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diff --git a/theories7/IntMap/Mapaxioms.v b/theories7/IntMap/Mapaxioms.v deleted file mode 100644 index 085afd69..00000000 --- a/theories7/IntMap/Mapaxioms.v +++ /dev/null @@ -1,670 +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: Mapaxioms.v,v 1.1.2.1 2004/07/16 19:31:28 herbelin Exp $ i*) - -Require Bool. -Require Sumbool. -Require ZArith. -Require Addr. -Require Adist. -Require Addec. -Require Map. -Require Fset. - -Section MapAxioms. - - Variable A, B, C : Set. - - Lemma eqm_sym : (f,f':ad->(option A)) (eqm A f f') -> (eqm A f' f). - Proof. - Unfold eqm. Intros. Rewrite H. Reflexivity. - Qed. - - Lemma eqm_refl : (f:ad->(option A)) (eqm A f f). - Proof. - Unfold eqm. Trivial. - Qed. - - Lemma eqm_trans : (f,f',f'':ad->(option A)) (eqm A f f') -> (eqm A f' f'') -> (eqm A f f''). - Proof. - Unfold eqm. Intros. Rewrite H. Exact (H0 a). - Qed. - - Definition eqmap := [m,m':(Map A)] (eqm A (MapGet A m) (MapGet A m')). - - Lemma eqmap_sym : (m,m':(Map A)) (eqmap m m') -> (eqmap m' m). - Proof. - Intros. Unfold eqmap. Apply eqm_sym. Assumption. - Qed. - - Lemma eqmap_refl : (m:(Map A)) (eqmap m m). - Proof. - Intros. Unfold eqmap. Apply eqm_refl. - Qed. - - Lemma eqmap_trans : (m,m',m'':(Map A)) (eqmap m m') -> (eqmap m' m'') -> (eqmap m m''). - Proof. - Intros. Exact (eqm_trans (MapGet A m) (MapGet A m') (MapGet A m'') H H0). - Qed. - - Lemma MapPut_as_Merge : (m:(Map A)) (a:ad) (y:A) - (eqmap (MapPut A m a y) (MapMerge A m (M1 A a y))). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapPut_semantics A m a y a0). - Rewrite (MapMerge_semantics A m (M1 A a y) a0). Unfold 2 MapGet. - Elim (sumbool_of_bool (ad_eq a a0)); Intro H; Rewrite H; Reflexivity. - Qed. - - Lemma MapPut_ext : (m,m':(Map A)) (eqmap m m') -> - (a:ad) (y:A) (eqmap (MapPut A m a y) (MapPut A m' a y)). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapPut_semantics A m' a y a0). - Rewrite (MapPut_semantics A m a y a0). - Case (ad_eq a a0); [ Reflexivity | Apply H ]. - Qed. - - Lemma MapPut_behind_as_Merge : (m:(Map A)) (a:ad) (y:A) - (eqmap (MapPut_behind A m a y) (MapMerge A (M1 A a y) m)). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapPut_behind_semantics A m a y a0). - Rewrite (MapMerge_semantics A (M1 A a y) m a0). Reflexivity. - Qed. - - Lemma MapPut_behind_ext : (m,m':(Map A)) (eqmap m m') -> - (a:ad) (y:A) (eqmap (MapPut_behind A m a y) (MapPut_behind A m' a y)). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapPut_behind_semantics A m' a y a0). - Rewrite (MapPut_behind_semantics A m a y a0). Rewrite (H a0). Reflexivity. - Qed. - - Lemma MapMerge_empty_m_1 : (m:(Map A)) (MapMerge A (M0 A) m)=m. - Proof. - Trivial. - Qed. - - Lemma MapMerge_empty_m : (m:(Map A)) (eqmap (MapMerge A (M0 A) m) m). - Proof. - Unfold eqmap eqm. Trivial. - Qed. - - Lemma MapMerge_m_empty_1 : (m:(Map A)) (MapMerge A m (M0 A))=m. - Proof. - Induction m;Trivial. - Qed. - - Lemma MapMerge_m_empty : (m:(Map A)) (eqmap (MapMerge A m (M0 A)) m). - Proof. - Unfold eqmap eqm. Intros. Rewrite MapMerge_m_empty_1. Reflexivity. - Qed. - - Lemma MapMerge_empty_l : (m,m':(Map A)) (eqmap (MapMerge A m m') (M0 A)) -> - (eqmap m (M0 A)). - Proof. - Unfold eqmap eqm. Intros. Cut (MapGet A (MapMerge A m m') a)=(MapGet A (M0 A) a). - Rewrite (MapMerge_semantics A m m' a). Case (MapGet A m' a). Trivial. - Intros. Discriminate H0. - Exact (H a). - Qed. - - Lemma MapMerge_empty_r : (m,m':(Map A)) (eqmap (MapMerge A m m') (M0 A)) -> - (eqmap m' (M0 A)). - Proof. - Unfold eqmap eqm. Intros. Cut (MapGet A (MapMerge A m m') a)=(MapGet A (M0 A) a). - Rewrite (MapMerge_semantics A m m' a). Case (MapGet A m' a). Trivial. - Intros. Discriminate H0. - Exact (H a). - Qed. - - Lemma MapMerge_assoc : (m,m',m'':(Map A)) (eqmap - (MapMerge A (MapMerge A m m') m'') - (MapMerge A m (MapMerge A m' m''))). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapMerge_semantics A (MapMerge A m m') m'' a). - Rewrite (MapMerge_semantics A m (MapMerge A m' m'') a). Rewrite (MapMerge_semantics A m m' a). - Rewrite (MapMerge_semantics A m' m'' a). - Case (MapGet A m'' a); Case (MapGet A m' a); Trivial. - Qed. - - Lemma MapMerge_idempotent : (m:(Map A)) (eqmap (MapMerge A m m) m). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapMerge_semantics A m m a). - Case (MapGet A m a); Trivial. - Qed. - - Lemma MapMerge_ext : (m1,m2,m'1,m'2:(Map A)) - (eqmap m1 m'1) -> (eqmap m2 m'2) -> - (eqmap (MapMerge A m1 m2) (MapMerge A m'1 m'2)). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapMerge_semantics A m1 m2 a). - Rewrite (MapMerge_semantics A m'1 m'2 a). Rewrite (H a). Rewrite (H0 a). Reflexivity. - Qed. - - Lemma MapMerge_ext_l : (m1,m'1,m2:(Map A)) - (eqmap m1 m'1) -> (eqmap (MapMerge A m1 m2) (MapMerge A m'1 m2)). - Proof. - Intros. Apply MapMerge_ext. Assumption. - Apply eqmap_refl. - Qed. - - Lemma MapMerge_ext_r : (m1,m2,m'2:(Map A)) - (eqmap m2 m'2) -> (eqmap (MapMerge A m1 m2) (MapMerge A m1 m'2)). - Proof. - Intros. Apply MapMerge_ext. Apply eqmap_refl. - Assumption. - Qed. - - Lemma MapMerge_RestrTo_l : (m,m',m'':(Map A)) - (eqmap (MapMerge A (MapDomRestrTo A A m m') m'') - (MapDomRestrTo A A (MapMerge A m m'') (MapMerge A m' m''))). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapMerge_semantics A (MapDomRestrTo A A m m') m'' a). - Rewrite (MapDomRestrTo_semantics A A m m' a). - Rewrite (MapDomRestrTo_semantics A A (MapMerge A m m'') (MapMerge A m' m'') a). - Rewrite (MapMerge_semantics A m' m'' a). Rewrite (MapMerge_semantics A m m'' a). - Case (MapGet A m'' a); Case (MapGet A m' a); Reflexivity. - Qed. - - Lemma MapRemove_as_RestrBy : (m:(Map A)) (a:ad) (y:B) - (eqmap (MapRemove A m a) (MapDomRestrBy A B m (M1 B a y))). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapRemove_semantics A m a a0). - Rewrite (MapDomRestrBy_semantics A B m (M1 B a y) a0). Elim (sumbool_of_bool (ad_eq a a0)). - Intro H. Rewrite H. Rewrite (ad_eq_complete a a0 H). Rewrite (M1_semantics_1 B a0 y). - Reflexivity. - Intro H. Rewrite H. Rewrite (M1_semantics_2 B a a0 y H). Reflexivity. - Qed. - - Lemma MapRemove_ext : (m,m':(Map A)) (eqmap m m') -> - (a:ad) (eqmap (MapRemove A m a) (MapRemove A m' a)). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapRemove_semantics A m' a a0). - Rewrite (MapRemove_semantics A m a a0). - Case (ad_eq a a0); [ Reflexivity | Apply H ]. - Qed. - - Lemma MapDomRestrTo_empty_m_1 : - (m:(Map B)) (MapDomRestrTo A B (M0 A) m)=(M0 A). - Proof. - Trivial. - Qed. - - Lemma MapDomRestrTo_empty_m : - (m:(Map B)) (eqmap (MapDomRestrTo A B (M0 A) m) (M0 A)). - Proof. - Unfold eqmap eqm. Trivial. - Qed. - - Lemma MapDomRestrTo_m_empty_1 : - (m:(Map A)) (MapDomRestrTo A B m (M0 B))=(M0 A). - Proof. - Induction m;Trivial. - Qed. - - Lemma MapDomRestrTo_m_empty : - (m:(Map A)) (eqmap (MapDomRestrTo A B m (M0 B)) (M0 A)). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapDomRestrTo_m_empty_1 m). Reflexivity. - Qed. - - Lemma MapDomRestrTo_assoc : (m:(Map A)) (m':(Map B)) (m'':(Map C)) - (eqmap (MapDomRestrTo A C (MapDomRestrTo A B m m') m'') - (MapDomRestrTo A B m (MapDomRestrTo B C m' m''))). - Proof. - Unfold eqmap eqm. Intros. - Rewrite (MapDomRestrTo_semantics A C (MapDomRestrTo A B m m') m'' a). - Rewrite (MapDomRestrTo_semantics A B m m' a). - Rewrite (MapDomRestrTo_semantics A B m (MapDomRestrTo B C m' m'') a). - Rewrite (MapDomRestrTo_semantics B C m' m'' a). - Case (MapGet C m'' a); Case (MapGet B m' a); Trivial. - Qed. - - Lemma MapDomRestrTo_idempotent : (m:(Map A)) (eqmap (MapDomRestrTo A A m m) m). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapDomRestrTo_semantics A A m m a). - Case (MapGet A m a); Trivial. - Qed. - - Lemma MapDomRestrTo_Dom : (m:(Map A)) (m':(Map B)) - (eqmap (MapDomRestrTo A B m m') (MapDomRestrTo A unit m (MapDom B m'))). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapDomRestrTo_semantics A B m m' a). - Rewrite (MapDomRestrTo_semantics A unit m (MapDom B m') a). - Elim (sumbool_of_bool (in_FSet a (MapDom B m'))). Intro H. - Elim (MapDom_semantics_2 B m' a H). Intros y H0. Rewrite H0. Unfold in_FSet in_dom in H. - Generalize H. Case (MapGet unit (MapDom B m') a); Trivial. Intro H1. Discriminate H1. - Intro H. Rewrite (MapDom_semantics_4 B m' a H). Unfold in_FSet in_dom in H. - Generalize H. Case (MapGet unit (MapDom B m') a). Trivial. - Intros H0 H1. Discriminate H1. - Qed. - - Lemma MapDomRestrBy_empty_m_1 : - (m:(Map B)) (MapDomRestrBy A B (M0 A) m)=(M0 A). - Proof. - Trivial. - Qed. - - Lemma MapDomRestrBy_empty_m : - (m:(Map B)) (eqmap (MapDomRestrBy A B (M0 A) m) (M0 A)). - Proof. - Unfold eqmap eqm. Trivial. - Qed. - - Lemma MapDomRestrBy_m_empty_1 : (m:(Map A)) (MapDomRestrBy A B m (M0 B))=m. - Proof. - Induction m;Trivial. - Qed. - - Lemma MapDomRestrBy_m_empty : (m:(Map A)) (eqmap (MapDomRestrBy A B m (M0 B)) m). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapDomRestrBy_m_empty_1 m). Reflexivity. - Qed. - - Lemma MapDomRestrBy_Dom : (m:(Map A)) (m':(Map B)) - (eqmap (MapDomRestrBy A B m m') (MapDomRestrBy A unit m (MapDom B m'))). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapDomRestrBy_semantics A B m m' a). - Rewrite (MapDomRestrBy_semantics A unit m (MapDom B m') a). - Elim (sumbool_of_bool (in_FSet a (MapDom B m'))). Intro H. - Elim (MapDom_semantics_2 B m' a H). Intros y H0. Rewrite H0. - Unfold in_FSet in_dom in H. Generalize H. Case (MapGet unit (MapDom B m') a); Trivial. - Intro H1. Discriminate H1. - Intro H. Rewrite (MapDom_semantics_4 B m' a H). Unfold in_FSet in_dom in H. - Generalize H. Case (MapGet unit (MapDom B m') a). Trivial. - Intros H0 H1. Discriminate H1. - Qed. - - Lemma MapDomRestrBy_m_m_1 : (m:(Map A)) (eqmap (MapDomRestrBy A A m m) (M0 A)). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapDomRestrBy_semantics A A m m a). - Case (MapGet A m a); Trivial. - Qed. - - Lemma MapDomRestrBy_By : (m:(Map A)) (m':(Map B)) (m'':(Map B)) - (eqmap (MapDomRestrBy A B (MapDomRestrBy A B m m') m'') - (MapDomRestrBy A B m (MapMerge B m' m''))). - Proof. - Unfold eqmap eqm. Intros. - Rewrite (MapDomRestrBy_semantics A B (MapDomRestrBy A B m m') m'' a). - Rewrite (MapDomRestrBy_semantics A B m m' a). - Rewrite (MapDomRestrBy_semantics A B m (MapMerge B m' m'') a). - Rewrite (MapMerge_semantics B m' m'' a). - Case (MapGet B m'' a); Case (MapGet B m' a); Trivial. - Qed. - - Lemma MapDomRestrBy_By_comm : (m:(Map A)) (m':(Map B)) (m'':(Map C)) - (eqmap (MapDomRestrBy A C (MapDomRestrBy A B m m') m'') - (MapDomRestrBy A B (MapDomRestrBy A C m m'') m')). - Proof. - Unfold eqmap eqm. Intros. - Rewrite (MapDomRestrBy_semantics A C (MapDomRestrBy A B m m') m'' a). - Rewrite (MapDomRestrBy_semantics A B m m' a). - Rewrite (MapDomRestrBy_semantics A B (MapDomRestrBy A C m m'') m' a). - Rewrite (MapDomRestrBy_semantics A C m m'' a). - Case (MapGet C m'' a); Case (MapGet B m' a); Trivial. - Qed. - - Lemma MapDomRestrBy_To : (m:(Map A)) (m':(Map B)) (m'':(Map C)) - (eqmap (MapDomRestrBy A C (MapDomRestrTo A B m m') m'') - (MapDomRestrTo A B m (MapDomRestrBy B C m' m''))). - Proof. - Unfold eqmap eqm. Intros. - Rewrite (MapDomRestrBy_semantics A C (MapDomRestrTo A B m m') m'' a). - Rewrite (MapDomRestrTo_semantics A B m m' a). - Rewrite (MapDomRestrTo_semantics A B m (MapDomRestrBy B C m' m'') a). - Rewrite (MapDomRestrBy_semantics B C m' m'' a). - Case (MapGet C m'' a); Case (MapGet B m' a); Trivial. - Qed. - - Lemma MapDomRestrBy_To_comm : (m:(Map A)) (m':(Map B)) (m'':(Map C)) - (eqmap (MapDomRestrBy A C (MapDomRestrTo A B m m') m'') - (MapDomRestrTo A B (MapDomRestrBy A C m m'') m')). - Proof. - Unfold eqmap eqm. Intros. - Rewrite (MapDomRestrBy_semantics A C (MapDomRestrTo A B m m') m'' a). - Rewrite (MapDomRestrTo_semantics A B m m' a). - Rewrite (MapDomRestrTo_semantics A B (MapDomRestrBy A C m m'') m' a). - Rewrite (MapDomRestrBy_semantics A C m m'' a). - Case (MapGet C m'' a); Case (MapGet B m' a); Trivial. - Qed. - - Lemma MapDomRestrTo_By : (m:(Map A)) (m':(Map B)) (m'':(Map C)) - (eqmap (MapDomRestrTo A C (MapDomRestrBy A B m m') m'') - (MapDomRestrTo A C m (MapDomRestrBy C B m'' m'))). - Proof. - Unfold eqmap eqm. Intros. - Rewrite (MapDomRestrTo_semantics A C (MapDomRestrBy A B m m') m'' a). - Rewrite (MapDomRestrBy_semantics A B m m' a). - Rewrite (MapDomRestrTo_semantics A C m (MapDomRestrBy C B m'' m') a). - Rewrite (MapDomRestrBy_semantics C B m'' m' a). - Case (MapGet C m'' a); Case (MapGet B m' a); Trivial. - Qed. - - Lemma MapDomRestrTo_By_comm : (m:(Map A)) (m':(Map B)) (m'':(Map C)) - (eqmap (MapDomRestrTo A C (MapDomRestrBy A B m m') m'') - (MapDomRestrBy A B (MapDomRestrTo A C m m'') m')). - Proof. - Unfold eqmap eqm. Intros. - Rewrite (MapDomRestrTo_semantics A C (MapDomRestrBy A B m m') m'' a). - Rewrite (MapDomRestrBy_semantics A B m m' a). - Rewrite (MapDomRestrBy_semantics A B (MapDomRestrTo A C m m'') m' a). - Rewrite (MapDomRestrTo_semantics A C m m'' a). - Case (MapGet C m'' a); Case (MapGet B m' a); Trivial. - Qed. - - Lemma MapDomRestrTo_To_comm : (m:(Map A)) (m':(Map B)) (m'':(Map C)) - (eqmap (MapDomRestrTo A C (MapDomRestrTo A B m m') m'') - (MapDomRestrTo A B (MapDomRestrTo A C m m'') m')). - Proof. - Unfold eqmap eqm. Intros. - Rewrite (MapDomRestrTo_semantics A C (MapDomRestrTo A B m m') m'' a). - Rewrite (MapDomRestrTo_semantics A B m m' a). - Rewrite (MapDomRestrTo_semantics A B (MapDomRestrTo A C m m'') m' a). - Rewrite (MapDomRestrTo_semantics A C m m'' a). - Case (MapGet C m'' a); Case (MapGet B m' a); Trivial. - Qed. - - Lemma MapMerge_DomRestrTo : (m,m':(Map A)) (m'':(Map B)) - (eqmap (MapDomRestrTo A B (MapMerge A m m') m'') - (MapMerge A (MapDomRestrTo A B m m'') (MapDomRestrTo A B m' m''))). - Proof. - Unfold eqmap eqm. Intros. - Rewrite (MapDomRestrTo_semantics A B (MapMerge A m m') m'' a). - Rewrite (MapMerge_semantics A m m' a). - Rewrite (MapMerge_semantics A (MapDomRestrTo A B m m'') (MapDomRestrTo A B m' m'') a). - Rewrite (MapDomRestrTo_semantics A B m' m'' a). - Rewrite (MapDomRestrTo_semantics A B m m'' a). - Case (MapGet B m'' a); Case (MapGet A m' a); Trivial. - Qed. - - Lemma MapMerge_DomRestrBy : (m,m':(Map A)) (m'':(Map B)) - (eqmap (MapDomRestrBy A B (MapMerge A m m') m'') - (MapMerge A (MapDomRestrBy A B m m'') (MapDomRestrBy A B m' m''))). - Proof. - Unfold eqmap eqm. Intros. - Rewrite (MapDomRestrBy_semantics A B (MapMerge A m m') m'' a). - Rewrite (MapMerge_semantics A m m' a). - Rewrite (MapMerge_semantics A (MapDomRestrBy A B m m'') (MapDomRestrBy A B m' m'') a). - Rewrite (MapDomRestrBy_semantics A B m' m'' a). - Rewrite (MapDomRestrBy_semantics A B m m'' a). - Case (MapGet B m'' a); Case (MapGet A m' a); Trivial. - Qed. - - Lemma MapDelta_empty_m_1 : (m:(Map A)) (MapDelta A (M0 A) m)=m. - Proof. - Trivial. - Qed. - - Lemma MapDelta_empty_m : (m:(Map A)) (eqmap (MapDelta A (M0 A) m) m). - Proof. - Unfold eqmap eqm. Trivial. - Qed. - - Lemma MapDelta_m_empty_1 : (m:(Map A)) (MapDelta A m (M0 A))=m. - Proof. - Induction m;Trivial. - Qed. - - Lemma MapDelta_m_empty : (m:(Map A)) (eqmap (MapDelta A m (M0 A)) m). - Proof. - Unfold eqmap eqm. Intros. Rewrite MapDelta_m_empty_1. Reflexivity. - Qed. - - Lemma MapDelta_nilpotent : (m:(Map A)) (eqmap (MapDelta A m m) (M0 A)). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapDelta_semantics A m m a). - Case (MapGet A m a); Trivial. - Qed. - - Lemma MapDelta_as_Merge : (m,m':(Map A)) (eqmap (MapDelta A m m') - (MapMerge A (MapDomRestrBy A A m m') (MapDomRestrBy A A m' m))). - Proof. - Unfold eqmap eqm. Intros. - Rewrite (MapDelta_semantics A m m' a). - Rewrite (MapMerge_semantics A (MapDomRestrBy A A m m') (MapDomRestrBy A A m' m) a). - Rewrite (MapDomRestrBy_semantics A A m' m a). - Rewrite (MapDomRestrBy_semantics A A m m' a). - Case (MapGet A m a); Case (MapGet A m' a); Trivial. - Qed. - - Lemma MapDelta_as_DomRestrBy : (m,m':(Map A)) (eqmap (MapDelta A m m') - (MapDomRestrBy A A (MapMerge A m m') (MapDomRestrTo A A m m'))). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapDelta_semantics A m m' a). - Rewrite (MapDomRestrBy_semantics A A (MapMerge A m m') (MapDomRestrTo A A m m') a). - Rewrite (MapDomRestrTo_semantics A A m m' a). Rewrite (MapMerge_semantics A m m' a). - Case (MapGet A m a); Case (MapGet A m' a); Trivial. - Qed. - - Lemma MapDelta_as_DomRestrBy_2 : (m,m':(Map A)) (eqmap (MapDelta A m m') - (MapDomRestrBy A A (MapMerge A m m') (MapDomRestrTo A A m' m))). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapDelta_semantics A m m' a). - Rewrite (MapDomRestrBy_semantics A A (MapMerge A m m') (MapDomRestrTo A A m' m) a). - Rewrite (MapDomRestrTo_semantics A A m' m a). Rewrite (MapMerge_semantics A m m' a). - Case (MapGet A m a); Case (MapGet A m' a); Trivial. - Qed. - - Lemma MapDelta_sym : (m,m':(Map A)) (eqmap (MapDelta A m m') (MapDelta A m' m)). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapDelta_semantics A m m' a). - Rewrite (MapDelta_semantics A m' m a). - Case (MapGet A m a); Case (MapGet A m' a); Trivial. - Qed. - - Lemma MapDelta_ext : (m1,m2,m'1,m'2:(Map A)) - (eqmap m1 m'1) -> (eqmap m2 m'2) -> - (eqmap (MapDelta A m1 m2) (MapDelta A m'1 m'2)). - Proof. - Unfold eqmap eqm. Intros. Rewrite (MapDelta_semantics A m1 m2 a). - Rewrite (MapDelta_semantics A m'1 m'2 a). Rewrite (H a). Rewrite (H0 a). Reflexivity. - Qed. - - Lemma MapDelta_ext_l : (m1,m'1,m2:(Map A)) - (eqmap m1 m'1) -> (eqmap (MapDelta A m1 m2) (MapDelta A m'1 m2)). - Proof. - Intros. Apply MapDelta_ext. Assumption. - Apply eqmap_refl. - Qed. - - Lemma MapDelta_ext_r : (m1,m2,m'2:(Map A)) - (eqmap m2 m'2) -> (eqmap (MapDelta A m1 m2) (MapDelta A m1 m'2)). - Proof. - Intros. Apply MapDelta_ext. Apply eqmap_refl. - Assumption. - Qed. - - Lemma MapDom_Split_1 : (m:(Map A)) (m':(Map B)) - (eqmap m (MapMerge A (MapDomRestrTo A B m m') (MapDomRestrBy A B m m'))). - Proof. - Unfold eqmap eqm. Intros. - Rewrite (MapMerge_semantics A (MapDomRestrTo A B m m') (MapDomRestrBy A B m m') a). - Rewrite (MapDomRestrBy_semantics A B m m' a). - Rewrite (MapDomRestrTo_semantics A B m m' a). - Case (MapGet B m' a); Case (MapGet A m a); Trivial. - Qed. - - Lemma MapDom_Split_2 : (m:(Map A)) (m':(Map B)) - (eqmap m (MapMerge A (MapDomRestrBy A B m m') (MapDomRestrTo A B m m'))). - Proof. - Unfold eqmap eqm. Intros. - Rewrite (MapMerge_semantics A (MapDomRestrBy A B m m') (MapDomRestrTo A B m m') a). - Rewrite (MapDomRestrBy_semantics A B m m' a). - Rewrite (MapDomRestrTo_semantics A B m m' a). - Case (MapGet B m' a); Case (MapGet A m a); Trivial. - Qed. - - Lemma MapDom_Split_3 : (m:(Map A)) (m':(Map B)) - (eqmap (MapDomRestrTo A A (MapDomRestrTo A B m m') (MapDomRestrBy A B m m')) - (M0 A)). - Proof. - Unfold eqmap eqm. Intros. - Rewrite (MapDomRestrTo_semantics A A (MapDomRestrTo A B m m') (MapDomRestrBy A B m m') a). - Rewrite (MapDomRestrBy_semantics A B m m' a). - Rewrite (MapDomRestrTo_semantics A B m m' a). - Case (MapGet B m' a); Case (MapGet A m a); Trivial. - Qed. - -End MapAxioms. - -Lemma MapDomRestrTo_ext : (A,B:Set) - (m1:(Map A)) (m2:(Map B)) (m'1:(Map A)) (m'2:(Map B)) - (eqmap A m1 m'1) -> (eqmap B m2 m'2) -> - (eqmap A (MapDomRestrTo A B m1 m2) (MapDomRestrTo A B m'1 m'2)). -Proof. - Unfold eqmap eqm. Intros. Rewrite (MapDomRestrTo_semantics A B m1 m2 a). - Rewrite (MapDomRestrTo_semantics A B m'1 m'2 a). Rewrite (H a). Rewrite (H0 a). Reflexivity. -Qed. - -Lemma MapDomRestrTo_ext_l : (A,B:Set) (m1:(Map A)) (m2:(Map B)) (m'1:(Map A)) - (eqmap A m1 m'1) -> - (eqmap A (MapDomRestrTo A B m1 m2) (MapDomRestrTo A B m'1 m2)). -Proof. - Intros. Apply MapDomRestrTo_ext; [ Assumption | Apply eqmap_refl ]. -Qed. - -Lemma MapDomRestrTo_ext_r : (A,B:Set) (m1:(Map A)) (m2:(Map B)) (m'2:(Map B)) - (eqmap B m2 m'2) -> - (eqmap A (MapDomRestrTo A B m1 m2) (MapDomRestrTo A B m1 m'2)). -Proof. - Intros. Apply MapDomRestrTo_ext; [ Apply eqmap_refl | Assumption ]. -Qed. - -Lemma MapDomRestrBy_ext : (A,B:Set) - (m1:(Map A)) (m2:(Map B)) (m'1:(Map A)) (m'2:(Map B)) - (eqmap A m1 m'1) -> (eqmap B m2 m'2) -> - (eqmap A (MapDomRestrBy A B m1 m2) (MapDomRestrBy A B m'1 m'2)). -Proof. - Unfold eqmap eqm. Intros. Rewrite (MapDomRestrBy_semantics A B m1 m2 a). - Rewrite (MapDomRestrBy_semantics A B m'1 m'2 a). Rewrite (H a). Rewrite (H0 a). Reflexivity. -Qed. - -Lemma MapDomRestrBy_ext_l : (A,B:Set) (m1:(Map A)) (m2:(Map B)) (m'1:(Map A)) - (eqmap A m1 m'1) -> - (eqmap A (MapDomRestrBy A B m1 m2) (MapDomRestrBy A B m'1 m2)). -Proof. - Intros. Apply MapDomRestrBy_ext; [ Assumption | Apply eqmap_refl ]. -Qed. - -Lemma MapDomRestrBy_ext_r : (A,B:Set) (m1:(Map A)) (m2:(Map B)) (m'2:(Map B)) - (eqmap B m2 m'2) -> - (eqmap A (MapDomRestrBy A B m1 m2) (MapDomRestrBy A B m1 m'2)). -Proof. - Intros. Apply MapDomRestrBy_ext; [ Apply eqmap_refl | Assumption ]. -Qed. - -Lemma MapDomRestrBy_m_m : (A:Set) (m:(Map A)) - (eqmap A (MapDomRestrBy A unit m (MapDom A m)) (M0 A)). -Proof. - Intros. Apply eqmap_trans with m':=(MapDomRestrBy A A m m). Apply eqmap_sym. - Apply MapDomRestrBy_Dom. - Apply MapDomRestrBy_m_m_1. -Qed. - -Lemma FSetDelta_assoc : (s,s',s'':FSet) - (eqmap unit (MapDelta ? (MapDelta ? s s') s'') (MapDelta ? s (MapDelta ? s' s''))). -Proof. - Unfold eqmap eqm. Intros. Rewrite (MapDelta_semantics unit (MapDelta unit s s') s'' a). - Rewrite (MapDelta_semantics unit s s' a). - Rewrite (MapDelta_semantics unit s (MapDelta unit s' s'') a). - Rewrite (MapDelta_semantics unit s' s'' a). - Case (MapGet ? s a); Case (MapGet ? s' a); Case (MapGet ? s'' a); Trivial. - Intros. Elim u. Elim u1. Reflexivity. -Qed. - -Lemma FSet_ext : (s,s':FSet) ((a:ad) (in_FSet a s)=(in_FSet a s')) -> (eqmap unit s s'). -Proof. - Unfold in_FSet eqmap eqm. Intros. Elim (sumbool_of_bool (in_dom ? a s)). Intro H0. - Elim (in_dom_some ? s a H0). Intros y H1. Rewrite (H a) in H0. Elim (in_dom_some ? s' a H0). - Intros y' H2. Rewrite H1. Rewrite H2. Elim y. Elim y'. Reflexivity. - Intro H0. Rewrite (in_dom_none ? s a H0). Rewrite (H a) in H0. Rewrite (in_dom_none ? s' a H0). - Reflexivity. -Qed. - -Lemma FSetUnion_comm : (s,s':FSet) (eqmap unit (FSetUnion s s') (FSetUnion s' s)). -Proof. - Intros. Apply FSet_ext. Intro. Rewrite in_FSet_union. Rewrite in_FSet_union. Apply orb_sym. -Qed. - -Lemma FSetUnion_assoc : (s,s',s'':FSet) (eqmap unit - (FSetUnion (FSetUnion s s') s'') (FSetUnion s (FSetUnion s' s''))). -Proof. - Exact (MapMerge_assoc unit). -Qed. - -Lemma FSetUnion_M0_s : (s:FSet) (eqmap unit (FSetUnion (M0 unit) s) s). -Proof. - Exact (MapMerge_empty_m unit). -Qed. - -Lemma FSetUnion_s_M0 : (s:FSet) (eqmap unit (FSetUnion s (M0 unit)) s). -Proof. - Exact (MapMerge_m_empty unit). -Qed. - -Lemma FSetUnion_idempotent : (s:FSet) (eqmap unit (FSetUnion s s) s). -Proof. - Exact (MapMerge_idempotent unit). -Qed. - -Lemma FSetInter_comm : (s,s':FSet) (eqmap unit (FSetInter s s') (FSetInter s' s)). -Proof. - Intros. Apply FSet_ext. Intro. Rewrite in_FSet_inter. Rewrite in_FSet_inter. Apply andb_sym. -Qed. - -Lemma FSetInter_assoc : (s,s',s'':FSet) (eqmap unit - (FSetInter (FSetInter s s') s'') (FSetInter s (FSetInter s' s''))). -Proof. - Exact (MapDomRestrTo_assoc unit unit unit). -Qed. - -Lemma FSetInter_M0_s : (s:FSet) (eqmap unit (FSetInter (M0 unit) s) (M0 unit)). -Proof. - Exact (MapDomRestrTo_empty_m unit unit). -Qed. - -Lemma FSetInter_s_M0 : (s:FSet) (eqmap unit (FSetInter s (M0 unit)) (M0 unit)). -Proof. - Exact (MapDomRestrTo_m_empty unit unit). -Qed. - -Lemma FSetInter_idempotent : (s:FSet) (eqmap unit (FSetInter s s) s). -Proof. - Exact (MapDomRestrTo_idempotent unit). -Qed. - -Lemma FSetUnion_Inter_l : (s,s',s'':FSet) (eqmap unit - (FSetUnion (FSetInter s s') s'') (FSetInter (FSetUnion s s'') (FSetUnion s' s''))). -Proof. - Intros. Apply FSet_ext. Intro. Rewrite in_FSet_union. Rewrite in_FSet_inter. - Rewrite in_FSet_inter. Rewrite in_FSet_union. Rewrite in_FSet_union. - Case (in_FSet a s); Case (in_FSet a s'); Case (in_FSet a s''); Reflexivity. -Qed. - -Lemma FSetUnion_Inter_r : (s,s',s'':FSet) (eqmap unit - (FSetUnion s (FSetInter s' s'')) (FSetInter (FSetUnion s s') (FSetUnion s s''))). -Proof. - Intros. Apply FSet_ext. Intro. Rewrite in_FSet_union. Rewrite in_FSet_inter. - Rewrite in_FSet_inter. Rewrite in_FSet_union. Rewrite in_FSet_union. - Case (in_FSet a s); Case (in_FSet a s'); Case (in_FSet a s''); Reflexivity. -Qed. - -Lemma FSetInter_Union_l : (s,s',s'':FSet) (eqmap unit - (FSetInter (FSetUnion s s') s'') (FSetUnion (FSetInter s s'') (FSetInter s' s''))). -Proof. - Intros. Apply FSet_ext. Intro. Rewrite in_FSet_inter. Rewrite in_FSet_union. - Rewrite in_FSet_union. Rewrite in_FSet_inter. Rewrite in_FSet_inter. - Case (in_FSet a s); Case (in_FSet a s'); Case (in_FSet a s''); Reflexivity. -Qed. - -Lemma FSetInter_Union_r : (s,s',s'':FSet) (eqmap unit - (FSetInter s (FSetUnion s' s'')) (FSetUnion (FSetInter s s') (FSetInter s s''))). -Proof. - Intros. Apply FSet_ext. Intro. Rewrite in_FSet_inter. Rewrite in_FSet_union. - Rewrite in_FSet_union. Rewrite in_FSet_inter. Rewrite in_FSet_inter. - Case (in_FSet a s); Case (in_FSet a s'); Case (in_FSet a s''); Reflexivity. -Qed. |