diff options
Diffstat (limited to 'theories/Numbers/Integer/SpecViaZ')
| -rw-r--r-- | theories/Numbers/Integer/SpecViaZ/ZSig.v | 117 | ||||
| -rw-r--r-- | theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v | 306 |
2 files changed, 423 insertions, 0 deletions
diff --git a/theories/Numbers/Integer/SpecViaZ/ZSig.v b/theories/Numbers/Integer/SpecViaZ/ZSig.v new file mode 100644 index 00000000..0af98c74 --- /dev/null +++ b/theories/Numbers/Integer/SpecViaZ/ZSig.v @@ -0,0 +1,117 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) +(************************************************************************) + +(*i $Id: ZSig.v 11027 2008-06-01 13:28:59Z letouzey $ i*) + +Require Import ZArith Znumtheory. + +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. + Notation "[ x ]" := (to_Z x). + + Definition eq x y := ([x] = [y]). + + Parameter of_Z : Z -> t. + Parameter spec_of_Z: forall x, to_Z (of_Z x) = x. + + Parameter zero : t. + Parameter one : t. + Parameter minus_one : t. + + Parameter spec_0: [zero] = 0. + Parameter spec_1: [one] = 1. + Parameter spec_m1: [minus_one] = -1. + + Parameter compare : t -> t -> comparison. + + Parameter spec_compare: forall x y, + match compare x y with + | Eq => [x] = [y] + | Lt => [x] < [y] + | Gt => [x] > [y] + end. + + Definition lt n m := compare n m = Lt. + Definition le n m := compare n m <> Gt. + 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. + + Parameter eq_bool : t -> t -> bool. + + Parameter spec_eq_bool: forall x y, + if eq_bool x y then [x] = [y] else [x] <> [y]. + + Parameter succ : t -> t. + + Parameter spec_succ: forall n, [succ n] = [n] + 1. + + Parameter add : t -> t -> t. + + Parameter spec_add: forall x y, [add x y] = [x] + [y]. + + Parameter pred : t -> t. + + Parameter spec_pred: forall x, [pred x] = [x] - 1. + + Parameter sub : t -> t -> t. + + Parameter spec_sub: forall x y, [sub x y] = [x] - [y]. + + Parameter opp : t -> t. + + Parameter spec_opp: forall x, [opp x] = - [x]. + + Parameter mul : t -> t -> t. + + Parameter spec_mul: forall x y, [mul x y] = [x] * [y]. + + Parameter square : t -> t. + + Parameter spec_square: forall x, [square x] = [x] * [x]. + + Parameter power_pos : t -> positive -> t. + + Parameter spec_power_pos: forall x n, [power_pos x n] = [x] ^ Zpos n. + + Parameter sqrt : t -> t. + + Parameter spec_sqrt: forall x, 0 <= [x] -> + [sqrt x] ^ 2 <= [x] < ([sqrt x] + 1) ^ 2. + + Parameter div_eucl : t -> t -> t * t. + + Parameter spec_div_eucl: forall x y, [y] <> 0 -> + let (q,r) := div_eucl x y in ([q], [r]) = Zdiv_eucl [x] [y]. + + Parameter div : t -> t -> t. + + Parameter spec_div: forall x y, [y] <> 0 -> [div x y] = [x] / [y]. + + Parameter modulo : t -> t -> t. + + Parameter spec_modulo: forall x y, [y] <> 0 -> + [modulo x y] = [x] mod [y]. + + Parameter gcd : t -> t -> t. + + Parameter spec_gcd: forall a b, [gcd a b] = Zgcd (to_Z a) (to_Z b). + +End ZType. diff --git a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v new file mode 100644 index 00000000..d7c56267 --- /dev/null +++ b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v @@ -0,0 +1,306 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: ZSigZAxioms.v 11040 2008-06-03 00:04:16Z letouzey $ i*) + +Require Import ZArith. +Require Import ZAxioms. +Require Import ZSig. + +(** * The interface [ZSig.ZType] implies the interface [ZAxiomsSig] *) + +Module ZSig_ZAxioms (Z:ZType) <: ZAxiomsSig. + +Delimit Scope IntScope with Int. +Bind Scope IntScope with Z.t. +Open Local Scope IntScope. +Notation "[ x ]" := (Z.to_Z x) : IntScope. +Infix "==" := Z.eq (at level 70) : IntScope. +Notation "0" := Z.zero : IntScope. +Infix "+" := Z.add : IntScope. +Infix "-" := Z.sub : IntScope. +Infix "*" := Z.mul : IntScope. +Notation "- x" := (Z.opp x) : IntScope. + +Hint Rewrite + Z.spec_0 Z.spec_1 Z.spec_add Z.spec_sub Z.spec_pred Z.spec_succ + Z.spec_mul Z.spec_opp Z.spec_of_Z : Zspec. + +Ltac zsimpl := unfold Z.eq in *; autorewrite with Zspec. + +Module Export NZOrdAxiomsMod <: NZOrdAxiomsSig. +Module Export NZAxiomsMod <: NZAxiomsSig. + +Definition NZ := Z.t. +Definition NZeq := Z.eq. +Definition NZ0 := Z.zero. +Definition NZsucc := Z.succ. +Definition NZpred := Z.pred. +Definition NZadd := Z.add. +Definition NZsub := Z.sub. +Definition NZmul := Z.mul. + +Theorem NZeq_equiv : equiv Z.t Z.eq. +Proof. +repeat split; repeat red; intros; auto; congruence. +Qed. + +Add Relation Z.t Z.eq + reflexivity proved by (proj1 NZeq_equiv) + symmetry proved by (proj2 (proj2 NZeq_equiv)) + transitivity proved by (proj1 (proj2 NZeq_equiv)) + as NZeq_rel. + +Add Morphism NZsucc with signature Z.eq ==> Z.eq as NZsucc_wd. +Proof. +intros; zsimpl; f_equal; assumption. +Qed. + +Add Morphism NZpred with signature Z.eq ==> Z.eq as NZpred_wd. +Proof. +intros; zsimpl; f_equal; assumption. +Qed. + +Add Morphism NZadd with signature Z.eq ==> Z.eq ==> Z.eq as NZadd_wd. +Proof. +intros; zsimpl; f_equal; assumption. +Qed. + +Add Morphism NZsub with signature Z.eq ==> Z.eq ==> Z.eq as NZsub_wd. +Proof. +intros; zsimpl; f_equal; assumption. +Qed. + +Add Morphism NZmul with signature Z.eq ==> Z.eq ==> Z.eq as NZmul_wd. +Proof. +intros; zsimpl; f_equal; assumption. +Qed. + +Theorem NZpred_succ : forall n, Z.pred (Z.succ n) == n. +Proof. +intros; zsimpl; auto with zarith. +Qed. + +Section Induction. + +Variable A : Z.t -> Prop. +Hypothesis A_wd : predicate_wd Z.eq A. +Hypothesis A0 : A 0. +Hypothesis AS : forall n, A n <-> A (Z.succ n). + +Add Morphism A with signature Z.eq ==> iff as A_morph. +Proof. apply A_wd. Qed. + +Let B (z : Z) := A (Z.of_Z z). + +Lemma B0 : B 0. +Proof. +unfold B; simpl. +rewrite <- (A_wd 0); auto. +zsimpl; 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 (Z.of_Z (z + 1)) with (Z.succ (Z.of_Z z)); auto. +zsimpl; auto. +Qed. + +Lemma BP : forall z : Z, B z -> B (z - 1). +Proof. +intros z H. +unfold B in *. rewrite AS. +setoid_replace (Z.succ (Z.of_Z (z - 1))) with (Z.of_Z z); auto. +zsimpl; 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 Zopp_succ; unfold Zpred; apply BP; auto. +subst z'; auto with zarith. +Qed. + +Theorem NZinduction : forall n, A n. +Proof. +intro n. setoid_replace n with (Z.of_Z (Z.to_Z n)). +apply B_holds. +zsimpl; auto. +Qed. + +End Induction. + +Theorem NZadd_0_l : forall n, 0 + n == n. +Proof. +intros; zsimpl; auto with zarith. +Qed. + +Theorem NZadd_succ_l : forall n m, (Z.succ n) + m == Z.succ (n + m). +Proof. +intros; zsimpl; auto with zarith. +Qed. + +Theorem NZsub_0_r : forall n, n - 0 == n. +Proof. +intros; zsimpl; auto with zarith. +Qed. + +Theorem NZsub_succ_r : forall n m, n - (Z.succ m) == Z.pred (n - m). +Proof. +intros; zsimpl; auto with zarith. +Qed. + +Theorem NZmul_0_l : forall n, 0 * n == 0. +Proof. +intros; zsimpl; auto with zarith. +Qed. + +Theorem NZmul_succ_l : forall n m, (Z.succ n) * m == n * m + m. +Proof. +intros; zsimpl; ring. +Qed. + +End NZAxiomsMod. + +Definition NZlt := Z.lt. +Definition NZle := Z.le. +Definition NZmin := Z.min. +Definition NZmax := Z.max. + +Infix "<=" := Z.le : IntScope. +Infix "<" := Z.lt : IntScope. + +Lemma spec_compare_alt : forall x y, Z.compare x y = ([x] ?= [y])%Z. +Proof. + intros; generalize (Z.spec_compare x y). + destruct (Z.compare x y); auto. + intros H; rewrite H; symmetry; apply Zcompare_refl. +Qed. + +Lemma spec_lt : forall x y, (x<y) <-> ([x]<[y])%Z. +Proof. + intros; unfold Z.lt, Zlt; rewrite spec_compare_alt; intuition. +Qed. + +Lemma spec_le : forall x y, (x<=y) <-> ([x]<=[y])%Z. +Proof. + intros; unfold Z.le, Zle; rewrite spec_compare_alt; intuition. +Qed. + +Lemma spec_min : forall x y, [Z.min x y] = Zmin [x] [y]. +Proof. + intros; unfold Z.min, Zmin. + rewrite spec_compare_alt; destruct Zcompare; auto. +Qed. + +Lemma spec_max : forall x y, [Z.max x y] = Zmax [x] [y]. +Proof. + intros; unfold Z.max, Zmax. + rewrite spec_compare_alt; destruct Zcompare; auto. +Qed. + +Add Morphism Z.compare with signature Z.eq ==> Z.eq ==> (@eq comparison) as compare_wd. +Proof. +intros x x' Hx y y' Hy. +rewrite 2 spec_compare_alt; rewrite Hx, Hy; intuition. +Qed. + +Add Morphism Z.lt with signature Z.eq ==> Z.eq ==> iff as NZlt_wd. +Proof. +intros x x' Hx y y' Hy; unfold Z.lt; rewrite Hx, Hy; intuition. +Qed. + +Add Morphism Z.le with signature Z.eq ==> Z.eq ==> iff as NZle_wd. +Proof. +intros x x' Hx y y' Hy; unfold Z.le; rewrite Hx, Hy; intuition. +Qed. + +Add Morphism Z.min with signature Z.eq ==> Z.eq ==> Z.eq as NZmin_wd. +Proof. +intros; red; rewrite 2 spec_min; congruence. +Qed. + +Add Morphism Z.max with signature Z.eq ==> Z.eq ==> Z.eq as NZmax_wd. +Proof. +intros; red; rewrite 2 spec_max; congruence. +Qed. + +Theorem NZlt_eq_cases : forall n m, n <= m <-> n < m \/ n == m. +Proof. +intros. +unfold Z.eq; rewrite spec_lt, spec_le; omega. +Qed. + +Theorem NZlt_irrefl : forall n, ~ n < n. +Proof. +intros; rewrite spec_lt; auto with zarith. +Qed. + +Theorem NZlt_succ_r : forall n m, n < (Z.succ m) <-> n <= m. +Proof. +intros; rewrite spec_lt, spec_le, Z.spec_succ; omega. +Qed. + +Theorem NZmin_l : forall n m, n <= m -> Z.min n m == n. +Proof. +intros n m; unfold Z.eq; rewrite spec_le, spec_min. +generalize (Zmin_spec [n] [m]); omega. +Qed. + +Theorem NZmin_r : forall n m, m <= n -> Z.min n m == m. +Proof. +intros n m; unfold Z.eq; rewrite spec_le, spec_min. +generalize (Zmin_spec [n] [m]); omega. +Qed. + +Theorem NZmax_l : forall n m, m <= n -> Z.max n m == n. +Proof. +intros n m; unfold Z.eq; rewrite spec_le, spec_max. +generalize (Zmax_spec [n] [m]); omega. +Qed. + +Theorem NZmax_r : forall n m, n <= m -> Z.max n m == m. +Proof. +intros n m; unfold Z.eq; rewrite spec_le, spec_max. +generalize (Zmax_spec [n] [m]); omega. +Qed. + +End NZOrdAxiomsMod. + +Definition Zopp := Z.opp. + +Add Morphism Z.opp with signature Z.eq ==> Z.eq as Zopp_wd. +Proof. +intros; zsimpl; auto with zarith. +Qed. + +Theorem Zsucc_pred : forall n, Z.succ (Z.pred n) == n. +Proof. +red; intros; zsimpl; auto with zarith. +Qed. + +Theorem Zopp_0 : - 0 == 0. +Proof. +red; intros; zsimpl; auto with zarith. +Qed. + +Theorem Zopp_succ : forall n, - (Z.succ n) == Z.pred (- n). +Proof. +intros; zsimpl; auto with zarith. +Qed. + +End ZSig_ZAxioms. |