Definition eqmap (m m':Map A) := eqm A (MapGet A m) (MapGet A m').

(* Specifications: *)
Lemma eqmap_sym : forall m m':Map A, eqmap m m' -> eqmap m' m.
Lemma eqmap_refl : forall m:Map A, eqmap m m.
Lemma eqmap_trans : forall m m' m'':Map A, eqmap m m' -> eqmap m' m'' -> eqmap m m''.
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)).
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).
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).
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).
Lemma MapMerge_empty_m_1 : forall m:Map A, MapMerge A (M0 A) m = m.
Lemma MapMerge_empty_m : forall m:Map A, eqmap (MapMerge A (M0 A) m) m.
Lemma MapMerge_m_empty_1 : forall m:Map A, MapMerge A m (M0 A) = m.
Lemma MapMerge_m_empty : forall m:Map A, eqmap (MapMerge A m (M0 A)) m.
Lemma MapMerge_empty_l : forall m m':Map A, eqmap (MapMerge A m m') (M0 A) -> eqmap m (M0 A).
Lemma MapMerge_empty_r : forall m m':Map A, eqmap (MapMerge A m m') (M0 A) -> eqmap m' (M0 A).
Lemma MapMerge_assoc : forall m m' m'':Map A, eqmap (MapMerge A (MapMerge A m m') m'') (MapMerge A m (MapMerge A m' m'')).
Lemma MapMerge_idempotent : forall m:Map A, eqmap (MapMerge A m m) m.
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).
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).
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).
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'')).
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)).
Lemma MapRemove_ext : forall m m':Map A, eqmap m m' -> forall a:ad, eqmap (MapRemove A m a) (MapRemove A m' a).
Lemma MapDomRestrTo_empty_m_1 : forall m:Map B, MapDomRestrTo A B (M0 A) m = M0 A.
Lemma MapDomRestrTo_empty_m : forall m:Map B, eqmap (MapDomRestrTo A B (M0 A) m) (M0 A).
Lemma MapDomRestrTo_m_empty_1 : forall m:Map A, MapDomRestrTo A B m (M0 B) = M0 A.
Lemma MapDomRestrTo_m_empty : forall m:Map A, eqmap (MapDomRestrTo A B m (M0 B)) (M0 A).
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'')).
Lemma MapDomRestrTo_idempotent : forall m:Map A, eqmap (MapDomRestrTo A A m m) m.
Lemma MapDomRestrTo_Dom : forall (m:Map A) (m':Map B), eqmap (MapDomRestrTo A B m m') (MapDomRestrTo A unit m (MapDom B m')).
Lemma MapDomRestrBy_empty_m_1 : forall m:Map B, MapDomRestrBy A B (M0 A) m = M0 A.
Lemma MapDomRestrBy_empty_m : forall m:Map B, eqmap (MapDomRestrBy A B (M0 A) m) (M0 A).
Lemma MapDomRestrBy_m_empty_1 : forall m:Map A, MapDomRestrBy A B m (M0 B) = m.
Lemma MapDomRestrBy_m_empty : forall m:Map A, eqmap (MapDomRestrBy A B m (M0 B)) m.
Lemma MapDomRestrBy_Dom : forall (m:Map A) (m':Map B), eqmap (MapDomRestrBy A B m m') (MapDomRestrBy A unit m (MapDom B m')).
Lemma MapDomRestrBy_m_m_1 : forall m:Map A, eqmap (MapDomRestrBy A A m m) (M0 A).
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'')).
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').
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'')).
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').
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')).
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').
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').
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'')).
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'')).
Lemma MapDelta_empty_m_1 : forall m:Map A, MapDelta A (M0 A) m = m.
Lemma MapDelta_empty_m : forall m:Map A, eqmap (MapDelta A (M0 A) m) m.
Lemma MapDelta_m_empty_1 : forall m:Map A, MapDelta A m (M0 A) = m.
Lemma MapDelta_m_empty : forall m:Map A, eqmap (MapDelta A m (M0 A)) m.
Lemma MapDelta_nilpotent : forall m:Map A, eqmap (MapDelta A m m) (M0 A).
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)).
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')).
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)).
Lemma MapDelta_sym : forall m m':Map A, eqmap (MapDelta A m m') (MapDelta A m' m).
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).
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).
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).
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')).
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')).
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).
Lemma MapDomRestrTo_ext : forall (A B:Type) (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).
Lemma MapDomRestrTo_ext_l : forall (A B:Type) (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).
Lemma MapDomRestrTo_ext_r : forall (A B:Type) (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).
Lemma MapDomRestrBy_ext : forall (A B:Type) (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).
Lemma MapDomRestrBy_ext_l : forall (A B:Type) (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).
Lemma MapDomRestrBy_ext_r : forall (A B:Type) (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).
Lemma MapDomRestrBy_m_m : forall (A:Type) (m:Map A), eqmap A (MapDomRestrBy A unit m (MapDom A m)) (M0 A).
Lemma FSetDelta_assoc : forall s s' s'':FSet, eqmap unit (MapDelta _ (MapDelta _ s s') s'') (MapDelta _ s (MapDelta _ s' s'')).
Lemma FSet_ext : forall s s':FSet, (forall a:ad, in_FSet a s = in_FSet a s') -> eqmap unit s s'.
Lemma FSetUnion_comm : forall s s':FSet, eqmap unit (FSetUnion s s') (FSetUnion s' s).
Lemma FSetUnion_assoc : forall s s' s'':FSet, eqmap unit (FSetUnion (FSetUnion s s') s'') (FSetUnion s (FSetUnion s' s'')).
Lemma FSetUnion_M0_s : forall s:FSet, eqmap unit (FSetUnion (M0 unit) s) s.
Lemma FSetUnion_s_M0 : forall s:FSet, eqmap unit (FSetUnion s (M0 unit)) s.
Lemma FSetUnion_idempotent : forall s:FSet, eqmap unit (FSetUnion s s) s.
Lemma FSetInter_comm : forall s s':FSet, eqmap unit (FSetInter s s') (FSetInter s' s).
Lemma FSetInter_assoc : forall s s' s'':FSet, eqmap unit (FSetInter (FSetInter s s') s'') (FSetInter s (FSetInter s' s'')).
Lemma FSetInter_M0_s : forall s:FSet, eqmap unit (FSetInter (M0 unit) s) (M0 unit).
Lemma FSetInter_s_M0 : forall s:FSet, eqmap unit (FSetInter s (M0 unit)) (M0 unit).
Lemma FSetInter_idempotent : forall s:FSet, eqmap unit (FSetInter s s) s.
Lemma FSetUnion_Inter_l : forall s s' s'':FSet, eqmap unit (FSetUnion (FSetInter s s') s'') (FSetInter (FSetUnion s s'') (FSetUnion s' s'')).
Lemma FSetUnion_Inter_r : forall s s' s'':FSet, eqmap unit (FSetUnion s (FSetInter s' s'')) (FSetInter (FSetUnion s s') (FSetUnion s s'')).
Lemma FSetInter_Union_l : forall s s' s'':FSet, eqmap unit (FSetInter (FSetUnion s s') s'') (FSetUnion (FSetInter s s'') (FSetInter s' s'')).
Lemma FSetInter_Union_r : forall s s' s'':FSet, eqmap unit (FSetInter s (FSetUnion s' s'')) (FSetUnion (FSetInter s s') (FSetInter s s'')).