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Diffstat (limited to 'theories/IntMap/Adalloc.v')
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diff --git a/theories/IntMap/Adalloc.v b/theories/IntMap/Adalloc.v new file mode 100644 index 00000000..9fde8f5f --- /dev/null +++ b/theories/IntMap/Adalloc.v @@ -0,0 +1,365 @@ +(************************************************************************) +(* 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: Adalloc.v,v 1.10.2.1 2004/07/16 19:31:04 herbelin Exp $ i*) + +Require Import Bool. +Require Import Sumbool. +Require Import ZArith. +Require Import Arith. +Require Import Addr. +Require Import Adist. +Require Import Addec. +Require Import Map. +Require Import Fset. + +Section AdAlloc. + + Variable A : Set. + + Definition nat_of_ad (a:ad) := + match a with + | ad_z => 0 + | ad_x p => nat_of_P p + end. + + Fixpoint nat_le (m:nat) : nat -> bool := + match m with + | O => fun _:nat => true + | S m' => + fun n:nat => match n with + | O => false + | S n' => nat_le m' n' + end + end. + + Lemma nat_le_correct : forall m n:nat, m <= n -> nat_le 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. + Qed. + + Lemma nat_le_complete : forall m n:nat, nat_le m n = true -> m <= n. + Proof. + induction m. trivial with arith. + destruct n. intro H. discriminate H. + auto with arith. + Qed. + + Lemma nat_le_correct_conv : forall m n:nat, m < n -> nat_le n m = false. + Proof. + intros. elim (sumbool_of_bool (nat_le n m)). intro H0. + elim (lt_irrefl _ (lt_le_trans _ _ _ H (nat_le_complete _ _ H0))). + trivial. + Qed. + + Lemma nat_le_complete_conv : forall m n:nat, nat_le n m = false -> m < n. + Proof. + intros. elim (le_or_lt n m). intro. conditional trivial rewrite nat_le_correct in H. discriminate H. + trivial. + Qed. + + Definition ad_of_nat (n:nat) := + match n with + | O => ad_z + | S n' => ad_x (P_of_succ_nat n') + end. + + Lemma ad_of_nat_of_ad : forall a:ad, ad_of_nat (nat_of_ad a) = a. + Proof. + destruct a as [| p]. reflexivity. + simpl in |- *. elim (ZL4 p). intros n H. rewrite H. simpl in |- *. rewrite <- nat_of_P_o_P_of_succ_nat_eq_succ in H. + rewrite nat_of_P_inj with (1 := H). reflexivity. + Qed. + + Lemma nat_of_ad_of_nat : forall n:nat, nat_of_ad (ad_of_nat n) = n. + Proof. + induction n. trivial. + intros. simpl in |- *. apply nat_of_P_o_P_of_succ_nat_eq_succ. + Qed. + + Definition ad_le (a b:ad) := nat_le (nat_of_ad a) (nat_of_ad b). + + Lemma ad_le_refl : forall a:ad, ad_le a a = true. + Proof. + intro. unfold ad_le in |- *. apply nat_le_correct. apply le_n. + Qed. + + Lemma ad_le_antisym : + forall a b:ad, ad_le a b = true -> ad_le b a = true -> a = b. + Proof. + unfold ad_le in |- *. intros. rewrite <- (ad_of_nat_of_ad a). rewrite <- (ad_of_nat_of_ad b). + rewrite (le_antisym _ _ (nat_le_complete _ _ H) (nat_le_complete _ _ H0)). reflexivity. + Qed. + + Lemma ad_le_trans : + forall a b c:ad, ad_le a b = true -> ad_le b c = true -> ad_le a c = true. + Proof. + unfold ad_le in |- *. intros. apply nat_le_correct. apply le_trans with (m := nat_of_ad b). + apply nat_le_complete. assumption. + apply nat_le_complete. assumption. + Qed. + + Lemma ad_le_lt_trans : + forall a b c:ad, + ad_le a b = true -> ad_le c b = false -> ad_le c a = false. + Proof. + unfold ad_le in |- *. intros. apply nat_le_correct_conv. apply le_lt_trans with (m := nat_of_ad b). + apply nat_le_complete. assumption. + apply nat_le_complete_conv. assumption. + Qed. + + Lemma ad_lt_le_trans : + forall a b c:ad, + ad_le b a = false -> ad_le b c = true -> ad_le c a = false. + Proof. + unfold ad_le in |- *. intros. apply nat_le_correct_conv. apply lt_le_trans with (m := nat_of_ad b). + apply nat_le_complete_conv. assumption. + apply nat_le_complete. assumption. + Qed. + + Lemma ad_lt_trans : + forall a b c:ad, + ad_le b a = false -> ad_le c b = false -> ad_le c a = false. + Proof. + unfold ad_le in |- *. intros. apply nat_le_correct_conv. apply lt_trans with (m := nat_of_ad b). + apply nat_le_complete_conv. assumption. + apply nat_le_complete_conv. assumption. + Qed. + + Lemma ad_lt_le_weak : forall a b:ad, ad_le b a = false -> ad_le a b = true. + Proof. + unfold ad_le in |- *. intros. apply nat_le_correct. apply lt_le_weak. + apply nat_le_complete_conv. assumption. + Qed. + + Definition ad_min (a b:ad) := if ad_le a b then a else b. + + Lemma ad_min_choice : forall a b:ad, {ad_min a b = a} + {ad_min a b = b}. + Proof. + unfold ad_min in |- *. intros. elim (sumbool_of_bool (ad_le a b)). intro H. left. rewrite H. + reflexivity. + intro H. right. rewrite H. reflexivity. + Qed. + + Lemma ad_min_le_1 : forall a b:ad, ad_le (ad_min a b) a = true. + Proof. + unfold ad_min in |- *. intros. elim (sumbool_of_bool (ad_le a b)). intro H. rewrite H. + apply ad_le_refl. + intro H. rewrite H. apply ad_lt_le_weak. assumption. + Qed. + + Lemma ad_min_le_2 : forall a b:ad, ad_le (ad_min a b) b = true. + Proof. + unfold ad_min in |- *. intros. elim (sumbool_of_bool (ad_le a b)). intro H. rewrite H. assumption. + intro H. rewrite H. apply ad_le_refl. + Qed. + + Lemma ad_min_le_3 : + forall a b c:ad, ad_le a (ad_min b c) = true -> ad_le a b = true. + Proof. + unfold ad_min in |- *. intros. elim (sumbool_of_bool (ad_le b c)). intro H0. rewrite H0 in H. + assumption. + intro H0. rewrite H0 in H. apply ad_lt_le_weak. apply ad_le_lt_trans with (b := c); assumption. + Qed. + + Lemma ad_min_le_4 : + forall a b c:ad, ad_le a (ad_min b c) = true -> ad_le a c = true. + Proof. + unfold ad_min in |- *. intros. elim (sumbool_of_bool (ad_le b c)). intro H0. rewrite H0 in H. + apply ad_le_trans with (b := b); assumption. + intro H0. rewrite H0 in H. assumption. + Qed. + + Lemma ad_min_le_5 : + forall a b c:ad, + ad_le a b = true -> ad_le a c = true -> ad_le a (ad_min b c) = true. + Proof. + intros. elim (ad_min_choice b c). intro H1. rewrite H1. assumption. + intro H1. rewrite H1. assumption. + Qed. + + Lemma ad_min_lt_3 : + forall a b c:ad, ad_le (ad_min b c) a = false -> ad_le b a = false. + Proof. + unfold ad_min in |- *. intros. elim (sumbool_of_bool (ad_le b c)). intro H0. rewrite H0 in H. + assumption. + intro H0. rewrite H0 in H. apply ad_lt_trans with (b := c); assumption. + Qed. + + Lemma ad_min_lt_4 : + forall a b c:ad, ad_le (ad_min b c) a = false -> ad_le c a = false. + Proof. + unfold ad_min in |- *. intros. elim (sumbool_of_bool (ad_le b c)). intro H0. rewrite H0 in H. + apply ad_lt_le_trans with (b := b); assumption. + intro H0. rewrite H0 in H. assumption. + Qed. + + (** Allocator: returns an address not in the domain of [m]. + This allocator is optimal in that it returns the lowest possible address, + in the usual ordering on integers. It is not the most efficient, however. *) + Fixpoint ad_alloc_opt (m:Map A) : ad := + match m with + | M0 => ad_z + | M1 a _ => if ad_eq a ad_z then ad_x 1 else ad_z + | M2 m1 m2 => + ad_min (ad_double (ad_alloc_opt m1)) + (ad_double_plus_un (ad_alloc_opt m2)) + end. + + Lemma ad_alloc_opt_allocates_1 : + forall m:Map A, MapGet A m (ad_alloc_opt m) = NONE A. + Proof. + induction m as [| a| m0 H m1 H0]. reflexivity. + simpl in |- *. elim (sumbool_of_bool (ad_eq a ad_z)). intro H. rewrite H. + rewrite (ad_eq_complete _ _ H). reflexivity. + intro H. rewrite H. rewrite H. reflexivity. + intros. change + (ad_alloc_opt (M2 A m0 m1)) with (ad_min (ad_double (ad_alloc_opt m0)) + (ad_double_plus_un (ad_alloc_opt m1))) + in |- *. + elim + (ad_min_choice (ad_double (ad_alloc_opt m0)) + (ad_double_plus_un (ad_alloc_opt m1))). + intro H1. rewrite H1. rewrite MapGet_M2_bit_0_0. rewrite ad_double_div_2. assumption. + apply ad_double_bit_0. + intro H1. rewrite H1. rewrite MapGet_M2_bit_0_1. rewrite ad_double_plus_un_div_2. assumption. + apply ad_double_plus_un_bit_0. + Qed. + + Lemma ad_alloc_opt_allocates : + forall m:Map A, in_dom A (ad_alloc_opt m) m = false. + Proof. + unfold in_dom in |- *. intro. rewrite (ad_alloc_opt_allocates_1 m). reflexivity. + Qed. + + (** Moreover, this is optimal: all addresses below [(ad_alloc_opt m)] + are in [dom m]: *) + + Lemma nat_of_ad_double : + forall a:ad, nat_of_ad (ad_double a) = 2 * nat_of_ad a. + Proof. + destruct a as [| p]. trivial. + exact (nat_of_P_xO p). + Qed. + + Lemma nat_of_ad_double_plus_un : + forall a:ad, nat_of_ad (ad_double_plus_un a) = S (2 * nat_of_ad a). + Proof. + destruct a as [| p]. trivial. + exact (nat_of_P_xI p). + Qed. + + Lemma ad_le_double_mono : + forall a b:ad, + ad_le a b = true -> ad_le (ad_double a) (ad_double b) = true. + Proof. + unfold ad_le in |- *. intros. rewrite nat_of_ad_double. rewrite nat_of_ad_double. apply nat_le_correct. + simpl in |- *. apply plus_le_compat. apply nat_le_complete. assumption. + apply plus_le_compat. apply nat_le_complete. assumption. + apply le_n. + Qed. + + Lemma ad_le_double_plus_un_mono : + forall a b:ad, + ad_le a b = true -> + ad_le (ad_double_plus_un a) (ad_double_plus_un b) = true. + Proof. + unfold ad_le in |- *. intros. rewrite nat_of_ad_double_plus_un. rewrite nat_of_ad_double_plus_un. + apply nat_le_correct. apply le_n_S. simpl in |- *. apply plus_le_compat. apply nat_le_complete. + assumption. + apply plus_le_compat. apply nat_le_complete. assumption. + apply le_n. + Qed. + + Lemma ad_le_double_mono_conv : + forall a b:ad, + ad_le (ad_double a) (ad_double b) = true -> ad_le a b = true. + Proof. + unfold ad_le in |- *. intros a b. rewrite nat_of_ad_double. rewrite nat_of_ad_double. intro. + apply nat_le_correct. apply (mult_S_le_reg_l 1). apply nat_le_complete. assumption. + Qed. + + Lemma ad_le_double_plus_un_mono_conv : + forall a b:ad, + ad_le (ad_double_plus_un a) (ad_double_plus_un b) = true -> + ad_le a b = true. + Proof. + unfold ad_le in |- *. intros a b. rewrite nat_of_ad_double_plus_un. rewrite nat_of_ad_double_plus_un. + intro. apply nat_le_correct. apply (mult_S_le_reg_l 1). apply le_S_n. apply nat_le_complete. + assumption. + Qed. + + Lemma ad_lt_double_mono : + forall a b:ad, + ad_le a b = false -> ad_le (ad_double a) (ad_double b) = false. + Proof. + intros. elim (sumbool_of_bool (ad_le (ad_double a) (ad_double b))). intro H0. + rewrite (ad_le_double_mono_conv _ _ H0) in H. discriminate H. + trivial. + Qed. + + Lemma ad_lt_double_plus_un_mono : + forall a b:ad, + ad_le a b = false -> + ad_le (ad_double_plus_un a) (ad_double_plus_un b) = false. + Proof. + intros. elim (sumbool_of_bool (ad_le (ad_double_plus_un a) (ad_double_plus_un b))). intro H0. + rewrite (ad_le_double_plus_un_mono_conv _ _ H0) in H. discriminate H. + trivial. + Qed. + + Lemma ad_lt_double_mono_conv : + forall a b:ad, + ad_le (ad_double a) (ad_double b) = false -> ad_le a b = false. + Proof. + intros. elim (sumbool_of_bool (ad_le a b)). intro H0. rewrite (ad_le_double_mono _ _ H0) in H. + discriminate H. + trivial. + Qed. + + Lemma ad_lt_double_plus_un_mono_conv : + forall a b:ad, + ad_le (ad_double_plus_un a) (ad_double_plus_un b) = false -> + ad_le a b = false. + Proof. + intros. elim (sumbool_of_bool (ad_le a b)). intro H0. + rewrite (ad_le_double_plus_un_mono _ _ H0) in H. discriminate H. + trivial. + Qed. + + Lemma ad_alloc_opt_optimal_1 : + forall (m:Map A) (a:ad), + ad_le (ad_alloc_opt m) a = false -> {y : A | MapGet A m a = SOME A y}. + Proof. + induction m as [| a y| m0 H m1 H0]. simpl in |- *. unfold ad_le in |- *. simpl in |- *. intros. discriminate H. + simpl in |- *. intros b H. elim (sumbool_of_bool (ad_eq a ad_z)). intro H0. rewrite H0 in H. + unfold ad_le in H. cut (ad_z = b). intro. split with y. rewrite <- H1. rewrite H0. reflexivity. + rewrite <- (ad_of_nat_of_ad b). + rewrite <- (le_n_O_eq _ (le_S_n _ _ (nat_le_complete_conv _ _ H))). reflexivity. + intro H0. rewrite H0 in H. discriminate H. + intros. simpl in H1. elim (ad_double_or_double_plus_un a). intro H2. elim H2. intros a0 H3. + rewrite H3 in H1. elim (H _ (ad_lt_double_mono_conv _ _ (ad_min_lt_3 _ _ _ H1))). intros y H4. + split with y. rewrite H3. rewrite MapGet_M2_bit_0_0. rewrite ad_double_div_2. assumption. + apply ad_double_bit_0. + intro H2. elim H2. intros a0 H3. rewrite H3 in H1. + elim (H0 _ (ad_lt_double_plus_un_mono_conv _ _ (ad_min_lt_4 _ _ _ H1))). intros y H4. + split with y. rewrite H3. rewrite MapGet_M2_bit_0_1. rewrite ad_double_plus_un_div_2. + assumption. + apply ad_double_plus_un_bit_0. + Qed. + + Lemma ad_alloc_opt_optimal : + forall (m:Map A) (a:ad), + ad_le (ad_alloc_opt m) a = false -> in_dom A a m = true. + Proof. + intros. unfold in_dom in |- *. elim (ad_alloc_opt_optimal_1 m a H). intros y H0. rewrite H0. + reflexivity. + Qed. + +End AdAlloc. |