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Diffstat (limited to 'theories/IntMap/Addr.v')
| -rw-r--r-- | theories/IntMap/Addr.v | 491 |
1 files changed, 491 insertions, 0 deletions
diff --git a/theories/IntMap/Addr.v b/theories/IntMap/Addr.v new file mode 100644 index 00000000..1370d72d --- /dev/null +++ b/theories/IntMap/Addr.v @@ -0,0 +1,491 @@ +(************************************************************************) +(* 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 1.8.2.1 2004/07/16 19:31:04 herbelin Exp $ i*) + +(** Representation of adresses by the [positive] type of binary numbers *) + +Require Import Bool. +Require Import ZArith. + +Inductive ad : Set := + | ad_z : ad + | ad_x : positive -> ad. + +Lemma ad_sum : forall a:ad, {p : positive | a = ad_x p} + {a = ad_z}. +Proof. + destruct a; auto. + left; exists p; trivial. +Qed. + +Fixpoint p_xor (p p2:positive) {struct p} : ad := + match p with + | xH => + match p2 with + | xH => ad_z + | xO p'2 => ad_x (xI p'2) + | xI p'2 => ad_x (xO p'2) + end + | xO p' => + match p2 with + | xH => ad_x (xI p') + | xO p'2 => + match p_xor p' p'2 with + | ad_z => ad_z + | ad_x p'' => ad_x (xO p'') + end + | xI p'2 => + match p_xor p' p'2 with + | ad_z => ad_x 1 + | ad_x p'' => ad_x (xI p'') + end + end + | xI p' => + match p2 with + | xH => ad_x (xO p') + | xO p'2 => + match p_xor p' p'2 with + | ad_z => ad_x 1 + | ad_x p'' => ad_x (xI p'') + end + | xI p'2 => + match p_xor p' p'2 with + | ad_z => ad_z + | ad_x p'' => ad_x (xO p'') + end + end + end. + +Definition ad_xor (a a':ad) := + match a with + | ad_z => a' + | ad_x p => match a' with + | ad_z => a + | ad_x p' => p_xor p p' + end + end. + +Lemma ad_xor_neutral_left : forall a:ad, ad_xor ad_z a = a. +Proof. + trivial. +Qed. + +Lemma ad_xor_neutral_right : forall a:ad, ad_xor a ad_z = a. +Proof. + destruct a; trivial. +Qed. + +Lemma ad_xor_comm : forall a a':ad, ad_xor a a' = ad_xor a' a. +Proof. + destruct a; destruct a'; simpl in |- *; auto. + generalize p0; clear p0; induction p as [p Hrecp| p Hrecp| ]; simpl in |- *; + auto. + destruct p0; simpl in |- *; trivial; intros. + rewrite Hrecp; trivial. + rewrite Hrecp; trivial. + destruct p0; simpl in |- *; trivial; intros. + rewrite Hrecp; trivial. + rewrite Hrecp; trivial. + destruct p0 as [p| p| ]; simpl in |- *; auto. +Qed. + +Lemma ad_xor_nilpotent : forall a:ad, ad_xor a a = ad_z. +Proof. + destruct a; trivial. + simpl in |- *. induction p as [p IHp| p IHp| ]; trivial. + simpl in |- *. rewrite IHp; reflexivity. + simpl in |- *. rewrite IHp; reflexivity. +Qed. + +Fixpoint ad_bit_1 (p:positive) : nat -> bool := + match p with + | xH => fun n:nat => match n with + | O => true + | S _ => false + end + | xO p => + fun n:nat => match n with + | O => false + | S n' => ad_bit_1 p n' + end + | xI p => fun n:nat => match n with + | O => true + | S n' => ad_bit_1 p n' + end + end. + +Definition ad_bit (a:ad) := + match a with + | ad_z => fun _:nat => false + | ad_x p => ad_bit_1 p + end. + +Definition eqf (f g:nat -> bool) := forall n:nat, f n = g n. + +Lemma ad_faithful_1 : forall a:ad, eqf (ad_bit ad_z) (ad_bit a) -> ad_z = a. +Proof. + destruct a. trivial. + induction p as [p IHp| p IHp| ]; intro H. absurd (ad_z = ad_x p). discriminate. + exact (IHp (fun n:nat => H (S n))). + absurd (ad_z = ad_x p). discriminate. + exact (IHp (fun n:nat => H (S n))). + absurd (false = true). discriminate. + exact (H 0). +Qed. + +Lemma ad_faithful_2 : + forall a:ad, eqf (ad_bit (ad_x 1)) (ad_bit a) -> ad_x 1 = a. +Proof. + destruct a. intros. absurd (true = false). discriminate. + exact (H 0). + destruct p. intro H. absurd (ad_z = ad_x p). discriminate. + exact (ad_faithful_1 (ad_x p) (fun n:nat => H (S n))). + intros. absurd (true = false). discriminate. + exact (H 0). + trivial. +Qed. + +Lemma ad_faithful_3 : + forall (a:ad) (p:positive), + (forall 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. + destruct 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 in |- *. intro. unfold eqf in H0. rewrite H0. reflexivity. + case p. intros. absurd (false = true). discriminate. + exact (H0 0). + intros. rewrite (H p0 (fun n:nat => H0 (S n))). reflexivity. + intros. absurd (false = true). discriminate. + exact (H0 0). +Qed. + +Lemma ad_faithful_4 : + forall (a:ad) (p:positive), + (forall 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. + destruct 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 in |- *. intro. unfold eqf in H0. rewrite H0. reflexivity. + case p. intros. rewrite (H p0 (fun n:nat => H0 (S n))). reflexivity. + intros. absurd (true = false). discriminate. + exact (H0 0). + intros. absurd (ad_z = ad_x p0). discriminate. + cut (eqf (ad_bit (ad_x 1)) (ad_bit (ad_x (xI p0)))). + intro. exact (ad_faithful_1 (ad_x p0) (fun n:nat => H1 (S n))). + unfold eqf in |- *. unfold eqf in H0. intro. rewrite H0. reflexivity. +Qed. + +Lemma ad_faithful : forall a a':ad, eqf (ad_bit a) (ad_bit a') -> a = a'. +Proof. + destruct a. exact ad_faithful_1. + induction 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 : forall a':ad, ad_bit (ad_xor ad_z a') 0 = ad_bit a' 0. +Proof. + trivial. +Qed. + +Lemma ad_xor_sem_2 : + forall a':ad, ad_bit (ad_xor (ad_x 1) a') 0 = negb (ad_bit a' 0). +Proof. + intro. case a'. trivial. + simpl in |- *. intro. + case p; trivial. +Qed. + +Lemma ad_xor_sem_3 : + forall (p:positive) (a':ad), + ad_bit (ad_xor (ad_x (xO p)) a') 0 = ad_bit a' 0. +Proof. + intros. case a'. trivial. + simpl in |- *. intro. + case p0; trivial. intro. + case (p_xor p p1); trivial. + intro. case (p_xor p p1); trivial. +Qed. + +Lemma ad_xor_sem_4 : + forall (p:positive) (a':ad), + ad_bit (ad_xor (ad_x (xI p)) a') 0 = negb (ad_bit a' 0). +Proof. + intros. case a'. trivial. + simpl in |- *. intro. case p0; trivial. intro. + case (p_xor p p1); trivial. + intro. + case (p_xor p p1); trivial. +Qed. + +Lemma ad_xor_sem_5 : + forall a a':ad, ad_bit (ad_xor a a') 0 = adf_xor (ad_bit a) (ad_bit a') 0. +Proof. + destruct a. intro. change (ad_bit a' 0 = xorb false (ad_bit a' 0)) in |- *. rewrite false_xorb. trivial. + case p. exact ad_xor_sem_4. + intros. change (ad_bit (ad_xor (ad_x (xO p0)) a') 0 = xorb false (ad_bit a' 0)) + in |- *. + rewrite false_xorb. apply ad_xor_sem_3. exact ad_xor_sem_2. +Qed. + +Lemma ad_xor_sem_6 : + forall n:nat, + (forall a a':ad, ad_bit (ad_xor a a') n = adf_xor (ad_bit a) (ad_bit a') n) -> + forall 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 in |- *. unfold ad_bit at 2 in |- *. rewrite false_xorb. reflexivity. + case a'. unfold adf_xor in |- *. unfold ad_bit at 3 in |- *. 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) + in |- *. + rewrite <- H. simpl in |- *. + 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) + in |- *. + rewrite <- H. simpl in |- *. + case (p_xor p2 p1); trivial. + intro. unfold adf_xor in |- *. unfold ad_bit at 3 in |- *. unfold ad_bit_1 in |- *. 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) + in |- *. + rewrite <- H. simpl in |- *. + 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) + in |- *. + rewrite <- H. simpl in |- *. + case (p_xor p2 p1); trivial. + intro. unfold adf_xor in |- *. unfold ad_bit at 3 in |- *. unfold ad_bit_1 in |- *. rewrite xorb_false. reflexivity. + unfold adf_xor in |- *. unfold ad_bit at 2 in |- *. unfold ad_bit_1 in |- *. rewrite false_xorb. simpl in |- *. case p; trivial. +Qed. + +Lemma ad_xor_semantics : + forall a a':ad, eqf (ad_bit (ad_xor a a')) (adf_xor (ad_bit a) (ad_bit a')). +Proof. + unfold eqf in |- *. intros. generalize a a'. elim n. exact ad_xor_sem_5. + exact ad_xor_sem_6. +Qed. + +Lemma eqf_sym : forall f f':nat -> bool, eqf f f' -> eqf f' f. +Proof. + unfold eqf in |- *. intros. rewrite H. reflexivity. +Qed. + +Lemma eqf_refl : forall f:nat -> bool, eqf f f. +Proof. + unfold eqf in |- *. trivial. +Qed. + +Lemma eqf_trans : + forall f f' f'':nat -> bool, eqf f f' -> eqf f' f'' -> eqf f f''. +Proof. + unfold eqf in |- *. intros. rewrite H. exact (H0 n). +Qed. + +Lemma adf_xor_eq : + forall f f':nat -> bool, eqf (adf_xor f f') (fun n:nat => false) -> eqf f f'. +Proof. + unfold eqf in |- *. unfold adf_xor in |- *. intros. apply xorb_eq. apply H. +Qed. + +Lemma ad_xor_eq : forall 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 in |- *. trivial. +Qed. + +Lemma adf_xor_assoc : + forall f f' f'':nat -> bool, + eqf (adf_xor (adf_xor f f') f'') (adf_xor f (adf_xor f' f'')). +Proof. + unfold eqf in |- *. unfold adf_xor in |- *. intros. apply xorb_assoc. +Qed. + +Lemma eqf_xor_1 : + forall f f' f'' f''':nat -> bool, + eqf f f' -> eqf f'' f''' -> eqf (adf_xor f f'') (adf_xor f' f'''). +Proof. + unfold eqf in |- *. intros. unfold adf_xor in |- *. rewrite H. rewrite H0. reflexivity. +Qed. + +Lemma ad_xor_assoc : + forall 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) := + match a with + | ad_z => ad_z + | ad_x p => ad_x (xO p) + end. + +Definition ad_double_plus_un (a:ad) := + match a with + | ad_z => ad_x 1 + | ad_x p => ad_x (xI p) + end. + +Definition ad_div_2 (a:ad) := + match a with + | 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 : forall a:ad, ad_div_2 (ad_double a) = a. +Proof. + destruct a; trivial. +Qed. + +Lemma ad_double_plus_un_div_2 : + forall a:ad, ad_div_2 (ad_double_plus_un a) = a. +Proof. + destruct a; trivial. +Qed. + +Lemma ad_double_inj : forall 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 : + forall 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) := + match a with + | ad_z => false + | ad_x (xO _) => false + | _ => true + end. + +Lemma ad_double_bit_0 : forall a:ad, ad_bit_0 (ad_double a) = false. +Proof. + destruct a; trivial. +Qed. + +Lemma ad_double_plus_un_bit_0 : + forall a:ad, ad_bit_0 (ad_double_plus_un a) = true. +Proof. + destruct a; trivial. +Qed. + +Lemma ad_div_2_double : + forall a:ad, ad_bit_0 a = false -> ad_double (ad_div_2 a) = a. +Proof. + destruct a. trivial. destruct p. intro H. discriminate H. + intros. reflexivity. + intro H. discriminate H. +Qed. + +Lemma ad_div_2_double_plus_un : + forall a:ad, ad_bit_0 a = true -> ad_double_plus_un (ad_div_2 a) = a. +Proof. + destruct a. intro. discriminate H. + destruct p. intros. reflexivity. + intro H. discriminate H. + intro. reflexivity. +Qed. + +Lemma ad_bit_0_correct : forall a:ad, ad_bit a 0 = ad_bit_0 a. +Proof. + destruct a; trivial. + destruct p; trivial. +Qed. + +Lemma ad_div_2_correct : + forall (a:ad) (n:nat), ad_bit (ad_div_2 a) n = ad_bit a (S n). +Proof. + destruct a; trivial. + destruct p; trivial. +Qed. + +Lemma ad_xor_bit_0 : + forall 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' 0). + unfold adf_xor in |- *. rewrite ad_bit_0_correct. rewrite ad_bit_0_correct. reflexivity. +Qed. + +Lemma ad_xor_div_2 : + forall 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 in |- *. 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 in |- *. rewrite ad_div_2_correct. rewrite ad_div_2_correct. + reflexivity. +Qed. + +Lemma ad_neg_bit_0 : + forall 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 : + forall a a':ad, ad_xor a a' = ad_x 1 -> 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 : + forall (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 : + forall (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.
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