1 z) - | C1 z1 => C1 (WW z1 z) - end - end - end. - -End AddS. - - Fixpoint length_pos x := - match x with xH => O | xO x1 => S (length_pos x1) | xI x1 => S (length_pos x1) end. - - Theorem length_pos_lt: forall x y, - (length_pos x < length_pos y)%nat -> Zpos x < Zpos y. - Proof. - intros x; elim x; clear x; [intros x1 Hrec | intros x1 Hrec | idtac]; - intros y; case y; clear y; intros y1 H || intros H; simpl length_pos; - try (rewrite (Pos2Z.inj_xI x1) || rewrite (Pos2Z.inj_xO x1)); - try (rewrite (Pos2Z.inj_xI y1) || rewrite (Pos2Z.inj_xO y1)); - try (inversion H; fail); - try (assert (Zpos x1 < Zpos y1); [apply Hrec; apply lt_S_n | idtac]; auto with zarith); - assert (0 < Zpos y1); auto with zarith; red; auto. - Qed. - - Theorem cancel_app: forall A B (f g: A -> B) x, f = g -> f x = g x. - Proof. - intros A B f g x H; rewrite H; auto. - Qed. - - - Section SimplOp. - - Variable w: Type. - - Theorem digits_zop: forall t (ops : ZnZ.Ops t), - ZnZ.digits (mk_zn2z_ops ops) = xO (ZnZ.digits ops). - Proof. - intros ww x; auto. - Qed. - - Theorem digits_kzop: forall t (ops : ZnZ.Ops t), - ZnZ.digits (mk_zn2z_ops_karatsuba ops) = xO (ZnZ.digits ops). - Proof. - intros ww x; auto. - Qed. - - Theorem make_zop: forall t (ops : ZnZ.Ops t), - @ZnZ.to_Z _ (mk_zn2z_ops ops) = - fun z => match z with - | W0 => 0 - | WW xh xl => ZnZ.to_Z xh * base (ZnZ.digits ops) - + ZnZ.to_Z xl - end. - Proof. - intros ww x; auto. - Qed. - - Theorem make_kzop: forall t (ops: ZnZ.Ops t), - @ZnZ.to_Z _ (mk_zn2z_ops_karatsuba ops) = - fun z => match z with - | W0 => 0 - | WW xh xl => ZnZ.to_Z xh * base (ZnZ.digits ops) - + ZnZ.to_Z xl - end. - Proof. - intros ww x; auto. - Qed. - - End SimplOp. - -(** Abstract vision of a datatype of arbitrary-large numbers. - Concrete operations can be derived from these generic - fonctions, in particular from [iter_t] and [same_level]. -*) - -Module Type NAbstract. - -(** The domains: a sequence of [Z/nZ] structures *) - -Parameter dom_t : nat -> Type. -Declare Instance dom_op n : ZnZ.Ops (dom_t n). -Declare Instance dom_spec n : ZnZ.Specs (dom_op n). - -Axiom digits_dom_op : forall n, - ZnZ.digits (dom_op n) = Pos.shiftl_nat (ZnZ.digits (dom_op 0)) n. - -(** The type [t] of arbitrary-large numbers, with abstract constructor [mk_t] - and destructor [destr_t] and iterator [iter_t] *) - -Parameter t : Type. - -Parameter mk_t : forall (n:nat), dom_t n -> t. - -Inductive View_t : t -> Prop := - Mk_t : forall n (x : dom_t n), View_t (mk_t n x). - -Axiom destr_t : forall x, View_t x. (* i.e. every x is a (mk_t n xw) *) - -Parameter iter_t : forall {A:Type}(f : forall n, dom_t n -> A), t -> A. - -Axiom iter_mk_t : forall A (f:forall n, dom_t n -> A), - forall n x, iter_t f (mk_t n x) = f n x. - -(** Conversion to [ZArith] *) - -Parameter to_Z : t -> Z. -Local Notation "[ x ]" := (to_Z x). - -Axiom spec_mk_t : forall n x, [mk_t n x] = ZnZ.to_Z x. - -(** [reduce] is like [mk_t], but try to minimise the level of the number *) - -Parameter reduce : forall (n:nat), dom_t n -> t. -Axiom spec_reduce : forall n x, [reduce n x] = ZnZ.to_Z x. - -(** Number of level in the tree representation of a number. - NB: This function isn't a morphism for setoid [eq]. *) - -Definition level := iter_t (fun n _ => n). - -(** [same_level] and its rich specification, indexed by [level] *) - -Parameter same_level : forall {A:Type} - (f : forall n, dom_t n -> dom_t n -> A), t -> t -> A. - -Axiom spec_same_level_dep : - forall res - (P : nat -> Z -> Z -> res -> Prop) - (Pantimon : forall n m z z' r, (n <= m)%nat -> P m z z' r -> P n z z' r) - (f : forall n, dom_t n -> dom_t n -> res) - (Pf: forall n x y, P n (ZnZ.to_Z x) (ZnZ.to_Z y) (f n x y)), - forall x y, P (level x) [x] [y] (same_level f x y). - -(** [mk_t_S] : building a number of the next level *) - -Parameter mk_t_S : forall (n:nat), zn2z (dom_t n) -> t. - -Axiom spec_mk_t_S : forall n (x:zn2z (dom_t n)), - [mk_t_S n x] = zn2z_to_Z (base (ZnZ.digits (dom_op n))) ZnZ.to_Z x. - -Axiom mk_t_S_level : forall n x, level (mk_t_S n x) = S n. - -End NAbstract. diff --git a/theories/Numbers/Natural/Binary/NBinary.v b/theories/Numbers/Natural/Binary/NBinary.v index 037b10d9..c9e1c640 100644 --- a/theories/Numbers/Natural/Binary/NBinary.v +++ b/theories/Numbers/Natural/Binary/NBinary.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Natural/Peano/NPeano.v b/theories/Numbers/Natural/Peano/NPeano.v index d0168bd9..6000bdcf 100644 --- a/theories/Numbers/Natural/Peano/NPeano.v +++ b/theories/Numbers/Natural/Peano/NPeano.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) @@ -18,74 +20,74 @@ Module Nat <: NAxiomsSig := Nat. (** Compat notations for stuff that used to be at the beginning of NPeano. *) -Notation leb := Nat.leb (compat "8.4"). -Notation ltb := Nat.ltb (compat "8.4"). -Notation leb_le := Nat.leb_le (compat "8.4"). -Notation ltb_lt := Nat.ltb_lt (compat "8.4"). -Notation pow := Nat.pow (compat "8.4"). -Notation pow_0_r := Nat.pow_0_r (compat "8.4"). -Notation pow_succ_r := Nat.pow_succ_r (compat "8.4"). -Notation square := Nat.square (compat "8.4"). -Notation square_spec := Nat.square_spec (compat "8.4"). -Notation Even := Nat.Even (compat "8.4"). -Notation Odd := Nat.Odd (compat "8.4"). -Notation even := Nat.even (compat "8.4"). -Notation odd := Nat.odd (compat "8.4"). -Notation even_spec := Nat.even_spec (compat "8.4"). -Notation odd_spec := Nat.odd_spec (compat "8.4"). +Notation leb := Nat.leb (only parsing). +Notation ltb := Nat.ltb (only parsing). +Notation leb_le := Nat.leb_le (only parsing). +Notation ltb_lt := Nat.ltb_lt (only parsing). +Notation pow := Nat.pow (only parsing). +Notation pow_0_r := Nat.pow_0_r (only parsing). +Notation pow_succ_r := Nat.pow_succ_r (only parsing). +Notation square := Nat.square (only parsing). +Notation square_spec := Nat.square_spec (only parsing). +Notation Even := Nat.Even (only parsing). +Notation Odd := Nat.Odd (only parsing). +Notation even := Nat.even (only parsing). +Notation odd := Nat.odd (only parsing). +Notation even_spec := Nat.even_spec (only parsing). +Notation odd_spec := Nat.odd_spec (only parsing). Lemma Even_equiv n : Even n <-> Even.even n. Proof. symmetry. apply Even.even_equiv. Qed. Lemma Odd_equiv n : Odd n <-> Even.odd n. Proof. symmetry. apply Even.odd_equiv. Qed. -Notation divmod := Nat.divmod (compat "8.4"). -Notation div := Nat.div (compat "8.4"). -Notation modulo := Nat.modulo (compat "8.4"). -Notation divmod_spec := Nat.divmod_spec (compat "8.4"). -Notation div_mod := Nat.div_mod (compat "8.4"). -Notation mod_bound_pos := Nat.mod_bound_pos (compat "8.4"). -Notation sqrt_iter := Nat.sqrt_iter (compat "8.4"). -Notation sqrt := Nat.sqrt (compat "8.4"). -Notation sqrt_iter_spec := Nat.sqrt_iter_spec (compat "8.4"). -Notation sqrt_spec := Nat.sqrt_spec (compat "8.4"). -Notation log2_iter := Nat.log2_iter (compat "8.4"). -Notation log2 := Nat.log2 (compat "8.4"). -Notation log2_iter_spec := Nat.log2_iter_spec (compat "8.4"). -Notation log2_spec := Nat.log2_spec (compat "8.4"). -Notation log2_nonpos := Nat.log2_nonpos (compat "8.4"). -Notation gcd := Nat.gcd (compat "8.4"). -Notation divide := Nat.divide (compat "8.4"). -Notation gcd_divide := Nat.gcd_divide (compat "8.4"). -Notation gcd_divide_l := Nat.gcd_divide_l (compat "8.4"). -Notation gcd_divide_r := Nat.gcd_divide_r (compat "8.4"). -Notation gcd_greatest := Nat.gcd_greatest (compat "8.4"). -Notation testbit := Nat.testbit (compat "8.4"). -Notation shiftl := Nat.shiftl (compat "8.4"). -Notation shiftr := Nat.shiftr (compat "8.4"). -Notation bitwise := Nat.bitwise (compat "8.4"). -Notation land := Nat.land (compat "8.4"). -Notation lor := Nat.lor (compat "8.4"). -Notation ldiff := Nat.ldiff (compat "8.4"). -Notation lxor := Nat.lxor (compat "8.4"). -Notation double_twice := Nat.double_twice (compat "8.4"). -Notation testbit_0_l := Nat.testbit_0_l (compat "8.4"). -Notation testbit_odd_0 := Nat.testbit_odd_0 (compat "8.4"). -Notation testbit_even_0 := Nat.testbit_even_0 (compat "8.4"). -Notation testbit_odd_succ := Nat.testbit_odd_succ (compat "8.4"). -Notation testbit_even_succ := Nat.testbit_even_succ (compat "8.4"). -Notation shiftr_spec := Nat.shiftr_spec (compat "8.4"). -Notation shiftl_spec_high := Nat.shiftl_spec_high (compat "8.4"). -Notation shiftl_spec_low := Nat.shiftl_spec_low (compat "8.4"). -Notation div2_bitwise := Nat.div2_bitwise (compat "8.4"). -Notation odd_bitwise := Nat.odd_bitwise (compat "8.4"). -Notation div2_decr := Nat.div2_decr (compat "8.4"). -Notation testbit_bitwise_1 := Nat.testbit_bitwise_1 (compat "8.4"). -Notation testbit_bitwise_2 := Nat.testbit_bitwise_2 (compat "8.4"). -Notation land_spec := Nat.land_spec (compat "8.4"). -Notation ldiff_spec := Nat.ldiff_spec (compat "8.4"). -Notation lor_spec := Nat.lor_spec (compat "8.4"). -Notation lxor_spec := Nat.lxor_spec (compat "8.4"). +Notation divmod := Nat.divmod (only parsing). +Notation div := Nat.div (only parsing). +Notation modulo := Nat.modulo (only parsing). +Notation divmod_spec := Nat.divmod_spec (only parsing). +Notation div_mod := Nat.div_mod (only parsing). +Notation mod_bound_pos := Nat.mod_bound_pos (only parsing). +Notation sqrt_iter := Nat.sqrt_iter (only parsing). +Notation sqrt := Nat.sqrt (only parsing). +Notation sqrt_iter_spec := Nat.sqrt_iter_spec (only parsing). +Notation sqrt_spec := Nat.sqrt_spec (only parsing). +Notation log2_iter := Nat.log2_iter (only parsing). +Notation log2 := Nat.log2 (only parsing). +Notation log2_iter_spec := Nat.log2_iter_spec (only parsing). +Notation log2_spec := Nat.log2_spec (only parsing). +Notation log2_nonpos := Nat.log2_nonpos (only parsing). +Notation gcd := Nat.gcd (only parsing). +Notation divide := Nat.divide (only parsing). +Notation gcd_divide := Nat.gcd_divide (only parsing). +Notation gcd_divide_l := Nat.gcd_divide_l (only parsing). +Notation gcd_divide_r := Nat.gcd_divide_r (only parsing). +Notation gcd_greatest := Nat.gcd_greatest (only parsing). +Notation testbit := Nat.testbit (only parsing). +Notation shiftl := Nat.shiftl (only parsing). +Notation shiftr := Nat.shiftr (only parsing). +Notation bitwise := Nat.bitwise (only parsing). +Notation land := Nat.land (only parsing). +Notation lor := Nat.lor (only parsing). +Notation ldiff := Nat.ldiff (only parsing). +Notation lxor := Nat.lxor (only parsing). +Notation double_twice := Nat.double_twice (only parsing). +Notation testbit_0_l := Nat.testbit_0_l (only parsing). +Notation testbit_odd_0 := Nat.testbit_odd_0 (only parsing). +Notation testbit_even_0 := Nat.testbit_even_0 (only parsing). +Notation testbit_odd_succ := Nat.testbit_odd_succ (only parsing). +Notation testbit_even_succ := Nat.testbit_even_succ (only parsing). +Notation shiftr_spec := Nat.shiftr_spec (only parsing). +Notation shiftl_spec_high := Nat.shiftl_spec_high (only parsing). +Notation shiftl_spec_low := Nat.shiftl_spec_low (only parsing). +Notation div2_bitwise := Nat.div2_bitwise (only parsing). +Notation odd_bitwise := Nat.odd_bitwise (only parsing). +Notation div2_decr := Nat.div2_decr (only parsing). +Notation testbit_bitwise_1 := Nat.testbit_bitwise_1 (only parsing). +Notation testbit_bitwise_2 := Nat.testbit_bitwise_2 (only parsing). +Notation land_spec := Nat.land_spec (only parsing). +Notation ldiff_spec := Nat.ldiff_spec (only parsing). +Notation lor_spec := Nat.lor_spec (only parsing). +Notation lxor_spec := Nat.lxor_spec (only parsing). Infix "<=?" := Nat.leb (at level 70) : nat_scope. Infix "<?" := Nat.ltb (at level 70) : nat_scope. diff --git a/theories/Numbers/Natural/SpecViaZ/NSig.v b/theories/Numbers/Natural/SpecViaZ/NSig.v deleted file mode 100644 index 258e0315..00000000 --- a/theories/Numbers/Natural/SpecViaZ/NSig.v +++ /dev/null @@ -1,124 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Require Import BinInt. - -Open Scope Z_scope. - -(** * NSig *) - -(** Interface of a rich structure about natural numbers. - Specifications are written via translation to Z. -*) - -Module Type NType. - - Parameter t : Type. - - Parameter to_Z : t -> Z. - Local Notation "[ x ]" := (to_Z x). - Parameter spec_pos: forall x, 0 <= [x]. - - Parameter of_N : N -> t. - Parameter spec_of_N: forall x, to_Z (of_N x) = Z.of_N x. - Definition to_N n := Z.to_N (to_Z n). - - Definition eq n m := [n] = [m]. - Definition lt n m := [n] < [m]. - Definition le n m := [n] <= [m]. - - Parameter compare : t -> t -> comparison. - Parameter eqb : t -> t -> bool. - Parameter ltb : t -> t -> bool. - Parameter leb : t -> t -> bool. - Parameter max : t -> t -> t. - Parameter min : t -> t -> t. - Parameter zero : t. - Parameter one : t. - Parameter two : t. - Parameter succ : t -> t. - Parameter pred : t -> t. - Parameter add : t -> t -> t. - Parameter sub : t -> t -> t. - Parameter mul : t -> t -> t. - Parameter square : t -> t. - Parameter pow_pos : t -> positive -> t. - Parameter pow_N : t -> N -> t. - Parameter pow : t -> t -> t. - Parameter sqrt : t -> t. - Parameter log2 : t -> t. - Parameter div_eucl : t -> t -> t * t. - Parameter div : t -> t -> t. - Parameter modulo : t -> t -> t. - Parameter gcd : t -> t -> t. - Parameter even : t -> bool. - Parameter odd : t -> bool. - Parameter testbit : t -> t -> bool. - Parameter shiftr : t -> t -> t. - Parameter shiftl : t -> t -> t. - Parameter land : t -> t -> t. - Parameter lor : t -> t -> t. - Parameter ldiff : t -> t -> t. - Parameter lxor : t -> t -> t. - Parameter div2 : t -> t. - - Parameter spec_compare: forall x y, compare x y = ([x] ?= [y]). - Parameter spec_eqb : forall x y, eqb x y = ([x] =? [y]). - Parameter spec_ltb : forall x y, ltb x y = ([x] <? [y]). - Parameter spec_leb : forall x y, leb x y = ([x] <=? [y]). - Parameter spec_max : forall x y, [max x y] = Z.max [x] [y]. - Parameter spec_min : forall x y, [min x y] = Z.min [x] [y]. - Parameter spec_0: [zero] = 0. - Parameter spec_1: [one] = 1. - Parameter spec_2: [two] = 2. - Parameter spec_succ: forall n, [succ n] = [n] + 1. - Parameter spec_add: forall x y, [add x y] = [x] + [y]. - Parameter spec_pred: forall x, [pred x] = Z.max 0 ([x] - 1). - Parameter spec_sub: forall x y, [sub x y] = Z.max 0 ([x] - [y]). - Parameter spec_mul: forall x y, [mul x y] = [x] * [y]. - Parameter spec_square: forall x, [square x] = [x] * [x]. - Parameter spec_pow_pos: forall x n, [pow_pos x n] = [x] ^ Zpos n. - Parameter spec_pow_N: forall x n, [pow_N x n] = [x] ^ Z.of_N n. - Parameter spec_pow: forall x n, [pow x n] = [x] ^ [n]. - Parameter spec_sqrt: forall x, [sqrt x] = Z.sqrt [x]. - Parameter spec_log2: forall x, [log2 x] = Z.log2 [x]. - Parameter spec_div_eucl: forall x y, - let (q,r) := div_eucl x y in ([q], [r]) = Z.div_eucl [x] [y]. - Parameter spec_div: forall x y, [div x y] = [x] / [y]. - Parameter spec_modulo: forall x y, [modulo x y] = [x] mod [y]. - Parameter spec_gcd: forall a b, [gcd a b] = Z.gcd [a] [b]. - Parameter spec_even: forall x, even x = Z.even [x]. - Parameter spec_odd: forall x, odd x = Z.odd [x]. - Parameter spec_testbit: forall x p, testbit x p = Z.testbit [x] [p]. - Parameter spec_shiftr: forall x p, [shiftr x p] = Z.shiftr [x] [p]. - Parameter spec_shiftl: forall x p, [shiftl x p] = Z.shiftl [x] [p]. - Parameter spec_land: forall x y, [land x y] = Z.land [x] [y]. - Parameter spec_lor: forall x y, [lor x y] = Z.lor [x] [y]. - Parameter spec_ldiff: forall x y, [ldiff x y] = Z.ldiff [x] [y]. - Parameter spec_lxor: forall x y, [lxor x y] = Z.lxor [x] [y]. - Parameter spec_div2: forall x, [div2 x] = Z.div2 [x]. - -End NType. - -Module Type NType_Notation (Import N:NType). - Notation "[ x ]" := (to_Z x). - Infix "==" := eq (at level 70). - Notation "0" := zero. - Notation "1" := one. - Notation "2" := two. - Infix "+" := add. - Infix "-" := sub. - Infix "*" := mul. - Infix "^" := pow. - Infix "<=" := le. - Infix "<" := lt. -End NType_Notation. - -Module Type NType' := NType <+ NType_Notation. diff --git a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v deleted file mode 100644 index 355da4cc..00000000 --- a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v +++ /dev/null @@ -1,487 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -Require Import ZArith OrdersFacts Nnat NAxioms NSig. - -(** * The interface [NSig.NType] implies the interface [NAxiomsSig] *) - -Module NTypeIsNAxioms (Import NN : NType'). - -Hint Rewrite - spec_0 spec_1 spec_2 spec_succ spec_add spec_mul spec_pred spec_sub - spec_div spec_modulo spec_gcd spec_compare spec_eqb spec_ltb spec_leb - spec_square spec_sqrt spec_log2 spec_max spec_min spec_pow_pos spec_pow_N - spec_pow spec_even spec_odd spec_testbit spec_shiftl spec_shiftr - spec_land spec_lor spec_ldiff spec_lxor spec_div2 spec_of_N - : nsimpl. -Ltac nsimpl := autorewrite with nsimpl. -Ltac ncongruence := unfold eq, to_N; repeat red; intros; nsimpl; congruence. -Ltac zify := unfold eq, lt, le, to_N in *; nsimpl. -Ltac omega_pos n := generalize (spec_pos n); omega with *. - -Local Obligation Tactic := ncongruence. - -Instance eq_equiv : Equivalence eq. -Proof. unfold eq. firstorder. Qed. - -Program Instance succ_wd : Proper (eq==>eq) succ. -Program Instance pred_wd : Proper (eq==>eq) pred. -Program Instance add_wd : Proper (eq==>eq==>eq) add. -Program Instance sub_wd : Proper (eq==>eq==>eq) sub. -Program Instance mul_wd : Proper (eq==>eq==>eq) mul. - -Theorem pred_succ : forall n, pred (succ n) == n. -Proof. -intros. zify. omega_pos n. -Qed. - -Theorem one_succ : 1 == succ 0. -Proof. -now zify. -Qed. - -Theorem two_succ : 2 == succ 1. -Proof. -now zify. -Qed. - -Definition N_of_Z z := of_N (Z.to_N z). - -Lemma spec_N_of_Z z : (0<=z)%Z -> [N_of_Z z] = z. -Proof. - unfold N_of_Z. zify. apply Z2N.id. -Qed. - -Section Induction. - -Variable A : NN.t -> Prop. -Hypothesis A_wd : Proper (eq==>iff) A. -Hypothesis A0 : A 0. -Hypothesis AS : forall n, A n <-> A (succ n). - -Let B (z : Z) := A (N_of_Z z). - -Lemma B0 : B 0. -Proof. -unfold B, N_of_Z; simpl. -rewrite <- (A_wd 0); auto. -red; rewrite spec_0, spec_of_N; auto. -Qed. - -Lemma BS : forall z : Z, (0 <= z)%Z -> B z -> B (z + 1). -Proof. -intros z H1 H2. -unfold B in *. apply -> AS in H2. -setoid_replace (N_of_Z (z + 1)) with (succ (N_of_Z z)); auto. -unfold eq. rewrite spec_succ, 2 spec_N_of_Z; auto with zarith. -Qed. - -Lemma B_holds : forall z : Z, (0 <= z)%Z -> B z. -Proof. -exact (natlike_ind B B0 BS). -Qed. - -Theorem bi_induction : forall n, A n. -Proof. -intro n. setoid_replace n with (N_of_Z (to_Z n)). -apply B_holds. apply spec_pos. -red. now rewrite spec_N_of_Z by apply spec_pos. -Qed. - -End Induction. - -Theorem add_0_l : forall n, 0 + n == n. -Proof. -intros. zify. auto with zarith. -Qed. - -Theorem add_succ_l : forall n m, (succ n) + m == succ (n + m). -Proof. -intros. zify. auto with zarith. -Qed. - -Theorem sub_0_r : forall n, n - 0 == n. -Proof. -intros. zify. omega_pos n. -Qed. - -Theorem sub_succ_r : forall n m, n - (succ m) == pred (n - m). -Proof. -intros. zify. omega with *. -Qed. - -Theorem mul_0_l : forall n, 0 * n == 0. -Proof. -intros. zify. auto with zarith. -Qed. - -Theorem mul_succ_l : forall n m, (succ n) * m == n * m + m. -Proof. -intros. zify. ring. -Qed. - -(** Order *) - -Lemma eqb_eq x y : eqb x y = true <-> x == y. -Proof. - zify. apply Z.eqb_eq. -Qed. - -Lemma leb_le x y : leb x y = true <-> x <= y. -Proof. - zify. apply Z.leb_le. -Qed. - -Lemma ltb_lt x y : ltb x y = true <-> x < y. -Proof. - zify. apply Z.ltb_lt. -Qed. - -Lemma compare_eq_iff n m : compare n m = Eq <-> n == m. -Proof. - intros. zify. apply Z.compare_eq_iff. -Qed. - -Lemma compare_lt_iff n m : compare n m = Lt <-> n < m. -Proof. - intros. zify. reflexivity. -Qed. - -Lemma compare_le_iff n m : compare n m <> Gt <-> n <= m. -Proof. - intros. zify. reflexivity. -Qed. - -Lemma compare_antisym n m : compare m n = CompOpp (compare n m). -Proof. - intros. zify. apply Z.compare_antisym. -Qed. - -Include BoolOrderFacts NN NN NN [no inline]. - -Instance compare_wd : Proper (eq ==> eq ==> Logic.eq) compare. -Proof. -intros x x' Hx y y' Hy. zify. now rewrite Hx, Hy. -Qed. - -Instance eqb_wd : Proper (eq ==> eq ==> Logic.eq) eqb. -Proof. -intros x x' Hx y y' Hy. zify. now rewrite Hx, Hy. -Qed. - -Instance ltb_wd : Proper (eq ==> eq ==> Logic.eq) ltb. -Proof. -intros x x' Hx y y' Hy. zify. now rewrite Hx, Hy. -Qed. - -Instance leb_wd : Proper (eq ==> eq ==> Logic.eq) leb. -Proof. -intros x x' Hx y y' Hy. zify. now rewrite Hx, Hy. -Qed. - -Instance lt_wd : Proper (eq ==> eq ==> iff) lt. -Proof. -intros x x' Hx y y' Hy; unfold lt; rewrite Hx, Hy; intuition. -Qed. - -Theorem lt_succ_r : forall n m, n < succ m <-> n <= m. -Proof. -intros. zify. omega. -Qed. - -Theorem min_l : forall n m, n <= m -> min n m == n. -Proof. -intros n m. zify. omega with *. -Qed. - -Theorem min_r : forall n m, m <= n -> min n m == m. -Proof. -intros n m. zify. omega with *. -Qed. - -Theorem max_l : forall n m, m <= n -> max n m == n. -Proof. -intros n m. zify. omega with *. -Qed. - -Theorem max_r : forall n m, n <= m -> max n m == m. -Proof. -intros n m. zify. omega with *. -Qed. - -(** Properties specific to natural numbers, not integers. *) - -Theorem pred_0 : pred 0 == 0. -Proof. -zify. auto. -Qed. - -(** Power *) - -Program Instance pow_wd : Proper (eq==>eq==>eq) pow. - -Lemma pow_0_r : forall a, a^0 == 1. -Proof. - intros. now zify. -Qed. - -Lemma pow_succ_r : forall a b, 0<=b -> a^(succ b) == a * a^b. -Proof. - intros a b. zify. intros. now Z.nzsimpl. -Qed. - -Lemma pow_neg_r : forall a b, b<0 -> a^b == 0. -Proof. - intros a b. zify. intro Hb. exfalso. omega_pos b. -Qed. - -Lemma pow_pow_N : forall a b, a^b == pow_N a (to_N b). -Proof. - intros. zify. f_equal. - now rewrite Z2N.id by apply spec_pos. -Qed. - -Lemma pow_N_pow : forall a b, pow_N a b == a^(of_N b). -Proof. - intros. now zify. -Qed. - -Lemma pow_pos_N : forall a p, pow_pos a p == pow_N a (Npos p). -Proof. - intros. now zify. -Qed. - -(** Square *) - -Lemma square_spec n : square n == n * n. -Proof. - now zify. -Qed. - -(** Sqrt *) - -Lemma sqrt_spec : forall n, 0<=n -> - (sqrt n)*(sqrt n) <= n /\ n < (succ (sqrt n))*(succ (sqrt n)). -Proof. - intros n. zify. apply Z.sqrt_spec. -Qed. - -Lemma sqrt_neg : forall n, n<0 -> sqrt n == 0. -Proof. - intros n. zify. intro H. exfalso. omega_pos n. -Qed. - -(** Log2 *) - -Lemma log2_spec : forall n, 0<n -> - 2^(log2 n) <= n /\ n < 2^(succ (log2 n)). -Proof. - intros n. zify. change (Z.log2 [n]+1)%Z with (Z.succ (Z.log2 [n])). - apply Z.log2_spec. -Qed. - -Lemma log2_nonpos : forall n, n<=0 -> log2 n == 0. -Proof. - intros n. zify. apply Z.log2_nonpos. -Qed. - -(** Even / Odd *) - -Definition Even n := exists m, n == 2*m. -Definition Odd n := exists m, n == 2*m+1. - -Lemma even_spec n : even n = true <-> Even n. -Proof. - unfold Even. zify. rewrite Z.even_spec. - split; intros (m,Hm). - - exists (N_of_Z m). zify. rewrite spec_N_of_Z; trivial. omega_pos n. - - exists [m]. revert Hm; now zify. -Qed. - -Lemma odd_spec n : odd n = true <-> Odd n. -Proof. - unfold Odd. zify. rewrite Z.odd_spec. - split; intros (m,Hm). - - exists (N_of_Z m). zify. rewrite spec_N_of_Z; trivial. omega_pos n. - - exists [m]. revert Hm; now zify. -Qed. - -(** Div / Mod *) - -Program Instance div_wd : Proper (eq==>eq==>eq) div. -Program Instance mod_wd : Proper (eq==>eq==>eq) modulo. - -Theorem div_mod : forall a b, ~b==0 -> a == b*(div a b) + (modulo a b). -Proof. -intros a b. zify. intros. apply Z.div_mod; auto. -Qed. - -Theorem mod_bound_pos : forall a b, 0<=a -> 0<b -> - 0 <= modulo a b /\ modulo a b < b. -Proof. -intros a b. zify. apply Z.mod_bound_pos. -Qed. - -(** Gcd *) - -Definition divide n m := exists p, m == p*n. -Local Notation "( x | y )" := (divide x y) (at level 0). - -Lemma spec_divide : forall n m, (n|m) <-> Z.divide [n] [m]. -Proof. - intros n m. split. - - intros (p,H). exists [p]. revert H; now zify. - - intros (z,H). exists (of_N (Z.abs_N z)). zify. - rewrite N2Z.inj_abs_N. - rewrite <- (Z.abs_eq [m]), <- (Z.abs_eq [n]) by apply spec_pos. - now rewrite H, Z.abs_mul. -Qed. - -Lemma gcd_divide_l : forall n m, (gcd n m | n). -Proof. - intros n m. apply spec_divide. zify. apply Z.gcd_divide_l. -Qed. - -Lemma gcd_divide_r : forall n m, (gcd n m | m). -Proof. - intros n m. apply spec_divide. zify. apply Z.gcd_divide_r. -Qed. - -Lemma gcd_greatest : forall n m p, (p|n) -> (p|m) -> (p|gcd n m). -Proof. - intros n m p. rewrite !spec_divide. zify. apply Z.gcd_greatest. -Qed. - -Lemma gcd_nonneg : forall n m, 0 <= gcd n m. -Proof. - intros. zify. apply Z.gcd_nonneg. -Qed. - -(** Bitwise operations *) - -Program Instance testbit_wd : Proper (eq==>eq==>Logic.eq) testbit. - -Lemma testbit_odd_0 : forall a, testbit (2*a+1) 0 = true. -Proof. - intros. zify. apply Z.testbit_odd_0. -Qed. - -Lemma testbit_even_0 : forall a, testbit (2*a) 0 = false. -Proof. - intros. zify. apply Z.testbit_even_0. -Qed. - -Lemma testbit_odd_succ : forall a n, 0<=n -> - testbit (2*a+1) (succ n) = testbit a n. -Proof. - intros a n. zify. apply Z.testbit_odd_succ. -Qed. - -Lemma testbit_even_succ : forall a n, 0<=n -> - testbit (2*a) (succ n) = testbit a n. -Proof. - intros a n. zify. apply Z.testbit_even_succ. -Qed. - -Lemma testbit_neg_r : forall a n, n<0 -> testbit a n = false. -Proof. - intros a n. zify. apply Z.testbit_neg_r. -Qed. - -Lemma shiftr_spec : forall a n m, 0<=m -> - testbit (shiftr a n) m = testbit a (m+n). -Proof. - intros a n m. zify. apply Z.shiftr_spec. -Qed. - -Lemma shiftl_spec_high : forall a n m, 0<=m -> n<=m -> - testbit (shiftl a n) m = testbit a (m-n). -Proof. - intros a n m. zify. intros Hn H. rewrite Z.max_r by auto with zarith. - now apply Z.shiftl_spec_high. -Qed. - -Lemma shiftl_spec_low : forall a n m, m<n -> - testbit (shiftl a n) m = false. -Proof. - intros a n m. zify. intros H. now apply Z.shiftl_spec_low. -Qed. - -Lemma land_spec : forall a b n, - testbit (land a b) n = testbit a n && testbit b n. -Proof. - intros a n m. zify. now apply Z.land_spec. -Qed. - -Lemma lor_spec : forall a b n, - testbit (lor a b) n = testbit a n || testbit b n. -Proof. - intros a n m. zify. now apply Z.lor_spec. -Qed. - -Lemma ldiff_spec : forall a b n, - testbit (ldiff a b) n = testbit a n && negb (testbit b n). -Proof. - intros a n m. zify. now apply Z.ldiff_spec. -Qed. - -Lemma lxor_spec : forall a b n, - testbit (lxor a b) n = xorb (testbit a n) (testbit b n). -Proof. - intros a n m. zify. now apply Z.lxor_spec. -Qed. - -Lemma div2_spec : forall a, div2 a == shiftr a 1. -Proof. - intros a. zify. now apply Z.div2_spec. -Qed. - -(** Recursion *) - -Definition recursion (A : Type) (a : A) (f : NN.t -> A -> A) (n : NN.t) := - N.peano_rect (fun _ => A) a (fun n a => f (NN.of_N n) a) (NN.to_N n). -Arguments recursion [A] a f n. - -Instance recursion_wd (A : Type) (Aeq : relation A) : - Proper (Aeq ==> (eq==>Aeq==>Aeq) ==> eq ==> Aeq) (@recursion A). -Proof. -unfold eq. -intros a a' Eaa' f f' Eff' x x' Exx'. -unfold recursion. -unfold NN.to_N. -rewrite <- Exx'; clear x' Exx'. -induction (Z.to_N [x]) using N.peano_ind. -simpl; auto. -rewrite 2 N.peano_rect_succ. now apply Eff'. -Qed. - -Theorem recursion_0 : - forall (A : Type) (a : A) (f : NN.t -> A -> A), recursion a f 0 = a. -Proof. -intros A a f; unfold recursion, NN.to_N; rewrite NN.spec_0; simpl; auto. -Qed. - -Theorem recursion_succ : - forall (A : Type) (Aeq : relation A) (a : A) (f : NN.t -> A -> A), - Aeq a a -> Proper (eq==>Aeq==>Aeq) f -> - forall n, Aeq (recursion a f (succ n)) (f n (recursion a f n)). -Proof. -unfold eq, recursion; intros A Aeq a f EAaa f_wd n. -replace (to_N (succ n)) with (N.succ (to_N n)) by - (zify; now rewrite <- Z2N.inj_succ by apply spec_pos). -rewrite N.peano_rect_succ. -apply f_wd; auto. -zify. now rewrite Z2N.id by apply spec_pos. -fold (recursion a f n). apply recursion_wd; auto. red; auto. -Qed. - -End NTypeIsNAxioms. - -Module NType_NAxioms (NN : NType) - <: NAxiomsSig <: OrderFunctions NN <: HasMinMax NN - := NN <+ NTypeIsNAxioms. diff --git a/theories/Numbers/NumPrelude.v b/theories/Numbers/NumPrelude.v index f323aaeb..7cf13fea 100644 --- a/theories/Numbers/NumPrelude.v +++ b/theories/Numbers/NumPrelude.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) diff --git a/theories/Numbers/Rational/BigQ/BigQ.v b/theories/Numbers/Rational/BigQ/BigQ.v deleted file mode 100644 index 850afe53..00000000 --- a/theories/Numbers/Rational/BigQ/BigQ.v +++ /dev/null @@ -1,162 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -(** * BigQ: an efficient implementation of rational numbers *) - -(** Initial authors: Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) - -Require Export BigZ. -Require Import Field Qfield QSig QMake Orders GenericMinMax. - -(** We choose for BigQ an implemention with - multiple representation of 0: 0, 1/0, 2/0 etc. - See [QMake.v] *) - -(** First, we provide translations functions between [BigN] and [BigZ] *) - -Module BigN_BigZ <: NType_ZType BigN.BigN BigZ. - Definition Z_of_N := BigZ.Pos. - Lemma spec_Z_of_N : forall n, BigZ.to_Z (Z_of_N n) = BigN.to_Z n. - Proof. - reflexivity. - Qed. - Definition Zabs_N := BigZ.to_N. - Lemma spec_Zabs_N : forall z, BigN.to_Z (Zabs_N z) = Z.abs (BigZ.to_Z z). - Proof. - unfold Zabs_N; intros. - rewrite BigZ.spec_to_Z, Z.mul_comm; apply Z.sgn_abs. - Qed. -End BigN_BigZ. - -(** This allows building [BigQ] out of [BigN] and [BigQ] via [QMake] *) - -Delimit Scope bigQ_scope with bigQ. - -Module BigQ <: QType <: OrderedTypeFull <: TotalOrder. - Include QMake.Make BigN BigZ BigN_BigZ - <+ !QProperties <+ HasEqBool2Dec - <+ !MinMaxLogicalProperties <+ !MinMaxDecProperties. - Ltac order := Private_Tac.order. -End BigQ. - -(** Notations about [BigQ] *) - -Local Open Scope bigQ_scope. - -Notation bigQ := BigQ.t. -Bind Scope bigQ_scope with bigQ BigQ.t BigQ.t_. -(** As in QArith, we use [#] to denote fractions *) -Notation "p # q" := (BigQ.Qq p q) (at level 55, no associativity) : bigQ_scope. -Local Notation "0" := BigQ.zero : bigQ_scope. -Local Notation "1" := BigQ.one : bigQ_scope. -Infix "+" := BigQ.add : bigQ_scope. -Infix "-" := BigQ.sub : bigQ_scope. -Notation "- x" := (BigQ.opp x) : bigQ_scope. -Infix "*" := BigQ.mul : bigQ_scope. -Infix "/" := BigQ.div : bigQ_scope. -Infix "^" := BigQ.power : bigQ_scope. -Infix "?=" := BigQ.compare : bigQ_scope. -Infix "==" := BigQ.eq : bigQ_scope. -Notation "x != y" := (~x==y) (at level 70, no associativity) : bigQ_scope. -Infix "<" := BigQ.lt : bigQ_scope. -Infix "<=" := BigQ.le : bigQ_scope. -Notation "x > y" := (BigQ.lt y x) (only parsing) : bigQ_scope. -Notation "x >= y" := (BigQ.le y x) (only parsing) : bigQ_scope. -Notation "x < y < z" := (x<y /\ y<z) : bigQ_scope. -Notation "x < y <= z" := (x<y /\ y<=z) : bigQ_scope. -Notation "x <= y < z" := (x<=y /\ y<z) : bigQ_scope. -Notation "x <= y <= z" := (x<=y /\ y<=z) : bigQ_scope. -Notation "[ q ]" := (BigQ.to_Q q) : bigQ_scope. - -(** [BigQ] is a field *) - -Lemma BigQfieldth : - field_theory 0 1 BigQ.add BigQ.mul BigQ.sub BigQ.opp - BigQ.div BigQ.inv BigQ.eq. -Proof. -constructor. -constructor. -exact BigQ.add_0_l. exact BigQ.add_comm. exact BigQ.add_assoc. -exact BigQ.mul_1_l. exact BigQ.mul_comm. exact BigQ.mul_assoc. -exact BigQ.mul_add_distr_r. exact BigQ.sub_add_opp. -exact BigQ.add_opp_diag_r. exact BigQ.neq_1_0. -exact BigQ.div_mul_inv. exact BigQ.mul_inv_diag_l. -Qed. - -Declare Equivalent Keys pow_N pow_pos. - -Lemma BigQpowerth : - power_theory 1 BigQ.mul BigQ.eq Z.of_N BigQ.power. -Proof. -constructor. intros. BigQ.qify. -replace ([r] ^ Z.of_N n)%Q with (pow_N 1 Qmult [r] n)%Q by (now destruct n). -destruct n. reflexivity. -induction p; simpl; auto; rewrite ?BigQ.spec_mul, ?IHp; reflexivity. -Qed. - -Ltac isBigQcst t := - match t with - | BigQ.Qz ?t => isBigZcst t - | BigQ.Qq ?n ?d => match isBigZcst n with - | true => isBigNcst d - | false => constr:(false) - end - | BigQ.zero => constr:(true) - | BigQ.one => constr:(true) - | BigQ.minus_one => constr:(true) - | _ => constr:(false) - end. - -Ltac BigQcst t := - match isBigQcst t with - | true => constr:(t) - | false => constr:(NotConstant) - end. - -Add Field BigQfield : BigQfieldth - (decidable BigQ.eqb_correct, - completeness BigQ.eqb_complete, - constants [BigQcst], - power_tac BigQpowerth [Qpow_tac]). - -Section TestField. - -Let ex1 : forall x y z, (x+y)*z == (x*z)+(y*z). - intros. - ring. -Qed. - -Let ex8 : forall x, x ^ 2 == x*x. - intro. - ring. -Qed. - -Let ex10 : forall x y, y!=0 -> (x/y)*y == x. -intros. -field. -auto. -Qed. - -End TestField. - -(** [BigQ] can also benefit from an "order" tactic *) - -Ltac bigQ_order := BigQ.order. - -Section TestOrder. -Let test : forall x y : bigQ, x<=y -> y<=x -> x==y. -Proof. bigQ_order. Qed. -End TestOrder. - -(** We can also reason by switching to QArith thanks to tactic - BigQ.qify. *) - -Section TestQify. -Let test : forall x : bigQ, 0+x == 1*x. -Proof. intro x. BigQ.qify. ring. Qed. -End TestQify. diff --git a/theories/Numbers/Rational/BigQ/QMake.v b/theories/Numbers/Rational/BigQ/QMake.v deleted file mode 100644 index b9fed9d5..00000000 --- a/theories/Numbers/Rational/BigQ/QMake.v +++ /dev/null @@ -1,1283 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -(** * QMake : a generic efficient implementation of rational numbers *) - -(** Initial authors : Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) - -Require Import BigNumPrelude ROmega. -Require Import QArith Qcanon Qpower Qminmax. -Require Import NSig ZSig QSig. - -(** We will build rationals out of an implementation of integers [ZType] - for numerators and an implementation of natural numbers [NType] for - denominators. But first we will need some glue between [NType] and - [ZType]. *) - -Module Type NType_ZType (NN:NType)(ZZ:ZType). - Parameter Z_of_N : NN.t -> ZZ.t. - Parameter spec_Z_of_N : forall n, ZZ.to_Z (Z_of_N n) = NN.to_Z n. - Parameter Zabs_N : ZZ.t -> NN.t. - Parameter spec_Zabs_N : forall z, NN.to_Z (Zabs_N z) = Z.abs (ZZ.to_Z z). -End NType_ZType. - -Module Make (NN:NType)(ZZ:ZType)(Import NZ:NType_ZType NN ZZ) <: QType. - - (** The notation of a rational number is either an integer x, - interpreted as itself or a pair (x,y) of an integer x and a natural - number y interpreted as x/y. The pairs (x,0) and (0,y) are all - interpreted as 0. *) - - Inductive t_ := - | Qz : ZZ.t -> t_ - | Qq : ZZ.t -> NN.t -> t_. - - Definition t := t_. - - (** Specification with respect to [QArith] *) - - Local Open Scope Q_scope. - - Definition of_Z x: t := Qz (ZZ.of_Z x). - - Definition of_Q (q:Q) : t := - let (x,y) := q in - match y with - | 1%positive => Qz (ZZ.of_Z x) - | _ => Qq (ZZ.of_Z x) (NN.of_N (Npos y)) - end. - - Definition to_Q (q: t) := - match q with - | Qz x => ZZ.to_Z x # 1 - | Qq x y => if NN.eqb y NN.zero then 0 - else ZZ.to_Z x # Z.to_pos (NN.to_Z y) - end. - - Notation "[ x ]" := (to_Q x). - - Lemma N_to_Z_pos : - forall x, (NN.to_Z x <> NN.to_Z NN.zero)%Z -> (0 < NN.to_Z x)%Z. - Proof. - intros x; rewrite NN.spec_0; generalize (NN.spec_pos x). romega. - Qed. - - Ltac destr_zcompare := case Z.compare_spec; intros ?H. - - Ltac destr_eqb := - match goal with - | |- context [ZZ.eqb ?x ?y] => - rewrite (ZZ.spec_eqb x y); - case (Z.eqb_spec (ZZ.to_Z x) (ZZ.to_Z y)); - destr_eqb - | |- context [NN.eqb ?x ?y] => - rewrite (NN.spec_eqb x y); - case (Z.eqb_spec (NN.to_Z x) (NN.to_Z y)); - [ | let H:=fresh "H" in - try (intro H;generalize (N_to_Z_pos _ H); clear H)]; - destr_eqb - | _ => idtac - end. - - Hint Rewrite - Z.add_0_r Z.add_0_l Z.mul_0_r Z.mul_0_l Z.mul_1_r Z.mul_1_l - ZZ.spec_0 NN.spec_0 ZZ.spec_1 NN.spec_1 ZZ.spec_m1 ZZ.spec_opp - ZZ.spec_compare NN.spec_compare - ZZ.spec_add NN.spec_add ZZ.spec_mul NN.spec_mul ZZ.spec_div NN.spec_div - ZZ.spec_gcd NN.spec_gcd Z.gcd_abs_l Z.gcd_1_r - spec_Z_of_N spec_Zabs_N - : nz. - - Ltac nzsimpl := autorewrite with nz in *. - - Ltac qsimpl := try red; unfold to_Q; simpl; intros; - destr_eqb; simpl; nzsimpl; intros; - rewrite ?Z2Pos.id by auto; - auto. - - Theorem strong_spec_of_Q: forall q: Q, [of_Q q] = q. - Proof. - intros(x,y); destruct y; simpl; rewrite ?ZZ.spec_of_Z; auto; - destr_eqb; now rewrite ?NN.spec_0, ?NN.spec_of_N. - Qed. - - Theorem spec_of_Q: forall q: Q, [of_Q q] == q. - Proof. - intros; rewrite strong_spec_of_Q; red; auto. - Qed. - - Definition eq x y := [x] == [y]. - - Definition zero: t := Qz ZZ.zero. - Definition one: t := Qz ZZ.one. - Definition minus_one: t := Qz ZZ.minus_one. - - Lemma spec_0: [zero] == 0. - Proof. - simpl. nzsimpl. reflexivity. - Qed. - - Lemma spec_1: [one] == 1. - Proof. - simpl. nzsimpl. reflexivity. - Qed. - - Lemma spec_m1: [minus_one] == -(1). - Proof. - simpl. nzsimpl. reflexivity. - Qed. - - Definition compare (x y: t) := - match x, y with - | Qz zx, Qz zy => ZZ.compare zx zy - | Qz zx, Qq ny dy => - if NN.eqb dy NN.zero then ZZ.compare zx ZZ.zero - else ZZ.compare (ZZ.mul zx (Z_of_N dy)) ny - | Qq nx dx, Qz zy => - if NN.eqb dx NN.zero then ZZ.compare ZZ.zero zy - else ZZ.compare nx (ZZ.mul zy (Z_of_N dx)) - | Qq nx dx, Qq ny dy => - match NN.eqb dx NN.zero, NN.eqb dy NN.zero with - | true, true => Eq - | true, false => ZZ.compare ZZ.zero ny - | false, true => ZZ.compare nx ZZ.zero - | false, false => ZZ.compare (ZZ.mul nx (Z_of_N dy)) - (ZZ.mul ny (Z_of_N dx)) - end - end. - - Theorem spec_compare: forall q1 q2, (compare q1 q2) = ([q1] ?= [q2]). - Proof. - intros [z1 | x1 y1] [z2 | x2 y2]; - unfold Qcompare, compare; qsimpl. - Qed. - - Definition lt n m := [n] < [m]. - Definition le n m := [n] <= [m]. - - Definition min n m := match compare n m with Gt => m | _ => n end. - Definition max n m := match compare n m with Lt => m | _ => n end. - - Lemma spec_min : forall n m, [min n m] == Qmin [n] [m]. - Proof. - unfold min, Qmin, GenericMinMax.gmin. intros. - rewrite spec_compare; destruct Qcompare; auto with qarith. - Qed. - - Lemma spec_max : forall n m, [max n m] == Qmax [n] [m]. - Proof. - unfold max, Qmax, GenericMinMax.gmax. intros. - rewrite spec_compare; destruct Qcompare; auto with qarith. - Qed. - - Definition eq_bool n m := - match compare n m with Eq => true | _ => false end. - - Theorem spec_eq_bool: forall x y, eq_bool x y = Qeq_bool [x] [y]. - Proof. - intros. unfold eq_bool. rewrite spec_compare. reflexivity. - Qed. - - (** [check_int] : is a reduced fraction [n/d] in fact a integer ? *) - - Definition check_int n d := - match NN.compare NN.one d with - | Lt => Qq n d - | Eq => Qz n - | Gt => zero (* n/0 encodes 0 *) - end. - - Theorem strong_spec_check_int : forall n d, [check_int n d] = [Qq n d]. - Proof. - intros; unfold check_int. - nzsimpl. - destr_zcompare. - simpl. rewrite <- H; qsimpl. congruence. - reflexivity. - qsimpl. exfalso; romega. - Qed. - - (** Normalisation function *) - - Definition norm n d : t := - let gcd := NN.gcd (Zabs_N n) d in - match NN.compare NN.one gcd with - | Lt => check_int (ZZ.div n (Z_of_N gcd)) (NN.div d gcd) - | Eq => check_int n d - | Gt => zero (* gcd = 0 => both numbers are 0 *) - end. - - Theorem spec_norm: forall n q, [norm n q] == [Qq n q]. - Proof. - intros p q; unfold norm. - assert (Hp := NN.spec_pos (Zabs_N p)). - assert (Hq := NN.spec_pos q). - nzsimpl. - destr_zcompare. - (* Eq *) - rewrite strong_spec_check_int; reflexivity. - (* Lt *) - rewrite strong_spec_check_int. - qsimpl. - generalize (Zgcd_div_pos (ZZ.to_Z p) (NN.to_Z q)). romega. - replace (NN.to_Z q) with 0%Z in * by assumption. - rewrite Zdiv_0_l in *; auto with zarith. - apply Zgcd_div_swap0; romega. - (* Gt *) - qsimpl. - assert (H' : Z.gcd (ZZ.to_Z p) (NN.to_Z q) = 0%Z). - generalize (Z.gcd_nonneg (ZZ.to_Z p) (NN.to_Z q)); romega. - symmetry; apply (Z.gcd_eq_0_l _ _ H'); auto. - Qed. - - Theorem strong_spec_norm : forall p q, [norm p q] = Qred [Qq p q]. - Proof. - intros. - replace (Qred [Qq p q]) with (Qred [norm p q]) by - (apply Qred_complete; apply spec_norm). - symmetry; apply Qred_identity. - unfold norm. - assert (Hp := NN.spec_pos (Zabs_N p)). - assert (Hq := NN.spec_pos q). - nzsimpl. - destr_zcompare; rewrite ?strong_spec_check_int. - (* Eq *) - qsimpl. - (* Lt *) - qsimpl. - rewrite Zgcd_1_rel_prime. - destruct (Z_lt_le_dec 0 (NN.to_Z q)). - apply Zis_gcd_rel_prime; auto with zarith. - apply Zgcd_is_gcd. - replace (NN.to_Z q) with 0%Z in * by romega. - rewrite Zdiv_0_l in *; romega. - (* Gt *) - simpl; auto with zarith. - Qed. - - (** Reduction function : producing irreducible fractions *) - - Definition red (x : t) : t := - match x with - | Qz z => x - | Qq n d => norm n d - end. - - Class Reduced x := is_reduced : [red x] = [x]. - - Theorem spec_red : forall x, [red x] == [x]. - Proof. - intros [ z | n d ]. - auto with qarith. - unfold red. - apply spec_norm. - Qed. - - Theorem strong_spec_red : forall x, [red x] = Qred [x]. - Proof. - intros [ z | n d ]. - unfold red. - symmetry; apply Qred_identity; simpl; auto with zarith. - unfold red; apply strong_spec_norm. - Qed. - - Definition add (x y: t): t := - match x with - | Qz zx => - match y with - | Qz zy => Qz (ZZ.add zx zy) - | Qq ny dy => - if NN.eqb dy NN.zero then x - else Qq (ZZ.add (ZZ.mul zx (Z_of_N dy)) ny) dy - end - | Qq nx dx => - if NN.eqb dx NN.zero then y - else match y with - | Qz zy => Qq (ZZ.add nx (ZZ.mul zy (Z_of_N dx))) dx - | Qq ny dy => - if NN.eqb dy NN.zero then x - else - let n := ZZ.add (ZZ.mul nx (Z_of_N dy)) (ZZ.mul ny (Z_of_N dx)) in - let d := NN.mul dx dy in - Qq n d - end - end. - - Theorem spec_add : forall x y, [add x y] == [x] + [y]. - Proof. - intros [x | nx dx] [y | ny dy]; unfold Qplus; qsimpl; - auto with zarith. - rewrite Pos.mul_1_r, Z2Pos.id; auto. - rewrite Pos.mul_1_r, Z2Pos.id; auto. - rewrite Z.mul_eq_0 in *; intuition. - rewrite Pos2Z.inj_mul, 2 Z2Pos.id; auto. - Qed. - - Definition add_norm (x y: t): t := - match x with - | Qz zx => - match y with - | Qz zy => Qz (ZZ.add zx zy) - | Qq ny dy => - if NN.eqb dy NN.zero then x - else norm (ZZ.add (ZZ.mul zx (Z_of_N dy)) ny) dy - end - | Qq nx dx => - if NN.eqb dx NN.zero then y - else match y with - | Qz zy => norm (ZZ.add nx (ZZ.mul zy (Z_of_N dx))) dx - | Qq ny dy => - if NN.eqb dy NN.zero then x - else - let n := ZZ.add (ZZ.mul nx (Z_of_N dy)) (ZZ.mul ny (Z_of_N dx)) in - let d := NN.mul dx dy in - norm n d - end - end. - - Theorem spec_add_norm : forall x y, [add_norm x y] == [x] + [y]. - Proof. - intros x y; rewrite <- spec_add. - destruct x; destruct y; unfold add_norm, add; - destr_eqb; auto using Qeq_refl, spec_norm. - Qed. - - Instance strong_spec_add_norm x y - `(Reduced x, Reduced y) : Reduced (add_norm x y). - Proof. - unfold Reduced; intros. - rewrite strong_spec_red. - rewrite <- (Qred_complete [add x y]); - [ | rewrite spec_add, spec_add_norm; apply Qeq_refl ]. - rewrite <- strong_spec_red. - destruct x as [zx|nx dx]; destruct y as [zy|ny dy]; - simpl; destr_eqb; nzsimpl; simpl; auto. - Qed. - - Definition opp (x: t): t := - match x with - | Qz zx => Qz (ZZ.opp zx) - | Qq nx dx => Qq (ZZ.opp nx) dx - end. - - Theorem strong_spec_opp: forall q, [opp q] = -[q]. - Proof. - intros [z | x y]; simpl. - rewrite ZZ.spec_opp; auto. - match goal with |- context[NN.eqb ?X ?Y] => - generalize (NN.spec_eqb X Y); case NN.eqb - end; auto; rewrite NN.spec_0. - rewrite ZZ.spec_opp; auto. - Qed. - - Theorem spec_opp : forall q, [opp q] == -[q]. - Proof. - intros; rewrite strong_spec_opp; red; auto. - Qed. - - Instance strong_spec_opp_norm q `(Reduced q) : Reduced (opp q). - Proof. - unfold Reduced; intros. - rewrite strong_spec_opp, <- H, !strong_spec_red, <- Qred_opp. - apply Qred_complete; apply spec_opp. - Qed. - - Definition sub x y := add x (opp y). - - Theorem spec_sub : forall x y, [sub x y] == [x] - [y]. - Proof. - intros x y; unfold sub; rewrite spec_add; auto. - rewrite spec_opp; ring. - Qed. - - Definition sub_norm x y := add_norm x (opp y). - - Theorem spec_sub_norm : forall x y, [sub_norm x y] == [x] - [y]. - Proof. - intros x y; unfold sub_norm; rewrite spec_add_norm; auto. - rewrite spec_opp; ring. - Qed. - - Instance strong_spec_sub_norm x y - `(Reduced x, Reduced y) : Reduced (sub_norm x y). - Proof. - intros. - unfold sub_norm. - apply strong_spec_add_norm; auto. - apply strong_spec_opp_norm; auto. - Qed. - - Definition mul (x y: t): t := - match x, y with - | Qz zx, Qz zy => Qz (ZZ.mul zx zy) - | Qz zx, Qq ny dy => Qq (ZZ.mul zx ny) dy - | Qq nx dx, Qz zy => Qq (ZZ.mul nx zy) dx - | Qq nx dx, Qq ny dy => Qq (ZZ.mul nx ny) (NN.mul dx dy) - end. - - Ltac nsubst := - match goal with E : NN.to_Z _ = _ |- _ => rewrite E in * end. - - Theorem spec_mul : forall x y, [mul x y] == [x] * [y]. - Proof. - intros [x | nx dx] [y | ny dy]; unfold Qmult; simpl; qsimpl. - rewrite Pos.mul_1_r, Z2Pos.id; auto. - rewrite Z.mul_eq_0 in *; intuition. - nsubst; auto with zarith. - nsubst; auto with zarith. - nsubst; nzsimpl; auto with zarith. - rewrite Pos2Z.inj_mul, 2 Z2Pos.id; auto. - Qed. - - Definition norm_denum n d := - if NN.eqb d NN.one then Qz n else Qq n d. - - Lemma spec_norm_denum : forall n d, - [norm_denum n d] == [Qq n d]. - Proof. - unfold norm_denum; intros; simpl; qsimpl. - congruence. - nsubst; auto with zarith. - Qed. - - Definition irred n d := - let gcd := NN.gcd (Zabs_N n) d in - match NN.compare gcd NN.one with - | Gt => (ZZ.div n (Z_of_N gcd), NN.div d gcd) - | _ => (n, d) - end. - - Lemma spec_irred : forall n d, exists g, - let (n',d') := irred n d in - (ZZ.to_Z n' * g = ZZ.to_Z n)%Z /\ (NN.to_Z d' * g = NN.to_Z d)%Z. - Proof. - intros. - unfold irred; nzsimpl; simpl. - destr_zcompare. - exists 1%Z; nzsimpl; auto. - exists 0%Z; nzsimpl. - assert (Z.gcd (ZZ.to_Z n) (NN.to_Z d) = 0%Z). - generalize (Z.gcd_nonneg (ZZ.to_Z n) (NN.to_Z d)); romega. - clear H. - split. - symmetry; apply (Z.gcd_eq_0_l _ _ H0). - symmetry; apply (Z.gcd_eq_0_r _ _ H0). - exists (Z.gcd (ZZ.to_Z n) (NN.to_Z d)). - simpl. - split. - nzsimpl. - destruct (Zgcd_is_gcd (ZZ.to_Z n) (NN.to_Z d)). - rewrite Z.mul_comm; symmetry; apply Zdivide_Zdiv_eq; auto with zarith. - nzsimpl. - destruct (Zgcd_is_gcd (ZZ.to_Z n) (NN.to_Z d)). - rewrite Z.mul_comm; symmetry; apply Zdivide_Zdiv_eq; auto with zarith. - Qed. - - Lemma spec_irred_zero : forall n d, - (NN.to_Z d = 0)%Z <-> (NN.to_Z (snd (irred n d)) = 0)%Z. - Proof. - intros. - unfold irred. - split. - nzsimpl; intros. - destr_zcompare; auto. - simpl. - nzsimpl. - rewrite H, Zdiv_0_l; auto. - nzsimpl; destr_zcompare; simpl; auto. - nzsimpl. - intros. - generalize (NN.spec_pos d); intros. - destruct (NN.to_Z d); auto. - assert (0 < 0)%Z. - rewrite <- H0 at 2. - apply Zgcd_div_pos; auto with zarith. - compute; auto. - discriminate. - compute in H1; elim H1; auto. - Qed. - - Lemma strong_spec_irred : forall n d, - (NN.to_Z d <> 0%Z) -> - let (n',d') := irred n d in Z.gcd (ZZ.to_Z n') (NN.to_Z d') = 1%Z. - Proof. - unfold irred; intros. - nzsimpl. - destr_zcompare; simpl; auto. - elim H. - apply (Z.gcd_eq_0_r (ZZ.to_Z n)). - generalize (Z.gcd_nonneg (ZZ.to_Z n) (NN.to_Z d)); romega. - - nzsimpl. - rewrite Zgcd_1_rel_prime. - apply Zis_gcd_rel_prime. - generalize (NN.spec_pos d); romega. - generalize (Z.gcd_nonneg (ZZ.to_Z n) (NN.to_Z d)); romega. - apply Zgcd_is_gcd; auto. - Qed. - - Definition mul_norm_Qz_Qq z n d := - if ZZ.eqb z ZZ.zero then zero - else - let gcd := NN.gcd (Zabs_N z) d in - match NN.compare gcd NN.one with - | Gt => - let z := ZZ.div z (Z_of_N gcd) in - let d := NN.div d gcd in - norm_denum (ZZ.mul z n) d - | _ => Qq (ZZ.mul z n) d - end. - - Definition mul_norm (x y: t): t := - match x, y with - | Qz zx, Qz zy => Qz (ZZ.mul zx zy) - | Qz zx, Qq ny dy => mul_norm_Qz_Qq zx ny dy - | Qq nx dx, Qz zy => mul_norm_Qz_Qq zy nx dx - | Qq nx dx, Qq ny dy => - let (nx, dy) := irred nx dy in - let (ny, dx) := irred ny dx in - norm_denum (ZZ.mul ny nx) (NN.mul dx dy) - end. - - Lemma spec_mul_norm_Qz_Qq : forall z n d, - [mul_norm_Qz_Qq z n d] == [Qq (ZZ.mul z n) d]. - Proof. - intros z n d; unfold mul_norm_Qz_Qq; nzsimpl; rewrite Zcompare_gt. - destr_eqb; nzsimpl; intros Hz. - qsimpl; rewrite Hz; auto. - destruct Z_le_gt_dec as [LE|GT]. - qsimpl. - rewrite spec_norm_denum. - qsimpl. - rewrite Zdiv_gcd_zero in GT; auto with zarith. - nsubst. rewrite Zdiv_0_l in *; discriminate. - rewrite <- Z.mul_assoc, (Z.mul_comm (ZZ.to_Z n)), Z.mul_assoc. - rewrite Zgcd_div_swap0; try romega. - ring. - Qed. - - Instance strong_spec_mul_norm_Qz_Qq z n d : - forall `(Reduced (Qq n d)), Reduced (mul_norm_Qz_Qq z n d). - Proof. - unfold Reduced. - rewrite 2 strong_spec_red, 2 Qred_iff. - simpl; nzsimpl. - destr_eqb; intros Hd H; simpl in *; nzsimpl. - - unfold mul_norm_Qz_Qq; nzsimpl; rewrite Zcompare_gt. - destr_eqb; intros Hz; simpl; nzsimpl; simpl; auto. - destruct Z_le_gt_dec. - simpl; nzsimpl. - destr_eqb; simpl; nzsimpl; auto with zarith. - unfold norm_denum. destr_eqb; simpl; nzsimpl. - rewrite Hd, Zdiv_0_l; discriminate. - intros _. - destr_eqb; simpl; nzsimpl; auto. - nzsimpl; rewrite Hd, Zdiv_0_l; auto with zarith. - - rewrite Z2Pos.id in H; auto. - unfold mul_norm_Qz_Qq; nzsimpl; rewrite Zcompare_gt. - destr_eqb; intros Hz; simpl; nzsimpl; simpl; auto. - destruct Z_le_gt_dec as [H'|H']. - simpl; nzsimpl. - destr_eqb; simpl; nzsimpl; auto. - intros. - rewrite Z2Pos.id; auto. - apply Zgcd_mult_rel_prime; auto. - generalize (Z.gcd_eq_0_l (ZZ.to_Z z) (NN.to_Z d)) - (Z.gcd_nonneg (ZZ.to_Z z) (NN.to_Z d)); romega. - destr_eqb; simpl; nzsimpl; auto. - unfold norm_denum. - destr_eqb; nzsimpl; simpl; destr_eqb; simpl; auto. - intros; nzsimpl. - rewrite Z2Pos.id; auto. - apply Zgcd_mult_rel_prime. - rewrite Zgcd_1_rel_prime. - apply Zis_gcd_rel_prime. - generalize (NN.spec_pos d); romega. - generalize (Z.gcd_nonneg (ZZ.to_Z z) (NN.to_Z d)); romega. - apply Zgcd_is_gcd. - destruct (Zgcd_is_gcd (ZZ.to_Z z) (NN.to_Z d)) as [ (z0,Hz0) (d0,Hd0) Hzd]. - replace (NN.to_Z d / Z.gcd (ZZ.to_Z z) (NN.to_Z d))%Z with d0. - rewrite Zgcd_1_rel_prime in *. - apply bezout_rel_prime. - destruct (rel_prime_bezout _ _ H) as [u v Huv]. - apply Bezout_intro with u (v*(Z.gcd (ZZ.to_Z z) (NN.to_Z d)))%Z. - rewrite <- Huv; rewrite Hd0 at 2; ring. - rewrite Hd0 at 1. - symmetry; apply Z_div_mult_full; auto with zarith. - Qed. - - Theorem spec_mul_norm : forall x y, [mul_norm x y] == [x] * [y]. - Proof. - intros x y; rewrite <- spec_mul; auto. - unfold mul_norm, mul; destruct x; destruct y. - apply Qeq_refl. - apply spec_mul_norm_Qz_Qq. - rewrite spec_mul_norm_Qz_Qq; qsimpl; ring. - - rename t0 into nx, t3 into dy, t2 into ny, t1 into dx. - destruct (spec_irred nx dy) as (g & Hg). - destruct (spec_irred ny dx) as (g' & Hg'). - assert (Hz := spec_irred_zero nx dy). - assert (Hz':= spec_irred_zero ny dx). - destruct irred as (n1,d1); destruct irred as (n2,d2). - simpl @snd in *; destruct Hg as [Hg1 Hg2]; destruct Hg' as [Hg1' Hg2']. - rewrite spec_norm_denum. - qsimpl. - - match goal with E : (_ * _ = 0)%Z |- _ => - rewrite Z.mul_eq_0 in E; destruct E as [Eq|Eq] end. - rewrite Eq in *; simpl in *. - rewrite <- Hg2' in *; auto with zarith. - rewrite Eq in *; simpl in *. - rewrite <- Hg2 in *; auto with zarith. - - match goal with E : (_ * _ = 0)%Z |- _ => - rewrite Z.mul_eq_0 in E; destruct E as [Eq|Eq] end. - rewrite Hz' in Eq; rewrite Eq in *; auto with zarith. - rewrite Hz in Eq; rewrite Eq in *; auto with zarith. - - rewrite <- Hg1, <- Hg2, <- Hg1', <- Hg2'; ring. - Qed. - - Instance strong_spec_mul_norm x y : - forall `(Reduced x, Reduced y), Reduced (mul_norm x y). - Proof. - unfold Reduced; intros. - rewrite strong_spec_red, Qred_iff. - destruct x as [zx|nx dx]; destruct y as [zy|ny dy]. - simpl in *; auto with zarith. - simpl. - rewrite <- Qred_iff, <- strong_spec_red, strong_spec_mul_norm_Qz_Qq; auto. - simpl. - rewrite <- Qred_iff, <- strong_spec_red, strong_spec_mul_norm_Qz_Qq; auto. - simpl. - destruct (spec_irred nx dy) as [g Hg]. - destruct (spec_irred ny dx) as [g' Hg']. - assert (Hz := spec_irred_zero nx dy). - assert (Hz':= spec_irred_zero ny dx). - assert (Hgc := strong_spec_irred nx dy). - assert (Hgc' := strong_spec_irred ny dx). - destruct irred as (n1,d1); destruct irred as (n2,d2). - simpl @snd in *; destruct Hg as [Hg1 Hg2]; destruct Hg' as [Hg1' Hg2']. - - unfold norm_denum; qsimpl. - - assert (NEQ : NN.to_Z dy <> 0%Z) by - (rewrite Hz; intros EQ; rewrite EQ in *; romega). - specialize (Hgc NEQ). - - assert (NEQ' : NN.to_Z dx <> 0%Z) by - (rewrite Hz'; intro EQ; rewrite EQ in *; romega). - specialize (Hgc' NEQ'). - - revert H H0. - rewrite 2 strong_spec_red, 2 Qred_iff; simpl. - destr_eqb; simpl; nzsimpl; try romega; intros. - rewrite Z2Pos.id in *; auto. - - apply Zgcd_mult_rel_prime; rewrite Z.gcd_comm; - apply Zgcd_mult_rel_prime; rewrite Z.gcd_comm; auto. - - rewrite Zgcd_1_rel_prime in *. - apply bezout_rel_prime. - destruct (rel_prime_bezout (ZZ.to_Z ny) (NN.to_Z dy)) as [u v Huv]; trivial. - apply Bezout_intro with (u*g')%Z (v*g)%Z. - rewrite <- Huv, <- Hg1', <- Hg2. ring. - - rewrite Zgcd_1_rel_prime in *. - apply bezout_rel_prime. - destruct (rel_prime_bezout (ZZ.to_Z nx) (NN.to_Z dx)) as [u v Huv]; trivial. - apply Bezout_intro with (u*g)%Z (v*g')%Z. - rewrite <- Huv, <- Hg2', <- Hg1. ring. - Qed. - - Definition inv (x: t): t := - match x with - | Qz z => - match ZZ.compare ZZ.zero z with - | Eq => zero - | Lt => Qq ZZ.one (Zabs_N z) - | Gt => Qq ZZ.minus_one (Zabs_N z) - end - | Qq n d => - match ZZ.compare ZZ.zero n with - | Eq => zero - | Lt => Qq (Z_of_N d) (Zabs_N n) - | Gt => Qq (ZZ.opp (Z_of_N d)) (Zabs_N n) - end - end. - - Theorem spec_inv : forall x, [inv x] == /[x]. - Proof. - destruct x as [ z | n d ]. - (* Qz z *) - simpl. - rewrite ZZ.spec_compare; destr_zcompare. - (* 0 = z *) - rewrite <- H. - simpl; nzsimpl; compute; auto. - (* 0 < z *) - simpl. - destr_eqb; nzsimpl; [ intros; rewrite Z.abs_eq in *; romega | intros _ ]. - set (z':=ZZ.to_Z z) in *; clearbody z'. - red; simpl. - rewrite Z.abs_eq by romega. - rewrite Z2Pos.id by auto. - unfold Qinv; simpl; destruct z'; simpl; auto; discriminate. - (* 0 > z *) - simpl. - destr_eqb; nzsimpl; [ intros; rewrite Z.abs_neq in *; romega | intros _ ]. - set (z':=ZZ.to_Z z) in *; clearbody z'. - red; simpl. - rewrite Z.abs_neq by romega. - rewrite Z2Pos.id by romega. - unfold Qinv; simpl; destruct z'; simpl; auto; discriminate. - (* Qq n d *) - simpl. - rewrite ZZ.spec_compare; destr_zcompare. - (* 0 = n *) - rewrite <- H. - simpl; nzsimpl. - destr_eqb; intros; compute; auto. - (* 0 < n *) - simpl. - destr_eqb; nzsimpl; intros. - intros; rewrite Z.abs_eq in *; romega. - intros; rewrite Z.abs_eq in *; romega. - nsubst; compute; auto. - set (n':=ZZ.to_Z n) in *; clearbody n'. - rewrite Z.abs_eq by romega. - red; simpl. - rewrite Z2Pos.id by auto. - unfold Qinv; simpl; destruct n'; simpl; auto; try discriminate. - rewrite Pos2Z.inj_mul, Z2Pos.id; auto. - (* 0 > n *) - simpl. - destr_eqb; nzsimpl; intros. - intros; rewrite Z.abs_neq in *; romega. - intros; rewrite Z.abs_neq in *; romega. - nsubst; compute; auto. - set (n':=ZZ.to_Z n) in *; clearbody n'. - red; simpl; nzsimpl. - rewrite Z.abs_neq by romega. - rewrite Z2Pos.id by romega. - unfold Qinv; simpl; destruct n'; simpl; auto; try discriminate. - assert (T : forall x, Zneg x = Z.opp (Zpos x)) by auto. - rewrite T, Pos2Z.inj_mul, Z2Pos.id; auto; ring. - Qed. - - Definition inv_norm (x: t): t := - match x with - | Qz z => - match ZZ.compare ZZ.zero z with - | Eq => zero - | Lt => Qq ZZ.one (Zabs_N z) - | Gt => Qq ZZ.minus_one (Zabs_N z) - end - | Qq n d => - if NN.eqb d NN.zero then zero else - match ZZ.compare ZZ.zero n with - | Eq => zero - | Lt => - match ZZ.compare n ZZ.one with - | Gt => Qq (Z_of_N d) (Zabs_N n) - | _ => Qz (Z_of_N d) - end - | Gt => - match ZZ.compare n ZZ.minus_one with - | Lt => Qq (ZZ.opp (Z_of_N d)) (Zabs_N n) - | _ => Qz (ZZ.opp (Z_of_N d)) - end - end - end. - - Theorem spec_inv_norm : forall x, [inv_norm x] == /[x]. - Proof. - intros. - rewrite <- spec_inv. - destruct x as [ z | n d ]. - (* Qz z *) - simpl. - rewrite ZZ.spec_compare; destr_zcompare; auto with qarith. - (* Qq n d *) - simpl; nzsimpl; destr_eqb. - destr_zcompare; simpl; auto with qarith. - destr_eqb; nzsimpl; auto with qarith. - intros _ Hd; rewrite Hd; auto with qarith. - destr_eqb; nzsimpl; auto with qarith. - intros _ Hd; rewrite Hd; auto with qarith. - (* 0 < n *) - destr_zcompare; auto with qarith. - destr_zcompare; nzsimpl; simpl; auto with qarith; intros. - destr_eqb; nzsimpl; [ intros; rewrite Z.abs_eq in *; romega | intros _ ]. - rewrite H0; auto with qarith. - romega. - (* 0 > n *) - destr_zcompare; nzsimpl; simpl; auto with qarith. - destr_eqb; nzsimpl; [ intros; rewrite Z.abs_neq in *; romega | intros _ ]. - rewrite H0; auto with qarith. - romega. - Qed. - - Instance strong_spec_inv_norm x : Reduced x -> Reduced (inv_norm x). - Proof. - unfold Reduced. - intros. - destruct x as [ z | n d ]. - (* Qz *) - simpl; nzsimpl. - rewrite strong_spec_red, Qred_iff. - destr_zcompare; simpl; nzsimpl; auto. - destr_eqb; nzsimpl; simpl; auto. - destr_eqb; nzsimpl; simpl; auto. - (* Qq n d *) - rewrite strong_spec_red, Qred_iff in H; revert H. - simpl; nzsimpl. - destr_eqb; nzsimpl; auto with qarith. - destr_zcompare; simpl; nzsimpl; auto; intros. - (* 0 < n *) - destr_zcompare; simpl; nzsimpl; auto. - destr_eqb; nzsimpl; simpl; auto. - rewrite Z.abs_eq; romega. - intros _. - rewrite strong_spec_norm; simpl; nzsimpl. - destr_eqb; nzsimpl. - rewrite Z.abs_eq; romega. - intros _. - rewrite Qred_iff. - simpl. - rewrite Z.abs_eq; auto with zarith. - rewrite Z2Pos.id in *; auto. - rewrite Z.gcd_comm; auto. - (* 0 > n *) - destr_eqb; nzsimpl; simpl; auto; intros. - destr_zcompare; simpl; nzsimpl; auto. - destr_eqb; nzsimpl. - rewrite Z.abs_neq; romega. - intros _. - rewrite strong_spec_norm; simpl; nzsimpl. - destr_eqb; nzsimpl. - rewrite Z.abs_neq; romega. - intros _. - rewrite Qred_iff. - simpl. - rewrite Z2Pos.id in *; auto. - intros. - rewrite Z.gcd_comm, Z.gcd_abs_l, Z.gcd_comm. - apply Zis_gcd_gcd; auto with zarith. - apply Zis_gcd_minus. - rewrite Z.opp_involutive, <- H1; apply Zgcd_is_gcd. - rewrite Z.abs_neq; romega. - Qed. - - Definition div x y := mul x (inv y). - - Theorem spec_div x y: [div x y] == [x] / [y]. - Proof. - unfold div; rewrite spec_mul; auto. - unfold Qdiv; apply Qmult_comp. - apply Qeq_refl. - apply spec_inv; auto. - Qed. - - Definition div_norm x y := mul_norm x (inv_norm y). - - Theorem spec_div_norm x y: [div_norm x y] == [x] / [y]. - Proof. - unfold div_norm; rewrite spec_mul_norm; auto. - unfold Qdiv; apply Qmult_comp. - apply Qeq_refl. - apply spec_inv_norm; auto. - Qed. - - Instance strong_spec_div_norm x y - `(Reduced x, Reduced y) : Reduced (div_norm x y). - Proof. - intros; unfold div_norm. - apply strong_spec_mul_norm; auto. - apply strong_spec_inv_norm; auto. - Qed. - - Definition square (x: t): t := - match x with - | Qz zx => Qz (ZZ.square zx) - | Qq nx dx => Qq (ZZ.square nx) (NN.square dx) - end. - - Theorem spec_square : forall x, [square x] == [x] ^ 2. - Proof. - destruct x as [ z | n d ]. - simpl; rewrite ZZ.spec_square; red; auto. - simpl. - destr_eqb; nzsimpl; intros. - apply Qeq_refl. - rewrite NN.spec_square in *; nzsimpl. - rewrite Z.mul_eq_0 in *; romega. - rewrite NN.spec_square in *; nzsimpl; nsubst; romega. - rewrite ZZ.spec_square, NN.spec_square. - red; simpl. - rewrite Pos2Z.inj_mul; rewrite !Z2Pos.id; auto. - apply Z.mul_pos_pos; auto. - Qed. - - Definition power_pos (x : t) p : t := - match x with - | Qz zx => Qz (ZZ.pow_pos zx p) - | Qq nx dx => Qq (ZZ.pow_pos nx p) (NN.pow_pos dx p) - end. - - Theorem spec_power_pos : forall x p, [power_pos x p] == [x] ^ Zpos p. - Proof. - intros [ z | n d ] p; unfold power_pos. - (* Qz *) - simpl. - rewrite ZZ.spec_pow_pos, Qpower_decomp. - red; simpl; f_equal. - now rewrite Pos2Z.inj_pow, Z.pow_1_l. - (* Qq *) - simpl. - rewrite ZZ.spec_pow_pos. - destr_eqb; nzsimpl; intros. - - apply Qeq_sym; apply Qpower_positive_0. - - rewrite NN.spec_pow_pos in *. - assert (0 < NN.to_Z d ^ ' p)%Z by - (apply Z.pow_pos_nonneg; auto with zarith). - romega. - - exfalso. - rewrite NN.spec_pow_pos in *. nsubst. - rewrite Z.pow_0_l' in *; [romega|discriminate]. - - rewrite Qpower_decomp. - red; simpl; do 3 f_equal. - apply Pos2Z.inj. rewrite Pos2Z.inj_pow. - rewrite 2 Z2Pos.id by (generalize (NN.spec_pos d); romega). - now rewrite NN.spec_pow_pos. - Qed. - - Instance strong_spec_power_pos x p `(Reduced x) : Reduced (power_pos x p). - Proof. - destruct x as [z | n d]; simpl; intros. - red; simpl; auto. - red; simpl; intros. - rewrite strong_spec_norm; simpl. - destr_eqb; nzsimpl; intros. - simpl; auto. - rewrite Qred_iff. - revert H. - unfold Reduced; rewrite strong_spec_red, Qred_iff; simpl. - destr_eqb; nzsimpl; simpl; intros. - exfalso. - rewrite NN.spec_pow_pos in *. nsubst. - rewrite Z.pow_0_l' in *; [romega|discriminate]. - rewrite Z2Pos.id in *; auto. - rewrite NN.spec_pow_pos, ZZ.spec_pow_pos; auto. - rewrite Zgcd_1_rel_prime in *. - apply rel_prime_Zpower; auto with zarith. - Qed. - - Definition power (x : t) (z : Z) : t := - match z with - | Z0 => one - | Zpos p => power_pos x p - | Zneg p => inv (power_pos x p) - end. - - Theorem spec_power : forall x z, [power x z] == [x]^z. - Proof. - destruct z. - simpl; nzsimpl; red; auto. - apply spec_power_pos. - simpl. - rewrite spec_inv, spec_power_pos; apply Qeq_refl. - Qed. - - Definition power_norm (x : t) (z : Z) : t := - match z with - | Z0 => one - | Zpos p => power_pos x p - | Zneg p => inv_norm (power_pos x p) - end. - - Theorem spec_power_norm : forall x z, [power_norm x z] == [x]^z. - Proof. - destruct z. - simpl; nzsimpl; red; auto. - apply spec_power_pos. - simpl. - rewrite spec_inv_norm, spec_power_pos; apply Qeq_refl. - Qed. - - Instance strong_spec_power_norm x z : - Reduced x -> Reduced (power_norm x z). - Proof. - destruct z; simpl. - intros _; unfold Reduced; rewrite strong_spec_red. - unfold one. - simpl to_Q; nzsimpl; auto. - intros; apply strong_spec_power_pos; auto. - intros; apply strong_spec_inv_norm; apply strong_spec_power_pos; auto. - Qed. - - - (** Interaction with [Qcanon.Qc] *) - - Open Scope Qc_scope. - - Definition of_Qc q := of_Q (this q). - - Definition to_Qc q := Q2Qc [q]. - - Notation "[[ x ]]" := (to_Qc x). - - Theorem strong_spec_of_Qc : forall q, [of_Qc q] = q. - Proof. - intros (q,Hq); intros. - unfold of_Qc; rewrite strong_spec_of_Q; auto. - Qed. - - Instance strong_spec_of_Qc_bis q : Reduced (of_Qc q). - Proof. - intros; red; rewrite strong_spec_red, strong_spec_of_Qc. - destruct q; simpl; auto. - Qed. - - Theorem spec_of_Qc: forall q, [[of_Qc q]] = q. - Proof. - intros; apply Qc_decomp; simpl; intros. - rewrite strong_spec_of_Qc. apply canon. - Qed. - - Theorem spec_oppc: forall q, [[opp q]] = -[[q]]. - Proof. - intros q; unfold Qcopp, to_Qc, Q2Qc. - apply Qc_decomp; unfold this. - apply Qred_complete. - rewrite spec_opp, <- Qred_opp, Qred_correct. - apply Qeq_refl. - Qed. - - Theorem spec_oppc_bis : forall q : Qc, [opp (of_Qc q)] = - q. - Proof. - intros. - rewrite <- strong_spec_opp_norm by apply strong_spec_of_Qc_bis. - rewrite strong_spec_red. - symmetry; apply (Qred_complete (-q)%Q). - rewrite spec_opp, strong_spec_of_Qc; auto with qarith. - Qed. - - Theorem spec_comparec: forall q1 q2, - compare q1 q2 = ([[q1]] ?= [[q2]]). - Proof. - unfold Qccompare, to_Qc. - intros q1 q2; rewrite spec_compare; simpl; auto. - apply Qcompare_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Theorem spec_addc x y: - [[add x y]] = [[x]] + [[y]]. - Proof. - unfold to_Qc. - transitivity (Q2Qc ([x] + [y])). - unfold Q2Qc. - apply Qc_decomp; unfold this. - apply Qred_complete; apply spec_add; auto. - unfold Qcplus, Q2Qc. - apply Qc_decomp; unfold this. - apply Qred_complete. - apply Qplus_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Theorem spec_add_normc x y: - [[add_norm x y]] = [[x]] + [[y]]. - Proof. - unfold to_Qc. - transitivity (Q2Qc ([x] + [y])). - unfold Q2Qc. - apply Qc_decomp; unfold this. - apply Qred_complete; apply spec_add_norm; auto. - unfold Qcplus, Q2Qc. - apply Qc_decomp; unfold this. - apply Qred_complete. - apply Qplus_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Theorem spec_add_normc_bis : forall x y : Qc, - [add_norm (of_Qc x) (of_Qc y)] = x+y. - Proof. - intros. - rewrite <- strong_spec_add_norm by apply strong_spec_of_Qc_bis. - rewrite strong_spec_red. - symmetry; apply (Qred_complete (x+y)%Q). - rewrite spec_add_norm, ! strong_spec_of_Qc; auto with qarith. - Qed. - - Theorem spec_subc x y: [[sub x y]] = [[x]] - [[y]]. - Proof. - unfold sub; rewrite spec_addc; auto. - rewrite spec_oppc; ring. - Qed. - - Theorem spec_sub_normc x y: - [[sub_norm x y]] = [[x]] - [[y]]. - Proof. - unfold sub_norm; rewrite spec_add_normc; auto. - rewrite spec_oppc; ring. - Qed. - - Theorem spec_sub_normc_bis : forall x y : Qc, - [sub_norm (of_Qc x) (of_Qc y)] = x-y. - Proof. - intros. - rewrite <- strong_spec_sub_norm by apply strong_spec_of_Qc_bis. - rewrite strong_spec_red. - symmetry; apply (Qred_complete (x+(-y)%Qc)%Q). - rewrite spec_sub_norm, ! strong_spec_of_Qc. - unfold Qcopp, Q2Qc, this. rewrite Qred_correct ; auto with qarith. - Qed. - - Theorem spec_mulc x y: - [[mul x y]] = [[x]] * [[y]]. - Proof. - unfold to_Qc. - transitivity (Q2Qc ([x] * [y])). - unfold Q2Qc. - apply Qc_decomp; unfold this. - apply Qred_complete; apply spec_mul; auto. - unfold Qcmult, Q2Qc. - apply Qc_decomp; unfold this. - apply Qred_complete. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Theorem spec_mul_normc x y: - [[mul_norm x y]] = [[x]] * [[y]]. - Proof. - unfold to_Qc. - transitivity (Q2Qc ([x] * [y])). - unfold Q2Qc. - apply Qc_decomp; unfold this. - apply Qred_complete; apply spec_mul_norm; auto. - unfold Qcmult, Q2Qc. - apply Qc_decomp; unfold this. - apply Qred_complete. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Theorem spec_mul_normc_bis : forall x y : Qc, - [mul_norm (of_Qc x) (of_Qc y)] = x*y. - Proof. - intros. - rewrite <- strong_spec_mul_norm by apply strong_spec_of_Qc_bis. - rewrite strong_spec_red. - symmetry; apply (Qred_complete (x*y)%Q). - rewrite spec_mul_norm, ! strong_spec_of_Qc; auto with qarith. - Qed. - - Theorem spec_invc x: - [[inv x]] = /[[x]]. - Proof. - unfold to_Qc. - transitivity (Q2Qc (/[x])). - unfold Q2Qc. - apply Qc_decomp; unfold this. - apply Qred_complete; apply spec_inv; auto. - unfold Qcinv, Q2Qc. - apply Qc_decomp; unfold this. - apply Qred_complete. - apply Qinv_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Theorem spec_inv_normc x: - [[inv_norm x]] = /[[x]]. - Proof. - unfold to_Qc. - transitivity (Q2Qc (/[x])). - unfold Q2Qc. - apply Qc_decomp; unfold this. - apply Qred_complete; apply spec_inv_norm; auto. - unfold Qcinv, Q2Qc. - apply Qc_decomp; unfold this. - apply Qred_complete. - apply Qinv_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Theorem spec_inv_normc_bis : forall x : Qc, - [inv_norm (of_Qc x)] = /x. - Proof. - intros. - rewrite <- strong_spec_inv_norm by apply strong_spec_of_Qc_bis. - rewrite strong_spec_red. - symmetry; apply (Qred_complete (/x)%Q). - rewrite spec_inv_norm, ! strong_spec_of_Qc; auto with qarith. - Qed. - - Theorem spec_divc x y: [[div x y]] = [[x]] / [[y]]. - Proof. - unfold div; rewrite spec_mulc; auto. - unfold Qcdiv; apply f_equal2 with (f := Qcmult); auto. - apply spec_invc; auto. - Qed. - - Theorem spec_div_normc x y: [[div_norm x y]] = [[x]] / [[y]]. - Proof. - unfold div_norm; rewrite spec_mul_normc; auto. - unfold Qcdiv; apply f_equal2 with (f := Qcmult); auto. - apply spec_inv_normc; auto. - Qed. - - Theorem spec_div_normc_bis : forall x y : Qc, - [div_norm (of_Qc x) (of_Qc y)] = x/y. - Proof. - intros. - rewrite <- strong_spec_div_norm by apply strong_spec_of_Qc_bis. - rewrite strong_spec_red. - symmetry; apply (Qred_complete (x*(/y)%Qc)%Q). - rewrite spec_div_norm, ! strong_spec_of_Qc. - unfold Qcinv, Q2Qc, this; rewrite Qred_correct; auto with qarith. - Qed. - - Theorem spec_squarec x: [[square x]] = [[x]]^2. - Proof. - unfold to_Qc. - transitivity (Q2Qc ([x]^2)). - unfold Q2Qc. - apply Qc_decomp; unfold this. - apply Qred_complete; apply spec_square; auto. - simpl Qcpower. - replace (Q2Qc [x] * 1) with (Q2Qc [x]); try ring. - simpl. - unfold Qcmult, Q2Qc. - apply Qc_decomp; unfold this. - apply Qred_complete. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Theorem spec_power_posc x p: - [[power_pos x p]] = [[x]] ^ Pos.to_nat p. - Proof. - unfold to_Qc. - transitivity (Q2Qc ([x]^Zpos p)). - unfold Q2Qc. - apply Qc_decomp; unfold this. - apply Qred_complete; apply spec_power_pos; auto. - induction p using Pos.peano_ind. - simpl; ring. - rewrite Pos2Nat.inj_succ; simpl Qcpower. - rewrite <- IHp; clear IHp. - unfold Qcmult, Q2Qc. - apply Qc_decomp; unfold this. - apply Qred_complete. - setoid_replace ([x] ^ ' Pos.succ p)%Q with ([x] * [x] ^ ' p)%Q. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - simpl. - rewrite <- Pos.add_1_l. - rewrite Qpower_plus_positive; simpl; apply Qeq_refl. - Qed. - -End Make. diff --git a/theories/Numbers/Rational/SpecViaQ/QSig.v b/theories/Numbers/Rational/SpecViaQ/QSig.v deleted file mode 100644 index 8e20fd06..00000000 --- a/theories/Numbers/Rational/SpecViaQ/QSig.v +++ /dev/null @@ -1,229 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -Require Import QArith Qpower Qminmax Orders RelationPairs GenericMinMax. - -Open Scope Q_scope. - -(** * QSig *) - -(** Interface of a rich structure about rational numbers. - Specifications are written via translation to Q. -*) - -Module Type QType. - - Parameter t : Type. - - Parameter to_Q : t -> Q. - Local Notation "[ x ]" := (to_Q x). - - Definition eq x y := [x] == [y]. - Definition lt x y := [x] < [y]. - Definition le x y := [x] <= [y]. - - Parameter of_Q : Q -> t. - Parameter spec_of_Q: forall x, to_Q (of_Q x) == x. - - Parameter red : t -> t. - Parameter compare : t -> t -> comparison. - Parameter eq_bool : t -> t -> bool. - Parameter max : t -> t -> t. - Parameter min : t -> t -> t. - Parameter zero : t. - Parameter one : t. - Parameter minus_one : t. - Parameter add : t -> t -> t. - Parameter sub : t -> t -> t. - Parameter opp : t -> t. - Parameter mul : t -> t -> t. - Parameter square : t -> t. - Parameter inv : t -> t. - Parameter div : t -> t -> t. - Parameter power : t -> Z -> t. - - Parameter spec_red : forall x, [red x] == [x]. - Parameter strong_spec_red : forall x, [red x] = Qred [x]. - Parameter spec_compare : forall x y, compare x y = ([x] ?= [y]). - Parameter spec_eq_bool : forall x y, eq_bool x y = Qeq_bool [x] [y]. - Parameter spec_max : forall x y, [max x y] == Qmax [x] [y]. - Parameter spec_min : forall x y, [min x y] == Qmin [x] [y]. - Parameter spec_0: [zero] == 0. - Parameter spec_1: [one] == 1. - Parameter spec_m1: [minus_one] == -(1). - Parameter spec_add: forall x y, [add x y] == [x] + [y]. - Parameter spec_sub: forall x y, [sub x y] == [x] - [y]. - Parameter spec_opp: forall x, [opp x] == - [x]. - Parameter spec_mul: forall x y, [mul x y] == [x] * [y]. - Parameter spec_square: forall x, [square x] == [x] ^ 2. - Parameter spec_inv : forall x, [inv x] == / [x]. - Parameter spec_div: forall x y, [div x y] == [x] / [y]. - Parameter spec_power: forall x z, [power x z] == [x] ^ z. - -End QType. - -(** NB: several of the above functions come with [..._norm] variants - that expect reduced arguments and return reduced results. *) - -(** TODO : also speak of specifications via Qcanon ... *) - -Module Type QType_Notation (Import Q : QType). - Notation "[ x ]" := (to_Q x). - Infix "==" := eq (at level 70). - Notation "x != y" := (~x==y) (at level 70). - Infix "<=" := le. - Infix "<" := lt. - Notation "0" := zero. - Notation "1" := one. - Infix "+" := add. - Infix "-" := sub. - Infix "*" := mul. - Notation "- x" := (opp x). - Infix "/" := div. - Notation "/ x" := (inv x). - Infix "^" := power. -End QType_Notation. - -Module Type QType' := QType <+ QType_Notation. - - -Module QProperties (Import Q : QType'). - -(** Conversion to Q *) - -Hint Rewrite - spec_red spec_compare spec_eq_bool spec_min spec_max - spec_add spec_sub spec_opp spec_mul spec_square spec_inv spec_div - spec_power : qsimpl. -Ltac qify := unfold eq, lt, le in *; autorewrite with qsimpl; - try rewrite spec_0 in *; try rewrite spec_1 in *; try rewrite spec_m1 in *. - -(** NB: do not add [spec_0] in the autorewrite database. Otherwise, - after instantiation in BigQ, this lemma become convertible to 0=0, - and autorewrite loops. Idem for [spec_1] and [spec_m1] *) - -(** Morphisms *) - -Ltac solve_wd1 := intros x x' Hx; qify; now rewrite Hx. -Ltac solve_wd2 := intros x x' Hx y y' Hy; qify; now rewrite Hx, Hy. - -Local Obligation Tactic := solve_wd2 || solve_wd1. - -Instance : Measure to_Q. -Instance eq_equiv : Equivalence eq. -Proof. - change eq with (RelCompFun Qeq to_Q); apply _. -Defined. - -Program Instance lt_wd : Proper (eq==>eq==>iff) lt. -Program Instance le_wd : Proper (eq==>eq==>iff) le. -Program Instance red_wd : Proper (eq==>eq) red. -Program Instance compare_wd : Proper (eq==>eq==>Logic.eq) compare. -Program Instance eq_bool_wd : Proper (eq==>eq==>Logic.eq) eq_bool. -Program Instance min_wd : Proper (eq==>eq==>eq) min. -Program Instance max_wd : Proper (eq==>eq==>eq) max. -Program Instance add_wd : Proper (eq==>eq==>eq) add. -Program Instance sub_wd : Proper (eq==>eq==>eq) sub. -Program Instance opp_wd : Proper (eq==>eq) opp. -Program Instance mul_wd : Proper (eq==>eq==>eq) mul. -Program Instance square_wd : Proper (eq==>eq) square. -Program Instance inv_wd : Proper (eq==>eq) inv. -Program Instance div_wd : Proper (eq==>eq==>eq) div. -Program Instance power_wd : Proper (eq==>Logic.eq==>eq) power. - -(** Let's implement [HasCompare] *) - -Lemma compare_spec : forall x y, CompareSpec (x==y) (x<y) (y<x) (compare x y). -Proof. intros. qify. destruct (Qcompare_spec [x] [y]); auto. Qed. - -(** Let's implement [TotalOrder] *) - -Definition lt_compat := lt_wd. -Instance lt_strorder : StrictOrder lt. -Proof. - change lt with (RelCompFun Qlt to_Q); apply _. -Qed. - -Lemma le_lteq : forall x y, x<=y <-> x<y \/ x==y. -Proof. intros. qify. apply Qle_lteq. Qed. - -Lemma lt_total : forall x y, x<y \/ x==y \/ y<x. -Proof. intros. destruct (compare_spec x y); auto. Qed. - -(** Let's implement [HasEqBool] *) - -Definition eqb := eq_bool. - -Lemma eqb_eq : forall x y, eq_bool x y = true <-> x == y. -Proof. intros. qify. apply Qeq_bool_iff. Qed. - -Lemma eqb_correct : forall x y, eq_bool x y = true -> x == y. -Proof. now apply eqb_eq. Qed. - -Lemma eqb_complete : forall x y, x == y -> eq_bool x y = true. -Proof. now apply eqb_eq. Qed. - -(** Let's implement [HasMinMax] *) - -Lemma max_l : forall x y, y<=x -> max x y == x. -Proof. intros x y. qify. apply Qminmax.Q.max_l. Qed. - -Lemma max_r : forall x y, x<=y -> max x y == y. -Proof. intros x y. qify. apply Qminmax.Q.max_r. Qed. - -Lemma min_l : forall x y, x<=y -> min x y == x. -Proof. intros x y. qify. apply Qminmax.Q.min_l. Qed. - -Lemma min_r : forall x y, y<=x -> min x y == y. -Proof. intros x y. qify. apply Qminmax.Q.min_r. Qed. - -(** Q is a ring *) - -Lemma add_0_l : forall x, 0+x == x. -Proof. intros. qify. apply Qplus_0_l. Qed. - -Lemma add_comm : forall x y, x+y == y+x. -Proof. intros. qify. apply Qplus_comm. Qed. - -Lemma add_assoc : forall x y z, x+(y+z) == x+y+z. -Proof. intros. qify. apply Qplus_assoc. Qed. - -Lemma mul_1_l : forall x, 1*x == x. -Proof. intros. qify. apply Qmult_1_l. Qed. - -Lemma mul_comm : forall x y, x*y == y*x. -Proof. intros. qify. apply Qmult_comm. Qed. - -Lemma mul_assoc : forall x y z, x*(y*z) == x*y*z. -Proof. intros. qify. apply Qmult_assoc. Qed. - -Lemma mul_add_distr_r : forall x y z, (x+y)*z == x*z + y*z. -Proof. intros. qify. apply Qmult_plus_distr_l. Qed. - -Lemma sub_add_opp : forall x y, x-y == x+(-y). -Proof. intros. qify. now unfold Qminus. Qed. - -Lemma add_opp_diag_r : forall x, x+(-x) == 0. -Proof. intros. qify. apply Qplus_opp_r. Qed. - -(** Q is a field *) - -Lemma neq_1_0 : 1!=0. -Proof. intros. qify. apply Q_apart_0_1. Qed. - -Lemma div_mul_inv : forall x y, x/y == x*(/y). -Proof. intros. qify. now unfold Qdiv. Qed. - -Lemma mul_inv_diag_l : forall x, x!=0 -> /x * x == 1. -Proof. intros x. qify. rewrite Qmult_comm. apply Qmult_inv_r. Qed. - -End QProperties. - -Module QTypeExt (Q : QType) - <: QType <: TotalOrder <: HasCompare Q <: HasMinMax Q <: HasEqBool Q - := Q <+ QProperties. diff --git a/theories/Numbers/vo.itarget b/theories/Numbers/vo.itarget deleted file mode 100644 index c69af03f..00000000 --- a/theories/Numbers/vo.itarget +++ /dev/null @@ -1,91 +0,0 @@ -BinNums.vo -BigNumPrelude.vo -Cyclic/Abstract/CyclicAxioms.vo -Cyclic/Abstract/NZCyclic.vo -Cyclic/DoubleCyclic/DoubleAdd.vo -Cyclic/DoubleCyclic/DoubleBase.vo -Cyclic/DoubleCyclic/DoubleCyclic.vo -Cyclic/DoubleCyclic/DoubleDivn1.vo -Cyclic/DoubleCyclic/DoubleDiv.vo -Cyclic/DoubleCyclic/DoubleLift.vo -Cyclic/DoubleCyclic/DoubleMul.vo -Cyclic/DoubleCyclic/DoubleSqrt.vo -Cyclic/DoubleCyclic/DoubleSub.vo -Cyclic/DoubleCyclic/DoubleType.vo -Cyclic/Int31/Int31.vo -Cyclic/Int31/Cyclic31.vo -Cyclic/Int31/Ring31.vo -Cyclic/ZModulo/ZModulo.vo -Integer/Abstract/ZAddOrder.vo -Integer/Abstract/ZAdd.vo -Integer/Abstract/ZAxioms.vo -Integer/Abstract/ZBase.vo -Integer/Abstract/ZLt.vo -Integer/Abstract/ZMulOrder.vo -Integer/Abstract/ZMul.vo -Integer/Abstract/ZSgnAbs.vo -Integer/Abstract/ZDivFloor.vo -Integer/Abstract/ZDivTrunc.vo -Integer/Abstract/ZDivEucl.vo -Integer/Abstract/ZMaxMin.vo -Integer/Abstract/ZParity.vo -Integer/Abstract/ZPow.vo -Integer/Abstract/ZGcd.vo -Integer/Abstract/ZLcm.vo -Integer/Abstract/ZBits.vo -Integer/Abstract/ZProperties.vo -Integer/BigZ/BigZ.vo -Integer/BigZ/ZMake.vo -Integer/Binary/ZBinary.vo -Integer/NatPairs/ZNatPairs.vo -Integer/SpecViaZ/ZSig.vo -Integer/SpecViaZ/ZSigZAxioms.vo -NaryFunctions.vo -NatInt/NZAddOrder.vo -NatInt/NZAdd.vo -NatInt/NZAxioms.vo -NatInt/NZBase.vo -NatInt/NZMulOrder.vo -NatInt/NZMul.vo -NatInt/NZOrder.vo -NatInt/NZProperties.vo -NatInt/NZDomain.vo -NatInt/NZParity.vo -NatInt/NZDiv.vo -NatInt/NZPow.vo -NatInt/NZSqrt.vo -NatInt/NZLog.vo -NatInt/NZGcd.vo -NatInt/NZBits.vo -Natural/Abstract/NAddOrder.vo -Natural/Abstract/NAdd.vo -Natural/Abstract/NAxioms.vo -Natural/Abstract/NBase.vo -Natural/Abstract/NDefOps.vo -Natural/Abstract/NIso.vo -Natural/Abstract/NMulOrder.vo -Natural/Abstract/NOrder.vo -Natural/Abstract/NStrongRec.vo -Natural/Abstract/NSub.vo -Natural/Abstract/NProperties.vo -Natural/Abstract/NDiv.vo -Natural/Abstract/NMaxMin.vo -Natural/Abstract/NParity.vo -Natural/Abstract/NPow.vo -Natural/Abstract/NSqrt.vo -Natural/Abstract/NLog.vo -Natural/Abstract/NGcd.vo -Natural/Abstract/NLcm.vo -Natural/Abstract/NBits.vo -Natural/BigN/BigN.vo -Natural/BigN/Nbasic.vo -Natural/BigN/NMake_gen.vo -Natural/BigN/NMake.vo -Natural/Binary/NBinary.vo -Natural/Peano/NPeano.vo -Natural/SpecViaZ/NSigNAxioms.vo -Natural/SpecViaZ/NSig.vo -NumPrelude.vo -Rational/BigQ/BigQ.vo -Rational/BigQ/QMake.vo -Rational/SpecViaQ/QSig.vo diff --git a/theories/PArith/BinPos.v b/theories/PArith/BinPos.v index 7baf102a..000d895e 100644 --- a/theories/PArith/BinPos.v +++ b/theories/PArith/BinPos.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export BinNums. @@ -813,6 +815,34 @@ Proof. rewrite compare_cont_spec. unfold ge. destruct (p ?= q); easy'. Qed. +Lemma compare_cont_Lt_not_Lt p q : + compare_cont Lt p q <> Lt <-> p > q. +Proof. + rewrite compare_cont_Lt_Lt. + unfold gt, le, compare. + now destruct compare_cont; split; try apply comparison_eq_stable. +Qed. + +Lemma compare_cont_Lt_not_Gt p q : + compare_cont Lt p q <> Gt <-> p <= q. +Proof. + apply not_iff_compat, compare_cont_Lt_Gt. +Qed. + +Lemma compare_cont_Gt_not_Lt p q : + compare_cont Gt p q <> Lt <-> p >= q. +Proof. + apply not_iff_compat, compare_cont_Gt_Lt. +Qed. + +Lemma compare_cont_Gt_not_Gt p q : + compare_cont Gt p q <> Gt <-> p < q. +Proof. + rewrite compare_cont_Gt_Gt. + unfold ge, lt, compare. + now destruct compare_cont; split; try apply comparison_eq_stable. +Qed. + (** We can express recursive equations for [compare] *) Lemma compare_xO_xO p q : (p~0 ?= q~0) = (p ?= q). @@ -1625,7 +1655,7 @@ Qed. Lemma sqrtrem_spec p : SqrtSpec (sqrtrem p) p. Proof. -revert p. fix 1. +revert p. fix sqrtrem_spec 1. destruct p; try destruct p; try (constructor; easy); apply sqrtrem_step_spec; auto. Qed. @@ -1875,180 +1905,180 @@ Notation IsNul := Pos.IsNul (only parsing). Notation IsPos := Pos.IsPos (only parsing). Notation IsNeg := Pos.IsNeg (only parsing). -Notation Psucc := Pos.succ (compat "8.3"). -Notation Pplus := Pos.add (compat "8.3"). -Notation Pplus_carry := Pos.add_carry (compat "8.3"). -Notation Ppred := Pos.pred (compat "8.3"). -Notation Piter_op := Pos.iter_op (compat "8.3"). -Notation Piter_op_succ := Pos.iter_op_succ (compat "8.3"). -Notation Pmult_nat := (Pos.iter_op plus) (compat "8.3"). -Notation nat_of_P := Pos.to_nat (compat "8.3"). -Notation P_of_succ_nat := Pos.of_succ_nat (compat "8.3"). -Notation Pdouble_minus_one := Pos.pred_double (compat "8.3"). -Notation positive_mask := Pos.mask (compat "8.3"). -Notation positive_mask_rect := Pos.mask_rect (compat "8.3"). -Notation positive_mask_ind := Pos.mask_ind (compat "8.3"). -Notation positive_mask_rec := Pos.mask_rec (compat "8.3"). -Notation Pdouble_plus_one_mask := Pos.succ_double_mask (compat "8.3"). -Notation Pdouble_mask := Pos.double_mask (compat "8.3"). -Notation Pdouble_minus_two := Pos.double_pred_mask (compat "8.3"). -Notation Pminus_mask := Pos.sub_mask (compat "8.3"). -Notation Pminus_mask_carry := Pos.sub_mask_carry (compat "8.3"). -Notation Pminus := Pos.sub (compat "8.3"). -Notation Pmult := Pos.mul (compat "8.3"). -Notation iter_pos := @Pos.iter (compat "8.3"). -Notation Ppow := Pos.pow (compat "8.3"). -Notation Pdiv2 := Pos.div2 (compat "8.3"). -Notation Pdiv2_up := Pos.div2_up (compat "8.3"). -Notation Psize := Pos.size_nat (compat "8.3"). -Notation Psize_pos := Pos.size (compat "8.3"). -Notation Pcompare x y m := (Pos.compare_cont m x y) (compat "8.3"). -Notation Plt := Pos.lt (compat "8.3"). -Notation Pgt := Pos.gt (compat "8.3"). -Notation Ple := Pos.le (compat "8.3"). -Notation Pge := Pos.ge (compat "8.3"). -Notation Pmin := Pos.min (compat "8.3"). -Notation Pmax := Pos.max (compat "8.3"). -Notation Peqb := Pos.eqb (compat "8.3"). -Notation positive_eq_dec := Pos.eq_dec (compat "8.3"). -Notation xI_succ_xO := Pos.xI_succ_xO (compat "8.3"). -Notation Psucc_discr := Pos.succ_discr (compat "8.3"). +Notation Psucc := Pos.succ (compat "8.6"). +Notation Pplus := Pos.add (only parsing). +Notation Pplus_carry := Pos.add_carry (only parsing). +Notation Ppred := Pos.pred (compat "8.6"). +Notation Piter_op := Pos.iter_op (compat "8.6"). +Notation Piter_op_succ := Pos.iter_op_succ (compat "8.6"). +Notation Pmult_nat := (Pos.iter_op plus) (only parsing). +Notation nat_of_P := Pos.to_nat (only parsing). +Notation P_of_succ_nat := Pos.of_succ_nat (only parsing). +Notation Pdouble_minus_one := Pos.pred_double (only parsing). +Notation positive_mask := Pos.mask (only parsing). +Notation positive_mask_rect := Pos.mask_rect (only parsing). +Notation positive_mask_ind := Pos.mask_ind (only parsing). +Notation positive_mask_rec := Pos.mask_rec (only parsing). +Notation Pdouble_plus_one_mask := Pos.succ_double_mask (only parsing). +Notation Pdouble_mask := Pos.double_mask (compat "8.6"). +Notation Pdouble_minus_two := Pos.double_pred_mask (only parsing). +Notation Pminus_mask := Pos.sub_mask (only parsing). +Notation Pminus_mask_carry := Pos.sub_mask_carry (only parsing). +Notation Pminus := Pos.sub (only parsing). +Notation Pmult := Pos.mul (only parsing). +Notation iter_pos := @Pos.iter (only parsing). +Notation Ppow := Pos.pow (compat "8.6"). +Notation Pdiv2 := Pos.div2 (compat "8.6"). +Notation Pdiv2_up := Pos.div2_up (compat "8.6"). +Notation Psize := Pos.size_nat (only parsing). +Notation Psize_pos := Pos.size (only parsing). +Notation Pcompare x y m := (Pos.compare_cont m x y) (only parsing). +Notation Plt := Pos.lt (compat "8.6"). +Notation Pgt := Pos.gt (compat "8.6"). +Notation Ple := Pos.le (compat "8.6"). +Notation Pge := Pos.ge (compat "8.6"). +Notation Pmin := Pos.min (compat "8.6"). +Notation Pmax := Pos.max (compat "8.6"). +Notation Peqb := Pos.eqb (compat "8.6"). +Notation positive_eq_dec := Pos.eq_dec (only parsing). +Notation xI_succ_xO := Pos.xI_succ_xO (only parsing). +Notation Psucc_discr := Pos.succ_discr (compat "8.6"). Notation Psucc_o_double_minus_one_eq_xO := - Pos.succ_pred_double (compat "8.3"). + Pos.succ_pred_double (only parsing). Notation Pdouble_minus_one_o_succ_eq_xI := - Pos.pred_double_succ (compat "8.3"). -Notation xO_succ_permute := Pos.double_succ (compat "8.3"). + Pos.pred_double_succ (only parsing). +Notation xO_succ_permute := Pos.double_succ (only parsing). Notation double_moins_un_xO_discr := - Pos.pred_double_xO_discr (compat "8.3"). -Notation Psucc_not_one := Pos.succ_not_1 (compat "8.3"). -Notation Ppred_succ := Pos.pred_succ (compat "8.3"). -Notation Psucc_pred := Pos.succ_pred_or (compat "8.3"). -Notation Psucc_inj := Pos.succ_inj (compat "8.3"). -Notation Pplus_carry_spec := Pos.add_carry_spec (compat "8.3"). -Notation Pplus_comm := Pos.add_comm (compat "8.3"). -Notation Pplus_succ_permute_r := Pos.add_succ_r (compat "8.3"). -Notation Pplus_succ_permute_l := Pos.add_succ_l (compat "8.3"). -Notation Pplus_no_neutral := Pos.add_no_neutral (compat "8.3"). -Notation Pplus_carry_plus := Pos.add_carry_add (compat "8.3"). -Notation Pplus_reg_r := Pos.add_reg_r (compat "8.3"). -Notation Pplus_reg_l := Pos.add_reg_l (compat "8.3"). -Notation Pplus_carry_reg_r := Pos.add_carry_reg_r (compat "8.3"). -Notation Pplus_carry_reg_l := Pos.add_carry_reg_l (compat "8.3"). -Notation Pplus_assoc := Pos.add_assoc (compat "8.3"). -Notation Pplus_xO := Pos.add_xO (compat "8.3"). -Notation Pplus_xI_double_minus_one := Pos.add_xI_pred_double (compat "8.3"). -Notation Pplus_xO_double_minus_one := Pos.add_xO_pred_double (compat "8.3"). -Notation Pplus_diag := Pos.add_diag (compat "8.3"). -Notation PeanoView := Pos.PeanoView (compat "8.3"). -Notation PeanoOne := Pos.PeanoOne (compat "8.3"). -Notation PeanoSucc := Pos.PeanoSucc (compat "8.3"). -Notation PeanoView_rect := Pos.PeanoView_rect (compat "8.3"). -Notation PeanoView_ind := Pos.PeanoView_ind (compat "8.3"). -Notation PeanoView_rec := Pos.PeanoView_rec (compat "8.3"). -Notation peanoView_xO := Pos.peanoView_xO (compat "8.3"). -Notation peanoView_xI := Pos.peanoView_xI (compat "8.3"). -Notation peanoView := Pos.peanoView (compat "8.3"). -Notation PeanoView_iter := Pos.PeanoView_iter (compat "8.3"). -Notation eq_dep_eq_positive := Pos.eq_dep_eq_positive (compat "8.3"). -Notation PeanoViewUnique := Pos.PeanoViewUnique (compat "8.3"). -Notation Prect := Pos.peano_rect (compat "8.3"). -Notation Prect_succ := Pos.peano_rect_succ (compat "8.3"). -Notation Prect_base := Pos.peano_rect_base (compat "8.3"). -Notation Prec := Pos.peano_rec (compat "8.3"). -Notation Pind := Pos.peano_ind (compat "8.3"). -Notation Pcase := Pos.peano_case (compat "8.3"). -Notation Pmult_1_r := Pos.mul_1_r (compat "8.3"). -Notation Pmult_Sn_m := Pos.mul_succ_l (compat "8.3"). -Notation Pmult_xO_permute_r := Pos.mul_xO_r (compat "8.3"). -Notation Pmult_xI_permute_r := Pos.mul_xI_r (compat "8.3"). -Notation Pmult_comm := Pos.mul_comm (compat "8.3"). -Notation Pmult_plus_distr_l := Pos.mul_add_distr_l (compat "8.3"). -Notation Pmult_plus_distr_r := Pos.mul_add_distr_r (compat "8.3"). -Notation Pmult_assoc := Pos.mul_assoc (compat "8.3"). -Notation Pmult_xI_mult_xO_discr := Pos.mul_xI_mul_xO_discr (compat "8.3"). -Notation Pmult_xO_discr := Pos.mul_xO_discr (compat "8.3"). -Notation Pmult_reg_r := Pos.mul_reg_r (compat "8.3"). -Notation Pmult_reg_l := Pos.mul_reg_l (compat "8.3"). -Notation Pmult_1_inversion_l := Pos.mul_eq_1_l (compat "8.3"). -Notation Psquare_xO := Pos.square_xO (compat "8.3"). -Notation Psquare_xI := Pos.square_xI (compat "8.3"). -Notation iter_pos_swap_gen := Pos.iter_swap_gen (compat "8.3"). -Notation iter_pos_swap := Pos.iter_swap (compat "8.3"). -Notation iter_pos_succ := Pos.iter_succ (compat "8.3"). -Notation iter_pos_plus := Pos.iter_add (compat "8.3"). -Notation iter_pos_invariant := Pos.iter_invariant (compat "8.3"). -Notation Ppow_1_r := Pos.pow_1_r (compat "8.3"). -Notation Ppow_succ_r := Pos.pow_succ_r (compat "8.3"). -Notation Peqb_refl := Pos.eqb_refl (compat "8.3"). -Notation Peqb_eq := Pos.eqb_eq (compat "8.3"). -Notation Pcompare_refl_id := Pos.compare_cont_refl (compat "8.3"). -Notation Pcompare_eq_iff := Pos.compare_eq_iff (compat "8.3"). -Notation Pcompare_Gt_Lt := Pos.compare_cont_Gt_Lt (compat "8.3"). -Notation Pcompare_eq_Lt := Pos.compare_lt_iff (compat "8.3"). -Notation Pcompare_Lt_Gt := Pos.compare_cont_Lt_Gt (compat "8.3"). - -Notation Pcompare_antisym := Pos.compare_cont_antisym (compat "8.3"). -Notation ZC1 := Pos.gt_lt (compat "8.3"). -Notation ZC2 := Pos.lt_gt (compat "8.3"). -Notation Pcompare_spec := Pos.compare_spec (compat "8.3"). -Notation Pcompare_p_Sp := Pos.lt_succ_diag_r (compat "8.3"). -Notation Pcompare_succ_succ := Pos.compare_succ_succ (compat "8.3"). -Notation Pcompare_1 := Pos.nlt_1_r (compat "8.3"). -Notation Plt_1 := Pos.nlt_1_r (compat "8.3"). -Notation Plt_1_succ := Pos.lt_1_succ (compat "8.3"). -Notation Plt_lt_succ := Pos.lt_lt_succ (compat "8.3"). -Notation Plt_irrefl := Pos.lt_irrefl (compat "8.3"). -Notation Plt_trans := Pos.lt_trans (compat "8.3"). -Notation Plt_ind := Pos.lt_ind (compat "8.3"). -Notation Ple_lteq := Pos.le_lteq (compat "8.3"). -Notation Ple_refl := Pos.le_refl (compat "8.3"). -Notation Ple_lt_trans := Pos.le_lt_trans (compat "8.3"). -Notation Plt_le_trans := Pos.lt_le_trans (compat "8.3"). -Notation Ple_trans := Pos.le_trans (compat "8.3"). -Notation Plt_succ_r := Pos.lt_succ_r (compat "8.3"). -Notation Ple_succ_l := Pos.le_succ_l (compat "8.3"). -Notation Pplus_compare_mono_l := Pos.add_compare_mono_l (compat "8.3"). -Notation Pplus_compare_mono_r := Pos.add_compare_mono_r (compat "8.3"). -Notation Pplus_lt_mono_l := Pos.add_lt_mono_l (compat "8.3"). -Notation Pplus_lt_mono_r := Pos.add_lt_mono_r (compat "8.3"). -Notation Pplus_lt_mono := Pos.add_lt_mono (compat "8.3"). -Notation Pplus_le_mono_l := Pos.add_le_mono_l (compat "8.3"). -Notation Pplus_le_mono_r := Pos.add_le_mono_r (compat "8.3"). -Notation Pplus_le_mono := Pos.add_le_mono (compat "8.3"). -Notation Pmult_compare_mono_l := Pos.mul_compare_mono_l (compat "8.3"). -Notation Pmult_compare_mono_r := Pos.mul_compare_mono_r (compat "8.3"). -Notation Pmult_lt_mono_l := Pos.mul_lt_mono_l (compat "8.3"). -Notation Pmult_lt_mono_r := Pos.mul_lt_mono_r (compat "8.3"). -Notation Pmult_lt_mono := Pos.mul_lt_mono (compat "8.3"). -Notation Pmult_le_mono_l := Pos.mul_le_mono_l (compat "8.3"). -Notation Pmult_le_mono_r := Pos.mul_le_mono_r (compat "8.3"). -Notation Pmult_le_mono := Pos.mul_le_mono (compat "8.3"). -Notation Plt_plus_r := Pos.lt_add_r (compat "8.3"). -Notation Plt_not_plus_l := Pos.lt_not_add_l (compat "8.3"). -Notation Ppow_gt_1 := Pos.pow_gt_1 (compat "8.3"). -Notation Ppred_mask := Pos.pred_mask (compat "8.3"). -Notation Pminus_mask_succ_r := Pos.sub_mask_succ_r (compat "8.3"). -Notation Pminus_mask_carry_spec := Pos.sub_mask_carry_spec (compat "8.3"). -Notation Pminus_succ_r := Pos.sub_succ_r (compat "8.3"). -Notation Pminus_mask_diag := Pos.sub_mask_diag (compat "8.3"). - -Notation Pplus_minus_eq := Pos.add_sub (compat "8.3"). -Notation Pmult_minus_distr_l := Pos.mul_sub_distr_l (compat "8.3"). -Notation Pminus_lt_mono_l := Pos.sub_lt_mono_l (compat "8.3"). -Notation Pminus_compare_mono_l := Pos.sub_compare_mono_l (compat "8.3"). -Notation Pminus_compare_mono_r := Pos.sub_compare_mono_r (compat "8.3"). -Notation Pminus_lt_mono_r := Pos.sub_lt_mono_r (compat "8.3"). -Notation Pminus_decr := Pos.sub_decr (compat "8.3"). -Notation Pminus_xI_xI := Pos.sub_xI_xI (compat "8.3"). -Notation Pplus_minus_assoc := Pos.add_sub_assoc (compat "8.3"). -Notation Pminus_plus_distr := Pos.sub_add_distr (compat "8.3"). -Notation Pminus_minus_distr := Pos.sub_sub_distr (compat "8.3"). -Notation Pminus_mask_Lt := Pos.sub_mask_neg (compat "8.3"). -Notation Pminus_Lt := Pos.sub_lt (compat "8.3"). -Notation Pminus_Eq := Pos.sub_diag (compat "8.3"). -Notation Psize_monotone := Pos.size_nat_monotone (compat "8.3"). -Notation Psize_pos_gt := Pos.size_gt (compat "8.3"). -Notation Psize_pos_le := Pos.size_le (compat "8.3"). + Pos.pred_double_xO_discr (only parsing). +Notation Psucc_not_one := Pos.succ_not_1 (only parsing). +Notation Ppred_succ := Pos.pred_succ (compat "8.6"). +Notation Psucc_pred := Pos.succ_pred_or (only parsing). +Notation Psucc_inj := Pos.succ_inj (compat "8.6"). +Notation Pplus_carry_spec := Pos.add_carry_spec (only parsing). +Notation Pplus_comm := Pos.add_comm (only parsing). +Notation Pplus_succ_permute_r := Pos.add_succ_r (only parsing). +Notation Pplus_succ_permute_l := Pos.add_succ_l (only parsing). +Notation Pplus_no_neutral := Pos.add_no_neutral (only parsing). +Notation Pplus_carry_plus := Pos.add_carry_add (only parsing). +Notation Pplus_reg_r := Pos.add_reg_r (only parsing). +Notation Pplus_reg_l := Pos.add_reg_l (only parsing). +Notation Pplus_carry_reg_r := Pos.add_carry_reg_r (only parsing). +Notation Pplus_carry_reg_l := Pos.add_carry_reg_l (only parsing). +Notation Pplus_assoc := Pos.add_assoc (only parsing). +Notation Pplus_xO := Pos.add_xO (only parsing). +Notation Pplus_xI_double_minus_one := Pos.add_xI_pred_double (only parsing). +Notation Pplus_xO_double_minus_one := Pos.add_xO_pred_double (only parsing). +Notation Pplus_diag := Pos.add_diag (only parsing). +Notation PeanoView := Pos.PeanoView (only parsing). +Notation PeanoOne := Pos.PeanoOne (only parsing). +Notation PeanoSucc := Pos.PeanoSucc (only parsing). +Notation PeanoView_rect := Pos.PeanoView_rect (only parsing). +Notation PeanoView_ind := Pos.PeanoView_ind (only parsing). +Notation PeanoView_rec := Pos.PeanoView_rec (only parsing). +Notation peanoView_xO := Pos.peanoView_xO (only parsing). +Notation peanoView_xI := Pos.peanoView_xI (only parsing). +Notation peanoView := Pos.peanoView (only parsing). +Notation PeanoView_iter := Pos.PeanoView_iter (only parsing). +Notation eq_dep_eq_positive := Pos.eq_dep_eq_positive (only parsing). +Notation PeanoViewUnique := Pos.PeanoViewUnique (only parsing). +Notation Prect := Pos.peano_rect (only parsing). +Notation Prect_succ := Pos.peano_rect_succ (only parsing). +Notation Prect_base := Pos.peano_rect_base (only parsing). +Notation Prec := Pos.peano_rec (only parsing). +Notation Pind := Pos.peano_ind (only parsing). +Notation Pcase := Pos.peano_case (only parsing). +Notation Pmult_1_r := Pos.mul_1_r (only parsing). +Notation Pmult_Sn_m := Pos.mul_succ_l (only parsing). +Notation Pmult_xO_permute_r := Pos.mul_xO_r (only parsing). +Notation Pmult_xI_permute_r := Pos.mul_xI_r (only parsing). +Notation Pmult_comm := Pos.mul_comm (only parsing). +Notation Pmult_plus_distr_l := Pos.mul_add_distr_l (only parsing). +Notation Pmult_plus_distr_r := Pos.mul_add_distr_r (only parsing). +Notation Pmult_assoc := Pos.mul_assoc (only parsing). +Notation Pmult_xI_mult_xO_discr := Pos.mul_xI_mul_xO_discr (only parsing). +Notation Pmult_xO_discr := Pos.mul_xO_discr (only parsing). +Notation Pmult_reg_r := Pos.mul_reg_r (only parsing). +Notation Pmult_reg_l := Pos.mul_reg_l (only parsing). +Notation Pmult_1_inversion_l := Pos.mul_eq_1_l (only parsing). +Notation Psquare_xO := Pos.square_xO (compat "8.6"). +Notation Psquare_xI := Pos.square_xI (compat "8.6"). +Notation iter_pos_swap_gen := Pos.iter_swap_gen (only parsing). +Notation iter_pos_swap := Pos.iter_swap (only parsing). +Notation iter_pos_succ := Pos.iter_succ (only parsing). +Notation iter_pos_plus := Pos.iter_add (only parsing). +Notation iter_pos_invariant := Pos.iter_invariant (only parsing). +Notation Ppow_1_r := Pos.pow_1_r (compat "8.6"). +Notation Ppow_succ_r := Pos.pow_succ_r (compat "8.6"). +Notation Peqb_refl := Pos.eqb_refl (compat "8.6"). +Notation Peqb_eq := Pos.eqb_eq (compat "8.6"). +Notation Pcompare_refl_id := Pos.compare_cont_refl (only parsing). +Notation Pcompare_eq_iff := Pos.compare_eq_iff (only parsing). +Notation Pcompare_Gt_Lt := Pos.compare_cont_Gt_Lt (only parsing). +Notation Pcompare_eq_Lt := Pos.compare_lt_iff (only parsing). +Notation Pcompare_Lt_Gt := Pos.compare_cont_Lt_Gt (only parsing). + +Notation Pcompare_antisym := Pos.compare_cont_antisym (only parsing). +Notation ZC1 := Pos.gt_lt (only parsing). +Notation ZC2 := Pos.lt_gt (only parsing). +Notation Pcompare_spec := Pos.compare_spec (compat "8.6"). +Notation Pcompare_p_Sp := Pos.lt_succ_diag_r (only parsing). +Notation Pcompare_succ_succ := Pos.compare_succ_succ (compat "8.6"). +Notation Pcompare_1 := Pos.nlt_1_r (only parsing). +Notation Plt_1 := Pos.nlt_1_r (only parsing). +Notation Plt_1_succ := Pos.lt_1_succ (compat "8.6"). +Notation Plt_lt_succ := Pos.lt_lt_succ (compat "8.6"). +Notation Plt_irrefl := Pos.lt_irrefl (compat "8.6"). +Notation Plt_trans := Pos.lt_trans (compat "8.6"). +Notation Plt_ind := Pos.lt_ind (compat "8.6"). +Notation Ple_lteq := Pos.le_lteq (compat "8.6"). +Notation Ple_refl := Pos.le_refl (compat "8.6"). +Notation Ple_lt_trans := Pos.le_lt_trans (compat "8.6"). +Notation Plt_le_trans := Pos.lt_le_trans (compat "8.6"). +Notation Ple_trans := Pos.le_trans (compat "8.6"). +Notation Plt_succ_r := Pos.lt_succ_r (compat "8.6"). +Notation Ple_succ_l := Pos.le_succ_l (compat "8.6"). +Notation Pplus_compare_mono_l := Pos.add_compare_mono_l (only parsing). +Notation Pplus_compare_mono_r := Pos.add_compare_mono_r (only parsing). +Notation Pplus_lt_mono_l := Pos.add_lt_mono_l (only parsing). +Notation Pplus_lt_mono_r := Pos.add_lt_mono_r (only parsing). +Notation Pplus_lt_mono := Pos.add_lt_mono (only parsing). +Notation Pplus_le_mono_l := Pos.add_le_mono_l (only parsing). +Notation Pplus_le_mono_r := Pos.add_le_mono_r (only parsing). +Notation Pplus_le_mono := Pos.add_le_mono (only parsing). +Notation Pmult_compare_mono_l := Pos.mul_compare_mono_l (only parsing). +Notation Pmult_compare_mono_r := Pos.mul_compare_mono_r (only parsing). +Notation Pmult_lt_mono_l := Pos.mul_lt_mono_l (only parsing). +Notation Pmult_lt_mono_r := Pos.mul_lt_mono_r (only parsing). +Notation Pmult_lt_mono := Pos.mul_lt_mono (only parsing). +Notation Pmult_le_mono_l := Pos.mul_le_mono_l (only parsing). +Notation Pmult_le_mono_r := Pos.mul_le_mono_r (only parsing). +Notation Pmult_le_mono := Pos.mul_le_mono (only parsing). +Notation Plt_plus_r := Pos.lt_add_r (only parsing). +Notation Plt_not_plus_l := Pos.lt_not_add_l (only parsing). +Notation Ppow_gt_1 := Pos.pow_gt_1 (compat "8.6"). +Notation Ppred_mask := Pos.pred_mask (compat "8.6"). +Notation Pminus_mask_succ_r := Pos.sub_mask_succ_r (only parsing). +Notation Pminus_mask_carry_spec := Pos.sub_mask_carry_spec (only parsing). +Notation Pminus_succ_r := Pos.sub_succ_r (only parsing). +Notation Pminus_mask_diag := Pos.sub_mask_diag (only parsing). + +Notation Pplus_minus_eq := Pos.add_sub (only parsing). +Notation Pmult_minus_distr_l := Pos.mul_sub_distr_l (only parsing). +Notation Pminus_lt_mono_l := Pos.sub_lt_mono_l (only parsing). +Notation Pminus_compare_mono_l := Pos.sub_compare_mono_l (only parsing). +Notation Pminus_compare_mono_r := Pos.sub_compare_mono_r (only parsing). +Notation Pminus_lt_mono_r := Pos.sub_lt_mono_r (only parsing). +Notation Pminus_decr := Pos.sub_decr (only parsing). +Notation Pminus_xI_xI := Pos.sub_xI_xI (only parsing). +Notation Pplus_minus_assoc := Pos.add_sub_assoc (only parsing). +Notation Pminus_plus_distr := Pos.sub_add_distr (only parsing). +Notation Pminus_minus_distr := Pos.sub_sub_distr (only parsing). +Notation Pminus_mask_Lt := Pos.sub_mask_neg (only parsing). +Notation Pminus_Lt := Pos.sub_lt (only parsing). +Notation Pminus_Eq := Pos.sub_diag (only parsing). +Notation Psize_monotone := Pos.size_nat_monotone (only parsing). +Notation Psize_pos_gt := Pos.size_gt (only parsing). +Notation Psize_pos_le := Pos.size_le (only parsing). (** More complex compatibility facts, expressed as lemmas (to preserve scopes for instance) *) diff --git a/theories/PArith/BinPosDef.v b/theories/PArith/BinPosDef.v index 74a292c6..07031474 100644 --- a/theories/PArith/BinPosDef.v +++ b/theories/PArith/BinPosDef.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (**********************************************************************) @@ -557,4 +559,59 @@ Fixpoint of_succ_nat (n:nat) : positive := | S x => succ (of_succ_nat x) end. +(** ** Conversion with a decimal representation for printing/parsing *) + +Local Notation ten := 1~0~1~0. + +Fixpoint of_uint_acc (d:Decimal.uint)(acc:positive) := + match d with + | Decimal.Nil => acc + | Decimal.D0 l => of_uint_acc l (mul ten acc) + | Decimal.D1 l => of_uint_acc l (add 1 (mul ten acc)) + | Decimal.D2 l => of_uint_acc l (add 1~0 (mul ten acc)) + | Decimal.D3 l => of_uint_acc l (add 1~1 (mul ten acc)) + | Decimal.D4 l => of_uint_acc l (add 1~0~0 (mul ten acc)) + | Decimal.D5 l => of_uint_acc l (add 1~0~1 (mul ten acc)) + | Decimal.D6 l => of_uint_acc l (add 1~1~0 (mul ten acc)) + | Decimal.D7 l => of_uint_acc l (add 1~1~1 (mul ten acc)) + | Decimal.D8 l => of_uint_acc l (add 1~0~0~0 (mul ten acc)) + | Decimal.D9 l => of_uint_acc l (add 1~0~0~1 (mul ten acc)) + end. + +Fixpoint of_uint (d:Decimal.uint) : N := + match d with + | Decimal.Nil => N0 + | Decimal.D0 l => of_uint l + | Decimal.D1 l => Npos (of_uint_acc l 1) + | Decimal.D2 l => Npos (of_uint_acc l 1~0) + | Decimal.D3 l => Npos (of_uint_acc l 1~1) + | Decimal.D4 l => Npos (of_uint_acc l 1~0~0) + | Decimal.D5 l => Npos (of_uint_acc l 1~0~1) + | Decimal.D6 l => Npos (of_uint_acc l 1~1~0) + | Decimal.D7 l => Npos (of_uint_acc l 1~1~1) + | Decimal.D8 l => Npos (of_uint_acc l 1~0~0~0) + | Decimal.D9 l => Npos (of_uint_acc l 1~0~0~1) + end. + +Definition of_int (d:Decimal.int) : option positive := + match d with + | Decimal.Pos d => + match of_uint d with + | N0 => None + | Npos p => Some p + end + | Decimal.Neg _ => None + end. + +Fixpoint to_little_uint p := + match p with + | 1 => Decimal.D1 Decimal.Nil + | p~1 => Decimal.Little.succ_double (to_little_uint p) + | p~0 => Decimal.Little.double (to_little_uint p) + end. + +Definition to_uint p := Decimal.rev (to_little_uint p). + +Definition to_int n := Decimal.Pos (to_uint n). + End Pos. diff --git a/theories/PArith/PArith.v b/theories/PArith/PArith.v index 6ee8d6d7..2be3d07c 100644 --- a/theories/PArith/PArith.v +++ b/theories/PArith/PArith.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Library for positive natural numbers *) diff --git a/theories/PArith/POrderedType.v b/theories/PArith/POrderedType.v index 7619f639..c454e8af 100644 --- a/theories/PArith/POrderedType.v +++ b/theories/PArith/POrderedType.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import BinPos Equalities Orders OrdersTac. diff --git a/theories/PArith/Pnat.v b/theories/PArith/Pnat.v index 9c2608f4..26aba87f 100644 --- a/theories/PArith/Pnat.v +++ b/theories/PArith/Pnat.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import BinPos PeanoNat. @@ -382,36 +384,36 @@ End SuccNat2Pos. (** For compatibility, old names and old-style lemmas *) -Notation Psucc_S := Pos2Nat.inj_succ (compat "8.3"). -Notation Pplus_plus := Pos2Nat.inj_add (compat "8.3"). -Notation Pmult_mult := Pos2Nat.inj_mul (compat "8.3"). -Notation Pcompare_nat_compare := Pos2Nat.inj_compare (compat "8.3"). -Notation nat_of_P_xH := Pos2Nat.inj_1 (compat "8.3"). -Notation nat_of_P_xO := Pos2Nat.inj_xO (compat "8.3"). -Notation nat_of_P_xI := Pos2Nat.inj_xI (compat "8.3"). -Notation nat_of_P_is_S := Pos2Nat.is_succ (compat "8.3"). -Notation nat_of_P_pos := Pos2Nat.is_pos (compat "8.3"). -Notation nat_of_P_inj_iff := Pos2Nat.inj_iff (compat "8.3"). -Notation nat_of_P_inj := Pos2Nat.inj (compat "8.3"). -Notation Plt_lt := Pos2Nat.inj_lt (compat "8.3"). -Notation Pgt_gt := Pos2Nat.inj_gt (compat "8.3"). -Notation Ple_le := Pos2Nat.inj_le (compat "8.3"). -Notation Pge_ge := Pos2Nat.inj_ge (compat "8.3"). -Notation Pminus_minus := Pos2Nat.inj_sub (compat "8.3"). -Notation iter_nat_of_P := @Pos2Nat.inj_iter (compat "8.3"). - -Notation nat_of_P_of_succ_nat := SuccNat2Pos.id_succ (compat "8.3"). -Notation P_of_succ_nat_of_P := Pos2SuccNat.id_succ (compat "8.3"). - -Notation nat_of_P_succ_morphism := Pos2Nat.inj_succ (compat "8.3"). -Notation nat_of_P_plus_morphism := Pos2Nat.inj_add (compat "8.3"). -Notation nat_of_P_mult_morphism := Pos2Nat.inj_mul (compat "8.3"). -Notation nat_of_P_compare_morphism := Pos2Nat.inj_compare (compat "8.3"). -Notation lt_O_nat_of_P := Pos2Nat.is_pos (compat "8.3"). -Notation ZL4 := Pos2Nat.is_succ (compat "8.3"). -Notation nat_of_P_o_P_of_succ_nat_eq_succ := SuccNat2Pos.id_succ (compat "8.3"). -Notation P_of_succ_nat_o_nat_of_P_eq_succ := Pos2SuccNat.id_succ (compat "8.3"). -Notation pred_o_P_of_succ_nat_o_nat_of_P_eq_id := Pos2SuccNat.pred_id (compat "8.3"). +Notation Psucc_S := Pos2Nat.inj_succ (only parsing). +Notation Pplus_plus := Pos2Nat.inj_add (only parsing). +Notation Pmult_mult := Pos2Nat.inj_mul (only parsing). +Notation Pcompare_nat_compare := Pos2Nat.inj_compare (only parsing). +Notation nat_of_P_xH := Pos2Nat.inj_1 (only parsing). +Notation nat_of_P_xO := Pos2Nat.inj_xO (only parsing). +Notation nat_of_P_xI := Pos2Nat.inj_xI (only parsing). +Notation nat_of_P_is_S := Pos2Nat.is_succ (only parsing). +Notation nat_of_P_pos := Pos2Nat.is_pos (only parsing). +Notation nat_of_P_inj_iff := Pos2Nat.inj_iff (only parsing). +Notation nat_of_P_inj := Pos2Nat.inj (only parsing). +Notation Plt_lt := Pos2Nat.inj_lt (only parsing). +Notation Pgt_gt := Pos2Nat.inj_gt (only parsing). +Notation Ple_le := Pos2Nat.inj_le (only parsing). +Notation Pge_ge := Pos2Nat.inj_ge (only parsing). +Notation Pminus_minus := Pos2Nat.inj_sub (only parsing). +Notation iter_nat_of_P := @Pos2Nat.inj_iter (only parsing). + +Notation nat_of_P_of_succ_nat := SuccNat2Pos.id_succ (only parsing). +Notation P_of_succ_nat_of_P := Pos2SuccNat.id_succ (only parsing). + +Notation nat_of_P_succ_morphism := Pos2Nat.inj_succ (only parsing). +Notation nat_of_P_plus_morphism := Pos2Nat.inj_add (only parsing). +Notation nat_of_P_mult_morphism := Pos2Nat.inj_mul (only parsing). +Notation nat_of_P_compare_morphism := Pos2Nat.inj_compare (only parsing). +Notation lt_O_nat_of_P := Pos2Nat.is_pos (only parsing). +Notation ZL4 := Pos2Nat.is_succ (only parsing). +Notation nat_of_P_o_P_of_succ_nat_eq_succ := SuccNat2Pos.id_succ (only parsing). +Notation P_of_succ_nat_o_nat_of_P_eq_succ := Pos2SuccNat.id_succ (only parsing). +Notation pred_o_P_of_succ_nat_o_nat_of_P_eq_id := Pos2SuccNat.pred_id (only parsing). Lemma nat_of_P_minus_morphism p q : Pos.compare_cont Eq p q = Gt -> diff --git a/theories/PArith/vo.itarget b/theories/PArith/vo.itarget deleted file mode 100644 index 73044e2c..00000000 --- a/theories/PArith/vo.itarget +++ /dev/null @@ -1,5 +0,0 @@ -BinPosDef.vo -BinPos.vo -Pnat.vo -POrderedType.vo -PArith.vo
\ No newline at end of file diff --git a/theories/Program/Basics.v b/theories/Program/Basics.v index 644b9b5a..f55093ed 100644 --- a/theories/Program/Basics.v +++ b/theories/Program/Basics.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Standard functions and combinators. diff --git a/theories/Program/Combinators.v b/theories/Program/Combinators.v index 772018aa..f78d06b1 100644 --- a/theories/Program/Combinators.v +++ b/theories/Program/Combinators.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Proofs about standard combinators, exports functional extensionality. @@ -22,15 +24,13 @@ Open Scope program_scope. Lemma compose_id_left : forall A B (f : A -> B), id ∘ f = f. Proof. intros. - unfold id, compose. - symmetry. apply eta_expansion. + reflexivity. Qed. Lemma compose_id_right : forall A B (f : A -> B), f ∘ id = f. Proof. intros. - unfold id, compose. - symmetry ; apply eta_expansion. + reflexivity. Qed. Lemma compose_assoc : forall A B C D (f : A -> B) (g : B -> C) (h : C -> D), @@ -47,9 +47,7 @@ Hint Rewrite <- @compose_assoc : core. Lemma flip_flip : forall A B C, @flip A B C ∘ flip = id. Proof. - unfold flip, compose. intros. - extensionality x ; extensionality y ; extensionality z. reflexivity. Qed. @@ -57,9 +55,7 @@ Qed. Lemma prod_uncurry_curry : forall A B C, @prod_uncurry A B C ∘ prod_curry = id. Proof. - simpl ; intros. - unfold prod_uncurry, prod_curry, compose. - extensionality x ; extensionality y ; extensionality z. + intros. reflexivity. Qed. diff --git a/theories/Program/Equality.v b/theories/Program/Equality.v index d6f9bb9d..cf42ed18 100644 --- a/theories/Program/Equality.v +++ b/theories/Program/Equality.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Tactics related to (dependent) equality and proof irrelevance. *) diff --git a/theories/Program/Program.v b/theories/Program/Program.v index 4a6f2786..de0a6d5d 100644 --- a/theories/Program/Program.v +++ b/theories/Program/Program.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export Coq.Program.Utils. Require Export Coq.Program.Wf. diff --git a/theories/Program/Subset.v b/theories/Program/Subset.v index 2a3ec926..1c89b6c3 100644 --- a/theories/Program/Subset.v +++ b/theories/Program/Subset.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Tactics related to subsets and proof irrelevance. *) diff --git a/theories/Program/Syntax.v b/theories/Program/Syntax.v index 2fccf624..785b9437 100644 --- a/theories/Program/Syntax.v +++ b/theories/Program/Syntax.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Custom notations and implicits for Coq prelude definitions. diff --git a/theories/Program/Tactics.v b/theories/Program/Tactics.v index dfd6b0ea..bc838818 100644 --- a/theories/Program/Tactics.v +++ b/theories/Program/Tactics.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** This module implements various tactics used to simplify the goals produced by Program, @@ -328,4 +330,4 @@ Ltac program_simpl := program_simplify ; try typeclasses eauto with program ; tr Obligation Tactic := program_simpl. -Definition obligation (A : Type) {a : A} := a.
\ No newline at end of file +Definition obligation (A : Type) {a : A} := a. diff --git a/theories/Program/Utils.v b/theories/Program/Utils.v index 396c9618..78c36dc7 100644 --- a/theories/Program/Utils.v +++ b/theories/Program/Utils.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Various syntactic shorthands that are useful with [Program]. *) diff --git a/theories/Program/Wf.v b/theories/Program/Wf.v index c490ea51..62787985 100644 --- a/theories/Program/Wf.v +++ b/theories/Program/Wf.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Reformulation of the Wf module using subsets where possible, providing the support for [Program]'s treatment of well-founded definitions. *) @@ -69,6 +71,7 @@ Section Well_founded. End Well_founded. +Require Coq.extraction.Extraction. Extraction Inline Fix_F_sub Fix_sub. Set Implicit Arguments. diff --git a/theories/Program/vo.itarget b/theories/Program/vo.itarget deleted file mode 100644 index 864c815a..00000000 --- a/theories/Program/vo.itarget +++ /dev/null @@ -1,9 +0,0 @@ -Basics.vo -Combinators.vo -Equality.vo -Program.vo -Subset.vo -Syntax.vo -Tactics.vo -Utils.vo -Wf.vo diff --git a/theories/QArith/QArith.v b/theories/QArith/QArith.v index 5ad08b65..81390082 100644 --- a/theories/QArith/QArith.v +++ b/theories/QArith/QArith.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export QArith_base. diff --git a/theories/QArith/QArith_base.v b/theories/QArith/QArith_base.v index 62304876..35706e7f 100644 --- a/theories/QArith/QArith_base.v +++ b/theories/QArith/QArith_base.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export ZArith. @@ -171,6 +173,26 @@ Proof. auto with qarith. Qed. +Lemma Qeq_bool_comm x y: Qeq_bool x y = Qeq_bool y x. +Proof. + apply eq_true_iff_eq. rewrite !Qeq_bool_iff. now symmetry. +Qed. + +Lemma Qeq_bool_refl x: Qeq_bool x x = true. +Proof. + rewrite Qeq_bool_iff. now reflexivity. +Qed. + +Lemma Qeq_bool_sym x y: Qeq_bool x y = true -> Qeq_bool y x = true. +Proof. + rewrite !Qeq_bool_iff. now symmetry. +Qed. + +Lemma Qeq_bool_trans x y z: Qeq_bool x y = true -> Qeq_bool y z = true -> Qeq_bool x z = true. +Proof. + rewrite !Qeq_bool_iff; apply Qeq_trans. +Qed. + Hint Resolve Qnot_eq_sym : qarith. (** * Addition, multiplication and opposite *) @@ -205,9 +227,7 @@ Infix "/" := Qdiv : Q_scope. (** A light notation for [Zpos] *) -Notation " ' x " := (Zpos x) (at level 20, no associativity) : Z_scope. - -Lemma Qmake_Qdiv a b : a#b==inject_Z a/inject_Z ('b). +Lemma Qmake_Qdiv a b : a#b==inject_Z a/inject_Z (Zpos b). Proof. unfold Qeq. simpl. ring. Qed. @@ -220,9 +240,9 @@ Proof. Open Scope Z_scope. intros (p1, p2) (q1, q2) H (r1, r2) (s1, s2) H0; simpl in *. simpl_mult; ring_simplify. - replace (p1 * 'r2 * 'q2) with (p1 * 'q2 * 'r2) by ring. + replace (p1 * Zpos r2 * Zpos q2) with (p1 * Zpos q2 * Zpos r2) by ring. rewrite H. - replace (r1 * 'p2 * 'q2 * 's2) with (r1 * 's2 * 'p2 * 'q2) by ring. + replace (r1 * Zpos p2 * Zpos q2 * Zpos s2) with (r1 * Zpos s2 * Zpos p2 * Zpos q2) by ring. rewrite H0. ring. Close Scope Z_scope. @@ -233,7 +253,7 @@ Proof. unfold Qeq, Qopp; simpl. Open Scope Z_scope. intros x y H; simpl. - replace (- Qnum x * ' Qden y) with (- (Qnum x * ' Qden y)) by ring. + replace (- Qnum x * Zpos (Qden y)) with (- (Qnum x * Zpos (Qden y))) by ring. rewrite H; ring. Close Scope Z_scope. Qed. @@ -250,9 +270,9 @@ Proof. Open Scope Z_scope. intros (p1, p2) (q1, q2) H (r1, r2) (s1, s2) H0; simpl in *. intros; simpl_mult; ring_simplify. - replace (q1 * s1 * 'p2) with (q1 * 'p2 * s1) by ring. + replace (q1 * s1 * Zpos p2) with (q1 * Zpos p2 * s1) by ring. rewrite <- H. - replace (p1 * r1 * 'q2 * 's2) with (r1 * 's2 * p1 * 'q2) by ring. + replace (p1 * r1 * Zpos q2 * Zpos s2) with (r1 * Zpos s2 * p1 * Zpos q2) by ring. rewrite H0. ring. Close Scope Z_scope. @@ -283,13 +303,13 @@ Proof. unfold Qeq, Qcompare. Open Scope Z_scope. intros (p1,p2) (q1,q2) H (r1,r2) (s1,s2) H'; simpl in *. - rewrite <- (Zcompare_mult_compat (q2*s2) (p1*'r2)). - rewrite <- (Zcompare_mult_compat (p2*r2) (q1*'s2)). - change ('(q2*s2)) with ('q2 * 's2). - change ('(p2*r2)) with ('p2 * 'r2). - replace ('q2 * 's2 * (p1*'r2)) with ((p1*'q2)*'r2*'s2) by ring. + rewrite <- (Zcompare_mult_compat (q2*s2) (p1*Zpos r2)). + rewrite <- (Zcompare_mult_compat (p2*r2) (q1*Zpos s2)). + change (Zpos (q2*s2)) with (Zpos q2 * Zpos s2). + change (Zpos (p2*r2)) with (Zpos p2 * Zpos r2). + replace (Zpos q2 * Zpos s2 * (p1*Zpos r2)) with ((p1*Zpos q2)*Zpos r2*Zpos s2) by ring. rewrite H. - replace ('q2 * 's2 * (r1*'p2)) with ((r1*'s2)*'q2*'p2) by ring. + replace (Zpos q2 * Zpos s2 * (r1*Zpos p2)) with ((r1*Zpos s2)*Zpos q2*Zpos p2) by ring. rewrite H'. f_equal; ring. Close Scope Z_scope. @@ -550,8 +570,8 @@ Lemma Qle_trans : forall x y z, x<=y -> y<=z -> x<=z. Proof. unfold Qle; intros (x1, x2) (y1, y2) (z1, z2); simpl; intros. Open Scope Z_scope. - apply Z.mul_le_mono_pos_r with ('y2); [easy|]. - apply Z.le_trans with (y1 * 'x2 * 'z2). + apply Z.mul_le_mono_pos_r with (Zpos y2); [easy|]. + apply Z.le_trans with (y1 * Zpos x2 * Zpos z2). - rewrite Z.mul_shuffle0. now apply Z.mul_le_mono_pos_r. - rewrite Z.mul_shuffle0, (Z.mul_shuffle0 z1). now apply Z.mul_le_mono_pos_r. @@ -598,8 +618,8 @@ Lemma Qle_lt_trans : forall x y z, x<=y -> y<z -> x<z. Proof. unfold Qle, Qlt; intros (x1, x2) (y1, y2) (z1, z2); simpl; intros. Open Scope Z_scope. - apply Z.mul_lt_mono_pos_r with ('y2); [easy|]. - apply Z.le_lt_trans with (y1 * 'x2 * 'z2). + apply Z.mul_lt_mono_pos_r with (Zpos y2); [easy|]. + apply Z.le_lt_trans with (y1 * Zpos x2 * Zpos z2). - rewrite Z.mul_shuffle0. now apply Z.mul_le_mono_pos_r. - rewrite Z.mul_shuffle0, (Z.mul_shuffle0 z1). now apply Z.mul_lt_mono_pos_r. @@ -610,8 +630,8 @@ Lemma Qlt_le_trans : forall x y z, x<y -> y<=z -> x<z. Proof. unfold Qle, Qlt; intros (x1, x2) (y1, y2) (z1, z2); simpl; intros. Open Scope Z_scope. - apply Z.mul_lt_mono_pos_r with ('y2); [easy|]. - apply Z.lt_le_trans with (y1 * 'x2 * 'z2). + apply Z.mul_lt_mono_pos_r with (Zpos y2); [easy|]. + apply Z.lt_le_trans with (y1 * Zpos x2 * Zpos z2). - rewrite Z.mul_shuffle0. now apply Z.mul_lt_mono_pos_r. - rewrite Z.mul_shuffle0, (Z.mul_shuffle0 z1). now apply Z.mul_le_mono_pos_r. @@ -701,9 +721,9 @@ Proof. match goal with |- ?a <= ?b => ring_simplify a b end. rewrite Z.add_comm. apply Z.add_le_mono. - match goal with |- ?a <= ?b => ring_simplify z1 t1 ('z2) ('t2) a b end. + match goal with |- ?a <= ?b => ring_simplify z1 t1 (Zpos z2) (Zpos t2) a b end. auto with zarith. - match goal with |- ?a <= ?b => ring_simplify x1 y1 ('x2) ('y2) a b end. + match goal with |- ?a <= ?b => ring_simplify x1 y1 (Zpos x2) (Zpos y2) a b end. auto with zarith. Close Scope Z_scope. Qed. @@ -718,9 +738,9 @@ Proof. match goal with |- ?a < ?b => ring_simplify a b end. rewrite Z.add_comm. apply Z.add_le_lt_mono. - match goal with |- ?a <= ?b => ring_simplify z1 t1 ('z2) ('t2) a b end. + match goal with |- ?a <= ?b => ring_simplify z1 t1 (Zpos z2) (Zpos t2) a b end. auto with zarith. - match goal with |- ?a < ?b => ring_simplify x1 y1 ('x2) ('y2) a b end. + match goal with |- ?a < ?b => ring_simplify x1 y1 (Zpos x2) (Zpos y2) a b end. do 2 (apply Z.mul_lt_mono_pos_r;try easy). Close Scope Z_scope. Qed. diff --git a/theories/QArith/QOrderedType.v b/theories/QArith/QOrderedType.v index 25e98f0b..37b4b298 100644 --- a/theories/QArith/QOrderedType.v +++ b/theories/QArith/QOrderedType.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import QArith_base Equalities Orders OrdersTac. diff --git a/theories/QArith/Qabs.v b/theories/QArith/Qabs.v index c60d0451..31eb41bc 100644 --- a/theories/QArith/Qabs.v +++ b/theories/QArith/Qabs.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export QArith. @@ -26,8 +28,8 @@ intros [xn xd] [yn yd] H. simpl. unfold Qeq in *. simpl in *. -change (' yd)%Z with (Z.abs (' yd)). -change (' xd)%Z with (Z.abs (' xd)). +change (Zpos yd)%Z with (Z.abs (Zpos yd)). +change (Zpos xd)%Z with (Z.abs (Zpos xd)). repeat rewrite <- Z.abs_mul. congruence. Qed. @@ -86,8 +88,8 @@ unfold Qplus. unfold Qle. simpl. apply Z.mul_le_mono_nonneg_r;auto with *. -change (' yd)%Z with (Z.abs (' yd)). -change (' xd)%Z with (Z.abs (' xd)). +change (Zpos yd)%Z with (Z.abs (Zpos yd)). +change (Zpos xd)%Z with (Z.abs (Zpos xd)). repeat rewrite <- Z.abs_mul. apply Z.abs_triangle. Qed. @@ -100,6 +102,13 @@ rewrite Z.abs_mul. reflexivity. Qed. +Lemma Qabs_Qinv : forall q, Qabs (/ q) == / (Qabs q). +Proof. + intros [n d]; simpl. + unfold Qinv. + case_eq n; intros; simpl in *; apply Qeq_refl. +Qed. + Lemma Qabs_Qminus x y: Qabs (x - y) = Qabs (y - x). Proof. unfold Qminus, Qopp. simpl. diff --git a/theories/QArith/Qcabs.v b/theories/QArith/Qcabs.v index c0ababff..f45868a7 100644 --- a/theories/QArith/Qcabs.v +++ b/theories/QArith/Qcabs.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * An absolute value for normalized rational numbers. *) @@ -22,7 +24,7 @@ Lemma Qcabs_canon (x : Q) : Qred x = x -> Qred (Qabs x) = Qabs x. Proof. intros H; now rewrite (Qred_abs x), H. Qed. Definition Qcabs (x:Qc) : Qc := {| canon := Qcabs_canon x (canon x) |}. -Notation "[ q ]" := (Qcabs q) (q at next level, format "[ q ]") : Qc_scope. +Notation "[ q ]" := (Qcabs q) : Qc_scope. Ltac Qc_unfolds := unfold Qcabs, Qcminus, Qcopp, Qcplus, Qcmult, Qcle, Q2Qc, this. @@ -126,4 +128,4 @@ Proof. destruct (proj1 (Qcabs_Qcle_condition x 0)) as [A B]. + rewrite H; apply Qcle_refl. + apply Qcle_antisym; auto. -Qed.
\ No newline at end of file +Qed. diff --git a/theories/QArith/Qcanon.v b/theories/QArith/Qcanon.v index 9f9651d8..1510a7b8 100644 --- a/theories/QArith/Qcanon.v +++ b/theories/QArith/Qcanon.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Field. @@ -28,7 +30,7 @@ Lemma Qred_identity : Proof. intros (a,b) H; simpl in *. rewrite <- Z.ggcd_gcd in H. - generalize (Z.ggcd_correct_divisors a ('b)). + generalize (Z.ggcd_correct_divisors a (Zpos b)). destruct Z.ggcd as (g,(aa,bb)); simpl in *; subst. rewrite !Z.mul_1_l. now intros (<-,<-). Qed. @@ -37,7 +39,7 @@ Lemma Qred_identity2 : forall q:Q, Qred q = q -> Z.gcd (Qnum q) (QDen q) = 1%Z. Proof. intros (a,b) H; simpl in *. - generalize (Z.gcd_nonneg a ('b)) (Z.ggcd_correct_divisors a ('b)). + generalize (Z.gcd_nonneg a (Zpos b)) (Z.ggcd_correct_divisors a (Zpos b)). rewrite <- Z.ggcd_gcd. destruct Z.ggcd as (g,(aa,bb)); simpl in *. injection H as <- <-. intros H (_,H'). diff --git a/theories/QArith/Qfield.v b/theories/QArith/Qfield.v index bbaf6027..6cbb491b 100644 --- a/theories/QArith/Qfield.v +++ b/theories/QArith/Qfield.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export Field. diff --git a/theories/QArith/Qminmax.v b/theories/QArith/Qminmax.v index 86584d9e..264b2f92 100644 --- a/theories/QArith/Qminmax.v +++ b/theories/QArith/Qminmax.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import QArith_base Orders QOrderedType GenericMinMax. diff --git a/theories/QArith/Qpower.v b/theories/QArith/Qpower.v index af89d300..01078220 100644 --- a/theories/QArith/Qpower.v +++ b/theories/QArith/Qpower.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Zpow_facts Qfield Qreduction. @@ -88,7 +90,7 @@ rewrite Qinv_power. reflexivity. Qed. -Lemma Qinv_power_n : forall n p, (1#p)^n == /(inject_Z ('p))^n. +Lemma Qinv_power_n : forall n p, (1#p)^n == /(inject_Z (Zpos p))^n. Proof. intros n p. rewrite Qmake_Qdiv. @@ -188,7 +190,7 @@ unfold Z.succ. rewrite Zpower_exp; auto with *; try discriminate. rewrite Qpower_plus' by discriminate. rewrite <- IHn by discriminate. -replace (a^'n*a^1)%Z with (a^'n*a)%Z by ring. +replace (a^Zpos n*a^1)%Z with (a^Zpos n*a)%Z by ring. ring_simplify. reflexivity. Qed. diff --git a/theories/QArith/Qreals.v b/theories/QArith/Qreals.v index 048e409c..c8329625 100644 --- a/theories/QArith/Qreals.v +++ b/theories/QArith/Qreals.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export Rbase. @@ -15,7 +17,8 @@ Definition Q2R (x : Q) : R := (IZR (Qnum x) * / IZR (QDen x))%R. Lemma IZR_nz : forall p : positive, IZR (Zpos p) <> 0%R. Proof. -intros; apply not_O_IZR; auto with qarith. +intros. +now apply not_O_IZR. Qed. Hint Resolve IZR_nz Rmult_integral_contrapositive. @@ -48,8 +51,7 @@ assert ((X1 * Y2)%R = (Y1 * X2)%R). apply IZR_eq; auto. clear H. field_simplify_eq; auto. -ring_simplify X1 Y2 (Y2 * X1)%R. -rewrite H0; ring. +rewrite H0; ring. Qed. Lemma Rle_Qle : forall x y : Q, (Q2R x <= Q2R y)%R -> x<=y. @@ -66,10 +68,8 @@ replace (X1 * Y2)%R with (X1 * / X2 * (X2 * Y2))%R; try (field; auto). replace (Y1 * X2)%R with (Y1 * / Y2 * (X2 * Y2))%R; try (field; auto). apply Rmult_le_compat_r; auto. apply Rmult_le_pos. -unfold X2; replace 0%R with (IZR 0); auto; apply IZR_le; - auto with zarith. -unfold Y2; replace 0%R with (IZR 0); auto; apply IZR_le; - auto with zarith. +now apply IZR_le. +now apply IZR_le. Qed. Lemma Qle_Rle : forall x y : Q, x<=y -> (Q2R x <= Q2R y)%R. @@ -88,10 +88,8 @@ replace (X1 * / X2)%R with (X1 * Y2 * (/ X2 * / Y2))%R; try (field; auto). replace (Y1 * / Y2)%R with (Y1 * X2 * (/ X2 * / Y2))%R; try (field; auto). apply Rmult_le_compat_r; auto. apply Rmult_le_pos; apply Rlt_le; apply Rinv_0_lt_compat. -unfold X2; replace 0%R with (IZR 0); auto; apply IZR_lt; red; - auto with zarith. -unfold Y2; replace 0%R with (IZR 0); auto; apply IZR_lt; red; - auto with zarith. +now apply IZR_lt. +now apply IZR_lt. Qed. Lemma Rlt_Qlt : forall x y : Q, (Q2R x < Q2R y)%R -> x<y. @@ -108,10 +106,8 @@ replace (X1 * Y2)%R with (X1 * / X2 * (X2 * Y2))%R; try (field; auto). replace (Y1 * X2)%R with (Y1 * / Y2 * (X2 * Y2))%R; try (field; auto). apply Rmult_lt_compat_r; auto. apply Rmult_lt_0_compat. -unfold X2; replace 0%R with (IZR 0); auto; apply IZR_lt; red; - auto with zarith. -unfold Y2; replace 0%R with (IZR 0); auto; apply IZR_lt; red; - auto with zarith. +now apply IZR_lt. +now apply IZR_lt. Qed. Lemma Qlt_Rlt : forall x y : Q, x<y -> (Q2R x < Q2R y)%R. @@ -130,10 +126,8 @@ replace (X1 * / X2)%R with (X1 * Y2 * (/ X2 * / Y2))%R; try (field; auto). replace (Y1 * / Y2)%R with (Y1 * X2 * (/ X2 * / Y2))%R; try (field; auto). apply Rmult_lt_compat_r; auto. apply Rmult_lt_0_compat; apply Rinv_0_lt_compat. -unfold X2; replace 0%R with (IZR 0); auto; apply IZR_lt; red; - auto with zarith. -unfold Y2; replace 0%R with (IZR 0); auto; apply IZR_lt; red; - auto with zarith. +now apply IZR_lt. +now apply IZR_lt. Qed. Lemma Q2R_plus : forall x y : Q, Q2R (x+y) = (Q2R x + Q2R y)%R. @@ -173,8 +167,8 @@ unfold Qinv, Q2R, Qeq; intros (x1, x2). case x1; unfold Qnum, Qden. simpl; intros; elim H; trivial. intros; field; auto. intros; - change (IZR (Zneg x2)) with (- IZR (' x2))%R; - change (IZR (Zneg p)) with (- IZR (' p))%R; + change (IZR (Zneg x2)) with (- IZR (Zpos x2))%R; + change (IZR (Zneg p)) with (- IZR (Zpos p))%R; simpl; field; (*auto 8 with real.*) repeat split; auto; auto with real. Qed. diff --git a/theories/QArith/Qreduction.v b/theories/QArith/Qreduction.v index 131214f5..17307c82 100644 --- a/theories/QArith/Qreduction.v +++ b/theories/QArith/Qreduction.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Normalisation functions for rational numbers. *) @@ -11,22 +13,22 @@ Require Export QArith_base. Require Import Znumtheory. -Notation Z2P := Z.to_pos (compat "8.3"). -Notation Z2P_correct := Z2Pos.id (compat "8.3"). +Notation Z2P := Z.to_pos (only parsing). +Notation Z2P_correct := Z2Pos.id (only parsing). (** Simplification of fractions using [Z.gcd]. This version can compute within Coq. *) Definition Qred (q:Q) := let (q1,q2) := q in - let (r1,r2) := snd (Z.ggcd q1 ('q2)) + let (r1,r2) := snd (Z.ggcd q1 (Zpos q2)) in r1#(Z.to_pos r2). Lemma Qred_correct : forall q, (Qred q) == q. Proof. unfold Qred, Qeq; intros (n,d); simpl. - generalize (Z.ggcd_gcd n ('d)) (Z.gcd_nonneg n ('d)) - (Z.ggcd_correct_divisors n ('d)). + generalize (Z.ggcd_gcd n (Zpos d)) (Z.gcd_nonneg n (Zpos d)) + (Z.ggcd_correct_divisors n (Zpos d)). destruct (Z.ggcd n (Zpos d)) as (g,(nn,dd)); simpl. Open Scope Z_scope. intros Hg LE (Hn,Hd). rewrite Hd, Hn. @@ -43,13 +45,13 @@ Proof. unfold Qred, Qeq in *; simpl in *. Open Scope Z_scope. intros H. - generalize (Z.ggcd_gcd a ('b)) (Zgcd_is_gcd a ('b)) - (Z.gcd_nonneg a ('b)) (Z.ggcd_correct_divisors a ('b)). + generalize (Z.ggcd_gcd a (Zpos b)) (Zgcd_is_gcd a (Zpos b)) + (Z.gcd_nonneg a (Zpos b)) (Z.ggcd_correct_divisors a (Zpos b)). destruct (Z.ggcd a (Zpos b)) as (g,(aa,bb)). simpl. intros <- Hg1 Hg2 (Hg3,Hg4). assert (Hg0 : g <> 0) by (intro; now subst g). - generalize (Z.ggcd_gcd c ('d)) (Zgcd_is_gcd c ('d)) - (Z.gcd_nonneg c ('d)) (Z.ggcd_correct_divisors c ('d)). + generalize (Z.ggcd_gcd c (Zpos d)) (Zgcd_is_gcd c (Zpos d)) + (Z.gcd_nonneg c (Zpos d)) (Z.ggcd_correct_divisors c (Zpos d)). destruct (Z.ggcd c (Zpos d)) as (g',(cc,dd)). simpl. intros <- Hg'1 Hg'2 (Hg'3,Hg'4). assert (Hg'0 : g' <> 0) by (intro; now subst g'). @@ -101,7 +103,7 @@ Proof. - apply Qred_complete. Qed. -Add Morphism Qred : Qred_comp. +Add Morphism Qred with signature (Qeq ==> Qeq) as Qred_comp. Proof. intros. now rewrite !Qred_correct. Qed. @@ -125,19 +127,19 @@ Proof. intros; unfold Qminus'; apply Qred_correct; auto. Qed. -Add Morphism Qplus' : Qplus'_comp. +Add Morphism Qplus' with signature (Qeq ==> Qeq ==> Qeq) as Qplus'_comp. Proof. intros; unfold Qplus'. rewrite H, H0; auto with qarith. Qed. -Add Morphism Qmult' : Qmult'_comp. +Add Morphism Qmult' with signature (Qeq ==> Qeq ==> Qeq) as Qmult'_comp. Proof. intros; unfold Qmult'. rewrite H, H0; auto with qarith. Qed. -Add Morphism Qminus' : Qminus'_comp. +Add Morphism Qminus' with signature (Qeq ==> Qeq ==> Qeq) as Qminus'_comp. Proof. intros; unfold Qminus'. rewrite H, H0; auto with qarith. diff --git a/theories/QArith/Qring.v b/theories/QArith/Qring.v index da11c2b1..7f972d56 100644 --- a/theories/QArith/Qring.v +++ b/theories/QArith/Qring.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export Qfield. diff --git a/theories/QArith/Qround.v b/theories/QArith/Qround.v index 0ed6d557..7c5ddbb6 100644 --- a/theories/QArith/Qround.v +++ b/theories/QArith/Qround.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import QArith. @@ -78,11 +80,11 @@ unfold Qlt. simpl. replace (n*1)%Z with n by ring. ring_simplify. -replace (n / ' d * ' d + ' d)%Z with - (('d * (n / 'd) + n mod 'd) + 'd - n mod 'd)%Z by ring. +replace (n / Zpos d * Zpos d + Zpos d)%Z with + ((Zpos d * (n / Zpos d) + n mod Zpos d) + Zpos d - n mod Zpos d)%Z by ring. rewrite <- Z_div_mod_eq; auto with*. rewrite <- Z.lt_add_lt_sub_r. -destruct (Z_mod_lt n ('d)); auto with *. +destruct (Z_mod_lt n (Zpos d)); auto with *. Qed. Hint Resolve Qlt_floor : qarith. @@ -105,9 +107,9 @@ Proof. intros [xn xd] [yn yd] Hxy. unfold Qle in *. simpl in *. -rewrite <- (Zdiv_mult_cancel_r xn ('xd) ('yd)); auto with *. -rewrite <- (Zdiv_mult_cancel_r yn ('yd) ('xd)); auto with *. -rewrite (Z.mul_comm ('yd) ('xd)). +rewrite <- (Zdiv_mult_cancel_r xn (Zpos xd) (Zpos yd)); auto with *. +rewrite <- (Zdiv_mult_cancel_r yn (Zpos yd) (Zpos xd)); auto with *. +rewrite (Z.mul_comm (Zpos yd) (Zpos xd)). apply Z_div_le; auto with *. Qed. @@ -141,7 +143,7 @@ Qed. Lemma Zdiv_Qdiv (n m: Z): (n / m)%Z = Qfloor (n / m). Proof. unfold Qfloor. intros. simpl. - destruct m as [?|?|p]; simpl. + destruct m as [ | | p]; simpl. now rewrite Zdiv_0_r, Z.mul_0_r. now rewrite Z.mul_1_r. rewrite <- Z.opp_eq_mul_m1. diff --git a/theories/QArith/vo.itarget b/theories/QArith/vo.itarget deleted file mode 100644 index b550b471..00000000 --- a/theories/QArith/vo.itarget +++ /dev/null @@ -1,13 +0,0 @@ -Qabs.vo -QArith_base.vo -QArith.vo -Qcanon.vo -Qcabs.vo -Qfield.vo -Qpower.vo -Qreals.vo -Qreduction.vo -Qring.vo -Qround.vo -QOrderedType.vo -Qminmax.vo
\ No newline at end of file diff --git a/theories/Reals/Alembert.v b/theories/Reals/Alembert.v index a98d529f..09aad1ec 100644 --- a/theories/Reals/Alembert.v +++ b/theories/Reals/Alembert.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -78,7 +80,7 @@ Proof. ring. discrR. discrR. - pattern 1 at 3; replace 1 with (/ 1); + replace 1 with (/ 1); [ apply tech7; discrR | apply Rinv_1 ]. replace (An (S x)) with (An (S x + 0)%nat). apply (tech6 (fun i:nat => An (S x + i)%nat) (/ 2)). diff --git a/theories/Reals/AltSeries.v b/theories/Reals/AltSeries.v index c3ab8edc..c17ad0cf 100644 --- a/theories/Reals/AltSeries.v +++ b/theories/Reals/AltSeries.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -339,51 +341,24 @@ Proof. symmetry ; apply S_pred with 0%nat. assumption. apply Rle_lt_trans with (/ INR (2 * N)). - apply Rmult_le_reg_l with (INR (2 * N)). + apply Rinv_le_contravar. rewrite mult_INR; apply Rmult_lt_0_compat; [ simpl; prove_sup0 | apply lt_INR_0; assumption ]. - rewrite <- Rinv_r_sym. - apply Rmult_le_reg_l with (INR (2 * n)). - rewrite mult_INR; apply Rmult_lt_0_compat; - [ simpl; prove_sup0 | apply lt_INR_0; assumption ]. - rewrite (Rmult_comm (INR (2 * n))); rewrite Rmult_assoc; - rewrite <- Rinv_l_sym. - do 2 rewrite Rmult_1_r; apply le_INR. - apply (fun m n p:nat => mult_le_compat_l p n m); assumption. - replace n with (S (pred n)). - apply not_O_INR; discriminate. - symmetry ; apply S_pred with 0%nat. - assumption. - replace N with (S (pred N)). - apply not_O_INR; discriminate. - symmetry ; apply S_pred with 0%nat. - assumption. + apply le_INR. + now apply mult_le_compat_l. rewrite mult_INR. - rewrite Rinv_mult_distr. - replace (INR 2) with 2; [ idtac | reflexivity ]. - apply Rmult_lt_reg_l with 2. - prove_sup0. - rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; [ idtac | discrR ]. - rewrite Rmult_1_l; apply Rmult_lt_reg_l with (INR N). - apply lt_INR_0; assumption. - rewrite <- Rinv_r_sym. - apply Rmult_lt_reg_l with (/ (2 * eps)). - apply Rinv_0_lt_compat; assumption. - rewrite Rmult_1_r; - replace (/ (2 * eps) * (INR N * (2 * eps))) with - (INR N * (2 * eps * / (2 * eps))); [ idtac | ring ]. - rewrite <- Rinv_r_sym. - rewrite Rmult_1_r; replace (INR N) with (IZR (Z.of_nat N)). - rewrite <- H4. - elim H1; intros; assumption. - symmetry ; apply INR_IZR_INZ. - apply prod_neq_R0; - [ discrR | red; intro; rewrite H8 in H; elim (Rlt_irrefl _ H) ]. - apply not_O_INR. - red; intro; rewrite H8 in H5; elim (lt_irrefl _ H5). - replace (INR 2) with 2; [ discrR | reflexivity ]. - apply not_O_INR. - red; intro; rewrite H8 in H5; elim (lt_irrefl _ H5). + apply Rmult_lt_reg_l with (INR N / eps). + apply Rdiv_lt_0_compat with (2 := H). + now apply (lt_INR 0). + replace (_ */ _) with (/(2 * eps)). + replace (_ / _ * _) with (INR N). + rewrite INR_IZR_INZ. + now rewrite <- H4. + field. + now apply Rgt_not_eq. + simpl (INR 2); field; split. + now apply Rgt_not_eq, (lt_INR 0). + now apply Rgt_not_eq. apply Rle_ge; apply PI_tg_pos. apply lt_le_trans with N; assumption. elim H1; intros H5 _. @@ -395,7 +370,6 @@ Proof. elim (Rlt_irrefl _ (Rlt_trans _ _ _ H6 H5)). elim (lt_n_O _ H6). apply le_IZR. - simpl. left; apply Rlt_trans with (/ (2 * eps)). apply Rinv_0_lt_compat; assumption. elim H1; intros; assumption. diff --git a/theories/Reals/ArithProp.v b/theories/Reals/ArithProp.v index 6fca9c8a..37240eb7 100644 --- a/theories/Reals/ArithProp.v +++ b/theories/Reals/ArithProp.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -143,7 +145,7 @@ Proof. assert (H0 := archimed (x / y)); rewrite <- Z_R_minus; simpl; cut (0 < y). intro; unfold Rminus; - replace (- ((IZR (up (x / y)) + -1) * y)) with ((1 - IZR (up (x / y))) * y); + replace (- ((IZR (up (x / y)) + -(1)) * y)) with ((1 - IZR (up (x / y))) * y); [ idtac | ring ]. split. apply Rmult_le_reg_l with (/ y). diff --git a/theories/Reals/Binomial.v b/theories/Reals/Binomial.v index f878abfa..271100a5 100644 --- a/theories/Reals/Binomial.v +++ b/theories/Reals/Binomial.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. diff --git a/theories/Reals/Cauchy_prod.v b/theories/Reals/Cauchy_prod.v index 5cf6f17d..306b09dc 100644 --- a/theories/Reals/Cauchy_prod.v +++ b/theories/Reals/Cauchy_prod.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. diff --git a/theories/Reals/Cos_plus.v b/theories/Reals/Cos_plus.v index b14d807d..d046ecf1 100644 --- a/theories/Reals/Cos_plus.v +++ b/theories/Reals/Cos_plus.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -289,11 +291,9 @@ Proof. apply INR_fact_lt_0. rewrite <- Rinv_r_sym. rewrite Rmult_1_r. - replace 1 with (INR 1). - apply le_INR. + apply (le_INR 1). apply lt_le_S. apply INR_lt; apply INR_fact_lt_0. - reflexivity. apply INR_fact_neq_0. apply Rmult_le_reg_l with (INR (fact (S (N + n)))). apply INR_fact_lt_0. @@ -576,11 +576,9 @@ Proof. apply INR_fact_lt_0. rewrite <- Rinv_r_sym. rewrite Rmult_1_r. - replace 1 with (INR 1). - apply le_INR. + apply (le_INR 1). apply lt_le_S. apply INR_lt; apply INR_fact_lt_0. - reflexivity. apply INR_fact_neq_0. apply Rmult_le_reg_l with (INR (fact (S (S (N + n))))). apply INR_fact_lt_0. diff --git a/theories/Reals/Cos_rel.v b/theories/Reals/Cos_rel.v index f5fcac47..f9919278 100644 --- a/theories/Reals/Cos_rel.v +++ b/theories/Reals/Cos_rel.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. diff --git a/theories/Reals/DiscrR.v b/theories/Reals/DiscrR.v index 4e2a7c3c..f3bc2f22 100644 --- a/theories/Reals/DiscrR.v +++ b/theories/Reals/DiscrR.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import RIneq. @@ -22,18 +24,10 @@ Proof. intros; rewrite H; reflexivity. Qed. -Lemma IZR_neq : forall z1 z2:Z, z1 <> z2 -> IZR z1 <> IZR z2. -Proof. -intros; red; intro; elim H; apply eq_IZR; assumption. -Qed. - Ltac discrR := try match goal with | |- (?X1 <> ?X2) => - change 2 with (IZR 2); - change 1 with (IZR 1); - change 0 with (IZR 0); repeat rewrite <- plus_IZR || rewrite <- mult_IZR || @@ -52,9 +46,6 @@ Ltac prove_sup0 := end. Ltac omega_sup := - change 2 with (IZR 2); - change 1 with (IZR 1); - change 0 with (IZR 0); repeat rewrite <- plus_IZR || rewrite <- mult_IZR || rewrite <- Ropp_Ropp_IZR || rewrite Z_R_minus; @@ -72,9 +63,6 @@ Ltac prove_sup := end. Ltac Rcompute := - change 2 with (IZR 2); - change 1 with (IZR 1); - change 0 with (IZR 0); repeat rewrite <- plus_IZR || rewrite <- mult_IZR || rewrite <- Ropp_Ropp_IZR || rewrite Z_R_minus; diff --git a/theories/Reals/Exp_prop.v b/theories/Reals/Exp_prop.v index 569518f7..3de131ea 100644 --- a/theories/Reals/Exp_prop.v +++ b/theories/Reals/Exp_prop.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -439,20 +441,16 @@ Proof. repeat rewrite <- Rmult_assoc. rewrite <- Rinv_r_sym. rewrite Rmult_1_l. - replace (INR N * INR N) with (Rsqr (INR N)); [ idtac | reflexivity ]. - rewrite Rmult_assoc. - rewrite Rmult_comm. - replace 4 with (Rsqr 2); [ idtac | ring_Rsqr ]. + change 4 with (Rsqr 2). rewrite <- Rsqr_mult. apply Rsqr_incr_1. - replace 2 with (INR 2). - rewrite <- mult_INR; apply H1. - reflexivity. + change 2 with (INR 2). + rewrite Rmult_comm, <- mult_INR; apply H1. left; apply lt_INR_0; apply H. left; apply Rmult_lt_0_compat. - prove_sup0. apply lt_INR_0; apply div2_not_R0. apply lt_n_S; apply H. + now apply IZR_lt. cut (1 < S N)%nat. intro; unfold Rsqr; apply prod_neq_R0; apply not_O_INR; intro; assert (H4 := div2_not_R0 _ H2); rewrite H3 in H4; @@ -536,7 +534,7 @@ Proof. apply Rmult_le_reg_l with (INR (fact (div2 (pred n)))). apply INR_fact_lt_0. rewrite Rmult_1_r; rewrite <- Rinv_r_sym. - replace 1 with (INR 1); [ apply le_INR | reflexivity ]. + apply (le_INR 1). apply lt_le_S. apply INR_lt. apply INR_fact_lt_0. diff --git a/theories/Reals/Integration.v b/theories/Reals/Integration.v index e3760e01..1f4fd576 100644 --- a/theories/Reals/Integration.v +++ b/theories/Reals/Integration.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export NewtonInt. diff --git a/theories/Reals/MVT.v b/theories/Reals/MVT.v index 26c51583..717df1b1 100644 --- a/theories/Reals/MVT.v +++ b/theories/Reals/MVT.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. diff --git a/theories/Reals/Machin.v b/theories/Reals/Machin.v index 19db476f..cdf98cbd 100644 --- a/theories/Reals/Machin.v +++ b/theories/Reals/Machin.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Fourier. @@ -53,7 +55,7 @@ assert (-(PI/4) <= atan x). destruct xm1 as [xm1 | xm1]. rewrite <- atan_1, <- atan_opp; apply Rlt_le, atan_increasing. assumption. - solve[rewrite <- xm1, atan_opp, atan_1; apply Rle_refl]. + solve[rewrite <- xm1; change (-1) with (-(1)); rewrite atan_opp, atan_1; apply Rle_refl]. assert (-(PI/4) < atan y). rewrite <- atan_1, <- atan_opp; apply atan_increasing. assumption. diff --git a/theories/Reals/NewtonInt.v b/theories/Reals/NewtonInt.v index ed5ae90c..66918eee 100644 --- a/theories/Reals/NewtonInt.v +++ b/theories/Reals/NewtonInt.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. diff --git a/theories/Reals/PSeries_reg.v b/theories/Reals/PSeries_reg.v index 03ac6582..61d1b5af 100644 --- a/theories/Reals/PSeries_reg.v +++ b/theories/Reals/PSeries_reg.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. diff --git a/theories/Reals/PartSum.v b/theories/Reals/PartSum.v index 37d54a6d..33feeac0 100644 --- a/theories/Reals/PartSum.v +++ b/theories/Reals/PartSum.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. diff --git a/theories/Reals/RIneq.v b/theories/Reals/RIneq.v index 379fee6f..59a10496 100644 --- a/theories/Reals/RIneq.v +++ b/theories/Reals/RIneq.v @@ -1,10 +1,12 @@ -(* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (************************************************************************) (*********************************************************) @@ -1611,6 +1613,9 @@ Proof. Qed. Hint Resolve mult_INR: real. +Lemma pow_INR (m n: nat) : INR (m ^ n) = pow (INR m) n. +Proof. now induction n as [|n IHn];[ | simpl; rewrite mult_INR, IHn]. Qed. + (*********) Lemma lt_0_INR : forall n:nat, (0 < n)%nat -> 0 < INR n. Proof. @@ -1629,7 +1634,7 @@ Hint Resolve lt_INR: real. Lemma lt_1_INR : forall n:nat, (1 < n)%nat -> 1 < INR n. Proof. - intros; replace 1 with (INR 1); auto with real. + apply lt_INR. Qed. Hint Resolve lt_1_INR: real. @@ -1653,17 +1658,16 @@ Hint Resolve pos_INR: real. Lemma INR_lt : forall n m:nat, INR n < INR m -> (n < m)%nat. Proof. - double induction n m; intros. - simpl; exfalso; apply (Rlt_irrefl 0); auto. - auto with arith. - generalize (pos_INR (S n0)); intro; cut (INR 0 = 0); - [ intro H2; rewrite H2 in H0; idtac | simpl; trivial ]. - generalize (Rle_lt_trans 0 (INR (S n0)) 0 H1 H0); intro; exfalso; - apply (Rlt_irrefl 0); auto. - do 2 rewrite S_INR in H1; cut (INR n1 < INR n0). - intro H2; generalize (H0 n0 H2); intro; auto with arith. - apply (Rplus_lt_reg_l 1 (INR n1) (INR n0)). - rewrite Rplus_comm; rewrite (Rplus_comm 1 (INR n0)); trivial. + intros n m. revert n. + induction m ; intros n H. + - elim (Rlt_irrefl 0). + apply Rle_lt_trans with (2 := H). + apply pos_INR. + - destruct n as [|n]. + apply Nat.lt_0_succ. + apply lt_n_S, IHm. + rewrite 2!S_INR in H. + apply Rplus_lt_reg_r with (1 := H). Qed. Hint Resolve INR_lt: real. @@ -1707,14 +1711,10 @@ Hint Resolve not_INR: real. Lemma INR_eq : forall n m:nat, INR n = INR m -> n = m. Proof. - intros; case (le_or_lt n m); intros H1. - case (le_lt_or_eq _ _ H1); intros H2; auto. - cut (n <> m). - intro H3; generalize (not_INR n m H3); intro H4; exfalso; auto. - omega. - symmetry ; cut (m <> n). - intro H3; generalize (not_INR m n H3); intro H4; exfalso; auto. - omega. + intros n m HR. + destruct (dec_eq_nat n m) as [H|H]. + exact H. + now apply not_INR in H. Qed. Hint Resolve INR_eq: real. @@ -1728,7 +1728,8 @@ Hint Resolve INR_le: real. Lemma not_1_INR : forall n:nat, n <> 1%nat -> INR n <> 1. Proof. - replace 1 with (INR 1); auto with real. + intros n. + apply not_INR. Qed. Hint Resolve not_1_INR: real. @@ -1743,24 +1744,40 @@ Proof. intros z; idtac; apply Z_of_nat_complete; assumption. Qed. +Lemma INR_IPR : forall p, INR (Pos.to_nat p) = IPR p. +Proof. + assert (H: forall p, 2 * INR (Pos.to_nat p) = IPR_2 p). + induction p as [p|p|] ; simpl IPR_2. + rewrite Pos2Nat.inj_xI, S_INR, mult_INR, <- IHp. + now rewrite (Rplus_comm (2 * _)). + now rewrite Pos2Nat.inj_xO, mult_INR, <- IHp. + apply Rmult_1_r. + intros [p|p|] ; unfold IPR. + rewrite Pos2Nat.inj_xI, S_INR, mult_INR, <- H. + apply Rplus_comm. + now rewrite Pos2Nat.inj_xO, mult_INR, <- H. + easy. +Qed. + (**********) Lemma INR_IZR_INZ : forall n:nat, INR n = IZR (Z.of_nat n). Proof. - simple induction n; auto with real. - intros; simpl; rewrite SuccNat2Pos.id_succ; - auto with real. + intros [|n]. + easy. + simpl Z.of_nat. unfold IZR. + now rewrite <- INR_IPR, SuccNat2Pos.id_succ. Qed. Lemma plus_IZR_NEG_POS : forall p q:positive, IZR (Zpos p + Zneg q) = IZR (Zpos p) + IZR (Zneg q). Proof. intros p q; simpl. rewrite Z.pos_sub_spec. - case Pos.compare_spec; intros H; simpl. + case Pos.compare_spec; intros H; unfold IZR. subst. ring. - rewrite Pos2Nat.inj_sub by trivial. + rewrite <- 3!INR_IPR, Pos2Nat.inj_sub by trivial. rewrite minus_INR by (now apply lt_le_weak, Pos2Nat.inj_lt). ring. - rewrite Pos2Nat.inj_sub by trivial. + rewrite <- 3!INR_IPR, Pos2Nat.inj_sub by trivial. rewrite minus_INR by (now apply lt_le_weak, Pos2Nat.inj_lt). ring. Qed. @@ -1769,26 +1786,18 @@ Qed. Lemma plus_IZR : forall n m:Z, IZR (n + m) = IZR n + IZR m. Proof. intro z; destruct z; intro t; destruct t; intros; auto with real. - simpl; intros; rewrite Pos2Nat.inj_add; auto with real. + simpl. unfold IZR. rewrite <- 3!INR_IPR, Pos2Nat.inj_add. apply plus_INR. apply plus_IZR_NEG_POS. rewrite Z.add_comm; rewrite Rplus_comm; apply plus_IZR_NEG_POS. - simpl; intros; rewrite Pos2Nat.inj_add; rewrite plus_INR; - auto with real. + simpl. unfold IZR. rewrite <- 3!INR_IPR, Pos2Nat.inj_add, plus_INR. + apply Ropp_plus_distr. Qed. (**********) Lemma mult_IZR : forall n m:Z, IZR (n * m) = IZR n * IZR m. Proof. - intros z t; case z; case t; simpl; auto with real. - intros t1 z1; rewrite Pos2Nat.inj_mul; auto with real. - intros t1 z1; rewrite Pos2Nat.inj_mul; auto with real. - rewrite Rmult_comm. - rewrite Ropp_mult_distr_l_reverse; auto with real. - apply Ropp_eq_compat; rewrite mult_comm; auto with real. - intros t1 z1; rewrite Pos2Nat.inj_mul; auto with real. - rewrite Ropp_mult_distr_l_reverse; auto with real. - intros t1 z1; rewrite Pos2Nat.inj_mul; auto with real. - rewrite Rmult_opp_opp; auto with real. + intros z t; case z; case t; simpl; auto with real; + unfold IZR; intros m n; rewrite <- 3!INR_IPR, Pos2Nat.inj_mul, mult_INR; ring. Qed. Lemma pow_IZR : forall z n, pow (IZR z) n = IZR (Z.pow z (Z.of_nat n)). @@ -1804,13 +1813,13 @@ Qed. (**********) Lemma succ_IZR : forall n:Z, IZR (Z.succ n) = IZR n + 1. Proof. - intro; change 1 with (IZR 1); unfold Z.succ; apply plus_IZR. + intro; unfold Z.succ; apply plus_IZR. Qed. (**********) Lemma opp_IZR : forall n:Z, IZR (- n) = - IZR n. Proof. - intro z; case z; simpl; auto with real. + intros [|z|z]; unfold IZR; simpl; auto with real. Qed. Definition Ropp_Ropp_IZR := opp_IZR. @@ -1833,10 +1842,12 @@ Qed. Lemma lt_0_IZR : forall n:Z, 0 < IZR n -> (0 < n)%Z. Proof. intro z; case z; simpl; intros. - absurd (0 < 0); auto with real. - unfold Z.lt; simpl; trivial. - case Rlt_not_le with (1 := H). - replace 0 with (-0); auto with real. + elim (Rlt_irrefl _ H). + easy. + elim (Rlt_not_le _ _ H). + unfold IZR. + rewrite <- INR_IPR. + auto with real. Qed. (**********) @@ -1852,9 +1863,12 @@ Qed. Lemma eq_IZR_R0 : forall n:Z, IZR n = 0 -> n = 0%Z. Proof. intro z; destruct z; simpl; intros; auto with zarith. - case (Rlt_not_eq 0 (INR (Pos.to_nat p))); auto with real. - case (Rlt_not_eq (- INR (Pos.to_nat p)) 0); auto with real. - apply Ropp_lt_gt_0_contravar. unfold Rgt; apply pos_INR_nat_of_P. + elim Rgt_not_eq with (2 := H). + unfold IZR. rewrite <- INR_IPR. + apply lt_0_INR, Pos2Nat.is_pos. + elim Rlt_not_eq with (2 := H). + unfold IZR. rewrite <- INR_IPR. + apply Ropp_lt_gt_0_contravar, lt_0_INR, Pos2Nat.is_pos. Qed. (**********) @@ -1892,8 +1906,8 @@ Qed. (**********) Lemma le_IZR_R1 : forall n:Z, IZR n <= 1 -> (n <= 1)%Z. Proof. - pattern 1 at 1; replace 1 with (IZR 1); intros; auto. - apply le_IZR; trivial. + intros n. + apply le_IZR. Qed. (**********) @@ -1917,12 +1931,23 @@ Proof. omega. Qed. +Lemma IZR_neq : forall z1 z2:Z, z1 <> z2 -> IZR z1 <> IZR z2. +Proof. +intros; red; intro; elim H; apply eq_IZR; assumption. +Qed. + +Hint Extern 0 (IZR _ <= IZR _) => apply IZR_le, Zle_bool_imp_le, eq_refl : real. +Hint Extern 0 (IZR _ >= IZR _) => apply Rle_ge, IZR_le, Zle_bool_imp_le, eq_refl : real. +Hint Extern 0 (IZR _ < IZR _) => apply IZR_lt, eq_refl : real. +Hint Extern 0 (IZR _ > IZR _) => apply IZR_lt, eq_refl : real. +Hint Extern 0 (IZR _ <> IZR _) => apply IZR_neq, Zeq_bool_neq, eq_refl : real. + Lemma one_IZR_lt1 : forall n:Z, -1 < IZR n < 1 -> n = 0%Z. Proof. intros z [H1 H2]. apply Z.le_antisymm. apply Z.lt_succ_r; apply lt_IZR; trivial. - replace 0%Z with (Z.succ (-1)); trivial. + change 0%Z with (Z.succ (-1)). apply Z.le_succ_l; apply lt_IZR; trivial. Qed. @@ -1999,10 +2024,34 @@ Lemma double_var : forall r1, r1 = r1 / 2 + r1 / 2. Proof. intro; rewrite <- double; unfold Rdiv; rewrite <- Rmult_assoc; symmetry ; apply Rinv_r_simpl_m. - replace 2 with (INR 2); - [ apply not_0_INR; discriminate | unfold INR; ring ]. + now apply not_0_IZR. +Qed. + +Lemma R_rm : ring_morph + 0%R 1%R Rplus Rmult Rminus Ropp eq + 0%Z 1%Z Zplus Zmult Zminus Z.opp Zeq_bool IZR. +Proof. +constructor ; try easy. +exact plus_IZR. +exact minus_IZR. +exact mult_IZR. +exact opp_IZR. +intros x y H. +apply f_equal. +now apply Zeq_bool_eq. +Qed. + +Lemma Zeq_bool_IZR x y : + IZR x = IZR y -> Zeq_bool x y = true. +Proof. +intros H. +apply Zeq_is_eq_bool. +now apply eq_IZR. Qed. +Add Field RField : Rfield + (completeness Zeq_bool_IZR, morphism R_rm, constants [IZR_tac], power_tac R_power_theory [Rpow_tac]). + (*********************************************************) (** ** Other rules about < and <= *) (*********************************************************) @@ -2017,42 +2066,18 @@ Qed. Lemma le_epsilon : forall r1 r2, (forall eps:R, 0 < eps -> r1 <= r2 + eps) -> r1 <= r2. Proof. - intros x y; intros; elim (Rtotal_order x y); intro. - left; assumption. - elim H0; intro. - right; assumption. - clear H0; generalize (Rgt_minus x y H1); intro H2; change (0 < x - y) in H2. - cut (0 < 2). - intro. - generalize (Rmult_lt_0_compat (x - y) (/ 2) H2 (Rinv_0_lt_compat 2 H0)); - intro H3; generalize (H ((x - y) * / 2) H3); - replace (y + (x - y) * / 2) with ((y + x) * / 2). - intro H4; - generalize (Rmult_le_compat_l 2 x ((y + x) * / 2) (Rlt_le 0 2 H0) H4); - rewrite <- (Rmult_comm ((y + x) * / 2)); rewrite Rmult_assoc; - rewrite <- Rinv_l_sym. - rewrite Rmult_1_r; replace (2 * x) with (x + x). - rewrite (Rplus_comm y); intro H5; apply Rplus_le_reg_l with x; assumption. - ring. - replace 2 with (INR 2); [ apply not_0_INR; discriminate | reflexivity ]. - pattern y at 2; replace y with (y / 2 + y / 2). - unfold Rminus, Rdiv. - repeat rewrite Rmult_plus_distr_r. - ring. - cut (forall z:R, 2 * z = z + z). - intro. - rewrite <- (H4 (y / 2)). - unfold Rdiv. - rewrite <- Rmult_assoc; apply Rinv_r_simpl_m. - replace 2 with (INR 2). - apply not_0_INR. - discriminate. - unfold INR; reflexivity. - intro; ring. - cut (0%nat <> 2%nat); - [ intro H0; generalize (lt_0_INR 2 (neq_O_lt 2 H0)); unfold INR; - intro; assumption - | discriminate ]. + intros x y H. + destruct (Rle_or_lt x y) as [H1|H1]. + exact H1. + apply Rplus_le_reg_r with x. + replace (y + x) with (2 * (y + (x - y) * / 2)) by field. + replace (x + x) with (2 * x) by ring. + apply Rmult_le_compat_l. + now apply (IZR_le 0 2). + apply H. + apply Rmult_lt_0_compat. + now apply Rgt_minus. + apply Rinv_0_lt_compat, Rlt_0_2. Qed. (**********) diff --git a/theories/Reals/RList.v b/theories/Reals/RList.v index 924d5117..e12937c7 100644 --- a/theories/Reals/RList.v +++ b/theories/Reals/RList.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. diff --git a/theories/Reals/ROrderedType.v b/theories/Reals/ROrderedType.v index f2dc7fd0..ee65ee1d 100644 --- a/theories/Reals/ROrderedType.v +++ b/theories/Reals/ROrderedType.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase Equalities Orders OrdersTac. diff --git a/theories/Reals/R_Ifp.v b/theories/Reals/R_Ifp.v index b6d07283..77e2a1e0 100644 --- a/theories/Reals/R_Ifp.v +++ b/theories/Reals/R_Ifp.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (**********************************************************) @@ -42,28 +44,23 @@ Qed. Lemma up_tech : forall (r:R) (z:Z), IZR z <= r -> r < IZR (z + 1) -> (z + 1)%Z = up r. Proof. - intros; generalize (Rplus_le_compat_l 1 (IZR z) r H); intro; clear H; - rewrite (Rplus_comm 1 (IZR z)) in H1; rewrite (Rplus_comm 1 r) in H1; - cut (1 = IZR 1); auto with zarith real. - intro; generalize H1; pattern 1 at 1; rewrite H; intro; clear H H1; - rewrite <- (plus_IZR z 1) in H2; apply (tech_up r (z + 1)); - auto with zarith real. + intros. + apply tech_up with (1 := H0). + rewrite plus_IZR. + now apply Rplus_le_compat_r. Qed. (**********) Lemma fp_R0 : frac_part 0 = 0. Proof. - unfold frac_part; unfold Int_part; elim (archimed 0); intros; - unfold Rminus; elim (Rplus_ne (- IZR (up 0 - 1))); - intros a b; rewrite b; clear a b; rewrite <- Z_R_minus; - cut (up 0 = 1%Z). - intro; rewrite H1; - rewrite (Rminus_diag_eq (IZR 1) (IZR 1) (eq_refl (IZR 1))); - apply Ropp_0. - elim (archimed 0); intros; clear H2; unfold Rgt in H1; - rewrite (Rminus_0_r (IZR (up 0))) in H0; generalize (lt_O_IZR (up 0) H1); - intro; clear H1; generalize (le_IZR_R1 (up 0) H0); - intro; clear H H0; omega. + unfold frac_part, Int_part. + replace (up 0) with 1%Z. + now rewrite <- minus_IZR. + destruct (archimed 0) as [H1 H2]. + apply lt_IZR in H1. + rewrite <- minus_IZR in H2. + apply le_IZR in H2. + omega. Qed. (**********) @@ -112,21 +109,12 @@ Lemma base_Int_part : Proof. intro; unfold Int_part; elim (archimed r); intros. split; rewrite <- (Z_R_minus (up r) 1); simpl. - generalize (Rle_minus (IZR (up r) - r) 1 H0); intro; unfold Rminus in H1; - rewrite (Rplus_assoc (IZR (up r)) (- r) (-1)) in H1; - rewrite (Rplus_comm (- r) (-1)) in H1; - rewrite <- (Rplus_assoc (IZR (up r)) (-1) (- r)) in H1; - fold (IZR (up r) - 1) in H1; fold (IZR (up r) - 1 - r) in H1; - apply Rminus_le; auto with zarith real. - generalize (Rplus_gt_compat_l (-1) (IZR (up r)) r H); intro; - rewrite (Rplus_comm (-1) (IZR (up r))) in H1; - generalize (Rplus_gt_compat_l (- r) (IZR (up r) + -1) (-1 + r) H1); - intro; clear H H0 H1; rewrite (Rplus_comm (- r) (IZR (up r) + -1)) in H2; - fold (IZR (up r) - 1) in H2; fold (IZR (up r) - 1 - r) in H2; - rewrite (Rplus_comm (- r) (-1 + r)) in H2; - rewrite (Rplus_assoc (-1) r (- r)) in H2; rewrite (Rplus_opp_r r) in H2; - elim (Rplus_ne (-1)); intros a b; rewrite a in H2; - clear a b; auto with zarith real. + apply Rminus_le. + replace (IZR (up r) - 1 - r) with (IZR (up r) - r - 1) by ring. + now apply Rle_minus. + apply Rminus_gt. + replace (IZR (up r) - 1 - r - -1) with (IZR (up r) - r) by ring. + now apply Rgt_minus. Qed. (**********) @@ -238,9 +226,7 @@ Proof. rewrite (Rplus_opp_r (IZR (Int_part r1) - IZR (Int_part r2))) in H; elim (Rplus_ne (r1 - r2)); intros a b; rewrite b in H; clear a b; rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H0; - rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H; - cut (1 = IZR 1); auto with zarith real. - intro; rewrite H1 in H; clear H1; + rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H. rewrite <- (plus_IZR (Int_part r1 - Int_part r2) 1) in H; generalize (up_tech (r1 - r2) (Int_part r1 - Int_part r2) H0 H); intros; clear H H0; unfold Int_part at 1; @@ -324,12 +310,12 @@ Proof. rewrite (Rplus_opp_r (IZR (Int_part r1) - IZR (Int_part r2))) in H0; elim (Rplus_ne (r1 - r2)); intros a b; rewrite b in H0; clear a b; rewrite <- (Rplus_opp_l 1) in H0; - rewrite <- (Rplus_assoc (IZR (Int_part r1) - IZR (Int_part r2)) (-1) 1) + rewrite <- (Rplus_assoc (IZR (Int_part r1) - IZR (Int_part r2)) (-(1)) 1) in H0; fold (IZR (Int_part r1) - IZR (Int_part r2) - 1) in H0; rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H0; rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H; - cut (1 = IZR 1); auto with zarith real. - intro; rewrite H1 in H; rewrite H1 in H0; clear H1; + auto with zarith real. + change (_ + -1) with (IZR (Int_part r1 - Int_part r2) - 1) in H; rewrite (Z_R_minus (Int_part r1 - Int_part r2) 1) in H; rewrite (Z_R_minus (Int_part r1 - Int_part r2) 1) in H0; rewrite <- (plus_IZR (Int_part r1 - Int_part r2 - 1) 1) in H0; @@ -442,9 +428,9 @@ Proof. in H0; rewrite (Rplus_opp_r (IZR (Int_part r1) + IZR (Int_part r2))) in H0; elim (Rplus_ne (r1 + r2)); intros a b; rewrite b in H0; clear a b; + change 2 with (1 + 1) in H0; rewrite <- (Rplus_assoc (IZR (Int_part r1) + IZR (Int_part r2)) 1 1) in H0; - cut (1 = IZR 1); auto with zarith real. - intro; rewrite H1 in H0; rewrite H1 in H; clear H1; + auto with zarith real. rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H; rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H0; rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H; @@ -507,9 +493,7 @@ Proof. in H0; rewrite (Rplus_opp_r (IZR (Int_part r1) + IZR (Int_part r2))) in H0; elim (Rplus_ne (IZR (Int_part r1) + IZR (Int_part r2))); intros a b; rewrite a in H0; clear a b; elim (Rplus_ne (r1 + r2)); - intros a b; rewrite b in H0; clear a b; cut (1 = IZR 1); - auto with zarith real. - intro; rewrite H in H1; clear H; + intros a b; rewrite b in H0; clear a b. rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H0; rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H1; rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H1; @@ -536,7 +520,7 @@ Proof. rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2))); unfold Rminus; rewrite - (Rplus_assoc (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2))) (-1)) + (Rplus_assoc (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2))) (-(1))) ; rewrite <- (Ropp_plus_distr (IZR (Int_part r1) + IZR (Int_part r2)) 1); trivial with zarith real. Qed. diff --git a/theories/Reals/R_sqr.v b/theories/Reals/R_sqr.v index 445ffcb2..a60bb7cf 100644 --- a/theories/Reals/R_sqr.v +++ b/theories/Reals/R_sqr.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -296,56 +298,9 @@ Lemma canonical_Rsqr : a * Rsqr (x + b / (2 * a)) + (4 * a * c - Rsqr b) / (4 * a). Proof. intros. - rewrite Rsqr_plus. - repeat rewrite Rmult_plus_distr_l. - repeat rewrite Rplus_assoc. - apply Rplus_eq_compat_l. - unfold Rdiv, Rminus. - replace (2 * 1 + 2 * 1) with 4; [ idtac | ring ]. - rewrite (Rmult_plus_distr_r (4 * a * c) (- Rsqr b) (/ (4 * a))). - rewrite Rsqr_mult. - repeat rewrite Rinv_mult_distr. - repeat rewrite (Rmult_comm a). - repeat rewrite Rmult_assoc. - rewrite <- Rinv_l_sym. - rewrite Rmult_1_r. - rewrite (Rmult_comm 2). - repeat rewrite Rmult_assoc. - rewrite <- Rinv_l_sym. - rewrite Rmult_1_r. - rewrite (Rmult_comm (/ 2)). - rewrite (Rmult_comm 2). - repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. - rewrite Rmult_1_r. - rewrite (Rmult_comm a). - repeat rewrite Rmult_assoc. - rewrite <- Rinv_l_sym. - rewrite Rmult_1_r. - rewrite (Rmult_comm 2). - repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. - rewrite Rmult_1_r. - repeat rewrite Rplus_assoc. - rewrite (Rplus_comm (Rsqr b * (Rsqr (/ a * / 2) * a))). - repeat rewrite Rplus_assoc. - rewrite (Rmult_comm x). - apply Rplus_eq_compat_l. - rewrite (Rmult_comm (/ a)). - unfold Rsqr; repeat rewrite Rmult_assoc. - rewrite <- Rinv_l_sym. - rewrite Rmult_1_r. - ring. - apply (cond_nonzero a). - discrR. - apply (cond_nonzero a). - discrR. - discrR. - apply (cond_nonzero a). - discrR. - discrR. - discrR. - apply (cond_nonzero a). - discrR. - apply (cond_nonzero a). + unfold Rsqr. + field. + apply a. Qed. Lemma Rsqr_eq : forall x y:R, Rsqr x = Rsqr y -> x = y \/ x = - y. diff --git a/theories/Reals/R_sqrt.v b/theories/Reals/R_sqrt.v index a6b1a26e..d4035fad 100644 --- a/theories/Reals/R_sqrt.v +++ b/theories/Reals/R_sqrt.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -359,107 +361,22 @@ Lemma Rsqr_sol_eq_0_1 : x = sol_x1 a b c \/ x = sol_x2 a b c -> a * Rsqr x + b * x + c = 0. Proof. intros; elim H0; intro. - unfold sol_x1 in H1; unfold Delta in H1; rewrite H1; unfold Rdiv; - repeat rewrite Rsqr_mult; rewrite Rsqr_plus; rewrite <- Rsqr_neg; - rewrite Rsqr_sqrt. - rewrite Rsqr_inv. - unfold Rsqr; repeat rewrite Rinv_mult_distr. - repeat rewrite Rmult_assoc; rewrite (Rmult_comm a). - repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. - rewrite Rmult_1_r; rewrite Rmult_plus_distr_r. - repeat rewrite Rmult_assoc. - pattern 2 at 2; rewrite (Rmult_comm 2). - repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. - rewrite Rmult_1_r. - rewrite - (Rmult_plus_distr_r (- b) (sqrt (b * b - 2 * (2 * (a * c)))) (/ 2 * / a)) - . - rewrite Rmult_plus_distr_l; repeat rewrite Rplus_assoc. - replace - (- b * (sqrt (b * b - 2 * (2 * (a * c))) * (/ 2 * / a)) + - (b * (- b * (/ 2 * / a)) + - (b * (sqrt (b * b - 2 * (2 * (a * c))) * (/ 2 * / a)) + c))) with - (b * (- b * (/ 2 * / a)) + c). - unfold Rminus; repeat rewrite <- Rplus_assoc. - replace (b * b + b * b) with (2 * (b * b)). - rewrite Rmult_plus_distr_r; repeat rewrite Rmult_assoc. - rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc. - rewrite <- Rinv_l_sym. - rewrite Rmult_1_r. - rewrite Ropp_mult_distr_l_reverse; repeat rewrite Rmult_assoc; - rewrite (Rmult_comm 2). - repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. - rewrite Rmult_1_r; rewrite (Rmult_comm (/ 2)); repeat rewrite Rmult_assoc; - rewrite (Rmult_comm 2). - repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. - rewrite Rmult_1_r; repeat rewrite Rmult_assoc. - rewrite (Rmult_comm a); rewrite Rmult_assoc. - rewrite <- Rinv_l_sym. - rewrite Rmult_1_r; rewrite <- Rmult_opp_opp. - ring. - apply (cond_nonzero a). - discrR. - discrR. - discrR. - ring. - ring. - discrR. - apply (cond_nonzero a). - discrR. - apply (cond_nonzero a). - apply prod_neq_R0; [ discrR | apply (cond_nonzero a) ]. - apply prod_neq_R0; [ discrR | apply (cond_nonzero a) ]. - apply prod_neq_R0; [ discrR | apply (cond_nonzero a) ]. - assumption. - unfold sol_x2 in H1; unfold Delta in H1; rewrite H1; unfold Rdiv; - repeat rewrite Rsqr_mult; rewrite Rsqr_minus; rewrite <- Rsqr_neg; - rewrite Rsqr_sqrt. - rewrite Rsqr_inv. - unfold Rsqr; repeat rewrite Rinv_mult_distr; - repeat rewrite Rmult_assoc. - rewrite (Rmult_comm a); repeat rewrite Rmult_assoc. - rewrite <- Rinv_l_sym. - rewrite Rmult_1_r; unfold Rminus; rewrite Rmult_plus_distr_r. - rewrite Ropp_mult_distr_l_reverse; repeat rewrite Rmult_assoc; - pattern 2 at 2; rewrite (Rmult_comm 2). - repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. - rewrite Rmult_1_r; - rewrite - (Rmult_plus_distr_r (- b) (- sqrt (b * b + - (2 * (2 * (a * c))))) - (/ 2 * / a)). - rewrite Rmult_plus_distr_l; repeat rewrite Rplus_assoc. - rewrite Ropp_mult_distr_l_reverse; rewrite Ropp_involutive. - replace - (b * (sqrt (b * b + - (2 * (2 * (a * c)))) * (/ 2 * / a)) + - (b * (- b * (/ 2 * / a)) + - (b * (- sqrt (b * b + - (2 * (2 * (a * c)))) * (/ 2 * / a)) + c))) with - (b * (- b * (/ 2 * / a)) + c). - repeat rewrite <- Rplus_assoc; replace (b * b + b * b) with (2 * (b * b)). - rewrite Rmult_plus_distr_r; repeat rewrite Rmult_assoc; - rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; - rewrite <- Rinv_l_sym. - rewrite Ropp_mult_distr_l_reverse; repeat rewrite Rmult_assoc. - rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. - rewrite Rmult_1_r; rewrite (Rmult_comm (/ 2)); repeat rewrite Rmult_assoc. - rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. - rewrite Rmult_1_r; repeat rewrite Rmult_assoc; rewrite (Rmult_comm a); - rewrite Rmult_assoc; rewrite <- Rinv_l_sym. - rewrite Rmult_1_r; rewrite <- Rmult_opp_opp; ring. - apply (cond_nonzero a). - discrR. - discrR. - discrR. - ring. - ring. - discrR. - apply (cond_nonzero a). - discrR. - discrR. - apply (cond_nonzero a). - apply prod_neq_R0; discrR || apply (cond_nonzero a). - apply prod_neq_R0; discrR || apply (cond_nonzero a). - apply prod_neq_R0; discrR || apply (cond_nonzero a). - assumption. + rewrite H1. + unfold sol_x1, Delta, Rsqr. + field_simplify. + rewrite <- (Rsqr_pow2 (sqrt _)), Rsqr_sqrt. + field. + apply a. + apply H. + apply a. + rewrite H1. + unfold sol_x2, Delta, Rsqr. + field_simplify. + rewrite <- (Rsqr_pow2 (sqrt _)), Rsqr_sqrt. + field. + apply a. + apply H. + apply a. Qed. Lemma Rsqr_sol_eq_0_0 : @@ -505,10 +422,10 @@ Proof. rewrite (Rmult_comm (/ a)). rewrite Rmult_assoc. rewrite <- Rinv_mult_distr. - replace (2 * (2 * a) * a) with (Rsqr (2 * a)). + replace (4 * a * a) with (Rsqr (2 * a)). reflexivity. ring_Rsqr. - rewrite <- Rmult_assoc; apply prod_neq_R0; + apply prod_neq_R0; [ discrR | apply (cond_nonzero a) ]. apply (cond_nonzero a). assumption. diff --git a/theories/Reals/Ranalysis.v b/theories/Reals/Ranalysis.v index 88ebb88b..4bde9b60 100644 --- a/theories/Reals/Ranalysis.v +++ b/theories/Reals/Ranalysis.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -26,4 +28,4 @@ Require Export RList. Require Export Sqrt_reg. Require Export Ranalysis4. Require Export Rpower. -Require Export Ranalysis_reg.
\ No newline at end of file +Require Export Ranalysis_reg. diff --git a/theories/Reals/Ranalysis1.v b/theories/Reals/Ranalysis1.v index 9e3abd81..36ac738c 100644 --- a/theories/Reals/Ranalysis1.v +++ b/theories/Reals/Ranalysis1.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. diff --git a/theories/Reals/Ranalysis2.v b/theories/Reals/Ranalysis2.v index 0254218c..7a97ca63 100644 --- a/theories/Reals/Ranalysis2.v +++ b/theories/Reals/Ranalysis2.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -88,17 +90,11 @@ Proof. right; unfold Rdiv. repeat rewrite Rabs_mult. rewrite Rabs_Rinv; discrR. - replace (Rabs 8) with 8. - replace 8 with 8; [ idtac | ring ]. - rewrite Rinv_mult_distr; [ idtac | discrR | discrR ]. - replace (2 * / Rabs (f2 x) * (Rabs eps * Rabs (f2 x) * (/ 2 * / 4))) with - (Rabs eps * / 4 * (2 * / 2) * (Rabs (f2 x) * / Rabs (f2 x))); - [ idtac | ring ]. - replace (Rabs eps) with eps. - repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption). - ring. - symmetry ; apply Rabs_right; left; assumption. - symmetry ; apply Rabs_right; left; prove_sup. + rewrite (Rabs_pos_eq 8) by now apply IZR_le. + rewrite (Rabs_pos_eq eps). + field. + now apply Rabs_no_R0. + now apply Rlt_le. Qed. Lemma maj_term2 : @@ -429,10 +425,7 @@ Proof. intro; rewrite H11 in H10; assert (H12 := Rmult_lt_compat_l 2 _ _ Hyp H10); rewrite Rmult_1_r in H12; rewrite <- Rinv_r_sym in H12; [ idtac | discrR ]. - cut (IZR 1 < IZR 2). - unfold IZR; unfold INR, Pos.to_nat; simpl; intro; - elim (Rlt_irrefl 1 (Rlt_trans _ _ _ H13 H12)). - apply IZR_lt; omega. + now apply lt_IZR in H12. unfold Rabs; case (Rcase_abs (/ 2)) as [Hlt|Hge]. assert (Hyp : 0 < 2). prove_sup0. diff --git a/theories/Reals/Ranalysis3.v b/theories/Reals/Ranalysis3.v index 4e88714d..301d6d2c 100644 --- a/theories/Reals/Ranalysis3.v +++ b/theories/Reals/Ranalysis3.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -201,8 +203,8 @@ Proof. apply Rabs_pos_lt. unfold Rdiv, Rsqr; repeat rewrite Rmult_assoc. repeat apply prod_neq_R0; try assumption. - red; intro; rewrite H15 in H6; elim (Rlt_irrefl _ H6). - apply Rinv_neq_0_compat; repeat apply prod_neq_R0; discrR || assumption. + now apply Rgt_not_eq. + apply Rinv_neq_0_compat; apply prod_neq_R0; [discrR | assumption]. apply H13. split. apply D_x_no_cond; assumption. @@ -213,8 +215,7 @@ Proof. red; intro; rewrite H11 in H6; elim (Rlt_irrefl _ H6). assumption. assumption. - apply Rinv_neq_0_compat; repeat apply prod_neq_R0; - [ discrR | discrR | discrR | assumption ]. + apply Rinv_neq_0_compat; apply prod_neq_R0; [discrR | assumption]. (***********************************) (* Third case *) (* (f1 x)<>0 l1=0 l2=0 *) @@ -224,11 +225,11 @@ Proof. elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x)))); [ idtac | apply Rabs_pos_lt; unfold Rdiv, Rsqr; repeat rewrite Rmult_assoc; - repeat apply prod_neq_R0; + repeat apply prod_neq_R0 ; [ assumption | assumption - | red; intro; rewrite H11 in H6; elim (Rlt_irrefl _ H6) - | apply Rinv_neq_0_compat; repeat apply prod_neq_R0; discrR || assumption ] ]. + | now apply Rgt_not_eq + | apply Rinv_neq_0_compat; apply prod_neq_R0; discrR || assumption ] ]. intros alp_f2d H12. cut (0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)). intro. @@ -295,8 +296,10 @@ Proof. elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x)))); [ idtac | apply Rabs_pos_lt; unfold Rsqr, Rdiv; - repeat rewrite Rinv_mult_distr; repeat apply prod_neq_R0; - try assumption || discrR ]. + repeat apply prod_neq_R0 ; + [ assumption.. + | now apply Rgt_not_eq + | apply Rinv_neq_0_compat; apply prod_neq_R0; discrR || assumption ] ]. intros alp_f2d H11. assert (H12 := derivable_continuous_pt _ _ X). unfold continuity_pt in H12. @@ -380,15 +383,9 @@ Proof. repeat apply prod_neq_R0; try assumption. red; intro H18; rewrite H18 in H6; elim (Rlt_irrefl _ H6). apply Rinv_neq_0_compat; discrR. - apply Rinv_neq_0_compat; discrR. - apply Rinv_neq_0_compat; discrR. apply Rinv_neq_0_compat; assumption. apply Rinv_neq_0_compat; assumption. discrR. - discrR. - discrR. - discrR. - discrR. apply prod_neq_R0; [ discrR | assumption ]. elim H13; intros. apply H19. @@ -408,16 +405,9 @@ Proof. repeat apply prod_neq_R0; try assumption. red; intro H13; rewrite H13 in H6; elim (Rlt_irrefl _ H6). apply Rinv_neq_0_compat; discrR. - apply Rinv_neq_0_compat; discrR. - apply Rinv_neq_0_compat; discrR. apply Rinv_neq_0_compat; assumption. apply Rinv_neq_0_compat; assumption. apply prod_neq_R0; [ discrR | assumption ]. - red; intro H11; rewrite H11 in H6; elim (Rlt_irrefl _ H6). - apply Rinv_neq_0_compat; discrR. - apply Rinv_neq_0_compat; discrR. - apply Rinv_neq_0_compat; discrR. - apply Rinv_neq_0_compat; assumption. (***********************************) (* Fifth case *) (* (f1 x)<>0 l1<>0 l2=0 *) diff --git a/theories/Reals/Ranalysis4.v b/theories/Reals/Ranalysis4.v index 661bc8c7..94f1757a 100644 --- a/theories/Reals/Ranalysis4.v +++ b/theories/Reals/Ranalysis4.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -130,15 +132,8 @@ Proof. intro; exists (mkposreal (- x) H1); intros. rewrite (Rabs_left x). rewrite (Rabs_left (x + h)). - rewrite Rplus_comm. - rewrite Ropp_plus_distr. - unfold Rminus; rewrite Ropp_involutive; rewrite Rplus_assoc; - rewrite Rplus_opp_l. - rewrite Rplus_0_r; unfold Rdiv. - rewrite Ropp_mult_distr_l_reverse. - rewrite <- Rinv_r_sym. - rewrite Ropp_involutive; rewrite Rplus_opp_l; rewrite Rabs_R0; apply H0. - apply H2. + replace ((-(x + h) - - x) / h - -1) with 0 by now field. + rewrite Rabs_R0; apply H0. destruct (Rcase_abs h) as [Hlt|Hgt]. apply Ropp_lt_cancel. rewrite Ropp_0; rewrite Ropp_plus_distr; apply Rplus_lt_0_compat. diff --git a/theories/Reals/Ranalysis5.v b/theories/Reals/Ranalysis5.v index d172139f..afb78e1c 100644 --- a/theories/Reals/Ranalysis5.v +++ b/theories/Reals/Ranalysis5.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -15,6 +17,7 @@ Require Import RiemannInt. Require Import SeqProp. Require Import Max. Require Import Omega. +Require Import Lra. Local Open Scope R_scope. (** * Preliminaries lemmas *) @@ -26,46 +29,34 @@ Lemma f_incr_implies_g_incr_interv : forall f g:R->R, forall lb ub, (forall x , f lb <= x -> x <= f ub -> lb <= g x <= ub) -> (forall x y, f lb <= x -> x < y -> y <= f ub -> g x < g y). Proof. -intros f g lb ub lb_lt_ub f_incr f_eq_g g_ok x y lb_le_x x_lt_y y_le_ub. - assert (x_encad : f lb <= x <= f ub). - split ; [assumption | apply Rle_trans with (r2:=y) ; [apply Rlt_le|] ; assumption]. - assert (y_encad : f lb <= y <= f ub). - split ; [apply Rle_trans with (r2:=x) ; [|apply Rlt_le] ; assumption | assumption]. - assert (Temp1 : lb <= lb) by intuition ; assert (Temp2 : ub <= ub) by intuition. - assert (gx_encad := g_ok _ (proj1 x_encad) (proj2 x_encad)). - assert (gy_encad := g_ok _ (proj1 y_encad) (proj2 y_encad)). - clear Temp1 Temp2. - case (Rlt_dec (g x) (g y)). - intuition. + intros f g lb ub lb_lt_ub f_incr f_eq_g g_ok x y lb_le_x x_lt_y y_le_ub. + assert (x_encad : f lb <= x <= f ub) by lra. + assert (y_encad : f lb <= y <= f ub) by lra. + assert (gx_encad := g_ok _ (proj1 x_encad) (proj2 x_encad)). + assert (gy_encad := g_ok _ (proj1 y_encad) (proj2 y_encad)). + case (Rlt_dec (g x) (g y)); [ easy |]. intros Hfalse. - assert (Temp := Rnot_lt_le _ _ Hfalse). - assert (Hcontradiction : y <= x). - replace y with (id y) by intuition ; replace x with (id x) by intuition ; - rewrite <- f_eq_g. rewrite <- f_eq_g. - assert (f_incr2 : forall x y, lb <= x -> x <= y -> y < ub -> f x <= f y). + assert (Temp := Rnot_lt_le _ _ Hfalse). + enough (y <= x) by lra. + replace y with (id y) by easy. + replace x with (id x) by easy. + rewrite <- f_eq_g by easy. + rewrite <- f_eq_g by easy. + assert (f_incr2 : forall x y, lb <= x -> x <= y -> y < ub -> f x <= f y). { intros m n lb_le_m m_le_n n_lt_ub. case (m_le_n). - intros ; apply Rlt_le ; apply f_incr ; [| | apply Rlt_le] ; assumption. - intros Hyp ; rewrite Hyp ; apply Req_le ; reflexivity. - apply f_incr2. - intuition. intuition. - Focus 3. intuition. - Focus 2. intuition. - Focus 2. intuition. Focus 2. intuition. - assert (Temp2 : g x <> ub). - intro Hf. - assert (Htemp : (comp f g) x = f ub). - unfold comp ; rewrite Hf ; reflexivity. - rewrite f_eq_g in Htemp ; unfold id in Htemp. - assert (Htemp2 : x < f ub). - apply Rlt_le_trans with (r2:=y) ; intuition. - clear -Htemp Htemp2. fourier. - intuition. intuition. - clear -Temp2 gx_encad. - case (proj2 gx_encad). - intuition. - intro Hfalse ; apply False_ind ; apply Temp2 ; assumption. - apply False_ind. clear - Hcontradiction x_lt_y. fourier. + - intros; apply Rlt_le, f_incr, Rlt_le; assumption. + - intros Hyp; rewrite Hyp; apply Req_le; reflexivity. + } + apply f_incr2; intuition. + enough (g x <> ub) by lra. + intro Hf. + assert (Htemp : (comp f g) x = f ub). { + unfold comp; rewrite Hf; reflexivity. + } + rewrite f_eq_g in Htemp by easy. + unfold id in Htemp. + fourier. Qed. Lemma derivable_pt_id_interv : forall (lb ub x:R), @@ -245,12 +236,8 @@ Lemma IVT_interv_prelim0 : forall (x y:R) (P:R->bool) (N:nat), x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= y. Proof. assert (Sublemma : forall x y lb ub, lb <= x <= ub /\ lb <= y <= ub -> lb <= (x+y) / 2 <= ub). - intros x y lb ub Hyp. - split. - replace lb with ((lb + lb) * /2) by field. - unfold Rdiv ; apply Rmult_le_compat_r ; intuition. - replace ub with ((ub + ub) * /2) by field. - unfold Rdiv ; apply Rmult_le_compat_r ; intuition. + intros x y lb ub Hyp. + lra. intros x y P N x_lt_y. induction N. simpl ; intuition. @@ -1027,9 +1014,7 @@ Qed. Lemma ub_lt_2_pos : forall x ub lb, lb < x -> x < ub -> 0 < (ub-lb)/2. Proof. intros x ub lb lb_lt_x x_lt_ub. - assert (T : 0 < ub - lb). - fourier. - unfold Rdiv ; apply Rlt_mult_inv_pos ; intuition. +lra. Qed. Definition mkposreal_lb_ub (x lb ub:R) (lb_lt_x:lb<x) (x_lt_ub:x<ub) : posreal. @@ -1102,7 +1087,7 @@ assert (Main : Rabs ((f (x+h) - fn N (x+h)) - (f x - fn N x) + (fn N (x+h) - fn rewrite <- Rmult_1_r ; replace 1 with (derive_pt id c (pr2 c P)) by reg. replace (- (fn N (x + h) - fn N x)) with (fn N x - fn N (x + h)) by field. assumption. - solve[apply Rlt_not_eq ; intuition]. + now apply Rlt_not_eq, IZR_lt. rewrite <- Hc'; clear Hc Hc'. replace (derive_pt (fn N) c (pr1 c P)) with (fn' N c). replace (h * fn' N c - h * g x) with (h * (fn' N c - g x)) by field. diff --git a/theories/Reals/Ranalysis_reg.v b/theories/Reals/Ranalysis_reg.v index 0c27d407..e1d4781b 100644 --- a/theories/Reals/Ranalysis_reg.v +++ b/theories/Reals/Ranalysis_reg.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. diff --git a/theories/Reals/Ratan.v b/theories/Reals/Ratan.v index e13ef1f2..ce39d5ff 100644 --- a/theories/Reals/Ratan.v +++ b/theories/Reals/Ratan.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Fourier. @@ -132,7 +134,7 @@ intros [ | N] Npos n decr to0 cv nN. unfold Rminus; apply Rplus_le_compat_l, Ropp_le_contravar. solve[apply Rge_le, (growing_prop _ _ _ (CV_ALT_step0 f decr) dist)]. unfold Rminus; rewrite tech5, Ropp_plus_distr, <- Rplus_assoc. - unfold tg_alt at 2; rewrite pow_1_odd, Ropp_mult_distr_l_reverse; fourier. + unfold tg_alt at 2; rewrite pow_1_odd; fourier. rewrite Nodd; destruct (alternated_series_ineq _ _ p decr to0 cv) as [B _]. destruct (alternated_series_ineq _ _ (S p) decr to0 cv) as [_ C]. assert (keep : (2 * S p = S (S ( 2 * p)))%nat) by ring. @@ -161,7 +163,6 @@ clear WLOG; intros Hyp [ | n] decr to0 cv _. generalize (alternated_series_ineq f l 0 decr to0 cv). unfold R_dist, tg_alt; simpl; rewrite !Rmult_1_l, !Rmult_1_r. assert (f 1%nat <= f 0%nat) by apply decr. - rewrite Ropp_mult_distr_l_reverse. intros [A B]; rewrite Rabs_pos_eq; fourier. apply Rle_trans with (f 1%nat). apply (Hyp 1%nat (le_n 1) (S n) decr to0 cv). @@ -320,31 +321,12 @@ apply PI2_lower_bound;[split; fourier | ]. destruct (pre_cos_bound (3/2) 1) as [t _]; [fourier | fourier | ]. apply Rlt_le_trans with (2 := t); clear t. unfold cos_approx; simpl; unfold cos_term. -simpl mult; replace ((-1)^ 0) with 1 by ring; replace ((-1)^2) with 1 by ring; - replace ((-1)^4) with 1 by ring; replace ((-1)^1) with (-1) by ring; - replace ((-1)^3) with (-1) by ring; replace 3 with (IZR 3) by (simpl; ring); - replace 2 with (IZR 2) by (simpl; ring); simpl Z.of_nat; - rewrite !INR_IZR_INZ, Ropp_mult_distr_l_reverse, Rmult_1_l. -match goal with |- _ < ?a => -replace a with ((- IZR 3 ^ 6 * IZR (Z.of_nat (fact 0)) * IZR (Z.of_nat (fact 2)) * - IZR (Z.of_nat (fact 4)) + - IZR 3 ^ 4 * IZR 2 ^ 2 * IZR (Z.of_nat (fact 0)) * IZR (Z.of_nat (fact 2)) * - IZR (Z.of_nat (fact 6)) - - IZR 3 ^ 2 * IZR 2 ^ 4 * IZR (Z.of_nat (fact 0)) * IZR (Z.of_nat (fact 4)) * - IZR (Z.of_nat (fact 6)) + - IZR 2 ^ 6 * IZR (Z.of_nat (fact 2)) * IZR (Z.of_nat (fact 4)) * - IZR (Z.of_nat (fact 6))) / - (IZR 2 ^ 6 * IZR (Z.of_nat (fact 0)) * IZR (Z.of_nat (fact 2)) * - IZR (Z.of_nat (fact 4)) * IZR (Z.of_nat (fact 6))));[ | field; - repeat apply conj;((rewrite <- INR_IZR_INZ; apply INR_fact_neq_0) || - (apply Rgt_not_eq; apply (IZR_lt 0); reflexivity)) ] -end. -rewrite !fact_simpl, !Nat2Z.inj_mul; simpl Z.of_nat. -unfold Rdiv; apply Rmult_lt_0_compat. -unfold Rminus; rewrite !pow_IZR, <- !opp_IZR, <- !mult_IZR, <- !opp_IZR, - <- !plus_IZR; apply (IZR_lt 0); reflexivity. -apply Rinv_0_lt_compat; rewrite !pow_IZR, <- !mult_IZR; apply (IZR_lt 0). -reflexivity. +rewrite !INR_IZR_INZ. +simpl. +field_simplify. +unfold Rdiv. +rewrite Rmult_0_l. +apply Rdiv_lt_0_compat ; now apply IZR_lt. Qed. Lemma PI2_1 : 1 < PI/2. @@ -502,11 +484,11 @@ split. rewrite (Rmult_comm (-1)); simpl ((/(Rabs y + 1)) ^ 0). unfold Rdiv; rewrite Rinv_1, !Rmult_assoc, <- !Rmult_plus_distr_l. apply tmp;[assumption | ]. - rewrite Rplus_assoc, Rmult_1_l; pattern 1 at 3; rewrite <- Rplus_0_r. + rewrite Rplus_assoc, Rmult_1_l; pattern 1 at 2; rewrite <- Rplus_0_r. apply Rplus_lt_compat_l. rewrite <- Rmult_assoc. match goal with |- (?a * (-1)) + _ < 0 => - rewrite <- (Rplus_opp_l a), Ropp_mult_distr_r_reverse, Rmult_1_r + rewrite <- (Rplus_opp_l a); change (-1) with (-(1)); rewrite Ropp_mult_distr_r_reverse, Rmult_1_r end. apply Rplus_lt_compat_l. assert (0 < u ^ 2) by (apply pow_lt; assumption). @@ -853,6 +835,8 @@ intros x Hx eps Heps. apply Rlt_trans with (2 := H). apply Rinv_0_lt_compat. exact Heps. + unfold N. + rewrite INR_IZR_INZ, positive_nat_Z. exact HN. apply lt_INR. omega. @@ -1076,8 +1060,9 @@ apply Rlt_not_eq; apply Rle_lt_trans with 0;[ | apply Rlt_0_1]. assert (t := pow2_ge_0 x); fourier. replace (1 + x ^ 2) with (1 - - (x ^ 2)) by ring; rewrite <- (tech3 _ n dif). apply sum_eq; unfold tg_alt, Datan_seq; intros i _. -rewrite pow_mult, <- Rpow_mult_distr, Ropp_mult_distr_l_reverse, Rmult_1_l. -reflexivity. +rewrite pow_mult, <- Rpow_mult_distr. +f_equal. +ring. Qed. Lemma Datan_seq_increasing : forall x y n, (n > 0)%nat -> 0 <= x < y -> Datan_seq x n < Datan_seq y n. @@ -1165,6 +1150,7 @@ assert (tool : forall a b, a / b - /b = (-1 + a) /b). reflexivity. set (u := 1 + x ^ 2); rewrite tool; unfold Rminus; rewrite <- Rplus_assoc. unfold Rdiv, u. +change (-1) with (-(1)). rewrite Rplus_opp_l, Rplus_0_l, Ropp_mult_distr_l_reverse, Rabs_Ropp. rewrite Rabs_mult; clear tool u. assert (tool : forall k, Rabs ((-x ^ 2) ^ k) = Rabs ((x ^ 2) ^ k)). diff --git a/theories/Reals/Raxioms.v b/theories/Reals/Raxioms.v index 9fbda92a..6019d4fa 100644 --- a/theories/Reals/Raxioms.v +++ b/theories/Reals/Raxioms.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (*********************************************************) @@ -115,19 +117,6 @@ Arguments INR n%nat. (**********************************************************) -(** * Injection from [Z] to [R] *) -(**********************************************************) - -(**********) -Definition IZR (z:Z) : R := - match z with - | Z0 => 0 - | Zpos n => INR (Pos.to_nat n) - | Zneg n => - INR (Pos.to_nat n) - end. -Arguments IZR z%Z. - -(**********************************************************) (** * [R] Archimedean *) (**********************************************************) diff --git a/theories/Reals/Rbase.v b/theories/Reals/Rbase.v index e56ce28d..b63c8e1c 100644 --- a/theories/Reals/Rbase.v +++ b/theories/Reals/Rbase.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export Rdefinitions. diff --git a/theories/Reals/Rbasic_fun.v b/theories/Reals/Rbasic_fun.v index c889d734..aa886cee 100644 --- a/theories/Reals/Rbasic_fun.v +++ b/theories/Reals/Rbasic_fun.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (*********************************************************) @@ -451,20 +453,16 @@ Qed. Lemma Rabs_Ropp : forall x:R, Rabs (- x) = Rabs x. Proof. - intro; cut (- x = -1 * x). - intros; rewrite H. + intro; replace (-x) with (-1 * x) by ring. rewrite Rabs_mult. - cut (Rabs (-1) = 1). - intros; rewrite H0. - ring. + replace (Rabs (-1)) with 1. + apply Rmult_1_l. unfold Rabs; case (Rcase_abs (-1)). intro; ring. - intro H0; generalize (Rge_le (-1) 0 H0); intros. - generalize (Ropp_le_ge_contravar 0 (-1) H1). - rewrite Ropp_involutive; rewrite Ropp_0. - intro; generalize (Rgt_not_le 1 0 Rlt_0_1); intro; generalize (Rge_le 0 1 H2); - intro; exfalso; auto. - ring. + rewrite <- Ropp_0. + intro H0; apply Ropp_ge_cancel in H0. + elim (Rge_not_lt _ _ H0). + apply Rlt_0_1. Qed. (*********) @@ -613,11 +611,12 @@ Qed. Lemma Rabs_Zabs : forall z:Z, Rabs (IZR z) = IZR (Z.abs z). Proof. - intros z; case z; simpl; auto with real. - apply Rabs_right; auto with real. - intros p0; apply Rabs_right; auto with real zarith. + intros z; case z; unfold Z.abs. + apply Rabs_R0. + now intros p0; apply Rabs_pos_eq, (IZR_le 0). + unfold IZR at 1. intros p0; rewrite Rabs_Ropp. - apply Rabs_right; auto with real zarith. + now apply Rabs_pos_eq, (IZR_le 0). Qed. Lemma abs_IZR : forall z, IZR (Z.abs z) = Rabs (IZR z). diff --git a/theories/Reals/Rcomplete.v b/theories/Reals/Rcomplete.v index 3520c26c..19cbbeca 100644 --- a/theories/Reals/Rcomplete.v +++ b/theories/Reals/Rcomplete.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. diff --git a/theories/Reals/Rdefinitions.v b/theories/Reals/Rdefinitions.v index f3f8f740..857b4ec3 100644 --- a/theories/Reals/Rdefinitions.v +++ b/theories/Reals/Rdefinitions.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (*********************************************************) @@ -69,3 +71,32 @@ Notation "x <= y <= z" := (x <= y /\ y <= z) : R_scope. Notation "x <= y < z" := (x <= y /\ y < z) : R_scope. Notation "x < y < z" := (x < y /\ y < z) : R_scope. Notation "x < y <= z" := (x < y /\ y <= z) : R_scope. + +(**********************************************************) +(** * Injection from [Z] to [R] *) +(**********************************************************) + +(* compact representation for 2*p *) +Fixpoint IPR_2 (p:positive) : R := + match p with + | xH => R1 + R1 + | xO p => (R1 + R1) * IPR_2 p + | xI p => (R1 + R1) * (R1 + IPR_2 p) + end. + +Definition IPR (p:positive) : R := + match p with + | xH => R1 + | xO p => IPR_2 p + | xI p => R1 + IPR_2 p + end. +Arguments IPR p%positive : simpl never. + +(**********) +Definition IZR (z:Z) : R := + match z with + | Z0 => R0 + | Zpos n => IPR n + | Zneg n => - IPR n + end. +Arguments IZR z%Z : simpl never. diff --git a/theories/Reals/Rderiv.v b/theories/Reals/Rderiv.v index bd330ac9..dfa5c710 100644 --- a/theories/Reals/Rderiv.v +++ b/theories/Reals/Rderiv.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (*********************************************************) @@ -296,14 +298,10 @@ Proof. intros; generalize (H0 eps H1); clear H0; intro; elim H0; clear H0; intros; elim H0; clear H0; simpl; intros; split with x; split; auto. - intros; generalize (H2 x1 H3); clear H2; intro; - rewrite Ropp_mult_distr_l_reverse in H2; - rewrite Ropp_mult_distr_l_reverse in H2; - rewrite Ropp_mult_distr_l_reverse in H2; - rewrite (let (H1, H2) := Rmult_ne (f x1) in H2) in H2; - rewrite (let (H1, H2) := Rmult_ne (f x0) in H2) in H2; - rewrite (let (H1, H2) := Rmult_ne (df x0) in H2) in H2; - assumption. + intros; generalize (H2 x1 H3); clear H2; intro. + replace (- f x1 - - f x0) with (-1 * f x1 - -1 * f x0) by ring. + replace (- df x0) with (-1 * df x0) by ring. + exact H2. Qed. (*********) diff --git a/theories/Reals/Reals.v b/theories/Reals/Reals.v index 8265f65a..b249b519 100644 --- a/theories/Reals/Reals.v +++ b/theories/Reals/Reals.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** The library REALS is divided in 6 parts : diff --git a/theories/Reals/Rfunctions.v b/theories/Reals/Rfunctions.v index 0a49d498..77e53147 100644 --- a/theories/Reals/Rfunctions.v +++ b/theories/Reals/Rfunctions.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (*i Some properties about pow and sum have been made with John Harrison i*) @@ -416,8 +418,9 @@ Proof. simpl; apply Rabs_R1. replace (S n) with (n + 1)%nat; [ rewrite pow_add | ring ]. rewrite Rabs_mult. - rewrite Hrecn; rewrite Rmult_1_l; simpl; rewrite Rmult_1_r; - rewrite Rabs_Ropp; apply Rabs_R1. + rewrite Hrecn; rewrite Rmult_1_l; simpl; rewrite Rmult_1_r. + change (-1) with (-(1)). + rewrite Rabs_Ropp; apply Rabs_R1. Qed. Lemma pow_mult : forall (x:R) (n1 n2:nat), x ^ (n1 * n2) = (x ^ n1) ^ n2. @@ -531,6 +534,36 @@ Qed. (*******************************) (*i Due to L.Thery i*) +Section PowerRZ. + +Local Coercion Z_of_nat : nat >-> Z. + +(* the following section should probably be somewhere else, but not sure where *) +Section Z_compl. + +Local Open Scope Z_scope. + +(* Provides a way to reason directly on Z in terms of nats instead of positive *) +Inductive Z_spec (x : Z) : Z -> Type := +| ZintNull : x = 0 -> Z_spec x 0 +| ZintPos (n : nat) : x = n -> Z_spec x n +| ZintNeg (n : nat) : x = - n -> Z_spec x (- n). + +Lemma intP (x : Z) : Z_spec x x. +Proof. + destruct x as [|p|p]. + - now apply ZintNull. + - rewrite <-positive_nat_Z at 2. + apply ZintPos. + now rewrite positive_nat_Z. + - rewrite <-Pos2Z.opp_pos. + rewrite <-positive_nat_Z at 2. + apply ZintNeg. + now rewrite positive_nat_Z. +Qed. + +End Z_compl. + Definition powerRZ (x:R) (n:Z) := match n with | Z0 => 1 @@ -657,6 +690,74 @@ Proof. auto with real. Qed. +Local Open Scope Z_scope. + +Lemma pow_powerRZ (r : R) (n : nat) : + (r ^ n)%R = powerRZ r (Z_of_nat n). +Proof. + induction n; [easy|simpl]. + now rewrite SuccNat2Pos.id_succ. +Qed. + +Lemma powerRZ_ind (P : Z -> R -> R -> Prop) : + (forall x, P 0 x 1%R) -> + (forall x n, P (Z.of_nat n) x (x ^ n)%R) -> + (forall x n, P ((-(Z.of_nat n))%Z) x (Rinv (x ^ n))) -> + forall x (m : Z), P m x (powerRZ x m)%R. +Proof. + intros ? ? ? x m. + destruct (intP m) as [Hm|n Hm|n Hm]. + - easy. + - now rewrite <- pow_powerRZ. + - unfold powerRZ. + destruct n as [|n]; [ easy |]. + rewrite Nat2Z.inj_succ, <- Zpos_P_of_succ_nat, Pos2Z.opp_pos. + now rewrite <- Pos2Z.opp_pos, <- positive_nat_Z. +Qed. + +Lemma powerRZ_inv x alpha : (x <> 0)%R -> powerRZ (/ x) alpha = Rinv (powerRZ x alpha). +Proof. + intros; destruct (intP alpha). + - now simpl; rewrite Rinv_1. + - now rewrite <-!pow_powerRZ, ?Rinv_pow, ?pow_powerRZ. + - unfold powerRZ. + destruct (- n). + + now rewrite Rinv_1. + + now rewrite Rinv_pow. + + now rewrite <-Rinv_pow. +Qed. + +Lemma powerRZ_neg x : forall alpha, x <> R0 -> powerRZ x (- alpha) = powerRZ (/ x) alpha. +Proof. + intros [|n|n] H ; simpl. + - easy. + - now rewrite Rinv_pow. + - rewrite Rinv_pow by now apply Rinv_neq_0_compat. + now rewrite Rinv_involutive. +Qed. + +Lemma powerRZ_mult_distr : + forall m x y, ((0 <= m)%Z \/ (x * y <> 0)%R) -> + (powerRZ (x*y) m = powerRZ x m * powerRZ y m)%R. +Proof. + intros m x0 y0 Hmxy. + destruct (intP m) as [ | | n Hm ]. + - now simpl; rewrite Rmult_1_l. + - now rewrite <- !pow_powerRZ, Rpow_mult_distr. + - destruct Hmxy as [H|H]. + + assert(m = 0) as -> by now omega. + now rewrite <- Hm, Rmult_1_l. + + assert(x0 <> 0)%R by now intros ->; apply H; rewrite Rmult_0_l. + assert(y0 <> 0)%R by now intros ->; apply H; rewrite Rmult_0_r. + rewrite !powerRZ_neg by assumption. + rewrite Rinv_mult_distr by assumption. + now rewrite <- !pow_powerRZ, Rpow_mult_distr. +Qed. + +End PowerRZ. + +Local Infix "^Z" := powerRZ (at level 30, right associativity) : R_scope. + (*******************************) (* For easy interface *) (*******************************) diff --git a/theories/Reals/Rgeom.v b/theories/Reals/Rgeom.v index 7423ffce..6c2f3ac6 100644 --- a/theories/Reals/Rgeom.v +++ b/theories/Reals/Rgeom.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. diff --git a/theories/Reals/RiemannInt.v b/theories/Reals/RiemannInt.v index 4c0466ac..f7d98fca 100644 --- a/theories/Reals/RiemannInt.v +++ b/theories/Reals/RiemannInt.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rfunctions. diff --git a/theories/Reals/RiemannInt_SF.v b/theories/Reals/RiemannInt_SF.v index 7885d697..ceac021e 100644 --- a/theories/Reals/RiemannInt_SF.v +++ b/theories/Reals/RiemannInt_SF.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -83,11 +85,10 @@ Proof. cut (x = INR (pred x0)). intro H19; rewrite H19; apply le_INR; apply lt_le_S; apply INR_lt; rewrite H18; rewrite <- H19; assumption. - rewrite H10; rewrite H8; rewrite <- INR_IZR_INZ; replace 1 with (INR 1); - [ idtac | reflexivity ]; rewrite <- minus_INR. - replace (x0 - 1)%nat with (pred x0); - [ reflexivity - | case x0; [ reflexivity | intro; simpl; apply minus_n_O ] ]. + rewrite H10; rewrite H8; rewrite <- INR_IZR_INZ; + rewrite <- (minus_INR _ 1). + apply f_equal; + case x0; [ reflexivity | intro; apply sym_eq, minus_n_O ]. induction x0 as [|x0 Hrecx0]. rewrite H8 in H3. rewrite <- INR_IZR_INZ in H3; simpl in H3. elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H6 H3)). diff --git a/theories/Reals/Rlimit.v b/theories/Reals/Rlimit.v index e424a732..b14fcc4d 100644 --- a/theories/Reals/Rlimit.v +++ b/theories/Reals/Rlimit.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (*********************************************************) @@ -29,59 +31,28 @@ Qed. Lemma eps2 : forall eps:R, eps * / 2 + eps * / 2 = eps. Proof. intro esp. - assert (H := double_var esp). - unfold Rdiv in H. - symmetry ; exact H. + apply eq_sym, double_var. Qed. (*********) Lemma eps4 : forall eps:R, eps * / (2 + 2) + eps * / (2 + 2) = eps * / 2. Proof. intro eps. - replace (2 + 2) with 4. - pattern eps at 3; rewrite double_var. - rewrite (Rmult_plus_distr_r (eps / 2) (eps / 2) (/ 2)). - unfold Rdiv. - repeat rewrite Rmult_assoc. - rewrite <- Rinv_mult_distr. - reflexivity. - discrR. - discrR. - ring. + field. Qed. (*********) Lemma Rlt_eps2_eps : forall eps:R, eps > 0 -> eps * / 2 < eps. Proof. intros. - pattern eps at 2; rewrite <- Rmult_1_r. - repeat rewrite (Rmult_comm eps). - apply Rmult_lt_compat_r. - exact H. - apply Rmult_lt_reg_l with 2. fourier. - rewrite Rmult_1_r; rewrite <- Rinv_r_sym. - fourier. - discrR. Qed. (*********) Lemma Rlt_eps4_eps : forall eps:R, eps > 0 -> eps * / (2 + 2) < eps. Proof. intros. - replace (2 + 2) with 4. - pattern eps at 2; rewrite <- Rmult_1_r. - repeat rewrite (Rmult_comm eps). - apply Rmult_lt_compat_r. - exact H. - apply Rmult_lt_reg_l with 4. - replace 4 with 4. - apply Rmult_lt_0_compat; fourier. - ring. - rewrite Rmult_1_r; rewrite <- Rinv_r_sym. fourier. - discrR. - ring. Qed. (*********) @@ -407,8 +378,7 @@ Proof. generalize (Rplus_lt_compat (R_dist (f x2) l) eps (R_dist (f x2) l') eps H H0); unfold R_dist; intros; rewrite (Rabs_minus_sym (f x2) l) in H1; - rewrite (Rmult_comm 2 eps); rewrite (Rmult_plus_distr_l eps 1 1); - elim (Rmult_ne eps); intros a b; rewrite a; clear a b; + rewrite (Rmult_comm 2 eps); replace (eps *2) with (eps + eps) by ring; generalize (R_dist_tri l l' (f x2)); unfold R_dist; intros; apply diff --git a/theories/Reals/Rlogic.v b/theories/Reals/Rlogic.v index b9a9458c..04f13477 100644 --- a/theories/Reals/Rlogic.v +++ b/theories/Reals/Rlogic.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** This module proves some logical properties of the axiomatic of Reals. @@ -63,7 +65,7 @@ destruct (Rle_lt_dec l 0) as [Hl|Hl]. now apply Rinv_0_lt_compat. now apply Hnp. left. -set (N := Zabs_nat (up (/l) - 2)). +set (N := Z.abs_nat (up (/l) - 2)). assert (H1l: (1 <= /l)%R). rewrite <- Rinv_1. apply Rinv_le_contravar with (1 := Hl). @@ -75,7 +77,7 @@ assert (HN: (INR N + 1 = IZR (up (/ l)) - 1)%R). rewrite inj_Zabs_nat. replace (IZR (up (/ l)) - 1)%R with (IZR (up (/ l) - 2) + 1)%R. apply (f_equal (fun v => IZR v + 1)%R). - apply Zabs_eq. + apply Z.abs_eq. apply Zle_minus_le_0. apply (Zlt_le_succ 1). apply lt_IZR. diff --git a/theories/Reals/Rminmax.v b/theories/Reals/Rminmax.v index 152988dc..7f73f7c1 100644 --- a/theories/Reals/Rminmax.v +++ b/theories/Reals/Rminmax.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Orders Rbase Rbasic_fun ROrderedType GenericMinMax. diff --git a/theories/Reals/Rpow_def.v b/theories/Reals/Rpow_def.v index 791718a4..0d921303 100644 --- a/theories/Reals/Rpow_def.v +++ b/theories/Reals/Rpow_def.v @@ -1,15 +1,17 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rdefinitions. Fixpoint pow (r:R) (n:nat) : R := match n with - | O => R1 + | O => 1 | S n => Rmult r (pow r n) end. diff --git a/theories/Reals/Rpower.v b/theories/Reals/Rpower.v index b3ce6fa3..c6fac951 100644 --- a/theories/Reals/Rpower.v +++ b/theories/Reals/Rpower.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (*i Due to L.Thery i*) @@ -55,25 +57,8 @@ Proof. simpl in H0. replace (/ 3) with (1 * / 1 + -1 * 1 * / 1 + -1 * (-1 * 1) * / 2 + - -1 * (-1 * (-1 * 1)) * / (2 + 1 + 1 + 1 + 1)). + -1 * (-1 * (-1 * 1)) * / (2 + 1 + 1 + 1 + 1)) by field. apply H0. - repeat rewrite Rinv_1; repeat rewrite Rmult_1_r; - rewrite Ropp_mult_distr_l_reverse; rewrite Rmult_1_l; - rewrite Ropp_involutive; rewrite Rplus_opp_r; rewrite Rmult_1_r; - rewrite Rplus_0_l; rewrite Rmult_1_l; apply Rmult_eq_reg_l with 6. - rewrite Rmult_plus_distr_l; replace (2 + 1 + 1 + 1 + 1) with 6. - rewrite <- (Rmult_comm (/ 6)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. - rewrite Rmult_1_l; replace 6 with 6. - do 2 rewrite Rmult_assoc; rewrite <- Rinv_r_sym. - rewrite Rmult_1_r; rewrite (Rmult_comm 3); rewrite <- Rmult_assoc; - rewrite <- Rinv_r_sym. - ring. - discrR. - discrR. - ring. - discrR. - ring. - discrR. apply H. unfold Un_decreasing; intros; apply Rmult_le_reg_l with (INR (fact n)). @@ -448,9 +433,9 @@ Proof. Qed. Theorem Rpower_lt : - forall x y z:R, 1 < x -> 0 <= y -> y < z -> x ^R y < x ^R z. + forall x y z:R, 1 < x -> y < z -> x ^R y < x ^R z. Proof. - intros x y z H H0 H1. + intros x y z H H1. unfold Rpower. apply exp_increasing. apply Rmult_lt_compat_r. @@ -473,7 +458,7 @@ Proof. unfold Rpower; auto. rewrite Rpower_mult. rewrite Rinv_l. - replace 1 with (INR 1); auto. + change 1 with (INR 1). repeat rewrite Rpower_pow; simpl. pattern x at 1; rewrite <- (sqrt_sqrt x (Rlt_le _ _ H)). ring. @@ -490,12 +475,28 @@ Proof. apply exp_Ropp. Qed. +Lemma powerRZ_Rpower x z : (0 < x)%R -> powerRZ x z = Rpower x (IZR z). +Proof. + intros Hx. + assert (x <> 0)%R + by now intros Habs; rewrite Habs in Hx; apply (Rlt_irrefl 0). + destruct (intP z). + - now rewrite Rpower_O. + - rewrite <- pow_powerRZ, <- Rpower_pow by assumption. + now rewrite INR_IZR_INZ. + - rewrite opp_IZR, Rpower_Ropp. + rewrite powerRZ_neg, powerRZ_inv by assumption. + now rewrite <- pow_powerRZ, <- INR_IZR_INZ, Rpower_pow. +Qed. + Theorem Rle_Rpower : - forall e n m:R, 1 < e -> 0 <= n -> n <= m -> e ^R n <= e ^R m. + forall e n m:R, 1 <= e -> n <= m -> e ^R n <= e ^R m. Proof. - intros e n m H H0 H1; case H1. - intros H2; left; apply Rpower_lt; assumption. - intros H2; rewrite H2; right; reflexivity. + intros e n m [H | H]; intros H1. + case H1. + intros H2; left; apply Rpower_lt; assumption. + intros H2; rewrite H2; right; reflexivity. + now rewrite <- H; unfold Rpower; rewrite ln_1, !Rmult_0_r; apply Rle_refl. Qed. Theorem ln_lt_2 : / 2 < ln 2. @@ -505,12 +506,9 @@ Proof. rewrite Rinv_r. apply exp_lt_inv. apply Rle_lt_trans with (1 := exp_le_3). - change (3 < 2 ^R 2). + change (3 < 2 ^R (1 + 1)). repeat rewrite Rpower_plus; repeat rewrite Rpower_1. - repeat rewrite Rmult_plus_distr_r; repeat rewrite Rmult_plus_distr_l; - repeat rewrite Rmult_1_l. - pattern 3 at 1; rewrite <- Rplus_0_r; replace (2 + 2) with (3 + 1); - [ apply Rplus_lt_compat_l; apply Rlt_0_1 | ring ]. + now apply (IZR_lt 3 4). prove_sup0. discrR. Qed. @@ -713,13 +711,18 @@ intros x y z x0 y0; unfold Rpower. rewrite <- exp_plus, ln_mult, Rmult_plus_distr_l; auto. Qed. -Lemma Rle_Rpower_l a b c: 0 <= c -> 0 < a <= b -> Rpower a c <= Rpower b c. +Lemma Rlt_Rpower_l a b c: 0 < c -> 0 < a < b -> a ^R c < b ^R c. +Proof. +intros c0 [a0 ab]; apply exp_increasing. +now apply Rmult_lt_compat_l; auto; apply ln_increasing; fourier. +Qed. + +Lemma Rle_Rpower_l a b c: 0 <= c -> 0 < a <= b -> a ^R c <= b ^R c. Proof. intros [c0 | c0]; [ | intros; rewrite <- c0, !Rpower_O; [apply Rle_refl | |] ]. intros [a0 [ab|ab]]. - left; apply exp_increasing. - now apply Rmult_lt_compat_l; auto; apply ln_increasing; fourier. + now apply Rlt_le, Rlt_Rpower_l;[ | split]; fourier. rewrite ab; apply Rle_refl. apply Rlt_le_trans with a; tauto. tauto. @@ -732,7 +735,7 @@ Definition arcsinh x := ln (x + sqrt (x ^ 2 + 1)). Lemma arcsinh_sinh : forall x, arcsinh (sinh x) = x. intros x; unfold sinh, arcsinh. assert (Rminus_eq_0 : forall r, r - r = 0) by (intros; ring). -pattern 1 at 5; rewrite <- exp_0, <- (Rminus_eq_0 x); unfold Rminus. +rewrite <- exp_0, <- (Rminus_eq_0 x); unfold Rminus. rewrite exp_plus. match goal with |- context[sqrt ?a] => replace a with (((exp x + exp(-x))/2)^2) by field diff --git a/theories/Reals/Rprod.v b/theories/Reals/Rprod.v index 883e4e1a..17736af6 100644 --- a/theories/Reals/Rprod.v +++ b/theories/Reals/Rprod.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Compare. diff --git a/theories/Reals/Rseries.v b/theories/Reals/Rseries.v index 744fd664..3521a476 100644 --- a/theories/Reals/Rseries.v +++ b/theories/Reals/Rseries.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -207,7 +209,7 @@ Section sequence. assert (Rabs (/2) < 1). rewrite Rabs_pos_eq. - rewrite <- Rinv_1 at 3. + rewrite <- Rinv_1. apply Rinv_lt_contravar. rewrite Rmult_1_l. now apply (IZR_lt 0 2). diff --git a/theories/Reals/Rsigma.v b/theories/Reals/Rsigma.v index ced2b3da..83c60751 100644 --- a/theories/Reals/Rsigma.v +++ b/theories/Reals/Rsigma.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. diff --git a/theories/Reals/Rsqrt_def.v b/theories/Reals/Rsqrt_def.v index b3c9c744..6a3dd976 100644 --- a/theories/Reals/Rsqrt_def.v +++ b/theories/Reals/Rsqrt_def.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Sumbool. @@ -648,7 +650,7 @@ Proof. Qed. (** We can now define the square root function as the reciprocal - transformation of the square root function *) + transformation of the square function *) Lemma Rsqrt_exists : forall y:R, 0 <= y -> { z:R | 0 <= z /\ y = Rsqr z }. Proof. diff --git a/theories/Reals/Rtopology.v b/theories/Reals/Rtopology.v index df3b95be..171dba55 100644 --- a/theories/Reals/Rtopology.v +++ b/theories/Reals/Rtopology.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. diff --git a/theories/Reals/Rtrigo.v b/theories/Reals/Rtrigo.v index ecef0d68..ffc0adf5 100644 --- a/theories/Reals/Rtrigo.v +++ b/theories/Reals/Rtrigo.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. diff --git a/theories/Reals/Rtrigo1.v b/theories/Reals/Rtrigo1.v index 4d241863..bf00f736 100644 --- a/theories/Reals/Rtrigo1.v +++ b/theories/Reals/Rtrigo1.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -182,13 +184,10 @@ destruct (pre_cos_bound _ 0 lo up) as [_ upper]. apply Rle_lt_trans with (1 := upper). apply Rlt_le_trans with (2 := lower). unfold cos_approx, sin_approx. -simpl sum_f_R0; replace 7 with (IZR 7) by (simpl; field). -replace 8 with (IZR 8) by (simpl; field). +simpl sum_f_R0. unfold cos_term, sin_term; simpl fact; rewrite !INR_IZR_INZ. -simpl plus; simpl mult. -field_simplify; - try (repeat apply conj; apply not_eq_sym, Rlt_not_eq, (IZR_lt 0); reflexivity). -unfold Rminus; rewrite !pow_IZR, <- !mult_IZR, <- !opp_IZR, <- ?plus_IZR. +simpl plus; simpl mult; simpl Z_of_nat. +field_simplify. match goal with |- IZR ?a / ?b < ?c / ?d => apply Rmult_lt_reg_r with d;[apply (IZR_lt 0); reflexivity | @@ -198,7 +197,7 @@ match goal with end. unfold Rdiv; rewrite !Rmult_assoc, Rinv_l, Rmult_1_r; [ | apply not_eq_sym, Rlt_not_eq, (IZR_lt 0); reflexivity]. -repeat (rewrite <- !plus_IZR || rewrite <- !mult_IZR). +rewrite <- !mult_IZR. apply IZR_lt; reflexivity. Qed. @@ -323,6 +322,7 @@ Lemma sin_PI : sin PI = 0. Proof. assert (H := sin2_cos2 PI). rewrite cos_PI in H. + change (-1) with (-(1)) in H. rewrite <- Rsqr_neg in H. rewrite Rsqr_1 in H. cut (Rsqr (sin PI) = 0). @@ -533,9 +533,8 @@ Qed. Lemma sin_PI_x : forall x:R, sin (PI - x) = sin x. Proof. - intro x; rewrite sin_minus; rewrite sin_PI; rewrite cos_PI; rewrite Rmult_0_l; - unfold Rminus in |- *; rewrite Rplus_0_l; rewrite Ropp_mult_distr_l_reverse; - rewrite Ropp_involutive; apply Rmult_1_l. + intro x; rewrite sin_minus; rewrite sin_PI; rewrite cos_PI. + ring. Qed. Lemma sin_period : forall (x:R) (k:nat), sin (x + 2 * INR k * PI) = sin x. @@ -593,9 +592,9 @@ Proof. generalize (Rsqr_incrst_1 1 (sin x) H (Rlt_le 0 1 Rlt_0_1) (Rlt_le 0 (sin x) (Rlt_trans 0 1 (sin x) Rlt_0_1 H))); - rewrite Rsqr_1; intro; rewrite sin2 in H0; unfold Rminus in H0; + rewrite Rsqr_1; intro; rewrite sin2 in H0; unfold Rminus in H0. generalize (Rplus_lt_compat_l (-1) 1 (1 + - Rsqr (cos x)) H0); - repeat rewrite <- Rplus_assoc; repeat rewrite Rplus_opp_l; + repeat rewrite <- Rplus_assoc; change (-1) with (-(1)); rewrite Rplus_opp_l; rewrite Rplus_0_l; intro; rewrite <- Ropp_0 in H1; generalize (Ropp_lt_gt_contravar (-0) (- Rsqr (cos x)) H1); repeat rewrite Ropp_involutive; intro; generalize (Rle_0_sqr (cos x)); @@ -603,6 +602,7 @@ Proof. auto with real. cut (sin x < -1). intro; generalize (Ropp_lt_gt_contravar (sin x) (-1) H); + change (-1) with (-(1)); rewrite Ropp_involutive; clear H; intro; generalize (Rsqr_incrst_1 1 (- sin x) H (Rlt_le 0 1 Rlt_0_1) @@ -610,7 +610,7 @@ Proof. rewrite Rsqr_1; intro; rewrite <- Rsqr_neg in H0; rewrite sin2 in H0; unfold Rminus in H0; generalize (Rplus_lt_compat_l (-1) 1 (1 + - Rsqr (cos x)) H0); - repeat rewrite <- Rplus_assoc; repeat rewrite Rplus_opp_l; + rewrite <- Rplus_assoc; change (-1) with (-(1)); rewrite Rplus_opp_l; rewrite Rplus_0_l; intro; rewrite <- Ropp_0 in H1; generalize (Ropp_lt_gt_contravar (-0) (- Rsqr (cos x)) H1); repeat rewrite Ropp_involutive; intro; generalize (Rle_0_sqr (cos x)); @@ -696,41 +696,38 @@ Proof. rewrite <- Rinv_l_sym. do 2 rewrite Rmult_1_r; apply Rle_lt_trans with (INR (fact (2 * n + 1)) * 4). apply Rmult_le_compat_l. - replace 0 with (INR 0); [ idtac | reflexivity ]; apply le_INR; apply le_O_n. - simpl in |- *; rewrite Rmult_1_r; replace 4 with (Rsqr 2); - [ idtac | ring_Rsqr ]; replace (a * a) with (Rsqr a); - [ idtac | reflexivity ]; apply Rsqr_incr_1. + apply pos_INR. + simpl in |- *; rewrite Rmult_1_r; change 4 with (Rsqr 2); + apply Rsqr_incr_1. apply Rle_trans with (PI / 2); [ assumption | unfold Rdiv in |- *; apply Rmult_le_reg_l with 2; [ prove_sup0 | rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m; - [ replace 4 with 4; [ apply PI_4 | ring ] | discrR ] ] ]. + [ apply PI_4 | discrR ] ] ]. left; assumption. left; prove_sup0. rewrite H1; replace (2 * n + 1 + 2)%nat with (S (S (2 * n + 1))). do 2 rewrite fact_simpl; do 2 rewrite mult_INR. repeat rewrite <- Rmult_assoc. rewrite <- (Rmult_comm (INR (fact (2 * n + 1)))). - rewrite Rmult_assoc. apply Rmult_lt_compat_l. apply lt_INR_0; apply neq_O_lt. assert (H2 := fact_neq_0 (2 * n + 1)). red in |- *; intro; elim H2; symmetry in |- *; assumption. do 2 rewrite S_INR; rewrite plus_INR; rewrite mult_INR; set (x := INR n); unfold INR in |- *. - replace ((2 * x + 1 + 1 + 1) * (2 * x + 1 + 1)) with (4 * x * x + 10 * x + 6); + replace (((1 + 1) * x + 1 + 1 + 1) * ((1 + 1) * x + 1 + 1)) with (4 * x * x + 10 * x + 6); [ idtac | ring ]. - apply Rplus_lt_reg_l with (-4); rewrite Rplus_opp_l; - replace (-4 + (4 * x * x + 10 * x + 6)) with (4 * x * x + 10 * x + 2); + apply Rplus_lt_reg_l with (-(4)); rewrite Rplus_opp_l; + replace (-(4) + (4 * x * x + 10 * x + 6)) with (4 * x * x + 10 * x + 2); [ idtac | ring ]. apply Rplus_le_lt_0_compat. cut (0 <= x). intro; apply Rplus_le_le_0_compat; repeat apply Rmult_le_pos; assumption || left; prove_sup. - unfold x in |- *; replace 0 with (INR 0); - [ apply le_INR; apply le_O_n | reflexivity ]. - prove_sup0. + apply pos_INR. + now apply IZR_lt. ring. apply INR_fact_neq_0. apply INR_fact_neq_0. @@ -738,39 +735,33 @@ Proof. Qed. Lemma SIN : forall a:R, 0 <= a -> a <= PI -> sin_lb a <= sin a <= sin_ub a. +Proof. intros; unfold sin_lb, sin_ub in |- *; apply (sin_bound a 1 H H0). Qed. Lemma COS : forall a:R, - PI / 2 <= a -> a <= PI / 2 -> cos_lb a <= cos a <= cos_ub a. +Proof. intros; unfold cos_lb, cos_ub in |- *; apply (cos_bound a 1 H H0). Qed. (**********) Lemma _PI2_RLT_0 : - (PI / 2) < 0. Proof. - rewrite <- Ropp_0; apply Ropp_lt_contravar; apply PI2_RGT_0. + assert (H := PI_RGT_0). + fourier. Qed. Lemma PI4_RLT_PI2 : PI / 4 < PI / 2. Proof. - unfold Rdiv in |- *; apply Rmult_lt_compat_l. - apply PI_RGT_0. - apply Rinv_lt_contravar. - apply Rmult_lt_0_compat; prove_sup0. - pattern 2 at 1 in |- *; rewrite <- Rplus_0_r. - replace 4 with (2 + 2); [ apply Rplus_lt_compat_l; prove_sup0 | ring ]. + assert (H := PI_RGT_0). + fourier. Qed. Lemma PI2_Rlt_PI : PI / 2 < PI. Proof. - unfold Rdiv in |- *; pattern PI at 2 in |- *; rewrite <- Rmult_1_r. - apply Rmult_lt_compat_l. - apply PI_RGT_0. - pattern 1 at 3 in |- *; rewrite <- Rinv_1; apply Rinv_lt_contravar. - rewrite Rmult_1_l; prove_sup0. - pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; - apply Rlt_0_1. + assert (H := PI_RGT_0). + fourier. Qed. (***************************************************) @@ -787,12 +778,10 @@ Proof. rewrite H3; rewrite sin_PI2; apply Rlt_0_1. rewrite <- sin_PI_x; generalize (Ropp_gt_lt_contravar x (PI / 2) H3); intro H4; generalize (Rplus_lt_compat_l PI (- x) (- (PI / 2)) H4). - replace (PI + - x) with (PI - x). replace (PI + - (PI / 2)) with (PI / 2). intro H5; generalize (Ropp_lt_gt_contravar x PI H0); intro H6; change (- PI < - x) in H6; generalize (Rplus_lt_compat_l PI (- PI) (- x) H6). rewrite Rplus_opp_r. - replace (PI + - x) with (PI - x). intro H7; elim (SIN (PI - x) (Rlt_le 0 (PI - x) H7) @@ -800,9 +789,7 @@ Proof. intros H8 _; generalize (sin_lb_gt_0 (PI - x) H7 (Rlt_le (PI - x) (PI / 2) H5)); intro H9; apply (Rlt_le_trans 0 (sin_lb (PI - x)) (sin (PI - x)) H9 H8). - reflexivity. - pattern PI at 2 in |- *; rewrite double_var; ring. - reflexivity. + field. Qed. Theorem cos_gt_0 : forall x:R, - (PI / 2) < x -> x < PI / 2 -> 0 < cos x. @@ -855,16 +842,12 @@ Proof. rewrite <- (Ropp_involutive (cos x)); apply Ropp_le_ge_contravar; rewrite <- neg_cos; replace (x + PI) with (x - PI + 2 * INR 1 * PI). rewrite cos_period; apply cos_ge_0. - replace (- (PI / 2)) with (- PI + PI / 2). + replace (- (PI / 2)) with (- PI + PI / 2) by field. unfold Rminus in |- *; rewrite (Rplus_comm x); apply Rplus_le_compat_l; assumption. - pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr; - ring. unfold Rminus in |- *; rewrite Rplus_comm; - replace (PI / 2) with (- PI + 3 * (PI / 2)). + replace (PI / 2) with (- PI + 3 * (PI / 2)) by field. apply Rplus_le_compat_l; assumption. - pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr; - ring. unfold INR in |- *; ring. Qed. @@ -905,16 +888,12 @@ Proof. apply Ropp_lt_gt_contravar; rewrite <- neg_cos; replace (x + PI) with (x - PI + 2 * INR 1 * PI). rewrite cos_period; apply cos_gt_0. - replace (- (PI / 2)) with (- PI + PI / 2). + replace (- (PI / 2)) with (- PI + PI / 2) by field. unfold Rminus in |- *; rewrite (Rplus_comm x); apply Rplus_lt_compat_l; assumption. - pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr; - ring. unfold Rminus in |- *; rewrite Rplus_comm; - replace (PI / 2) with (- PI + 3 * (PI / 2)). + replace (PI / 2) with (- PI + 3 * (PI / 2)) by field. apply Rplus_lt_compat_l; assumption. - pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr; - ring. unfold INR in |- *; ring. Qed. @@ -951,7 +930,7 @@ Lemma cos_ge_0_3PI2 : forall x:R, 3 * (PI / 2) <= x -> x <= 2 * PI -> 0 <= cos x. Proof. intros; rewrite <- cos_neg; rewrite <- (cos_period (- x) 1); - unfold INR in |- *; replace (- x + 2 * 1 * PI) with (2 * PI - x). + unfold INR in |- *; replace (- x + 2 * 1 * PI) with (2 * PI - x) by ring. generalize (Ropp_le_ge_contravar x (2 * PI) H0); intro H1; generalize (Rge_le (- x) (- (2 * PI)) H1); clear H1; intro H1; generalize (Rplus_le_compat_l (2 * PI) (- (2 * PI)) (- x) H1). @@ -960,36 +939,30 @@ Proof. generalize (Rge_le (- (3 * (PI / 2))) (- x) H3); clear H3; intro H3; generalize (Rplus_le_compat_l (2 * PI) (- x) (- (3 * (PI / 2))) H3). - replace (2 * PI + - (3 * (PI / 2))) with (PI / 2). + replace (2 * PI + - (3 * (PI / 2))) with (PI / 2) by field. intro H4; apply (cos_ge_0 (2 * PI - x) (Rlt_le (- (PI / 2)) (2 * PI - x) (Rlt_le_trans (- (PI / 2)) 0 (2 * PI - x) _PI2_RLT_0 H2)) H4). - rewrite double; pattern PI at 2 3 in |- *; rewrite double_var; ring. - ring. Qed. Lemma form1 : forall p q:R, cos p + cos q = 2 * cos ((p - q) / 2) * cos ((p + q) / 2). Proof. intros p q; pattern p at 1 in |- *; - replace p with ((p - q) / 2 + (p + q) / 2). - rewrite <- (cos_neg q); replace (- q) with ((p - q) / 2 - (p + q) / 2). + replace p with ((p - q) / 2 + (p + q) / 2) by field. + rewrite <- (cos_neg q); replace (- q) with ((p - q) / 2 - (p + q) / 2) by field. rewrite cos_plus; rewrite cos_minus; ring. - pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. - pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. Qed. Lemma form2 : forall p q:R, cos p - cos q = -2 * sin ((p - q) / 2) * sin ((p + q) / 2). Proof. intros p q; pattern p at 1 in |- *; - replace p with ((p - q) / 2 + (p + q) / 2). - rewrite <- (cos_neg q); replace (- q) with ((p - q) / 2 - (p + q) / 2). + replace p with ((p - q) / 2 + (p + q) / 2) by field. + rewrite <- (cos_neg q); replace (- q) with ((p - q) / 2 - (p + q) / 2) by field. rewrite cos_plus; rewrite cos_minus; ring. - pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. - pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. Qed. Lemma form3 : @@ -1007,11 +980,9 @@ Lemma form4 : forall p q:R, sin p - sin q = 2 * cos ((p + q) / 2) * sin ((p - q) / 2). Proof. intros p q; pattern p at 1 in |- *; - replace p with ((p - q) / 2 + (p + q) / 2). - pattern q at 3 in |- *; replace q with ((p + q) / 2 - (p - q) / 2). + replace p with ((p - q) / 2 + (p + q) / 2) by field. + pattern q at 3 in |- *; replace q with ((p + q) / 2 - (p - q) / 2) by field. rewrite sin_plus; rewrite sin_minus; ring. - pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. - pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. Qed. @@ -1067,13 +1038,13 @@ Proof. repeat rewrite (Rmult_comm (/ 2)). clear H4; intro H4; generalize (Rplus_le_compat (- (PI / 2)) x (- (PI / 2)) y H H1); - replace (- (PI / 2) + - (PI / 2)) with (- PI). + replace (- (PI / 2) + - (PI / 2)) with (- PI) by field. intro H5; generalize (Rmult_le_compat_l (/ 2) (- PI) (x + y) (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H5). - replace (/ 2 * (x + y)) with ((x + y) / 2). - replace (/ 2 * - PI) with (- (PI / 2)). + replace (/ 2 * (x + y)) with ((x + y) / 2) by apply Rmult_comm. + replace (/ 2 * - PI) with (- (PI / 2)) by field. clear H5; intro H5; elim H4; intro H40. elim H5; intro H50. generalize (cos_gt_0 ((x + y) / 2) H50 H40); intro H6; @@ -1095,13 +1066,6 @@ Proof. rewrite H40 in H3; assert (H50 := cos_PI2); unfold Rdiv in H50; rewrite H50 in H3; rewrite Rmult_0_r in H3; rewrite Rmult_0_l in H3; elim (Rlt_irrefl 0 H3). - unfold Rdiv in |- *. - rewrite <- Ropp_mult_distr_l_reverse. - apply Rmult_comm. - unfold Rdiv in |- *; apply Rmult_comm. - pattern PI at 1 in |- *; rewrite double_var. - rewrite Ropp_plus_distr. - reflexivity. Qed. Lemma sin_increasing_1 : @@ -1111,43 +1075,42 @@ Lemma sin_increasing_1 : Proof. intros; generalize (Rplus_lt_compat_l x x y H3); intro H4; generalize (Rplus_le_compat (- (PI / 2)) x (- (PI / 2)) x H H); - replace (- (PI / 2) + - (PI / 2)) with (- PI). + replace (- (PI / 2) + - (PI / 2)) with (- PI) by field. assert (Hyp : 0 < 2). prove_sup0. intro H5; generalize (Rle_lt_trans (- PI) (x + x) (x + y) H5 H4); intro H6; generalize (Rmult_lt_compat_l (/ 2) (- PI) (x + y) (Rinv_0_lt_compat 2 Hyp) H6); - replace (/ 2 * - PI) with (- (PI / 2)). - replace (/ 2 * (x + y)) with ((x + y) / 2). + replace (/ 2 * - PI) with (- (PI / 2)) by field. + replace (/ 2 * (x + y)) with ((x + y) / 2) by apply Rmult_comm. clear H4 H5 H6; intro H4; generalize (Rplus_lt_compat_l y x y H3); intro H5; rewrite Rplus_comm in H5; generalize (Rplus_le_compat y (PI / 2) y (PI / 2) H2 H2). rewrite <- double_var. intro H6; generalize (Rlt_le_trans (x + y) (y + y) PI H5 H6); intro H7; generalize (Rmult_lt_compat_l (/ 2) (x + y) PI (Rinv_0_lt_compat 2 Hyp) H7); - replace (/ 2 * PI) with (PI / 2). - replace (/ 2 * (x + y)) with ((x + y) / 2). + replace (/ 2 * PI) with (PI / 2) by apply Rmult_comm. + replace (/ 2 * (x + y)) with ((x + y) / 2) by apply Rmult_comm. clear H5 H6 H7; intro H5; generalize (Ropp_le_ge_contravar (- (PI / 2)) y H1); rewrite Ropp_involutive; clear H1; intro H1; generalize (Rge_le (PI / 2) (- y) H1); clear H1; intro H1; generalize (Ropp_le_ge_contravar y (PI / 2) H2); clear H2; intro H2; generalize (Rge_le (- y) (- (PI / 2)) H2); clear H2; intro H2; generalize (Rplus_lt_compat_l (- y) x y H3); - replace (- y + x) with (x - y). + replace (- y + x) with (x - y) by apply Rplus_comm. rewrite Rplus_opp_l. intro H6; generalize (Rmult_lt_compat_l (/ 2) (x - y) 0 (Rinv_0_lt_compat 2 Hyp) H6); - rewrite Rmult_0_r; replace (/ 2 * (x - y)) with ((x - y) / 2). + rewrite Rmult_0_r; replace (/ 2 * (x - y)) with ((x - y) / 2) by apply Rmult_comm. clear H6; intro H6; generalize (Rplus_le_compat (- (PI / 2)) x (- (PI / 2)) (- y) H H2); - replace (- (PI / 2) + - (PI / 2)) with (- PI). - replace (x + - y) with (x - y). + replace (- (PI / 2) + - (PI / 2)) with (- PI) by field. intro H7; generalize (Rmult_le_compat_l (/ 2) (- PI) (x - y) (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H7); - replace (/ 2 * - PI) with (- (PI / 2)). - replace (/ 2 * (x - y)) with ((x - y) / 2). + replace (/ 2 * - PI) with (- (PI / 2)) by field. + replace (/ 2 * (x - y)) with ((x - y) / 2) by apply Rmult_comm. clear H7; intro H7; clear H H0 H1 H2; apply Rminus_lt; rewrite form4; generalize (cos_gt_0 ((x + y) / 2) H4 H5); intro H8; generalize (Rmult_lt_0_compat 2 (cos ((x + y) / 2)) Hyp H8); @@ -1162,23 +1125,6 @@ Proof. 2 * cos ((x + y) / 2)) H10 H8); intro H11; rewrite Rmult_0_r in H11; rewrite Rmult_comm; assumption. apply Ropp_lt_gt_contravar; apply PI2_Rlt_PI. - unfold Rdiv in |- *; apply Rmult_comm. - unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse; apply Rmult_comm. - reflexivity. - pattern PI at 1 in |- *; rewrite double_var. - rewrite Ropp_plus_distr. - reflexivity. - unfold Rdiv in |- *; apply Rmult_comm. - unfold Rminus in |- *; apply Rplus_comm. - unfold Rdiv in |- *; apply Rmult_comm. - unfold Rdiv in |- *; apply Rmult_comm. - unfold Rdiv in |- *; apply Rmult_comm. - unfold Rdiv in |- *. - rewrite <- Ropp_mult_distr_l_reverse. - apply Rmult_comm. - pattern PI at 1 in |- *; rewrite double_var. - rewrite Ropp_plus_distr. - reflexivity. Qed. Lemma sin_decreasing_0 : @@ -1193,33 +1139,16 @@ Proof. generalize (Rplus_le_compat_l (- PI) (PI / 2) x H0); generalize (Rplus_le_compat_l (- PI) y (3 * (PI / 2)) H1); generalize (Rplus_le_compat_l (- PI) (PI / 2) y H2); - replace (- PI + x) with (x - PI). - replace (- PI + PI / 2) with (- (PI / 2)). - replace (- PI + y) with (y - PI). - replace (- PI + 3 * (PI / 2)) with (PI / 2). - replace (- (PI - x)) with (x - PI). - replace (- (PI - y)) with (y - PI). + replace (- PI + x) with (x - PI) by apply Rplus_comm. + replace (- PI + PI / 2) with (- (PI / 2)) by field. + replace (- PI + y) with (y - PI) by apply Rplus_comm. + replace (- PI + 3 * (PI / 2)) with (PI / 2) by field. + replace (- (PI - x)) with (x - PI) by ring. + replace (- (PI - y)) with (y - PI) by ring. intros; change (sin (y - PI) < sin (x - PI)) in H8; - apply Rplus_lt_reg_l with (- PI); rewrite Rplus_comm; - replace (y + - PI) with (y - PI). - rewrite Rplus_comm; replace (x + - PI) with (x - PI). + apply Rplus_lt_reg_l with (- PI); rewrite Rplus_comm. + rewrite (Rplus_comm _ x). apply (sin_increasing_0 (y - PI) (x - PI) H4 H5 H6 H7 H8). - reflexivity. - reflexivity. - unfold Rminus in |- *; rewrite Ropp_plus_distr. - rewrite Ropp_involutive. - apply Rplus_comm. - unfold Rminus in |- *; rewrite Ropp_plus_distr. - rewrite Ropp_involutive. - apply Rplus_comm. - pattern PI at 2 in |- *; rewrite double_var. - rewrite Ropp_plus_distr. - ring. - unfold Rminus in |- *; apply Rplus_comm. - pattern PI at 2 in |- *; rewrite double_var. - rewrite Ropp_plus_distr. - ring. - unfold Rminus in |- *; apply Rplus_comm. Qed. Lemma sin_decreasing_1 : @@ -1233,24 +1162,14 @@ Proof. generalize (Rplus_le_compat_l (- PI) y (3 * (PI / 2)) H1); generalize (Rplus_le_compat_l (- PI) (PI / 2) y H2); generalize (Rplus_lt_compat_l (- PI) x y H3); - replace (- PI + PI / 2) with (- (PI / 2)). - replace (- PI + y) with (y - PI). - replace (- PI + 3 * (PI / 2)) with (PI / 2). - replace (- PI + x) with (x - PI). + replace (- PI + PI / 2) with (- (PI / 2)) by field. + replace (- PI + y) with (y - PI) by apply Rplus_comm. + replace (- PI + 3 * (PI / 2)) with (PI / 2) by field. + replace (- PI + x) with (x - PI) by apply Rplus_comm. intros; apply Ropp_lt_cancel; repeat rewrite <- sin_neg; - replace (- (PI - x)) with (x - PI). - replace (- (PI - y)) with (y - PI). + replace (- (PI - x)) with (x - PI) by ring. + replace (- (PI - y)) with (y - PI) by ring. apply (sin_increasing_1 (x - PI) (y - PI) H7 H8 H5 H6 H4). - unfold Rminus in |- *; rewrite Ropp_plus_distr. - rewrite Ropp_involutive. - apply Rplus_comm. - unfold Rminus in |- *; rewrite Ropp_plus_distr. - rewrite Ropp_involutive. - apply Rplus_comm. - unfold Rminus in |- *; apply Rplus_comm. - pattern PI at 2 in |- *; rewrite double_var; ring. - unfold Rminus in |- *; apply Rplus_comm. - pattern PI at 2 in |- *; rewrite double_var; ring. Qed. Lemma cos_increasing_0 : @@ -1260,44 +1179,22 @@ Proof. intros x y H1 H2 H3 H4; rewrite <- (cos_neg x); rewrite <- (cos_neg y); rewrite <- (cos_period (- x) 1); rewrite <- (cos_period (- y) 1); unfold INR in |- *; - replace (- x + 2 * 1 * PI) with (PI / 2 - (x - 3 * (PI / 2))). - replace (- y + 2 * 1 * PI) with (PI / 2 - (y - 3 * (PI / 2))). + replace (- x + 2 * 1 * PI) with (PI / 2 - (x - 3 * (PI / 2))) by field. + replace (- y + 2 * 1 * PI) with (PI / 2 - (y - 3 * (PI / 2))) by field. repeat rewrite cos_shift; intro H5; generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI x H1); generalize (Rplus_le_compat_l (-3 * (PI / 2)) x (2 * PI) H2); generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI y H3); generalize (Rplus_le_compat_l (-3 * (PI / 2)) y (2 * PI) H4). - replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)). - replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)). - replace (-3 * (PI / 2) + 2 * PI) with (PI / 2). - replace (-3 * (PI / 2) + PI) with (- (PI / 2)). + replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)) by ring. + replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)) by ring. + replace (-3 * (PI / 2) + 2 * PI) with (PI / 2) by field. + replace (-3 * (PI / 2) + PI) with (- (PI / 2)) by field. clear H1 H2 H3 H4; intros H1 H2 H3 H4; apply Rplus_lt_reg_l with (-3 * (PI / 2)); - replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)). - replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)). + replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)) by ring. + replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)) by ring. apply (sin_increasing_0 (x - 3 * (PI / 2)) (y - 3 * (PI / 2)) H4 H3 H2 H1 H5). - unfold Rminus in |- *. - rewrite Ropp_mult_distr_l_reverse. - apply Rplus_comm. - unfold Rminus in |- *. - rewrite Ropp_mult_distr_l_reverse. - apply Rplus_comm. - pattern PI at 3 in |- *; rewrite double_var. - ring. - rewrite double; pattern PI at 3 4 in |- *; rewrite double_var. - ring. - unfold Rminus in |- *. - rewrite Ropp_mult_distr_l_reverse. - apply Rplus_comm. - unfold Rminus in |- *. - rewrite Ropp_mult_distr_l_reverse. - apply Rplus_comm. - rewrite Rmult_1_r. - rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var. - ring. - rewrite Rmult_1_r. - rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var. - ring. Qed. Lemma cos_increasing_1 : @@ -1312,31 +1209,16 @@ Proof. generalize (Rplus_lt_compat_l (-3 * (PI / 2)) x y H5); rewrite <- (cos_neg x); rewrite <- (cos_neg y); rewrite <- (cos_period (- x) 1); rewrite <- (cos_period (- y) 1); - unfold INR in |- *; replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)). - replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)). - replace (-3 * (PI / 2) + PI) with (- (PI / 2)). - replace (-3 * (PI / 2) + 2 * PI) with (PI / 2). + unfold INR in |- *; replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)) by ring. + replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)) by ring. + replace (-3 * (PI / 2) + PI) with (- (PI / 2)) by field. + replace (-3 * (PI / 2) + 2 * PI) with (PI / 2) by field. clear H1 H2 H3 H4 H5; intros H1 H2 H3 H4 H5; - replace (- x + 2 * 1 * PI) with (PI / 2 - (x - 3 * (PI / 2))). - replace (- y + 2 * 1 * PI) with (PI / 2 - (y - 3 * (PI / 2))). + replace (- x + 2 * 1 * PI) with (PI / 2 - (x - 3 * (PI / 2))) by field. + replace (- y + 2 * 1 * PI) with (PI / 2 - (y - 3 * (PI / 2))) by field. repeat rewrite cos_shift; apply (sin_increasing_1 (x - 3 * (PI / 2)) (y - 3 * (PI / 2)) H5 H4 H3 H2 H1). - rewrite Rmult_1_r. - rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var. - ring. - rewrite Rmult_1_r. - rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var. - ring. - rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var. - ring. - pattern PI at 3 in |- *; rewrite double_var; ring. - unfold Rminus in |- *. - rewrite <- Ropp_mult_distr_l_reverse. - apply Rplus_comm. - unfold Rminus in |- *. - rewrite <- Ropp_mult_distr_l_reverse. - apply Rplus_comm. Qed. Lemma cos_decreasing_0 : @@ -1375,31 +1257,8 @@ Lemma tan_diff : cos x <> 0 -> cos y <> 0 -> tan x - tan y = sin (x - y) / (cos x * cos y). Proof. intros; unfold tan in |- *; rewrite sin_minus. - unfold Rdiv in |- *. - unfold Rminus in |- *. - rewrite Rmult_plus_distr_r. - rewrite Rinv_mult_distr. - repeat rewrite (Rmult_comm (sin x)). - repeat rewrite Rmult_assoc. - rewrite (Rmult_comm (cos y)). - repeat rewrite Rmult_assoc. - rewrite <- Rinv_l_sym. - rewrite Rmult_1_r. - rewrite (Rmult_comm (sin x)). - apply Rplus_eq_compat_l. - rewrite <- Ropp_mult_distr_l_reverse. - rewrite <- Ropp_mult_distr_r_reverse. - rewrite (Rmult_comm (/ cos x)). - repeat rewrite Rmult_assoc. - rewrite (Rmult_comm (cos x)). - repeat rewrite Rmult_assoc. - rewrite <- Rinv_l_sym. - rewrite Rmult_1_r. - reflexivity. - assumption. - assumption. - assumption. - assumption. + field. + now split. Qed. Lemma tan_increasing_0 : @@ -1436,10 +1295,9 @@ Proof. intro H11; generalize (Rge_le (- y) (- (PI / 4)) H11); clear H11; intro H11; generalize (Rplus_le_compat (- (PI / 4)) x (- (PI / 4)) (- y) H H11); - generalize (Rplus_le_compat x (PI / 4) (- y) (PI / 4) H0 H10); - replace (x + - y) with (x - y). - replace (PI / 4 + PI / 4) with (PI / 2). - replace (- (PI / 4) + - (PI / 4)) with (- (PI / 2)). + generalize (Rplus_le_compat x (PI / 4) (- y) (PI / 4) H0 H10). + replace (PI / 4 + PI / 4) with (PI / 2) by field. + replace (- (PI / 4) + - (PI / 4)) with (- (PI / 2)) by field. intros; case (Rtotal_order 0 (x - y)); intro H14. generalize (sin_gt_0 (x - y) H14 (Rle_lt_trans (x - y) (PI / 2) PI H12 PI2_Rlt_PI)); @@ -1447,28 +1305,6 @@ Proof. elim H14; intro H15. rewrite <- H15 in H9; rewrite sin_0 in H9; elim (Rlt_irrefl 0 H9). apply Rminus_lt; assumption. - pattern PI at 1 in |- *; rewrite double_var. - unfold Rdiv in |- *. - rewrite Rmult_plus_distr_r. - repeat rewrite Rmult_assoc. - rewrite <- Rinv_mult_distr. - rewrite Ropp_plus_distr. - replace 4 with 4. - reflexivity. - ring. - discrR. - discrR. - pattern PI at 1 in |- *; rewrite double_var. - unfold Rdiv in |- *. - rewrite Rmult_plus_distr_r. - repeat rewrite Rmult_assoc. - rewrite <- Rinv_mult_distr. - replace 4 with 4. - reflexivity. - ring. - discrR. - discrR. - reflexivity. case (Rcase_abs (sin (x - y))); intro H9. assumption. generalize (Rge_le (sin (x - y)) 0 H9); clear H9; intro H9; @@ -1482,8 +1318,7 @@ Proof. (Rlt_le 0 (/ (cos x * cos y)) H12)); intro H13; elim (Rlt_irrefl 0 (Rle_lt_trans 0 (sin (x - y) * / (cos x * cos y)) 0 H13 H3)). - rewrite Rinv_mult_distr. - reflexivity. + apply Rinv_mult_distr. assumption. assumption. Qed. @@ -1521,9 +1356,8 @@ Proof. clear H10 H11; intro H8; generalize (Ropp_le_ge_contravar y (PI / 4) H2); intro H11; generalize (Rge_le (- y) (- (PI / 4)) H11); clear H11; intro H11; - generalize (Rplus_le_compat (- (PI / 4)) x (- (PI / 4)) (- y) H H11); - replace (x + - y) with (x - y). - replace (- (PI / 4) + - (PI / 4)) with (- (PI / 2)). + generalize (Rplus_le_compat (- (PI / 4)) x (- (PI / 4)) (- y) H H11). + replace (- (PI / 4) + - (PI / 4)) with (- (PI / 2)) by field. clear H11; intro H9; generalize (Rlt_minus x y H3); clear H3; intro H3; clear H H0 H1 H2 H4 H5 HP1 HP2; generalize PI2_Rlt_PI; intro H1; generalize (Ropp_lt_gt_contravar (PI / 2) PI H1); @@ -1534,18 +1368,6 @@ Proof. generalize (Rmult_lt_gt_compat_neg_l (sin (x - y)) 0 (/ (cos x * cos y)) H2 H8); rewrite Rmult_0_r; intro H4; assumption. - pattern PI at 1 in |- *; rewrite double_var. - unfold Rdiv in |- *. - rewrite Rmult_plus_distr_r. - repeat rewrite Rmult_assoc. - rewrite <- Rinv_mult_distr. - replace 4 with 4. - rewrite Ropp_plus_distr. - reflexivity. - ring. - discrR. - discrR. - reflexivity. apply Rinv_mult_distr; assumption. Qed. @@ -1737,7 +1559,7 @@ Proof. rewrite H5. rewrite mult_INR. simpl in |- *. - rewrite <- (Rplus_0_l (2 * INR x2 * PI)). + rewrite <- (Rplus_0_l ((1 + 1) * INR x2 * PI)). rewrite sin_period. apply sin_0. rewrite H5. @@ -1747,7 +1569,7 @@ Proof. rewrite Rmult_1_l; rewrite sin_plus. rewrite sin_PI. rewrite Rmult_0_r. - rewrite <- (Rplus_0_l (2 * INR x2 * PI)). + rewrite <- (Rplus_0_l ((1 + 1) * INR x2 * PI)). rewrite sin_period. rewrite sin_0; ring. apply le_IZR. @@ -1769,7 +1591,7 @@ Proof. rewrite H5. rewrite mult_INR. simpl in |- *. - rewrite <- (Rplus_0_l (2 * INR x2 * PI)). + rewrite <- (Rplus_0_l ((1 + 1) * INR x2 * PI)). rewrite sin_period. rewrite sin_0; ring. rewrite H5. @@ -1779,7 +1601,7 @@ Proof. rewrite Rmult_1_l; rewrite sin_plus. rewrite sin_PI. rewrite Rmult_0_r. - rewrite <- (Rplus_0_l (2 * INR x2 * PI)). + rewrite <- (Rplus_0_l ((1 + 1) * INR x2 * PI)). rewrite sin_period. rewrite sin_0; ring. apply le_IZR. @@ -1787,8 +1609,7 @@ Proof. rewrite Rplus_0_r. rewrite Ropp_Ropp_IZR. rewrite Rplus_opp_r. - left; replace 0 with (IZR 0); [ apply IZR_lt | reflexivity ]. - assumption. + now apply Rlt_le, IZR_lt. rewrite <- sin_neg. rewrite Ropp_mult_distr_l_reverse. rewrite Ropp_involutive. @@ -1858,7 +1679,7 @@ Proof. - right; left; auto. - left. clear Hi. subst. - replace 0 with (IZR 0 * PI) by (simpl; ring). f_equal. f_equal. + replace 0 with (IZR 0 * PI) by apply Rmult_0_l. f_equal. f_equal. apply one_IZR_lt1. split. + apply Rlt_le_trans with 0; diff --git a/theories/Reals/Rtrigo_alt.v b/theories/Reals/Rtrigo_alt.v index a5092d22..71b90fb4 100644 --- a/theories/Reals/Rtrigo_alt.v +++ b/theories/Reals/Rtrigo_alt.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -99,24 +101,22 @@ Proof. apply Rle_trans with 20. apply Rle_trans with 16. replace 16 with (Rsqr 4); [ idtac | ring_Rsqr ]. - replace (a * a) with (Rsqr a); [ idtac | reflexivity ]. apply Rsqr_incr_1. assumption. assumption. - left; prove_sup0. - rewrite <- (Rplus_0_r 16); replace 20 with (16 + 4); - [ apply Rplus_le_compat_l; left; prove_sup0 | ring ]. - rewrite <- (Rplus_comm 20); pattern 20 at 1; rewrite <- Rplus_0_r; - apply Rplus_le_compat_l. + now apply IZR_le. + now apply IZR_le. + rewrite <- (Rplus_0_l 20) at 1; + apply Rplus_le_compat_r. apply Rplus_le_le_0_compat. - repeat apply Rmult_le_pos. - left; prove_sup0. - left; prove_sup0. - replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. - replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. apply Rmult_le_pos. - left; prove_sup0. - replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. + apply Rmult_le_pos. + now apply IZR_le. + apply pos_INR. + apply pos_INR. + apply Rmult_le_pos. + now apply IZR_le. + apply pos_INR. apply INR_fact_neq_0. apply INR_fact_neq_0. simpl; ring. @@ -182,16 +182,14 @@ Proof. replace (- sum_f_R0 (tg_alt Un) (S (2 * n))) with (-1 * sum_f_R0 (tg_alt Un) (S (2 * n))); [ rewrite scal_sum | ring ]. apply sum_eq; intros; unfold sin_term, Un, tg_alt; - replace ((-1) ^ S i) with (-1 * (-1) ^ i). + change ((-1) ^ S i) with (-1 * (-1) ^ i). unfold Rdiv; ring. - reflexivity. replace (- sum_f_R0 (tg_alt Un) (2 * n)) with (-1 * sum_f_R0 (tg_alt Un) (2 * n)); [ rewrite scal_sum | ring ]. apply sum_eq; intros. unfold sin_term, Un, tg_alt; - replace ((-1) ^ S i) with (-1 * (-1) ^ i). + change ((-1) ^ S i) with (-1 * (-1) ^ i). unfold Rdiv; ring. - reflexivity. replace (2 * (n + 1))%nat with (S (S (2 * n))). reflexivity. ring. @@ -279,26 +277,23 @@ Proof. with (4 * INR n1 * INR n1 + 14 * INR n1 + 12); [ idtac | ring ]. apply Rle_trans with 12. apply Rle_trans with 4. - replace 4 with (Rsqr 2); [ idtac | ring_Rsqr ]. - replace (a0 * a0) with (Rsqr a0); [ idtac | reflexivity ]. + change 4 with (Rsqr 2). apply Rsqr_incr_1. assumption. - discrR. assumption. - left; prove_sup0. - pattern 4 at 1; rewrite <- Rplus_0_r; replace 12 with (4 + 8); - [ apply Rplus_le_compat_l; left; prove_sup0 | ring ]. - rewrite <- (Rplus_comm 12); pattern 12 at 1; rewrite <- Rplus_0_r; - apply Rplus_le_compat_l. + now apply IZR_le. + now apply IZR_le. + rewrite <- (Rplus_0_l 12) at 1; + apply Rplus_le_compat_r. apply Rplus_le_le_0_compat. - repeat apply Rmult_le_pos. - left; prove_sup0. - left; prove_sup0. - replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. - replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. apply Rmult_le_pos. - left; prove_sup0. - replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. + apply Rmult_le_pos. + now apply IZR_le. + apply pos_INR. + apply pos_INR. + apply Rmult_le_pos. + now apply IZR_le. + apply pos_INR. apply INR_fact_neq_0. apply INR_fact_neq_0. simpl; ring. @@ -320,7 +315,7 @@ Proof. (1 - sum_f_R0 (fun i:nat => cos_n i * Rsqr a0 ^ i) (S n1)). unfold Rminus; rewrite Ropp_plus_distr; rewrite Ropp_involutive; repeat rewrite Rplus_assoc; rewrite (Rplus_comm 1); - rewrite (Rplus_comm (-1)); repeat rewrite Rplus_assoc; + rewrite (Rplus_comm (-(1))); repeat rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; rewrite <- Rabs_Ropp; rewrite Ropp_plus_distr; rewrite Ropp_involutive; unfold Rminus in H6; apply H6. @@ -351,15 +346,13 @@ Proof. replace (- sum_f_R0 (tg_alt Un) (S (2 * n0))) with (-1 * sum_f_R0 (tg_alt Un) (S (2 * n0))); [ rewrite scal_sum | ring ]. apply sum_eq; intros; unfold cos_term, Un, tg_alt; - replace ((-1) ^ S i) with (-1 * (-1) ^ i). + change ((-1) ^ S i) with (-1 * (-1) ^ i). unfold Rdiv; ring. - reflexivity. replace (- sum_f_R0 (tg_alt Un) (2 * n0)) with (-1 * sum_f_R0 (tg_alt Un) (2 * n0)); [ rewrite scal_sum | ring ]; apply sum_eq; intros; unfold cos_term, Un, tg_alt; - replace ((-1) ^ S i) with (-1 * (-1) ^ i). + change ((-1) ^ S i) with (-1 * (-1) ^ i). unfold Rdiv; ring. - reflexivity. replace (2 * (n0 + 1))%nat with (S (S (2 * n0))). reflexivity. ring. @@ -367,10 +360,10 @@ Proof. reflexivity. ring. intro; elim H2; intros; split. - apply Rplus_le_reg_l with (-1). + apply Rplus_le_reg_l with (-(1)). rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l; rewrite (Rplus_comm (-1)); apply H3. - apply Rplus_le_reg_l with (-1). + apply Rplus_le_reg_l with (-(1)). rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l; rewrite (Rplus_comm (-1)); apply H4. unfold cos_term; simpl; unfold Rdiv; rewrite Rinv_1; diff --git a/theories/Reals/Rtrigo_calc.v b/theories/Reals/Rtrigo_calc.v index 9ba14ee7..7cbfc630 100644 --- a/theories/Reals/Rtrigo_calc.v +++ b/theories/Reals/Rtrigo_calc.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -32,48 +34,22 @@ Proof. Qed. Lemma sin_cos_PI4 : sin (PI / 4) = cos (PI / 4). -Proof with trivial. - rewrite cos_sin... - replace (PI / 2 + PI / 4) with (- (PI / 4) + PI)... - rewrite neg_sin; rewrite sin_neg; ring... - cut (PI = PI / 2 + PI / 2); [ intro | apply double_var ]... - pattern PI at 2 3; rewrite H; pattern PI at 2 3; rewrite H... - assert (H0 : 2 <> 0); - [ discrR | unfold Rdiv; rewrite Rinv_mult_distr; try ring ]... +Proof. + rewrite cos_sin. + replace (PI / 2 + PI / 4) with (- (PI / 4) + PI) by field. + rewrite neg_sin, sin_neg; ring. Qed. Lemma sin_PI3_cos_PI6 : sin (PI / 3) = cos (PI / 6). -Proof with trivial. - replace (PI / 6) with (PI / 2 - PI / 3)... - rewrite cos_shift... - assert (H0 : 6 <> 0); [ discrR | idtac ]... - assert (H1 : 3 <> 0); [ discrR | idtac ]... - assert (H2 : 2 <> 0); [ discrR | idtac ]... - apply Rmult_eq_reg_l with 6... - rewrite Rmult_minus_distr_l; repeat rewrite (Rmult_comm 6)... - unfold Rdiv; repeat rewrite Rmult_assoc... - rewrite <- Rinv_l_sym... - rewrite (Rmult_comm (/ 3)); repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym... - rewrite (Rmult_comm PI); repeat rewrite Rmult_1_r; - repeat rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym... - ring... +Proof. + replace (PI / 6) with (PI / 2 - PI / 3) by field. + now rewrite cos_shift. Qed. Lemma sin_PI6_cos_PI3 : cos (PI / 3) = sin (PI / 6). -Proof with trivial. - replace (PI / 6) with (PI / 2 - PI / 3)... - rewrite sin_shift... - assert (H0 : 6 <> 0); [ discrR | idtac ]... - assert (H1 : 3 <> 0); [ discrR | idtac ]... - assert (H2 : 2 <> 0); [ discrR | idtac ]... - apply Rmult_eq_reg_l with 6... - rewrite Rmult_minus_distr_l; repeat rewrite (Rmult_comm 6)... - unfold Rdiv; repeat rewrite Rmult_assoc... - rewrite <- Rinv_l_sym... - rewrite (Rmult_comm (/ 3)); repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym... - rewrite (Rmult_comm PI); repeat rewrite Rmult_1_r; - repeat rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym... - ring... +Proof. + replace (PI / 6) with (PI / 2 - PI / 3) by field. + now rewrite sin_shift. Qed. Lemma PI6_RGT_0 : 0 < PI / 6. @@ -90,29 +66,20 @@ Proof. Qed. Lemma sin_PI6 : sin (PI / 6) = 1 / 2. -Proof with trivial. - assert (H : 2 <> 0); [ discrR | idtac ]... - apply Rmult_eq_reg_l with (2 * cos (PI / 6))... +Proof. + apply Rmult_eq_reg_l with (2 * cos (PI / 6)). replace (2 * cos (PI / 6) * sin (PI / 6)) with - (2 * sin (PI / 6) * cos (PI / 6))... - rewrite <- sin_2a; replace (2 * (PI / 6)) with (PI / 3)... - rewrite sin_PI3_cos_PI6... - unfold Rdiv; rewrite Rmult_1_l; rewrite Rmult_assoc; - pattern 2 at 2; rewrite (Rmult_comm 2); rewrite Rmult_assoc; - rewrite <- Rinv_l_sym... - rewrite Rmult_1_r... - unfold Rdiv; rewrite Rinv_mult_distr... - rewrite (Rmult_comm (/ 2)); rewrite (Rmult_comm 2); - repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... - rewrite Rmult_1_r... - discrR... - ring... - apply prod_neq_R0... + (2 * sin (PI / 6) * cos (PI / 6)) by ring. + rewrite <- sin_2a; replace (2 * (PI / 6)) with (PI / 3) by field. + rewrite sin_PI3_cos_PI6. + field. + apply prod_neq_R0. + discrR. cut (0 < cos (PI / 6)); [ intro H1; auto with real | apply cos_gt_0; [ apply (Rlt_trans (- (PI / 2)) 0 (PI / 6) _PI2_RLT_0 PI6_RGT_0) - | apply PI6_RLT_PI2 ] ]... + | apply PI6_RLT_PI2 ] ]. Qed. Lemma sqrt2_neq_0 : sqrt 2 <> 0. @@ -188,20 +155,13 @@ Proof with trivial. apply Rinv_0_lt_compat; apply Rlt_sqrt2_0... rewrite Rsqr_div... rewrite Rsqr_1; rewrite Rsqr_sqrt... - assert (H : 2 <> 0); [ discrR | idtac ]... unfold Rsqr; pattern (cos (PI / 4)) at 1; rewrite <- sin_cos_PI4; replace (sin (PI / 4) * cos (PI / 4)) with - (1 / 2 * (2 * sin (PI / 4) * cos (PI / 4)))... - rewrite <- sin_2a; replace (2 * (PI / 4)) with (PI / 2)... + (1 / 2 * (2 * sin (PI / 4) * cos (PI / 4))) by field. + rewrite <- sin_2a; replace (2 * (PI / 4)) with (PI / 2) by field. rewrite sin_PI2... - apply Rmult_1_r... - unfold Rdiv; rewrite (Rmult_comm 2); rewrite Rinv_mult_distr... - repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... - rewrite Rmult_1_r... - unfold Rdiv; rewrite Rmult_1_l; repeat rewrite <- Rmult_assoc... - rewrite <- Rinv_l_sym... - rewrite Rmult_1_l... + field. left; prove_sup... apply sqrt2_neq_0... Qed. @@ -219,24 +179,17 @@ Proof. Qed. Lemma cos3PI4 : cos (3 * (PI / 4)) = -1 / sqrt 2. -Proof with trivial. - replace (3 * (PI / 4)) with (PI / 2 - - (PI / 4))... - rewrite cos_shift; rewrite sin_neg; rewrite sin_PI4... - unfold Rdiv; rewrite Ropp_mult_distr_l_reverse... - unfold Rminus; rewrite Ropp_involutive; pattern PI at 1; - rewrite double_var; unfold Rdiv; rewrite Rmult_plus_distr_r; - repeat rewrite Rmult_assoc; rewrite <- Rinv_mult_distr; - [ ring | discrR | discrR ]... +Proof. + replace (3 * (PI / 4)) with (PI / 2 - - (PI / 4)) by field. + rewrite cos_shift; rewrite sin_neg; rewrite sin_PI4. + unfold Rdiv. + ring. Qed. Lemma sin3PI4 : sin (3 * (PI / 4)) = 1 / sqrt 2. -Proof with trivial. - replace (3 * (PI / 4)) with (PI / 2 - - (PI / 4))... - rewrite sin_shift; rewrite cos_neg; rewrite cos_PI4... - unfold Rminus; rewrite Ropp_involutive; pattern PI at 1; - rewrite double_var; unfold Rdiv; rewrite Rmult_plus_distr_r; - repeat rewrite Rmult_assoc; rewrite <- Rinv_mult_distr; - [ ring | discrR | discrR ]... +Proof. + replace (3 * (PI / 4)) with (PI / 2 - - (PI / 4)) by field. + now rewrite sin_shift, cos_neg, cos_PI4. Qed. Lemma cos_PI6 : cos (PI / 6) = sqrt 3 / 2. @@ -248,19 +201,11 @@ Proof with trivial. left; apply (Rmult_lt_0_compat (sqrt 3) (/ 2))... apply Rlt_sqrt3_0... apply Rinv_0_lt_compat; prove_sup0... - assert (H : 2 <> 0); [ discrR | idtac ]... - assert (H1 : 4 <> 0); [ apply prod_neq_R0 | idtac ]... rewrite Rsqr_div... rewrite cos2; unfold Rsqr; rewrite sin_PI6; rewrite sqrt_def... - unfold Rdiv; rewrite Rmult_1_l; apply Rmult_eq_reg_l with 4... - rewrite Rmult_minus_distr_l; rewrite (Rmult_comm 3); - repeat rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym... - rewrite Rmult_1_l; rewrite Rmult_1_r... - rewrite <- (Rmult_comm (/ 2)); repeat rewrite <- Rmult_assoc... - rewrite <- Rinv_l_sym... - rewrite Rmult_1_l; rewrite <- Rinv_r_sym... - ring... - left; prove_sup0... + field. + left ; prove_sup0. + discrR. Qed. Lemma tan_PI6 : tan (PI / 6) = 1 / sqrt 3. @@ -306,56 +251,32 @@ Proof. Qed. Lemma cos_2PI3 : cos (2 * (PI / 3)) = -1 / 2. -Proof with trivial. - assert (H : 2 <> 0); [ discrR | idtac ]... - assert (H0 : 4 <> 0); [ apply prod_neq_R0 | idtac ]... - rewrite double; rewrite cos_plus; rewrite sin_PI3; rewrite cos_PI3; - unfold Rdiv; rewrite Rmult_1_l; apply Rmult_eq_reg_l with 4... - rewrite Rmult_minus_distr_l; repeat rewrite Rmult_assoc; - rewrite (Rmult_comm 2)... - repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... - rewrite Rmult_1_r; rewrite <- Rinv_r_sym... - pattern 2 at 4; rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; - rewrite <- Rinv_l_sym... - rewrite Rmult_1_r; rewrite Ropp_mult_distr_r_reverse; rewrite Rmult_1_r... - rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... - rewrite Rmult_1_r; rewrite (Rmult_comm 2); rewrite (Rmult_comm (/ 2))... - repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... - rewrite Rmult_1_r; rewrite sqrt_def... - ring... - left; prove_sup... +Proof. + rewrite cos_2a, sin_PI3, cos_PI3. + replace (sqrt 3 / 2 * (sqrt 3 / 2)) with ((sqrt 3 * sqrt 3) / 4) by field. + rewrite sqrt_sqrt. + field. + left ; prove_sup0. Qed. Lemma tan_2PI3 : tan (2 * (PI / 3)) = - sqrt 3. -Proof with trivial. - assert (H : 2 <> 0); [ discrR | idtac ]... - unfold tan; rewrite sin_2PI3; rewrite cos_2PI3; unfold Rdiv; - rewrite Ropp_mult_distr_l_reverse; rewrite Rmult_1_l; - rewrite <- Ropp_inv_permute... - rewrite Rinv_involutive... - rewrite Rmult_assoc; rewrite Ropp_mult_distr_r_reverse; rewrite <- Rinv_l_sym... - ring... - apply Rinv_neq_0_compat... +Proof. + unfold tan; rewrite sin_2PI3, cos_2PI3. + field. Qed. Lemma cos_5PI4 : cos (5 * (PI / 4)) = -1 / sqrt 2. -Proof with trivial. - replace (5 * (PI / 4)) with (PI / 4 + PI)... - rewrite neg_cos; rewrite cos_PI4; unfold Rdiv; - rewrite Ropp_mult_distr_l_reverse... - pattern PI at 2; rewrite double_var; pattern PI at 2 3; - rewrite double_var; assert (H : 2 <> 0); - [ discrR | unfold Rdiv; repeat rewrite Rinv_mult_distr; try ring ]... +Proof. + replace (5 * (PI / 4)) with (PI / 4 + PI) by field. + rewrite neg_cos; rewrite cos_PI4; unfold Rdiv. + ring. Qed. Lemma sin_5PI4 : sin (5 * (PI / 4)) = -1 / sqrt 2. -Proof with trivial. - replace (5 * (PI / 4)) with (PI / 4 + PI)... - rewrite neg_sin; rewrite sin_PI4; unfold Rdiv; - rewrite Ropp_mult_distr_l_reverse... - pattern PI at 2; rewrite double_var; pattern PI at 2 3; - rewrite double_var; assert (H : 2 <> 0); - [ discrR | unfold Rdiv; repeat rewrite Rinv_mult_distr; try ring ]... +Proof. + replace (5 * (PI / 4)) with (PI / 4 + PI) by field. + rewrite neg_sin; rewrite sin_PI4; unfold Rdiv. + ring. Qed. Lemma sin_cos5PI4 : cos (5 * (PI / 4)) = sin (5 * (PI / 4)). diff --git a/theories/Reals/Rtrigo_def.v b/theories/Reals/Rtrigo_def.v index 0d2a9a8b..d2faf95b 100644 --- a/theories/Reals/Rtrigo_def.v +++ b/theories/Reals/Rtrigo_def.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase Rfunctions SeqSeries Rtrigo_fun Max. @@ -157,7 +159,7 @@ Proof. apply Rinv_0_lt_compat; assumption. rewrite H3 in H0; assumption. apply lt_le_trans with 1%nat; [ apply lt_O_Sn | apply le_max_r ]. - apply le_IZR; replace (IZR 0) with 0; [ idtac | reflexivity ]; left; + apply le_IZR; left; apply Rlt_trans with (/ eps); [ apply Rinv_0_lt_compat; assumption | assumption ]. assert (H0 := archimed (/ eps)). @@ -194,30 +196,27 @@ Proof. elim H1; intros; assumption. apply lt_le_trans with (S n). unfold ge in H2; apply le_lt_n_Sm; assumption. - replace (2 * n + 1)%nat with (S (2 * n)); [ idtac | ring ]. + replace (2 * n + 1)%nat with (S (2 * n)) by ring. apply le_n_S; apply le_n_2n. apply Rmult_lt_reg_l with (INR (2 * S n)). apply lt_INR_0; replace (2 * S n)%nat with (S (S (2 * n))). apply lt_O_Sn. - replace (S n) with (n + 1)%nat; [ idtac | ring ]. + replace (S n) with (n + 1)%nat by ring. ring. rewrite <- Rinv_r_sym. - rewrite Rmult_1_r; replace 1 with (INR 1); [ apply lt_INR | reflexivity ]. + rewrite Rmult_1_r. + apply (lt_INR 1). replace (2 * S n)%nat with (S (S (2 * n))). apply lt_n_S; apply lt_O_Sn. - replace (S n) with (n + 1)%nat; [ ring | ring ]. + ring. apply not_O_INR; discriminate. apply not_O_INR; discriminate. replace (2 * n + 1)%nat with (S (2 * n)); [ apply not_O_INR; discriminate | ring ]. apply Rle_ge; left; apply Rinv_0_lt_compat. apply lt_INR_0. - replace (2 * S n * (2 * n + 1))%nat with (S (S (4 * (n * n) + 6 * n))). + replace (2 * S n * (2 * n + 1))%nat with (2 + (4 * (n * n) + 6 * n))%nat by ring. apply lt_O_Sn. - apply INR_eq. - repeat rewrite S_INR; rewrite plus_INR; repeat rewrite mult_INR; - rewrite plus_INR; rewrite mult_INR; repeat rewrite S_INR; - replace (INR 0) with 0; [ ring | reflexivity ]. Qed. Lemma cosn_no_R0 : forall n:nat, cos_n n <> 0. @@ -318,28 +317,25 @@ Proof. elim H1; intros; assumption. apply lt_le_trans with (S n). unfold ge in H2; apply le_lt_n_Sm; assumption. - replace (2 * S n + 1)%nat with (S (2 * S n)); [ idtac | ring ]. + replace (2 * S n + 1)%nat with (S (2 * S n)) by ring. apply le_S; apply le_n_2n. apply Rmult_lt_reg_l with (INR (2 * S n)). apply lt_INR_0; replace (2 * S n)%nat with (S (S (2 * n))); - [ apply lt_O_Sn | replace (S n) with (n + 1)%nat; [ idtac | ring ]; ring ]. + [ apply lt_O_Sn | ring ]. rewrite <- Rinv_r_sym. - rewrite Rmult_1_r; replace 1 with (INR 1); [ apply lt_INR | reflexivity ]. + rewrite Rmult_1_r. + apply (lt_INR 1). replace (2 * S n)%nat with (S (S (2 * n))). apply lt_n_S; apply lt_O_Sn. - replace (S n) with (n + 1)%nat; [ ring | ring ]. + ring. apply not_O_INR; discriminate. apply not_O_INR; discriminate. apply not_O_INR; discriminate. - left; change (0 < / INR ((2 * S n + 1) * (2 * S n))); - apply Rinv_0_lt_compat. + left; apply Rinv_0_lt_compat. apply lt_INR_0. replace ((2 * S n + 1) * (2 * S n))%nat with - (S (S (S (S (S (S (4 * (n * n) + 10 * n))))))). + (6 + (4 * (n * n) + 10 * n))%nat by ring. apply lt_O_Sn. - apply INR_eq; repeat rewrite S_INR; rewrite plus_INR; repeat rewrite mult_INR; - rewrite plus_INR; rewrite mult_INR; repeat rewrite S_INR; - replace (INR 0) with 0; [ ring | reflexivity ]. Qed. Lemma sin_no_R0 : forall n:nat, sin_n n <> 0. diff --git a/theories/Reals/Rtrigo_fun.v b/theories/Reals/Rtrigo_fun.v index f395f9ae..744a99a1 100644 --- a/theories/Reals/Rtrigo_fun.v +++ b/theories/Reals/Rtrigo_fun.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. diff --git a/theories/Reals/Rtrigo_reg.v b/theories/Reals/Rtrigo_reg.v index eed612d9..456fb6a7 100644 --- a/theories/Reals/Rtrigo_reg.v +++ b/theories/Reals/Rtrigo_reg.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -251,6 +253,7 @@ Proof. exists delta; intros. rewrite Rplus_0_l; replace (cos h - cos 0) with (-2 * Rsqr (sin (h / 2))). unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r. + change (-2) with (-(2)). unfold Rdiv; do 2 rewrite Ropp_mult_distr_l_reverse. rewrite Rabs_Ropp. replace (2 * Rsqr (sin (h * / 2)) * / h) with @@ -266,7 +269,7 @@ Proof. apply Rabs_pos. assert (H9 := SIN_bound (h / 2)). unfold Rabs; case (Rcase_abs (sin (h / 2))); intro. - pattern 1 at 3; rewrite <- (Ropp_involutive 1). + rewrite <- (Ropp_involutive 1). apply Ropp_le_contravar. elim H9; intros; assumption. elim H9; intros; assumption. @@ -395,15 +398,8 @@ Proof. apply Rlt_le_trans with alp. apply H7. unfold alp; apply Rmin_l. - rewrite sin_plus; unfold Rminus, Rdiv; - repeat rewrite Rmult_plus_distr_r; repeat rewrite Rmult_plus_distr_l; - repeat rewrite Rmult_assoc; repeat rewrite Rplus_assoc; - apply Rplus_eq_compat_l. - rewrite (Rplus_comm (sin x * (-1 * / h))); repeat rewrite Rplus_assoc; - apply Rplus_eq_compat_l. - rewrite Ropp_mult_distr_r_reverse; rewrite Ropp_mult_distr_l_reverse; - rewrite Rmult_1_r; rewrite Rmult_1_l; rewrite Ropp_mult_distr_r_reverse; - rewrite <- Ropp_mult_distr_l_reverse; apply Rplus_comm. + rewrite sin_plus. + now field. unfold alp; unfold Rmin; case (Rle_dec alp1 alp2); intro. apply (cond_pos alp1). apply (cond_pos alp2). diff --git a/theories/Reals/SeqProp.v b/theories/Reals/SeqProp.v index 5a2a07c4..38b0b3c4 100644 --- a/theories/Reals/SeqProp.v +++ b/theories/Reals/SeqProp.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -150,7 +152,7 @@ Definition sequence_lb (Un:nat -> R) (pr:has_lb Un) (* Compatibility *) Notation sequence_majorant := sequence_ub (only parsing). Notation sequence_minorant := sequence_lb (only parsing). -Unset Standard Proposition Elimination Names. + Lemma Wn_decreasing : forall (Un:nat -> R) (pr:has_ub Un), Un_decreasing (sequence_ub Un pr). Proof. @@ -1167,7 +1169,7 @@ Proof. assert (H6 := archimed (Rabs x)); fold M in H6; elim H6; intros. rewrite H4 in H7; rewrite <- INR_IZR_INZ in H7. simpl in H7; elim (Rlt_irrefl _ (Rlt_trans _ _ _ H2 H7)). - replace 1 with (INR 1); [ apply le_INR | reflexivity ]; apply le_n_S; + apply (le_INR 1); apply le_n_S; apply le_O_n. apply le_IZR; simpl; left; apply Rlt_trans with (Rabs x). assumption. diff --git a/theories/Reals/SeqSeries.v b/theories/Reals/SeqSeries.v index 1123e7ee..ccd205e2 100644 --- a/theories/Reals/SeqSeries.v +++ b/theories/Reals/SeqSeries.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. diff --git a/theories/Reals/SplitAbsolu.v b/theories/Reals/SplitAbsolu.v index a78a6e19..aa67b677 100644 --- a/theories/Reals/SplitAbsolu.v +++ b/theories/Reals/SplitAbsolu.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbasic_fun. diff --git a/theories/Reals/SplitRmult.v b/theories/Reals/SplitRmult.v index 074a7631..a8ff60b0 100644 --- a/theories/Reals/SplitRmult.v +++ b/theories/Reals/SplitRmult.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (*i Lemma mult_non_zero :(r1,r2:R)``r1<>0`` /\ ``r2<>0`` -> ``r1*r2<>0``. i*) diff --git a/theories/Reals/Sqrt_reg.v b/theories/Reals/Sqrt_reg.v index d43baee8..d6b386f1 100644 --- a/theories/Reals/Sqrt_reg.v +++ b/theories/Reals/Sqrt_reg.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -21,6 +23,7 @@ Proof. destruct (total_order_T h 0) as [[Hlt|Heq]|Hgt]. repeat rewrite Rabs_left. unfold Rminus; do 2 rewrite <- (Rplus_comm (-1)). + change (-1) with (-(1)). do 2 rewrite Ropp_plus_distr; rewrite Ropp_involutive; apply Rplus_le_compat_l. apply Ropp_le_contravar; apply sqrt_le_1. diff --git a/theories/Reals/vo.itarget b/theories/Reals/vo.itarget deleted file mode 100644 index 0c8f0b97..00000000 --- a/theories/Reals/vo.itarget +++ /dev/null @@ -1,62 +0,0 @@ -Alembert.vo -AltSeries.vo -ArithProp.vo -Binomial.vo -Cauchy_prod.vo -Cos_plus.vo -Cos_rel.vo -DiscrR.vo -Exp_prop.vo -Integration.vo -Machin.vo -MVT.vo -NewtonInt.vo -PartSum.vo -PSeries_reg.vo -Ranalysis1.vo -Ranalysis2.vo -Ranalysis3.vo -Ranalysis4.vo -Ranalysis5.vo -Ranalysis.vo -Ranalysis_reg.vo -Ratan.vo -Raxioms.vo -Rbase.vo -Rbasic_fun.vo -Rcomplete.vo -Rdefinitions.vo -Rderiv.vo -Reals.vo -Rfunctions.vo -Rgeom.vo -RiemannInt_SF.vo -RiemannInt.vo -R_Ifp.vo -RIneq.vo -Rlimit.vo -RList.vo -Rlogic.vo -Rpow_def.vo -Rpower.vo -Rprod.vo -Rseries.vo -Rsigma.vo -Rsqrt_def.vo -R_sqrt.vo -R_sqr.vo -Rtopology.vo -Rtrigo_alt.vo -Rtrigo_calc.vo -Rtrigo_def.vo -Rtrigo_fun.vo -Rtrigo_reg.vo -Rtrigo1.vo -Rtrigo.vo -SeqProp.vo -SeqSeries.vo -SplitAbsolu.vo -SplitRmult.vo -Sqrt_reg.vo -ROrderedType.vo -Rminmax.vo diff --git a/theories/Relations/Operators_Properties.v b/theories/Relations/Operators_Properties.v index fe8a96ac..e82a6734 100644 --- a/theories/Relations/Operators_Properties.v +++ b/theories/Relations/Operators_Properties.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (************************************************************************) diff --git a/theories/Relations/Relation_Definitions.v b/theories/Relations/Relation_Definitions.v index 9c98879c..53def474 100644 --- a/theories/Relations/Relation_Definitions.v +++ b/theories/Relations/Relation_Definitions.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Section Relation_Definition. diff --git a/theories/Relations/Relation_Operators.v b/theories/Relations/Relation_Operators.v index 88239475..529e4d08 100644 --- a/theories/Relations/Relation_Operators.v +++ b/theories/Relations/Relation_Operators.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (************************************************************************) diff --git a/theories/Relations/Relations.v b/theories/Relations/Relations.v index ce6bdbc6..61344974 100644 --- a/theories/Relations/Relations.v +++ b/theories/Relations/Relations.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export Relation_Definitions. diff --git a/theories/Relations/vo.itarget b/theories/Relations/vo.itarget deleted file mode 100644 index 9d81dd07..00000000 --- a/theories/Relations/vo.itarget +++ /dev/null @@ -1,4 +0,0 @@ -Operators_Properties.vo -Relation_Definitions.vo -Relation_Operators.vo -Relations.vo diff --git a/theories/Setoids/Setoid.v b/theories/Setoids/Setoid.v index 55b301c3..af06bcf4 100644 --- a/theories/Setoids/Setoid.v +++ b/theories/Setoids/Setoid.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export Coq.Classes.SetoidTactics. diff --git a/theories/Setoids/vo.itarget b/theories/Setoids/vo.itarget deleted file mode 100644 index 8d608cf7..00000000 --- a/theories/Setoids/vo.itarget +++ /dev/null @@ -1 +0,0 @@ -Setoid.vo
\ No newline at end of file diff --git a/theories/Sets/Classical_sets.v b/theories/Sets/Classical_sets.v index 837437a2..b68022f8 100644 --- a/theories/Sets/Classical_sets.v +++ b/theories/Sets/Classical_sets.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (****************************************************************************) (* *) diff --git a/theories/Sets/Constructive_sets.v b/theories/Sets/Constructive_sets.v index 6291248e..f2475af1 100644 --- a/theories/Sets/Constructive_sets.v +++ b/theories/Sets/Constructive_sets.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (****************************************************************************) (* *) diff --git a/theories/Sets/Cpo.v b/theories/Sets/Cpo.v index b5db9030..3977097e 100644 --- a/theories/Sets/Cpo.v +++ b/theories/Sets/Cpo.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (****************************************************************************) (* *) diff --git a/theories/Sets/Ensembles.v b/theories/Sets/Ensembles.v index 0fefb354..c3713207 100644 --- a/theories/Sets/Ensembles.v +++ b/theories/Sets/Ensembles.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (****************************************************************************) (* *) diff --git a/theories/Sets/Finite_sets.v b/theories/Sets/Finite_sets.v index edbc1efe..5c42cbe6 100644 --- a/theories/Sets/Finite_sets.v +++ b/theories/Sets/Finite_sets.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (****************************************************************************) (* *) diff --git a/theories/Sets/Finite_sets_facts.v b/theories/Sets/Finite_sets_facts.v index 31cc11e1..1c191613 100644 --- a/theories/Sets/Finite_sets_facts.v +++ b/theories/Sets/Finite_sets_facts.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (****************************************************************************) (* *) diff --git a/theories/Sets/Image.v b/theories/Sets/Image.v index e74ef41e..3e28bbe9 100644 --- a/theories/Sets/Image.v +++ b/theories/Sets/Image.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (****************************************************************************) (* *) diff --git a/theories/Sets/Infinite_sets.v b/theories/Sets/Infinite_sets.v index 98eeb7d7..bdeeb6a7 100644 --- a/theories/Sets/Infinite_sets.v +++ b/theories/Sets/Infinite_sets.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (****************************************************************************) (* *) diff --git a/theories/Sets/Integers.v b/theories/Sets/Integers.v index 057fc9b6..225e388b 100644 --- a/theories/Sets/Integers.v +++ b/theories/Sets/Integers.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (****************************************************************************) (* *) diff --git a/theories/Sets/Multiset.v b/theories/Sets/Multiset.v index 42d0c76d..a79ddead 100644 --- a/theories/Sets/Multiset.v +++ b/theories/Sets/Multiset.v @@ -1,14 +1,17 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* G. Huet 1-9-95 *) Require Import Permut Setoid. +Require Plus. (* comm. and ass. of plus *) Set Implicit Arguments. @@ -67,9 +70,6 @@ Section multiset_defs. unfold meq; unfold munion; simpl; auto. Qed. - - Require Plus. (* comm. and ass. of plus *) - Lemma munion_comm : forall x y:multiset, meq (munion x y) (munion y x). Proof. unfold meq; unfold multiplicity; unfold munion. diff --git a/theories/Sets/Partial_Order.v b/theories/Sets/Partial_Order.v index 335fec5b..17fc0ed2 100644 --- a/theories/Sets/Partial_Order.v +++ b/theories/Sets/Partial_Order.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (****************************************************************************) (* *) diff --git a/theories/Sets/Permut.v b/theories/Sets/Permut.v index adbde18e..86a500df 100644 --- a/theories/Sets/Permut.v +++ b/theories/Sets/Permut.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* G. Huet 1-9-95 *) diff --git a/theories/Sets/Powerset.v b/theories/Sets/Powerset.v index 7c2435da..88bcd655 100644 --- a/theories/Sets/Powerset.v +++ b/theories/Sets/Powerset.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (****************************************************************************) (* *) diff --git a/theories/Sets/Powerset_Classical_facts.v b/theories/Sets/Powerset_Classical_facts.v index e802beac..7975a026 100644 --- a/theories/Sets/Powerset_Classical_facts.v +++ b/theories/Sets/Powerset_Classical_facts.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (****************************************************************************) (* *) diff --git a/theories/Sets/Powerset_facts.v b/theories/Sets/Powerset_facts.v index e9696a1c..81b475ac 100644 --- a/theories/Sets/Powerset_facts.v +++ b/theories/Sets/Powerset_facts.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (****************************************************************************) (* *) @@ -40,6 +42,11 @@ Section Sets_as_an_algebra. auto 6 with sets. Qed. + Theorem Empty_set_zero_right : forall X:Ensemble U, Union U X (Empty_set U) = X. + Proof. + auto 6 with sets. + Qed. + Theorem Empty_set_zero' : forall x:U, Add U (Empty_set U) x = Singleton U x. Proof. unfold Add at 1; auto using Empty_set_zero with sets. @@ -131,6 +138,17 @@ Section Sets_as_an_algebra. elim H'; intros x0 H'0; elim H'0; auto with sets. Qed. + Lemma Distributivity_l + : forall (A B C : Ensemble U), + Intersection U (Union U A B) C = + Union U (Intersection U A C) (Intersection U B C). + Proof. + intros A B C. + rewrite Intersection_commutative. + rewrite Distributivity. + f_equal; apply Intersection_commutative. + Qed. + Theorem Distributivity' : forall A B C:Ensemble U, Union U A (Intersection U B C) = @@ -251,6 +269,81 @@ Section Sets_as_an_algebra. intros; apply Definition_of_covers; auto with sets. Qed. + Lemma Disjoint_Intersection: + forall A s1 s2, Disjoint A s1 s2 -> Intersection A s1 s2 = Empty_set A. + Proof. + intros. apply Extensionality_Ensembles. split. + * destruct H. + intros x H1. unfold In in *. exfalso. intuition. apply (H _ H1). + * intuition. + Qed. + + Lemma Intersection_Empty_set_l: + forall A s, Intersection A (Empty_set A) s = Empty_set A. + Proof. + intros. auto with sets. + Qed. + + Lemma Intersection_Empty_set_r: + forall A s, Intersection A s (Empty_set A) = Empty_set A. + Proof. + intros. auto with sets. + Qed. + + Lemma Seminus_Empty_set_l: + forall A s, Setminus A (Empty_set A) s = Empty_set A. + Proof. + intros. apply Extensionality_Ensembles. split. + * intros x H1. destruct H1. unfold In in *. assumption. + * intuition. + Qed. + + Lemma Seminus_Empty_set_r: + forall A s, Setminus A s (Empty_set A) = s. + Proof. + intros. apply Extensionality_Ensembles. split. + * intros x H1. destruct H1. unfold In in *. assumption. + * intuition. + Qed. + + Lemma Setminus_Union_l: + forall A s1 s2 s3, + Setminus A (Union A s1 s2) s3 = Union A (Setminus A s1 s3) (Setminus A s2 s3). + Proof. + intros. apply Extensionality_Ensembles. split. + * intros x H. inversion H. inversion H0; intuition. + * intros x H. constructor; inversion H; inversion H0; intuition. + Qed. + + Lemma Setminus_Union_r: + forall A s1 s2 s3, + Setminus A s1 (Union A s2 s3) = Setminus A (Setminus A s1 s2) s3. + Proof. + intros. apply Extensionality_Ensembles. split. + * intros x H. inversion H. constructor. intuition. contradict H1. intuition. + * intros x H. inversion H. inversion H0. constructor; intuition. inversion H4; intuition. + Qed. + + Lemma Setminus_Disjoint_noop: + forall A s1 s2, + Intersection A s1 s2 = Empty_set A -> Setminus A s1 s2 = s1. + Proof. + intros. apply Extensionality_Ensembles. split. + * intros x H1. inversion_clear H1. intuition. + * intros x H1. constructor; intuition. contradict H. + apply Inhabited_not_empty. + exists x. intuition. + Qed. + + Lemma Setminus_Included_empty: + forall A s1 s2, + Included A s1 s2 -> Setminus A s1 s2 = Empty_set A. + Proof. + intros. apply Extensionality_Ensembles. split. + * intros x H1. inversion_clear H1. contradiction H2. intuition. + * intuition. + Qed. + End Sets_as_an_algebra. Hint Resolve Empty_set_zero Empty_set_zero' Union_associative Union_add diff --git a/theories/Sets/Relations_1.v b/theories/Sets/Relations_1.v index 45fb8134..1ed74599 100644 --- a/theories/Sets/Relations_1.v +++ b/theories/Sets/Relations_1.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (****************************************************************************) (* *) diff --git a/theories/Sets/Relations_1_facts.v b/theories/Sets/Relations_1_facts.v index bf8a4261..296ec42a 100644 --- a/theories/Sets/Relations_1_facts.v +++ b/theories/Sets/Relations_1_facts.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (****************************************************************************) (* *) diff --git a/theories/Sets/Relations_2.v b/theories/Sets/Relations_2.v index 1e0b83fe..cc839506 100644 --- a/theories/Sets/Relations_2.v +++ b/theories/Sets/Relations_2.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (****************************************************************************) (* *) diff --git a/theories/Sets/Relations_2_facts.v b/theories/Sets/Relations_2_facts.v index da93e922..36da3684 100644 --- a/theories/Sets/Relations_2_facts.v +++ b/theories/Sets/Relations_2_facts.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (****************************************************************************) (* *) diff --git a/theories/Sets/Relations_3.v b/theories/Sets/Relations_3.v index c05b5ee7..de3212e2 100644 --- a/theories/Sets/Relations_3.v +++ b/theories/Sets/Relations_3.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (****************************************************************************) (* *) diff --git a/theories/Sets/Relations_3_facts.v b/theories/Sets/Relations_3_facts.v index 89fb900c..0c1f670d 100644 --- a/theories/Sets/Relations_3_facts.v +++ b/theories/Sets/Relations_3_facts.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (****************************************************************************) (* *) diff --git a/theories/Sets/Uniset.v b/theories/Sets/Uniset.v index e297d97e..7940bda1 100644 --- a/theories/Sets/Uniset.v +++ b/theories/Sets/Uniset.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Sets as characteristic functions *) @@ -11,7 +13,7 @@ (* G. Huet 1-9-95 *) (* Updated Papageno 12/98 *) -Require Import Bool. +Require Import Bool Permut. Set Implicit Arguments. @@ -138,8 +140,6 @@ Hint Resolve seq_right. (** Here we should make uniset an abstract datatype, by hiding [Charac], [union], [charac]; all further properties are proved abstractly *) -Require Import Permut. - Lemma union_rotate : forall x y z:uniset, seq (union x (union y z)) (union z (union x y)). Proof. diff --git a/theories/Sets/vo.itarget b/theories/Sets/vo.itarget deleted file mode 100644 index 9ebe92f5..00000000 --- a/theories/Sets/vo.itarget +++ /dev/null @@ -1,22 +0,0 @@ -Classical_sets.vo -Constructive_sets.vo -Cpo.vo -Ensembles.vo -Finite_sets_facts.vo -Finite_sets.vo -Image.vo -Infinite_sets.vo -Integers.vo -Multiset.vo -Partial_Order.vo -Permut.vo -Powerset_Classical_facts.vo -Powerset_facts.vo -Powerset.vo -Relations_1_facts.vo -Relations_1.vo -Relations_2_facts.vo -Relations_2.vo -Relations_3_facts.vo -Relations_3.vo -Uniset.vo diff --git a/theories/Sorting/Heap.v b/theories/Sorting/Heap.v index 20c6feb9..2ef162be 100644 --- a/theories/Sorting/Heap.v +++ b/theories/Sorting/Heap.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** This file is deprecated, for a tree on list, use [Mergesort.v]. *) @@ -134,7 +136,7 @@ Section defs. (munion (list_contents _ eqA_dec l1) (list_contents _ eqA_dec l2)) -> (forall a, HdRel leA a l1 -> HdRel leA a l2 -> HdRel leA a l) -> merge_lem l1 l2. - Require Import Morphisms. + Import Morphisms. Instance: Equivalence (@meq A). Proof. constructor; auto with datatypes. red. apply meq_trans. Defined. @@ -146,10 +148,10 @@ Section defs. forall l1:list A, Sorted leA l1 -> forall l2:list A, Sorted leA l2 -> merge_lem l1 l2. Proof. - fix 1; intros; destruct l1. + fix merge 1; intros; destruct l1. apply merge_exist with l2; auto with datatypes. rename l1 into l. - revert l2 H0. fix 1. intros. + revert l2 H0. fix merge0 1. intros. destruct l2 as [|a0 l0]. apply merge_exist with (a :: l); simpl; auto with datatypes. induction (leA_dec a a0) as [Hle|Hle]. diff --git a/theories/Sorting/Mergesort.v b/theories/Sorting/Mergesort.v index 4b967c16..824d000d 100644 --- a/theories/Sorting/Mergesort.v +++ b/theories/Sorting/Mergesort.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** A modular implementation of mergesort (the complexity is O(n.log n) in diff --git a/theories/Sorting/PermutEq.v b/theories/Sorting/PermutEq.v index 45d27e35..97297f7e 100644 --- a/theories/Sorting/PermutEq.v +++ b/theories/Sorting/PermutEq.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Relations Setoid SetoidList List Multiset PermutSetoid Permutation Omega. diff --git a/theories/Sorting/PermutSetoid.v b/theories/Sorting/PermutSetoid.v index 0697a5e4..08bc400f 100644 --- a/theories/Sorting/PermutSetoid.v +++ b/theories/Sorting/PermutSetoid.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Omega Relations Multiset SetoidList. diff --git a/theories/Sorting/Permutation.v b/theories/Sorting/Permutation.v index 44f8ff6a..7b99b362 100644 --- a/theories/Sorting/Permutation.v +++ b/theories/Sorting/Permutation.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (*********************************************************************) diff --git a/theories/Sorting/Sorted.v b/theories/Sorting/Sorted.v index df03ff1c..89e9c7f3 100644 --- a/theories/Sorting/Sorted.v +++ b/theories/Sorting/Sorted.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Made by Hugo Herbelin *) diff --git a/theories/Sorting/Sorting.v b/theories/Sorting/Sorting.v index 6e9702f2..c2be1461 100644 --- a/theories/Sorting/Sorting.v +++ b/theories/Sorting/Sorting.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export Sorted. diff --git a/theories/Sorting/vo.itarget b/theories/Sorting/vo.itarget deleted file mode 100644 index 079eaad1..00000000 --- a/theories/Sorting/vo.itarget +++ /dev/null @@ -1,7 +0,0 @@ -Heap.vo -Permutation.vo -PermutSetoid.vo -PermutEq.vo -Sorted.vo -Sorting.vo -Mergesort.vo diff --git a/theories/Strings/Ascii.v b/theories/Strings/Ascii.v index 55a533c5..5154b75b 100644 --- a/theories/Strings/Ascii.v +++ b/theories/Strings/Ascii.v @@ -1,10 +1,12 @@ -(* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (************************************************************************) (** Contributed by Laurent Théry (INRIA); diff --git a/theories/Strings/String.v b/theories/Strings/String.v index 451b65cb..2be6618a 100644 --- a/theories/Strings/String.v +++ b/theories/Strings/String.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Contributed by Laurent Théry (INRIA); @@ -163,6 +165,18 @@ intros n0 H; apply Rec; simpl; auto. apply Le.le_S_n; auto. Qed. +(** *** Concatenating lists of strings *) + +(** [concat sep sl] concatenates the list of strings [sl], inserting + the separator string [sep] between each. *) + +Fixpoint concat (sep : string) (ls : list string) := + match ls with + | nil => EmptyString + | cons x nil => x + | cons x xs => x ++ sep ++ concat sep xs + end. + (** *** Test functions *) (** Test if [s1] is a prefix of [s2] *) diff --git a/theories/Strings/vo.itarget b/theories/Strings/vo.itarget deleted file mode 100644 index 20813b42..00000000 --- a/theories/Strings/vo.itarget +++ /dev/null @@ -1,2 +0,0 @@ -Ascii.vo -String.vo diff --git a/theories/Structures/DecidableType.v b/theories/Structures/DecidableType.v index f85222df..24333ad8 100644 --- a/theories/Structures/DecidableType.v +++ b/theories/Structures/DecidableType.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) Require Export SetoidList. Require Equalities. @@ -86,7 +88,7 @@ Module KeyDecidableType(D:DecidableType). Lemma InA_eqk : forall p q m, eqk p q -> InA eqk p m -> InA eqk q m. Proof. - intros; apply InA_eqA with p; auto with *. + intros; apply InA_eqA with p; auto using eqk_equiv. Qed. Definition MapsTo (k:key)(e:elt):= InA eqke (k,e). @@ -109,7 +111,7 @@ Module KeyDecidableType(D:DecidableType). Lemma MapsTo_eq : forall l x y e, eq x y -> MapsTo x e l -> MapsTo y e l. Proof. - intros; unfold MapsTo in *; apply InA_eqA with (x,e); eauto with *. + intros; unfold MapsTo in *; apply InA_eqA with (x,e); auto using eqke_equiv. Qed. Lemma In_eq : forall l x y, eq x y -> In x l -> In y l. diff --git a/theories/Structures/DecidableTypeEx.v b/theories/Structures/DecidableTypeEx.v index 163a40f2..8dd2e710 100644 --- a/theories/Structures/DecidableTypeEx.v +++ b/theories/Structures/DecidableTypeEx.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) Require Import DecidableType OrderedType OrderedTypeEx. Set Implicit Arguments. diff --git a/theories/Structures/Equalities.v b/theories/Structures/Equalities.v index 747d03f8..5f60a979 100644 --- a/theories/Structures/Equalities.v +++ b/theories/Structures/Equalities.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) Require Export RelationClasses. Require Import Bool Morphisms Setoid. diff --git a/theories/Structures/EqualitiesFacts.v b/theories/Structures/EqualitiesFacts.v index cee3d63f..7b6ee2ea 100644 --- a/theories/Structures/EqualitiesFacts.v +++ b/theories/Structures/EqualitiesFacts.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) Require Import Equalities Bool SetoidList RelationPairs. diff --git a/theories/Structures/GenericMinMax.v b/theories/Structures/GenericMinMax.v index ac52d1bb..05edc6cc 100644 --- a/theories/Structures/GenericMinMax.v +++ b/theories/Structures/GenericMinMax.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) Require Import Orders OrdersTac OrdersFacts Setoid Morphisms Basics. diff --git a/theories/Structures/OrderedType.v b/theories/Structures/OrderedType.v index 93ca383b..f6fc247d 100644 --- a/theories/Structures/OrderedType.v +++ b/theories/Structures/OrderedType.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) Require Export SetoidList Morphisms OrdersTac. Set Implicit Arguments. diff --git a/theories/Structures/OrderedTypeAlt.v b/theories/Structures/OrderedTypeAlt.v index b054496e..278046a8 100644 --- a/theories/Structures/OrderedTypeAlt.v +++ b/theories/Structures/OrderedTypeAlt.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) Require Import OrderedType. (** * An alternative (but equivalent) presentation for an Ordered Type diff --git a/theories/Structures/OrderedTypeEx.v b/theories/Structures/OrderedTypeEx.v index 3c6afc7b..f2a9a569 100644 --- a/theories/Structures/OrderedTypeEx.v +++ b/theories/Structures/OrderedTypeEx.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) Require Import OrderedType. Require Import ZArith. @@ -55,7 +57,7 @@ Module Nat_as_OT <: UsualOrderedType. Definition compare x y : Compare lt eq x y. Proof. - case_eq (nat_compare x y); intro. + case_eq (Nat.compare x y); intro. - apply EQ. now apply nat_compare_eq. - apply LT. now apply nat_compare_Lt_lt. - apply GT. now apply nat_compare_Gt_gt. diff --git a/theories/Structures/Orders.v b/theories/Structures/Orders.v index 724690b4..42756ad3 100644 --- a/theories/Structures/Orders.v +++ b/theories/Structures/Orders.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) Require Export Relations Morphisms Setoid Equalities. Set Implicit Arguments. diff --git a/theories/Structures/OrdersAlt.v b/theories/Structures/OrdersAlt.v index 5dd917a7..ad6a3876 100644 --- a/theories/Structures/OrdersAlt.v +++ b/theories/Structures/OrdersAlt.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (* Finite sets library. * Authors: Pierre Letouzey and Jean-Christophe Filliâtre diff --git a/theories/Structures/OrdersEx.v b/theories/Structures/OrdersEx.v index 89c56388..93168f7d 100644 --- a/theories/Structures/OrdersEx.v +++ b/theories/Structures/OrdersEx.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (* Finite sets library. * Authors: Pierre Letouzey and Jean-Christophe Filliâtre diff --git a/theories/Structures/OrdersFacts.v b/theories/Structures/OrdersFacts.v index 0115d8a5..87df6b47 100644 --- a/theories/Structures/OrdersFacts.v +++ b/theories/Structures/OrdersFacts.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) Require Import Bool Basics OrdersTac. Require Export Orders. diff --git a/theories/Structures/OrdersLists.v b/theories/Structures/OrdersLists.v index bf8529bc..abdb9eff 100644 --- a/theories/Structures/OrdersLists.v +++ b/theories/Structures/OrdersLists.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) Require Export RelationPairs SetoidList Orders EqualitiesFacts. diff --git a/theories/Structures/OrdersTac.v b/theories/Structures/OrdersTac.v index 475a25a4..ebd8ee8f 100644 --- a/theories/Structures/OrdersTac.v +++ b/theories/Structures/OrdersTac.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) Require Import Setoid Morphisms Basics Equalities Orders. Set Implicit Arguments. diff --git a/theories/Structures/vo.itarget b/theories/Structures/vo.itarget deleted file mode 100644 index 674e9fba..00000000 --- a/theories/Structures/vo.itarget +++ /dev/null @@ -1,14 +0,0 @@ -Equalities.vo -EqualitiesFacts.vo -Orders.vo -OrdersEx.vo -OrdersFacts.vo -OrdersLists.vo -OrdersTac.vo -OrdersAlt.vo -GenericMinMax.vo -DecidableType.vo -DecidableTypeEx.vo -OrderedTypeAlt.vo -OrderedTypeEx.vo -OrderedType.vo diff --git a/theories/Unicode/Utf8.v b/theories/Unicode/Utf8.v index d5c2fa73..6cc9ea09 100644 --- a/theories/Unicode/Utf8.v +++ b/theories/Unicode/Utf8.v @@ -1,10 +1,12 @@ (* -*- coding:utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export Utf8_core. diff --git a/theories/Unicode/Utf8_core.v b/theories/Unicode/Utf8_core.v index 44a0700a..5a8931a8 100644 --- a/theories/Unicode/Utf8_core.v +++ b/theories/Unicode/Utf8_core.v @@ -1,19 +1,23 @@ (* -*- coding:utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Logic *) -Notation "∀ x .. y , P" := (forall x, .. (forall y, P) ..) - (at level 200, x binder, y binder, right associativity) : type_scope. -Notation "∃ x .. y , P" := (exists x, .. (exists y, P) ..) - (at level 200, x binder, y binder, right associativity) : type_scope. +Notation "∀ x .. y , P" := (forall x, .. (forall y, P) ..) + (at level 200, x binder, y binder, right associativity, + format "'[ ' ∀ x .. y ']' , P") : type_scope. +Notation "∃ x .. y , P" := (exists x, .. (exists y, P) ..) + (at level 200, x binder, y binder, right associativity, + format "'[ ' ∃ x .. y ']' , P") : type_scope. Notation "x ∨ y" := (x \/ y) (at level 85, right associativity) : type_scope. Notation "x ∧ y" := (x /\ y) (at level 80, right associativity) : type_scope. @@ -25,5 +29,6 @@ Notation "¬ x" := (~x) (at level 75, right associativity) : type_scope. Notation "x ≠ y" := (x <> y) (at level 70) : type_scope. (* Abstraction *) -Notation "'λ' x .. y , t" := (fun x => .. (fun y => t) ..) - (at level 200, x binder, y binder, right associativity). +Notation "'λ' x .. y , t" := (fun x => .. (fun y => t) ..) + (at level 200, x binder, y binder, right associativity, + format "'[ ' 'λ' x .. y ']' , t"). diff --git a/theories/Unicode/vo.itarget b/theories/Unicode/vo.itarget deleted file mode 100644 index 7be1b996..00000000 --- a/theories/Unicode/vo.itarget +++ /dev/null @@ -1,2 +0,0 @@ -Utf8.vo -Utf8_core.vo diff --git a/theories/Vectors/Fin.v b/theories/Vectors/Fin.v index 2955184f..4088843a 100644 --- a/theories/Vectors/Fin.v +++ b/theories/Vectors/Fin.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Arith_base. diff --git a/theories/Vectors/Vector.v b/theories/Vectors/Vector.v index 672858fa..08158769 100644 --- a/theories/Vectors/Vector.v +++ b/theories/Vectors/Vector.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Vectors. @@ -21,4 +23,4 @@ Require VectorSpec. Require VectorEq. Include VectorDef. Include VectorSpec. -Include VectorEq.
\ No newline at end of file +Include VectorEq. diff --git a/theories/Vectors/VectorDef.v b/theories/Vectors/VectorDef.v index 1f8b76cb..f6f3cafa 100644 --- a/theories/Vectors/VectorDef.v +++ b/theories/Vectors/VectorDef.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Definitions of Vectors and functions to use them @@ -147,6 +149,16 @@ Definition shiftrepeat {A} := @rectS _ (fun n _ => t A (S (S n))) (fun h => h :: h :: []) (fun h _ _ H => h :: H). Global Arguments shiftrepeat {A} {n} v. +(** Take first [p] elements of a vector *) +Fixpoint take {A} {n} (p:nat) (le:p <= n) (v:t A n) : t A p := + match p as p return p <= n -> t A p with + | 0 => fun _ => [] + | S p' => match v in t _ n return S p' <= n -> t A (S p') with + | []=> fun le => False_rect _ (Nat.nle_succ_0 p' le) + | x::xs => fun le => x::take p' (le_S_n p' _ le) xs + end + end le. + (** Remove [p] last elements of a vector *) Lemma trunc : forall {A} {n} (p:nat), n > p -> t A n -> t A (n - p). @@ -295,12 +307,10 @@ End VECTORLIST. Module VectorNotations. Delimit Scope vector_scope with vector. Notation "[ ]" := [] (format "[ ]") : vector_scope. -Notation "[]" := [] (compat "8.5") : vector_scope. Notation "h :: t" := (h :: t) (at level 60, right associativity) : vector_scope. Notation "[ x ]" := (x :: []) : vector_scope. Notation "[ x ; y ; .. ; z ]" := (cons _ x _ (cons _ y _ .. (cons _ z _ (nil _)) ..)) : vector_scope. -Notation "[ x ; .. ; y ]" := (cons _ x _ .. (cons _ y _ (nil _)) ..) (compat "8.4") : vector_scope. Notation "v [@ p ]" := (nth v p) (at level 1, format "v [@ p ]") : vector_scope. Open Scope vector_scope. End VectorNotations. diff --git a/theories/Vectors/VectorEq.v b/theories/Vectors/VectorEq.v index 04c57073..317f3f1c 100644 --- a/theories/Vectors/VectorEq.v +++ b/theories/Vectors/VectorEq.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Equalities and Vector relations diff --git a/theories/Vectors/VectorSpec.v b/theories/Vectors/VectorSpec.v index c5278b91..34dbaf36 100644 --- a/theories/Vectors/VectorSpec.v +++ b/theories/Vectors/VectorSpec.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Proofs of specification for functions defined over Vector @@ -122,3 +124,32 @@ induction l. - reflexivity. - unfold to_list; simpl. now f_equal. Qed. + +Lemma take_O : forall {A} {n} le (v:t A n), take 0 le v = []. +Proof. + reflexivity. +Qed. + +Lemma take_idem : forall {A} p n (v:t A n) le le', + take p le' (take p le v) = take p le v. +Proof. + induction p; intros n v le le'. + - auto. + - destruct v. inversion le. simpl. apply f_equal. apply IHp. +Qed. + +Lemma take_app : forall {A} {n} (v:t A n) {m} (w:t A m) le, take n le (append v w) = v. +Proof. + induction v; intros m w le. + - reflexivity. + - simpl. apply f_equal. apply IHv. +Qed. + +(* Proof is irrelevant for [take] *) +Lemma take_prf_irr : forall {A} p {n} (v:t A n) le le', take p le v = take p le' v. +Proof. + induction p; intros n v le le'. + - reflexivity. + - destruct v. inversion le. simpl. apply f_equal. apply IHp. +Qed. + diff --git a/theories/Vectors/vo.itarget b/theories/Vectors/vo.itarget deleted file mode 100644 index 779b1821..00000000 --- a/theories/Vectors/vo.itarget +++ /dev/null @@ -1,5 +0,0 @@ -Fin.vo -VectorDef.vo -VectorSpec.vo -VectorEq.vo -Vector.vo diff --git a/theories/Wellfounded/Disjoint_Union.v b/theories/Wellfounded/Disjoint_Union.v index ebfc27b3..10d90274 100644 --- a/theories/Wellfounded/Disjoint_Union.v +++ b/theories/Wellfounded/Disjoint_Union.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Author: Cristina Cornes diff --git a/theories/Wellfounded/Inclusion.v b/theories/Wellfounded/Inclusion.v index 1ff9b005..ff233ef9 100644 --- a/theories/Wellfounded/Inclusion.v +++ b/theories/Wellfounded/Inclusion.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Author: Bruno Barras *) diff --git a/theories/Wellfounded/Inverse_Image.v b/theories/Wellfounded/Inverse_Image.v index 7786c8b3..4a4ab87d 100644 --- a/theories/Wellfounded/Inverse_Image.v +++ b/theories/Wellfounded/Inverse_Image.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Author: Bruno Barras *) diff --git a/theories/Wellfounded/Lexicographic_Exponentiation.v b/theories/Wellfounded/Lexicographic_Exponentiation.v index d90f9956..684efeeb 100644 --- a/theories/Wellfounded/Lexicographic_Exponentiation.v +++ b/theories/Wellfounded/Lexicographic_Exponentiation.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Author: Cristina Cornes diff --git a/theories/Wellfounded/Lexicographic_Product.v b/theories/Wellfounded/Lexicographic_Product.v index b2ada2f9..37fd2fb2 100644 --- a/theories/Wellfounded/Lexicographic_Product.v +++ b/theories/Wellfounded/Lexicographic_Product.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Authors: Bruno Barras, Cristina Cornes *) diff --git a/theories/Wellfounded/Transitive_Closure.v b/theories/Wellfounded/Transitive_Closure.v index eb12d5d7..59068623 100644 --- a/theories/Wellfounded/Transitive_Closure.v +++ b/theories/Wellfounded/Transitive_Closure.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Author: Bruno Barras *) diff --git a/theories/Wellfounded/Union.v b/theories/Wellfounded/Union.v index 61355c8d..9e671651 100644 --- a/theories/Wellfounded/Union.v +++ b/theories/Wellfounded/Union.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Author: Bruno Barras *) diff --git a/theories/Wellfounded/Well_Ordering.v b/theories/Wellfounded/Well_Ordering.v index b5acc287..fd363d02 100644 --- a/theories/Wellfounded/Well_Ordering.v +++ b/theories/Wellfounded/Well_Ordering.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Author: Cristina Cornes. diff --git a/theories/Wellfounded/Wellfounded.v b/theories/Wellfounded/Wellfounded.v index 397f35aa..bfe09e40 100644 --- a/theories/Wellfounded/Wellfounded.v +++ b/theories/Wellfounded/Wellfounded.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export Disjoint_Union. diff --git a/theories/Wellfounded/vo.itarget b/theories/Wellfounded/vo.itarget deleted file mode 100644 index 034d5310..00000000 --- a/theories/Wellfounded/vo.itarget +++ /dev/null @@ -1,9 +0,0 @@ -Disjoint_Union.vo -Inclusion.vo -Inverse_Image.vo -Lexicographic_Exponentiation.vo -Lexicographic_Product.vo -Transitive_Closure.vo -Union.vo -Wellfounded.vo -Well_Ordering.vo diff --git a/theories/ZArith/BinInt.v b/theories/ZArith/BinInt.v index 5aa397f8..cf7397b5 100644 --- a/theories/ZArith/BinInt.v +++ b/theories/ZArith/BinInt.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export BinNums BinPos Pnat. @@ -1367,7 +1369,7 @@ Lemma inj_testbit a n : 0<=n -> Z.testbit (Z.pos a) n = N.testbit (N.pos a) (Z.to_N n). Proof. apply Z.testbit_Zpos. Qed. -(** Some results concerning Z.neg *) +(** Some results concerning Z.neg and Z.pos *) Lemma inj_neg p q : Z.neg p = Z.neg q -> p = q. Proof. now injection 1. Qed. @@ -1375,18 +1377,54 @@ Proof. now injection 1. Qed. Lemma inj_neg_iff p q : Z.neg p = Z.neg q <-> p = q. Proof. split. apply inj_neg. intros; now f_equal. Qed. +Lemma inj_pos p q : Z.pos p = Z.pos q -> p = q. +Proof. now injection 1. Qed. + +Lemma inj_pos_iff p q : Z.pos p = Z.pos q <-> p = q. +Proof. split. apply inj_pos. intros; now f_equal. Qed. + Lemma neg_is_neg p : Z.neg p < 0. Proof. reflexivity. Qed. Lemma neg_is_nonpos p : Z.neg p <= 0. Proof. easy. Qed. +Lemma pos_is_pos p : 0 < Z.pos p. +Proof. reflexivity. Qed. + +Lemma pos_is_nonneg p : 0 <= Z.pos p. +Proof. easy. Qed. + +Lemma neg_le_pos p q : Zneg p <= Zpos q. +Proof. easy. Qed. + +Lemma neg_lt_pos p q : Zneg p < Zpos q. +Proof. easy. Qed. + +Lemma neg_le_neg p q : (q <= p)%positive -> Zneg p <= Zneg q. +Proof. intros; unfold Z.le; simpl. now rewrite <- Pos.compare_antisym. Qed. + +Lemma neg_lt_neg p q : (q < p)%positive -> Zneg p < Zneg q. +Proof. intros; unfold Z.lt; simpl. now rewrite <- Pos.compare_antisym. Qed. + +Lemma pos_le_pos p q : (p <= q)%positive -> Zpos p <= Zpos q. +Proof. easy. Qed. + +Lemma pos_lt_pos p q : (p < q)%positive -> Zpos p < Zpos q. +Proof. easy. Qed. + Lemma neg_xO p : Z.neg p~0 = 2 * Z.neg p. Proof. reflexivity. Qed. Lemma neg_xI p : Z.neg p~1 = 2 * Z.neg p - 1. Proof. reflexivity. Qed. +Lemma pos_xO p : Z.pos p~0 = 2 * Z.pos p. +Proof. reflexivity. Qed. + +Lemma pos_xI p : Z.pos p~1 = 2 * Z.pos p + 1. +Proof. reflexivity. Qed. + Lemma opp_neg p : - Z.neg p = Z.pos p. Proof. reflexivity. Qed. @@ -1402,6 +1440,9 @@ Proof. reflexivity. Qed. Lemma add_neg_pos p q : Z.neg p + Z.pos q = Z.pos_sub q p. Proof. reflexivity. Qed. +Lemma add_pos_pos p q : Z.pos p + Z.pos q = Z.pos (p+q). +Proof. reflexivity. Qed. + Lemma divide_pos_neg_r n p : (n|Z.pos p) <-> (n|Z.neg p). Proof. apply Z.divide_Zpos_Zneg_r. Qed. @@ -1412,6 +1453,10 @@ Lemma testbit_neg a n : 0<=n -> Z.testbit (Z.neg a) n = negb (N.testbit (Pos.pred_N a) (Z.to_N n)). Proof. apply Z.testbit_Zneg. Qed. +Lemma testbit_pos a n : 0<=n -> + Z.testbit (Z.pos a) n = N.testbit (N.pos a) (Z.to_N n). +Proof. apply Z.testbit_Zpos. Qed. + End Pos2Z. Module Z2Pos. @@ -1522,94 +1567,94 @@ End Z2Pos. (** Compatibility Notations *) -Notation Zdouble_plus_one := Z.succ_double (compat "8.3"). -Notation Zdouble_minus_one := Z.pred_double (compat "8.3"). -Notation Zdouble := Z.double (compat "8.3"). -Notation ZPminus := Z.pos_sub (compat "8.3"). -Notation Zsucc' := Z.succ (compat "8.3"). -Notation Zpred' := Z.pred (compat "8.3"). -Notation Zplus' := Z.add (compat "8.3"). -Notation Zplus := Z.add (compat "8.3"). (* Slightly incompatible *) -Notation Zopp := Z.opp (compat "8.3"). -Notation Zsucc := Z.succ (compat "8.3"). -Notation Zpred := Z.pred (compat "8.3"). -Notation Zminus := Z.sub (compat "8.3"). -Notation Zmult := Z.mul (compat "8.3"). -Notation Zcompare := Z.compare (compat "8.3"). -Notation Zsgn := Z.sgn (compat "8.3"). -Notation Zle := Z.le (compat "8.3"). -Notation Zge := Z.ge (compat "8.3"). -Notation Zlt := Z.lt (compat "8.3"). -Notation Zgt := Z.gt (compat "8.3"). -Notation Zmax := Z.max (compat "8.3"). -Notation Zmin := Z.min (compat "8.3"). -Notation Zabs := Z.abs (compat "8.3"). -Notation Zabs_nat := Z.abs_nat (compat "8.3"). -Notation Zabs_N := Z.abs_N (compat "8.3"). -Notation Z_of_nat := Z.of_nat (compat "8.3"). -Notation Z_of_N := Z.of_N (compat "8.3"). - -Notation Zind := Z.peano_ind (compat "8.3"). -Notation Zopp_0 := Z.opp_0 (compat "8.3"). -Notation Zopp_involutive := Z.opp_involutive (compat "8.3"). -Notation Zopp_inj := Z.opp_inj (compat "8.3"). -Notation Zplus_0_l := Z.add_0_l (compat "8.3"). -Notation Zplus_0_r := Z.add_0_r (compat "8.3"). -Notation Zplus_comm := Z.add_comm (compat "8.3"). -Notation Zopp_plus_distr := Z.opp_add_distr (compat "8.3"). -Notation Zopp_succ := Z.opp_succ (compat "8.3"). -Notation Zplus_opp_r := Z.add_opp_diag_r (compat "8.3"). -Notation Zplus_opp_l := Z.add_opp_diag_l (compat "8.3"). -Notation Zplus_assoc := Z.add_assoc (compat "8.3"). -Notation Zplus_permute := Z.add_shuffle3 (compat "8.3"). -Notation Zplus_reg_l := Z.add_reg_l (compat "8.3"). -Notation Zplus_succ_l := Z.add_succ_l (compat "8.3"). -Notation Zplus_succ_comm := Z.add_succ_comm (compat "8.3"). -Notation Zsucc_discr := Z.neq_succ_diag_r (compat "8.3"). -Notation Zsucc_inj := Z.succ_inj (compat "8.3"). -Notation Zsucc'_inj := Z.succ_inj (compat "8.3"). -Notation Zsucc'_pred' := Z.succ_pred (compat "8.3"). -Notation Zpred'_succ' := Z.pred_succ (compat "8.3"). -Notation Zpred'_inj := Z.pred_inj (compat "8.3"). -Notation Zsucc'_discr := Z.neq_succ_diag_r (compat "8.3"). -Notation Zminus_0_r := Z.sub_0_r (compat "8.3"). -Notation Zminus_diag := Z.sub_diag (compat "8.3"). -Notation Zminus_plus_distr := Z.sub_add_distr (compat "8.3"). -Notation Zminus_succ_r := Z.sub_succ_r (compat "8.3"). -Notation Zminus_plus := Z.add_simpl_l (compat "8.3"). -Notation Zmult_0_l := Z.mul_0_l (compat "8.3"). -Notation Zmult_0_r := Z.mul_0_r (compat "8.3"). -Notation Zmult_1_l := Z.mul_1_l (compat "8.3"). -Notation Zmult_1_r := Z.mul_1_r (compat "8.3"). -Notation Zmult_comm := Z.mul_comm (compat "8.3"). -Notation Zmult_assoc := Z.mul_assoc (compat "8.3"). -Notation Zmult_permute := Z.mul_shuffle3 (compat "8.3"). -Notation Zmult_1_inversion_l := Z.mul_eq_1 (compat "8.3"). -Notation Zdouble_mult := Z.double_spec (compat "8.3"). -Notation Zdouble_plus_one_mult := Z.succ_double_spec (compat "8.3"). -Notation Zopp_mult_distr_l_reverse := Z.mul_opp_l (compat "8.3"). -Notation Zmult_opp_opp := Z.mul_opp_opp (compat "8.3"). -Notation Zmult_opp_comm := Z.mul_opp_comm (compat "8.3"). -Notation Zopp_eq_mult_neg_1 := Z.opp_eq_mul_m1 (compat "8.3"). -Notation Zmult_plus_distr_r := Z.mul_add_distr_l (compat "8.3"). -Notation Zmult_plus_distr_l := Z.mul_add_distr_r (compat "8.3"). -Notation Zmult_minus_distr_r := Z.mul_sub_distr_r (compat "8.3"). -Notation Zmult_reg_l := Z.mul_reg_l (compat "8.3"). -Notation Zmult_reg_r := Z.mul_reg_r (compat "8.3"). -Notation Zmult_succ_l := Z.mul_succ_l (compat "8.3"). -Notation Zmult_succ_r := Z.mul_succ_r (compat "8.3"). - -Notation Zpos_xI := Pos2Z.inj_xI (compat "8.3"). -Notation Zpos_xO := Pos2Z.inj_xO (compat "8.3"). -Notation Zneg_xI := Pos2Z.neg_xI (compat "8.3"). -Notation Zneg_xO := Pos2Z.neg_xO (compat "8.3"). -Notation Zopp_neg := Pos2Z.opp_neg (compat "8.3"). -Notation Zpos_succ_morphism := Pos2Z.inj_succ (compat "8.3"). -Notation Zpos_mult_morphism := Pos2Z.inj_mul (compat "8.3"). -Notation Zpos_minus_morphism := Pos2Z.inj_sub (compat "8.3"). -Notation Zpos_eq_rev := Pos2Z.inj (compat "8.3"). -Notation Zpos_plus_distr := Pos2Z.inj_add (compat "8.3"). -Notation Zneg_plus_distr := Pos2Z.add_neg_neg (compat "8.3"). +Notation Zdouble_plus_one := Z.succ_double (only parsing). +Notation Zdouble_minus_one := Z.pred_double (only parsing). +Notation Zdouble := Z.double (compat "8.6"). +Notation ZPminus := Z.pos_sub (only parsing). +Notation Zsucc' := Z.succ (compat "8.6"). +Notation Zpred' := Z.pred (compat "8.6"). +Notation Zplus' := Z.add (compat "8.6"). +Notation Zplus := Z.add (only parsing). (* Slightly incompatible *) +Notation Zopp := Z.opp (compat "8.6"). +Notation Zsucc := Z.succ (compat "8.6"). +Notation Zpred := Z.pred (compat "8.6"). +Notation Zminus := Z.sub (only parsing). +Notation Zmult := Z.mul (only parsing). +Notation Zcompare := Z.compare (compat "8.6"). +Notation Zsgn := Z.sgn (compat "8.6"). +Notation Zle := Z.le (compat "8.6"). +Notation Zge := Z.ge (compat "8.6"). +Notation Zlt := Z.lt (compat "8.6"). +Notation Zgt := Z.gt (compat "8.6"). +Notation Zmax := Z.max (compat "8.6"). +Notation Zmin := Z.min (compat "8.6"). +Notation Zabs := Z.abs (compat "8.6"). +Notation Zabs_nat := Z.abs_nat (compat "8.6"). +Notation Zabs_N := Z.abs_N (compat "8.6"). +Notation Z_of_nat := Z.of_nat (only parsing). +Notation Z_of_N := Z.of_N (only parsing). + +Notation Zind := Z.peano_ind (only parsing). +Notation Zopp_0 := Z.opp_0 (compat "8.6"). +Notation Zopp_involutive := Z.opp_involutive (compat "8.6"). +Notation Zopp_inj := Z.opp_inj (compat "8.6"). +Notation Zplus_0_l := Z.add_0_l (only parsing). +Notation Zplus_0_r := Z.add_0_r (only parsing). +Notation Zplus_comm := Z.add_comm (only parsing). +Notation Zopp_plus_distr := Z.opp_add_distr (only parsing). +Notation Zopp_succ := Z.opp_succ (compat "8.6"). +Notation Zplus_opp_r := Z.add_opp_diag_r (only parsing). +Notation Zplus_opp_l := Z.add_opp_diag_l (only parsing). +Notation Zplus_assoc := Z.add_assoc (only parsing). +Notation Zplus_permute := Z.add_shuffle3 (only parsing). +Notation Zplus_reg_l := Z.add_reg_l (only parsing). +Notation Zplus_succ_l := Z.add_succ_l (only parsing). +Notation Zplus_succ_comm := Z.add_succ_comm (only parsing). +Notation Zsucc_discr := Z.neq_succ_diag_r (only parsing). +Notation Zsucc_inj := Z.succ_inj (compat "8.6"). +Notation Zsucc'_inj := Z.succ_inj (compat "8.6"). +Notation Zsucc'_pred' := Z.succ_pred (compat "8.6"). +Notation Zpred'_succ' := Z.pred_succ (compat "8.6"). +Notation Zpred'_inj := Z.pred_inj (compat "8.6"). +Notation Zsucc'_discr := Z.neq_succ_diag_r (only parsing). +Notation Zminus_0_r := Z.sub_0_r (only parsing). +Notation Zminus_diag := Z.sub_diag (only parsing). +Notation Zminus_plus_distr := Z.sub_add_distr (only parsing). +Notation Zminus_succ_r := Z.sub_succ_r (only parsing). +Notation Zminus_plus := Z.add_simpl_l (only parsing). +Notation Zmult_0_l := Z.mul_0_l (only parsing). +Notation Zmult_0_r := Z.mul_0_r (only parsing). +Notation Zmult_1_l := Z.mul_1_l (only parsing). +Notation Zmult_1_r := Z.mul_1_r (only parsing). +Notation Zmult_comm := Z.mul_comm (only parsing). +Notation Zmult_assoc := Z.mul_assoc (only parsing). +Notation Zmult_permute := Z.mul_shuffle3 (only parsing). +Notation Zmult_1_inversion_l := Z.mul_eq_1 (only parsing). +Notation Zdouble_mult := Z.double_spec (only parsing). +Notation Zdouble_plus_one_mult := Z.succ_double_spec (only parsing). +Notation Zopp_mult_distr_l_reverse := Z.mul_opp_l (only parsing). +Notation Zmult_opp_opp := Z.mul_opp_opp (only parsing). +Notation Zmult_opp_comm := Z.mul_opp_comm (only parsing). +Notation Zopp_eq_mult_neg_1 := Z.opp_eq_mul_m1 (only parsing). +Notation Zmult_plus_distr_r := Z.mul_add_distr_l (only parsing). +Notation Zmult_plus_distr_l := Z.mul_add_distr_r (only parsing). +Notation Zmult_minus_distr_r := Z.mul_sub_distr_r (only parsing). +Notation Zmult_reg_l := Z.mul_reg_l (only parsing). +Notation Zmult_reg_r := Z.mul_reg_r (only parsing). +Notation Zmult_succ_l := Z.mul_succ_l (only parsing). +Notation Zmult_succ_r := Z.mul_succ_r (only parsing). + +Notation Zpos_xI := Pos2Z.inj_xI (only parsing). +Notation Zpos_xO := Pos2Z.inj_xO (only parsing). +Notation Zneg_xI := Pos2Z.neg_xI (only parsing). +Notation Zneg_xO := Pos2Z.neg_xO (only parsing). +Notation Zopp_neg := Pos2Z.opp_neg (only parsing). +Notation Zpos_succ_morphism := Pos2Z.inj_succ (only parsing). +Notation Zpos_mult_morphism := Pos2Z.inj_mul (only parsing). +Notation Zpos_minus_morphism := Pos2Z.inj_sub (only parsing). +Notation Zpos_eq_rev := Pos2Z.inj (only parsing). +Notation Zpos_plus_distr := Pos2Z.inj_add (only parsing). +Notation Zneg_plus_distr := Pos2Z.add_neg_neg (only parsing). Notation Z := Z (only parsing). Notation Z_rect := Z_rect (only parsing). diff --git a/theories/ZArith/BinIntDef.v b/theories/ZArith/BinIntDef.v index 8c2e7d94..db4de0b9 100644 --- a/theories/ZArith/BinIntDef.v +++ b/theories/ZArith/BinIntDef.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export BinNums. @@ -299,6 +301,23 @@ Definition to_pos (z:Z) : positive := | _ => 1%positive end. +(** Conversion with a decimal representation for printing/parsing *) + +Definition of_uint (d:Decimal.uint) := of_N (Pos.of_uint d). + +Definition of_int (d:Decimal.int) := + match d with + | Decimal.Pos d => of_uint d + | Decimal.Neg d => opp (of_uint d) + end. + +Definition to_int n := + match n with + | 0 => Decimal.Pos Decimal.zero + | pos p => Decimal.Pos (Pos.to_uint p) + | neg p => Decimal.Neg (Pos.to_uint p) + end. + (** ** Iteration of a function By convention, iterating a negative number of times is identity. @@ -616,4 +635,4 @@ Definition lxor a b := | neg a, neg b => of_N (N.lxor (Pos.pred_N a) (Pos.pred_N b)) end. -End Z.
\ No newline at end of file +End Z. diff --git a/theories/ZArith/Int.v b/theories/ZArith/Int.v index 72021f2e..2f3bf9a3 100644 --- a/theories/ZArith/Int.v +++ b/theories/ZArith/Int.v @@ -1,10 +1,12 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) (** * An light axiomatization of integers (used in MSetAVL). *) @@ -452,7 +454,7 @@ Module Z_as_Int <: Int. Proof. reflexivity. Qed. (** Compatibility notations for Coq v8.4 *) - Notation plus := add (compat "8.4"). - Notation minus := sub (compat "8.4"). - Notation mult := mul (compat "8.4"). + Notation plus := add (only parsing). + Notation minus := sub (only parsing). + Notation mult := mul (only parsing). End Z_as_Int. diff --git a/theories/ZArith/Wf_Z.v b/theories/ZArith/Wf_Z.v index 4fbbac26..86434208 100644 --- a/theories/ZArith/Wf_Z.v +++ b/theories/ZArith/Wf_Z.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import BinInt. diff --git a/theories/ZArith/ZArith.v b/theories/ZArith/ZArith.v index f86a7e52..0842920b 100644 --- a/theories/ZArith/ZArith.v +++ b/theories/ZArith/ZArith.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Library for manipulating integers based on binary encoding *) diff --git a/theories/ZArith/ZArith_base.v b/theories/ZArith/ZArith_base.v index a1da544d..3d6bcddc 100644 --- a/theories/ZArith/ZArith_base.v +++ b/theories/ZArith/ZArith_base.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Library for manipulating integers based on binary encoding. diff --git a/theories/ZArith/ZArith_dec.v b/theories/ZArith/ZArith_dec.v index 8947295e..9bcdb73a 100644 --- a/theories/ZArith/ZArith_dec.v +++ b/theories/ZArith/ZArith_dec.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Sumbool. @@ -32,7 +34,7 @@ Lemma Zcompare_rec (P:Set) (n m:Z) : ((n ?= m) = Eq -> P) -> ((n ?= m) = Lt -> P) -> ((n ?= m) = Gt -> P) -> P. Proof. apply Zcompare_rect. Defined. -Notation Z_eq_dec := Z.eq_dec (compat "8.3"). +Notation Z_eq_dec := Z.eq_dec (compat "8.6"). Section decidability. diff --git a/theories/ZArith/Zabs.v b/theories/ZArith/Zabs.v index df9db5b2..0d8450e3 100644 --- a/theories/ZArith/Zabs.v +++ b/theories/ZArith/Zabs.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Binary Integers : properties of absolute value *) @@ -27,17 +29,17 @@ Local Open Scope Z_scope. (**********************************************************************) (** * Properties of absolute value *) -Notation Zabs_eq := Z.abs_eq (compat "8.3"). -Notation Zabs_non_eq := Z.abs_neq (compat "8.3"). -Notation Zabs_Zopp := Z.abs_opp (compat "8.3"). -Notation Zabs_pos := Z.abs_nonneg (compat "8.3"). -Notation Zabs_involutive := Z.abs_involutive (compat "8.3"). -Notation Zabs_eq_case := Z.abs_eq_cases (compat "8.3"). -Notation Zabs_triangle := Z.abs_triangle (compat "8.3"). -Notation Zsgn_Zabs := Z.sgn_abs (compat "8.3"). -Notation Zabs_Zsgn := Z.abs_sgn (compat "8.3"). -Notation Zabs_Zmult := Z.abs_mul (compat "8.3"). -Notation Zabs_square := Z.abs_square (compat "8.3"). +Notation Zabs_eq := Z.abs_eq (compat "8.6"). +Notation Zabs_non_eq := Z.abs_neq (only parsing). +Notation Zabs_Zopp := Z.abs_opp (only parsing). +Notation Zabs_pos := Z.abs_nonneg (only parsing). +Notation Zabs_involutive := Z.abs_involutive (compat "8.6"). +Notation Zabs_eq_case := Z.abs_eq_cases (only parsing). +Notation Zabs_triangle := Z.abs_triangle (compat "8.6"). +Notation Zsgn_Zabs := Z.sgn_abs (only parsing). +Notation Zabs_Zsgn := Z.abs_sgn (only parsing). +Notation Zabs_Zmult := Z.abs_mul (only parsing). +Notation Zabs_square := Z.abs_square (compat "8.6"). (** * Proving a property of the absolute value by cases *) @@ -68,11 +70,11 @@ Qed. (** * Some results about the sign function. *) -Notation Zsgn_Zmult := Z.sgn_mul (compat "8.3"). -Notation Zsgn_Zopp := Z.sgn_opp (compat "8.3"). -Notation Zsgn_pos := Z.sgn_pos_iff (compat "8.3"). -Notation Zsgn_neg := Z.sgn_neg_iff (compat "8.3"). -Notation Zsgn_null := Z.sgn_null_iff (compat "8.3"). +Notation Zsgn_Zmult := Z.sgn_mul (only parsing). +Notation Zsgn_Zopp := Z.sgn_opp (only parsing). +Notation Zsgn_pos := Z.sgn_pos_iff (only parsing). +Notation Zsgn_neg := Z.sgn_neg_iff (only parsing). +Notation Zsgn_null := Z.sgn_null_iff (only parsing). (** A characterization of the sign function: *) @@ -86,13 +88,13 @@ Qed. (** Compatibility *) -Notation inj_Zabs_nat := Zabs2Nat.id_abs (compat "8.3"). -Notation Zabs_nat_Z_of_nat := Zabs2Nat.id (compat "8.3"). -Notation Zabs_nat_mult := Zabs2Nat.inj_mul (compat "8.3"). -Notation Zabs_nat_Zsucc := Zabs2Nat.inj_succ (compat "8.3"). -Notation Zabs_nat_Zplus := Zabs2Nat.inj_add (compat "8.3"). -Notation Zabs_nat_Zminus := (fun n m => Zabs2Nat.inj_sub m n) (compat "8.3"). -Notation Zabs_nat_compare := Zabs2Nat.inj_compare (compat "8.3"). +Notation inj_Zabs_nat := Zabs2Nat.id_abs (only parsing). +Notation Zabs_nat_Z_of_nat := Zabs2Nat.id (only parsing). +Notation Zabs_nat_mult := Zabs2Nat.inj_mul (only parsing). +Notation Zabs_nat_Zsucc := Zabs2Nat.inj_succ (only parsing). +Notation Zabs_nat_Zplus := Zabs2Nat.inj_add (only parsing). +Notation Zabs_nat_Zminus := (fun n m => Zabs2Nat.inj_sub m n) (only parsing). +Notation Zabs_nat_compare := Zabs2Nat.inj_compare (only parsing). Lemma Zabs_nat_le n m : 0 <= n <= m -> (Z.abs_nat n <= Z.abs_nat m)%nat. Proof. diff --git a/theories/ZArith/Zbool.v b/theories/ZArith/Zbool.v index 41d1b2b5..632d41b6 100644 --- a/theories/ZArith/Zbool.v +++ b/theories/ZArith/Zbool.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import BinInt. @@ -33,10 +35,10 @@ Definition Zeven_odd_bool (x:Z) := bool_of_sumbool (Zeven_odd_dec x). (**********************************************************************) (** * Boolean comparisons of binary integers *) -Notation Zle_bool := Z.leb (compat "8.3"). -Notation Zge_bool := Z.geb (compat "8.3"). -Notation Zlt_bool := Z.ltb (compat "8.3"). -Notation Zgt_bool := Z.gtb (compat "8.3"). +Notation Zle_bool := Z.leb (only parsing). +Notation Zge_bool := Z.geb (only parsing). +Notation Zlt_bool := Z.ltb (only parsing). +Notation Zgt_bool := Z.gtb (only parsing). (** We now provide a direct [Z.eqb] that doesn't refer to [Z.compare]. The old [Zeq_bool] is kept for compatibility. *) @@ -87,7 +89,7 @@ Proof. apply Z.leb_le. Qed. -Notation Zle_bool_refl := Z.leb_refl (compat "8.3"). +Notation Zle_bool_refl := Z.leb_refl (only parsing). Lemma Zle_bool_antisym n m : (n <=? m) = true -> (m <=? n) = true -> n = m. diff --git a/theories/ZArith/Zcompare.v b/theories/ZArith/Zcompare.v index 2627f174..c8432e27 100644 --- a/theories/ZArith/Zcompare.v +++ b/theories/ZArith/Zcompare.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Binary Integers : results about Z.compare *) @@ -181,18 +183,18 @@ Qed. (** Compatibility notations *) -Notation Zcompare_refl := Z.compare_refl (compat "8.3"). -Notation Zcompare_Eq_eq := Z.compare_eq (compat "8.3"). -Notation Zcompare_Eq_iff_eq := Z.compare_eq_iff (compat "8.3"). -Notation Zcompare_spec := Z.compare_spec (compat "8.3"). -Notation Zmin_l := Z.min_l (compat "8.3"). -Notation Zmin_r := Z.min_r (compat "8.3"). -Notation Zmax_l := Z.max_l (compat "8.3"). -Notation Zmax_r := Z.max_r (compat "8.3"). -Notation Zabs_eq := Z.abs_eq (compat "8.3"). -Notation Zabs_non_eq := Z.abs_neq (compat "8.3"). -Notation Zsgn_0 := Z.sgn_null (compat "8.3"). -Notation Zsgn_1 := Z.sgn_pos (compat "8.3"). -Notation Zsgn_m1 := Z.sgn_neg (compat "8.3"). +Notation Zcompare_refl := Z.compare_refl (compat "8.6"). +Notation Zcompare_Eq_eq := Z.compare_eq (only parsing). +Notation Zcompare_Eq_iff_eq := Z.compare_eq_iff (only parsing). +Notation Zcompare_spec := Z.compare_spec (compat "8.6"). +Notation Zmin_l := Z.min_l (compat "8.6"). +Notation Zmin_r := Z.min_r (compat "8.6"). +Notation Zmax_l := Z.max_l (compat "8.6"). +Notation Zmax_r := Z.max_r (compat "8.6"). +Notation Zabs_eq := Z.abs_eq (compat "8.6"). +Notation Zabs_non_eq := Z.abs_neq (only parsing). +Notation Zsgn_0 := Z.sgn_null (only parsing). +Notation Zsgn_1 := Z.sgn_pos (only parsing). +Notation Zsgn_m1 := Z.sgn_neg (only parsing). (** Not kept: Zcompare_egal_dec *) diff --git a/theories/ZArith/Zcomplements.v b/theories/ZArith/Zcomplements.v index 5bdb32fc..adf72a6a 100644 --- a/theories/ZArith/Zcomplements.v +++ b/theories/ZArith/Zcomplements.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import ZArithRing. diff --git a/theories/ZArith/Zdigits.v b/theories/ZArith/Zdigits.v index bc3694bc..d1eef613 100644 --- a/theories/ZArith/Zdigits.v +++ b/theories/ZArith/Zdigits.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Bit vectors interpreted as integers. diff --git a/theories/ZArith/Zdiv.v b/theories/ZArith/Zdiv.v index 2ba865bd..15d0e487 100644 --- a/theories/ZArith/Zdiv.v +++ b/theories/ZArith/Zdiv.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Euclidean Division *) @@ -18,16 +20,16 @@ Local Open Scope Z_scope. (** The definition of the division is now in [BinIntDef], the initial specifications and properties are in [BinInt]. *) -Notation Zdiv_eucl_POS := Z.pos_div_eucl (compat "8.3"). -Notation Zdiv_eucl := Z.div_eucl (compat "8.3"). -Notation Zdiv := Z.div (compat "8.3"). -Notation Zmod := Z.modulo (compat "8.3"). +Notation Zdiv_eucl_POS := Z.pos_div_eucl (only parsing). +Notation Zdiv_eucl := Z.div_eucl (compat "8.6"). +Notation Zdiv := Z.div (compat "8.6"). +Notation Zmod := Z.modulo (only parsing). -Notation Zdiv_eucl_eq := Z.div_eucl_eq (compat "8.3"). -Notation Z_div_mod_eq_full := Z.div_mod (compat "8.3"). -Notation Zmod_POS_bound := Z.pos_div_eucl_bound (compat "8.3"). -Notation Zmod_pos_bound := Z.mod_pos_bound (compat "8.3"). -Notation Zmod_neg_bound := Z.mod_neg_bound (compat "8.3"). +Notation Zdiv_eucl_eq := Z.div_eucl_eq (compat "8.6"). +Notation Z_div_mod_eq_full := Z.div_mod (only parsing). +Notation Zmod_POS_bound := Z.pos_div_eucl_bound (only parsing). +Notation Zmod_pos_bound := Z.mod_pos_bound (only parsing). +Notation Zmod_neg_bound := Z.mod_neg_bound (only parsing). (** * Main division theorems *) @@ -508,6 +510,13 @@ Qed. (** Unfortunately, the previous result isn't always true on negative numbers. For instance: 3/(-2)/(-2) = 1 <> 0 = 3 / (-2*-2) *) +Lemma Zmod_div : forall a b, a mod b / b = 0. +Proof. + intros a b. + zero_or_not b. + auto using Z.mod_div. +Qed. + (** A last inequality: *) Theorem Zdiv_mult_le: diff --git a/theories/ZArith/Zeuclid.v b/theories/ZArith/Zeuclid.v index 38a824cd..dc75db13 100644 --- a/theories/ZArith/Zeuclid.v +++ b/theories/ZArith/Zeuclid.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Morphisms BinInt ZDivEucl. diff --git a/theories/ZArith/Zeven.v b/theories/ZArith/Zeven.v index d4051ef7..00a58b51 100644 --- a/theories/ZArith/Zeven.v +++ b/theories/ZArith/Zeven.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Binary Integers : Parity and Division by Two *) @@ -58,8 +60,8 @@ Proof (Zodd_equiv n). (** Boolean tests of parity (now in BinInt.Z) *) -Notation Zeven_bool := Z.even (compat "8.3"). -Notation Zodd_bool := Z.odd (compat "8.3"). +Notation Zeven_bool := Z.even (only parsing). +Notation Zodd_bool := Z.odd (only parsing). Lemma Zeven_bool_iff n : Z.even n = true <-> Zeven n. Proof. @@ -130,17 +132,17 @@ Qed. Hint Unfold Zeven Zodd: zarith. -Notation Zeven_bool_succ := Z.even_succ (compat "8.3"). -Notation Zeven_bool_pred := Z.even_pred (compat "8.3"). -Notation Zodd_bool_succ := Z.odd_succ (compat "8.3"). -Notation Zodd_bool_pred := Z.odd_pred (compat "8.3"). +Notation Zeven_bool_succ := Z.even_succ (only parsing). +Notation Zeven_bool_pred := Z.even_pred (only parsing). +Notation Zodd_bool_succ := Z.odd_succ (only parsing). +Notation Zodd_bool_pred := Z.odd_pred (only parsing). (******************************************************************) (** * Definition of [Z.quot2], [Z.div2] and properties wrt [Zeven] and [Zodd] *) -Notation Zdiv2 := Z.div2 (compat "8.3"). -Notation Zquot2 := Z.quot2 (compat "8.3"). +Notation Zdiv2 := Z.div2 (compat "8.6"). +Notation Zquot2 := Z.quot2 (compat "8.6"). (** Properties of [Z.div2] *) diff --git a/theories/ZArith/Zgcd_alt.v b/theories/ZArith/Zgcd_alt.v index 3977ca9d..cced1190 100644 --- a/theories/ZArith/Zgcd_alt.v +++ b/theories/ZArith/Zgcd_alt.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Zgcd_alt : an alternate version of Z.gcd, based on Euclid's algorithm *) diff --git a/theories/ZArith/Zhints.v b/theories/ZArith/Zhints.v index c4e201e3..bfcc60ed 100644 --- a/theories/ZArith/Zhints.v +++ b/theories/ZArith/Zhints.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** This file centralizes the lemmas about [Z], classifying them diff --git a/theories/ZArith/Zlogarithm.v b/theories/ZArith/Zlogarithm.v index fdfd71e1..24412e94 100644 --- a/theories/ZArith/Zlogarithm.v +++ b/theories/ZArith/Zlogarithm.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (**********************************************************************) diff --git a/theories/ZArith/Zmax.v b/theories/ZArith/Zmax.v index 529e9f1d..7f595fcf 100644 --- a/theories/ZArith/Zmax.v +++ b/theories/ZArith/Zmax.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** THIS FILE IS DEPRECATED. *) @@ -16,32 +18,32 @@ Local Open Scope Z_scope. (** Exact compatibility *) -Notation Zmax_case := Z.max_case (compat "8.3"). -Notation Zmax_case_strong := Z.max_case_strong (compat "8.3"). -Notation Zmax_right := Z.max_r (compat "8.3"). -Notation Zle_max_l := Z.le_max_l (compat "8.3"). -Notation Zle_max_r := Z.le_max_r (compat "8.3"). -Notation Zmax_lub := Z.max_lub (compat "8.3"). -Notation Zmax_lub_lt := Z.max_lub_lt (compat "8.3"). -Notation Zle_max_compat_r := Z.max_le_compat_r (compat "8.3"). -Notation Zle_max_compat_l := Z.max_le_compat_l (compat "8.3"). -Notation Zmax_idempotent := Z.max_id (compat "8.3"). -Notation Zmax_n_n := Z.max_id (compat "8.3"). -Notation Zmax_comm := Z.max_comm (compat "8.3"). -Notation Zmax_assoc := Z.max_assoc (compat "8.3"). -Notation Zmax_irreducible_dec := Z.max_dec (compat "8.3"). -Notation Zmax_le_prime := Z.max_le (compat "8.3"). -Notation Zsucc_max_distr := Z.succ_max_distr (compat "8.3"). -Notation Zmax_SS := Z.succ_max_distr (compat "8.3"). -Notation Zplus_max_distr_l := Z.add_max_distr_l (compat "8.3"). -Notation Zplus_max_distr_r := Z.add_max_distr_r (compat "8.3"). -Notation Zmax_plus := Z.add_max_distr_r (compat "8.3"). -Notation Zmax1 := Z.le_max_l (compat "8.3"). -Notation Zmax2 := Z.le_max_r (compat "8.3"). -Notation Zmax_irreducible_inf := Z.max_dec (compat "8.3"). -Notation Zmax_le_prime_inf := Z.max_le (compat "8.3"). -Notation Zpos_max := Pos2Z.inj_max (compat "8.3"). -Notation Zpos_minus := Pos2Z.inj_sub_max (compat "8.3"). +Notation Zmax_case := Z.max_case (compat "8.6"). +Notation Zmax_case_strong := Z.max_case_strong (compat "8.6"). +Notation Zmax_right := Z.max_r (only parsing). +Notation Zle_max_l := Z.le_max_l (compat "8.6"). +Notation Zle_max_r := Z.le_max_r (compat "8.6"). +Notation Zmax_lub := Z.max_lub (compat "8.6"). +Notation Zmax_lub_lt := Z.max_lub_lt (compat "8.6"). +Notation Zle_max_compat_r := Z.max_le_compat_r (only parsing). +Notation Zle_max_compat_l := Z.max_le_compat_l (only parsing). +Notation Zmax_idempotent := Z.max_id (only parsing). +Notation Zmax_n_n := Z.max_id (only parsing). +Notation Zmax_comm := Z.max_comm (compat "8.6"). +Notation Zmax_assoc := Z.max_assoc (compat "8.6"). +Notation Zmax_irreducible_dec := Z.max_dec (only parsing). +Notation Zmax_le_prime := Z.max_le (only parsing). +Notation Zsucc_max_distr := Z.succ_max_distr (compat "8.6"). +Notation Zmax_SS := Z.succ_max_distr (only parsing). +Notation Zplus_max_distr_l := Z.add_max_distr_l (only parsing). +Notation Zplus_max_distr_r := Z.add_max_distr_r (only parsing). +Notation Zmax_plus := Z.add_max_distr_r (only parsing). +Notation Zmax1 := Z.le_max_l (only parsing). +Notation Zmax2 := Z.le_max_r (only parsing). +Notation Zmax_irreducible_inf := Z.max_dec (only parsing). +Notation Zmax_le_prime_inf := Z.max_le (only parsing). +Notation Zpos_max := Pos2Z.inj_max (only parsing). +Notation Zpos_minus := Pos2Z.inj_sub_max (only parsing). (** Slightly different lemmas *) diff --git a/theories/ZArith/Zmin.v b/theories/ZArith/Zmin.v index 782a5158..6bc72227 100644 --- a/theories/ZArith/Zmin.v +++ b/theories/ZArith/Zmin.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** THIS FILE IS DEPRECATED. *) @@ -16,24 +18,24 @@ Local Open Scope Z_scope. (** Exact compatibility *) -Notation Zmin_case := Z.min_case (compat "8.3"). -Notation Zmin_case_strong := Z.min_case_strong (compat "8.3"). -Notation Zle_min_l := Z.le_min_l (compat "8.3"). -Notation Zle_min_r := Z.le_min_r (compat "8.3"). -Notation Zmin_glb := Z.min_glb (compat "8.3"). -Notation Zmin_glb_lt := Z.min_glb_lt (compat "8.3"). -Notation Zle_min_compat_r := Z.min_le_compat_r (compat "8.3"). -Notation Zle_min_compat_l := Z.min_le_compat_l (compat "8.3"). -Notation Zmin_idempotent := Z.min_id (compat "8.3"). -Notation Zmin_n_n := Z.min_id (compat "8.3"). -Notation Zmin_comm := Z.min_comm (compat "8.3"). -Notation Zmin_assoc := Z.min_assoc (compat "8.3"). -Notation Zmin_irreducible_inf := Z.min_dec (compat "8.3"). -Notation Zsucc_min_distr := Z.succ_min_distr (compat "8.3"). -Notation Zmin_SS := Z.succ_min_distr (compat "8.3"). -Notation Zplus_min_distr_r := Z.add_min_distr_r (compat "8.3"). -Notation Zmin_plus := Z.add_min_distr_r (compat "8.3"). -Notation Zpos_min := Pos2Z.inj_min (compat "8.3"). +Notation Zmin_case := Z.min_case (compat "8.6"). +Notation Zmin_case_strong := Z.min_case_strong (compat "8.6"). +Notation Zle_min_l := Z.le_min_l (compat "8.6"). +Notation Zle_min_r := Z.le_min_r (compat "8.6"). +Notation Zmin_glb := Z.min_glb (compat "8.6"). +Notation Zmin_glb_lt := Z.min_glb_lt (compat "8.6"). +Notation Zle_min_compat_r := Z.min_le_compat_r (only parsing). +Notation Zle_min_compat_l := Z.min_le_compat_l (only parsing). +Notation Zmin_idempotent := Z.min_id (only parsing). +Notation Zmin_n_n := Z.min_id (only parsing). +Notation Zmin_comm := Z.min_comm (compat "8.6"). +Notation Zmin_assoc := Z.min_assoc (compat "8.6"). +Notation Zmin_irreducible_inf := Z.min_dec (only parsing). +Notation Zsucc_min_distr := Z.succ_min_distr (compat "8.6"). +Notation Zmin_SS := Z.succ_min_distr (only parsing). +Notation Zplus_min_distr_r := Z.add_min_distr_r (only parsing). +Notation Zmin_plus := Z.add_min_distr_r (only parsing). +Notation Zpos_min := Pos2Z.inj_min (only parsing). (** Slightly different lemmas *) @@ -46,7 +48,7 @@ Qed. Lemma Zmin_irreducible n m : Z.min n m = n \/ Z.min n m = m. Proof. destruct (Z.min_dec n m); auto. Qed. -Notation Zmin_or := Zmin_irreducible (compat "8.3"). +Notation Zmin_or := Zmin_irreducible (only parsing). Lemma Zmin_le_prime_inf n m p : Z.min n m <= p -> {n <= p} + {m <= p}. Proof. apply Z.min_case; auto. Qed. diff --git a/theories/ZArith/Zminmax.v b/theories/ZArith/Zminmax.v index ea9a5f86..06919642 100644 --- a/theories/ZArith/Zminmax.v +++ b/theories/ZArith/Zminmax.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Orders BinInt Zcompare Zorder. @@ -12,11 +14,11 @@ Require Import Orders BinInt Zcompare Zorder. (*begin hide*) (* Compatibility with names of the old Zminmax file *) -Notation Zmin_max_absorption_r_r := Z.min_max_absorption (compat "8.3"). -Notation Zmax_min_absorption_r_r := Z.max_min_absorption (compat "8.3"). -Notation Zmax_min_distr_r := Z.max_min_distr (compat "8.3"). -Notation Zmin_max_distr_r := Z.min_max_distr (compat "8.3"). -Notation Zmax_min_modular_r := Z.max_min_modular (compat "8.3"). -Notation Zmin_max_modular_r := Z.min_max_modular (compat "8.3"). -Notation max_min_disassoc := Z.max_min_disassoc (compat "8.3"). +Notation Zmin_max_absorption_r_r := Z.min_max_absorption (only parsing). +Notation Zmax_min_absorption_r_r := Z.max_min_absorption (only parsing). +Notation Zmax_min_distr_r := Z.max_min_distr (only parsing). +Notation Zmin_max_distr_r := Z.min_max_distr (only parsing). +Notation Zmax_min_modular_r := Z.max_min_modular (only parsing). +Notation Zmin_max_modular_r := Z.min_max_modular (only parsing). +Notation max_min_disassoc := Z.max_min_disassoc (only parsing). (*end hide*) diff --git a/theories/ZArith/Zmisc.v b/theories/ZArith/Zmisc.v index 65831c78..c46241ce 100644 --- a/theories/ZArith/Zmisc.v +++ b/theories/ZArith/Zmisc.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Wf_nat. @@ -18,7 +20,7 @@ Local Open Scope Z_scope. (** [n]th iteration of the function [f] *) -Notation iter := @Z.iter (compat "8.3"). +Notation iter := @Z.iter (only parsing). Lemma iter_nat_of_Z : forall n A f x, 0 <= n -> Z.iter n f x = iter_nat (Z.abs_nat n) A f x. diff --git a/theories/ZArith/Znat.v b/theories/ZArith/Znat.v index 06428a7c..5c960da1 100644 --- a/theories/ZArith/Znat.v +++ b/theories/ZArith/Znat.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Binary Integers (Pierre Crégut, CNET, Lannion, France) *) @@ -951,65 +953,65 @@ Definition inj_gt n m := proj1 (Nat2Z.inj_gt n m). (** For the others, a Notation is fine *) -Notation inj_0 := Nat2Z.inj_0 (compat "8.3"). -Notation inj_S := Nat2Z.inj_succ (compat "8.3"). -Notation inj_compare := Nat2Z.inj_compare (compat "8.3"). -Notation inj_eq_rev := Nat2Z.inj (compat "8.3"). -Notation inj_eq_iff := (fun n m => iff_sym (Nat2Z.inj_iff n m)) (compat "8.3"). -Notation inj_le_iff := Nat2Z.inj_le (compat "8.3"). -Notation inj_lt_iff := Nat2Z.inj_lt (compat "8.3"). -Notation inj_ge_iff := Nat2Z.inj_ge (compat "8.3"). -Notation inj_gt_iff := Nat2Z.inj_gt (compat "8.3"). -Notation inj_le_rev := (fun n m => proj2 (Nat2Z.inj_le n m)) (compat "8.3"). -Notation inj_lt_rev := (fun n m => proj2 (Nat2Z.inj_lt n m)) (compat "8.3"). -Notation inj_ge_rev := (fun n m => proj2 (Nat2Z.inj_ge n m)) (compat "8.3"). -Notation inj_gt_rev := (fun n m => proj2 (Nat2Z.inj_gt n m)) (compat "8.3"). -Notation inj_plus := Nat2Z.inj_add (compat "8.3"). -Notation inj_mult := Nat2Z.inj_mul (compat "8.3"). -Notation inj_minus1 := Nat2Z.inj_sub (compat "8.3"). -Notation inj_minus := Nat2Z.inj_sub_max (compat "8.3"). -Notation inj_min := Nat2Z.inj_min (compat "8.3"). -Notation inj_max := Nat2Z.inj_max (compat "8.3"). - -Notation Z_of_nat_of_P := positive_nat_Z (compat "8.3"). +Notation inj_0 := Nat2Z.inj_0 (only parsing). +Notation inj_S := Nat2Z.inj_succ (only parsing). +Notation inj_compare := Nat2Z.inj_compare (only parsing). +Notation inj_eq_rev := Nat2Z.inj (only parsing). +Notation inj_eq_iff := (fun n m => iff_sym (Nat2Z.inj_iff n m)) (only parsing). +Notation inj_le_iff := Nat2Z.inj_le (only parsing). +Notation inj_lt_iff := Nat2Z.inj_lt (only parsing). +Notation inj_ge_iff := Nat2Z.inj_ge (only parsing). +Notation inj_gt_iff := Nat2Z.inj_gt (only parsing). +Notation inj_le_rev := (fun n m => proj2 (Nat2Z.inj_le n m)) (only parsing). +Notation inj_lt_rev := (fun n m => proj2 (Nat2Z.inj_lt n m)) (only parsing). +Notation inj_ge_rev := (fun n m => proj2 (Nat2Z.inj_ge n m)) (only parsing). +Notation inj_gt_rev := (fun n m => proj2 (Nat2Z.inj_gt n m)) (only parsing). +Notation inj_plus := Nat2Z.inj_add (only parsing). +Notation inj_mult := Nat2Z.inj_mul (only parsing). +Notation inj_minus1 := Nat2Z.inj_sub (only parsing). +Notation inj_minus := Nat2Z.inj_sub_max (only parsing). +Notation inj_min := Nat2Z.inj_min (only parsing). +Notation inj_max := Nat2Z.inj_max (only parsing). + +Notation Z_of_nat_of_P := positive_nat_Z (only parsing). Notation Zpos_eq_Z_of_nat_o_nat_of_P := - (fun p => eq_sym (positive_nat_Z p)) (compat "8.3"). - -Notation Z_of_nat_of_N := N_nat_Z (compat "8.3"). -Notation Z_of_N_of_nat := nat_N_Z (compat "8.3"). - -Notation Z_of_N_eq := (f_equal Z.of_N) (compat "8.3"). -Notation Z_of_N_eq_rev := N2Z.inj (compat "8.3"). -Notation Z_of_N_eq_iff := (fun n m => iff_sym (N2Z.inj_iff n m)) (compat "8.3"). -Notation Z_of_N_compare := N2Z.inj_compare (compat "8.3"). -Notation Z_of_N_le_iff := N2Z.inj_le (compat "8.3"). -Notation Z_of_N_lt_iff := N2Z.inj_lt (compat "8.3"). -Notation Z_of_N_ge_iff := N2Z.inj_ge (compat "8.3"). -Notation Z_of_N_gt_iff := N2Z.inj_gt (compat "8.3"). -Notation Z_of_N_le := (fun n m => proj1 (N2Z.inj_le n m)) (compat "8.3"). -Notation Z_of_N_lt := (fun n m => proj1 (N2Z.inj_lt n m)) (compat "8.3"). -Notation Z_of_N_ge := (fun n m => proj1 (N2Z.inj_ge n m)) (compat "8.3"). -Notation Z_of_N_gt := (fun n m => proj1 (N2Z.inj_gt n m)) (compat "8.3"). -Notation Z_of_N_le_rev := (fun n m => proj2 (N2Z.inj_le n m)) (compat "8.3"). -Notation Z_of_N_lt_rev := (fun n m => proj2 (N2Z.inj_lt n m)) (compat "8.3"). -Notation Z_of_N_ge_rev := (fun n m => proj2 (N2Z.inj_ge n m)) (compat "8.3"). -Notation Z_of_N_gt_rev := (fun n m => proj2 (N2Z.inj_gt n m)) (compat "8.3"). -Notation Z_of_N_pos := N2Z.inj_pos (compat "8.3"). -Notation Z_of_N_abs := N2Z.inj_abs_N (compat "8.3"). -Notation Z_of_N_le_0 := N2Z.is_nonneg (compat "8.3"). -Notation Z_of_N_plus := N2Z.inj_add (compat "8.3"). -Notation Z_of_N_mult := N2Z.inj_mul (compat "8.3"). -Notation Z_of_N_minus := N2Z.inj_sub_max (compat "8.3"). -Notation Z_of_N_succ := N2Z.inj_succ (compat "8.3"). -Notation Z_of_N_min := N2Z.inj_min (compat "8.3"). -Notation Z_of_N_max := N2Z.inj_max (compat "8.3"). -Notation Zabs_of_N := Zabs2N.id (compat "8.3"). -Notation Zabs_N_succ_abs := Zabs2N.inj_succ_abs (compat "8.3"). -Notation Zabs_N_succ := Zabs2N.inj_succ (compat "8.3"). -Notation Zabs_N_plus_abs := Zabs2N.inj_add_abs (compat "8.3"). -Notation Zabs_N_plus := Zabs2N.inj_add (compat "8.3"). -Notation Zabs_N_mult_abs := Zabs2N.inj_mul_abs (compat "8.3"). -Notation Zabs_N_mult := Zabs2N.inj_mul (compat "8.3"). + (fun p => eq_sym (positive_nat_Z p)) (only parsing). + +Notation Z_of_nat_of_N := N_nat_Z (only parsing). +Notation Z_of_N_of_nat := nat_N_Z (only parsing). + +Notation Z_of_N_eq := (f_equal Z.of_N) (only parsing). +Notation Z_of_N_eq_rev := N2Z.inj (only parsing). +Notation Z_of_N_eq_iff := (fun n m => iff_sym (N2Z.inj_iff n m)) (only parsing). +Notation Z_of_N_compare := N2Z.inj_compare (only parsing). +Notation Z_of_N_le_iff := N2Z.inj_le (only parsing). +Notation Z_of_N_lt_iff := N2Z.inj_lt (only parsing). +Notation Z_of_N_ge_iff := N2Z.inj_ge (only parsing). +Notation Z_of_N_gt_iff := N2Z.inj_gt (only parsing). +Notation Z_of_N_le := (fun n m => proj1 (N2Z.inj_le n m)) (only parsing). +Notation Z_of_N_lt := (fun n m => proj1 (N2Z.inj_lt n m)) (only parsing). +Notation Z_of_N_ge := (fun n m => proj1 (N2Z.inj_ge n m)) (only parsing). +Notation Z_of_N_gt := (fun n m => proj1 (N2Z.inj_gt n m)) (only parsing). +Notation Z_of_N_le_rev := (fun n m => proj2 (N2Z.inj_le n m)) (only parsing). +Notation Z_of_N_lt_rev := (fun n m => proj2 (N2Z.inj_lt n m)) (only parsing). +Notation Z_of_N_ge_rev := (fun n m => proj2 (N2Z.inj_ge n m)) (only parsing). +Notation Z_of_N_gt_rev := (fun n m => proj2 (N2Z.inj_gt n m)) (only parsing). +Notation Z_of_N_pos := N2Z.inj_pos (only parsing). +Notation Z_of_N_abs := N2Z.inj_abs_N (only parsing). +Notation Z_of_N_le_0 := N2Z.is_nonneg (only parsing). +Notation Z_of_N_plus := N2Z.inj_add (only parsing). +Notation Z_of_N_mult := N2Z.inj_mul (only parsing). +Notation Z_of_N_minus := N2Z.inj_sub_max (only parsing). +Notation Z_of_N_succ := N2Z.inj_succ (only parsing). +Notation Z_of_N_min := N2Z.inj_min (only parsing). +Notation Z_of_N_max := N2Z.inj_max (only parsing). +Notation Zabs_of_N := Zabs2N.id (only parsing). +Notation Zabs_N_succ_abs := Zabs2N.inj_succ_abs (only parsing). +Notation Zabs_N_succ := Zabs2N.inj_succ (only parsing). +Notation Zabs_N_plus_abs := Zabs2N.inj_add_abs (only parsing). +Notation Zabs_N_plus := Zabs2N.inj_add (only parsing). +Notation Zabs_N_mult_abs := Zabs2N.inj_mul_abs (only parsing). +Notation Zabs_N_mult := Zabs2N.inj_mul (only parsing). Theorem inj_minus2 : forall n m:nat, (m > n)%nat -> Z.of_nat (n - m) = 0. Proof. diff --git a/theories/ZArith/Znumtheory.v b/theories/ZArith/Znumtheory.v index ee6efb3c..f5444c31 100644 --- a/theories/ZArith/Znumtheory.v +++ b/theories/ZArith/Znumtheory.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import ZArith_base. @@ -25,20 +27,20 @@ Open Scope Z_scope. - properties of the efficient [Z.gcd] function *) -Notation Zgcd := Z.gcd (compat "8.3"). -Notation Zggcd := Z.ggcd (compat "8.3"). -Notation Zggcd_gcd := Z.ggcd_gcd (compat "8.3"). -Notation Zggcd_correct_divisors := Z.ggcd_correct_divisors (compat "8.3"). -Notation Zgcd_divide_l := Z.gcd_divide_l (compat "8.3"). -Notation Zgcd_divide_r := Z.gcd_divide_r (compat "8.3"). -Notation Zgcd_greatest := Z.gcd_greatest (compat "8.3"). -Notation Zgcd_nonneg := Z.gcd_nonneg (compat "8.3"). -Notation Zggcd_opp := Z.ggcd_opp (compat "8.3"). +Notation Zgcd := Z.gcd (compat "8.6"). +Notation Zggcd := Z.ggcd (compat "8.6"). +Notation Zggcd_gcd := Z.ggcd_gcd (compat "8.6"). +Notation Zggcd_correct_divisors := Z.ggcd_correct_divisors (compat "8.6"). +Notation Zgcd_divide_l := Z.gcd_divide_l (compat "8.6"). +Notation Zgcd_divide_r := Z.gcd_divide_r (compat "8.6"). +Notation Zgcd_greatest := Z.gcd_greatest (compat "8.6"). +Notation Zgcd_nonneg := Z.gcd_nonneg (compat "8.6"). +Notation Zggcd_opp := Z.ggcd_opp (compat "8.6"). (** The former specialized inductive predicate [Z.divide] is now a generic existential predicate. *) -Notation Zdivide := Z.divide (compat "8.3"). +Notation Zdivide := Z.divide (compat "8.6"). (** Its former constructor is now a pseudo-constructor. *) @@ -46,17 +48,17 @@ Definition Zdivide_intro a b q (H:b=q*a) : Z.divide a b := ex_intro _ q H. (** Results concerning divisibility*) -Notation Zdivide_refl := Z.divide_refl (compat "8.3"). -Notation Zone_divide := Z.divide_1_l (compat "8.3"). -Notation Zdivide_0 := Z.divide_0_r (compat "8.3"). -Notation Zmult_divide_compat_l := Z.mul_divide_mono_l (compat "8.3"). -Notation Zmult_divide_compat_r := Z.mul_divide_mono_r (compat "8.3"). -Notation Zdivide_plus_r := Z.divide_add_r (compat "8.3"). -Notation Zdivide_minus_l := Z.divide_sub_r (compat "8.3"). -Notation Zdivide_mult_l := Z.divide_mul_l (compat "8.3"). -Notation Zdivide_mult_r := Z.divide_mul_r (compat "8.3"). -Notation Zdivide_factor_r := Z.divide_factor_l (compat "8.3"). -Notation Zdivide_factor_l := Z.divide_factor_r (compat "8.3"). +Notation Zdivide_refl := Z.divide_refl (compat "8.6"). +Notation Zone_divide := Z.divide_1_l (only parsing). +Notation Zdivide_0 := Z.divide_0_r (only parsing). +Notation Zmult_divide_compat_l := Z.mul_divide_mono_l (only parsing). +Notation Zmult_divide_compat_r := Z.mul_divide_mono_r (only parsing). +Notation Zdivide_plus_r := Z.divide_add_r (only parsing). +Notation Zdivide_minus_l := Z.divide_sub_r (only parsing). +Notation Zdivide_mult_l := Z.divide_mul_l (only parsing). +Notation Zdivide_mult_r := Z.divide_mul_r (only parsing). +Notation Zdivide_factor_r := Z.divide_factor_l (only parsing). +Notation Zdivide_factor_l := Z.divide_factor_r (only parsing). Lemma Zdivide_opp_r a b : (a | b) -> (a | - b). Proof. apply Z.divide_opp_r. Qed. @@ -91,12 +93,12 @@ Qed. (** Only [1] and [-1] divide [1]. *) -Notation Zdivide_1 := Z.divide_1_r (compat "8.3"). +Notation Zdivide_1 := Z.divide_1_r (only parsing). (** If [a] divides [b] and [b] divides [a] then [a] is [b] or [-b]. *) -Notation Zdivide_antisym := Z.divide_antisym (compat "8.3"). -Notation Zdivide_trans := Z.divide_trans (compat "8.3"). +Notation Zdivide_antisym := Z.divide_antisym (compat "8.6"). +Notation Zdivide_trans := Z.divide_trans (compat "8.6"). (** If [a] divides [b] and [b<>0] then [|a| <= |b|]. *) @@ -734,7 +736,7 @@ Qed. (** we now prove that [Z.gcd] is indeed a gcd in the sense of [Zis_gcd]. *) -Notation Zgcd_is_pos := Z.gcd_nonneg (compat "8.3"). +Notation Zgcd_is_pos := Z.gcd_nonneg (only parsing). Lemma Zgcd_is_gcd : forall a b, Zis_gcd a b (Z.gcd a b). Proof. @@ -767,8 +769,8 @@ Proof. - subst. now case (Z.gcd a b). Qed. -Notation Zgcd_inv_0_l := Z.gcd_eq_0_l (compat "8.3"). -Notation Zgcd_inv_0_r := Z.gcd_eq_0_r (compat "8.3"). +Notation Zgcd_inv_0_l := Z.gcd_eq_0_l (only parsing). +Notation Zgcd_inv_0_r := Z.gcd_eq_0_r (only parsing). Theorem Zgcd_div_swap0 : forall a b : Z, 0 < Z.gcd a b -> @@ -798,16 +800,16 @@ Proof. rewrite <- Zdivide_Zdiv_eq; auto. Qed. -Notation Zgcd_comm := Z.gcd_comm (compat "8.3"). +Notation Zgcd_comm := Z.gcd_comm (compat "8.6"). Lemma Zgcd_ass a b c : Z.gcd (Z.gcd a b) c = Z.gcd a (Z.gcd b c). Proof. symmetry. apply Z.gcd_assoc. Qed. -Notation Zgcd_Zabs := Z.gcd_abs_l (compat "8.3"). -Notation Zgcd_0 := Z.gcd_0_r (compat "8.3"). -Notation Zgcd_1 := Z.gcd_1_r (compat "8.3"). +Notation Zgcd_Zabs := Z.gcd_abs_l (only parsing). +Notation Zgcd_0 := Z.gcd_0_r (only parsing). +Notation Zgcd_1 := Z.gcd_1_r (only parsing). Hint Resolve Z.gcd_0_r Z.gcd_1_r : zarith. diff --git a/theories/ZArith/Zorder.v b/theories/ZArith/Zorder.v index 73dee0cf..a1ec4b35 100644 --- a/theories/ZArith/Zorder.v +++ b/theories/ZArith/Zorder.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Binary Integers : results about order predicates *) @@ -38,9 +40,9 @@ Qed. (**********************************************************************) (** * Decidability of equality and order on Z *) -Notation dec_eq := Z.eq_decidable (compat "8.3"). -Notation dec_Zle := Z.le_decidable (compat "8.3"). -Notation dec_Zlt := Z.lt_decidable (compat "8.3"). +Notation dec_eq := Z.eq_decidable (only parsing). +Notation dec_Zle := Z.le_decidable (only parsing). +Notation dec_Zlt := Z.lt_decidable (only parsing). Theorem dec_Zne n m : decidable (Zne n m). Proof. @@ -64,12 +66,12 @@ Qed. (** * Relating strict and large orders *) -Notation Zgt_lt := Z.gt_lt (compat "8.3"). -Notation Zlt_gt := Z.lt_gt (compat "8.3"). -Notation Zge_le := Z.ge_le (compat "8.3"). -Notation Zle_ge := Z.le_ge (compat "8.3"). -Notation Zgt_iff_lt := Z.gt_lt_iff (compat "8.3"). -Notation Zge_iff_le := Z.ge_le_iff (compat "8.3"). +Notation Zgt_lt := Z.gt_lt (compat "8.6"). +Notation Zlt_gt := Z.lt_gt (compat "8.6"). +Notation Zge_le := Z.ge_le (compat "8.6"). +Notation Zle_ge := Z.le_ge (compat "8.6"). +Notation Zgt_iff_lt := Z.gt_lt_iff (only parsing). +Notation Zge_iff_le := Z.ge_le_iff (only parsing). Lemma Zle_not_lt n m : n <= m -> ~ m < n. Proof. @@ -121,18 +123,18 @@ Qed. (** Reflexivity *) -Notation Zle_refl := Z.le_refl (compat "8.3"). -Notation Zeq_le := Z.eq_le_incl (compat "8.3"). +Notation Zle_refl := Z.le_refl (compat "8.6"). +Notation Zeq_le := Z.eq_le_incl (only parsing). Hint Resolve Z.le_refl: zarith. (** Antisymmetry *) -Notation Zle_antisym := Z.le_antisymm (compat "8.3"). +Notation Zle_antisym := Z.le_antisymm (only parsing). (** Asymmetry *) -Notation Zlt_asym := Z.lt_asymm (compat "8.3"). +Notation Zlt_asym := Z.lt_asymm (only parsing). Lemma Zgt_asym n m : n > m -> ~ m > n. Proof. @@ -141,8 +143,8 @@ Qed. (** Irreflexivity *) -Notation Zlt_irrefl := Z.lt_irrefl (compat "8.3"). -Notation Zlt_not_eq := Z.lt_neq (compat "8.3"). +Notation Zlt_irrefl := Z.lt_irrefl (compat "8.6"). +Notation Zlt_not_eq := Z.lt_neq (only parsing). Lemma Zgt_irrefl n : ~ n > n. Proof. @@ -151,8 +153,8 @@ Qed. (** Large = strict or equal *) -Notation Zlt_le_weak := Z.lt_le_incl (compat "8.3"). -Notation Zle_lt_or_eq_iff := Z.lt_eq_cases (compat "8.3"). +Notation Zlt_le_weak := Z.lt_le_incl (only parsing). +Notation Zle_lt_or_eq_iff := Z.lt_eq_cases (only parsing). Lemma Zle_lt_or_eq n m : n <= m -> n < m \/ n = m. Proof. @@ -161,11 +163,11 @@ Qed. (** Dichotomy *) -Notation Zle_or_lt := Z.le_gt_cases (compat "8.3"). +Notation Zle_or_lt := Z.le_gt_cases (only parsing). (** Transitivity of strict orders *) -Notation Zlt_trans := Z.lt_trans (compat "8.3"). +Notation Zlt_trans := Z.lt_trans (compat "8.6"). Lemma Zgt_trans n m p : n > m -> m > p -> n > p. Proof. @@ -174,8 +176,8 @@ Qed. (** Mixed transitivity *) -Notation Zlt_le_trans := Z.lt_le_trans (compat "8.3"). -Notation Zle_lt_trans := Z.le_lt_trans (compat "8.3"). +Notation Zlt_le_trans := Z.lt_le_trans (compat "8.6"). +Notation Zle_lt_trans := Z.le_lt_trans (compat "8.6"). Lemma Zle_gt_trans n m p : m <= n -> m > p -> n > p. Proof. @@ -189,7 +191,7 @@ Qed. (** Transitivity of large orders *) -Notation Zle_trans := Z.le_trans (compat "8.3"). +Notation Zle_trans := Z.le_trans (compat "8.6"). Lemma Zge_trans n m p : n >= m -> m >= p -> n >= p. Proof. @@ -240,8 +242,8 @@ Qed. (** Special base instances of order *) -Notation Zlt_succ := Z.lt_succ_diag_r (compat "8.3"). -Notation Zlt_pred := Z.lt_pred_l (compat "8.3"). +Notation Zlt_succ := Z.lt_succ_diag_r (only parsing). +Notation Zlt_pred := Z.lt_pred_l (only parsing). Lemma Zgt_succ n : Z.succ n > n. Proof. @@ -255,8 +257,8 @@ Qed. (** Relating strict and large order using successor or predecessor *) -Notation Zlt_succ_r := Z.lt_succ_r (compat "8.3"). -Notation Zle_succ_l := Z.le_succ_l (compat "8.3"). +Notation Zlt_succ_r := Z.lt_succ_r (compat "8.6"). +Notation Zle_succ_l := Z.le_succ_l (compat "8.6"). Lemma Zgt_le_succ n m : m > n -> Z.succ n <= m. Proof. @@ -295,10 +297,10 @@ Qed. (** Weakening order *) -Notation Zle_succ := Z.le_succ_diag_r (compat "8.3"). -Notation Zle_pred := Z.le_pred_l (compat "8.3"). -Notation Zlt_lt_succ := Z.lt_lt_succ_r (compat "8.3"). -Notation Zle_le_succ := Z.le_le_succ_r (compat "8.3"). +Notation Zle_succ := Z.le_succ_diag_r (only parsing). +Notation Zle_pred := Z.le_pred_l (only parsing). +Notation Zlt_lt_succ := Z.lt_lt_succ_r (only parsing). +Notation Zle_le_succ := Z.le_le_succ_r (only parsing). Lemma Zle_succ_le n m : Z.succ n <= m -> n <= m. Proof. @@ -334,12 +336,12 @@ Qed. (** Special cases of ordered integers *) -Notation Zlt_0_1 := Z.lt_0_1 (compat "8.3"). -Notation Zle_0_1 := Z.le_0_1 (compat "8.3"). +Notation Zlt_0_1 := Z.lt_0_1 (compat "8.6"). +Notation Zle_0_1 := Z.le_0_1 (compat "8.6"). Lemma Zle_neg_pos : forall p q:positive, Zneg p <= Zpos q. Proof. - easy. + exact Pos2Z.neg_le_pos. Qed. Lemma Zgt_pos_0 : forall p:positive, Zpos p > 0. @@ -350,12 +352,12 @@ Qed. (* weaker but useful (in [Z.pow] for instance) *) Lemma Zle_0_pos : forall p:positive, 0 <= Zpos p. Proof. - easy. + exact Pos2Z.pos_is_nonneg. Qed. Lemma Zlt_neg_0 : forall p:positive, Zneg p < 0. Proof. - easy. + exact Pos2Z.neg_is_neg. Qed. Lemma Zle_0_nat : forall n:nat, 0 <= Z.of_nat n. @@ -375,10 +377,10 @@ Qed. (** ** Addition *) (** Compatibility of addition wrt to order *) -Notation Zplus_lt_le_compat := Z.add_lt_le_mono (compat "8.3"). -Notation Zplus_le_lt_compat := Z.add_le_lt_mono (compat "8.3"). -Notation Zplus_le_compat := Z.add_le_mono (compat "8.3"). -Notation Zplus_lt_compat := Z.add_lt_mono (compat "8.3"). +Notation Zplus_lt_le_compat := Z.add_lt_le_mono (only parsing). +Notation Zplus_le_lt_compat := Z.add_le_lt_mono (only parsing). +Notation Zplus_le_compat := Z.add_le_mono (only parsing). +Notation Zplus_lt_compat := Z.add_lt_mono (only parsing). Lemma Zplus_gt_compat_l n m p : n > m -> p + n > p + m. Proof. @@ -412,7 +414,7 @@ Qed. (** Compatibility of addition wrt to being positive *) -Notation Zplus_le_0_compat := Z.add_nonneg_nonneg (compat "8.3"). +Notation Zplus_le_0_compat := Z.add_nonneg_nonneg (only parsing). (** Simplification of addition wrt to order *) @@ -570,9 +572,9 @@ Qed. (** Compatibility of multiplication by a positive wrt to being positive *) -Notation Zmult_le_0_compat := Z.mul_nonneg_nonneg (compat "8.3"). -Notation Zmult_lt_0_compat := Z.mul_pos_pos (compat "8.3"). -Notation Zmult_lt_O_compat := Z.mul_pos_pos (compat "8.3"). +Notation Zmult_le_0_compat := Z.mul_nonneg_nonneg (only parsing). +Notation Zmult_lt_0_compat := Z.mul_pos_pos (only parsing). +Notation Zmult_lt_O_compat := Z.mul_pos_pos (only parsing). Lemma Zmult_gt_0_compat n m : n > 0 -> m > 0 -> n * m > 0. Proof. @@ -624,9 +626,9 @@ Qed. (** * Equivalence between inequalities *) -Notation Zle_plus_swap := Z.le_add_le_sub_r (compat "8.3"). -Notation Zlt_plus_swap := Z.lt_add_lt_sub_r (compat "8.3"). -Notation Zlt_minus_simpl_swap := Z.lt_sub_pos (compat "8.3"). +Notation Zle_plus_swap := Z.le_add_le_sub_r (only parsing). +Notation Zlt_plus_swap := Z.lt_add_lt_sub_r (only parsing). +Notation Zlt_minus_simpl_swap := Z.lt_sub_pos (only parsing). Lemma Zeq_plus_swap n m p : n + p = m <-> n = m - p. Proof. diff --git a/theories/ZArith/Zpow_alt.v b/theories/ZArith/Zpow_alt.v index 79a5a555..983405ac 100644 --- a/theories/ZArith/Zpow_alt.v +++ b/theories/ZArith/Zpow_alt.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import BinInt. diff --git a/theories/ZArith/Zpow_def.v b/theories/ZArith/Zpow_def.v index 9eafa076..2b099671 100644 --- a/theories/ZArith/Zpow_def.v +++ b/theories/ZArith/Zpow_def.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import BinInt Ring_theory. @@ -14,12 +16,12 @@ Local Open Scope Z_scope. (** Nota : this file is mostly deprecated. The definition of [Z.pow] and its usual properties are now provided by module [BinInt.Z]. *) -Notation Zpower_pos := Z.pow_pos (compat "8.3"). -Notation Zpower := Z.pow (compat "8.3"). -Notation Zpower_0_r := Z.pow_0_r (compat "8.3"). -Notation Zpower_succ_r := Z.pow_succ_r (compat "8.3"). -Notation Zpower_neg_r := Z.pow_neg_r (compat "8.3"). -Notation Zpower_Ppow := Pos2Z.inj_pow (compat "8.3"). +Notation Zpower_pos := Z.pow_pos (only parsing). +Notation Zpower := Z.pow (only parsing). +Notation Zpower_0_r := Z.pow_0_r (only parsing). +Notation Zpower_succ_r := Z.pow_succ_r (only parsing). +Notation Zpower_neg_r := Z.pow_neg_r (only parsing). +Notation Zpower_Ppow := Pos2Z.inj_pow (only parsing). Lemma Zpower_theory : power_theory 1 Z.mul (@eq Z) Z.of_N Z.pow. Proof. diff --git a/theories/ZArith/Zpow_facts.v b/theories/ZArith/Zpow_facts.v index 2930e677..a9bc5bd0 100644 --- a/theories/ZArith/Zpow_facts.v +++ b/theories/ZArith/Zpow_facts.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import ZArith_base ZArithRing Zcomplements Zdiv Znumtheory. @@ -29,17 +31,17 @@ Proof. now apply (Z.pow_0_l (Zpos p)). Qed. Lemma Zpower_pos_pos x p : 0 < x -> 0 < Z.pow_pos x p. Proof. intros. now apply (Z.pow_pos_nonneg x (Zpos p)). Qed. -Notation Zpower_1_r := Z.pow_1_r (compat "8.3"). -Notation Zpower_1_l := Z.pow_1_l (compat "8.3"). -Notation Zpower_0_l := Z.pow_0_l' (compat "8.3"). -Notation Zpower_0_r := Z.pow_0_r (compat "8.3"). -Notation Zpower_2 := Z.pow_2_r (compat "8.3"). -Notation Zpower_gt_0 := Z.pow_pos_nonneg (compat "8.3"). -Notation Zpower_ge_0 := Z.pow_nonneg (compat "8.3"). -Notation Zpower_Zabs := Z.abs_pow (compat "8.3"). -Notation Zpower_Zsucc := Z.pow_succ_r (compat "8.3"). -Notation Zpower_mult := Z.pow_mul_r (compat "8.3"). -Notation Zpower_le_monotone2 := Z.pow_le_mono_r (compat "8.3"). +Notation Zpower_1_r := Z.pow_1_r (only parsing). +Notation Zpower_1_l := Z.pow_1_l (only parsing). +Notation Zpower_0_l := Z.pow_0_l' (only parsing). +Notation Zpower_0_r := Z.pow_0_r (only parsing). +Notation Zpower_2 := Z.pow_2_r (only parsing). +Notation Zpower_gt_0 := Z.pow_pos_nonneg (only parsing). +Notation Zpower_ge_0 := Z.pow_nonneg (only parsing). +Notation Zpower_Zabs := Z.abs_pow (only parsing). +Notation Zpower_Zsucc := Z.pow_succ_r (only parsing). +Notation Zpower_mult := Z.pow_mul_r (only parsing). +Notation Zpower_le_monotone2 := Z.pow_le_mono_r (only parsing). Theorem Zpower_le_monotone a b c : 0 < a -> 0 <= b <= c -> a^b <= a^c. @@ -231,7 +233,7 @@ Qed. (** * Z.square: a direct definition of [z^2] *) -Notation Psquare := Pos.square (compat "8.3"). -Notation Zsquare := Z.square (compat "8.3"). -Notation Psquare_correct := Pos.square_spec (compat "8.3"). -Notation Zsquare_correct := Z.square_spec (compat "8.3"). +Notation Psquare := Pos.square (compat "8.6"). +Notation Zsquare := Z.square (compat "8.6"). +Notation Psquare_correct := Pos.square_spec (only parsing). +Notation Zsquare_correct := Z.square_spec (only parsing). diff --git a/theories/ZArith/Zpower.v b/theories/ZArith/Zpower.v index 6f3a89f1..fa690535 100644 --- a/theories/ZArith/Zpower.v +++ b/theories/ZArith/Zpower.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Wf_nat ZArith_base Omega Zcomplements. diff --git a/theories/ZArith/Zquot.v b/theories/ZArith/Zquot.v index 0d92f1d5..e93ebb1a 100644 --- a/theories/ZArith/Zquot.v +++ b/theories/ZArith/Zquot.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Nnat ZArith_base ROmega ZArithRing Zdiv Morphisms. @@ -31,21 +33,21 @@ Local Open Scope Z_scope. exploiting the arbitrary value of division by 0). *) -Notation Ndiv_Zquot := N2Z.inj_quot (compat "8.3"). -Notation Nmod_Zrem := N2Z.inj_rem (compat "8.3"). -Notation Z_quot_rem_eq := Z.quot_rem' (compat "8.3"). -Notation Zrem_lt := Z.rem_bound_abs (compat "8.3"). -Notation Zquot_unique := Z.quot_unique (compat "8.3"). -Notation Zrem_unique := Z.rem_unique (compat "8.3"). -Notation Zrem_1_r := Z.rem_1_r (compat "8.3"). -Notation Zquot_1_r := Z.quot_1_r (compat "8.3"). -Notation Zrem_1_l := Z.rem_1_l (compat "8.3"). -Notation Zquot_1_l := Z.quot_1_l (compat "8.3"). -Notation Z_quot_same := Z.quot_same (compat "8.3"). -Notation Z_quot_mult := Z.quot_mul (compat "8.3"). -Notation Zquot_small := Z.quot_small (compat "8.3"). -Notation Zrem_small := Z.rem_small (compat "8.3"). -Notation Zquot2_quot := Zquot2_quot (compat "8.3"). +Notation Ndiv_Zquot := N2Z.inj_quot (only parsing). +Notation Nmod_Zrem := N2Z.inj_rem (only parsing). +Notation Z_quot_rem_eq := Z.quot_rem' (only parsing). +Notation Zrem_lt := Z.rem_bound_abs (only parsing). +Notation Zquot_unique := Z.quot_unique (compat "8.6"). +Notation Zrem_unique := Z.rem_unique (compat "8.6"). +Notation Zrem_1_r := Z.rem_1_r (compat "8.6"). +Notation Zquot_1_r := Z.quot_1_r (compat "8.6"). +Notation Zrem_1_l := Z.rem_1_l (compat "8.6"). +Notation Zquot_1_l := Z.quot_1_l (compat "8.6"). +Notation Z_quot_same := Z.quot_same (compat "8.6"). +Notation Z_quot_mult := Z.quot_mul (only parsing). +Notation Zquot_small := Z.quot_small (compat "8.6"). +Notation Zrem_small := Z.rem_small (compat "8.6"). +Notation Zquot2_quot := Zquot2_quot (compat "8.6"). (** Particular values taken for [a÷0] and [(Z.rem a 0)]. We avise to not rely on these arbitrary values. *) diff --git a/theories/ZArith/Zsqrt_compat.v b/theories/ZArith/Zsqrt_compat.v index f4baba19..bd090454 100644 --- a/theories/ZArith/Zsqrt_compat.v +++ b/theories/ZArith/Zsqrt_compat.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import ZArithRing. @@ -229,4 +231,4 @@ Proof. symmetry. apply Z.sqrt_unique; trivial. now apply Zsqrt_interval. now destruct n. -Qed.
\ No newline at end of file +Qed. diff --git a/theories/ZArith/Zwf.v b/theories/ZArith/Zwf.v index 90754af3..a71ea4f3 100644 --- a/theories/ZArith/Zwf.v +++ b/theories/ZArith/Zwf.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import ZArith_base. diff --git a/theories/ZArith/auxiliary.v b/theories/ZArith/auxiliary.v index c6c38976..306a8563 100644 --- a/theories/ZArith/auxiliary.v +++ b/theories/ZArith/auxiliary.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Binary Integers (Pierre Crégut, CNET, Lannion, France) *) diff --git a/theories/ZArith/vo.itarget b/theories/ZArith/vo.itarget deleted file mode 100644 index 178111cd..00000000 --- a/theories/ZArith/vo.itarget +++ /dev/null @@ -1,33 +0,0 @@ -auxiliary.vo -BinIntDef.vo -BinInt.vo -Int.vo -Wf_Z.vo -Zabs.vo -ZArith_base.vo -ZArith_dec.vo -ZArith.vo -Zdigits.vo -Zbool.vo -Zcompare.vo -Zcomplements.vo -Zdiv.vo -Zeven.vo -Zgcd_alt.vo -Zpow_alt.vo -Zhints.vo -Zlogarithm.vo -Zmax.vo -Zminmax.vo -Zmin.vo -Zmisc.vo -Znat.vo -Znumtheory.vo -Zquot.vo -Zorder.vo -Zpow_def.vo -Zpower.vo -Zpow_facts.vo -Zsqrt_compat.vo -Zwf.vo -Zeuclid.vo |