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Diffstat (limited to 'theories7/IntMap/Adalloc.v')
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diff --git a/theories7/IntMap/Adalloc.v b/theories7/IntMap/Adalloc.v deleted file mode 100644 index 9e8dd1b3..00000000 --- a/theories7/IntMap/Adalloc.v +++ /dev/null @@ -1,339 +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: Adalloc.v,v 1.1.2.1 2004/07/16 19:31:26 herbelin Exp $ i*) - -Require Bool. -Require Sumbool. -Require ZArith. -Require Arith. -Require Addr. -Require Adist. -Require Addec. -Require Map. -Require Fset. - -Section AdAlloc. - - Variable A : Set. - - Definition nat_of_ad := [a:ad] Cases a of - ad_z => O - | (ad_x p) => (convert p) - end. - - Fixpoint nat_le [m:nat] : nat -> bool := - Cases m of - O => [_:nat] true - | (S m') => [n:nat] Cases n of - O => false - | (S n') => (nat_le m' n') - end - end. - - Lemma nat_le_correct : (m,n:nat) (le m n) -> (nat_le m n)=true. - Proof. - NewInduction m as [|m IHm]. Trivial. - NewDestruct n. Intro H. Elim (le_Sn_O ? H). - Intros. Simpl. Apply IHm. Apply le_S_n. Assumption. - Qed. - - Lemma nat_le_complete : (m,n:nat) (nat_le m n)=true -> (le m n). - Proof. - NewInduction m. Trivial with arith. - NewDestruct n. Intro H. Discriminate H. - Auto with arith. - Qed. - - Lemma nat_le_correct_conv : (m,n:nat) (lt m n) -> (nat_le n m)=false. - Proof. - Intros. Elim (sumbool_of_bool (nat_le n m)). Intro H0. - Elim (lt_n_n ? (lt_le_trans ? ? ? H (nat_le_complete ? ? H0))). - Trivial. - Qed. - - Lemma nat_le_complete_conv : (m,n:nat) (nat_le n m)=false -> (lt 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] Cases n of - O => ad_z - | (S n') => (ad_x (anti_convert n')) - end. - - Lemma ad_of_nat_of_ad : (a:ad) (ad_of_nat (nat_of_ad a))=a. - Proof. - NewDestruct a as [|p]. Reflexivity. - Simpl. Elim (ZL4 p). Intros n H. Rewrite H. Simpl. Rewrite <- bij1 in H. - Rewrite convert_intro with 1:=H. Reflexivity. - Qed. - - Lemma nat_of_ad_of_nat : (n:nat) (nat_of_ad (ad_of_nat n))=n. - Proof. - NewInduction n. Trivial. - Intros. Simpl. Apply bij1. - Qed. - - Definition ad_le := [a,b:ad] (nat_le (nat_of_ad a) (nat_of_ad b)). - - Lemma ad_le_refl : (a:ad) (ad_le a a)=true. - Proof. - Intro. Unfold ad_le. Apply nat_le_correct. Apply le_n. - Qed. - - Lemma ad_le_antisym : (a,b:ad) (ad_le a b)=true -> (ad_le b a)=true -> a=b. - Proof. - Unfold ad_le. 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 : (a,b,c:ad) (ad_le a b)=true -> (ad_le b c)=true -> - (ad_le a c)=true. - Proof. - Unfold ad_le. 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 : (a,b,c:ad) (ad_le a b)=true -> (ad_le c b)=false -> - (ad_le c a)=false. - Proof. - Unfold ad_le. 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 : (a,b,c:ad) (ad_le b a)=false -> (ad_le b c)=true -> - (ad_le c a)=false. - Proof. - Unfold ad_le. 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 : (a,b,c:ad) (ad_le b a)=false -> (ad_le c b)=false -> - (ad_le c a)=false. - Proof. - Unfold ad_le. 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 : (a,b:ad) (ad_le b a)=false -> (ad_le a b)=true. - Proof. - Unfold ad_le. 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 : (a,b:ad) {(ad_min a b)=a}+{(ad_min a b)=b}. - Proof. - Unfold ad_min. 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 : (a,b:ad) (ad_le (ad_min a b) a)=true. - Proof. - Unfold ad_min. 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 : (a,b:ad) (ad_le (ad_min a b) b)=true. - Proof. - Unfold ad_min. 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 : (a,b,c:ad) (ad_le a (ad_min b c))=true -> (ad_le a b)=true. - Proof. - Unfold ad_min. 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 : (a,b,c:ad) (ad_le a (ad_min b c))=true -> (ad_le a c)=true. - Proof. - Unfold ad_min. 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 : (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 : (a,b,c:ad) (ad_le (ad_min b c) a)=false -> (ad_le b a)=false. - Proof. - Unfold ad_min. 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 : (a,b,c:ad) (ad_le (ad_min b c) a)=false -> (ad_le c a)=false. - Proof. - Unfold ad_min. 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 := - Cases m of - M0 => ad_z - | (M1 a _) => if (ad_eq a ad_z) - then (ad_x xH) - 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 : (m:(Map A)) (MapGet A m (ad_alloc_opt m))=(NONE A). - Proof. - NewInduction m as [|a|m0 H m1 H0]. Reflexivity. - Simpl. 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))). - 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 : (m:(Map A)) (in_dom A (ad_alloc_opt m) m)=false. - Proof. - Unfold in_dom. 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 : (a:ad) (nat_of_ad (ad_double a))=(mult (2) (nat_of_ad a)). - Proof. - NewDestruct a as [|p]. Trivial. - Exact (convert_xO p). - Qed. - - Lemma nat_of_ad_double_plus_un : (a:ad) - (nat_of_ad (ad_double_plus_un a))=(S (mult (2) (nat_of_ad a))). - Proof. - NewDestruct a as [|p]. Trivial. - Exact (convert_xI p). - Qed. - - Lemma ad_le_double_mono : (a,b:ad) (ad_le a b)=true -> - (ad_le (ad_double a) (ad_double b))=true. - Proof. - Unfold ad_le. Intros. Rewrite nat_of_ad_double. Rewrite nat_of_ad_double. Apply nat_le_correct. - Simpl. Apply le_plus_plus. Apply nat_le_complete. Assumption. - Apply le_plus_plus. Apply nat_le_complete. Assumption. - Apply le_n. - Qed. - - Lemma ad_le_double_plus_un_mono : (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. Intros. Rewrite nat_of_ad_double_plus_un. Rewrite nat_of_ad_double_plus_un. - Apply nat_le_correct. Apply le_n_S. Simpl. Apply le_plus_plus. Apply nat_le_complete. - Assumption. - Apply le_plus_plus. Apply nat_le_complete. Assumption. - Apply le_n. - Qed. - - Lemma ad_le_double_mono_conv : (a,b:ad) (ad_le (ad_double a) (ad_double b))=true -> - (ad_le a b)=true. - Proof. - Unfold ad_le. Intros a b. Rewrite nat_of_ad_double. Rewrite nat_of_ad_double. Intro. - Apply nat_le_correct. Apply (mult_le_conv_1 (1)). Apply nat_le_complete. Assumption. - Qed. - - Lemma ad_le_double_plus_un_mono_conv : (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. Intros a b. Rewrite nat_of_ad_double_plus_un. Rewrite nat_of_ad_double_plus_un. - Intro. Apply nat_le_correct. Apply (mult_le_conv_1 (1)). Apply le_S_n. Apply nat_le_complete. - Assumption. - Qed. - - Lemma ad_lt_double_mono : (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 : (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 : (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 : (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 : (m:(Map A)) (a:ad) (ad_le (ad_alloc_opt m) a)=false -> - {y:A | (MapGet A m a)=(SOME A y)}. - Proof. - NewInduction m as [|a y|m0 H m1 H0]. Simpl. Unfold ad_le. Simpl. Intros. Discriminate H. - Simpl. 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 : (m:(Map A)) (a:ad) (ad_le (ad_alloc_opt m) a)=false -> - (in_dom A a m)=true. - Proof. - Intros. Unfold in_dom. Elim (ad_alloc_opt_optimal_1 m a H). Intros y H0. Rewrite H0. - Reflexivity. - Qed. - -End AdAlloc. - -V7only [ -(* Moved to NArith *) -Notation positive_to_nat_2 := positive_to_nat_2. -Notation positive_to_nat_4 := positive_to_nat_4. -]. |