Commit 24623c52 by Robbert Krebbers

### Tweak some proof using tweaks for setoid stuff.

parent 47e0f1c4
 ... ... @@ -119,13 +119,13 @@ Context `{FinMap K M}. (** ** Setoids *) Section setoid. Context `{Equiv A}. Lemma map_equiv_lookup_l (m1 m2 : M A) i x : m1 ≡ m2 → m1 !! i = Some x → ∃ y, m2 !! i = Some y ∧ x ≡ y. Proof. generalize (equiv_Some_inv_l (m1 !! i) (m2 !! i) x); naive_solver. Qed. Context `{!Equivalence ((≡) : relation A)}. Global Instance map_equivalence : Equivalence ((≡) : relation (M A)). Global Instance map_equivalence : Equivalence ((≡) : relation A) → Equivalence ((≡) : relation (M A)). Proof. split. - by intros m i. ... ... @@ -147,7 +147,10 @@ Section setoid. Proof. by intros ???; apply partial_alter_proper; [constructor|]. Qed. Global Instance singleton_proper k : Proper ((≡) ==> (≡)) (singletonM k : A → M A). Proof. by intros ???; apply insert_proper. Qed. Proof. intros ???; apply insert_proper; [done|]. intros ?. rewrite lookup_empty; constructor. Qed. Global Instance delete_proper (i : K) : Proper ((≡) ==> (≡)) (delete (M:=M A) i). Proof. by apply partial_alter_proper; [constructor|]. Qed. ... ... @@ -170,14 +173,12 @@ Section setoid. by do 2 destruct 1; first [apply Hf | constructor]. Qed. Global Instance map_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (M A). Proof. intros m1 m2 Hm; apply map_eq; intros i. by unfold_leibniz; apply lookup_proper. Qed. Proof. intros m1 m2 Hm; apply map_eq; intros i. apply leibniz_equiv, Hm. Qed. Lemma map_equiv_empty (m : M A) : m ≡ ∅ ↔ m = ∅. Proof. split; [intros Hm; apply map_eq; intros i|by intros ->]. by rewrite lookup_empty, <-equiv_None, Hm, lookup_empty. split; [intros Hm; apply map_eq; intros i|intros ->]. - generalize (Hm i). by rewrite lookup_empty, equiv_None. - intros ?. rewrite lookup_empty; constructor. Qed. Global Instance map_fmap_proper `{Equiv B} (f : A → B) : Proper ((≡) ==> (≡)) f → Proper ((≡) ==> (≡)) (fmap (M:=M) f). ... ...
 ... ... @@ -2753,9 +2753,8 @@ Section setoid. by setoid_rewrite equiv_option_Forall2. Qed. Context {Hequiv: Equivalence ((≡) : relation A)}. Global Instance list_equivalence : Equivalence ((≡) : relation (list A)). Global Instance list_equivalence : Equivalence ((≡) : relation A) → Equivalence ((≡) : relation (list A)). Proof. split. - intros l. by apply equiv_Forall2. ... ... @@ -2766,48 +2765,53 @@ Section setoid. Proof. induction 1; f_equal; fold_leibniz; auto. Qed. Global Instance cons_proper : Proper ((≡) ==> (≡) ==> (≡)) (@cons A). Proof using -(Hequiv). by constructor. Qed. Proof. by constructor. Qed. Global Instance app_proper : Proper ((≡) ==> (≡) ==> (≡)) (@app A). Proof using -(Hequiv). induction 1; intros ???; simpl; try constructor; auto. Qed. Proof. induction 1; intros ???; simpl; try constructor; auto. Qed. Global Instance length_proper : Proper ((≡) ==> (=)) (@length A). Proof using -(Hequiv). induction 1; f_equal/=; auto. Qed. Proof. induction 1; f_equal/=; auto. Qed. Global Instance tail_proper : Proper ((≡) ==> (≡)) (@tail A). Proof. by destruct 1. Qed. Proof. destruct 1; try constructor; auto. Qed. Global Instance take_proper n : Proper ((≡) ==> (≡)) (@take A n). Proof using -(Hequiv). induction n; destruct 1; constructor; auto. Qed. Proof. induction n; destruct 1; constructor; auto. Qed. Global Instance drop_proper n : Proper ((≡) ==> (≡)) (@drop A n). Proof using -(Hequiv). induction n; destruct 1; simpl; try constructor; auto. Qed. Proof. induction n; destruct 1; simpl; try constructor; auto. Qed. Global Instance list_lookup_proper i : Proper ((≡) ==> (≡)) (lookup (M:=list A) i). Proof. induction i; destruct 1; simpl; f_equiv; auto. Qed. Proof. induction i; destruct 1; simpl; try constructor; auto. Qed. Global Instance list_alter_proper f i : Proper ((≡) ==> (≡)) f → Proper ((≡) ==> (≡)) (alter (M:=list A) f i). Proof using -(Hequiv). intros. induction i; destruct 1; constructor; eauto. Qed. Proof. intros. induction i; destruct 1; constructor; eauto. Qed. Global Instance list_insert_proper i : Proper ((≡) ==> (≡) ==> (≡)) (insert (M:=list A) i). Proof using -(Hequiv). intros ???; induction i; destruct 1; constructor; eauto. Qed. Proof. intros ???; induction i; destruct 1; constructor; eauto. Qed. Global Instance list_inserts_proper i : Proper ((≡) ==> (≡) ==> (≡)) (@list_inserts A i). Proof using -(Hequiv). Proof. intros k1 k2 Hk; revert i. induction Hk; intros ????; simpl; try f_equiv; naive_solver. Qed. Global Instance list_delete_proper i : Proper ((≡) ==> (≡)) (delete (M:=list A) i). Proof using -(Hequiv). induction i; destruct 1; try constructor; eauto. Qed. Proof. induction i; destruct 1; try constructor; eauto. Qed. Global Instance option_list_proper : Proper ((≡) ==> (≡)) (@option_list A). Proof. destruct 1; by constructor. Qed. Proof. destruct 1; repeat constructor; auto. Qed. Global Instance list_filter_proper P `{∀ x, Decision (P x)} : Proper ((≡) ==> iff) P → Proper ((≡) ==> (≡)) (filter (B:=list A) P). Proof using -(Hequiv). intros ???. rewrite !equiv_Forall2. by apply Forall2_filter. Qed. Proof. intros ???. rewrite !equiv_Forall2. by apply Forall2_filter. Qed. Global Instance replicate_proper n : Proper ((≡) ==> (≡)) (@replicate A n). Proof using -(Hequiv). induction n; constructor; auto. Qed. Proof. induction n; constructor; auto. Qed. Global Instance reverse_proper : Proper ((≡) ==> (≡)) (@reverse A). Proof. induction 1; rewrite ?reverse_cons; repeat (done || f_equiv). Qed. Proof. induction 1; rewrite ?reverse_cons; simpl; [constructor|]. apply app_proper; repeat constructor; auto. Qed. Global Instance last_proper : Proper ((≡) ==> (≡)) (@last A). Proof. induction 1 as [|????? []]; simpl; repeat (done || f_equiv). Qed. Proof. induction 1 as [|????? []]; simpl; repeat constructor; auto. Qed. Global Instance resize_proper n : Proper ((≡) ==> (≡) ==> (≡)) (@resize A n). Proof. induction n; destruct 2; simpl; repeat (auto || f_equiv). Qed. Proof. induction n; destruct 2; simpl; repeat (constructor || f_equiv); auto. Qed. End setoid. (** * Properties of the monadic operations *) ... ...
 ... ... @@ -115,36 +115,38 @@ End Forall2. Instance option_equiv `{Equiv A} : Equiv (option A) := option_Forall2 (≡). Section setoids. Context `{Equiv A} {Hequiv: Equivalence ((≡) : relation A)}. Context `{Equiv A}. Implicit Types mx my : option A. Lemma equiv_option_Forall2 mx my : mx ≡ my ↔ option_Forall2 (≡) mx my. Proof using -(Hequiv). done. Qed. Proof. done. Qed. Global Instance option_equivalence : Equivalence ((≡) : relation (option A)). Global Instance option_equivalence : Equivalence ((≡) : relation A) → Equivalence ((≡) : relation (option A)). Proof. apply _. Qed. Global Instance Some_proper : Proper ((≡) ==> (≡)) (@Some A). Proof using -(Hequiv). by constructor. Qed. Proof. by constructor. Qed. Global Instance Some_equiv_inj : Inj (≡) (≡) (@Some A). Proof using -(Hequiv). by inversion_clear 1. Qed. Proof. by inversion_clear 1. Qed. Global Instance option_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (option A). Proof. intros x y; destruct 1; fold_leibniz; congruence. Qed. Proof. intros x y; destruct 1; f_equal; by apply leibniz_equiv. Qed. Lemma equiv_None mx : mx ≡ None ↔ mx = None. Proof. split; [by inversion_clear 1|by intros ->]. Qed. Proof. split; [by inversion_clear 1|intros ->; constructor]. Qed. Lemma equiv_Some_inv_l mx my x : mx ≡ my → mx = Some x → ∃ y, my = Some y ∧ x ≡ y. Proof using -(Hequiv). destruct 1; naive_solver. Qed. Proof. destruct 1; naive_solver. Qed. Lemma equiv_Some_inv_r mx my y : mx ≡ my → my = Some y → ∃ x, mx = Some x ∧ x ≡ y. Proof using -(Hequiv). destruct 1; naive_solver. Qed. Proof. destruct 1; naive_solver. Qed. Lemma equiv_Some_inv_l' my x : Some x ≡ my → ∃ x', Some x' = my ∧ x ≡ x'. Proof using -(Hequiv). intros ?%(equiv_Some_inv_l _ _ x); naive_solver. Qed. Lemma equiv_Some_inv_r' mx y : mx ≡ Some y → ∃ y', mx = Some y' ∧ y ≡ y'. Proof. intros ?%(equiv_Some_inv_l _ _ x); naive_solver. Qed. Lemma equiv_Some_inv_r' `{!Equivalence ((≡) : relation A)} mx y : mx ≡ Some y → ∃ y', mx = Some y' ∧ y ≡ y'. Proof. intros ?%(equiv_Some_inv_r _ _ y); naive_solver. Qed. Global Instance is_Some_proper : Proper ((≡) ==> iff) (@is_Some A). Proof using -(Hequiv). inversion_clear 1; split; eauto. Qed. Proof. inversion_clear 1; split; eauto. Qed. Global Instance from_option_proper {B} (R : relation B) (f : A → B) : Proper ((≡) ==> R) f → Proper (R ==> (≡) ==> R) (from_option f). Proof. destruct 3; simpl; auto. Qed. ... ...
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