diff --git a/theories/fin_map_dom.v b/theories/fin_map_dom.v index 3c430a6b59ff9b2c49a97c631e3a1991a6b89d25..5ba8bb788b4ee5ddcd9ba095e5759a7766d3358e 100644 --- a/theories/fin_map_dom.v +++ b/theories/fin_map_dom.v @@ -170,6 +170,56 @@ Section leibniz. Lemma dom_fmap_L {A B} (f : A → B) (m : M A) : dom D (f <$> m) = dom D m. Proof. unfold_leibniz; apply dom_fmap. Qed. End leibniz. + +(** * Set solver instances *) +Global Instance set_unfold_dom_empty {A} i : SetUnfoldElemOf i (dom D (∅:M A)) False. +Proof. constructor. by rewrite dom_empty, elem_of_empty. Qed. +Global Instance set_unfold_dom_alter {A} f i j (m : M A) Q : + SetUnfoldElemOf i (dom D m) Q → + SetUnfoldElemOf i (dom D (alter f j m)) Q. +Proof. constructor. by rewrite dom_alter, (set_unfold_elem_of _ (dom _ _) _). Qed. +Global Instance set_unfold_dom_insert {A} i j x (m : M A) Q : + SetUnfoldElemOf i (dom D m) Q → + SetUnfoldElemOf i (dom D (<[j:=x]> m)) (i = j ∨ Q). +Proof. + constructor. by rewrite dom_insert, elem_of_union, + (set_unfold_elem_of _ (dom _ _) _), elem_of_singleton. +Qed. +Global Instance set_unfold_dom_delete {A} i j (m : M A) Q : + SetUnfoldElemOf i (dom D m) Q → + SetUnfoldElemOf i (dom D (delete j m)) (Q ∧ i ≠j). +Proof. + constructor. by rewrite dom_delete, elem_of_difference, + (set_unfold_elem_of _ (dom _ _) _), elem_of_singleton. +Qed. +Global Instance set_unfold_dom_singleton {A} i j : + SetUnfoldElemOf i (dom D ({[ j := x ]} : M A)) (i = j). +Proof. constructor. by rewrite dom_singleton, elem_of_singleton. Qed. +Global Instance set_unfold_dom_union {A} i (m1 m2 : M A) Q1 Q2 : + SetUnfoldElemOf i (dom D m1) Q1 → SetUnfoldElemOf i (dom D m2) Q2 → + SetUnfoldElemOf i (dom D (m1 ∪ m2)) (Q1 ∨ Q2). +Proof. + constructor. by rewrite dom_union, elem_of_union, + !(set_unfold_elem_of _ (dom _ _) _). +Qed. +Global Instance set_unfold_dom_intersection {A} i (m1 m2 : M A) Q1 Q2 : + SetUnfoldElemOf i (dom D m1) Q1 → SetUnfoldElemOf i (dom D m2) Q2 → + SetUnfoldElemOf i (dom D (m1 ∩ m2)) (Q1 ∧ Q2). +Proof. + constructor. by rewrite dom_intersection, elem_of_intersection, + !(set_unfold_elem_of _ (dom _ _) _). +Qed. +Global Instance set_unfold_dom_difference {A} i (m1 m2 : M A) Q1 Q2 : + SetUnfoldElemOf i (dom D m1) Q1 → SetUnfoldElemOf i (dom D m2) Q2 → + SetUnfoldElemOf i (dom D (m1 ∖ m2)) (Q1 ∧ ¬Q2). +Proof. + constructor. by rewrite dom_difference, elem_of_difference, + !(set_unfold_elem_of _ (dom _ _) _). +Qed. +Global Instance set_unfold_dom_fmap {A B} (f : A → B) i (m : M A) Q : + SetUnfoldElemOf i (dom D m) Q → + SetUnfoldElemOf i (dom D (f <$> m)) Q. +Proof. constructor. by rewrite dom_fmap, (set_unfold_elem_of _ (dom _ _) _). Qed. End fin_map_dom. Lemma dom_seq `{FinMapDom nat M D} {A} start (xs : list A) : @@ -182,3 +232,7 @@ Qed. Lemma dom_seq_L `{FinMapDom nat M D, !LeibnizEquiv D} {A} start (xs : list A) : dom D (map_seq (M:=M A) start xs) = set_seq start (length xs). Proof. unfold_leibniz. apply dom_seq. Qed. + +Instance set_unfold_dom_seq `{FinMapDom nat M D} {A} start (xs : list A) : + SetUnfoldElemOf i (dom D (map_seq start (M:=M A) xs)) (start ≤ i < start + length xs). +Proof. constructor. by rewrite dom_seq, elem_of_set_seq. Qed.