diff options
Diffstat (limited to 'theories/Arith')
| -rw-r--r-- | theories/Arith/Arith.v | 4 | ||||
| -rw-r--r-- | theories/Arith/Arith_base.v | 4 | ||||
| -rw-r--r-- | theories/Arith/Between.v | 10 | ||||
| -rw-r--r-- | theories/Arith/Bool_nat.v | 8 | ||||
| -rw-r--r-- | theories/Arith/Compare.v | 12 | ||||
| -rw-r--r-- | theories/Arith/Compare_dec.v | 26 | ||||
| -rw-r--r-- | theories/Arith/Div2.v | 46 | ||||
| -rw-r--r-- | theories/Arith/EqNat.v | 20 | ||||
| -rw-r--r-- | theories/Arith/Euclid.v | 31 | ||||
| -rw-r--r-- | theories/Arith/Even.v | 8 | ||||
| -rw-r--r-- | theories/Arith/Factorial.v | 14 | ||||
| -rw-r--r-- | theories/Arith/Gt.v | 18 | ||||
| -rw-r--r-- | theories/Arith/Le.v | 17 | ||||
| -rw-r--r-- | theories/Arith/Lt.v | 21 | ||||
| -rw-r--r-- | theories/Arith/Max.v | 58 | ||||
| -rw-r--r-- | theories/Arith/Min.v | 54 | ||||
| -rw-r--r-- | theories/Arith/MinMax.v | 113 | ||||
| -rw-r--r-- | theories/Arith/Minus.v | 34 | ||||
| -rw-r--r-- | theories/Arith/Mult.v | 41 | ||||
| -rw-r--r-- | theories/Arith/NatOrderedType.v | 64 | ||||
| -rw-r--r-- | theories/Arith/Peano_dec.v | 30 | ||||
| -rw-r--r-- | theories/Arith/Plus.v | 63 | ||||
| -rw-r--r-- | theories/Arith/Wf_nat.v | 43 | ||||
| -rw-r--r-- | theories/Arith/vo.itarget | 2 |
24 files changed, 260 insertions, 481 deletions
diff --git a/theories/Arith/Arith.v b/theories/Arith/Arith.v index 2e9dc2de..fea10ce1 100644 --- a/theories/Arith/Arith.v +++ b/theories/Arith/Arith.v @@ -1,12 +1,10 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: Arith.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Export Arith_base. Require Export ArithRing. diff --git a/theories/Arith/Arith_base.v b/theories/Arith/Arith_base.v index e9953e54..9f0f05db 100644 --- a/theories/Arith/Arith_base.v +++ b/theories/Arith/Arith_base.v @@ -1,13 +1,11 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: Arith_base.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Export Le. Require Export Lt. Require Export Plus. diff --git a/theories/Arith/Between.v b/theories/Arith/Between.v index 65753e31..fb488526 100644 --- a/theories/Arith/Between.v +++ b/theories/Arith/Between.v @@ -1,17 +1,15 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: Between.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Import Le. Require Import Lt. -Open Local Scope nat_scope. +Local Open Scope nat_scope. Implicit Types k l p q r : nat. @@ -76,7 +74,7 @@ Section Between. Lemma in_int_intro : forall p q r, p <= r -> r < q -> in_int p q r. Proof. - red in |- *; auto with arith. + red; auto with arith. Qed. Hint Resolve in_int_intro: arith v62. @@ -151,7 +149,7 @@ Section Between. between k l -> (forall n:nat, in_int k l n -> P n -> ~ Q n) -> ~ exists_between k l. Proof. - induction 1; red in |- *; intros. + 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'. diff --git a/theories/Arith/Bool_nat.v b/theories/Arith/Bool_nat.v index b3dcd8ec..4c15a173 100644 --- a/theories/Arith/Bool_nat.v +++ b/theories/Arith/Bool_nat.v @@ -1,18 +1,16 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(* $Id: Bool_nat.v 14641 2011-11-06 11:59:10Z herbelin $ *) - Require Export Compare_dec. Require Export Peano_dec. Require Import Sumbool. -Open Local Scope nat_scope. +Local Open Scope nat_scope. Implicit Types m n x y : nat. @@ -36,4 +34,4 @@ Definition nat_noteq_bool x y := bool_of_sumbool (sumbool_not _ _ (eq_nat_dec x y)). Definition zerop_bool x := bool_of_sumbool (zerop x). -Definition notzerop_bool x := bool_of_sumbool (notzerop x).
\ No newline at end of file +Definition notzerop_bool x := bool_of_sumbool (notzerop x). diff --git a/theories/Arith/Compare.v b/theories/Arith/Compare.v index 2fe5c0d9..65219655 100644 --- a/theories/Arith/Compare.v +++ b/theories/Arith/Compare.v @@ -1,18 +1,16 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: Compare.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - (** Equality is decidable on [nat] *) -Open Local Scope nat_scope. +Local Open Scope nat_scope. -Notation not_eq_sym := sym_not_eq. +Notation not_eq_sym := not_eq_sym (only parsing). Implicit Types m n p q : nat. @@ -43,7 +41,7 @@ Proof. lapply (lt_le_S m n); auto with arith. intro H'; lapply (le_lt_or_eq (S m) n); auto with arith. induction 1; auto with arith. - right; exists (n - S (S m)); simpl in |- *. + right; exists (n - S (S m)); simpl. rewrite (plus_comm m (n - S (S m))). rewrite (plus_n_Sm (n - S (S m)) m). rewrite (plus_n_Sm (n - S (S m)) (S m)). @@ -52,4 +50,4 @@ Qed. Require Export Wf_nat. -Require Export Min Max.
\ No newline at end of file +Require Export Min Max. diff --git a/theories/Arith/Compare_dec.v b/theories/Arith/Compare_dec.v index 99c7415e..a90a9ce9 100644 --- a/theories/Arith/Compare_dec.v +++ b/theories/Arith/Compare_dec.v @@ -1,19 +1,17 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: Compare_dec.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Import Le. Require Import Lt. Require Import Gt. Require Import Decidable. -Open Local Scope nat_scope. +Local Open Scope nat_scope. Implicit Types m n x y : nat. @@ -22,21 +20,21 @@ Proof. destruct n; auto with arith. Defined. -Definition lt_eq_lt_dec : forall n m, {n < m} + {n = m} + {m < n}. +Definition lt_eq_lt_dec n m : {n < m} + {n = m} + {m < n}. Proof. - induction n; destruct m; auto with arith. + induction n in m |- *; destruct m; auto with arith. destruct (IHn m) as [H|H]; auto with arith. destruct H; auto with arith. Defined. -Definition gt_eq_gt_dec : forall n m, {m > n} + {n = m} + {n > m}. +Definition gt_eq_gt_dec n m : {m > n} + {n = m} + {n > m}. Proof. intros; apply lt_eq_lt_dec; assumption. Defined. -Definition le_lt_dec : forall n m, {n <= m} + {m < n}. +Definition le_lt_dec n m : {n <= m} + {m < n}. Proof. - induction n. + induction n in m |- *. auto with arith. destruct m. auto with arith. @@ -140,7 +138,7 @@ Proof. Qed. -(** A ternary comparison function in the spirit of [Zcompare]. *) +(** A ternary comparison function in the spirit of [Z.compare]. *) Fixpoint nat_compare n m := match n, m with @@ -200,16 +198,16 @@ Proof. apply -> nat_compare_lt; auto. Qed. -Lemma nat_compare_spec : forall x y, CompSpec eq lt x y (nat_compare x y). +Lemma nat_compare_spec : + forall x y, CompareSpec (x=y) (x<y) (y<x) (nat_compare x y). Proof. intros. - destruct (nat_compare x y) as [ ]_eqn; constructor. + destruct (nat_compare x y) eqn:?; constructor. apply nat_compare_eq; auto. apply <- nat_compare_lt; auto. apply <- nat_compare_gt; auto. Qed. - (** Some projections of the above equivalences. *) Lemma nat_compare_Lt_lt : forall n m, nat_compare n m = Lt -> n<m. @@ -258,7 +256,7 @@ Lemma leb_correct : forall m n, m <= n -> leb m n = true. Proof. induction m as [| m IHm]. trivial. destruct n. intro H. elim (le_Sn_O _ H). - intros. simpl in |- *. apply IHm. apply le_S_n. assumption. + intros. simpl. apply IHm. apply le_S_n. assumption. Qed. Lemma leb_complete : forall m n, leb m n = true -> m <= n. diff --git a/theories/Arith/Div2.v b/theories/Arith/Div2.v index 89620f5f..56115c7f 100644 --- a/theories/Arith/Div2.v +++ b/theories/Arith/Div2.v @@ -1,19 +1,17 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: Div2.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Import Lt. Require Import Plus. Require Import Compare_dec. Require Import Even. -Open Local Scope nat_scope. +Local Open Scope nat_scope. Implicit Type n : nat. @@ -45,7 +43,7 @@ Qed. Lemma lt_div2 : forall n, 0 < n -> div2 n < n. Proof. - intro n. pattern n in |- *. apply ind_0_1_SS. + intro n. pattern n. apply ind_0_1_SS. (* n = 0 *) inversion 1. (* n=1 *) @@ -71,24 +69,24 @@ Proof. (* S n *) inversion_clear H. apply even_div2 in H0 as <-. trivial. Qed. -Lemma div2_even : forall n, div2 n = div2 (S n) -> even n -with div2_odd : forall n, S (div2 n) = div2 (S n) -> odd n. +Lemma div2_even n : div2 n = div2 (S n) -> even n +with div2_odd n : S (div2 n) = div2 (S n) -> odd n. Proof. - destruct n; intro H. - (* 0 *) constructor. - (* S n *) constructor. apply div2_odd. rewrite H. trivial. - destruct n; intro H. - (* 0 *) discriminate. - (* S n *) constructor. apply div2_even. injection H as <-. trivial. +{ destruct n; intro H. + - constructor. + - constructor. apply div2_odd. rewrite H. trivial. } +{ destruct n; intro H. + - discriminate. + - constructor. apply div2_even. injection H as <-. trivial. } Qed. Hint Resolve even_div2 div2_even odd_div2 div2_odd: arith. -Lemma even_odd_div2 : - forall n, - (even n <-> div2 n = div2 (S n)) /\ (odd n <-> S (div2 n) = div2 (S n)). +Lemma even_odd_div2 n : + (even n <-> div2 n = div2 (S n)) /\ + (odd n <-> S (div2 n) = div2 (S n)). Proof. - auto decomp using div2_odd, div2_even, odd_div2, even_div2. + split; split; auto using div2_odd, div2_even, odd_div2, even_div2. Qed. @@ -101,12 +99,12 @@ Hint Unfold double: arith. Lemma double_S : forall n, double (S n) = S (S (double n)). Proof. - intro. unfold double in |- *. simpl in |- *. auto with arith. + intro. unfold double. simpl. auto with arith. Qed. Lemma double_plus : forall n (m:nat), double (n + m) = double n + double m. Proof. - intros m n. unfold double in |- *. + intros m n. unfold double. do 2 rewrite plus_assoc_reverse. rewrite (plus_permute n). reflexivity. Qed. @@ -117,7 +115,7 @@ Lemma even_odd_double : forall n, (even n <-> n = double (div2 n)) /\ (odd n <-> n = S (double (div2 n))). Proof. - intro n. pattern n in |- *. apply ind_0_1_SS. + intro n. pattern n. apply ind_0_1_SS. (* n = 0 *) split; split; auto with arith. intro H. inversion H. @@ -128,11 +126,11 @@ Proof. intros. destruct H as ((IH1,IH2),(IH3,IH4)). split; split. intro H. inversion H. inversion H1. - simpl in |- *. rewrite (double_S (div2 n0)). auto with arith. - simpl in |- *. rewrite (double_S (div2 n0)). intro H. injection H. auto with arith. + simpl. rewrite (double_S (div2 n0)). auto with arith. + simpl. rewrite (double_S (div2 n0)). intro H. injection H. auto with arith. intro H. inversion H. inversion H1. - simpl in |- *. rewrite (double_S (div2 n0)). auto with arith. - simpl in |- *. rewrite (double_S (div2 n0)). intro H. injection H. auto with arith. + simpl. rewrite (double_S (div2 n0)). auto with arith. + simpl. rewrite (double_S (div2 n0)). intro H. injection H. auto with arith. Qed. (** Specializations *) diff --git a/theories/Arith/EqNat.v b/theories/Arith/EqNat.v index 60575beb..ce8eb478 100644 --- a/theories/Arith/EqNat.v +++ b/theories/Arith/EqNat.v @@ -1,16 +1,14 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: EqNat.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - (** Equality on natural numbers *) -Open Local Scope nat_scope. +Local Open Scope nat_scope. Implicit Types m n x y : nat. @@ -25,7 +23,7 @@ Fixpoint eq_nat n m : Prop := end. Theorem eq_nat_refl : forall n, eq_nat n n. - induction n; simpl in |- *; auto. + induction n; simpl; auto. Qed. Hint Resolve eq_nat_refl: arith v62. @@ -37,7 +35,7 @@ Qed. Hint Immediate eq_eq_nat: arith v62. Lemma eq_nat_eq : forall n m, eq_nat n m -> n = m. - induction n; induction m; simpl in |- *; contradiction || auto with arith. + induction n; induction m; simpl; contradiction || auto with arith. Qed. Hint Immediate eq_nat_eq: arith v62. @@ -57,11 +55,11 @@ Proof. induction n. destruct m as [| n]. auto with arith. - intros; right; red in |- *; trivial with arith. + intros; right; red; trivial with arith. destruct m as [| n0]. - right; red in |- *; auto with arith. + right; red; auto with arith. intros. - simpl in |- *. + simpl. apply IHn. Defined. @@ -78,12 +76,12 @@ Fixpoint beq_nat n m : bool := Lemma beq_nat_refl : forall n, true = beq_nat n n. Proof. - intro x; induction x; simpl in |- *; auto. + intro x; induction x; simpl; auto. Qed. Definition beq_nat_eq : forall x y, true = beq_nat x y -> x = y. Proof. - double induction x y; simpl in |- *. + double induction x y; simpl. reflexivity. intros n H1 H2. discriminate H2. intros n H1 H2. discriminate H2. diff --git a/theories/Arith/Euclid.v b/theories/Arith/Euclid.v index f32e1ad4..3abdff98 100644 --- a/theories/Arith/Euclid.v +++ b/theories/Arith/Euclid.v @@ -1,71 +1,68 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: Euclid.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Import Mult. Require Import Compare_dec. Require Import Wf_nat. -Open Local Scope nat_scope. +Local Open Scope nat_scope. Implicit Types a b n q r : nat. Inductive diveucl a b : Set := divex : forall q r, b > r -> a = q * b + r -> diveucl a b. - Lemma eucl_dev : forall n, n > 0 -> forall m:nat, diveucl m n. Proof. - intros b H a; pattern a in |- *; apply gt_wf_rec; intros n H0. + intros b H a; pattern a; apply gt_wf_rec; intros n H0. elim (le_gt_dec b n). intro lebn. elim (H0 (n - b)); auto with arith. intros q r g e. - apply divex with (S q) r; simpl in |- *; auto with arith. + apply divex with (S q) r; simpl; auto with arith. elim plus_assoc. elim e; auto with arith. intros gtbn. - apply divex with 0 n; simpl in |- *; auto with arith. -Qed. + apply divex with 0 n; simpl; auto with arith. +Defined. Lemma quotient : forall n, n > 0 -> forall m:nat, {q : nat | exists r : nat, m = q * n + r /\ n > r}. Proof. - intros b H a; pattern a in |- *; apply gt_wf_rec; intros n H0. + intros b H a; pattern a; apply gt_wf_rec; intros n H0. elim (le_gt_dec b n). intro lebn. elim (H0 (n - b)); auto with arith. intros q Hq; exists (S q). elim Hq; intros r Hr. - exists r; simpl in |- *; elim Hr; intros. + exists r; simpl; elim Hr; intros. elim plus_assoc. elim H1; auto with arith. intros gtbn. - exists 0; exists n; simpl in |- *; auto with arith. -Qed. + exists 0; exists n; simpl; auto with arith. +Defined. Lemma modulo : forall n, n > 0 -> forall m:nat, {r : nat | exists q : nat, m = q * n + r /\ n > r}. Proof. - intros b H a; pattern a in |- *; apply gt_wf_rec; intros n H0. + intros b H a; pattern a; apply gt_wf_rec; intros n H0. elim (le_gt_dec b n). intro lebn. elim (H0 (n - b)); auto with arith. intros r Hr; exists r. elim Hr; intros q Hq. - elim Hq; intros; exists (S q); simpl in |- *. + elim Hq; intros; exists (S q); simpl. elim plus_assoc. elim H1; auto with arith. intros gtbn. - exists n; exists 0; simpl in |- *; auto with arith. -Qed. + exists n; exists 0; simpl; auto with arith. +Defined. diff --git a/theories/Arith/Even.v b/theories/Arith/Even.v index 5bab97c2..4f679fe2 100644 --- a/theories/Arith/Even.v +++ b/theories/Arith/Even.v @@ -1,18 +1,16 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: Even.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - (** Here we define the predicates [even] and [odd] by mutual induction and we prove the decidability and the exclusion of those predicates. The main results about parity are proved in the module Div2. *) -Open Local Scope nat_scope. +Local Open Scope nat_scope. Implicit Types m n : nat. @@ -147,7 +145,7 @@ Lemma even_mult_aux : forall n m, (odd (n * m) <-> odd n /\ odd m) /\ (even (n * m) <-> even n \/ even m). Proof. - intros n; elim n; simpl in |- *; auto with arith. + intros n; elim n; simpl; auto with arith. intros m; split; split; auto with arith. intros H'; inversion H'. intros H'; elim H'; auto. diff --git a/theories/Arith/Factorial.v b/theories/Arith/Factorial.v index 3b434b96..37aa1b2c 100644 --- a/theories/Arith/Factorial.v +++ b/theories/Arith/Factorial.v @@ -1,37 +1,35 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: Factorial.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Import Plus. Require Import Mult. Require Import Lt. -Open Local Scope nat_scope. +Local Open Scope nat_scope. (** Factorial *) -Boxed Fixpoint fact (n:nat) : nat := +Fixpoint fact (n:nat) : nat := match n with | O => 1 | S n => S n * fact n end. -Arguments Scope fact [nat_scope]. +Arguments fact n%nat. Lemma lt_O_fact : forall n:nat, 0 < fact n. Proof. - simple induction n; unfold lt in |- *; simpl in |- *; auto with arith. + simple induction n; unfold lt; simpl; auto with arith. Qed. Lemma fact_neq_0 : forall n:nat, fact n <> 0. Proof. intro. - apply sym_not_eq. + apply not_eq_sym. apply lt_O_neq. apply lt_O_fact. Qed. diff --git a/theories/Arith/Gt.v b/theories/Arith/Gt.v index 43df01c0..31b15507 100644 --- a/theories/Arith/Gt.v +++ b/theories/Arith/Gt.v @@ -1,13 +1,11 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: Gt.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - (** Theorems about [gt] in [nat]. [gt] is defined in [Init/Peano.v] as: << Definition gt (n m:nat) := m < n. @@ -17,7 +15,7 @@ Definition gt (n m:nat) := m < n. Require Import Le. Require Import Lt. Require Import Plus. -Open Local Scope nat_scope. +Local Open Scope nat_scope. Implicit Types m n p : nat. @@ -49,7 +47,7 @@ Hint Immediate gt_S_n: arith v62. Theorem gt_S : forall n m, S n > m -> n > m \/ m = n. Proof. - intros n m H; unfold gt in |- *; apply le_lt_or_eq; auto with arith. + intros n m H; unfold gt; apply le_lt_or_eq; auto with arith. Qed. Lemma gt_pred : forall n m, m > S n -> pred m > n. @@ -112,23 +110,23 @@ Hint Resolve le_gt_S: arith v62. Theorem le_gt_trans : forall n m p, m <= n -> m > p -> n > p. Proof. - red in |- *; intros; apply lt_le_trans with m; auto with arith. + red; intros; apply lt_le_trans with m; auto with arith. Qed. Theorem gt_le_trans : forall n m p, n > m -> p <= m -> n > p. Proof. - red in |- *; intros; apply le_lt_trans with m; auto with arith. + red; intros; apply le_lt_trans with m; auto with arith. Qed. Lemma gt_trans : forall n m p, n > m -> m > p -> n > p. Proof. - red in |- *; intros n m p H1 H2. + red; intros n m p H1 H2. apply lt_trans with m; auto with arith. Qed. Theorem gt_trans_S : forall n m p, S n > m -> m > p -> n > p. Proof. - red in |- *; intros; apply lt_le_trans with m; auto with arith. + red; intros; apply lt_le_trans with m; auto with arith. Qed. Hint Resolve gt_trans_S le_gt_trans gt_le_trans: arith v62. @@ -144,7 +142,7 @@ Qed. Lemma plus_gt_reg_l : forall n m p, p + n > p + m -> n > m. Proof. - red in |- *; intros n m p H; apply plus_lt_reg_l with p; auto with arith. + red; intros n m p H; apply plus_lt_reg_l with p; auto with arith. Qed. Lemma plus_gt_compat_l : forall n m p, n > m -> p + n > p + m. diff --git a/theories/Arith/Le.v b/theories/Arith/Le.v index b73959e7..1febb76b 100644 --- a/theories/Arith/Le.v +++ b/theories/Arith/Le.v @@ -1,13 +1,11 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: Le.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - (** Order on natural numbers. [le] is defined in [Init/Peano.v] as: << Inductive le (n:nat) : nat -> Prop := @@ -18,7 +16,7 @@ where "n <= m" := (le n m) : nat_scope. >> *) -Open Local Scope nat_scope. +Local Open Scope nat_scope. Implicit Types m n p : nat. @@ -48,8 +46,8 @@ Qed. Theorem le_Sn_0 : forall n, ~ S n <= 0. Proof. - red in |- *; intros n H. - change (IsSucc 0) in |- *; elim H; simpl in |- *; auto with arith. + red; intros n H. + change (IsSucc 0); elim H; simpl; auto with arith. Qed. Hint Resolve le_0_n le_Sn_0: arith v62. @@ -84,8 +82,7 @@ Hint Immediate le_Sn_le: arith v62. Theorem le_S_n : forall n m, S n <= S m -> n <= m. Proof. - intros n m H; change (pred (S n) <= pred (S m)) in |- *. - destruct H; simpl; auto with arith. + exact Peano.le_S_n. Qed. Hint Immediate le_S_n: arith v62. @@ -105,11 +102,9 @@ Hint Resolve le_pred_n: arith v62. Theorem le_pred : forall n m, n <= m -> pred n <= pred m. Proof. - destruct n; simpl; auto with arith. - destruct m; simpl; auto with arith. + exact Peano.le_pred. Qed. - (** * [le] is a order on [nat] *) (** Antisymmetry *) diff --git a/theories/Arith/Lt.v b/theories/Arith/Lt.v index 004274fe..8559b782 100644 --- a/theories/Arith/Lt.v +++ b/theories/Arith/Lt.v @@ -1,13 +1,11 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: Lt.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - (** Theorems about [lt] in nat. [lt] is defined in library [Init/Peano.v] as: << Definition lt (n m:nat) := S n <= m. @@ -16,7 +14,7 @@ Infix "<" := lt : nat_scope. *) Require Import Le. -Open Local Scope nat_scope. +Local Open Scope nat_scope. Implicit Types m n p : nat. @@ -53,7 +51,7 @@ Qed. Theorem lt_not_le : forall n m, n < m -> ~ m <= n. Proof. - red in |- *; intros n m Lt Le; exact (le_not_lt m n Le Lt). + red; intros n m Lt Le; exact (le_not_lt m n Le Lt). Qed. Hint Immediate le_not_lt lt_not_le: arith v62. @@ -96,9 +94,9 @@ Proof. Qed. Hint Resolve lt_0_Sn: arith v62. -Theorem lt_n_O : forall n, ~ n < 0. -Proof le_Sn_O. -Hint Resolve lt_n_O: arith v62. +Theorem lt_n_0 : forall n, ~ n < 0. +Proof le_Sn_0. +Hint Resolve lt_n_0: arith v62. (** * Predecessor *) @@ -109,12 +107,12 @@ Qed. Lemma lt_pred : forall n m, S n < m -> n < pred m. Proof. -induction 1; simpl in |- *; auto with arith. +induction 1; simpl; auto with arith. Qed. Hint Immediate lt_pred: arith v62. Lemma lt_pred_n_n : forall n, 0 < n -> pred n < n. -destruct 1; simpl in |- *; auto with arith. +destruct 1; simpl; auto with arith. Qed. Hint Resolve lt_pred_n_n: arith v62. @@ -161,7 +159,7 @@ Hint Immediate lt_le_weak: arith v62. Theorem le_or_lt : forall n m, n <= m \/ m < n. Proof. - intros n m; pattern n, m in |- *; apply nat_double_ind; auto with arith. + intros n m; pattern n, m; apply nat_double_ind; auto with arith. induction 1; auto with arith. Qed. @@ -192,4 +190,5 @@ Hint Immediate lt_0_neq: arith v62. Notation lt_O_Sn := lt_0_Sn (only parsing). Notation neq_O_lt := neq_0_lt (only parsing). Notation lt_O_neq := lt_0_neq (only parsing). +Notation lt_n_O := lt_n_0 (only parsing). (* end hide *) diff --git a/theories/Arith/Max.v b/theories/Arith/Max.v index d1b1b269..5623564a 100644 --- a/theories/Arith/Max.v +++ b/theories/Arith/Max.v @@ -1,44 +1,48 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: Max.v 14641 2011-11-06 11:59:10Z herbelin $ i*) +(** THIS FILE IS DEPRECATED. Use [NPeano.Nat] instead. *) -(** THIS FILE IS DEPRECATED. Use [MinMax] instead. *) - -Require Export MinMax. +Require Import NPeano. Local Open Scope nat_scope. Implicit Types m n p : nat. -Notation max := MinMax.max (only parsing). - -Definition max_0_l := max_0_l. -Definition max_0_r := max_0_r. -Definition succ_max_distr := succ_max_distr. -Definition plus_max_distr_l := plus_max_distr_l. -Definition plus_max_distr_r := plus_max_distr_r. -Definition max_case_strong := max_case_strong. -Definition max_spec := max_spec. -Definition max_dec := max_dec. -Definition max_case := max_case. -Definition max_idempotent := max_id. -Definition max_assoc := max_assoc. -Definition max_comm := max_comm. -Definition max_l := max_l. -Definition max_r := max_r. -Definition le_max_l := le_max_l. -Definition le_max_r := le_max_r. -Definition max_lub_l := max_lub_l. -Definition max_lub_r := max_lub_r. -Definition max_lub := max_lub. +Notation max := Peano.max (only parsing). + +Definition max_0_l := Nat.max_0_l. +Definition max_0_r := Nat.max_0_r. +Definition succ_max_distr := Nat.succ_max_distr. +Definition plus_max_distr_l := Nat.add_max_distr_l. +Definition plus_max_distr_r := Nat.add_max_distr_r. +Definition max_case_strong := Nat.max_case_strong. +Definition max_spec := Nat.max_spec. +Definition max_dec := Nat.max_dec. +Definition max_case := Nat.max_case. +Definition max_idempotent := Nat.max_id. +Definition max_assoc := Nat.max_assoc. +Definition max_comm := Nat.max_comm. +Definition max_l := Nat.max_l. +Definition max_r := Nat.max_r. +Definition le_max_l := Nat.le_max_l. +Definition le_max_r := Nat.le_max_r. +Definition max_lub_l := Nat.max_lub_l. +Definition max_lub_r := Nat.max_lub_r. +Definition max_lub := Nat.max_lub. (* begin hide *) (* Compatibility *) Notation max_case2 := max_case (only parsing). -Notation max_SS := succ_max_distr (only parsing). +Notation max_SS := Nat.succ_max_distr (only parsing). (* end hide *) + +Hint Resolve + Nat.max_l Nat.max_r Nat.le_max_l Nat.le_max_r : arith v62. + +Hint Resolve + Nat.min_l Nat.min_r Nat.le_min_l Nat.le_min_r : arith v62. diff --git a/theories/Arith/Min.v b/theories/Arith/Min.v index 0c8b5669..a2a7930d 100644 --- a/theories/Arith/Min.v +++ b/theories/Arith/Min.v @@ -1,44 +1,42 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: Min.v 14641 2011-11-06 11:59:10Z herbelin $ i*) +(** THIS FILE IS DEPRECATED. Use [NPeano.Nat] instead. *) -(** THIS FILE IS DEPRECATED. Use [MinMax] instead. *) +Require Import NPeano. -Require Export MinMax. - -Open Local Scope nat_scope. +Local Open Scope nat_scope. Implicit Types m n p : nat. -Notation min := MinMax.min (only parsing). +Notation min := Peano.min (only parsing). -Definition min_0_l := min_0_l. -Definition min_0_r := min_0_r. -Definition succ_min_distr := succ_min_distr. -Definition plus_min_distr_l := plus_min_distr_l. -Definition plus_min_distr_r := plus_min_distr_r. -Definition min_case_strong := min_case_strong. -Definition min_spec := min_spec. -Definition min_dec := min_dec. -Definition min_case := min_case. -Definition min_idempotent := min_id. -Definition min_assoc := min_assoc. -Definition min_comm := min_comm. -Definition min_l := min_l. -Definition min_r := min_r. -Definition le_min_l := le_min_l. -Definition le_min_r := le_min_r. -Definition min_glb_l := min_glb_l. -Definition min_glb_r := min_glb_r. -Definition min_glb := min_glb. +Definition min_0_l := Nat.min_0_l. +Definition min_0_r := Nat.min_0_r. +Definition succ_min_distr := Nat.succ_min_distr. +Definition plus_min_distr_l := Nat.add_min_distr_l. +Definition plus_min_distr_r := Nat.add_min_distr_r. +Definition min_case_strong := Nat.min_case_strong. +Definition min_spec := Nat.min_spec. +Definition min_dec := Nat.min_dec. +Definition min_case := Nat.min_case. +Definition min_idempotent := Nat.min_id. +Definition min_assoc := Nat.min_assoc. +Definition min_comm := Nat.min_comm. +Definition min_l := Nat.min_l. +Definition min_r := Nat.min_r. +Definition le_min_l := Nat.le_min_l. +Definition le_min_r := Nat.le_min_r. +Definition min_glb_l := Nat.min_glb_l. +Definition min_glb_r := Nat.min_glb_r. +Definition min_glb := Nat.min_glb. (* begin hide *) (* Compatibility *) Notation min_case2 := min_case (only parsing). -Notation min_SS := succ_min_distr (only parsing). -(* end hide *)
\ No newline at end of file +Notation min_SS := Nat.succ_min_distr (only parsing). +(* end hide *) diff --git a/theories/Arith/MinMax.v b/theories/Arith/MinMax.v deleted file mode 100644 index 8a23c8f6..00000000 --- a/theories/Arith/MinMax.v +++ /dev/null @@ -1,113 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -Require Import Orders NatOrderedType GenericMinMax. - -(** * Maximum and Minimum of two natural numbers *) - -Fixpoint max n m : nat := - match n, m with - | O, _ => m - | S n', O => n - | S n', S m' => S (max n' m') - end. - -Fixpoint min n m : nat := - match n, m with - | O, _ => 0 - | S n', O => 0 - | S n', S m' => S (min n' m') - end. - -(** These functions implement indeed a maximum and a minimum *) - -Lemma max_l : forall x y, y<=x -> max x y = x. -Proof. - induction x; destruct y; simpl; auto with arith. -Qed. - -Lemma max_r : forall x y, x<=y -> max x y = y. -Proof. - induction x; destruct y; simpl; auto with arith. -Qed. - -Lemma min_l : forall x y, x<=y -> min x y = x. -Proof. - induction x; destruct y; simpl; auto with arith. -Qed. - -Lemma min_r : forall x y, y<=x -> min x y = y. -Proof. - induction x; destruct y; simpl; auto with arith. -Qed. - - -Module NatHasMinMax <: HasMinMax Nat_as_OT. - Definition max := max. - Definition min := min. - Definition max_l := max_l. - Definition max_r := max_r. - Definition min_l := min_l. - Definition min_r := min_r. -End NatHasMinMax. - -(** We obtain hence all the generic properties of [max] and [min], - see file [GenericMinMax] or use SearchAbout. *) - -Module Export MMP := UsualMinMaxProperties Nat_as_OT NatHasMinMax. - - -(** * Properties specific to the [nat] domain *) - -(** Simplifications *) - -Lemma max_0_l : forall n, max 0 n = n. -Proof. reflexivity. Qed. - -Lemma max_0_r : forall n, max n 0 = n. -Proof. destruct n; auto. Qed. - -Lemma min_0_l : forall n, min 0 n = 0. -Proof. reflexivity. Qed. - -Lemma min_0_r : forall n, min n 0 = 0. -Proof. destruct n; auto. Qed. - -(** Compatibilities (consequences of monotonicity) *) - -Lemma succ_max_distr : forall n m, S (max n m) = max (S n) (S m). -Proof. auto. Qed. - -Lemma succ_min_distr : forall n m, S (min n m) = min (S n) (S m). -Proof. auto. Qed. - -Lemma plus_max_distr_l : forall n m p, max (p + n) (p + m) = p + max n m. -Proof. -intros. apply max_monotone. repeat red; auto with arith. -Qed. - -Lemma plus_max_distr_r : forall n m p, max (n + p) (m + p) = max n m + p. -Proof. -intros. apply max_monotone with (f:=fun x => x + p). -repeat red; auto with arith. -Qed. - -Lemma plus_min_distr_l : forall n m p, min (p + n) (p + m) = p + min n m. -Proof. -intros. apply min_monotone. repeat red; auto with arith. -Qed. - -Lemma plus_min_distr_r : forall n m p, min (n + p) (m + p) = min n m + p. -Proof. -intros. apply min_monotone with (f:=fun x => x + p). -repeat red; auto with arith. -Qed. - -Hint Resolve - max_l max_r le_max_l le_max_r - min_l min_r le_min_l le_min_r : arith v62. diff --git a/theories/Arith/Minus.v b/theories/Arith/Minus.v index 1b36f236..48024331 100644 --- a/theories/Arith/Minus.v +++ b/theories/Arith/Minus.v @@ -1,13 +1,11 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: Minus.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - (** [minus] (difference between two natural numbers) is defined in [Init/Peano.v] as: << Fixpoint minus (n m:nat) : nat := @@ -23,7 +21,7 @@ where "n - m" := (minus n m) : nat_scope. Require Import Lt. Require Import Le. -Open Local Scope nat_scope. +Local Open Scope nat_scope. Implicit Types m n p : nat. @@ -31,7 +29,7 @@ Implicit Types m n p : nat. Lemma minus_n_O : forall n, n = n - 0. Proof. - induction n; simpl in |- *; auto with arith. + induction n; simpl; auto with arith. Qed. Hint Resolve minus_n_O: arith v62. @@ -39,21 +37,21 @@ Hint Resolve minus_n_O: arith v62. Lemma minus_Sn_m : forall n m, m <= n -> S (n - m) = S n - m. Proof. - intros n m Le; pattern m, n in |- *; apply le_elim_rel; simpl in |- *; + intros n m Le; pattern m, n; apply le_elim_rel; simpl; auto with arith. Qed. Hint Resolve minus_Sn_m: arith v62. Theorem pred_of_minus : forall n, pred n = n - 1. Proof. - intro x; induction x; simpl in |- *; auto with arith. + intro x; induction x; simpl; auto with arith. Qed. (** * Diagonal *) Lemma minus_diag : forall n, n - n = 0. Proof. - induction n; simpl in |- *; auto with arith. + induction n; simpl; auto with arith. Qed. Lemma minus_diag_reverse : forall n, 0 = n - n. @@ -68,7 +66,7 @@ Notation minus_n_n := minus_diag_reverse. Lemma minus_plus_simpl_l_reverse : forall n m p, n - m = p + n - (p + m). Proof. - induction p; simpl in |- *; auto with arith. + induction p; simpl; auto with arith. Qed. Hint Resolve minus_plus_simpl_l_reverse: arith v62. @@ -76,7 +74,7 @@ Hint Resolve minus_plus_simpl_l_reverse: arith v62. Lemma plus_minus : forall n m p, n = m + p -> p = n - m. Proof. - intros n m p; pattern m, n in |- *; apply nat_double_ind; simpl in |- *; + intros n m p; pattern m, n; apply nat_double_ind; simpl; intros. replace (n0 - 0) with n0; auto with arith. absurd (0 = S (n0 + p)); auto with arith. @@ -85,20 +83,20 @@ Qed. Hint Immediate plus_minus: arith v62. Lemma minus_plus : forall n m, n + m - n = m. - symmetry in |- *; auto with arith. + symmetry ; auto with arith. Qed. Hint Resolve minus_plus: arith v62. Lemma le_plus_minus : forall n m, n <= m -> m = n + (m - n). Proof. - intros n m Le; pattern n, m in |- *; apply le_elim_rel; simpl in |- *; + intros n m Le; pattern n, m; apply le_elim_rel; simpl; auto with arith. Qed. Hint Resolve le_plus_minus: arith v62. Lemma le_plus_minus_r : forall n m, n <= m -> n + (m - n) = m. Proof. - symmetry in |- *; auto with arith. + symmetry ; auto with arith. Qed. Hint Resolve le_plus_minus_r: arith v62. @@ -134,7 +132,7 @@ Qed. Lemma lt_minus : forall n m, m <= n -> 0 < m -> n - m < n. Proof. - intros n m Le; pattern m, n in |- *; apply le_elim_rel; simpl in |- *; + intros n m Le; pattern m, n; apply le_elim_rel; simpl; auto using le_minus with arith. intros; absurd (0 < 0); auto with arith. Qed. @@ -142,7 +140,7 @@ Hint Resolve lt_minus: arith v62. Lemma lt_O_minus_lt : forall n m, 0 < n - m -> m < n. Proof. - intros n m; pattern n, m in |- *; apply nat_double_ind; simpl in |- *; + intros n m; pattern n, m; apply nat_double_ind; simpl; auto with arith. intros; absurd (0 < 0); trivial with arith. Qed. @@ -150,9 +148,9 @@ Hint Immediate lt_O_minus_lt: arith v62. Theorem not_le_minus_0 : forall n m, ~ m <= n -> n - m = 0. Proof. - intros y x; pattern y, x in |- *; apply nat_double_ind; - [ simpl in |- *; trivial with arith + intros y x; pattern y, x; apply nat_double_ind; + [ simpl; trivial with arith | intros n H; absurd (0 <= S n); [ assumption | apply le_O_n ] - | simpl in |- *; intros n m H1 H2; apply H1; unfold not in |- *; intros H3; + | simpl; intros n m H1 H2; apply H1; unfold not; intros H3; apply H2; apply le_n_S; assumption ]. Qed. diff --git a/theories/Arith/Mult.v b/theories/Arith/Mult.v index 5dd61d67..cbb9b376 100644 --- a/theories/Arith/Mult.v +++ b/theories/Arith/Mult.v @@ -1,19 +1,17 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: Mult.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Export Plus. Require Export Minus. Require Export Lt. Require Export Le. -Open Local Scope nat_scope. +Local Open Scope nat_scope. Implicit Types m n p : nat. @@ -25,7 +23,7 @@ Implicit Types m n p : nat. Lemma mult_0_r : forall n, n * 0 = 0. Proof. - intro; symmetry in |- *; apply mult_n_O. + intro; symmetry ; apply mult_n_O. Qed. Lemma mult_0_l : forall n, 0 * n = 0. @@ -37,7 +35,7 @@ Qed. Lemma mult_1_l : forall n, 1 * n = n. Proof. - simpl in |- *; auto with arith. + simpl; auto with arith. Qed. Hint Resolve mult_1_l: arith v62. @@ -70,12 +68,12 @@ Hint Resolve mult_plus_distr_r: arith v62. Lemma mult_plus_distr_l : forall n m p, n * (m + p) = n * m + n * p. Proof. induction n. trivial. - intros. simpl in |- *. rewrite IHn. symmetry. apply plus_permute_2_in_4. + intros. simpl. rewrite IHn. symmetry. apply plus_permute_2_in_4. Qed. Lemma mult_minus_distr_r : forall n m p, (n - m) * p = n * p - m * p. Proof. - intros; induction n m using nat_double_ind; simpl; auto with arith. + intros; induction n, m using nat_double_ind; simpl; auto with arith. rewrite <- minus_plus_simpl_l_reverse; auto with arith. Qed. Hint Resolve mult_minus_distr_r: arith v62. @@ -139,13 +137,13 @@ Qed. Lemma mult_O_le : forall n m, m = 0 \/ n <= m * n. Proof. - induction m; simpl in |- *; auto with arith. + induction m; simpl; auto with arith. Qed. Hint Resolve mult_O_le: arith v62. Lemma mult_le_compat_l : forall n m p, n <= m -> p * n <= p * m. Proof. - induction p as [| p IHp]; intros; simpl in |- *. + induction p as [| p IHp]; intros; simpl. apply le_n. auto using plus_le_compat. Qed. @@ -169,7 +167,7 @@ Proof. assumption. apply le_plus_l. (* m*p<=m0*q -> m*p<=(S m0)*q *) - simpl in |- *; apply le_trans with (m0 * q). + simpl; apply le_trans with (m0 * q). assumption. apply le_plus_r. Qed. @@ -177,19 +175,22 @@ Qed. Lemma mult_S_lt_compat_l : forall n m p, m < p -> S n * m < S n * p. Proof. induction n; intros; simpl in *. - rewrite <- 2! plus_n_O; assumption. + rewrite <- 2 plus_n_O; assumption. auto using plus_lt_compat. Qed. Hint Resolve mult_S_lt_compat_l: arith. +Lemma mult_lt_compat_l : forall n m p, n < m -> 0 < p -> p * n < p * m. +Proof. + intros m n p H Hp. destruct p. elim (lt_irrefl _ Hp). + now apply mult_S_lt_compat_l. +Qed. + Lemma mult_lt_compat_r : forall n m p, n < m -> 0 < p -> n * p < m * p. Proof. - intros m n p H H0. - induction p. - elim (lt_irrefl _ H0). - rewrite mult_comm. - replace (n * S p) with (S p * n); auto with arith. + intros m n p H Hp. destruct p. elim (lt_irrefl _ Hp). + rewrite (mult_comm m), (mult_comm n). now apply mult_S_lt_compat_l. Qed. Lemma mult_S_le_reg_l : forall n m p, S n * m <= S n * p -> m <= p. @@ -231,7 +232,7 @@ Fixpoint mult_acc (s:nat) m n : nat := Lemma mult_acc_aux : forall n m p, m + n * p = mult_acc m p n. Proof. - induction n as [| p IHp]; simpl in |- *; auto. + induction n as [| p IHp]; simpl; auto. intros s m; rewrite <- plus_tail_plus; rewrite <- IHp. rewrite <- plus_assoc_reverse; apply f_equal2; auto. rewrite plus_comm; auto. @@ -241,7 +242,7 @@ Definition tail_mult n m := mult_acc 0 m n. Lemma mult_tail_mult : forall n m, n * m = tail_mult n m. Proof. - intros; unfold tail_mult in |- *; rewrite <- mult_acc_aux; auto. + intros; unfold tail_mult; rewrite <- mult_acc_aux; auto. Qed. (** [TailSimpl] transforms any [tail_plus] and [tail_mult] into [plus] @@ -249,4 +250,4 @@ Qed. Ltac tail_simpl := repeat rewrite <- plus_tail_plus; repeat rewrite <- mult_tail_mult; - simpl in |- *. + simpl. diff --git a/theories/Arith/NatOrderedType.v b/theories/Arith/NatOrderedType.v deleted file mode 100644 index fb4bf233..00000000 --- a/theories/Arith/NatOrderedType.v +++ /dev/null @@ -1,64 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -Require Import Lt Peano_dec Compare_dec EqNat - Equalities Orders OrdersTac. - - -(** * DecidableType structure for Peano numbers *) - -Module Nat_as_UBE <: UsualBoolEq. - Definition t := nat. - Definition eq := @eq nat. - Definition eqb := beq_nat. - Definition eqb_eq := beq_nat_true_iff. -End Nat_as_UBE. - -Module Nat_as_DT <: UsualDecidableTypeFull := Make_UDTF Nat_as_UBE. - -(** Note that the last module fulfills by subtyping many other - interfaces, such as [DecidableType] or [EqualityType]. *) - - - -(** * OrderedType structure for Peano numbers *) - -Module Nat_as_OT <: OrderedTypeFull. - Include Nat_as_DT. - Definition lt := lt. - Definition le := le. - Definition compare := nat_compare. - - Instance lt_strorder : StrictOrder lt. - Proof. split; [ exact lt_irrefl | exact lt_trans ]. Qed. - - Instance lt_compat : Proper (Logic.eq==>Logic.eq==>iff) lt. - Proof. repeat red; intros; subst; auto. Qed. - - Definition le_lteq := le_lt_or_eq_iff. - Definition compare_spec := nat_compare_spec. - -End Nat_as_OT. - -(** Note that [Nat_as_OT] can also be seen as a [UsualOrderedType] - and a [OrderedType] (and also as a [DecidableType]). *) - - - -(** * An [order] tactic for Peano numbers *) - -Module NatOrder := OTF_to_OrderTac Nat_as_OT. -Ltac nat_order := NatOrder.order. - -(** Note that [nat_order] is domain-agnostic: it will not prove - [1<=2] or [x<=x+x], but rather things like [x<=y -> y<=x -> x=y]. *) - -Section Test. -Let test : forall x y : nat, x<=y -> y<=x -> x=y. -Proof. nat_order. Qed. -End Test. diff --git a/theories/Arith/Peano_dec.v b/theories/Arith/Peano_dec.v index 5cceab8b..e0bed0d3 100644 --- a/theories/Arith/Peano_dec.v +++ b/theories/Arith/Peano_dec.v @@ -1,16 +1,15 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: Peano_dec.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Import Decidable. - -Open Local Scope nat_scope. +Require Eqdep_dec. +Require Import Le Lt. +Local Open Scope nat_scope. Implicit Types m n x y : nat. @@ -30,5 +29,24 @@ Defined. Hint Resolve O_or_S eq_nat_dec: arith. Theorem dec_eq_nat : forall n m, decidable (n = m). - intros x y; unfold decidable in |- *; elim (eq_nat_dec x y); auto with arith. + intros x y; unfold decidable; elim (eq_nat_dec x y); auto with arith. Defined. + +Definition UIP_nat:= Eqdep_dec.UIP_dec eq_nat_dec. + +Lemma le_unique: forall m n (h1 h2: m <= n), h1 = h2. +Proof. +fix 3. +refine (fun m _ h1 => match h1 as h' in _ <= k return forall hh: m <= k, h' = hh + with le_n => _ |le_S i H => _ end). +refine (fun hh => match hh as h' in _ <= k return forall eq: m = k, + le_n m = match eq in _ = p return m <= p -> m <= m with |eq_refl => fun bli => bli end h' with + |le_n => fun eq => _ |le_S j H' => fun eq => _ end eq_refl). +rewrite (UIP_nat _ _ eq eq_refl). reflexivity. +subst m. destruct (Lt.lt_irrefl j H'). +refine (fun hh => match hh as h' in _ <= k return match k as k' return m <= k' -> Prop + with |0 => fun _ => True |S i' => fun h'' => forall H':m <= i', le_S m i' H' = h'' end h' + with |le_n => _ |le_S j H2 => fun H' => _ end H). +destruct m. exact I. intros; destruct (Lt.lt_irrefl m H'). +f_equal. apply le_unique. +Qed. diff --git a/theories/Arith/Plus.v b/theories/Arith/Plus.v index 12f12300..5428ada3 100644 --- a/theories/Arith/Plus.v +++ b/theories/Arith/Plus.v @@ -1,13 +1,11 @@ -(************************************************************************) + (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: Plus.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - (** Properties of addition. [add] is defined in [Init/Peano.v] as: << Fixpoint plus (n m:nat) : nat := @@ -22,45 +20,32 @@ where "n + m" := (plus n m) : nat_scope. Require Import Le. Require Import Lt. -Open Local Scope nat_scope. +Local Open Scope nat_scope. Implicit Types m n p q : nat. -(** * Zero is neutral *) - -Lemma plus_0_l : forall n, 0 + n = n. -Proof. - reflexivity. -Qed. - -Lemma plus_0_r : forall n, n + 0 = n. -Proof. - intro; symmetry in |- *; apply plus_n_O. -Qed. +(** * Zero is neutral +Deprecated : Already in Init/Peano.v *) +Notation plus_0_l := plus_O_n (only parsing). +Definition plus_0_r n := eq_sym (plus_n_O n). (** * Commutativity *) Lemma plus_comm : forall n m, n + m = m + n. Proof. - intros n m; elim n; simpl in |- *; auto with arith. + intros n m; elim n; simpl; auto with arith. intros y H; elim (plus_n_Sm m y); auto with arith. Qed. Hint Immediate plus_comm: arith v62. (** * Associativity *) -Lemma plus_Snm_nSm : forall n m, S n + m = n + S m. -Proof. - intros. - simpl in |- *. - rewrite (plus_comm n m). - rewrite (plus_comm n (S m)). - trivial with arith. -Qed. +Definition plus_Snm_nSm : forall n m, S n + m = n + S m:= + plus_n_Sm. Lemma plus_assoc : forall n m p, n + (m + p) = n + m + p. Proof. - intros n m p; elim n; simpl in |- *; auto with arith. + intros n m p; elim n; simpl; auto with arith. Qed. Hint Resolve plus_assoc: arith v62. @@ -79,42 +64,42 @@ Hint Resolve plus_assoc_reverse: arith v62. Lemma plus_reg_l : forall n m p, p + n = p + m -> n = m. Proof. - intros m p n; induction n; simpl in |- *; auto with arith. + intros m p n; induction n; simpl; auto with arith. Qed. Lemma plus_le_reg_l : forall n m p, p + n <= p + m -> n <= m. Proof. - induction p; simpl in |- *; auto with arith. + induction p; simpl; auto with arith. Qed. Lemma plus_lt_reg_l : forall n m p, p + n < p + m -> n < m. Proof. - induction p; simpl in |- *; auto with arith. + induction p; simpl; auto with arith. Qed. (** * Compatibility with order *) Lemma plus_le_compat_l : forall n m p, n <= m -> p + n <= p + m. Proof. - induction p; simpl in |- *; auto with arith. + induction p; simpl; auto with arith. Qed. Hint Resolve plus_le_compat_l: arith v62. Lemma plus_le_compat_r : forall n m p, n <= m -> n + p <= m + p. Proof. - induction 1; simpl in |- *; auto with arith. + induction 1; simpl; auto with arith. Qed. Hint Resolve plus_le_compat_r: arith v62. Lemma le_plus_l : forall n m, n <= n + m. Proof. - induction n; simpl in |- *; auto with arith. + induction n; simpl; auto with arith. Qed. Hint Resolve le_plus_l: arith v62. Lemma le_plus_r : forall n m, m <= n + m. Proof. - intros n m; elim n; simpl in |- *; auto with arith. + intros n m; elim n; simpl; auto with arith. Qed. Hint Resolve le_plus_r: arith v62. @@ -132,7 +117,7 @@ Hint Immediate lt_plus_trans: arith v62. Lemma plus_lt_compat_l : forall n m p, n < m -> p + n < p + m. Proof. - induction p; simpl in |- *; auto with arith. + induction p; simpl; auto with arith. Qed. Hint Resolve plus_lt_compat_l: arith v62. @@ -146,18 +131,18 @@ Hint Resolve plus_lt_compat_r: arith v62. Lemma plus_le_compat : forall n m p q, n <= m -> p <= q -> n + p <= m + q. Proof. intros n m p q H H0. - elim H; simpl in |- *; auto with arith. + elim H; simpl; auto with arith. Qed. Lemma plus_le_lt_compat : forall n m p q, n <= m -> p < q -> n + p < m + q. Proof. - unfold lt in |- *. intros. change (S n + p <= m + q) in |- *. rewrite plus_Snm_nSm. + unfold lt. intros. change (S n + p <= m + q). rewrite plus_Snm_nSm. apply plus_le_compat; assumption. Qed. Lemma plus_lt_le_compat : forall n m p q, n < m -> p <= q -> n + p < m + q. Proof. - unfold lt in |- *. intros. change (S n + p <= m + q) in |- *. apply plus_le_compat; assumption. + unfold lt. intros. change (S n + p <= m + q). apply plus_le_compat; assumption. Qed. Lemma plus_lt_compat : forall n m p q, n < m -> p < q -> n + p < m + q. @@ -205,8 +190,8 @@ Fixpoint tail_plus n m : nat := end. Lemma plus_tail_plus : forall n m, n + m = tail_plus n m. -induction n as [| n IHn]; simpl in |- *; auto. -intro m; rewrite <- IHn; simpl in |- *; auto. +induction n as [| n IHn]; simpl; auto. +intro m; rewrite <- IHn; simpl; auto. Qed. (** * Discrimination *) diff --git a/theories/Arith/Wf_nat.v b/theories/Arith/Wf_nat.v index 23419531..b5545123 100644 --- a/theories/Arith/Wf_nat.v +++ b/theories/Arith/Wf_nat.v @@ -1,18 +1,16 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) (************************************************************************) -(*i $Id: Wf_nat.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - (** Well-founded relations and natural numbers *) Require Import Lt. -Open Local Scope nat_scope. +Local Open Scope nat_scope. Implicit Types m n p : nat. @@ -26,14 +24,14 @@ Definition gtof (a b:A) := f b > f a. Theorem well_founded_ltof : well_founded ltof. Proof. - red in |- *. + red. cut (forall n (a:A), f a < n -> Acc ltof a). intros H a; apply (H (S (f a))); auto with arith. induction n. intros; absurd (f a < 0); auto with arith. intros a ltSma. apply Acc_intro. - unfold ltof in |- *; intros b ltfafb. + unfold ltof; intros b ltfafb. apply IHn. apply lt_le_trans with (f a); auto with arith. Defined. @@ -75,7 +73,7 @@ Proof. intros; absurd (f a < 0); auto with arith. intros a ltSma. apply F. - unfold ltof in |- *; intros b ltfafb. + unfold ltof; intros b ltfafb. apply IHn. apply lt_le_trans with (f a); auto with arith. Defined. @@ -110,7 +108,7 @@ Hypothesis H_compat : forall x y:A, R x y -> f x < f y. Theorem well_founded_lt_compat : well_founded R. Proof. - red in |- *. + red. cut (forall n (a:A), f a < n -> Acc R a). intros H a; apply (H (S (f a))); auto with arith. induction n. @@ -163,8 +161,8 @@ Lemma lt_wf_double_rec : (forall p q, p < n -> P p q) -> (forall p, p < m -> P n p) -> P n m) -> forall n m, P n m. Proof. - intros P Hrec p; pattern p in |- *; apply lt_wf_rec. - intros n H q; pattern q in |- *; apply lt_wf_rec; auto with arith. + intros P Hrec p; pattern p; apply lt_wf_rec. + intros n H q; pattern q; apply lt_wf_rec; auto with arith. Defined. Lemma lt_wf_double_ind : @@ -173,8 +171,8 @@ Lemma lt_wf_double_ind : (forall p (q:nat), p < n -> P p q) -> (forall p, p < m -> P n p) -> P n m) -> forall n m, P n m. Proof. - intros P Hrec p; pattern p in |- *; apply lt_wf_ind. - intros n H q; pattern q in |- *; apply lt_wf_ind; auto with arith. + intros P Hrec p; pattern p; apply lt_wf_ind. + intros n H q; pattern q; apply lt_wf_ind; auto with arith. Qed. Hint Resolve lt_wf: arith. @@ -192,7 +190,7 @@ Section LT_WF_REL. Remark acc_lt_rel : forall x:A, (exists n, F x n) -> Acc R x. Proof. intros x [n fxn]; generalize dependent x. - pattern n in |- *; apply lt_wf_ind; intros. + pattern n; apply lt_wf_ind; intros. constructor; intros. destruct (F_compat y x) as (x0,H1,H2); trivial. apply (H x0); auto. @@ -260,19 +258,6 @@ Qed. Unset Implicit Arguments. -(** [n]th iteration of the function [f] *) - -Fixpoint iter_nat (n:nat) (A:Type) (f:A -> A) (x:A) : A := - match n with - | O => x - | S n' => f (iter_nat n' A f x) - end. - -Theorem iter_nat_plus : - forall (n m:nat) (A:Type) (f:A -> A) (x:A), - iter_nat (n + m) A f x = iter_nat n A f (iter_nat m A f x). -Proof. - simple induction n; - [ simpl in |- *; auto with arith - | intros; simpl in |- *; apply f_equal with (f := f); apply H ]. -Qed. +Notation iter_nat := @nat_iter (only parsing). +Notation iter_nat_plus := @nat_iter_plus (only parsing). +Notation iter_nat_invariant := @nat_iter_invariant (only parsing). diff --git a/theories/Arith/vo.itarget b/theories/Arith/vo.itarget index c3f29d21..0b6564e1 100644 --- a/theories/Arith/vo.itarget +++ b/theories/Arith/vo.itarget @@ -19,5 +19,3 @@ Mult.vo Peano_dec.vo Plus.vo Wf_nat.vo -NatOrderedType.vo -MinMax.vo |