Coq.IntMap.Mapc

# Module Coq.IntMap.Mapc

``Require` Bool.
`Require` Sumbool.
`Require` Arith.
`Require` ZArith.
`Require` Addr.
`Require` Adist.
`Require` Addec.
`Require` Map.
`Require` Mapaxioms.
`Require` Fset.
`Require` Mapiter.
`Require` Mapsubset.
`Require` PolyList.
`Require` Lsort.
`Require` Mapcard.
`Require` Mapcanon.

`Section` MapC.

`Variable` A, B, C : Set.

`Lemma` MapPut_as_Merge_c : (m:(Map A)) (mapcanon A m) ->
(a:ad) (y:A) (MapPut A m a y)=(MapMerge A m (M1 A a y)).
`Proof`.
Intros. Apply mapcanon_unique. Exact (MapPut_canon A m H a y).
Apply MapMerge_canon. Assumption.
Apply M1_canon.
Apply MapPut_as_Merge.
`Qed`.

`Lemma` MapPut_behind_as_Merge_c : (m:(Map A)) (mapcanon A m) ->
(a:ad) (y:A) (MapPut_behind A m a y)=(MapMerge A (M1 A a y) m).
`Proof`.
Intros. Apply mapcanon_unique. Exact (MapPut_behind_canon A m H a y).
Apply MapMerge_canon. Apply M1_canon.
Assumption.
Apply MapPut_behind_as_Merge.
`Qed`.

`Lemma` MapMerge_empty_m_c : (m:(Map A)) (MapMerge A (M0 A) m)=m.
`Proof`.
Trivial.
`Qed`.

`Lemma` MapMerge_assoc_c : (m,m',m'':(Map A))
(mapcanon A m) -> (mapcanon A m') -> (mapcanon A m'') ->
(MapMerge A (MapMerge A m m') m'')=(MapMerge A m (MapMerge A m' m'')).
`Proof`.
Intros. Apply mapcanon_unique.
(Apply MapMerge_canon; Try Assumption). (Apply MapMerge_canon; Try Assumption).
(Apply MapMerge_canon; Try Assumption). (Apply MapMerge_canon; Try Assumption).
Apply MapMerge_assoc.
`Qed`.

`Lemma` MapMerge_idempotent_c : (m:(Map A)) (mapcanon A m) -> (MapMerge A m m)=m.
`Proof`.
Intros. Apply mapcanon_unique. (Apply MapMerge_canon; Assumption).
Assumption.
Apply MapMerge_idempotent.
`Qed`.

`Lemma` MapMerge_RestrTo_l_c : (m,m',m'':(Map A))
(mapcanon A m) -> (mapcanon A m'') ->
(MapMerge A (MapDomRestrTo A A m m') m'')=
(MapDomRestrTo A A (MapMerge A m m'') (MapMerge A m' m'')).
`Proof`.
Intros. Apply mapcanon_unique. Apply MapMerge_canon. Apply MapDomRestrTo_canon; Assumption.
Assumption.
Apply MapDomRestrTo_canon; Apply MapMerge_canon; Assumption.
Apply MapMerge_RestrTo_l.
`Qed`.

`Lemma` MapRemove_as_RestrBy_c : (m:(Map A)) (mapcanon A m) ->
(a:ad) (y:B) (MapRemove A m a)=(MapDomRestrBy A B m (M1 B a y)).
`Proof`.
Intros. Apply mapcanon_unique. (Apply MapRemove_canon; Assumption).
(Apply MapDomRestrBy_canon; Assumption).
Apply MapRemove_as_RestrBy.
`Qed`.

`Lemma` MapDomRestrTo_assoc_c : (m:(Map A)) (m':(Map B)) (m'':(Map C))
(mapcanon A m) ->
(MapDomRestrTo A C (MapDomRestrTo A B m m') m'')=
(MapDomRestrTo A B m (MapDomRestrTo B C m' m'')).
`Proof`.
Intros. Apply mapcanon_unique. (Apply MapDomRestrTo_canon; Try Assumption).
(Apply MapDomRestrTo_canon; Try Assumption).
(Apply MapDomRestrTo_canon; Try Assumption).
Apply MapDomRestrTo_assoc.
`Qed`.

`Lemma` MapDomRestrTo_idempotent_c : (m:(Map A)) (mapcanon A m) ->
(MapDomRestrTo A A m m)=m.
`Proof`.
Intros. Apply mapcanon_unique. (Apply MapDomRestrTo_canon; Assumption).
Assumption.
Apply MapDomRestrTo_idempotent.
`Qed`.

`Lemma` MapDomRestrTo_Dom_c : (m:(Map A)) (m':(Map B)) (mapcanon A m) ->
(MapDomRestrTo A B m m')=(MapDomRestrTo A unit m (MapDom B m')).
`Proof`.
Intros. Apply mapcanon_unique. (Apply MapDomRestrTo_canon; Assumption).
(Apply MapDomRestrTo_canon; Assumption).
Apply MapDomRestrTo_Dom.
`Qed`.

`Lemma` MapDomRestrBy_Dom_c : (m:(Map A)) (m':(Map B)) (mapcanon A m) ->
(MapDomRestrBy A B m m')=(MapDomRestrBy A unit m (MapDom B m')).
`Proof`.
Intros. Apply mapcanon_unique. Apply MapDomRestrBy_canon; Assumption.
Apply MapDomRestrBy_canon; Assumption.
Apply MapDomRestrBy_Dom.
`Qed`.

`Lemma` MapDomRestrBy_By_c : (m:(Map A)) (m':(Map B)) (m'':(Map B))
(mapcanon A m) ->
(MapDomRestrBy A B (MapDomRestrBy A B m m') m'')=
(MapDomRestrBy A B m (MapMerge B m' m'')).
`Proof`.
Intros. Apply mapcanon_unique. (Apply MapDomRestrBy_canon; Try Assumption).
(Apply MapDomRestrBy_canon; Try Assumption).
(Apply MapDomRestrBy_canon; Try Assumption).
Apply MapDomRestrBy_By.
`Qed`.

`Lemma` MapDomRestrBy_By_comm_c : (m:(Map A)) (m':(Map B)) (m'':(Map C))
(mapcanon A m) ->
(MapDomRestrBy A C (MapDomRestrBy A B m m') m'')=
(MapDomRestrBy A B (MapDomRestrBy A C m m'') m').
`Proof`.
Intros. Apply mapcanon_unique. Apply MapDomRestrBy_canon.
(Apply MapDomRestrBy_canon; Assumption).
Apply MapDomRestrBy_canon. (Apply MapDomRestrBy_canon; Assumption).
Apply MapDomRestrBy_By_comm.
`Qed`.

`Lemma` MapDomRestrBy_To_c : (m:(Map A)) (m':(Map B)) (m'':(Map C))
(mapcanon A m) ->
(MapDomRestrBy A C (MapDomRestrTo A B m m') m'')=
(MapDomRestrTo A B m (MapDomRestrBy B C m' m'')).
`Proof`.
Intros. Apply mapcanon_unique. Apply MapDomRestrBy_canon.
(Apply MapDomRestrTo_canon; Assumption).
(Apply MapDomRestrTo_canon; Assumption).
Apply MapDomRestrBy_To.
`Qed`.

`Lemma` MapDomRestrBy_To_comm_c : (m:(Map A)) (m':(Map B)) (m'':(Map C))
(mapcanon A m) ->
(MapDomRestrBy A C (MapDomRestrTo A B m m') m'')=
(MapDomRestrTo A B (MapDomRestrBy A C m m'') m').
`Proof`.
Intros. Apply mapcanon_unique. Apply MapDomRestrBy_canon.
Apply MapDomRestrTo_canon; Assumption.
Apply MapDomRestrTo_canon. Apply MapDomRestrBy_canon; Assumption.
Apply MapDomRestrBy_To_comm.
`Qed`.

`Lemma` MapDomRestrTo_By_c : (m:(Map A)) (m':(Map B)) (m'':(Map C))
(mapcanon A m) ->
(MapDomRestrTo A C (MapDomRestrBy A B m m') m'')=
(MapDomRestrTo A C m (MapDomRestrBy C B m'' m')).
`Proof`.
Intros. Apply mapcanon_unique. Apply MapDomRestrTo_canon.
Apply MapDomRestrBy_canon; Assumption.
Apply MapDomRestrTo_canon; Assumption.
Apply MapDomRestrTo_By.
`Qed`.

`Lemma` MapDomRestrTo_By_comm_c : (m:(Map A)) (m':(Map B)) (m'':(Map C))
(mapcanon A m) ->
(MapDomRestrTo A C (MapDomRestrBy A B m m') m'')=
(MapDomRestrBy A B (MapDomRestrTo A C m m'') m').
`Proof`.
Intros. Apply mapcanon_unique. Apply MapDomRestrTo_canon.
(Apply MapDomRestrBy_canon; Assumption).
Apply MapDomRestrBy_canon. (Apply MapDomRestrTo_canon; Assumption).
Apply MapDomRestrTo_By_comm.
`Qed`.

`Lemma` MapDomRestrTo_To_comm_c : (m:(Map A)) (m':(Map B)) (m'':(Map C))
(mapcanon A m) ->
(MapDomRestrTo A C (MapDomRestrTo A B m m') m'')=
(MapDomRestrTo A B (MapDomRestrTo A C m m'') m').
`Proof`.
Intros. Apply mapcanon_unique. Apply MapDomRestrTo_canon.
Apply MapDomRestrTo_canon; Assumption.
Apply MapDomRestrTo_canon. Apply MapDomRestrTo_canon; Assumption.
Apply MapDomRestrTo_To_comm.
`Qed`.

`Lemma` MapMerge_DomRestrTo_c : (m,m':(Map A)) (m'':(Map B))
(mapcanon A m) -> (mapcanon A m') ->
(MapDomRestrTo A B (MapMerge A m m') m'')=
(MapMerge A (MapDomRestrTo A B m m'') (MapDomRestrTo A B m' m'')).
`Proof`.
Intros. Apply mapcanon_unique. Apply MapDomRestrTo_canon.
(Apply MapMerge_canon; Assumption).
Apply MapMerge_canon. (Apply MapDomRestrTo_canon; Assumption).
(Apply MapDomRestrTo_canon; Assumption).
Apply MapMerge_DomRestrTo.
`Qed`.

`Lemma` MapMerge_DomRestrBy_c : (m,m':(Map A)) (m'':(Map B))
(mapcanon A m) -> (mapcanon A m') ->
(MapDomRestrBy A B (MapMerge A m m') m'')=
(MapMerge A (MapDomRestrBy A B m m'') (MapDomRestrBy A B m' m'')).
`Proof`.
Intros. Apply mapcanon_unique. Apply MapDomRestrBy_canon. Apply MapMerge_canon; Assumption.
Apply MapMerge_canon. Apply MapDomRestrBy_canon; Assumption.
Apply MapDomRestrBy_canon; Assumption.
Apply MapMerge_DomRestrBy.
`Qed`.

`Lemma` MapDelta_nilpotent_c : (m:(Map A)) (mapcanon A m) ->
(MapDelta A m m)=(M0 A).
`Proof`.
Intros. Apply mapcanon_unique. (Apply MapDelta_canon; Assumption).
Apply M0_canon.
Apply MapDelta_nilpotent.
`Qed`.

`Lemma` MapDelta_as_Merge_c : (m,m':(Map A))
(mapcanon A m) -> (mapcanon A m') ->
(MapDelta A m m')=
(MapMerge A (MapDomRestrBy A A m m') (MapDomRestrBy A A m' m)).
`Proof`.
Intros. Apply mapcanon_unique. (Apply MapDelta_canon; Assumption).
(Apply MapMerge_canon; Apply MapDomRestrBy_canon; Assumption).
Apply MapDelta_as_Merge.
`Qed`.

`Lemma` MapDelta_as_DomRestrBy_c : (m,m':(Map A))
(mapcanon A m) -> (mapcanon A m') ->
(MapDelta A m m')=
(MapDomRestrBy A A (MapMerge A m m') (MapDomRestrTo A A m m')).
`Proof`.
Intros. Apply mapcanon_unique. Apply MapDelta_canon; Assumption.
Apply MapDomRestrBy_canon. (Apply MapMerge_canon; Assumption).
Apply MapDelta_as_DomRestrBy.
`Qed`.

`Lemma` MapDelta_as_DomRestrBy_2_c : (m,m':(Map A))
(mapcanon A m) -> (mapcanon A m') ->
(MapDelta A m m')=
(MapDomRestrBy A A (MapMerge A m m') (MapDomRestrTo A A m' m)).
`Proof`.
Intros. Apply mapcanon_unique. (Apply MapDelta_canon; Assumption).
Apply MapDomRestrBy_canon. Apply MapMerge_canon; Assumption.
Apply MapDelta_as_DomRestrBy_2.
`Qed`.

`Lemma` MapDelta_sym_c : (m,m':(Map A))
(mapcanon A m) -> (mapcanon A m') -> (MapDelta A m m')=(MapDelta A m' m).
`Proof`.
Intros. Apply mapcanon_unique. (Apply MapDelta_canon; Assumption).
(Apply MapDelta_canon; Assumption). Apply MapDelta_sym.
`Qed`.

`Lemma` MapDom_Split_1_c : (m:(Map A)) (m':(Map B)) (mapcanon A m) ->
m=(MapMerge A (MapDomRestrTo A B m m') (MapDomRestrBy A B m m')).
`Proof`.
Intros. Apply mapcanon_unique. Assumption.
Apply MapMerge_canon. Apply MapDomRestrTo_canon; Assumption.
Apply MapDomRestrBy_canon; Assumption.
Apply MapDom_Split_1.
`Qed`.

`Lemma` MapDom_Split_2_c : (m:(Map A)) (m':(Map B)) (mapcanon A m) ->
m=(MapMerge A (MapDomRestrBy A B m m') (MapDomRestrTo A B m m')).
`Proof`.
Intros. Apply mapcanon_unique. Assumption.
Apply MapMerge_canon. (Apply MapDomRestrBy_canon; Assumption).
(Apply MapDomRestrTo_canon; Assumption).
Apply MapDom_Split_2.
`Qed`.

`Lemma` MapDom_Split_3_c : (m:(Map A)) (m':(Map B)) (mapcanon A m) ->
(MapDomRestrTo A A (MapDomRestrTo A B m m') (MapDomRestrBy A B m m'))=
(M0 A).
`Proof`.
Intros. Apply mapcanon_unique. Apply MapDomRestrTo_canon.
Apply MapDomRestrTo_canon; Assumption.
Apply M0_canon.
Apply MapDom_Split_3.
`Qed`.

`Lemma` Map_of_alist_of_Map_c : (m:(Map A)) (mapcanon A m) ->
(Map_of_alist A (alist_of_Map A m))=m.
`Proof`.
Intros. (Apply mapcanon_unique; Try Assumption). Apply Map_of_alist_canon.
Apply Map_of_alist_of_Map.
`Qed`.

`Lemma` alist_of_Map_of_alist_c : (l:(alist A)) (alist_sorted_2 A l) ->
(alist_of_Map A (Map_of_alist A l))=l.
`Proof`.
Intros. Apply alist_canonical. Apply alist_of_Map_of_alist.
Apply alist_of_Map_sorts2.
Assumption.
`Qed`.

`Lemma` MapSubset_antisym_c : (m:(Map A)) (m':(Map B))
(mapcanon A m) -> (mapcanon B m') ->
(MapSubset A B m m') -> (MapSubset B A m' m) -> (MapDom A m)=(MapDom B m').
`Proof`.
Intros. Apply (mapcanon_unique unit). (Apply MapDom_canon; Assumption).
(Apply MapDom_canon; Assumption).
(Apply MapSubset_antisym; Assumption).
`Qed`.

`Lemma` FSubset_antisym_c : (s,s':FSet) (mapcanon unit s) -> (mapcanon unit s') ->
(MapSubset ? ? s s') -> (MapSubset ? ? s' s) -> s=s'.
`Proof`.
Intros. Apply (mapcanon_unique unit); Try Assumption. Apply FSubset_antisym; Assumption.
`Qed`.

`Lemma` MapDisjoint_empty_c : (m:(Map A)) (mapcanon A m) ->
(MapDisjoint A A m m) -> m=(M0 A).
`Proof`.
Intros. Apply mapcanon_unique; Try Assumption; Try Apply M0_canon.
Apply MapDisjoint_empty; Assumption.
`Qed`.

`Lemma` MapDelta_disjoint_c : (m,m':(Map A)) (mapcanon A m) -> (mapcanon A m') ->
(MapDisjoint A A m m') -> (MapDelta A m m')=(MapMerge A m m').
`Proof`.
Intros. Apply mapcanon_unique. (Apply MapDelta_canon; Assumption).
(Apply MapMerge_canon; Assumption). Apply MapDelta_disjoint; Assumption.
`Qed`.

`End` MapC.

`Lemma` FSetDelta_assoc_c : (s,s',s'':FSet)
(mapcanon unit s) -> (mapcanon unit s') -> (mapcanon unit s'') ->
(MapDelta ? (MapDelta ? s s') s'')=(MapDelta ? s (MapDelta ? s' s'')).
`Proof`.
Intros. Apply (mapcanon_unique unit). Apply MapDelta_canon. (Apply MapDelta_canon; Assumption).
Assumption.
Apply MapDelta_canon. Assumption.
(Apply MapDelta_canon; Assumption).
Apply FSetDelta_assoc; Assumption.
`Qed`.

`Lemma` FSet_ext_c : (s,s':FSet) (mapcanon unit s) -> (mapcanon unit s') ->
((a:ad) (in_FSet a s)=(in_FSet a s')) -> s=s'.
`Proof`.
Intros. (Apply (mapcanon_unique unit); Try Assumption). Apply FSet_ext. Assumption.
`Qed`.

`Lemma` FSetUnion_comm_c : (s,s':FSet) (mapcanon unit s) -> (mapcanon unit s') ->
(FSetUnion s s')=(FSetUnion s' s).
`Proof`.
Intros.
Apply (mapcanon_unique unit); Try (Unfold FSetUnion; Apply MapMerge_canon; Assumption).
Apply FSetUnion_comm.
`Qed`.

`Lemma` FSetUnion_assoc_c : (s,s',s'':FSet)
(mapcanon unit s) -> (mapcanon unit s') -> (mapcanon unit s'') ->
(FSetUnion (FSetUnion s s') s'')=(FSetUnion s (FSetUnion s' s'')).
`Proof`.
Exact (MapMerge_assoc_c unit).
`Qed`.

`Lemma` FSetUnion_M0_s_c : (s:FSet) (FSetUnion (M0 unit) s)=s.
`Proof`.
Exact (MapMerge_empty_m_c unit).
`Qed`.

`Lemma` FSetUnion_s_M0_c : (s:FSet) (FSetUnion s (M0 unit))=s.
`Proof`.
Exact (MapMerge_m_empty_1 unit).
`Qed`.

`Lemma` FSetUnion_idempotent : (s:FSet) (mapcanon unit s) -> (FSetUnion s s)=s.
`Proof`.
Exact (MapMerge_idempotent_c unit).
`Qed`.

`Lemma` FSetInter_comm_c : (s,s':FSet) (mapcanon unit s) -> (mapcanon unit s') ->
(FSetInter s s')=(FSetInter s' s).
`Proof`.
Intros.
Apply (mapcanon_unique unit); Try (Unfold FSetInter; Apply MapDomRestrTo_canon; Assumption).
Apply FSetInter_comm.
`Qed`.

`Lemma` FSetInter_assoc_c : (s,s',s'':FSet)
(mapcanon unit s) ->
(FSetInter (FSetInter s s') s'')=(FSetInter s (FSetInter s' s'')).
`Proof`.
Exact (MapDomRestrTo_assoc_c unit unit unit).
`Qed`.

`Lemma` FSetInter_M0_s_c : (s:FSet) (FSetInter (M0 unit) s)=(M0 unit).
`Proof`.
Trivial.
`Qed`.

`Lemma` FSetInter_s_M0_c : (s:FSet) (FSetInter s (M0 unit))=(M0 unit).
`Proof`.
Exact (MapDomRestrTo_m_empty_1 unit unit).
`Qed`.

`Lemma` FSetInter_idempotent : (s:FSet) (mapcanon unit s) -> (FSetInter s s)=s.
`Proof`.
Exact (MapDomRestrTo_idempotent_c unit).
`Qed`.

`Lemma` FSetUnion_Inter_l_c : (s,s',s'':FSet) (mapcanon unit s) -> (mapcanon unit s'') ->
(FSetUnion (FSetInter s s') s'')=(FSetInter (FSetUnion s s'') (FSetUnion s' s'')).
`Proof`.
Intros. Apply (mapcanon_unique unit). Unfold FSetUnion. (Apply MapMerge_canon; Try Assumption).
Unfold FSetInter. (Apply MapDomRestrTo_canon; Assumption).
Unfold FSetInter; Unfold FSetUnion; Apply MapDomRestrTo_canon; Apply MapMerge_canon; Assumption.
Apply FSetUnion_Inter_l.
`Qed`.

`Lemma` FSetUnion_Inter_r : (s,s',s'':FSet) (mapcanon unit s) -> (mapcanon unit s') ->
(FSetUnion s (FSetInter s' s''))=(FSetInter (FSetUnion s s') (FSetUnion s s'')).
`Proof`.
Intros. Apply (mapcanon_unique unit). Unfold FSetUnion. (Apply MapMerge_canon; Try Assumption).
Unfold FSetInter. (Apply MapDomRestrTo_canon; Assumption).
Unfold FSetInter; Unfold FSetUnion; Apply MapDomRestrTo_canon; Apply MapMerge_canon; Assumption.
Apply FSetUnion_Inter_r.
`Qed`.

`Lemma` FSetInter_Union_l_c : (s,s',s'':FSet) (mapcanon unit s) -> (mapcanon unit s') ->
(FSetInter (FSetUnion s s') s'')=(FSetUnion (FSetInter s s'') (FSetInter s' s'')).
`Proof`.
Intros. Apply (mapcanon_unique unit). Unfold FSetInter.
Apply MapDomRestrTo_canon; Try Assumption. Unfold FSetUnion.
Apply MapMerge_canon; Assumption.
Unfold FSetUnion; Unfold FSetInter; Apply MapMerge_canon; Apply MapDomRestrTo_canon;
Assumption.
Apply FSetInter_Union_l.
`Qed`.

`Lemma` FSetInter_Union_r : (s,s',s'':FSet) (mapcanon unit s) -> (mapcanon unit s') ->
(FSetInter s (FSetUnion s' s''))=(FSetUnion (FSetInter s s') (FSetInter s s'')).
`Proof`.
Intros. Apply (mapcanon_unique unit). Unfold FSetInter.
Apply MapDomRestrTo_canon; Try Assumption.
Unfold FSetUnion. Apply MapMerge_canon; Unfold FSetInter; Apply MapDomRestrTo_canon; Assumption.
Apply FSetInter_Union_r.
`Qed`.

`

Index