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diff --git a/theories7/IntMap/Mapsubset.v b/theories7/IntMap/Mapsubset.v deleted file mode 100644 index c0b1cccd..00000000 --- a/theories7/IntMap/Mapsubset.v +++ /dev/null @@ -1,554 +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: 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. |