From iris.algebra Require Export cmra. From iris.algebra Require Import upred updates local_updates. Local Arguments pcore _ _ !_ /. Local Arguments cmra_pcore _ !_ /. Local Arguments validN _ _ _ !_ /. Local Arguments valid _ _ !_ /. Local Arguments cmra_validN _ _ !_ /. Local Arguments cmra_valid _ !_ /. Inductive csum (A B : Type) := | Cinl : A → csum A B | Cinr : B → csum A B | CsumBot : csum A B. Arguments Cinl {_ _} _. Arguments Cinr {_ _} _. Arguments CsumBot {_ _}. Section cofe. Context {A B : cofeT}. Implicit Types a : A. Implicit Types b : B. (* Cofe *) Inductive csum_equiv : Equiv (csum A B) := | Cinl_equiv a a' : a ≡ a' → Cinl a ≡ Cinl a' | Cinlr_equiv b b' : b ≡ b' → Cinr b ≡ Cinr b' | CsumBot_equiv : CsumBot ≡ CsumBot. Existing Instance csum_equiv. Inductive csum_dist : Dist (csum A B) := | Cinl_dist n a a' : a ≡{n}≡ a' → Cinl a ≡{n}≡ Cinl a' | Cinlr_dist n b b' : b ≡{n}≡ b' → Cinr b ≡{n}≡ Cinr b' | CsumBot_dist n : CsumBot ≡{n}≡ CsumBot. Existing Instance csum_dist. Global Instance Cinl_ne n : Proper (dist n ==> dist n) (@Cinl A B). Proof. by constructor. Qed. Global Instance Cinl_proper : Proper ((≡) ==> (≡)) (@Cinl A B). Proof. by constructor. Qed. Global Instance Cinl_inj : Inj (≡) (≡) (@Cinl A B). Proof. by inversion_clear 1. Qed. Global Instance Cinl_inj_dist n : Inj (dist n) (dist n) (@Cinl A B). Proof. by inversion_clear 1. Qed. Global Instance Cinr_ne n : Proper (dist n ==> dist n) (@Cinr A B). Proof. by constructor. Qed. Global Instance Cinr_proper : Proper ((≡) ==> (≡)) (@Cinr A B). Proof. by constructor. Qed. Global Instance Cinr_inj : Inj (≡) (≡) (@Cinr A B). Proof. by inversion_clear 1. Qed. Global Instance Cinr_inj_dist n : Inj (dist n) (dist n) (@Cinr A B). Proof. by inversion_clear 1. Qed. Program Definition csum_chain_l (c : chain (csum A B)) (a : A) : chain A := {| chain_car n := match c n return _ with Cinl a' => a' | _ => a end |}. Next Obligation. intros c a n i ?; simpl. by destruct (chain_cauchy c n i). Qed. Program Definition csum_chain_r (c : chain (csum A B)) (b : B) : chain B := {| chain_car n := match c n return _ with Cinr b' => b' | _ => b end |}. Next Obligation. intros c b n i ?; simpl. by destruct (chain_cauchy c n i). Qed. Instance csum_compl : Compl (csum A B) := λ c, match c 0 with | Cinl a => Cinl (compl (csum_chain_l c a)) | Cinr b => Cinr (compl (csum_chain_r c b)) | CsumBot => CsumBot end. Definition csum_cofe_mixin : CofeMixin (csum A B). Proof. split. - intros mx my; split. + by destruct 1; constructor; try apply equiv_dist. + intros Hxy; feed inversion (Hxy 0); subst; constructor; try done; apply equiv_dist=> n; by feed inversion (Hxy n). - intros n; split. + by intros [|a|]; constructor. + by destruct 1; constructor. + destruct 1; inversion_clear 1; constructor; etrans; eauto. - by inversion_clear 1; constructor; apply dist_S. - intros n c; rewrite /compl /csum_compl. feed inversion (chain_cauchy c 0 n); first auto with lia; constructor. + rewrite (conv_compl n (csum_chain_l c a')) /=. destruct (c n); naive_solver. + rewrite (conv_compl n (csum_chain_r c b')) /=. destruct (c n); naive_solver. Qed. Canonical Structure csumC : cofeT := CofeT (csum A B) csum_cofe_mixin. Global Instance csum_discrete : Discrete A → Discrete B → Discrete csumC. Proof. by inversion_clear 3; constructor; apply (timeless _). Qed. Global Instance csum_leibniz : LeibnizEquiv A → LeibnizEquiv B → LeibnizEquiv (csumC A B). Proof. by destruct 3; f_equal; apply leibniz_equiv. Qed. Global Instance Cinl_timeless a : Timeless a → Timeless (Cinl a). Proof. by inversion_clear 2; constructor; apply (timeless _). Qed. Global Instance Cinr_timeless b : Timeless b → Timeless (Cinr b). Proof. by inversion_clear 2; constructor; apply (timeless _). Qed. End cofe. Arguments csumC : clear implicits. (* Functor on COFEs *) Definition csum_map {A A' B B'} (fA : A → A') (fB : B → B') (x : csum A B) : csum A' B' := match x with | Cinl a => Cinl (fA a) | Cinr b => Cinr (fB b) | CsumBot => CsumBot end. Instance: Params (@csum_map) 4. Lemma csum_map_id {A B} (x : csum A B) : csum_map id id x = x. Proof. by destruct x. Qed. Lemma csum_map_compose {A A' A'' B B' B''} (f : A → A') (f' : A' → A'') (g : B → B') (g' : B' → B'') (x : csum A B) : csum_map (f' ∘ f) (g' ∘ g) x = csum_map f' g' (csum_map f g x). Proof. by destruct x. Qed. Lemma csum_map_ext {A A' B B' : cofeT} (f f' : A → A') (g g' : B → B') x : (∀ x, f x ≡ f' x) → (∀ x, g x ≡ g' x) → csum_map f g x ≡ csum_map f' g' x. Proof. by destruct x; constructor. Qed. Instance csum_map_cmra_ne {A A' B B' : cofeT} n : Proper ((dist n ==> dist n) ==> (dist n ==> dist n) ==> dist n ==> dist n) (@csum_map A A' B B'). Proof. intros f f' Hf g g' Hg []; destruct 1; constructor; by apply Hf || apply Hg. Qed. Definition csumC_map {A A' B B'} (f : A -n> A') (g : B -n> B') : csumC A B -n> csumC A' B' := CofeMor (csum_map f g). Instance csumC_map_ne A A' B B' n : Proper (dist n ==> dist n ==> dist n) (@csumC_map A A' B B'). Proof. by intros f f' Hf g g' Hg []; constructor. Qed. Section cmra. Context {A B : cmraT}. Implicit Types a : A. Implicit Types b : B. (* CMRA *) Instance csum_valid : Valid (csum A B) := λ x, match x with | Cinl a => ✓ a | Cinr b => ✓ b | CsumBot => False end. Instance csum_validN : ValidN (csum A B) := λ n x, match x with | Cinl a => ✓{n} a | Cinr b => ✓{n} b | CsumBot => False end. Instance csum_pcore : PCore (csum A B) := λ x, match x with | Cinl a => Cinl <\$> pcore a | Cinr b => Cinr <\$> pcore b | CsumBot => Some CsumBot end. Instance csum_op : Op (csum A B) := λ x y, match x, y with | Cinl a, Cinl a' => Cinl (a ⋅ a') | Cinr b, Cinr b' => Cinr (b ⋅ b') | _, _ => CsumBot end. Lemma Cinl_op a a' : Cinl a ⋅ Cinl a' = Cinl (a ⋅ a'). Proof. done. Qed. Lemma Cinr_op b b' : Cinr b ⋅ Cinr b' = Cinr (b ⋅ b'). Proof. done. Qed. Lemma csum_included x y : x ≼ y ↔ y = CsumBot ∨ (∃ a a', x = Cinl a ∧ y = Cinl a' ∧ a ≼ a') ∨ (∃ b b', x = Cinr b ∧ y = Cinr b' ∧ b ≼ b'). Proof. split. - intros [z Hy]; destruct x as [a|b|], z as [a'|b'|]; inversion_clear Hy; auto. + right; left; eexists _, _; split_and!; eauto. eexists; eauto. + right; right; eexists _, _; split_and!; eauto. eexists; eauto. - intros [->|[(a&a'&->&->&c&?)|(b&b'&->&->&c&?)]]. + destruct x; exists CsumBot; constructor. + exists (Cinl c); by constructor. + exists (Cinr c); by constructor. Qed. Lemma csum_cmra_mixin : CMRAMixin (csum A B). Proof. split. - intros n []; destruct 1; constructor; by cofe_subst. - intros ???? [n a a' Ha|n b b' Hb|n] [=]; subst; eauto. + destruct (pcore a) as [ca|] eqn:?; simplify_option_eq. destruct (cmra_pcore_ne n a a' ca) as (ca'&->&?); auto. exists (Cinl ca'); by repeat constructor. + destruct (pcore b) as [cb|] eqn:?; simplify_option_eq. destruct (cmra_pcore_ne n b b' cb) as (cb'&->&?); auto. exists (Cinr cb'); by repeat constructor. - intros ? [a|b|] [a'|b'|] H; inversion_clear H; cofe_subst; done. - intros [a|b|]; rewrite /= ?cmra_valid_validN; naive_solver eauto using O. - intros n [a|b|]; simpl; auto using cmra_validN_S. - intros [a1|b1|] [a2|b2|] [a3|b3|]; constructor; by rewrite ?assoc. - intros [a1|b1|] [a2|b2|]; constructor; by rewrite 1?comm. - intros [a|b|] ? [=]; subst; auto. + destruct (pcore a) as [ca|] eqn:?; simplify_option_eq. constructor; eauto using cmra_pcore_l. + destruct (pcore b) as [cb|] eqn:?; simplify_option_eq. constructor; eauto using cmra_pcore_l. - intros [a|b|] ? [=]; subst; auto. + destruct (pcore a) as [ca|] eqn:?; simplify_option_eq. feed inversion (cmra_pcore_idemp a ca); repeat constructor; auto. + destruct (pcore b) as [cb|] eqn:?; simplify_option_eq. feed inversion (cmra_pcore_idemp b cb); repeat constructor; auto. - intros x y ? [->|[(a&a'&->&->&?)|(b&b'&->&->&?)]]%csum_included [=]. + exists CsumBot. rewrite csum_included; eauto. + destruct (pcore a) as [ca|] eqn:?; simplify_option_eq. destruct (cmra_pcore_mono a a' ca) as (ca'&->&?); auto. exists (Cinl ca'). rewrite csum_included; eauto 10. + destruct (pcore b) as [cb|] eqn:?; simplify_option_eq. destruct (cmra_pcore_mono b b' cb) as (cb'&->&?); auto. exists (Cinr cb'). rewrite csum_included; eauto 10. - intros n [a1|b1|] [a2|b2|]; simpl; eauto using cmra_validN_op_l; done. - intros n [a|b|] y1 y2 Hx Hx'. + destruct y1 as [a1|b1|], y2 as [a2|b2|]; try (exfalso; by inversion_clear Hx'). apply (inj Cinl) in Hx'. destruct (cmra_extend n a a1 a2) as ([z1 z2]&?&?&?); auto. exists (Cinl z1, Cinl z2). by repeat constructor. + destruct y1 as [a1|b1|], y2 as [a2|b2|]; try (exfalso; by inversion_clear Hx'). apply (inj Cinr) in Hx'. destruct (cmra_extend n b b1 b2) as ([z1 z2]&?&?&?); auto. exists (Cinr z1, Cinr z2). by repeat constructor. + by exists (CsumBot, CsumBot); destruct y1, y2; inversion_clear Hx'. Qed. Canonical Structure csumR := CMRAT (csum A B) csum_cofe_mixin csum_cmra_mixin. Global Instance csum_cmra_discrete : CMRADiscrete A → CMRADiscrete B → CMRADiscrete csumR. Proof. split; first apply _. by move=>[a|b|] HH /=; try apply cmra_discrete_valid. Qed. Global Instance Cinl_persistent a : Persistent a → Persistent (Cinl a). Proof. rewrite /Persistent /=. inversion_clear 1; by repeat constructor. Qed. Global Instance Cinr_persistent b : Persistent b → Persistent (Cinr b). Proof. rewrite /Persistent /=. inversion_clear 1; by repeat constructor. Qed. Global Instance Cinl_exclusive a : Exclusive a → Exclusive (Cinl a). Proof. by move=> H[]? =>[/H||]. Qed. Global Instance Cinr_exclusive b : Exclusive b → Exclusive (Cinr b). Proof. by move=> H[]? =>[|/H|]. Qed. (** Internalized properties *) Lemma csum_equivI {M} (x y : csum A B) : x ≡ y ⊣⊢ (match x, y with | Cinl a, Cinl a' => a ≡ a' | Cinr b, Cinr b' => b ≡ b' | CsumBot, CsumBot => True | _, _ => False end : uPred M). Proof. uPred.unseal; do 2 split; first by destruct 1. by destruct x, y; try destruct 1; try constructor. Qed. Lemma csum_validI {M} (x : csum A B) : ✓ x ⊣⊢ (match x with | Cinl a => ✓ a | Cinr b => ✓ b | CsumBot => False end : uPred M). Proof. uPred.unseal. by destruct x. Qed. (** Updates *) Lemma csum_update_l (a1 a2 : A) : a1 ~~> a2 → Cinl a1 ~~> Cinl a2. Proof. intros Ha n [[a|b|]|] ?; simpl in *; auto. - by apply (Ha n (Some a)). - by apply (Ha n None). Qed. Lemma csum_update_r (b1 b2 : B) : b1 ~~> b2 → Cinr b1 ~~> Cinr b2. Proof. intros Hb n [[a|b|]|] ?; simpl in *; auto. - by apply (Hb n (Some b)). - by apply (Hb n None). Qed. Lemma csum_updateP_l (P : A → Prop) (Q : csum A B → Prop) a : a ~~>: P → (∀ a', P a' → Q (Cinl a')) → Cinl a ~~>: Q. Proof. intros Hx HP n mf Hm. destruct mf as [[a'|b'|]|]; try by destruct Hm. - destruct (Hx n (Some a')) as (c&?&?); naive_solver. - destruct (Hx n None) as (c&?&?); naive_solver eauto using cmra_validN_op_l. Qed. Lemma csum_updateP_r (P : B → Prop) (Q : csum A B → Prop) b : b ~~>: P → (∀ b', P b' → Q (Cinr b')) → Cinr b ~~>: Q. Proof. intros Hx HP n mf Hm. destruct mf as [[a'|b'|]|]; try by destruct Hm. - destruct (Hx n (Some b')) as (c&?&?); naive_solver. - destruct (Hx n None) as (c&?&?); naive_solver eauto using cmra_validN_op_l. Qed. Lemma csum_updateP'_l (P : A → Prop) a : a ~~>: P → Cinl a ~~>: λ m', ∃ a', m' = Cinl a' ∧ P a'. Proof. eauto using csum_updateP_l. Qed. Lemma csum_updateP'_r (P : B → Prop) b : b ~~>: P → Cinr b ~~>: λ m', ∃ b', m' = Cinr b' ∧ P b'. Proof. eauto using csum_updateP_r. Qed. Lemma csum_local_update_l (a1 a2 : A) af : (∀ af', af = Cinl <\$> af' → a1 ~l~> a2 @ af') → Cinl a1 ~l~> Cinl a2 @ af. Proof. intros Ha. split; destruct af as [[af'| |]|]=>//=. - by eapply (Ha (Some af')). - by eapply (Ha None). - destruct (Ha (Some af') (reflexivity _)) as [_ Ha']. intros n [[mz|mz|]|] ?; inversion 1; subst; constructor. by apply (Ha' n (Some mz)). by apply (Ha' n None). - destruct (Ha None (reflexivity _)) as [_ Ha']. intros n [[mz|mz|]|] ?; inversion 1; subst; constructor. by apply (Ha' n (Some mz)). by apply (Ha' n None). Qed. Lemma csum_local_update_r (b1 b2 : B) bf : (∀ bf', bf = Cinr <\$> bf' → b1 ~l~> b2 @ bf') → Cinr b1 ~l~> Cinr b2 @ bf. Proof. intros Hb. split; destruct bf as [[|bf'|]|]=>//=. - by eapply (Hb (Some bf')). - by eapply (Hb None). - destruct (Hb (Some bf') (reflexivity _)) as [_ Hb']. intros n [[mz|mz|]|] ?; inversion 1; subst; constructor. by apply (Hb' n (Some mz)). by apply (Hb' n None). - destruct (Hb None (reflexivity _)) as [_ Hb']. intros n [[mz|mz|]|] ?; inversion 1; subst; constructor. by apply (Hb' n (Some mz)). by apply (Hb' n None). Qed. End cmra. Arguments csumR : clear implicits. (* Functor *) Instance csum_map_cmra_monotone {A A' B B' : cmraT} (f : A → A') (g : B → B') : CMRAMonotone f → CMRAMonotone g → CMRAMonotone (csum_map f g). Proof. split; try apply _. - intros n [a|b|]; simpl; auto using validN_preserving. - intros x y; rewrite !csum_included. intros [->|[(a&a'&->&->&?)|(b&b'&->&->&?)]]; simpl; eauto 10 using cmra_monotone. Qed. Program Definition csumRF (Fa Fb : rFunctor) : rFunctor := {| rFunctor_car A B := csumR (rFunctor_car Fa A B) (rFunctor_car Fb A B); rFunctor_map A1 A2 B1 B2 fg := csumC_map (rFunctor_map Fa fg) (rFunctor_map Fb fg) |}. Next Obligation. by intros Fa Fb A1 A2 B1 B2 n f g Hfg; apply csumC_map_ne; try apply rFunctor_ne. Qed. Next Obligation. intros Fa Fb A B x. rewrite /= -{2}(csum_map_id x). apply csum_map_ext=>y; apply rFunctor_id. Qed. Next Obligation. intros Fa Fb A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -csum_map_compose. apply csum_map_ext=>y; apply rFunctor_compose. Qed. Instance csumRF_contractive Fa Fb : rFunctorContractive Fa → rFunctorContractive Fb → rFunctorContractive (csumRF Fa Fb). Proof. by intros ?? A1 A2 B1 B2 n f g Hfg; apply csumC_map_ne; try apply rFunctor_contractive. Qed.