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diff --git a/theories/IntMap/Mapaxioms.v b/theories/IntMap/Mapaxioms.v new file mode 100644 index 00000000..9d09f2a9 --- /dev/null +++ b/theories/IntMap/Mapaxioms.v @@ -0,0 +1,763 @@ +(************************************************************************) +(* 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 1.4.2.1 2004/07/16 19:31:04 herbelin Exp $ i*) + +Require Import Bool. +Require Import Sumbool. +Require Import ZArith. +Require Import Addr. +Require Import Adist. +Require Import Addec. +Require Import Map. +Require Import Fset. + +Section MapAxioms. + + Variables A B C : Set. + + Lemma eqm_sym : forall f f':ad -> option A, eqm A f f' -> eqm A f' f. + Proof. + unfold eqm in |- *. intros. rewrite H. reflexivity. + Qed. + + Lemma eqm_refl : forall f:ad -> option A, eqm A f f. + Proof. + unfold eqm in |- *. trivial. + Qed. + + Lemma eqm_trans : + forall f f' f'':ad -> option A, eqm A f f' -> eqm A f' f'' -> eqm A f f''. + Proof. + unfold eqm in |- *. intros. rewrite H. exact (H0 a). + Qed. + + Definition eqmap (m m':Map A) := eqm A (MapGet A m) (MapGet A m'). + + Lemma eqmap_sym : forall m m':Map A, eqmap m m' -> eqmap m' m. + Proof. + intros. unfold eqmap in |- *. apply eqm_sym. assumption. + Qed. + + Lemma eqmap_refl : forall m:Map A, eqmap m m. + Proof. + intros. unfold eqmap in |- *. apply eqm_refl. + Qed. + + Lemma eqmap_trans : + forall 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 : + forall (m:Map A) (a:ad) (y:A), + eqmap (MapPut A m a y) (MapMerge A m (M1 A a y)). + Proof. + unfold eqmap, eqm in |- *. intros. rewrite (MapPut_semantics A m a y a0). + rewrite (MapMerge_semantics A m (M1 A a y) a0). unfold MapGet at 2 in |- *. + elim (sumbool_of_bool (ad_eq a a0)); intro H; rewrite H; reflexivity. + Qed. + + Lemma MapPut_ext : + forall m m':Map A, + eqmap m m' -> + forall (a:ad) (y:A), eqmap (MapPut A m a y) (MapPut A m' a y). + Proof. + unfold eqmap, eqm in |- *. 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 : + forall (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 in |- *. 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 : + forall m m':Map A, + eqmap m m' -> + forall (a:ad) (y:A), + eqmap (MapPut_behind A m a y) (MapPut_behind A m' a y). + Proof. + unfold eqmap, eqm in |- *. 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 : forall m:Map A, MapMerge A (M0 A) m = m. + Proof. + trivial. + Qed. + + Lemma MapMerge_empty_m : forall m:Map A, eqmap (MapMerge A (M0 A) m) m. + Proof. + unfold eqmap, eqm in |- *. trivial. + Qed. + + Lemma MapMerge_m_empty_1 : forall m:Map A, MapMerge A m (M0 A) = m. + Proof. + simple induction m; trivial. + Qed. + + Lemma MapMerge_m_empty : forall m:Map A, eqmap (MapMerge A m (M0 A)) m. + Proof. + unfold eqmap, eqm in |- *. intros. rewrite MapMerge_m_empty_1. reflexivity. + Qed. + + Lemma MapMerge_empty_l : + forall m m':Map A, eqmap (MapMerge A m m') (M0 A) -> eqmap m (M0 A). + Proof. + unfold eqmap, eqm in |- *. 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 : + forall m m':Map A, eqmap (MapMerge A m m') (M0 A) -> eqmap m' (M0 A). + Proof. + unfold eqmap, eqm in |- *. 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 : + forall m m' m'':Map A, + eqmap (MapMerge A (MapMerge A m m') m'') + (MapMerge A m (MapMerge A m' m'')). + Proof. + unfold eqmap, eqm in |- *. 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 : forall m:Map A, eqmap (MapMerge A m m) m. + Proof. + unfold eqmap, eqm in |- *. intros. rewrite (MapMerge_semantics A m m a). + case (MapGet A m a); trivial. + Qed. + + Lemma MapMerge_ext : + forall 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 in |- *. 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 : + forall 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 : + forall 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 : + forall 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 in |- *. 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 : + forall (m:Map A) (a:ad) (y:B), + eqmap (MapRemove A m a) (MapDomRestrBy A B m (M1 B a y)). + Proof. + unfold eqmap, eqm in |- *. 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 : + forall m m':Map A, + eqmap m m' -> forall a:ad, eqmap (MapRemove A m a) (MapRemove A m' a). + Proof. + unfold eqmap, eqm in |- *. 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 : + forall m:Map B, MapDomRestrTo A B (M0 A) m = M0 A. + Proof. + trivial. + Qed. + + Lemma MapDomRestrTo_empty_m : + forall m:Map B, eqmap (MapDomRestrTo A B (M0 A) m) (M0 A). + Proof. + unfold eqmap, eqm in |- *. trivial. + Qed. + + Lemma MapDomRestrTo_m_empty_1 : + forall m:Map A, MapDomRestrTo A B m (M0 B) = M0 A. + Proof. + simple induction m; trivial. + Qed. + + Lemma MapDomRestrTo_m_empty : + forall m:Map A, eqmap (MapDomRestrTo A B m (M0 B)) (M0 A). + Proof. + unfold eqmap, eqm in |- *. intros. rewrite (MapDomRestrTo_m_empty_1 m). reflexivity. + Qed. + + Lemma MapDomRestrTo_assoc : + forall (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 in |- *. 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 : + forall m:Map A, eqmap (MapDomRestrTo A A m m) m. + Proof. + unfold eqmap, eqm in |- *. intros. rewrite (MapDomRestrTo_semantics A A m m a). + case (MapGet A m a); trivial. + Qed. + + Lemma MapDomRestrTo_Dom : + forall (m:Map A) (m':Map B), + eqmap (MapDomRestrTo A B m m') (MapDomRestrTo A unit m (MapDom B m')). + Proof. + unfold eqmap, eqm in |- *. 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 : + forall m:Map B, MapDomRestrBy A B (M0 A) m = M0 A. + Proof. + trivial. + Qed. + + Lemma MapDomRestrBy_empty_m : + forall m:Map B, eqmap (MapDomRestrBy A B (M0 A) m) (M0 A). + Proof. + unfold eqmap, eqm in |- *. trivial. + Qed. + + Lemma MapDomRestrBy_m_empty_1 : + forall m:Map A, MapDomRestrBy A B m (M0 B) = m. + Proof. + simple induction m; trivial. + Qed. + + Lemma MapDomRestrBy_m_empty : + forall m:Map A, eqmap (MapDomRestrBy A B m (M0 B)) m. + Proof. + unfold eqmap, eqm in |- *. intros. rewrite (MapDomRestrBy_m_empty_1 m). reflexivity. + Qed. + + Lemma MapDomRestrBy_Dom : + forall (m:Map A) (m':Map B), + eqmap (MapDomRestrBy A B m m') (MapDomRestrBy A unit m (MapDom B m')). + Proof. + unfold eqmap, eqm in |- *. 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 : + forall m:Map A, eqmap (MapDomRestrBy A A m m) (M0 A). + Proof. + unfold eqmap, eqm in |- *. intros. rewrite (MapDomRestrBy_semantics A A m m a). + case (MapGet A m a); trivial. + Qed. + + Lemma MapDomRestrBy_By : + forall (m:Map A) (m' 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 in |- *. 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 : + forall (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 in |- *. 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 : + forall (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 in |- *. 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 : + forall (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 in |- *. 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 : + forall (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 in |- *. 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 : + forall (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 in |- *. 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 : + forall (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 in |- *. 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 : + forall (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 in |- *. 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 : + forall (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 in |- *. 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 : forall m:Map A, MapDelta A (M0 A) m = m. + Proof. + trivial. + Qed. + + Lemma MapDelta_empty_m : forall m:Map A, eqmap (MapDelta A (M0 A) m) m. + Proof. + unfold eqmap, eqm in |- *. trivial. + Qed. + + Lemma MapDelta_m_empty_1 : forall m:Map A, MapDelta A m (M0 A) = m. + Proof. + simple induction m; trivial. + Qed. + + Lemma MapDelta_m_empty : forall m:Map A, eqmap (MapDelta A m (M0 A)) m. + Proof. + unfold eqmap, eqm in |- *. intros. rewrite MapDelta_m_empty_1. reflexivity. + Qed. + + Lemma MapDelta_nilpotent : forall m:Map A, eqmap (MapDelta A m m) (M0 A). + Proof. + unfold eqmap, eqm in |- *. intros. rewrite (MapDelta_semantics A m m a). + case (MapGet A m a); trivial. + Qed. + + Lemma MapDelta_as_Merge : + forall 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 in |- *. 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 : + forall 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 in |- *. 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 : + forall 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 in |- *. 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 : + forall m m':Map A, eqmap (MapDelta A m m') (MapDelta A m' m). + Proof. + unfold eqmap, eqm in |- *. 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 : + forall 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 in |- *. 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 : + forall 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 : + forall 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 : + forall (m:Map A) (m':Map B), + eqmap m (MapMerge A (MapDomRestrTo A B m m') (MapDomRestrBy A B m m')). + Proof. + unfold eqmap, eqm in |- *. 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 : + forall (m:Map A) (m':Map B), + eqmap m (MapMerge A (MapDomRestrBy A B m m') (MapDomRestrTo A B m m')). + Proof. + unfold eqmap, eqm in |- *. 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 : + forall (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 in |- *. 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 : + forall (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 in |- *. 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 : + forall (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 : + forall (A B:Set) (m1:Map A) (m2 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 : + forall (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 in |- *. 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 : + forall (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 : + forall (A B:Set) (m1:Map A) (m2 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 : + forall (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 : + forall s s' s'':FSet, + eqmap unit (MapDelta _ (MapDelta _ s s') s'') + (MapDelta _ s (MapDelta _ s' s'')). +Proof. + unfold eqmap, eqm in |- *. 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 : + forall s s':FSet, + (forall a:ad, in_FSet a s = in_FSet a s') -> eqmap unit s s'. +Proof. + unfold in_FSet, eqmap, eqm in |- *. 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 : + forall 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_comm. +Qed. + +Lemma FSetUnion_assoc : + forall 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 : forall s:FSet, eqmap unit (FSetUnion (M0 unit) s) s. +Proof. + exact (MapMerge_empty_m unit). +Qed. + +Lemma FSetUnion_s_M0 : forall s:FSet, eqmap unit (FSetUnion s (M0 unit)) s. +Proof. + exact (MapMerge_m_empty unit). +Qed. + +Lemma FSetUnion_idempotent : forall s:FSet, eqmap unit (FSetUnion s s) s. +Proof. + exact (MapMerge_idempotent unit). +Qed. + +Lemma FSetInter_comm : + forall 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_comm. +Qed. + +Lemma FSetInter_assoc : + forall 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 : + forall s:FSet, eqmap unit (FSetInter (M0 unit) s) (M0 unit). +Proof. + exact (MapDomRestrTo_empty_m unit unit). +Qed. + +Lemma FSetInter_s_M0 : + forall s:FSet, eqmap unit (FSetInter s (M0 unit)) (M0 unit). +Proof. + exact (MapDomRestrTo_m_empty unit unit). +Qed. + +Lemma FSetInter_idempotent : forall s:FSet, eqmap unit (FSetInter s s) s. +Proof. + exact (MapDomRestrTo_idempotent unit). +Qed. + +Lemma FSetUnion_Inter_l : + forall 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 : + forall 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 : + forall 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 : + forall 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.
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