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diff --git a/theories7/IntMap/Mapaxioms.v b/theories7/IntMap/Mapaxioms.v new file mode 100644 index 00000000..085afd69 --- /dev/null +++ b/theories7/IntMap/Mapaxioms.v @@ -0,0 +1,670 @@ +(************************************************************************) +(* 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. |