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Diffstat (limited to 'theories7/IntMap/Addr.v')
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diff --git a/theories7/IntMap/Addr.v b/theories7/IntMap/Addr.v deleted file mode 100644 index 9f362772..00000000 --- a/theories7/IntMap/Addr.v +++ /dev/null @@ -1,456 +0,0 @@ -(************************************************************************) -(* 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: Addr.v,v 1.1.2.1 2004/07/16 19:31:27 herbelin Exp $ i*) - -(** Representation of adresses by the [positive] type of binary numbers *) - -Require Bool. -Require ZArith. - -Inductive ad : Set := - ad_z : ad - | ad_x : positive -> ad. - -Lemma ad_sum : (a:ad) {p:positive | a=(ad_x p)}+{a=ad_z}. -Proof. - NewDestruct a; Auto. - Left; Exists p; Trivial. -Qed. - -Fixpoint p_xor [p:positive] : positive -> ad := - [p2] Cases p of - xH => Cases p2 of - xH => ad_z - | (xO p'2) => (ad_x (xI p'2)) - | (xI p'2) => (ad_x (xO p'2)) - end - | (xO p') => Cases p2 of - xH => (ad_x (xI p')) - | (xO p'2) => Cases (p_xor p' p'2) of - ad_z => ad_z - | (ad_x p'') => (ad_x (xO p'')) - end - | (xI p'2) => Cases (p_xor p' p'2) of - ad_z => (ad_x xH) - | (ad_x p'') => (ad_x (xI p'')) - end - end - | (xI p') => Cases p2 of - xH => (ad_x (xO p')) - | (xO p'2) => Cases (p_xor p' p'2) of - ad_z => (ad_x xH) - | (ad_x p'') => (ad_x (xI p'')) - end - | (xI p'2) => Cases (p_xor p' p'2) of - ad_z => ad_z - | (ad_x p'') => (ad_x (xO p'')) - end - end - end. - -Definition ad_xor := [a,a':ad] - Cases a of - ad_z => a' - | (ad_x p) => Cases a' of - ad_z => a - | (ad_x p') => (p_xor p p') - end - end. - -Lemma ad_xor_neutral_left : (a:ad) (ad_xor ad_z a)=a. -Proof. - Trivial. -Qed. - -Lemma ad_xor_neutral_right : (a:ad) (ad_xor a ad_z)=a. -Proof. - NewDestruct a; Trivial. -Qed. - -Lemma ad_xor_comm : (a,a':ad) (ad_xor a a')=(ad_xor a' a). -Proof. - NewDestruct a; NewDestruct a'; Simpl; Auto. - Generalize p0; Clear p0; NewInduction p as [p Hrecp|p Hrecp|]; Simpl; Auto. - NewDestruct p0; Simpl; Trivial; Intros. - Rewrite Hrecp; Trivial. - Rewrite Hrecp; Trivial. - NewDestruct p0; Simpl; Trivial; Intros. - Rewrite Hrecp; Trivial. - Rewrite Hrecp; Trivial. - NewDestruct p0; Simpl; Auto. -Qed. - -Lemma ad_xor_nilpotent : (a:ad) (ad_xor a a)=ad_z. -Proof. - NewDestruct a; Trivial. - Simpl. NewInduction p as [p IHp|p IHp|]; Trivial. - Simpl. Rewrite IHp; Reflexivity. - Simpl. Rewrite IHp; Reflexivity. -Qed. - -Fixpoint ad_bit_1 [p:positive] : nat -> bool := - Cases p of - xH => [n:nat] Cases n of - O => true - | (S _) => false - end - | (xO p) => [n:nat] Cases n of - O => false - | (S n') => (ad_bit_1 p n') - end - | (xI p) => [n:nat] Cases n of - O => true - | (S n') => (ad_bit_1 p n') - end - end. - -Definition ad_bit := [a:ad] - Cases a of - ad_z => [_:nat] false - | (ad_x p) => (ad_bit_1 p) - end. - -Definition eqf := [f,g:nat->bool] (n:nat) (f n)=(g n). - -Lemma ad_faithful_1 : (a:ad) (eqf (ad_bit ad_z) (ad_bit a)) -> ad_z=a. -Proof. - NewDestruct a. Trivial. - NewInduction p as [p IHp|p IHp|];Intro H. Absurd ad_z=(ad_x p). Discriminate. - Exact (IHp [n:nat](H (S n))). - Absurd ad_z=(ad_x p). Discriminate. - Exact (IHp [n:nat](H (S n))). - Absurd false=true. Discriminate. - Exact (H O). -Qed. - -Lemma ad_faithful_2 : (a:ad) (eqf (ad_bit (ad_x xH)) (ad_bit a)) -> (ad_x xH)=a. -Proof. - NewDestruct a. Intros. Absurd true=false. Discriminate. - Exact (H O). - NewDestruct p. Intro H. Absurd ad_z=(ad_x p). Discriminate. - Exact (ad_faithful_1 (ad_x p) [n:nat](H (S n))). - Intros. Absurd true=false. Discriminate. - Exact (H O). - Trivial. -Qed. - -Lemma ad_faithful_3 : - (a:ad) (p:positive) - ((p':positive) (eqf (ad_bit (ad_x p)) (ad_bit (ad_x p'))) -> p=p') -> - (eqf (ad_bit (ad_x (xO p))) (ad_bit a)) -> - (ad_x (xO p))=a. -Proof. - NewDestruct a. Intros. Cut (eqf (ad_bit ad_z) (ad_bit (ad_x (xO p)))). - Intro. Rewrite (ad_faithful_1 (ad_x (xO p)) H1). Reflexivity. - Unfold eqf. Intro. Unfold eqf in H0. Rewrite H0. Reflexivity. - Case p. Intros. Absurd false=true. Discriminate. - Exact (H0 O). - Intros. Rewrite (H p0 [n:nat](H0 (S n))). Reflexivity. - Intros. Absurd false=true. Discriminate. - Exact (H0 O). -Qed. - -Lemma ad_faithful_4 : - (a:ad) (p:positive) - ((p':positive) (eqf (ad_bit (ad_x p)) (ad_bit (ad_x p'))) -> p=p') -> - (eqf (ad_bit (ad_x (xI p))) (ad_bit a)) -> - (ad_x (xI p))=a. -Proof. - NewDestruct a. Intros. Cut (eqf (ad_bit ad_z) (ad_bit (ad_x (xI p)))). - Intro. Rewrite (ad_faithful_1 (ad_x (xI p)) H1). Reflexivity. - Unfold eqf. Intro. Unfold eqf in H0. Rewrite H0. Reflexivity. - Case p. Intros. Rewrite (H p0 [n:nat](H0 (S n))). Reflexivity. - Intros. Absurd true=false. Discriminate. - Exact (H0 O). - Intros. Absurd ad_z=(ad_x p0). Discriminate. - Cut (eqf (ad_bit (ad_x xH)) (ad_bit (ad_x (xI p0)))). - Intro. Exact (ad_faithful_1 (ad_x p0) [n:nat](H1 (S n))). - Unfold eqf. Unfold eqf in H0. Intro. Rewrite H0. Reflexivity. -Qed. - -Lemma ad_faithful : (a,a':ad) (eqf (ad_bit a) (ad_bit a')) -> a=a'. -Proof. - NewDestruct a. Exact ad_faithful_1. - NewInduction p. Intros a' H. Apply ad_faithful_4. Intros. Cut (ad_x p)=(ad_x p'). - Intro. Inversion H1. Reflexivity. - Exact (IHp (ad_x p') H0). - Assumption. - Intros. Apply ad_faithful_3. Intros. Cut (ad_x p)=(ad_x p'). Intro. Inversion H1. Reflexivity. - Exact (IHp (ad_x p') H0). - Assumption. - Exact ad_faithful_2. -Qed. - -Definition adf_xor := [f,g:nat->bool; n:nat] (xorb (f n) (g n)). - -Lemma ad_xor_sem_1 : (a':ad) (ad_bit (ad_xor ad_z a') O)=(ad_bit a' O). -Proof. - Trivial. -Qed. - -Lemma ad_xor_sem_2 : (a':ad) (ad_bit (ad_xor (ad_x xH) a') O)=(negb (ad_bit a' O)). -Proof. - Intro. Case a'. Trivial. - Simpl. Intro. - Case p; Trivial. -Qed. - -Lemma ad_xor_sem_3 : - (p:positive) (a':ad) (ad_bit (ad_xor (ad_x (xO p)) a') O)=(ad_bit a' O). -Proof. - Intros. Case a'. Trivial. - Simpl. Intro. - Case p0; Trivial. Intro. - Case (p_xor p p1); Trivial. - Intro. Case (p_xor p p1); Trivial. -Qed. - -Lemma ad_xor_sem_4 : (p:positive) (a':ad) - (ad_bit (ad_xor (ad_x (xI p)) a') O)=(negb (ad_bit a' O)). -Proof. - Intros. Case a'. Trivial. - Simpl. Intro. Case p0; Trivial. Intro. - Case (p_xor p p1); Trivial. - Intro. - Case (p_xor p p1); Trivial. -Qed. - -Lemma ad_xor_sem_5 : - (a,a':ad) (ad_bit (ad_xor a a') O)=(adf_xor (ad_bit a) (ad_bit a') O). -Proof. - NewDestruct a. Intro. Change (ad_bit a' O)=(xorb false (ad_bit a' O)). Rewrite false_xorb. Trivial. - Case p. Exact ad_xor_sem_4. - Intros. Change (ad_bit (ad_xor (ad_x (xO p0)) a') O)=(xorb false (ad_bit a' O)). - Rewrite false_xorb. Apply ad_xor_sem_3. Exact ad_xor_sem_2. -Qed. - -Lemma ad_xor_sem_6 : (n:nat) - ((a,a':ad) (ad_bit (ad_xor a a') n)=(adf_xor (ad_bit a) (ad_bit a') n)) -> - (a,a':ad) (ad_bit (ad_xor a a') (S n))=(adf_xor (ad_bit a) (ad_bit a') (S n)). -Proof. - Intros. Case a. Unfold adf_xor. Unfold 2 ad_bit. Rewrite false_xorb. Reflexivity. - Case a'. Unfold adf_xor. Unfold 3 ad_bit. Intro. Rewrite xorb_false. Reflexivity. - Intros. Case p0. Case p. Intros. - Change (ad_bit (ad_xor (ad_x (xI p2)) (ad_x (xI p1))) (S n)) - =(adf_xor (ad_bit (ad_x p2)) (ad_bit (ad_x p1)) n). - Rewrite <- H. Simpl. - Case (p_xor p2 p1); Trivial. - Intros. - Change (ad_bit (ad_xor (ad_x (xI p2)) (ad_x (xO p1))) (S n)) - =(adf_xor (ad_bit (ad_x p2)) (ad_bit (ad_x p1)) n). - Rewrite <- H. Simpl. - Case (p_xor p2 p1); Trivial. - Intro. Unfold adf_xor. Unfold 3 ad_bit. Unfold ad_bit_1. Rewrite xorb_false. Reflexivity. - Case p. Intros. - Change (ad_bit (ad_xor (ad_x (xO p2)) (ad_x (xI p1))) (S n)) - =(adf_xor (ad_bit (ad_x p2)) (ad_bit (ad_x p1)) n). - Rewrite <- H. Simpl. - Case (p_xor p2 p1); Trivial. - Intros. - Change (ad_bit (ad_xor (ad_x (xO p2)) (ad_x (xO p1))) (S n)) - =(adf_xor (ad_bit (ad_x p2)) (ad_bit (ad_x p1)) n). - Rewrite <- H. Simpl. - Case (p_xor p2 p1); Trivial. - Intro. Unfold adf_xor. Unfold 3 ad_bit. Unfold ad_bit_1. Rewrite xorb_false. Reflexivity. - Unfold adf_xor. Unfold 2 ad_bit. Unfold ad_bit_1. Rewrite false_xorb. Simpl. Case p; Trivial. -Qed. - -Lemma ad_xor_semantics : - (a,a':ad) (eqf (ad_bit (ad_xor a a')) (adf_xor (ad_bit a) (ad_bit a'))). -Proof. - Unfold eqf. Intros. Generalize a a'. Elim n. Exact ad_xor_sem_5. - Exact ad_xor_sem_6. -Qed. - -Lemma eqf_sym : (f,f':nat->bool) (eqf f f') -> (eqf f' f). -Proof. - Unfold eqf. Intros. Rewrite H. Reflexivity. -Qed. - -Lemma eqf_refl : (f:nat->bool) (eqf f f). -Proof. - Unfold eqf. Trivial. -Qed. - -Lemma eqf_trans : (f,f',f'':nat->bool) (eqf f f') -> (eqf f' f'') -> (eqf f f''). -Proof. - Unfold eqf. Intros. Rewrite H. Exact (H0 n). -Qed. - -Lemma adf_xor_eq : (f,f':nat->bool) (eqf (adf_xor f f') [n:nat] false) -> (eqf f f'). -Proof. - Unfold eqf. Unfold adf_xor. Intros. Apply xorb_eq. Apply H. -Qed. - -Lemma ad_xor_eq : (a,a':ad) (ad_xor a a')=ad_z -> a=a'. -Proof. - Intros. Apply ad_faithful. Apply adf_xor_eq. Apply eqf_trans with f':=(ad_bit (ad_xor a a')). - Apply eqf_sym. Apply ad_xor_semantics. - Rewrite H. Unfold eqf. Trivial. -Qed. - -Lemma adf_xor_assoc : (f,f',f'':nat->bool) - (eqf (adf_xor (adf_xor f f') f'') (adf_xor f (adf_xor f' f''))). -Proof. - Unfold eqf. Unfold adf_xor. Intros. Apply xorb_assoc. -Qed. - -Lemma eqf_xor_1 : (f,f',f'',f''':nat->bool) (eqf f f') -> (eqf f'' f''') -> - (eqf (adf_xor f f'') (adf_xor f' f''')). -Proof. - Unfold eqf. Intros. Unfold adf_xor. Rewrite H. Rewrite H0. Reflexivity. -Qed. - -Lemma ad_xor_assoc : - (a,a',a'':ad) (ad_xor (ad_xor a a') a'')=(ad_xor a (ad_xor a' a'')). -Proof. - Intros. Apply ad_faithful. - Apply eqf_trans with f':=(adf_xor (adf_xor (ad_bit a) (ad_bit a')) (ad_bit a'')). - Apply eqf_trans with f':=(adf_xor (ad_bit (ad_xor a a')) (ad_bit a'')). - Apply ad_xor_semantics. - Apply eqf_xor_1. Apply ad_xor_semantics. - Apply eqf_refl. - Apply eqf_trans with f':=(adf_xor (ad_bit a) (adf_xor (ad_bit a') (ad_bit a''))). - Apply adf_xor_assoc. - Apply eqf_trans with f':=(adf_xor (ad_bit a) (ad_bit (ad_xor a' a''))). - Apply eqf_xor_1. Apply eqf_refl. - Apply eqf_sym. Apply ad_xor_semantics. - Apply eqf_sym. Apply ad_xor_semantics. -Qed. - -Definition ad_double := [a:ad] - Cases a of - ad_z => ad_z - | (ad_x p) => (ad_x (xO p)) - end. - -Definition ad_double_plus_un := [a:ad] - Cases a of - ad_z => (ad_x xH) - | (ad_x p) => (ad_x (xI p)) - end. - -Definition ad_div_2 := [a:ad] - Cases a of - ad_z => ad_z - | (ad_x xH) => ad_z - | (ad_x (xO p)) => (ad_x p) - | (ad_x (xI p)) => (ad_x p) - end. - -Lemma ad_double_div_2 : (a:ad) (ad_div_2 (ad_double a))=a. -Proof. - NewDestruct a; Trivial. -Qed. - -Lemma ad_double_plus_un_div_2 : (a:ad) (ad_div_2 (ad_double_plus_un a))=a. -Proof. - NewDestruct a; Trivial. -Qed. - -Lemma ad_double_inj : (a0,a1:ad) (ad_double a0)=(ad_double a1) -> a0=a1. -Proof. - Intros. Rewrite <- (ad_double_div_2 a0). Rewrite H. Apply ad_double_div_2. -Qed. - -Lemma ad_double_plus_un_inj : - (a0,a1:ad) (ad_double_plus_un a0)=(ad_double_plus_un a1) -> a0=a1. -Proof. - Intros. Rewrite <- (ad_double_plus_un_div_2 a0). Rewrite H. Apply ad_double_plus_un_div_2. -Qed. - -Definition ad_bit_0 := [a:ad] - Cases a of - ad_z => false - | (ad_x (xO _)) => false - | _ => true - end. - -Lemma ad_double_bit_0 : (a:ad) (ad_bit_0 (ad_double a))=false. -Proof. - NewDestruct a; Trivial. -Qed. - -Lemma ad_double_plus_un_bit_0 : (a:ad) (ad_bit_0 (ad_double_plus_un a))=true. -Proof. - NewDestruct a; Trivial. -Qed. - -Lemma ad_div_2_double : (a:ad) (ad_bit_0 a)=false -> (ad_double (ad_div_2 a))=a. -Proof. - NewDestruct a. Trivial. NewDestruct p. Intro H. Discriminate H. - Intros. Reflexivity. - Intro H. Discriminate H. -Qed. - -Lemma ad_div_2_double_plus_un : - (a:ad) (ad_bit_0 a)=true -> (ad_double_plus_un (ad_div_2 a))=a. -Proof. - NewDestruct a. Intro. Discriminate H. - NewDestruct p. Intros. Reflexivity. - Intro H. Discriminate H. - Intro. Reflexivity. -Qed. - -Lemma ad_bit_0_correct : (a:ad) (ad_bit a O)=(ad_bit_0 a). -Proof. - NewDestruct a; Trivial. - NewDestruct p; Trivial. -Qed. - -Lemma ad_div_2_correct : (a:ad) (n:nat) (ad_bit (ad_div_2 a) n)=(ad_bit a (S n)). -Proof. - NewDestruct a; Trivial. - NewDestruct p; Trivial. -Qed. - -Lemma ad_xor_bit_0 : - (a,a':ad) (ad_bit_0 (ad_xor a a'))=(xorb (ad_bit_0 a) (ad_bit_0 a')). -Proof. - Intros. Rewrite <- ad_bit_0_correct. Rewrite (ad_xor_semantics a a' O). - Unfold adf_xor. Rewrite ad_bit_0_correct. Rewrite ad_bit_0_correct. Reflexivity. -Qed. - -Lemma ad_xor_div_2 : - (a,a':ad) (ad_div_2 (ad_xor a a'))=(ad_xor (ad_div_2 a) (ad_div_2 a')). -Proof. - Intros. Apply ad_faithful. Unfold eqf. Intro. - Rewrite (ad_xor_semantics (ad_div_2 a) (ad_div_2 a') n). - Rewrite ad_div_2_correct. - Rewrite (ad_xor_semantics a a' (S n)). - Unfold adf_xor. Rewrite ad_div_2_correct. Rewrite ad_div_2_correct. - Reflexivity. -Qed. - -Lemma ad_neg_bit_0 : (a,a':ad) (ad_bit_0 (ad_xor a a'))=true -> - (ad_bit_0 a)=(negb (ad_bit_0 a')). -Proof. - Intros. Rewrite <- true_xorb. Rewrite <- H. Rewrite ad_xor_bit_0. - Rewrite xorb_assoc. Rewrite xorb_nilpotent. Rewrite xorb_false. Reflexivity. -Qed. - -Lemma ad_neg_bit_0_1 : - (a,a':ad) (ad_xor a a')=(ad_x xH) -> (ad_bit_0 a)=(negb (ad_bit_0 a')). -Proof. - Intros. Apply ad_neg_bit_0. Rewrite H. Reflexivity. -Qed. - -Lemma ad_neg_bit_0_2 : (a,a':ad) (p:positive) (ad_xor a a')=(ad_x (xI p)) -> - (ad_bit_0 a)=(negb (ad_bit_0 a')). -Proof. - Intros. Apply ad_neg_bit_0. Rewrite H. Reflexivity. -Qed. - -Lemma ad_same_bit_0 : (a,a':ad) (p:positive) (ad_xor a a')=(ad_x (xO p)) -> - (ad_bit_0 a)=(ad_bit_0 a'). -Proof. - Intros. Rewrite <- (xorb_false (ad_bit_0 a)). Cut (ad_bit_0 (ad_x (xO p)))=false. - Intro. Rewrite <- H0. Rewrite <- H. Rewrite ad_xor_bit_0. Rewrite <- xorb_assoc. - Rewrite xorb_nilpotent. Rewrite false_xorb. Reflexivity. - Reflexivity. -Qed. |