Commit 3f784972 by Robbert Krebbers

### Remove file cofe_instances.v: put general stuff in cofe.v and map stuff

`in cofe_maps.v.`
parent 51b04b25
 ... ... @@ -214,7 +214,7 @@ Proof. apply (conv_compl (snd_chain c) n). Qed. Canonical Structure prodC (A B : cofeT) : cofeT := CofeT (A * B). Local Instance prod_map_ne `{Dist A, Dist A', Dist B, Dist B'} n : Instance prod_map_ne `{Dist A, Dist A', Dist B, Dist B'} n : Proper ((dist n ==> dist n) ==> (dist n ==> dist n) ==> dist n ==> dist n) (@prod_map A A' B B'). Proof. by intros f f' Hf g g' Hg ?? [??]; split; [apply Hf|apply Hg]. Qed. ... ... @@ -229,3 +229,67 @@ Instance pair_ne `{Dist A, Dist B} : Instance fst_ne `{Dist A, Dist B} : Proper (dist n ==> dist n) (@fst A B) := _. Instance snd_ne `{Dist A, Dist B} : Proper (dist n ==> dist n) (@snd A B) := _. Typeclasses Opaque prod_dist. (** Discrete cofe *) Section discrete_cofe. Context `{Equiv A, @Equivalence A (≡)}. Instance discrete_dist : Dist A := λ n x y, match n with 0 => True | S n => x ≡ y end. Instance discrete_compl : Compl A := λ c, c 1. Instance discrete_cofe : Cofe A. Proof. split. * intros x y; split; [by intros ? []|intros Hn; apply (Hn 1)]. * intros [|n]; [done|apply _]. * by intros [|n]. * done. * intros c [|n]; [done|apply (chain_cauchy c 1 (S n)); lia]. Qed. Definition discrete_cofeC : cofeT := CofeT A. End discrete_cofe. Arguments discrete_cofeC _ {_ _}. (** Later *) Inductive later (A : Type) : Type := Later { later_car : A }. Arguments Later {_} _. Arguments later_car {_} _. Section later. Instance later_equiv `{Equiv A} : Equiv (later A) := λ x y, later_car x ≡ later_car y. Instance later_dist `{Dist A} : Dist (later A) := λ n x y, match n with 0 => True | S n => later_car x ={n}= later_car y end. Program Definition later_chain `{Dist A} (c : chain (later A)) : chain A := {| chain_car n := later_car (c (S n)) |}. Next Obligation. intros A ? c n i ?; apply (chain_cauchy c (S n)); lia. Qed. Instance later_compl `{Compl A} : Compl (later A) := λ c, Later (compl (later_chain c)). Instance later_cofe `{Cofe A} : Cofe (later A). Proof. split. * intros x y; unfold equiv, later_equiv; rewrite !equiv_dist. split. intros Hxy [|n]; [done|apply Hxy]. intros Hxy n; apply (Hxy (S n)). * intros [|n]; [by split|split]; unfold dist, later_dist. + by intros [x]. + by intros [x] [y]. + by intros [x] [y] [z] ??; transitivity y. * intros [|n] [x] [y] ?; [done|]; unfold dist, later_dist; by apply dist_S. * done. * intros c [|n]; [done|by apply (conv_compl (later_chain c) n)]. Qed. Canonical Structure laterC (A : cofeT) : cofeT := CofeT (later A). Instance later_fmap : FMap later := λ A B f x, Later (f (later_car x)). Instance later_fmap_ne `{Cofe A, Cofe B} (f : A → B) : (∀ n, Proper (dist n ==> dist n) f) → ∀ n, Proper (dist n ==> dist n) (fmap f : later A → later B). Proof. intros Hf [|n] [x] [y] ?; do 2 red; simpl. done. by apply Hf. Qed. Lemma later_fmap_id {A} (x : later A) : id <\$> x = x. Proof. by destruct x. Qed. Lemma later_fmap_compose {A B C} (f : A → B) (g : B → C) (x : later A) : g ∘ f <\$> x = g <\$> f <\$> x. Proof. by destruct x. Qed. Definition laterC_map {A B} (f : A -n> B) : laterC A -n> laterC B := CofeMor (fmap f : laterC A → laterC B). Instance laterC_contractive (A B : cofeT) : Contractive (@laterC_map A B). Proof. intros n f g Hf n'; apply Hf. Qed. End later.
 Require Export iris.cofe. Require Import prelude.fin_maps prelude.pmap prelude.nmap prelude.zmap prelude.stringmap. (** Discrete cofe *) Section discrete_cofe. Context `{Equiv A, @Equivalence A (≡)}. Instance discrete_dist : Dist A := λ n x y, match n with 0 => True | S n => x ≡ y end. Instance discrete_compl `{Equiv A} : Compl A := λ c, c 1. Instance discrete_cofe : Cofe A. Proof. split. * intros x y; split; [by intros ? []|intros Hn; apply (Hn 1)]. * intros [|n]; [done|apply _]. * by intros [|n]. * done. * intros c [|n]; [done|apply (chain_cauchy c 1 (S n)); lia]. Qed. Definition discreteC : cofeT := CofeT A. End discrete_cofe. Arguments discreteC _ {_ _}. (** Later *) Inductive later (A : Type) : Type := Later { later_car : A }. Arguments Later {_} _. Arguments later_car {_} _. Section later. Instance later_equiv `{Equiv A} : Equiv (later A) := λ x y, later_car x ≡ later_car y. Instance later_dist `{Dist A} : Dist (later A) := λ n x y, match n with 0 => True | S n => later_car x ={n}= later_car y end. Program Definition later_chain `{Dist A} (c : chain (later A)) : chain A := {| chain_car n := later_car (c (S n)) |}. Next Obligation. intros A ? c n i ?; apply (chain_cauchy c (S n)); lia. Qed. Instance later_compl `{Compl A} : Compl (later A) := λ c, Later (compl (later_chain c)). Instance later_cofe `{Cofe A} : Cofe (later A). Proof. split. * intros x y; unfold equiv, later_equiv; rewrite !equiv_dist. split. intros Hxy [|n]; [done|apply Hxy]. intros Hxy n; apply (Hxy (S n)). * intros [|n]; [by split|split]; unfold dist, later_dist. + by intros [x]. + by intros [x] [y]. + by intros [x] [y] [z] ??; transitivity y. * intros [|n] [x] [y] ?; [done|]; unfold dist, later_dist; by apply dist_S. * done. * intros c [|n]; [done|by apply (conv_compl (later_chain c) n)]. Qed. Canonical Structure laterC (A : cofeT) : cofeT := CofeT (later A). Instance later_fmap : FMap later := λ A B f x, Later (f (later_car x)). Instance later_fmap_ne `{Cofe A, Cofe B} (f : A → B) : (∀ n, Proper (dist n ==> dist n) f) → ∀ n, Proper (dist n ==> dist n) (fmap f : later A → later B). Proof. intros Hf [|n] [x] [y] ?; do 2 red; simpl. done. by apply Hf. Qed. Lemma later_fmap_id {A} (x : later A) : id <\$> x = x. Proof. by destruct x. Qed. Lemma later_fmap_compose {A B C} (f : A → B) (g : B → C) (x : later A) : g ∘ f <\$> x = g <\$> f <\$> x. Proof. by destruct x. Qed. Definition laterC_map {A B} (f : A -n> B) : laterC A -n> laterC B := CofeMor (fmap f : laterC A → laterC B). Instance laterC_contractive (A B : cofeT) : Contractive (@laterC_map A B). Proof. intros n f g Hf n'; apply Hf. Qed. End later. (* Option *) Instance option_dist `{Dist A} : Dist (option A) := λ n o1 o2, match n with 0 => True | S n => option_Forall2 (dist n) o1 o2 end. Program Definition option_chain `{Dist A} (c : chain (option A)) (x : A) (H : c 1 = Some x) : chain A := {| chain_car n := from_option x (c (S n)) |}. Next Obligation. intros A ? c x ? n i ?. feed inversion (chain_cauchy c 1 (S i)); auto with lia congruence. feed inversion (chain_cauchy c (S n) (S i)); simpl; auto with lia congruence. Qed. Instance option_compl `{Compl A} : Compl (option A) := λ c, match Some_dec (c 1) with | inleft (exist x H) => Some (compl (option_chain c x H)) | inright _ => None end. Instance option_cofe `{Cofe A} : Cofe (option A). Proof. split. * intros mx my; split. { by destruct 1; intros [|n]; constructor; apply equiv_dist. } intros Hxy; feed inversion (Hxy 1); constructor; apply equiv_dist. intros n. feed inversion (Hxy (S n)); congruence. * intros [|n]; [by split|split]. + by intros [x|]; constructor. + by destruct 1; constructor. + by intros [x|] [y|] [z|]; do 2 inversion 1; constructor; transitivity y. * by destruct n; [|destruct 1; constructor; apply dist_S]. * done. * intros c [|n]; unfold compl, option_compl; [constructor|]. destruct (Some_dec (c 1)) as [[x Hx]|]. + assert (is_Some (c (S n))) as [y Hy]. { feed inversion (chain_cauchy c 1 (S n)); try congruence; eauto with lia. } rewrite Hy; constructor. by rewrite (conv_compl (option_chain c x Hx) n); simpl; rewrite Hy. + feed inversion (chain_cauchy c 1 (S n)); auto with lia congruence. by constructor. Qed. Instance Some_ne `{Cofe A} : Proper (dist n ==> dist n) Some. Proof. by intros [|n];[done|constructor; apply dist_S]. Qed. Instance option_fmap_ne `{Cofe A, Cofe B} (f : A → B) : (∀ n, Proper (dist n ==> dist n) f) → ∀ n, Proper (dist n ==> dist n) (fmap f : option A → option B). Proof. intros Hf [|n];[done|destruct 1;constructor;by apply Hf]. Qed. (** Finite maps *) Section map. Context `{FinMap K M}. Instance map_dist `{Dist A} : Dist (M A) := λ n m1 m2, ∀ i, m1 !! i ={n}= m2 !! i. Program Definition map_chain `{Dist A} (c : chain (M A)) (k : K) : chain (option A) := {| chain_car n := c n !! k |}. Next Obligation. by intros A ? c k n i ?; apply (chain_cauchy c). Qed. Instance map_compl `{Compl A} : Compl (M A) := λ c, map_imap (λ i _, compl (map_chain c i)) (c 1). Instance map_cofe `{Cofe A} : Cofe (M A). Proof. split. * intros m1 m2; split. + by intros Hm n k; apply equiv_dist. + intros Hm k; apply equiv_dist; intros n; apply Hm. * intros n; split. + by intros m k. + by intros m1 m2 ? k. + by intros m1 m2 m3 ?? k; transitivity (m2 !! k). * by intros n m1 m2 ? k; apply dist_S. * done. * intros c [|n] k; unfold compl, map_compl; [apply dist_0|]. rewrite lookup_imap. assert ((∃ x y, c 1 !! k = Some x ∧ c (S n) !! k = Some y) ∨ c 1 !! k = None ∧ c (S n) !! k = None) as [(x&y&Hx&Hy)|[-> ->]] by (feed inversion (λ H, chain_cauchy c 1 (S n) H k); eauto with lia congruence); [|done]. by rewrite Hx; simpl; rewrite conv_compl; simpl; rewrite Hy. Qed. Instance lookup_ne `{Cofe A} n : Proper ((=) ==> dist n ==> dist n) (lookup : K → M A → option A). Proof. by intros k1 k2 -> m1 m2 Hm; apply Hm. Qed. Instance map_fmap_ne `{Cofe A, Cofe B} (f : A → B) : (∀ n, Proper (dist n ==> dist n) f) → ∀ n, Proper (dist n ==> dist n) (fmap f : M A → M B). Proof. by intros ? n m1 m2 Hm k; rewrite !lookup_fmap, Hm. Qed. Definition mapC (A : cofeT) : cofeT := CofeT (M A). Definition mapC_map {A B} (f: A -n> B) : mapC A -n> mapC B := CofeMor (fmap f : mapC A → mapC B). Global Instance mapC_map_ne (A B : cofeT) : Proper (dist n ==> dist n) (@mapC_map A B). Proof. intros [|n] f g Hf m k; simpl; rewrite !lookup_fmap; [apply dist_0|]. destruct (_ !! k) eqn:?; simpl; constructor; apply dist_S, Hf. Qed. End map. Arguments mapC {_} _ {_ _ _ _ _ _ _ _ _} _. Canonical Structure PmapC := mapC Pmap. Canonical Structure NmapC := mapC Nmap. Canonical Structure ZmapC := mapC Zmap. Canonical Structure stringmapC := mapC stringmap.
iris/cofe_maps.v 0 → 100644
 Require Export iris.cofe prelude.fin_maps. Require Import prelude.pmap prelude.nmap prelude.zmap. Require Import prelude.stringmap prelude.natmap. Local Obligation Tactic := idtac. (** option *) Inductive option_dist `{Dist A} : Dist (option A) := | option_0_dist (x y : option A) : x ={0}= y | Some_dist n x y : x ={n}= y → Some x ={n}= Some y | None_dist n : None ={n}= None. Existing Instance option_dist. Program Definition option_chain `{Cofe A} (c : chain (option A)) (x : A) (H : c 1 = Some x) : chain A := {| chain_car n := from_option x (c n) |}. Next Obligation. intros A ???? c x ? n i ?; simpl; destruct (decide (i = 0)) as [->|]. { by replace n with 0 by lia. } feed inversion (chain_cauchy c 1 i); auto with lia congruence. feed inversion (chain_cauchy c n i); simpl; auto with lia congruence. Qed. Instance option_compl `{Cofe A} : Compl (option A) := λ c, match Some_dec (c 1) with | inleft (exist x H) => Some (compl (option_chain c x H)) | inright _ => None end. Instance option_cofe `{Cofe A} : Cofe (option A). Proof. split. * intros mx my; split; [by destruct 1; constructor; apply equiv_dist|]. intros Hxy; feed inversion (Hxy 1); subst; constructor; apply equiv_dist. by intros n; feed inversion (Hxy n). * intros n; split. + by intros [x|]; constructor. + by destruct 1; constructor. + destruct 1; inversion_clear 1; constructor; etransitivity; eauto. * by inversion_clear 1; constructor; apply dist_S. * constructor. * intros c n; unfold compl, option_compl. destruct (decide (n = 0)) as [->|]; [constructor|]. destruct (Some_dec (c 1)) as [[x Hx]|]. { assert (is_Some (c n)) as [y Hy]. { feed inversion (chain_cauchy c 1 n); try congruence; eauto with lia. } rewrite Hy; constructor. by rewrite (conv_compl (option_chain c x Hx) n); simpl; rewrite Hy. } feed inversion (chain_cauchy c 1 n); auto with lia congruence; constructor. Qed. Instance Some_ne `{Dist A} : Proper (dist n ==> dist n) Some. Proof. by constructor. Qed. Instance option_fmap_ne `{Dist A, Dist B} (f : A → B) n: Proper (dist n ==> dist n) f → Proper (dist n==>dist n) (fmap (M:=option) f). Proof. by intros Hf; destruct 1; constructor; apply Hf. Qed. (** Finite maps *) Section map. Context `{FinMap K M}. Global Instance map_dist `{Dist A} : Dist (M A) := λ n m1 m2, ∀ i, m1 !! i ={n}= m2 !! i. Program Definition map_chain `{Dist A} (c : chain (M A)) (k : K) : chain (option A) := {| chain_car n := c n !! k |}. Next Obligation. by intros A ? c k n i ?; apply (chain_cauchy c). Qed. Global Instance map_compl `{Cofe A} : Compl (M A) := λ c, map_imap (λ i _, compl (map_chain c i)) (c 1). Global Instance map_cofe `{Cofe A} : Cofe (M A). Proof. split. * intros m1 m2; split. + by intros Hm n k; apply equiv_dist. + intros Hm k; apply equiv_dist; intros n; apply Hm. * intros n; split. + by intros m k. + by intros m1 m2 ? k. + by intros m1 m2 m3 ?? k; transitivity (m2 !! k). * by intros n m1 m2 ? k; apply dist_S. * by intros m1 m2 k. * intros c n k; unfold compl, map_compl; rewrite lookup_imap. destruct (decide (n = 0)) as [->|]; [constructor|]. feed inversion (λ H, chain_cauchy c 1 n H k); simpl; auto with lia. by rewrite conv_compl; simpl; apply reflexive_eq. Qed. Global Instance lookup_ne `{Dist A} n k : Proper (dist n ==> dist n) (lookup k : M A → option A). Proof. by intros m1 m2. Qed. Instance map_fmap_ne `{Dist A, Dist B} (f : A → B) n : Proper (dist n ==> dist n) f → Proper (dist n ==> dist n) (fmap (M:=M) f). Proof. by intros ? m m' Hm k; rewrite !lookup_fmap; apply option_fmap_ne. Qed. Definition mapC (A : cofeT) : cofeT := CofeT (M A). Definition mapC_map {A B} (f: A -n> B) : mapC A -n> mapC B := CofeMor (fmap f : mapC A → mapC B). Global Instance mapC_map_ne (A B : cofeT) : Proper (dist n ==> dist n) (@mapC_map A B). Proof. intros n f g Hf m k; simpl; rewrite !lookup_fmap. destruct (_ !! k) eqn:?; simpl; constructor; apply Hf. Qed. End map. Arguments mapC {_} _ {_ _ _ _ _ _ _ _ _} _. Canonical Structure natmapC := mapC natmap. Definition natmapC_map {A B} (f : A -n> B) : natmapC A -n> natmapC B := mapC_map f. Canonical Structure PmapC := mapC Pmap. Definition PmapC_map {A B} (f : A -n> B) : PmapC A -n> PmapC B := mapC_map f. Canonical Structure NmapC := mapC Nmap. Definition NmapC_map {A B} (f : A -n> B) : NmapC A -n> NmapC B := mapC_map f. Canonical Structure ZmapC := mapC Zmap. Definition ZmapC_map {A B} (f : A -n> B) : ZmapC A -n> ZmapC B := mapC_map f. Canonical Structure stringmapC := mapC stringmap. Definition stringmapC_map {A B} (f : A -n> B) : stringmapC A -n> stringmapC B := mapC_map f.
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