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Diffstat (limited to 'theories7/IntMap/Fset.v')
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diff --git a/theories7/IntMap/Fset.v b/theories7/IntMap/Fset.v deleted file mode 100644 index 545c1716..00000000 --- a/theories7/IntMap/Fset.v +++ /dev/null @@ -1,338 +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: Fset.v,v 1.1.2.1 2004/07/16 19:31:27 herbelin Exp $ i*) - -(*s Sets operations on maps *) - -Require Bool. -Require Sumbool. -Require ZArith. -Require Addr. -Require Adist. -Require Addec. -Require Map. - -Section Dom. - - Variable A, B : Set. - - Fixpoint MapDomRestrTo [m:(Map A)] : (Map B) -> (Map A) := - Cases m of - M0 => [_:(Map B)] (M0 A) - | (M1 a y) => [m':(Map B)] Cases (MapGet B m' a) of - NONE => (M0 A) - | _ => m - end - | (M2 m1 m2) => [m':(Map B)] Cases m' of - M0 => (M0 A) - | (M1 a' y') => Cases (MapGet A m a') of - NONE => (M0 A) - | (SOME y) => (M1 A a' y) - end - | (M2 m'1 m'2) => (makeM2 A (MapDomRestrTo m1 m'1) - (MapDomRestrTo m2 m'2)) - end - end. - - Lemma MapDomRestrTo_semantics : (m:(Map A)) (m':(Map B)) - (eqm A (MapGet A (MapDomRestrTo m m')) - [a0:ad] Cases (MapGet B m' a0) of - NONE => (NONE A) - | _ => (MapGet A m a0) - end). - Proof. - Unfold eqm. Induction m. Simpl. Intros. Case (MapGet B m' a); Trivial. - Intros. Simpl. Elim (sumbool_of_bool (ad_eq a a1)). Intro H. Rewrite H. - Rewrite <- (ad_eq_complete ? ? H). Case (MapGet B m' a). Reflexivity. - Intro. Apply M1_semantics_1. - Intro H. Rewrite H. Case (MapGet B m' a). - Case (MapGet B m' a1); Reflexivity. - Case (MapGet B m' a1); Intros; Exact (M1_semantics_2 A a a1 a0 H). - Induction m'. Trivial. - Unfold MapDomRestrTo. Intros. Elim (sumbool_of_bool (ad_eq a a1)). - Intro H1. - Rewrite (ad_eq_complete ? ? H1). Rewrite (M1_semantics_1 B a1 a0). - Case (MapGet A (M2 A m0 m1) a1). Reflexivity. - Intro. Apply M1_semantics_1. - Intro H1. Rewrite (M1_semantics_2 B a a1 a0 H1). Case (MapGet A (M2 A m0 m1) a). Reflexivity. - Intro. Exact (M1_semantics_2 A a a1 a2 H1). - Intros. Change (MapGet A (makeM2 A (MapDomRestrTo m0 m2) (MapDomRestrTo m1 m3)) a) - =(Cases (MapGet B (M2 B m2 m3) a) of - NONE => (NONE A) - | (SOME _) => (MapGet A (M2 A m0 m1) a) - end). - Rewrite (makeM2_M2 A (MapDomRestrTo m0 m2) (MapDomRestrTo m1 m3) a). - Rewrite MapGet_M2_bit_0_if. Rewrite (H0 m3 (ad_div_2 a)). Rewrite (H m2 (ad_div_2 a)). - Rewrite (MapGet_M2_bit_0_if B m2 m3 a). Rewrite (MapGet_M2_bit_0_if A m0 m1 a). - Case (ad_bit_0 a); Reflexivity. - Qed. - - Fixpoint MapDomRestrBy [m:(Map A)] : (Map B) -> (Map A) := - Cases m of - M0 => [_:(Map B)] (M0 A) - | (M1 a y) => [m':(Map B)] Cases (MapGet B m' a) of - NONE => m - | _ => (M0 A) - end - | (M2 m1 m2) => [m':(Map B)] Cases m' of - M0 => m - | (M1 a' y') => (MapRemove A m a') - | (M2 m'1 m'2) => (makeM2 A (MapDomRestrBy m1 m'1) - (MapDomRestrBy m2 m'2)) - end - end. - - Lemma MapDomRestrBy_semantics : (m:(Map A)) (m':(Map B)) - (eqm A (MapGet A (MapDomRestrBy m m')) - [a0:ad] Cases (MapGet B m' a0) of - NONE => (MapGet A m a0) - | _ => (NONE A) - end). - Proof. - Unfold eqm. Induction m. Simpl. Intros. Case (MapGet B m' a); Trivial. - Intros. Simpl. Elim (sumbool_of_bool (ad_eq a a1)). Intro H. Rewrite H. - Rewrite (ad_eq_complete ? ? H). Case (MapGet B m' a1). Apply M1_semantics_1. - Trivial. - Intro H. Rewrite H. Case (MapGet B m' a). Rewrite (M1_semantics_2 A a a1 a0 H). - Case (MapGet B m' a1); Trivial. - Case (MapGet B m' a1); Trivial. - Induction m'. Trivial. - Unfold MapDomRestrBy. Intros. Rewrite (MapRemove_semantics A (M2 A m0 m1) a a1). - Elim (sumbool_of_bool (ad_eq a a1)). Intro H1. Rewrite H1. Rewrite (ad_eq_complete ? ? H1). - Rewrite (M1_semantics_1 B a1 a0). Reflexivity. - Intro H1. Rewrite H1. Rewrite (M1_semantics_2 B a a1 a0 H1). Reflexivity. - Intros. Change (MapGet A (makeM2 A (MapDomRestrBy m0 m2) (MapDomRestrBy m1 m3)) a) - =(Cases (MapGet B (M2 B m2 m3) a) of - NONE => (MapGet A (M2 A m0 m1) a) - | (SOME _) => (NONE A) - end). - Rewrite (makeM2_M2 A (MapDomRestrBy m0 m2) (MapDomRestrBy m1 m3) a). - Rewrite MapGet_M2_bit_0_if. Rewrite (H0 m3 (ad_div_2 a)). Rewrite (H m2 (ad_div_2 a)). - Rewrite (MapGet_M2_bit_0_if B m2 m3 a). Rewrite (MapGet_M2_bit_0_if A m0 m1 a). - Case (ad_bit_0 a); Reflexivity. - Qed. - - Definition in_dom := [a:ad; m:(Map A)] - Cases (MapGet A m a) of - NONE => false - | _ => true - end. - - Lemma in_dom_M0 : (a:ad) (in_dom a (M0 A))=false. - Proof. - Trivial. - Qed. - - Lemma in_dom_M1 : (a,a0:ad) (y:A) (in_dom a0 (M1 A a y))=(ad_eq a a0). - Proof. - Unfold in_dom. Intros. Simpl. Case (ad_eq a a0); Reflexivity. - Qed. - - Lemma in_dom_M1_1 : (a:ad) (y:A) (in_dom a (M1 A a y))=true. - Proof. - Intros. Rewrite in_dom_M1. Apply ad_eq_correct. - Qed. - - Lemma in_dom_M1_2 : (a,a0:ad) (y:A) (in_dom a0 (M1 A a y))=true -> a=a0. - Proof. - Intros. Apply (ad_eq_complete a a0). Rewrite (in_dom_M1 a a0 y) in H. Assumption. - Qed. - - Lemma in_dom_some : (m:(Map A)) (a:ad) (in_dom a m)=true -> - {y:A | (MapGet A m a)=(SOME A y)}. - Proof. - Unfold in_dom. Intros. Elim (option_sum ? (MapGet A m a)). Trivial. - Intro H0. Rewrite H0 in H. Discriminate H. - Qed. - - Lemma in_dom_none : (m:(Map A)) (a:ad) (in_dom a m)=false -> - (MapGet A m a)=(NONE A). - Proof. - Unfold in_dom. Intros. Elim (option_sum ? (MapGet A m a)). Intro H0. Elim H0. - Intros y H1. Rewrite H1 in H. Discriminate H. - Trivial. - Qed. - - Lemma in_dom_put : (m:(Map A)) (a0:ad) (y0:A) (a:ad) - (in_dom a (MapPut A m a0 y0))=(orb (ad_eq a a0) (in_dom a m)). - Proof. - Unfold in_dom. Intros. Rewrite (MapPut_semantics A m a0 y0 a). - Elim (sumbool_of_bool (ad_eq a a0)). Intro H. Rewrite H. Rewrite (ad_eq_comm a a0) in H. - Rewrite H. Rewrite orb_true_b. Reflexivity. - Intro H. Rewrite H. Rewrite (ad_eq_comm a a0) in H. Rewrite H. Rewrite orb_false_b. - Reflexivity. - Qed. - - Lemma in_dom_put_behind : (m:(Map A)) (a0:ad) (y0:A) (a:ad) - (in_dom a (MapPut_behind A m a0 y0))=(orb (ad_eq a a0) (in_dom a m)). - Proof. - Unfold in_dom. Intros. Rewrite (MapPut_behind_semantics A m a0 y0 a). - Elim (sumbool_of_bool (ad_eq a a0)). Intro H. Rewrite H. Rewrite (ad_eq_comm a a0) in H. - Rewrite H. Case (MapGet A m a); Reflexivity. - Intro H. Rewrite H. Rewrite (ad_eq_comm a a0) in H. Rewrite H. Case (MapGet A m a); Trivial. - Qed. - - Lemma in_dom_remove : (m:(Map A)) (a0:ad) (a:ad) - (in_dom a (MapRemove A m a0))=(andb (negb (ad_eq a a0)) (in_dom a m)). - Proof. - Unfold in_dom. Intros. Rewrite (MapRemove_semantics A m a0 a). - Elim (sumbool_of_bool (ad_eq a a0)). Intro H. Rewrite H. Rewrite (ad_eq_comm a a0) in H. - Rewrite H. Reflexivity. - Intro H. Rewrite H. Rewrite (ad_eq_comm a a0) in H. Rewrite H. - Case (MapGet A m a); Reflexivity. - Qed. - - Lemma in_dom_merge : (m,m':(Map A)) (a:ad) - (in_dom a (MapMerge A m m'))=(orb (in_dom a m) (in_dom a m')). - Proof. - Unfold in_dom. Intros. Rewrite (MapMerge_semantics A m m' a). - Elim (option_sum A (MapGet A m' a)). Intro H. Elim H. Intros y H0. Rewrite H0. - Case (MapGet A m a); Reflexivity. - Intro H. Rewrite H. Rewrite orb_b_false. Reflexivity. - Qed. - - Lemma in_dom_delta : (m,m':(Map A)) (a:ad) - (in_dom a (MapDelta A m m'))=(xorb (in_dom a m) (in_dom a m')). - Proof. - Unfold in_dom. Intros. Rewrite (MapDelta_semantics A m m' a). - Elim (option_sum A (MapGet A m' a)). Intro H. Elim H. Intros y H0. Rewrite H0. - Case (MapGet A m a); Reflexivity. - Intro H. Rewrite H. Case (MapGet A m a); Reflexivity. - Qed. - -End Dom. - -Section InDom. - - Variable A, B : Set. - - Lemma in_dom_restrto : (m:(Map A)) (m':(Map B)) (a:ad) - (in_dom A a (MapDomRestrTo A B m m'))=(andb (in_dom A a m) (in_dom B a m')). - Proof. - Unfold in_dom. Intros. Rewrite (MapDomRestrTo_semantics A B m m' a). - Elim (option_sum B (MapGet B m' a)). Intro H. Elim H. Intros y H0. Rewrite H0. - Rewrite andb_b_true. Reflexivity. - Intro H. Rewrite H. Rewrite andb_b_false. Reflexivity. - Qed. - - Lemma in_dom_restrby : (m:(Map A)) (m':(Map B)) (a:ad) - (in_dom A a (MapDomRestrBy A B m m'))=(andb (in_dom A a m) (negb (in_dom B a m'))). - Proof. - Unfold in_dom. Intros. Rewrite (MapDomRestrBy_semantics A B m m' a). - Elim (option_sum B (MapGet B m' a)). Intro H. Elim H. Intros y H0. Rewrite H0. - Unfold negb. Rewrite andb_b_false. Reflexivity. - Intro H. Rewrite H. Unfold negb. Rewrite andb_b_true. Reflexivity. - Qed. - -End InDom. - -Definition FSet := (Map unit). - -Section FSetDefs. - - Variable A : Set. - - Definition in_FSet : ad -> FSet -> bool := (in_dom unit). - - Fixpoint MapDom [m:(Map A)] : FSet := - Cases m of - M0 => (M0 unit) - | (M1 a _) => (M1 unit a tt) - | (M2 m m') => (M2 unit (MapDom m) (MapDom m')) - end. - - Lemma MapDom_semantics_1 : (m:(Map A)) (a:ad) - (y:A) (MapGet A m a)=(SOME A y) -> (in_FSet a (MapDom m))=true. - Proof. - Induction m. Intros. Discriminate H. - Unfold MapDom. Unfold in_FSet. Unfold in_dom. Unfold MapGet. Intros a y a0 y0. - Case (ad_eq a a0). Trivial. - Intro. Discriminate H. - Intros m0 H m1 H0 a y. Rewrite (MapGet_M2_bit_0_if A m0 m1 a). Simpl. Unfold in_FSet. - Unfold in_dom. Rewrite (MapGet_M2_bit_0_if unit (MapDom m0) (MapDom m1) a). - Case (ad_bit_0 a). Unfold in_FSet in_dom in H0. Intro. Apply H0 with y:=y. Assumption. - Unfold in_FSet in_dom in H. Intro. Apply H with y:=y. Assumption. - Qed. - - Lemma MapDom_semantics_2 : (m:(Map A)) (a:ad) - (in_FSet a (MapDom m))=true -> {y:A | (MapGet A m a)=(SOME A y)}. - Proof. - Induction m. Intros. Discriminate H. - Unfold MapDom. Unfold in_FSet. Unfold in_dom. Unfold MapGet. Intros a y a0. Case (ad_eq a a0). - Intro. Split with y. Reflexivity. - Intro. Discriminate H. - Intros m0 H m1 H0 a. Rewrite (MapGet_M2_bit_0_if A m0 m1 a). Simpl. Unfold in_FSet. - Unfold in_dom. Rewrite (MapGet_M2_bit_0_if unit (MapDom m0) (MapDom m1) a). - Case (ad_bit_0 a). Unfold in_FSet in_dom in H0. Intro. Apply H0. Assumption. - Unfold in_FSet in_dom in H. Intro. Apply H. Assumption. - Qed. - - Lemma MapDom_semantics_3 : (m:(Map A)) (a:ad) - (MapGet A m a)=(NONE A) -> (in_FSet a (MapDom m))=false. - Proof. - Intros. Elim (sumbool_of_bool (in_FSet a (MapDom m))). Intro H0. - Elim (MapDom_semantics_2 m a H0). Intros y H1. Rewrite H in H1. Discriminate H1. - Trivial. - Qed. - - Lemma MapDom_semantics_4 : (m:(Map A)) (a:ad) - (in_FSet a (MapDom m))=false -> (MapGet A m a)=(NONE A). - Proof. - Intros. Elim (option_sum A (MapGet A m a)). Intro H0. Elim H0. Intros y H1. - Rewrite (MapDom_semantics_1 m a y H1) in H. Discriminate H. - Trivial. - Qed. - - Lemma MapDom_Dom : (m:(Map A)) (a:ad) (in_dom A a m)=(in_FSet a (MapDom m)). - Proof. - Intros. Elim (sumbool_of_bool (in_FSet a (MapDom m))). Intro H. - Elim (MapDom_semantics_2 m a H). Intros y H0. Rewrite H. Unfold in_dom. Rewrite H0. - Reflexivity. - Intro H. Rewrite H. Unfold in_dom. Rewrite (MapDom_semantics_4 m a H). Reflexivity. - Qed. - - Definition FSetUnion : FSet -> FSet -> FSet := [s,s':FSet] (MapMerge unit s s'). - - Lemma in_FSet_union : (s,s':FSet) (a:ad) - (in_FSet a (FSetUnion s s'))=(orb (in_FSet a s) (in_FSet a s')). - Proof. - Exact (in_dom_merge unit). - Qed. - - Definition FSetInter : FSet -> FSet -> FSet := [s,s':FSet] (MapDomRestrTo unit unit s s'). - - Lemma in_FSet_inter : (s,s':FSet) (a:ad) - (in_FSet a (FSetInter s s'))=(andb (in_FSet a s) (in_FSet a s')). - Proof. - Exact (in_dom_restrto unit unit). - Qed. - - Definition FSetDiff : FSet -> FSet -> FSet := [s,s':FSet] (MapDomRestrBy unit unit s s'). - - Lemma in_FSet_diff : (s,s':FSet) (a:ad) - (in_FSet a (FSetDiff s s'))=(andb (in_FSet a s) (negb (in_FSet a s'))). - Proof. - Exact (in_dom_restrby unit unit). - Qed. - - Definition FSetDelta : FSet -> FSet -> FSet := [s,s':FSet] (MapDelta unit s s'). - - Lemma in_FSet_delta : (s,s':FSet) (a:ad) - (in_FSet a (FSetDelta s s'))=(xorb (in_FSet a s) (in_FSet a s')). - Proof. - Exact (in_dom_delta unit). - Qed. - -End FSetDefs. - -Lemma FSet_Dom : (s:FSet) (MapDom unit s)=s. -Proof. - Induction s. Trivial. - Simpl. Intros a t. Elim t. Reflexivity. - Intros. Simpl. Rewrite H. Rewrite H0. Reflexivity. -Qed. |