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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 new file mode 100644 index 00000000..545c1716 --- /dev/null +++ b/theories7/IntMap/Fset.v @@ -0,0 +1,338 @@ +(************************************************************************) +(* 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. |