diff options
author | Benjamin Barenblat <bbaren@debian.org> | 2018-12-29 14:31:27 -0500 |
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committer | Benjamin Barenblat <bbaren@debian.org> | 2018-12-29 14:31:27 -0500 |
commit | 9043add656177eeac1491a73d2f3ab92bec0013c (patch) | |
tree | 2b0092c84bfbf718eca10c81f60b2640dc8cab05 /theories/Numbers/Cyclic/Abstract | |
parent | a4c7f8bd98be2a200489325ff7c5061cf80ab4f3 (diff) |
Imported Upstream version 8.8.2upstream/8.8.2
Diffstat (limited to 'theories/Numbers/Cyclic/Abstract')
-rw-r--r-- | theories/Numbers/Cyclic/Abstract/CyclicAxioms.v | 12 | ||||
-rw-r--r-- | theories/Numbers/Cyclic/Abstract/DoubleType.v | 71 | ||||
-rw-r--r-- | theories/Numbers/Cyclic/Abstract/NZCyclic.v | 33 |
3 files changed, 106 insertions, 10 deletions
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/Abstract/DoubleType.v b/theories/Numbers/Cyclic/Abstract/DoubleType.v new file mode 100644 index 00000000..fe0476e4 --- /dev/null +++ b/theories/Numbers/Cyclic/Abstract/DoubleType.v @@ -0,0 +1,71 @@ +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS 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) *) +(************************************************************************) +(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) +(************************************************************************) + +Set Implicit Arguments. + +Require Import ZArith. +Local Open Scope Z_scope. + +Definition base digits := Z.pow 2 (Zpos digits). +Arguments base digits: simpl never. + +Section Carry. + + Variable A : Type. + + Inductive carry := + | C0 : A -> carry + | C1 : A -> carry. + + Definition interp_carry (sign:Z)(B:Z)(interp:A -> Z) c := + match c with + | C0 x => interp x + | C1 x => sign*B + interp x + end. + +End Carry. + +Section Zn2Z. + + Variable znz : Type. + + (** From a type [znz] representing a cyclic structure Z/nZ, + we produce a representation of Z/2nZ by pairs of elements of [znz] + (plus a special case for zero). High half of the new number comes + first. + *) + + Inductive zn2z := + | W0 : zn2z + | WW : znz -> znz -> zn2z. + + Definition zn2z_to_Z (wB:Z) (w_to_Z:znz->Z) (x:zn2z) := + match x with + | W0 => 0 + | WW xh xl => w_to_Z xh * wB + w_to_Z xl + end. + +End Zn2Z. + +Arguments W0 {znz}. + +(** From a cyclic representation [w], we iterate the [zn2z] construct + [n] times, gaining the type of binary trees of depth at most [n], + whose leafs are either W0 (if depth < n) or elements of w + (if depth = n). +*) + +Fixpoint word (w:Type) (n:nat) : Type := + match n with + | 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]. |