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Diffstat (limited to 'theories/IntMap/Adalloc.v')
| -rw-r--r-- | theories/IntMap/Adalloc.v | 438 |
1 files changed, 232 insertions, 206 deletions
diff --git a/theories/IntMap/Adalloc.v b/theories/IntMap/Adalloc.v index 5dcd41c84a..550633a212 100644 --- a/theories/IntMap/Adalloc.v +++ b/theories/IntMap/Adalloc.v @@ -7,333 +7,359 @@ (***********************************************************************) (*i $Id$ i*) -Require Bool. -Require Sumbool. -Require ZArith. -Require Arith. -Require Addr. -Require Adist. -Require Addec. -Require Map. -Require Fset. +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] 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 + Definition nat_of_ad (a:ad) := + match a with + | ad_z => 0 + | ad_x p => nat_of_P p end. - Lemma nat_le_correct : (m,n:nat) (le m n) -> (nat_le m n)=true. + 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. - NewInduction m as [|m IHm]. Trivial. - NewDestruct n. Intro H. Elim (le_Sn_O ? H). - Intros. Simpl. Apply IHm. Apply le_S_n. Assumption. + 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 : (m,n:nat) (nat_le m n)=true -> (le m n). + Lemma nat_le_complete : forall m n:nat, nat_le m n = true -> m <= n. Proof. - NewInduction m. Trivial with arith. - NewDestruct n. Intro H. Discriminate H. - Auto with arith. + induction m. trivial with arith. + destruct 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. + 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_n_n ? (lt_le_trans ? ? ? H (nat_le_complete ? ? H0))). - Trivial. + 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 : (m,n:nat) (nat_le n m)=false -> (lt m n). + 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. + 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. + 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 : (a:ad) (ad_of_nat (nat_of_ad a))=a. + Lemma ad_of_nat_of_ad : forall 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. + 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 : (n:nat) (nat_of_ad (ad_of_nat n))=n. + Lemma nat_of_ad_of_nat : forall n:nat, nat_of_ad (ad_of_nat n) = n. Proof. - NewInduction n. Trivial. - Intros. Simpl. Apply bij1. + 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)). + 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. + Lemma ad_le_refl : forall a:ad, ad_le a a = true. Proof. - Intro. Unfold ad_le. Apply nat_le_correct. Apply le_n. + intro. unfold ad_le in |- *. 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. + Lemma ad_le_antisym : + forall 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. + 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 : (a,b,c:ad) (ad_le a b)=true -> (ad_le b c)=true -> - (ad_le a c)=true. + 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. Intros. Apply nat_le_correct. Apply le_trans with m:=(nat_of_ad b). - Apply nat_le_complete. Assumption. - Apply nat_le_complete. Assumption. + 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 : (a,b,c:ad) (ad_le a b)=true -> (ad_le c b)=false -> - (ad_le c a)=false. + 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. 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. + 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 : (a,b,c:ad) (ad_le b a)=false -> (ad_le b c)=true -> - (ad_le c a)=false. + 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. 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. + 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 : (a,b,c:ad) (ad_le b a)=false -> (ad_le c b)=false -> - (ad_le c a)=false. + 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. 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. + 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 : (a,b:ad) (ad_le b a)=false -> (ad_le a b)=true. + Lemma ad_lt_le_weak : forall 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. + 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. + 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}. + Lemma ad_min_choice : forall 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. + 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 : (a,b:ad) (ad_le (ad_min a b) a)=true. + Lemma ad_min_le_1 : forall 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. + 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 : (a,b:ad) (ad_le (ad_min a b) b)=true. + Lemma ad_min_le_2 : forall 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. + 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 : (a,b,c:ad) (ad_le a (ad_min b c))=true -> (ad_le a b)=true. + 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. 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. + 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 : (a,b,c:ad) (ad_le a (ad_min b c))=true -> (ad_le a c)=true. + 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. 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. + 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 : (a,b,c:ad) (ad_le a b)=true -> (ad_le a c)=true -> - (ad_le a (ad_min b c))=true. + 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. + 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. + 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. 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. + 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 : (a,b,c:ad) (ad_le (ad_min b c) a)=false -> (ad_le c a)=false. + 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. 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. + 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 := - 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))) + 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 : (m:(Map A)) (MapGet A m (ad_alloc_opt m))=(NONE A). + Lemma ad_alloc_opt_allocates_1 : + forall 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. + 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 : (m:(Map A)) (in_dom A (ad_alloc_opt m) m)=false. + Lemma ad_alloc_opt_allocates : + forall 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. + 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 : (a:ad) (nat_of_ad (ad_double a))=(mult (2) (nat_of_ad a)). + Lemma nat_of_ad_double : + forall a:ad, nat_of_ad (ad_double a) = 2 * nat_of_ad a. Proof. - NewDestruct a as [|p]. Trivial. - Exact (convert_xO p). + destruct a as [| p]. trivial. + exact (nat_of_P_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))). + 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. - NewDestruct a as [|p]. Trivial. - Exact (convert_xI p). + destruct a as [| p]. trivial. + exact (nat_of_P_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. + 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. 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. + 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 : (a,b:ad) (ad_le a b)=true -> - (ad_le (ad_double_plus_un a) (ad_double_plus_un b))=true. + 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. 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. + 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 : (a,b:ad) (ad_le (ad_double a) (ad_double b))=true -> - (ad_le a b)=true. + 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. 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. + 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 : (a,b:ad) - (ad_le (ad_double_plus_un a) (ad_double_plus_un b))=true -> (ad_le a b)=true. + 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. 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. + 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 : (a,b:ad) (ad_le a b)=false -> - (ad_le (ad_double a) (ad_double b))=false. + 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. + 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. + 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. + 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. + 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. + 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. + 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. + 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)}. + 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. - 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. + 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 : (m:(Map A)) (a:ad) (ad_le (ad_alloc_opt m) a)=false -> - (in_dom A a m)=true. + 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. Elim (ad_alloc_opt_optimal_1 m a H). Intros y H0. Rewrite H0. - Reflexivity. + intros. unfold in_dom in |- *. 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. -]. |