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diff --git a/theories7/IntMap/Mapsubset.v b/theories7/IntMap/Mapsubset.v new file mode 100644 index 00000000..c0b1cccd --- /dev/null +++ b/theories7/IntMap/Mapsubset.v @@ -0,0 +1,554 @@ +(************************************************************************) +(* 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: Mapsubset.v,v 1.1.2.1 2004/07/16 19:31:28 herbelin Exp $ i*) + +Require Bool. +Require Sumbool. +Require Arith. +Require ZArith. +Require Addr. +Require Adist. +Require Addec. +Require Map. +Require Fset. +Require Mapaxioms. +Require Mapiter. + +Section MapSubsetDef. + + Variable A, B : Set. + + Definition MapSubset := [m:(Map A)] [m':(Map B)] + (a:ad) (in_dom A a m)=true -> (in_dom B a m')=true. + + Definition MapSubset_1 := [m:(Map A)] [m':(Map B)] + Cases (MapSweep A [a:ad][_:A] (negb (in_dom B a m')) m) of + NONE => true + | _ => false + end. + + Definition MapSubset_2 := [m:(Map A)] [m':(Map B)] + (eqmap A (MapDomRestrBy A B m m') (M0 A)). + + Lemma MapSubset_imp_1 : (m:(Map A)) (m':(Map B)) + (MapSubset m m') -> (MapSubset_1 m m')=true. + Proof. + Unfold MapSubset MapSubset_1. Intros. + Elim (option_sum ? (MapSweep A [a:ad][_:A](negb (in_dom B a m')) m)). + Intro H0. Elim H0. Intro r. Elim r. Intros a y H1. Cut (negb (in_dom B a m'))=true. + Intro. Cut (in_dom A a m)=false. Intro. Unfold in_dom in H3. + Rewrite (MapSweep_semantics_2 ? ? m a y H1) in H3. Discriminate H3. + Elim (sumbool_of_bool (in_dom A a m)). Intro H3. Rewrite (H a H3) in H2. Discriminate H2. + Trivial. + Exact (MapSweep_semantics_1 ? ? m a y H1). + Intro H0. Rewrite H0. Reflexivity. + Qed. + + Lemma MapSubset_1_imp : (m:(Map A)) (m':(Map B)) + (MapSubset_1 m m')=true -> (MapSubset m m'). + Proof. + Unfold MapSubset MapSubset_1. Unfold 2 in_dom. Intros. Elim (option_sum ? (MapGet A m a)). + Intro H1. Elim H1. Intros y H2. + Elim (option_sum ? (MapSweep A [a:ad][_:A](negb (in_dom B a m')) m)). Intro H3. + Elim H3. Intro r. Elim r. Intros a' y' H4. Rewrite H4 in H. Discriminate H. + Intro H3. Cut (negb (in_dom B a m'))=false. Intro. Rewrite (negb_intro (in_dom B a m')). + Rewrite H4. Reflexivity. + Exact (MapSweep_semantics_3 ? ? m H3 a y H2). + Intro H1. Rewrite H1 in H0. Discriminate H0. + Qed. + + Lemma map_dom_empty_1 : + (m:(Map A)) (eqmap A m (M0 A)) -> (a:ad) (in_dom ? a m)=false. + Proof. + Unfold eqmap eqm in_dom. Intros. Rewrite (H a). Reflexivity. + Qed. + + Lemma map_dom_empty_2 : + (m:(Map A)) ((a:ad) (in_dom ? a m)=false) -> (eqmap A m (M0 A)). + Proof. + Unfold eqmap eqm in_dom. Intros. + Cut (Cases (MapGet A m a) of NONE => false | (SOME _) => true end)=false. + Case (MapGet A m a). Trivial. + Intros. Discriminate H0. + Exact (H a). + Qed. + + Lemma MapSubset_imp_2 : + (m:(Map A)) (m':(Map B)) (MapSubset m m') -> (MapSubset_2 m m'). + Proof. + Unfold MapSubset MapSubset_2. Intros. Apply map_dom_empty_2. Intro. Rewrite in_dom_restrby. + Elim (sumbool_of_bool (in_dom A a m)). Intro H0. Rewrite H0. Rewrite (H a H0). Reflexivity. + Intro H0. Rewrite H0. Reflexivity. + Qed. + + Lemma MapSubset_2_imp : + (m:(Map A)) (m':(Map B)) (MapSubset_2 m m') -> (MapSubset m m'). + Proof. + Unfold MapSubset MapSubset_2. Intros. Cut (in_dom ? a (MapDomRestrBy A B m m'))=false. + Rewrite in_dom_restrby. Intro. Elim (andb_false_elim ? ? H1). Rewrite H0. + Intro H2. Discriminate H2. + Intro H2. Rewrite (negb_intro (in_dom B a m')). Rewrite H2. Reflexivity. + Exact (map_dom_empty_1 ? H a). + Qed. + +End MapSubsetDef. + +Section MapSubsetOrder. + + Variable A, B, C : Set. + + Lemma MapSubset_refl : (m:(Map A)) (MapSubset A A m m). + Proof. + Unfold MapSubset. Trivial. + Qed. + + Lemma MapSubset_antisym : (m:(Map A)) (m':(Map B)) + (MapSubset A B m m') -> (MapSubset B A m' m) -> + (eqmap unit (MapDom A m) (MapDom B m')). + Proof. + Unfold MapSubset eqmap eqm. Intros. Elim (option_sum ? (MapGet ? (MapDom A m) a)). + Intro H1. Elim H1. Intro t. Elim t. Intro H2. Elim (option_sum ? (MapGet ? (MapDom B m') a)). + Intro H3. Elim H3. Intro t'. Elim t'. Intro H4. Rewrite H4. Exact H2. + Intro H3. Cut (in_dom B a m')=true. Intro. Rewrite (MapDom_Dom B m' a) in H4. + Unfold in_FSet in_dom in H4. Rewrite H3 in H4. Discriminate H4. + Apply H. Rewrite (MapDom_Dom A m a). Unfold in_FSet in_dom. Rewrite H2. Reflexivity. + Intro H1. Elim (option_sum ? (MapGet ? (MapDom B m') a)). Intro H2. Elim H2. Intros t H3. + Cut (in_dom A a m)=true. Intro. Rewrite (MapDom_Dom A m a) in H4. Unfold in_FSet in_dom in H4. + Rewrite H1 in H4. Discriminate H4. + Apply H0. Rewrite (MapDom_Dom B m' a). Unfold in_FSet in_dom. Rewrite H3. Reflexivity. + Intro H2. Rewrite H2. Exact H1. + Qed. + + Lemma MapSubset_trans : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (MapSubset A B m m') -> (MapSubset B C m' m'') -> (MapSubset A C m m''). + Proof. + Unfold MapSubset. Intros. Apply H0. Apply H. Assumption. + Qed. + +End MapSubsetOrder. + +Section FSubsetOrder. + + Lemma FSubset_refl : (s:FSet) (MapSubset ? ? s s). + Proof. + Exact (MapSubset_refl unit). + Qed. + + Lemma FSubset_antisym : (s,s':FSet) + (MapSubset ? ? s s') -> (MapSubset ? ? s' s) -> (eqmap unit s s'). + Proof. + Intros. Rewrite <- (FSet_Dom s). Rewrite <- (FSet_Dom s'). + Exact (MapSubset_antisym ? ? s s' H H0). + Qed. + + Lemma FSubset_trans : (s,s',s'':FSet) + (MapSubset ? ? s s') -> (MapSubset ? ? s' s'') -> (MapSubset ? ? s s''). + Proof. + Exact (MapSubset_trans unit unit unit). + Qed. + +End FSubsetOrder. + +Section MapSubsetExtra. + + Variable A, B : Set. + + Lemma MapSubset_Dom_1 : (m:(Map A)) (m':(Map B)) + (MapSubset A B m m') -> (MapSubset unit unit (MapDom A m) (MapDom B m')). + Proof. + Unfold MapSubset. Intros. Elim (MapDom_semantics_2 ? m a H0). Intros y H1. + Cut (in_dom A a m)=true->(in_dom B a m')=true. Intro. Unfold in_dom in H2. + Rewrite H1 in H2. Elim (option_sum ? (MapGet B m' a)). Intro H3. Elim H3. + Intros y' H4. Exact (MapDom_semantics_1 ? m' a y' H4). + Intro H3. Rewrite H3 in H2. Cut false=true. Intro. Discriminate H4. + Apply H2. Reflexivity. + Exact (H a). + Qed. + + Lemma MapSubset_Dom_2 : (m:(Map A)) (m':(Map B)) + (MapSubset unit unit (MapDom A m) (MapDom B m')) -> (MapSubset A B m m'). + Proof. + Unfold MapSubset. Intros. Unfold in_dom in H0. Elim (option_sum ? (MapGet A m a)). + Intro H1. Elim H1. Intros y H2. + Elim (MapDom_semantics_2 ? ? ? (H a (MapDom_semantics_1 ? ? ? ? H2))). Intros y' H3. + Unfold in_dom. Rewrite H3. Reflexivity. + Intro H1. Rewrite H1 in H0. Discriminate H0. + Qed. + + Lemma MapSubset_1_Dom : (m:(Map A)) (m':(Map B)) + (MapSubset_1 A B m m')=(MapSubset_1 unit unit (MapDom A m) (MapDom B m')). + Proof. + Intros. Elim (sumbool_of_bool (MapSubset_1 A B m m')). Intro H. Rewrite H. + Apply sym_eq. Apply MapSubset_imp_1. Apply MapSubset_Dom_1. Exact (MapSubset_1_imp ? ? ? ? H). + Intro H. Rewrite H. Elim (sumbool_of_bool (MapSubset_1 unit unit (MapDom A m) (MapDom B m'))). + Intro H0. + Rewrite (MapSubset_imp_1 ? ? ? ? (MapSubset_Dom_2 ? ? (MapSubset_1_imp ? ? ? ? H0))) in H. + Discriminate H. + Intro. Apply sym_eq. Assumption. + Qed. + + Lemma MapSubset_Put : (m:(Map A)) (a:ad) (y:A) (MapSubset A A m (MapPut A m a y)). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_put. Rewrite H. Apply orb_b_true. + Qed. + + Lemma MapSubset_Put_mono : (m:(Map A)) (m':(Map B)) (a:ad) (y:A) (y':B) + (MapSubset A B m m') -> (MapSubset A B (MapPut A m a y) (MapPut B m' a y')). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_put. Rewrite (in_dom_put A m a y a0) in H0. + Elim (orb_true_elim ? ? H0). Intro H1. Rewrite H1. Reflexivity. + Intro H1. Rewrite (H ? H1). Apply orb_b_true. + Qed. + + Lemma MapSubset_Put_behind : + (m:(Map A)) (a:ad) (y:A) (MapSubset A A m (MapPut_behind A m a y)). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_put_behind. Rewrite H. Apply orb_b_true. + Qed. + + Lemma MapSubset_Put_behind_mono : (m:(Map A)) (m':(Map B)) (a:ad) (y:A) (y':B) + (MapSubset A B m m') -> + (MapSubset A B (MapPut_behind A m a y) (MapPut_behind B m' a y')). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_put_behind. + Rewrite (in_dom_put_behind A m a y a0) in H0. + Elim (orb_true_elim ? ? H0). Intro H1. Rewrite H1. Reflexivity. + Intro H1. Rewrite (H ? H1). Apply orb_b_true. + Qed. + + Lemma MapSubset_Remove : (m:(Map A)) (a:ad) (MapSubset A A (MapRemove A m a) m). + Proof. + Unfold MapSubset. Intros. Unfold MapSubset. Intros. Rewrite (in_dom_remove ? m a a0) in H. + Elim (andb_prop ? ? H). Trivial. + Qed. + + Lemma MapSubset_Remove_mono : (m:(Map A)) (m':(Map B)) (a:ad) + (MapSubset A B m m') -> (MapSubset A B (MapRemove A m a) (MapRemove B m' a)). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_remove. Rewrite (in_dom_remove A m a a0) in H0. + Elim (andb_prop ? ? H0). Intros. Rewrite H1. Rewrite (H ? H2). Reflexivity. + Qed. + + Lemma MapSubset_Merge_l : (m,m':(Map A)) (MapSubset A A m (MapMerge A m m')). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_merge. Rewrite H. Reflexivity. + Qed. + + Lemma MapSubset_Merge_r : (m,m':(Map A)) (MapSubset A A m' (MapMerge A m m')). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_merge. Rewrite H. Apply orb_b_true. + Qed. + + Lemma MapSubset_Merge_mono : (m,m':(Map A)) (m'',m''':(Map B)) + (MapSubset A B m m'') -> (MapSubset A B m' m''') -> + (MapSubset A B (MapMerge A m m') (MapMerge B m'' m''')). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_merge. Rewrite (in_dom_merge A m m' a) in H1. + Elim (orb_true_elim ? ? H1). Intro H2. Rewrite (H ? H2). Reflexivity. + Intro H2. Rewrite (H0 ? H2). Apply orb_b_true. + Qed. + + Lemma MapSubset_DomRestrTo_l : (m:(Map A)) (m':(Map B)) + (MapSubset A A (MapDomRestrTo A B m m') m). + Proof. + Unfold MapSubset. Intros. Rewrite (in_dom_restrto ? ? m m' a) in H. Elim (andb_prop ? ? H). + Trivial. + Qed. + + Lemma MapSubset_DomRestrTo_r: (m:(Map A)) (m':(Map B)) + (MapSubset A B (MapDomRestrTo A B m m') m'). + Proof. + Unfold MapSubset. Intros. Rewrite (in_dom_restrto ? ? m m' a) in H. Elim (andb_prop ? ? H). + Trivial. + Qed. + + Lemma MapSubset_ext : (m0,m1:(Map A)) (m2,m3:(Map B)) + (eqmap A m0 m1) -> (eqmap B m2 m3) -> + (MapSubset A B m0 m2) -> (MapSubset A B m1 m3). + Proof. + Intros. Apply MapSubset_2_imp. Unfold MapSubset_2. + Apply eqmap_trans with m':=(MapDomRestrBy A B m0 m2). Apply MapDomRestrBy_ext. Apply eqmap_sym. + Assumption. + Apply eqmap_sym. Assumption. + Exact (MapSubset_imp_2 ? ? ? ? H1). + Qed. + + Variable C, D : Set. + + Lemma MapSubset_DomRestrTo_mono : + (m:(Map A)) (m':(Map B)) (m'':(Map C)) (m''':(Map D)) + (MapSubset ? ? m m'') -> (MapSubset ? ? m' m''') -> + (MapSubset ? ? (MapDomRestrTo ? ? m m') (MapDomRestrTo ? ? m'' m''')). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_restrto. Rewrite (in_dom_restrto A B m m' a) in H1. + Elim (andb_prop ? ? H1). Intros. Rewrite (H ? H2). Rewrite (H0 ? H3). Reflexivity. + Qed. + + Lemma MapSubset_DomRestrBy_l : (m:(Map A)) (m':(Map B)) + (MapSubset A A (MapDomRestrBy A B m m') m). + Proof. + Unfold MapSubset. Intros. Rewrite (in_dom_restrby ? ? m m' a) in H. Elim (andb_prop ? ? H). + Trivial. + Qed. + + Lemma MapSubset_DomRestrBy_mono : + (m:(Map A)) (m':(Map B)) (m'':(Map C)) (m''':(Map D)) + (MapSubset ? ? m m'') -> (MapSubset ? ? m''' m') -> + (MapSubset ? ? (MapDomRestrBy ? ? m m') (MapDomRestrBy ? ? m'' m''')). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_restrby. Rewrite (in_dom_restrby A B m m' a) in H1. + Elim (andb_prop ? ? H1). Intros. Rewrite (H ? H2). Elim (sumbool_of_bool (in_dom D a m''')). + Intro H4. Rewrite (H0 ? H4) in H3. Discriminate H3. + Intro H4. Rewrite H4. Reflexivity. + Qed. + +End MapSubsetExtra. + +Section MapDisjointDef. + + Variable A, B : Set. + + Definition MapDisjoint := [m:(Map A)] [m':(Map B)] + (a:ad) (in_dom A a m)=true -> (in_dom B a m')=true -> False. + + Definition MapDisjoint_1 := [m:(Map A)] [m':(Map B)] + Cases (MapSweep A [a:ad][_:A] (in_dom B a m') m) of + NONE => true + | _ => false + end. + + Definition MapDisjoint_2 := [m:(Map A)] [m':(Map B)] + (eqmap A (MapDomRestrTo A B m m') (M0 A)). + + Lemma MapDisjoint_imp_1 : (m:(Map A)) (m':(Map B)) + (MapDisjoint m m') -> (MapDisjoint_1 m m')=true. + Proof. + Unfold MapDisjoint MapDisjoint_1. Intros. + Elim (option_sum ? (MapSweep A [a:ad][_:A](in_dom B a m') m)). Intro H0. Elim H0. + Intro r. Elim r. Intros a y H1. Cut (in_dom A a m)=true->(in_dom B a m')=true->False. + Intro. Unfold 1 in_dom in H2. Rewrite (MapSweep_semantics_2 ? ? ? ? ? H1) in H2. + Rewrite (MapSweep_semantics_1 ? ? ? ? ? H1) in H2. Elim (H2 (refl_equal ? ?) (refl_equal ? ?)). + Exact (H a). + Intro H0. Rewrite H0. Reflexivity. + Qed. + + Lemma MapDisjoint_1_imp : (m:(Map A)) (m':(Map B)) + (MapDisjoint_1 m m')=true -> (MapDisjoint m m'). + Proof. + Unfold MapDisjoint MapDisjoint_1. Intros. + Elim (option_sum ? (MapSweep A [a:ad][_:A](in_dom B a m') m)). Intro H2. Elim H2. + Intro r. Elim r. Intros a' y' H3. Rewrite H3 in H. Discriminate H. + Intro H2. Unfold in_dom in H0. Elim (option_sum ? (MapGet A m a)). Intro H3. Elim H3. + Intros y H4. Rewrite (MapSweep_semantics_3 ? ? ? H2 a y H4) in H1. Discriminate H1. + Intro H3. Rewrite H3 in H0. Discriminate H0. + Qed. + + Lemma MapDisjoint_imp_2 : (m:(Map A)) (m':(Map B)) (MapDisjoint m m') -> + (MapDisjoint_2 m m'). + Proof. + Unfold MapDisjoint MapDisjoint_2. Unfold eqmap eqm. Intros. + Rewrite (MapDomRestrTo_semantics A B m m' a). + Cut (in_dom A a m)=true->(in_dom B a m')=true->False. Intro. + Elim (option_sum ? (MapGet A m a)). Intro H1. Elim H1. Intros y H2. Unfold 1 in_dom in H0. + Elim (option_sum ? (MapGet B m' a)). Intro H3. Elim H3. Intros y' H4. Unfold 1 in_dom in H0. + Rewrite H4 in H0. Rewrite H2 in H0. Elim (H0 (refl_equal ? ?) (refl_equal ? ?)). + Intro H3. Rewrite H3. Reflexivity. + Intro H1. Rewrite H1. Case (MapGet B m' a); Reflexivity. + Exact (H a). + Qed. + + Lemma MapDisjoint_2_imp : (m:(Map A)) (m':(Map B)) (MapDisjoint_2 m m') -> + (MapDisjoint m m'). + Proof. + Unfold MapDisjoint MapDisjoint_2. Unfold eqmap eqm. Intros. Elim (in_dom_some ? ? ? H0). + Intros y H2. Elim (in_dom_some ? ? ? H1). Intros y' H3. + Cut (MapGet A (MapDomRestrTo A B m m') a)=(NONE A). Intro. + Rewrite (MapDomRestrTo_semantics ? ? m m' a) in H4. Rewrite H3 in H4. Rewrite H2 in H4. + Discriminate H4. + Exact (H a). + Qed. + + Lemma Map_M0_disjoint : (m:(Map B)) (MapDisjoint (M0 A) m). + Proof. + Unfold MapDisjoint in_dom. Intros. Discriminate H. + Qed. + + Lemma Map_disjoint_M0 : (m:(Map A)) (MapDisjoint m (M0 B)). + Proof. + Unfold MapDisjoint in_dom. Intros. Discriminate H0. + Qed. + +End MapDisjointDef. + +Section MapDisjointExtra. + + Variable A, B : Set. + + Lemma MapDisjoint_ext : (m0,m1:(Map A)) (m2,m3:(Map B)) + (eqmap A m0 m1) -> (eqmap B m2 m3) -> + (MapDisjoint A B m0 m2) -> (MapDisjoint A B m1 m3). + Proof. + Intros. Apply MapDisjoint_2_imp. Unfold MapDisjoint_2. + Apply eqmap_trans with m':=(MapDomRestrTo A B m0 m2). Apply eqmap_sym. Apply MapDomRestrTo_ext. + Assumption. + Assumption. + Exact (MapDisjoint_imp_2 ? ? ? ? H1). + Qed. + + Lemma MapMerge_disjoint : (m,m':(Map A)) (MapDisjoint A A m m') -> + (a:ad) (in_dom A a (MapMerge A m m'))= + (orb (andb (in_dom A a m) (negb (in_dom A a m'))) + (andb (in_dom A a m') (negb (in_dom A a m)))). + Proof. + Unfold MapDisjoint. Intros. Rewrite in_dom_merge. Elim (sumbool_of_bool (in_dom A a m)). + Intro H0. Rewrite H0. Elim (sumbool_of_bool (in_dom A a m')). Intro H1. Elim (H a H0 H1). + Intro H1. Rewrite H1. Reflexivity. + Intro H0. Rewrite H0. Simpl. Rewrite andb_b_true. Reflexivity. + Qed. + + Lemma MapDisjoint_M2_l : (m0,m1:(Map A)) (m2,m3:(Map B)) + (MapDisjoint A B (M2 A m0 m1) (M2 B m2 m3)) -> (MapDisjoint A B m0 m2). + Proof. + Unfold MapDisjoint in_dom. Intros. Elim (option_sum ? (MapGet A m0 a)). Intro H2. + Elim H2. Intros y H3. Elim (option_sum ? (MapGet B m2 a)). Intro H4. Elim H4. + Intros y' H5. Apply (H (ad_double a)). + Rewrite (MapGet_M2_bit_0_0 ? (ad_double a) (ad_double_bit_0 a) m0 m1). + Rewrite (ad_double_div_2 a). Rewrite H3. Reflexivity. + Rewrite (MapGet_M2_bit_0_0 ? (ad_double a) (ad_double_bit_0 a) m2 m3). + Rewrite (ad_double_div_2 a). Rewrite H5. Reflexivity. + Intro H4. Rewrite H4 in H1. Discriminate H1. + Intro H2. Rewrite H2 in H0. Discriminate H0. + Qed. + + Lemma MapDisjoint_M2_r : (m0,m1:(Map A)) (m2,m3:(Map B)) + (MapDisjoint A B (M2 A m0 m1) (M2 B m2 m3)) -> (MapDisjoint A B m1 m3). + Proof. + Unfold MapDisjoint in_dom. Intros. Elim (option_sum ? (MapGet A m1 a)). Intro H2. + Elim H2. Intros y H3. Elim (option_sum ? (MapGet B m3 a)). Intro H4. Elim H4. + Intros y' H5. Apply (H (ad_double_plus_un a)). + Rewrite (MapGet_M2_bit_0_1 ? (ad_double_plus_un a) (ad_double_plus_un_bit_0 a) m0 m1). + Rewrite (ad_double_plus_un_div_2 a). Rewrite H3. Reflexivity. + Rewrite (MapGet_M2_bit_0_1 ? (ad_double_plus_un a) (ad_double_plus_un_bit_0 a) m2 m3). + Rewrite (ad_double_plus_un_div_2 a). Rewrite H5. Reflexivity. + Intro H4. Rewrite H4 in H1. Discriminate H1. + Intro H2. Rewrite H2 in H0. Discriminate H0. + Qed. + + Lemma MapDisjoint_M2 : (m0,m1:(Map A)) (m2,m3:(Map B)) + (MapDisjoint A B m0 m2) -> (MapDisjoint A B m1 m3) -> + (MapDisjoint A B (M2 A m0 m1) (M2 B m2 m3)). + Proof. + Unfold MapDisjoint in_dom. Intros. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H3. + Rewrite (MapGet_M2_bit_0_1 A a H3 m0 m1) in H1. + Rewrite (MapGet_M2_bit_0_1 B a H3 m2 m3) in H2. Exact (H0 (ad_div_2 a) H1 H2). + Intro H3. Rewrite (MapGet_M2_bit_0_0 A a H3 m0 m1) in H1. + Rewrite (MapGet_M2_bit_0_0 B a H3 m2 m3) in H2. Exact (H (ad_div_2 a) H1 H2). + Qed. + + Lemma MapDisjoint_M1_l : (m:(Map A)) (a:ad) (y:B) + (MapDisjoint B A (M1 B a y) m) -> (in_dom A a m)=false. + Proof. + Unfold MapDisjoint. Intros. Elim (sumbool_of_bool (in_dom A a m)). Intro H0. + Elim (H a (in_dom_M1_1 B a y) H0). + Trivial. + Qed. + + Lemma MapDisjoint_M1_r : (m:(Map A)) (a:ad) (y:B) + (MapDisjoint A B m (M1 B a y)) -> (in_dom A a m)=false. + Proof. + Unfold MapDisjoint. Intros. Elim (sumbool_of_bool (in_dom A a m)). Intro H0. + Elim (H a H0 (in_dom_M1_1 B a y)). + Trivial. + Qed. + + Lemma MapDisjoint_M1_conv_l : (m:(Map A)) (a:ad) (y:B) + (in_dom A a m)=false -> (MapDisjoint B A (M1 B a y) m). + Proof. + Unfold MapDisjoint. Intros. Rewrite (in_dom_M1_2 B a a0 y H0) in H. Rewrite H1 in H. + Discriminate H. + Qed. + + Lemma MapDisjoint_M1_conv_r : (m:(Map A)) (a:ad) (y:B) + (in_dom A a m)=false -> (MapDisjoint A B m (M1 B a y)). + Proof. + Unfold MapDisjoint. Intros. Rewrite (in_dom_M1_2 B a a0 y H1) in H. Rewrite H0 in H. + Discriminate H. + Qed. + + Lemma MapDisjoint_sym : (m:(Map A)) (m':(Map B)) + (MapDisjoint A B m m') -> (MapDisjoint B A m' m). + Proof. + Unfold MapDisjoint. Intros. Exact (H ? H1 H0). + Qed. + + Lemma MapDisjoint_empty : (m:(Map A)) (MapDisjoint A A m m) -> (eqmap A m (M0 A)). + Proof. + Unfold eqmap eqm. Intros. Rewrite <- (MapDomRestrTo_idempotent A m a). + Exact (MapDisjoint_imp_2 A A m m H a). + Qed. + + Lemma MapDelta_disjoint : (m,m':(Map A)) (MapDisjoint A A m m') -> + (eqmap A (MapDelta A m m') (MapMerge A m m')). + Proof. + Intros. + Apply eqmap_trans with m':=(MapDomRestrBy A A (MapMerge A m m') (MapDomRestrTo A A m m')). + Apply MapDelta_as_DomRestrBy. + Apply eqmap_trans with m':=(MapDomRestrBy A A (MapMerge A m m') (M0 A)). + Apply MapDomRestrBy_ext. Apply eqmap_refl. + Exact (MapDisjoint_imp_2 A A m m' H). + Apply MapDomRestrBy_m_empty. + Qed. + + Variable C : Set. + + Lemma MapDomRestr_disjoint : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (MapDisjoint A B (MapDomRestrTo A C m m'') (MapDomRestrBy B C m' m'')). + Proof. + Unfold MapDisjoint. Intros m m' m'' a. Rewrite in_dom_restrto. Rewrite in_dom_restrby. + Intros. Elim (andb_prop ? ? H). Elim (andb_prop ? ? H0). Intros. Rewrite H4 in H2. + Discriminate H2. + Qed. + + Lemma MapDelta_RestrTo_disjoint : (m,m':(Map A)) + (MapDisjoint A A (MapDelta A m m') (MapDomRestrTo A A m m')). + Proof. + Unfold MapDisjoint. Intros m m' a. Rewrite in_dom_delta. Rewrite in_dom_restrto. + Intros. Elim (andb_prop ? ? H0). Intros. Rewrite H1 in H. Rewrite H2 in H. Discriminate H. + Qed. + + Lemma MapDelta_RestrTo_disjoint_2 : (m,m':(Map A)) + (MapDisjoint A A (MapDelta A m m') (MapDomRestrTo A A m' m)). + Proof. + Unfold MapDisjoint. Intros m m' a. Rewrite in_dom_delta. Rewrite in_dom_restrto. + Intros. Elim (andb_prop ? ? H0). Intros. Rewrite H1 in H. Rewrite H2 in H. Discriminate H. + Qed. + + Variable D : Set. + + Lemma MapSubset_Disjoint : (m:(Map A)) (m':(Map B)) (m'':(Map C)) (m''':(Map D)) + (MapSubset ? ? m m') -> (MapSubset ? ? m'' m''') -> (MapDisjoint ? ? m' m''') -> + (MapDisjoint ? ? m m''). + Proof. + Unfold MapSubset MapDisjoint. Intros. Exact (H1 ? (H ? H2) (H0 ? H3)). + Qed. + + Lemma MapSubset_Disjoint_l : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (MapSubset ? ? m m') -> (MapDisjoint ? ? m' m'') -> + (MapDisjoint ? ? m m''). + Proof. + Unfold MapSubset MapDisjoint. Intros. Exact (H0 ? (H ? H1) H2). + Qed. + + Lemma MapSubset_Disjoint_r : (m:(Map A)) (m'':(Map C)) (m''':(Map D)) + (MapSubset ? ? m'' m''') -> (MapDisjoint ? ? m m''') -> + (MapDisjoint ? ? m m''). + Proof. + Unfold MapSubset MapDisjoint. Intros. Exact (H0 ? H1 (H ? H2)). + Qed. + +End MapDisjointExtra. |