From iris.algebra Require Export cmra. Local Hint Extern 1 (_ ≼ _) => etrans; [eassumption|]. Local Hint Extern 1 (_ ≼ _) => etrans; [|eassumption]. Local Hint Extern 10 (_ ≤ _) => omega. Record uPred (M : cmraT) : Type := IProp { uPred_holds :> nat → M → Prop; uPred_ne n x1 x2 : uPred_holds n x1 → x1 ≡{n}≡ x2 → uPred_holds n x2; uPred_weaken n1 n2 x1 x2 : uPred_holds n1 x1 → x1 ≼ x2 → n2 ≤ n1 → ✓{n2} x2 → uPred_holds n2 x2 }. Arguments uPred_holds {_} _ _ _ : simpl never. Add Printing Constructor uPred. Instance: Params (@uPred_holds) 3. Delimit Scope uPred_scope with I. Bind Scope uPred_scope with uPred. Arguments uPred_holds {_} _%I _ _. Section cofe. Context {M : cmraT}. Inductive uPred_equiv' (P Q : uPred M) : Prop := { uPred_in_equiv : ∀ n x, ✓{n} x → P n x ↔ Q n x }. Instance uPred_equiv : Equiv (uPred M) := uPred_equiv'. Inductive uPred_dist' (n : nat) (P Q : uPred M) : Prop := { uPred_in_dist : ∀ n' x, n' ≤ n → ✓{n'} x → P n' x ↔ Q n' x }. Instance uPred_dist : Dist (uPred M) := uPred_dist'. Program Instance uPred_compl : Compl (uPred M) := λ c, {| uPred_holds n x := c n n x |}. Next Obligation. by intros c n x y ??; simpl in *; apply uPred_ne with x. Qed. Next Obligation. intros c n1 n2 x1 x2 ????; simpl in *. apply (chain_cauchy c n2 n1); eauto using uPred_weaken. Qed. Definition uPred_cofe_mixin : CofeMixin (uPred M). Proof. split. - intros P Q; split. + by intros HPQ n; split=> i x ??; apply HPQ. + intros HPQ; split=> n x ?; apply HPQ with n; auto. - intros n; split. + by intros P; split=> x i. + by intros P Q HPQ; split=> x i ??; symmetry; apply HPQ. + intros P Q Q' HP HQ; split=> i x ??. by trans (Q i x);[apply HP|apply HQ]. - intros n P Q HPQ; split=> i x ??; apply HPQ; auto. - intros n c; split=>i x ??; symmetry; apply (chain_cauchy c i n); auto. Qed. Canonical Structure uPredC : cofeT := CofeT uPred_cofe_mixin. End cofe. Arguments uPredC : clear implicits. Instance uPred_ne' {M} (P : uPred M) n : Proper (dist n ==> iff) (P n). Proof. intros x1 x2 Hx; split; eauto using uPred_ne. Qed. Instance uPred_proper {M} (P : uPred M) n : Proper ((≡) ==> iff) (P n). Proof. by intros x1 x2 Hx; apply uPred_ne', equiv_dist. Qed. Lemma uPred_holds_ne {M} (P1 P2 : uPred M) n x : P1 ≡{n}≡ P2 → ✓{n} x → P1 n x → P2 n x. Proof. intros HP ?; apply HP; auto. Qed. Lemma uPred_weaken' {M} (P : uPred M) n1 n2 x1 x2 : x1 ≼ x2 → n2 ≤ n1 → ✓{n2} x2 → P n1 x1 → P n2 x2. Proof. eauto using uPred_weaken. Qed. (** functor *) Program Definition uPred_map {M1 M2 : cmraT} (f : M2 -n> M1) `{!CMRAMonotone f} (P : uPred M1) : uPred M2 := {| uPred_holds n x := P n (f x) |}. Next Obligation. by intros M1 M2 f ? P y1 y2 n ? Hy; rewrite /= -Hy. Qed. Next Obligation. naive_solver eauto using uPred_weaken, included_preserving, validN_preserving. Qed. Instance uPred_map_ne {M1 M2 : cmraT} (f : M2 -n> M1) `{!CMRAMonotone f} n : Proper (dist n ==> dist n) (uPred_map f). Proof. intros x1 x2 Hx; split=> n' y ??. split; apply Hx; auto using validN_preserving. Qed. Lemma uPred_map_id {M : cmraT} (P : uPred M): uPred_map cid P ≡ P. Proof. by split=> n x ?. Qed. Lemma uPred_map_compose {M1 M2 M3 : cmraT} (f : M1 -n> M2) (g : M2 -n> M3) `{!CMRAMonotone f, !CMRAMonotone g} (P : uPred M3): uPred_map (g ◎ f) P ≡ uPred_map f (uPred_map g P). Proof. by split=> n x Hx. Qed. Lemma uPred_map_ext {M1 M2 : cmraT} (f g : M1 -n> M2) `{!CMRAMonotone f} `{!CMRAMonotone g}: (∀ x, f x ≡ g x) → ∀ x, uPred_map f x ≡ uPred_map g x. Proof. intros Hf P; split=> n x Hx /=; by rewrite /uPred_holds /= Hf. Qed. Definition uPredC_map {M1 M2 : cmraT} (f : M2 -n> M1) `{!CMRAMonotone f} : uPredC M1 -n> uPredC M2 := CofeMor (uPred_map f : uPredC M1 → uPredC M2). Lemma uPredC_map_ne {M1 M2 : cmraT} (f g : M2 -n> M1) `{!CMRAMonotone f, !CMRAMonotone g} n : f ≡{n}≡ g → uPredC_map f ≡{n}≡ uPredC_map g. Proof. by intros Hfg P; split=> n' y ??; rewrite /uPred_holds /= (dist_le _ _ _ _(Hfg y)); last lia. Qed. Program Definition uPredCF (F : rFunctor) : cFunctor := {| cFunctor_car A B := uPredC (rFunctor_car F B A); cFunctor_map A1 A2 B1 B2 fg := uPredC_map (rFunctor_map F (fg.2, fg.1)) |}. Next Obligation. intros F A1 A2 B1 B2 n P Q HPQ. apply uPredC_map_ne, rFunctor_ne; split; by apply HPQ. Qed. Next Obligation. intros F A B P; simpl. rewrite -{2}(uPred_map_id P). apply uPred_map_ext=>y. by rewrite rFunctor_id. Qed. Next Obligation. intros F A1 A2 A3 B1 B2 B3 f g f' g' P; simpl. rewrite -uPred_map_compose. apply uPred_map_ext=>y; apply rFunctor_compose. Qed. Instance uPredCF_contractive F : rFunctorContractive F → cFunctorContractive (uPredCF F). Proof. intros ? A1 A2 B1 B2 n P Q HPQ. apply uPredC_map_ne, rFunctor_contractive=> i ?; split; by apply HPQ. Qed. (** logical entailement *) Inductive uPred_entails {M} (P Q : uPred M) : Prop := { uPred_in_entails : ∀ n x, ✓{n} x → P n x → Q n x }. Hint Extern 0 (uPred_entails _ _) => reflexivity. Instance uPred_entails_rewrite_relation M : RewriteRelation (@uPred_entails M). (** logical connectives *) Program Definition uPred_const_def {M} (φ : Prop) : uPred M := {| uPred_holds n x := φ |}. Solve Obligations with done. Definition uPred_const_aux : { x | x = @uPred_const_def }. by eexists. Qed. Definition uPred_const {M} := proj1_sig uPred_const_aux M. Definition uPred_const_eq : @uPred_const = @uPred_const_def := proj2_sig uPred_const_aux. Instance uPred_inhabited M : Inhabited (uPred M) := populate (uPred_const True). Program Definition uPred_and_def {M} (P Q : uPred M) : uPred M := {| uPred_holds n x := P n x ∧ Q n x |}. Solve Obligations with naive_solver eauto 2 using uPred_ne, uPred_weaken. Definition uPred_and_aux : { x | x = @uPred_and_def }. by eexists. Qed. Definition uPred_and {M} := proj1_sig uPred_and_aux M. Definition uPred_and_eq: @uPred_and = @uPred_and_def := proj2_sig uPred_and_aux. Program Definition uPred_or_def {M} (P Q : uPred M) : uPred M := {| uPred_holds n x := P n x ∨ Q n x |}. Solve Obligations with naive_solver eauto 2 using uPred_ne, uPred_weaken. Definition uPred_or_aux : { x | x = @uPred_or_def }. by eexists. Qed. Definition uPred_or {M} := proj1_sig uPred_or_aux M. Definition uPred_or_eq: @uPred_or = @uPred_or_def := proj2_sig uPred_or_aux. Program Definition uPred_impl_def {M} (P Q : uPred M) : uPred M := {| uPred_holds n x := ∀ n' x', x ≼ x' → n' ≤ n → ✓{n'} x' → P n' x' → Q n' x' |}. Next Obligation. intros M P Q n1 x1' x1 HPQ Hx1 n2 x2 ????. destruct (cmra_included_dist_l n1 x1 x2 x1') as (x2'&?&Hx2); auto. assert (x2' ≡{n2}≡ x2) as Hx2' by (by apply dist_le with n1). assert (✓{n2} x2') by (by rewrite Hx2'); rewrite -Hx2'. eauto using uPred_weaken, uPred_ne. Qed. Next Obligation. intros M P Q [|n] x1 x2; auto with lia. Qed. Definition uPred_impl_aux : { x | x = @uPred_impl_def }. by eexists. Qed. Definition uPred_impl {M} := proj1_sig uPred_impl_aux M. Definition uPred_impl_eq : @uPred_impl = @uPred_impl_def := proj2_sig uPred_impl_aux. Program Definition uPred_forall_def {M A} (Ψ : A → uPred M) : uPred M := {| uPred_holds n x := ∀ a, Ψ a n x |}. Solve Obligations with naive_solver eauto 2 using uPred_ne, uPred_weaken. Definition uPred_forall_aux : { x | x = @uPred_forall_def }. by eexists. Qed. Definition uPred_forall {M A} := proj1_sig uPred_forall_aux M A. Definition uPred_forall_eq : @uPred_forall = @uPred_forall_def := proj2_sig uPred_forall_aux. Program Definition uPred_exist_def {M A} (Ψ : A → uPred M) : uPred M := {| uPred_holds n x := ∃ a, Ψ a n x |}. Solve Obligations with naive_solver eauto 2 using uPred_ne, uPred_weaken. Definition uPred_exist_aux : { x | x = @uPred_exist_def }. by eexists. Qed. Definition uPred_exist {M A} := proj1_sig uPred_exist_aux M A. Definition uPred_exist_eq: @uPred_exist = @uPred_exist_def := proj2_sig uPred_exist_aux. Program Definition uPred_eq_def {M} {A : cofeT} (a1 a2 : A) : uPred M := {| uPred_holds n x := a1 ≡{n}≡ a2 |}. Solve Obligations with naive_solver eauto 2 using (dist_le (A:=A)). Definition uPred_eq_aux : { x | x = @uPred_eq_def }. by eexists. Qed. Definition uPred_eq {M A} := proj1_sig uPred_eq_aux M A. Definition uPred_eq_eq: @uPred_eq = @uPred_eq_def := proj2_sig uPred_eq_aux. Program Definition uPred_sep_def {M} (P Q : uPred M) : uPred M := {| uPred_holds n x := ∃ x1 x2, x ≡{n}≡ x1 ⋅ x2 ∧ P n x1 ∧ Q n x2 |}. Next Obligation. by intros M P Q n x y (x1&x2&?&?&?) Hxy; exists x1, x2; rewrite -Hxy. Qed. Next Obligation. intros M P Q n1 n2 x y (x1&x2&Hx&?&?) Hxy ??. assert (∃ x2', y ≡{n2}≡ x1 ⋅ x2' ∧ x2 ≼ x2') as (x2'&Hy&?). { destruct Hxy as [z Hy]; exists (x2 ⋅ z); split; eauto using cmra_included_l. apply dist_le with n1; auto. by rewrite (assoc op) -Hx Hy. } clear Hxy; cofe_subst y; exists x1, x2'; split_and?; [done| |]. - apply uPred_weaken with n1 x1; eauto using cmra_validN_op_l. - apply uPred_weaken with n1 x2; eauto using cmra_validN_op_r. Qed. Definition uPred_sep_aux : { x | x = @uPred_sep_def }. by eexists. Qed. Definition uPred_sep {M} := proj1_sig uPred_sep_aux M. Definition uPred_sep_eq: @uPred_sep = @uPred_sep_def := proj2_sig uPred_sep_aux. Program Definition uPred_wand_def {M} (P Q : uPred M) : uPred M := {| uPred_holds n x := ∀ n' x', n' ≤ n → ✓{n'} (x ⋅ x') → P n' x' → Q n' (x ⋅ x') |}. Next Obligation. intros M P Q n1 x1 x2 HPQ Hx n2 x3 ???; simpl in *. rewrite -(dist_le _ _ _ _ Hx) //; apply HPQ; auto. by rewrite (dist_le _ _ _ _ Hx). Qed. Next Obligation. intros M P Q n1 n2 x1 x2 HPQ ??? n3 x3 ???; simpl in *. apply uPred_weaken with n3 (x1 ⋅ x3); eauto using cmra_validN_included, cmra_preserving_r. Qed. Definition uPred_wand_aux : { x | x = @uPred_wand_def }. by eexists. Qed. Definition uPred_wand {M} := proj1_sig uPred_wand_aux M. Definition uPred_wand_eq : @uPred_wand = @uPred_wand_def := proj2_sig uPred_wand_aux. Program Definition uPred_always_def {M} (P : uPred M) : uPred M := {| uPred_holds n x := P n (core x) |}. Next Obligation. by intros M P x1 x2 n ? Hx; rewrite /= -Hx. Qed. Next Obligation. intros M P n1 n2 x1 x2 ????; eapply uPred_weaken with n1 (core x1); eauto using cmra_core_preserving, cmra_core_validN. Qed. Definition uPred_always_aux : { x | x = @uPred_always_def }. by eexists. Qed. Definition uPred_always {M} := proj1_sig uPred_always_aux M. Definition uPred_always_eq : @uPred_always = @uPred_always_def := proj2_sig uPred_always_aux. Program Definition uPred_later_def {M} (P : uPred M) : uPred M := {| uPred_holds n x := match n return _ with 0 => True | S n' => P n' x end |}. Next Obligation. intros M P [|n] ??; eauto using uPred_ne,(dist_le (A:=M)). Qed. Next Obligation. intros M P [|n1] [|n2] x1 x2; eauto using uPred_weaken,cmra_validN_S; try lia. Qed. Definition uPred_later_aux : { x | x = @uPred_later_def }. by eexists. Qed. Definition uPred_later {M} := proj1_sig uPred_later_aux M. Definition uPred_later_eq : @uPred_later = @uPred_later_def := proj2_sig uPred_later_aux. Program Definition uPred_ownM_def {M : cmraT} (a : M) : uPred M := {| uPred_holds n x := a ≼{n} x |}. Next Obligation. by intros M a n x1 x2 [a' ?] Hx; exists a'; rewrite -Hx. Qed. Next Obligation. intros M a n1 n2 x1 x [a' Hx1] [x2 Hx] ??. exists (a' ⋅ x2). by rewrite (assoc op) -(dist_le _ _ _ _ Hx1) // Hx. Qed. Definition uPred_ownM_aux : { x | x = @uPred_ownM_def }. by eexists. Qed. Definition uPred_ownM {M} := proj1_sig uPred_ownM_aux M. Definition uPred_ownM_eq : @uPred_ownM = @uPred_ownM_def := proj2_sig uPred_ownM_aux. Program Definition uPred_valid_def {M A : cmraT} (a : A) : uPred M := {| uPred_holds n x := ✓{n} a |}. Solve Obligations with naive_solver eauto 2 using cmra_validN_le. Definition uPred_valid_aux : { x | x = @uPred_valid_def }. by eexists. Qed. Definition uPred_valid {M A} := proj1_sig uPred_valid_aux M A. Definition uPred_valid_eq : @uPred_valid = @uPred_valid_def := proj2_sig uPred_valid_aux. Notation "P ⊢ Q" := (uPred_entails P%I Q%I) (at level 70) : C_scope. Notation "(⊢)" := uPred_entails (only parsing) : C_scope. Notation "P ⊣⊢ Q" := (equiv (A:=uPred _) P%I Q%I) (at level 70) : C_scope. Notation "(⊣⊢)" := (equiv (A:=uPred _)) (only parsing) : C_scope. Notation "■ φ" := (uPred_const φ%C%type) (at level 20, right associativity) : uPred_scope. Notation "x = y" := (uPred_const (x%C%type = y%C%type)) : uPred_scope. Notation "'False'" := (uPred_const False) : uPred_scope. Notation "'True'" := (uPred_const True) : uPred_scope. Infix "∧" := uPred_and : uPred_scope. Notation "(∧)" := uPred_and (only parsing) : uPred_scope. Infix "∨" := uPred_or : uPred_scope. Notation "(∨)" := uPred_or (only parsing) : uPred_scope. Infix "→" := uPred_impl : uPred_scope. Infix "★" := uPred_sep (at level 80, right associativity) : uPred_scope. Notation "(★)" := uPred_sep (only parsing) : uPred_scope. Notation "P -★ Q" := (uPred_wand P Q) (at level 199, Q at level 200, right associativity) : uPred_scope. Notation "∀ x .. y , P" := (uPred_forall (λ x, .. (uPred_forall (λ y, P)) ..)%I) : uPred_scope. Notation "∃ x .. y , P" := (uPred_exist (λ x, .. (uPred_exist (λ y, P)) ..)%I) : uPred_scope. Notation "□ P" := (uPred_always P) (at level 20, right associativity) : uPred_scope. Notation "▷ P" := (uPred_later P) (at level 20, right associativity) : uPred_scope. Infix "≡" := uPred_eq : uPred_scope. Notation "✓ x" := (uPred_valid x) (at level 20) : uPred_scope. Definition uPred_iff {M} (P Q : uPred M) : uPred M := ((P → Q) ∧ (Q → P))%I. Infix "↔" := uPred_iff : uPred_scope. Class TimelessP {M} (P : uPred M) := timelessP : ▷ P ⊢ (P ∨ ▷ False). Arguments timelessP {_} _ {_}. Class Persistent {M} (P : uPred M) := persistent : P ⊢ □ P. Arguments persistent {_} _ {_}. Module uPred. Definition unseal := (uPred_const_eq, uPred_and_eq, uPred_or_eq, uPred_impl_eq, uPred_forall_eq, uPred_exist_eq, uPred_eq_eq, uPred_sep_eq, uPred_wand_eq, uPred_always_eq, uPred_later_eq, uPred_ownM_eq, uPred_valid_eq). Ltac unseal := rewrite !unseal. Section uPred_logic. Context {M : cmraT}. Implicit Types φ : Prop. Implicit Types P Q : uPred M. Implicit Types A : Type. Notation "P ⊢ Q" := (@uPred_entails M P%I Q%I). (* Force implicit argument M *) Notation "P ⊣⊢ Q" := (equiv (A:=uPred M) P%I Q%I). (* Force implicit argument M *) Arguments uPred_holds {_} !_ _ _ /. Hint Immediate uPred_in_entails. Global Instance: PreOrder (@uPred_entails M). Proof. split. * by intros P; split=> x i. * by intros P Q Q' HP HQ; split=> x i ??; apply HQ, HP. Qed. Global Instance: AntiSymm (⊣⊢) (@uPred_entails M). Proof. intros P Q HPQ HQP; split=> x n; by split; [apply HPQ|apply HQP]. Qed. Lemma equiv_spec P Q : P ⊣⊢ Q ↔ P ⊢ Q ∧ Q ⊢ P. Proof. split; [|by intros [??]; apply (anti_symm (⊢))]. intros HPQ; split; split=> x i; apply HPQ. Qed. Lemma equiv_entails P Q : P ⊣⊢ Q → P ⊢ Q. Proof. apply equiv_spec. Qed. Lemma equiv_entails_sym P Q : Q ⊣⊢ P → P ⊢ Q. Proof. apply equiv_spec. Qed. Global Instance entails_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> iff) ((⊢) : relation (uPred M)). Proof. move => P1 P2 /equiv_spec [HP1 HP2] Q1 Q2 /equiv_spec [HQ1 HQ2]; split; intros. - by trans P1; [|trans Q1]. - by trans P2; [|trans Q2]. Qed. Lemma entails_equiv_l (P Q R : uPred M) : P ⊣⊢ Q → Q ⊢ R → P ⊢ R. Proof. by intros ->. Qed. Lemma entails_equiv_r (P Q R : uPred M) : P ⊢ Q → Q ⊣⊢ R → P ⊢ R. Proof. by intros ? <-. Qed. (** Non-expansiveness and setoid morphisms *) Global Instance const_proper : Proper (iff ==> (⊣⊢)) (@uPred_const M). Proof. intros φ1 φ2 Hφ. by unseal; split=> -[|n] ?; try apply Hφ. Qed. Global Instance and_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_and M). Proof. intros P P' HP Q Q' HQ; unseal; split=> x n' ??. split; (intros [??]; split; [by apply HP|by apply HQ]). Qed. Global Instance and_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@uPred_and M) := ne_proper_2 _. Global Instance or_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_or M). Proof. intros P P' HP Q Q' HQ; split=> x n' ??. unseal; split; (intros [?|?]; [left; by apply HP|right; by apply HQ]). Qed. Global Instance or_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@uPred_or M) := ne_proper_2 _. Global Instance impl_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_impl M). Proof. intros P P' HP Q Q' HQ; split=> x n' ??. unseal; split; intros HPQ x' n'' ????; apply HQ, HPQ, HP; auto. Qed. Global Instance impl_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@uPred_impl M) := ne_proper_2 _. Global Instance sep_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_sep M). Proof. intros P P' HP Q Q' HQ; split=> n' x ??. unseal; split; intros (x1&x2&?&?&?); cofe_subst x; exists x1, x2; split_and!; try (apply HP || apply HQ); eauto using cmra_validN_op_l, cmra_validN_op_r. Qed. Global Instance sep_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@uPred_sep M) := ne_proper_2 _. Global Instance wand_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_wand M). Proof. intros P P' HP Q Q' HQ; split=> n' x ??; unseal; split; intros HPQ x' n'' ???; apply HQ, HPQ, HP; eauto using cmra_validN_op_r. Qed. Global Instance wand_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@uPred_wand M) := ne_proper_2 _. Global Instance eq_ne (A : cofeT) n : Proper (dist n ==> dist n ==> dist n) (@uPred_eq M A). Proof. intros x x' Hx y y' Hy; split=> n' z; unseal; split; intros; simpl in *. * by rewrite -(dist_le _ _ _ _ Hx) -?(dist_le _ _ _ _ Hy); auto. * by rewrite (dist_le _ _ _ _ Hx) ?(dist_le _ _ _ _ Hy); auto. Qed. Global Instance eq_proper (A : cofeT) : Proper ((≡) ==> (≡) ==> (⊣⊢)) (@uPred_eq M A) := ne_proper_2 _. Global Instance forall_ne A n : Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_forall M A). Proof. by intros Ψ1 Ψ2 HΨ; unseal; split=> n' x; split; intros HP a; apply HΨ. Qed. Global Instance forall_proper A : Proper (pointwise_relation _ (⊣⊢) ==> (⊣⊢)) (@uPred_forall M A). Proof. by intros Ψ1 Ψ2 HΨ; unseal; split=> n' x; split; intros HP a; apply HΨ. Qed. Global Instance exist_ne A n : Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_exist M A). Proof. intros Ψ1 Ψ2 HΨ. unseal; split=> n' x ??; split; intros [a ?]; exists a; by apply HΨ. Qed. Global Instance exist_proper A : Proper (pointwise_relation _ (⊣⊢) ==> (⊣⊢)) (@uPred_exist M A). Proof. intros Ψ1 Ψ2 HΨ. unseal; split=> n' x ?; split; intros [a ?]; exists a; by apply HΨ. Qed. Global Instance later_contractive : Contractive (@uPred_later M). Proof. intros n P Q HPQ; unseal; split=> -[|n'] x ??; simpl; [done|]. apply (HPQ n'); eauto using cmra_validN_S. Qed. Global Instance later_proper : Proper ((⊣⊢) ==> (⊣⊢)) (@uPred_later M) := ne_proper _. Global Instance always_ne n : Proper (dist n ==> dist n) (@uPred_always M). Proof. intros P1 P2 HP. unseal; split=> n' x; split; apply HP; eauto using cmra_core_validN. Qed. Global Instance always_proper : Proper ((⊣⊢) ==> (⊣⊢)) (@uPred_always M) := ne_proper _. Global Instance ownM_ne n : Proper (dist n ==> dist n) (@uPred_ownM M). Proof. intros a b Ha. unseal; split=> n' x ? /=. by rewrite (dist_le _ _ _ _ Ha); last lia. Qed. Global Instance ownM_proper: Proper ((≡) ==> (⊣⊢)) (@uPred_ownM M) := ne_proper _. Global Instance valid_ne {A : cmraT} n : Proper (dist n ==> dist n) (@uPred_valid M A). Proof. intros a b Ha; unseal; split=> n' x ? /=. by rewrite (dist_le _ _ _ _ Ha); last lia. Qed. Global Instance valid_proper {A : cmraT} : Proper ((≡) ==> (⊣⊢)) (@uPred_valid M A) := ne_proper _. Global Instance iff_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_iff M). Proof. unfold uPred_iff; solve_proper. Qed. Global Instance iff_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@uPred_iff M) := ne_proper_2 _. (** Introduction and elimination rules *) Lemma const_intro φ P : φ → P ⊢ ■ φ. Proof. by intros ?; unseal; split. Qed. Lemma const_elim φ Q R : Q ⊢ ■ φ → (φ → Q ⊢ R) → Q ⊢ R. Proof. unseal; intros HQP HQR; split=> n x ??; apply HQR; first eapply HQP; eauto. Qed. Lemma False_elim P : False ⊢ P. Proof. by unseal; split=> n x ?. Qed. Lemma and_elim_l P Q : (P ∧ Q) ⊢ P. Proof. by unseal; split=> n x ? [??]. Qed. Lemma and_elim_r P Q : (P ∧ Q) ⊢ Q. Proof. by unseal; split=> n x ? [??]. Qed. Lemma and_intro P Q R : P ⊢ Q → P ⊢ R → P ⊢ (Q ∧ R). Proof. intros HQ HR; unseal; split=> n x ??; by split; [apply HQ|apply HR]. Qed. Lemma or_intro_l P Q : P ⊢ (P ∨ Q). Proof. unseal; split=> n x ??; left; auto. Qed. Lemma or_intro_r P Q : Q ⊢ (P ∨ Q). Proof. unseal; split=> n x ??; right; auto. Qed. Lemma or_elim P Q R : P ⊢ R → Q ⊢ R → (P ∨ Q) ⊢ R. Proof. intros HP HQ; unseal; split=> n x ? [?|?]. by apply HP. by apply HQ. Qed. Lemma impl_intro_r P Q R : (P ∧ Q) ⊢ R → P ⊢ (Q → R). Proof. unseal; intros HQ; split=> n x ?? n' x' ????. apply HQ; naive_solver eauto using uPred_weaken. Qed. Lemma impl_elim P Q R : P ⊢ (Q → R) → P ⊢ Q → P ⊢ R. Proof. by unseal; intros HP HP'; split=> n x ??; apply HP with n x, HP'. Qed. Lemma forall_intro {A} P (Ψ : A → uPred M): (∀ a, P ⊢ Ψ a) → P ⊢ (∀ a, Ψ a). Proof. unseal; intros HPΨ; split=> n x ?? a; by apply HPΨ. Qed. Lemma forall_elim {A} {Ψ : A → uPred M} a : (∀ a, Ψ a) ⊢ Ψ a. Proof. unseal; split=> n x ? HP; apply HP. Qed. Lemma exist_intro {A} {Ψ : A → uPred M} a : Ψ a ⊢ (∃ a, Ψ a). Proof. unseal; split=> n x ??; by exists a. Qed. Lemma exist_elim {A} (Φ : A → uPred M) Q : (∀ a, Φ a ⊢ Q) → (∃ a, Φ a) ⊢ Q. Proof. unseal; intros HΦΨ; split=> n x ? [a ?]; by apply HΦΨ with a. Qed. Lemma eq_refl {A : cofeT} (a : A) P : P ⊢ (a ≡ a). Proof. unseal; by split=> n x ??; simpl. Qed. Lemma eq_rewrite {A : cofeT} a b (Ψ : A → uPred M) P `{HΨ : ∀ n, Proper (dist n ==> dist n) Ψ} : P ⊢ (a ≡ b) → P ⊢ Ψ a → P ⊢ Ψ b. Proof. unseal; intros Hab Ha; split=> n x ??. apply HΨ with n a; auto. by symmetry; apply Hab with x. by apply Ha. Qed. Lemma eq_equiv `{Empty M, !CMRAUnit M} {A : cofeT} (a b : A) : True ⊢ (a ≡ b) → a ≡ b. Proof. unseal=> Hab; apply equiv_dist; intros n; apply Hab with ∅; last done. apply cmra_valid_validN, cmra_unit_valid. Qed. Lemma iff_equiv P Q : True ⊢ (P ↔ Q) → P ⊣⊢ Q. Proof. rewrite /uPred_iff; unseal=> HPQ. split=> n x ?; split; intros; by apply HPQ with n x. Qed. (* Derived logical stuff *) Lemma True_intro P : P ⊢ True. Proof. by apply const_intro. Qed. Lemma and_elim_l' P Q R : P ⊢ R → (P ∧ Q) ⊢ R. Proof. by rewrite and_elim_l. Qed. Lemma and_elim_r' P Q R : Q ⊢ R → (P ∧ Q) ⊢ R. Proof. by rewrite and_elim_r. Qed. Lemma or_intro_l' P Q R : P ⊢ Q → P ⊢ (Q ∨ R). Proof. intros ->; apply or_intro_l. Qed. Lemma or_intro_r' P Q R : P ⊢ R → P ⊢ (Q ∨ R). Proof. intros ->; apply or_intro_r. Qed. Lemma exist_intro' {A} P (Ψ : A → uPred M) a : P ⊢ Ψ a → P ⊢ (∃ a, Ψ a). Proof. intros ->; apply exist_intro. Qed. Lemma forall_elim' {A} P (Ψ : A → uPred M) : P ⊢ (∀ a, Ψ a) → (∀ a, P ⊢ Ψ a). Proof. move=> HP a. by rewrite HP forall_elim. Qed. Hint Resolve or_elim or_intro_l' or_intro_r'. Hint Resolve and_intro and_elim_l' and_elim_r'. Hint Immediate True_intro False_elim. Lemma impl_intro_l P Q R : (Q ∧ P) ⊢ R → P ⊢ (Q → R). Proof. intros HR; apply impl_intro_r; rewrite -HR; auto. Qed. Lemma impl_elim_l P Q : ((P → Q) ∧ P) ⊢ Q. Proof. apply impl_elim with P; auto. Qed. Lemma impl_elim_r P Q : (P ∧ (P → Q)) ⊢ Q. Proof. apply impl_elim with P; auto. Qed. Lemma impl_elim_l' P Q R : P ⊢ (Q → R) → (P ∧ Q) ⊢ R. Proof. intros; apply impl_elim with Q; auto. Qed. Lemma impl_elim_r' P Q R : Q ⊢ (P → R) → (P ∧ Q) ⊢ R. Proof. intros; apply impl_elim with P; auto. Qed. Lemma impl_entails P Q : True ⊢ (P → Q) → P ⊢ Q. Proof. intros HPQ; apply impl_elim with P; rewrite -?HPQ; auto. Qed. Lemma entails_impl P Q : (P ⊢ Q) → True ⊢ (P → Q). Proof. auto using impl_intro_l. Qed. Lemma const_mono φ1 φ2 : (φ1 → φ2) → ■ φ1 ⊢ ■ φ2. Proof. intros; apply const_elim with φ1; eauto using const_intro. Qed. Lemma and_mono P P' Q Q' : P ⊢ Q → P' ⊢ Q' → (P ∧ P') ⊢ (Q ∧ Q'). Proof. auto. Qed. Lemma and_mono_l P P' Q : P ⊢ Q → (P ∧ P') ⊢ (Q ∧ P'). Proof. by intros; apply and_mono. Qed. Lemma and_mono_r P P' Q' : P' ⊢ Q' → (P ∧ P') ⊢ (P ∧ Q'). Proof. by apply and_mono. Qed. Lemma or_mono P P' Q Q' : P ⊢ Q → P' ⊢ Q' → (P ∨ P') ⊢ (Q ∨ Q'). Proof. auto. Qed. Lemma or_mono_l P P' Q : P ⊢ Q → (P ∨ P') ⊢ (Q ∨ P'). Proof. by intros; apply or_mono. Qed. Lemma or_mono_r P P' Q' : P' ⊢ Q' → (P ∨ P') ⊢ (P ∨ Q'). Proof. by apply or_mono. Qed. Lemma impl_mono P P' Q Q' : Q ⊢ P → P' ⊢ Q' → (P → P') ⊢ (Q → Q'). Proof. intros HP HQ'; apply impl_intro_l; rewrite -HQ'. apply impl_elim with P; eauto. Qed. Lemma forall_mono {A} (Φ Ψ : A → uPred M) : (∀ a, Φ a ⊢ Ψ a) → (∀ a, Φ a) ⊢ (∀ a, Ψ a). Proof. intros HP. apply forall_intro=> a; rewrite -(HP a); apply forall_elim. Qed. Lemma exist_mono {A} (Φ Ψ : A → uPred M) : (∀ a, Φ a ⊢ Ψ a) → (∃ a, Φ a) ⊢ (∃ a, Ψ a). Proof. intros HΦ. apply exist_elim=> a; rewrite (HΦ a); apply exist_intro. Qed. Global Instance const_mono' : Proper (impl ==> (⊢)) (@uPred_const M). Proof. intros φ1 φ2; apply const_mono. Qed. Global Instance and_mono' : Proper ((⊢) ==> (⊢) ==> (⊢)) (@uPred_and M). Proof. by intros P P' HP Q Q' HQ; apply and_mono. Qed. Global Instance and_flip_mono' : Proper (flip (⊢) ==> flip (⊢) ==> flip (⊢)) (@uPred_and M). Proof. by intros P P' HP Q Q' HQ; apply and_mono. Qed. Global Instance or_mono' : Proper ((⊢) ==> (⊢) ==> (⊢)) (@uPred_or M). Proof. by intros P P' HP Q Q' HQ; apply or_mono. Qed. Global Instance or_flip_mono' : Proper (flip (⊢) ==> flip (⊢) ==> flip (⊢)) (@uPred_or M). Proof. by intros P P' HP Q Q' HQ; apply or_mono. Qed. Global Instance impl_mono' : Proper (flip (⊢) ==> (⊢) ==> (⊢)) (@uPred_impl M). Proof. by intros P P' HP Q Q' HQ; apply impl_mono. Qed. Global Instance forall_mono' A : Proper (pointwise_relation _ (⊢) ==> (⊢)) (@uPred_forall M A). Proof. intros P1 P2; apply forall_mono. Qed. Global Instance exist_mono' A : Proper (pointwise_relation _ (⊢) ==> (⊢)) (@uPred_exist M A). Proof. intros P1 P2; apply exist_mono. Qed. Global Instance and_idem : IdemP (⊣⊢) (@uPred_and M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance or_idem : IdemP (⊣⊢) (@uPred_or M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance and_comm : Comm (⊣⊢) (@uPred_and M). Proof. intros P Q; apply (anti_symm (⊢)); auto. Qed. Global Instance True_and : LeftId (⊣⊢) True%I (@uPred_and M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance and_True : RightId (⊣⊢) True%I (@uPred_and M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance False_and : LeftAbsorb (⊣⊢) False%I (@uPred_and M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance and_False : RightAbsorb (⊣⊢) False%I (@uPred_and M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance True_or : LeftAbsorb (⊣⊢) True%I (@uPred_or M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance or_True : RightAbsorb (⊣⊢) True%I (@uPred_or M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance False_or : LeftId (⊣⊢) False%I (@uPred_or M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance or_False : RightId (⊣⊢) False%I (@uPred_or M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance and_assoc : Assoc (⊣⊢) (@uPred_and M). Proof. intros P Q R; apply (anti_symm (⊢)); auto. Qed. Global Instance or_comm : Comm (⊣⊢) (@uPred_or M). Proof. intros P Q; apply (anti_symm (⊢)); auto. Qed. Global Instance or_assoc : Assoc (⊣⊢) (@uPred_or M). Proof. intros P Q R; apply (anti_symm (⊢)); auto. Qed. Global Instance True_impl : LeftId (⊣⊢) True%I (@uPred_impl M). Proof. intros P; apply (anti_symm (⊢)). - by rewrite -(left_id True%I uPred_and (_ → _)%I) impl_elim_r. - by apply impl_intro_l; rewrite left_id. Qed. Lemma iff_refl Q P : Q ⊢ (P ↔ P). Proof. rewrite /uPred_iff; apply and_intro; apply impl_intro_l; auto. Qed. Lemma or_and_l P Q R : (P ∨ Q ∧ R) ⊣⊢ ((P ∨ Q) ∧ (P ∨ R)). Proof. apply (anti_symm (⊢)); first auto. do 2 (apply impl_elim_l', or_elim; apply impl_intro_l); auto. Qed. Lemma or_and_r P Q R : (P ∧ Q ∨ R) ⊣⊢ ((P ∨ R) ∧ (Q ∨ R)). Proof. by rewrite -!(comm _ R) or_and_l. Qed. Lemma and_or_l P Q R : (P ∧ (Q ∨ R)) ⊣⊢ (P ∧ Q ∨ P ∧ R). Proof. apply (anti_symm (⊢)); last auto. apply impl_elim_r', or_elim; apply impl_intro_l; auto. Qed. Lemma and_or_r P Q R : ((P ∨ Q) ∧ R) ⊣⊢ (P ∧ R ∨ Q ∧ R). Proof. by rewrite -!(comm _ R) and_or_l. Qed. Lemma and_exist_l {A} P (Ψ : A → uPred M) : (P ∧ ∃ a, Ψ a) ⊣⊢ (∃ a, P ∧ Ψ a). Proof. apply (anti_symm (⊢)). - apply impl_elim_r'. apply exist_elim=>a. apply impl_intro_l. by rewrite -(exist_intro a). - apply exist_elim=>a. apply and_intro; first by rewrite and_elim_l. by rewrite -(exist_intro a) and_elim_r. Qed. Lemma and_exist_r {A} P (Φ: A → uPred M) : ((∃ a, Φ a) ∧ P) ⊣⊢ (∃ a, Φ a ∧ P). Proof. rewrite -(comm _ P) and_exist_l. apply exist_proper=>a. by rewrite comm. Qed. Lemma const_intro_l φ Q R : φ → (■ φ ∧ Q) ⊢ R → Q ⊢ R. Proof. intros ? <-; auto using const_intro. Qed. Lemma const_intro_r φ Q R : φ → (Q ∧ ■ φ) ⊢ R → Q ⊢ R. Proof. intros ? <-; auto using const_intro. Qed. Lemma const_intro_impl φ Q R : φ → Q ⊢ (■ φ → R) → Q ⊢ R. Proof. intros ? ->. eauto using const_intro_l, impl_elim_r. Qed. Lemma const_elim_l φ Q R : (φ → Q ⊢ R) → (■ φ ∧ Q) ⊢ R. Proof. intros; apply const_elim with φ; eauto. Qed. Lemma const_elim_r φ Q R : (φ → Q ⊢ R) → (Q ∧ ■ φ) ⊢ R. Proof. intros; apply const_elim with φ; eauto. Qed. Lemma const_equiv (φ : Prop) : φ → (■ φ) ⊣⊢ True. Proof. intros; apply (anti_symm _); auto using const_intro. Qed. Lemma equiv_eq {A : cofeT} P (a b : A) : a ≡ b → P ⊢ (a ≡ b). Proof. intros ->; apply eq_refl. Qed. Lemma eq_sym {A : cofeT} (a b : A) : (a ≡ b) ⊢ (b ≡ a). Proof. apply (eq_rewrite a b (λ b, b ≡ a)%I); auto using eq_refl. solve_proper. Qed. (* BI connectives *) Lemma sep_mono P P' Q Q' : P ⊢ Q → P' ⊢ Q' → (P ★ P') ⊢ (Q ★ Q'). Proof. intros HQ HQ'; unseal. split; intros n' x ? (x1&x2&?&?&?); exists x1,x2; cofe_subst x; eauto 7 using cmra_validN_op_l, cmra_validN_op_r, uPred_in_entails. Qed. Global Instance True_sep : LeftId (⊣⊢) True%I (@uPred_sep M). Proof. intros P; unseal; split=> n x Hvalid; split. - intros (x1&x2&?&_&?); cofe_subst; eauto using uPred_weaken, cmra_included_r. - by intros ?; exists (core x), x; rewrite cmra_core_l. Qed. Global Instance sep_comm : Comm (⊣⊢) (@uPred_sep M). Proof. by intros P Q; unseal; split=> n x ?; split; intros (x1&x2&?&?&?); exists x2, x1; rewrite (comm op). Qed. Global Instance sep_assoc : Assoc (⊣⊢) (@uPred_sep M). Proof. intros P Q R; unseal; split=> n x ?; split. - intros (x1&x2&Hx&?&y1&y2&Hy&?&?); exists (x1 ⋅ y1), y2; split_and?; auto. + by rewrite -(assoc op) -Hy -Hx. + by exists x1, y1. - intros (x1&x2&Hx&(y1&y2&Hy&?&?)&?); exists y1, (y2 ⋅ x2); split_and?; auto. + by rewrite (assoc op) -Hy -Hx. + by exists y2, x2. Qed. Lemma wand_intro_r P Q R : (P ★ Q) ⊢ R → P ⊢ (Q -★ R). Proof. unseal=> HPQR; split=> n x ?? n' x' ???; apply HPQR; auto. exists x, x'; split_and?; auto. eapply uPred_weaken with n x; eauto using cmra_validN_op_l. Qed. Lemma wand_elim_l' P Q R : P ⊢ (Q -★ R) → (P ★ Q) ⊢ R. Proof. unseal =>HPQR. split; intros n x ? (?&?&?&?&?). cofe_subst. eapply HPQR; eauto using cmra_validN_op_l. Qed. (* Derived BI Stuff *) Hint Resolve sep_mono. Lemma sep_mono_l P P' Q : P ⊢ Q → (P ★ P') ⊢ (Q ★ P'). Proof. by intros; apply sep_mono. Qed. Lemma sep_mono_r P P' Q' : P' ⊢ Q' → (P ★ P') ⊢ (P ★ Q'). Proof. by apply sep_mono. Qed. Global Instance sep_mono' : Proper ((⊢) ==> (⊢) ==> (⊢)) (@uPred_sep M). Proof. by intros P P' HP Q Q' HQ; apply sep_mono. Qed. Global Instance sep_flip_mono' : Proper (flip (⊢) ==> flip (⊢) ==> flip (⊢)) (@uPred_sep M). Proof. by intros P P' HP Q Q' HQ; apply sep_mono. Qed. Lemma wand_mono P P' Q Q' : Q ⊢ P → P' ⊢ Q' → (P -★ P') ⊢ (Q -★ Q'). Proof. intros HP HQ; apply wand_intro_r. rewrite HP -HQ. by apply wand_elim_l'. Qed. Global Instance wand_mono' : Proper (flip (⊢) ==> (⊢) ==> (⊢)) (@uPred_wand M). Proof. by intros P P' HP Q Q' HQ; apply wand_mono. Qed. Global Instance sep_True : RightId (⊣⊢) True%I (@uPred_sep M). Proof. by intros P; rewrite comm left_id. Qed. Lemma sep_elim_l P Q : (P ★ Q) ⊢ P. Proof. by rewrite (True_intro Q) right_id. Qed. Lemma sep_elim_r P Q : (P ★ Q) ⊢ Q. Proof. by rewrite (comm (★))%I; apply sep_elim_l. Qed. Lemma sep_elim_l' P Q R : P ⊢ R → (P ★ Q) ⊢ R. Proof. intros ->; apply sep_elim_l. Qed. Lemma sep_elim_r' P Q R : Q ⊢ R → (P ★ Q) ⊢ R. Proof. intros ->; apply sep_elim_r. Qed. Hint Resolve sep_elim_l' sep_elim_r'. Lemma sep_intro_True_l P Q R : True ⊢ P → R ⊢ Q → R ⊢ (P ★ Q). Proof. by intros; rewrite -(left_id True%I uPred_sep R); apply sep_mono. Qed. Lemma sep_intro_True_r P Q R : R ⊢ P → True ⊢ Q → R ⊢ (P ★ Q). Proof. by intros; rewrite -(right_id True%I uPred_sep R); apply sep_mono. Qed. Lemma sep_elim_True_l P Q R : True ⊢ P → (P ★ R) ⊢ Q → R ⊢ Q. Proof. by intros HP; rewrite -HP left_id. Qed. Lemma sep_elim_True_r P Q R : True ⊢ P → (R ★ P) ⊢ Q → R ⊢ Q. Proof. by intros HP; rewrite -HP right_id. Qed. Lemma wand_intro_l P Q R : (Q ★ P) ⊢ R → P ⊢ (Q -★ R). Proof. rewrite comm; apply wand_intro_r. Qed. Lemma wand_elim_l P Q : ((P -★ Q) ★ P) ⊢ Q. Proof. by apply wand_elim_l'. Qed. Lemma wand_elim_r P Q : (P ★ (P -★ Q)) ⊢ Q. Proof. rewrite (comm _ P); apply wand_elim_l. Qed. Lemma wand_elim_r' P Q R : Q ⊢ (P -★ R) → (P ★ Q) ⊢ R. Proof. intros ->; apply wand_elim_r. Qed. Lemma wand_apply_l P Q Q' R R' : P ⊢ (Q' -★ R') → R' ⊢ R → Q ⊢ Q' → (P ★ Q) ⊢ R. Proof. intros -> -> <-; apply wand_elim_l. Qed. Lemma wand_apply_r P Q Q' R R' : P ⊢ (Q' -★ R') → R' ⊢ R → Q ⊢ Q' → (Q ★ P) ⊢ R. Proof. intros -> -> <-; apply wand_elim_r. Qed. Lemma wand_apply_l' P Q Q' R : P ⊢ (Q' -★ R) → Q ⊢ Q' → (P ★ Q) ⊢ R. Proof. intros -> <-; apply wand_elim_l. Qed. Lemma wand_apply_r' P Q Q' R : P ⊢ (Q' -★ R) → Q ⊢ Q' → (Q ★ P) ⊢ R. Proof. intros -> <-; apply wand_elim_r. Qed. Lemma wand_frame_l P Q R : (Q -★ R) ⊢ (P ★ Q -★ P ★ R). Proof. apply wand_intro_l. rewrite -assoc. apply sep_mono_r, wand_elim_r. Qed. Lemma wand_frame_r P Q R : (Q -★ R) ⊢ (Q ★ P -★ R ★ P). Proof. apply wand_intro_l. rewrite ![(_ ★ P)%I]comm -assoc. apply sep_mono_r, wand_elim_r. Qed. Lemma wand_diag P : (P -★ P) ⊣⊢ True. Proof. apply (anti_symm _); auto. apply wand_intro_l; by rewrite right_id. Qed. Lemma wand_True P : (True -★ P) ⊣⊢ P. Proof. apply (anti_symm _); last by auto using wand_intro_l. eapply sep_elim_True_l; first reflexivity. by rewrite wand_elim_r. Qed. Lemma wand_entails P Q : True ⊢ (P -★ Q) → P ⊢ Q. Proof. intros HPQ. eapply sep_elim_True_r; first exact: HPQ. by rewrite wand_elim_r. Qed. Lemma entails_wand P Q : (P ⊢ Q) → True ⊢ (P -★ Q). Proof. auto using wand_intro_l. Qed. Lemma sep_and P Q : (P ★ Q) ⊢ (P ∧ Q). Proof. auto. Qed. Lemma impl_wand P Q : (P → Q) ⊢ (P -★ Q). Proof. apply wand_intro_r, impl_elim with P; auto. Qed. Lemma const_elim_sep_l φ Q R : (φ → Q ⊢ R) → (■ φ ★ Q) ⊢ R. Proof. intros; apply const_elim with φ; eauto. Qed. Lemma const_elim_sep_r φ Q R : (φ → Q ⊢ R) → (Q ★ ■ φ) ⊢ R. Proof. intros; apply const_elim with φ; eauto. Qed. Global Instance sep_False : LeftAbsorb (⊣⊢) False%I (@uPred_sep M). Proof. intros P; apply (anti_symm _); auto. Qed. Global Instance False_sep : RightAbsorb (⊣⊢) False%I (@uPred_sep M). Proof. intros P; apply (anti_symm _); auto. Qed. Lemma sep_and_l P Q R : (P ★ (Q ∧ R)) ⊢ ((P ★ Q) ∧ (P ★ R)). Proof. auto. Qed. Lemma sep_and_r P Q R : ((P ∧ Q) ★ R) ⊢ ((P ★ R) ∧ (Q ★ R)). Proof. auto. Qed. Lemma sep_or_l P Q R : (P ★ (Q ∨ R)) ⊣⊢ ((P ★ Q) ∨ (P ★ R)). Proof. apply (anti_symm (⊢)); last by eauto 8. apply wand_elim_r', or_elim; apply wand_intro_l. - by apply or_intro_l. - by apply or_intro_r. Qed. Lemma sep_or_r P Q R : ((P ∨ Q) ★ R) ⊣⊢ ((P ★ R) ∨ (Q ★ R)). Proof. by rewrite -!(comm _ R) sep_or_l. Qed. Lemma sep_exist_l {A} P (Ψ : A → uPred M) : (P ★ ∃ a, Ψ a) ⊣⊢ (∃ a, P ★ Ψ a). Proof. intros; apply (anti_symm (⊢)). - apply wand_elim_r', exist_elim=>a. apply wand_intro_l. by rewrite -(exist_intro a). - apply exist_elim=> a; apply sep_mono; auto using exist_intro. Qed. Lemma sep_exist_r {A} (Φ: A → uPred M) Q: ((∃ a, Φ a) ★ Q) ⊣⊢ (∃ a, Φ a ★ Q). Proof. setoid_rewrite (comm _ _ Q); apply sep_exist_l. Qed. Lemma sep_forall_l {A} P (Ψ : A → uPred M) : (P ★ ∀ a, Ψ a) ⊢ (∀ a, P ★ Ψ a). Proof. by apply forall_intro=> a; rewrite forall_elim. Qed. Lemma sep_forall_r {A} (Φ : A → uPred M) Q : ((∀ a, Φ a) ★ Q) ⊢ (∀ a, Φ a ★ Q). Proof. by apply forall_intro=> a; rewrite forall_elim. Qed. (* Always *) Lemma always_const φ : (□ ■ φ) ⊣⊢ (■ φ). Proof. by unseal. Qed. Lemma always_elim P : □ P ⊢ P. Proof. unseal; split=> n x ? /=; eauto using uPred_weaken, cmra_included_core. Qed. Lemma always_intro' P Q : □ P ⊢ Q → □ P ⊢ □ Q. Proof. unseal=> HPQ. split=> n x ??; apply HPQ; simpl; auto using cmra_core_validN. by rewrite cmra_core_idemp. Qed. Lemma always_and P Q : (□ (P ∧ Q)) ⊣⊢ (□ P ∧ □ Q). Proof. by unseal. Qed. Lemma always_or P Q : (□ (P ∨ Q)) ⊣⊢ (□ P ∨ □ Q). Proof. by unseal. Qed. Lemma always_forall {A} (Ψ : A → uPred M) : (□ ∀ a, Ψ a) ⊣⊢ (∀ a, □ Ψ a). Proof. by unseal. Qed. Lemma always_exist {A} (Ψ : A → uPred M) : (□ ∃ a, Ψ a) ⊣⊢ (∃ a, □ Ψ a). Proof. by unseal. Qed. Lemma always_and_sep_1 P Q : □ (P ∧ Q) ⊢ □ (P ★ Q). Proof. unseal; split=> n x ? [??]. exists (core x), (core x); rewrite cmra_core_core; auto. Qed. Lemma always_and_sep_l_1 P Q : (□ P ∧ Q) ⊢ (□ P ★ Q). Proof. unseal; split=> n x ? [??]; exists (core x), x; simpl in *. by rewrite cmra_core_l cmra_core_idemp. Qed. Lemma always_later P : (□ ▷ P) ⊣⊢ (▷ □ P). Proof. by unseal. Qed. (* Always derived *) Lemma always_mono P Q : P ⊢ Q → □ P ⊢ □ Q. Proof. intros. apply always_intro'. by rewrite always_elim. Qed. Hint Resolve always_mono. Global Instance always_mono' : Proper ((⊢) ==> (⊢)) (@uPred_always M). Proof. intros P Q; apply always_mono. Qed. Global Instance always_flip_mono' : Proper (flip (⊢) ==> flip (⊢)) (@uPred_always M). Proof. intros P Q; apply always_mono. Qed. Lemma always_impl P Q : □ (P → Q) ⊢ (□ P → □ Q). Proof. apply impl_intro_l; rewrite -always_and. apply always_mono, impl_elim with P; auto. Qed. Lemma always_eq {A:cofeT} (a b : A) : (□ (a ≡ b)) ⊣⊢ (a ≡ b). Proof. apply (anti_symm (⊢)); auto using always_elim. apply (eq_rewrite a b (λ b, □ (a ≡ b))%I); auto. { intros n; solve_proper. } rewrite -(eq_refl a True) always_const; auto. Qed. Lemma always_and_sep P Q : (□ (P ∧ Q)) ⊣⊢ (□ (P ★ Q)). Proof. apply (anti_symm (⊢)); auto using always_and_sep_1. Qed. Lemma always_and_sep_l' P Q : (□ P ∧ Q) ⊣⊢ (□ P ★ Q). Proof. apply (anti_symm (⊢)); auto using always_and_sep_l_1. Qed. Lemma always_and_sep_r' P Q : (P ∧ □ Q) ⊣⊢ (P ★ □ Q). Proof. by rewrite !(comm _ P) always_and_sep_l'. Qed. Lemma always_sep P Q : (□ (P ★ Q)) ⊣⊢ (□ P ★ □ Q). Proof. by rewrite -always_and_sep -always_and_sep_l' always_and. Qed. Lemma always_wand P Q : □ (P -★ Q) ⊢ (□ P -★ □ Q). Proof. by apply wand_intro_r; rewrite -always_sep wand_elim_l. Qed. Lemma always_sep_dup' P : (□ P) ⊣⊢ (□ P ★ □ P). Proof. by rewrite -always_sep -always_and_sep (idemp _). Qed. Lemma always_wand_impl P Q : (□ (P -★ Q)) ⊣⊢ (□ (P → Q)). Proof. apply (anti_symm (⊢)); [|by rewrite -impl_wand]. apply always_intro', impl_intro_r. by rewrite always_and_sep_l' always_elim wand_elim_l. Qed. Lemma always_entails_l' P Q : (P ⊢ □ Q) → P ⊢ (□ Q ★ P). Proof. intros; rewrite -always_and_sep_l'; auto. Qed. Lemma always_entails_r' P Q : (P ⊢ □ Q) → P ⊢ (P ★ □ Q). Proof. intros; rewrite -always_and_sep_r'; auto. Qed. (* Later *) Lemma later_mono P Q : P ⊢ Q → ▷ P ⊢ ▷ Q. Proof. unseal=> HP; split=>-[|n] x ??; [done|apply HP; eauto using cmra_validN_S]. Qed. Lemma later_intro P : P ⊢ ▷ P. Proof. unseal; split=> -[|n] x ??; simpl in *; [done|]. apply uPred_weaken with (S n) x; eauto using cmra_validN_S. Qed. Lemma löb P : (▷ P → P) ⊢ P. Proof. unseal; split=> n x ? HP; induction n as [|n IH]; [by apply HP|]. apply HP, IH, uPred_weaken with (S n) x; eauto using cmra_validN_S. Qed. Lemma later_and P Q : (▷ (P ∧ Q)) ⊣⊢ (▷ P ∧ ▷ Q). Proof. unseal; split=> -[|n] x; by split. Qed. Lemma later_or P Q : (▷ (P ∨ Q)) ⊣⊢ (▷ P ∨ ▷ Q). Proof. unseal; split=> -[|n] x; simpl; tauto. Qed. Lemma later_forall {A} (Φ : A → uPred M) : (▷ ∀ a, Φ a) ⊣⊢ (∀ a, ▷ Φ a). Proof. unseal; by split=> -[|n] x. Qed. Lemma later_exist_1 {A} (Φ : A → uPred M) : (∃ a, ▷ Φ a) ⊢ (▷ ∃ a, Φ a). Proof. unseal; by split=> -[|[|n]] x. Qed. Lemma later_exist' `{Inhabited A} (Φ : A → uPred M) : (▷ ∃ a, Φ a)%I ⊢ (∃ a, ▷ Φ a)%I. Proof. unseal; split=> -[|[|n]] x; done || by exists inhabitant. Qed. Lemma later_sep P Q : (▷ (P ★ Q)) ⊣⊢ (▷ P ★ ▷ Q). Proof. unseal; split=> n x ?; split. - destruct n as [|n]; simpl. { by exists x, (core x); rewrite cmra_core_r. } intros (x1&x2&Hx&?&?); destruct (cmra_extend n x x1 x2) as ([y1 y2]&Hx'&Hy1&Hy2); eauto using cmra_validN_S; simpl in *. exists y1, y2; split; [by rewrite Hx'|by rewrite Hy1 Hy2]. - destruct n as [|n]; simpl; [done|intros (x1&x2&Hx&?&?)]. exists x1, x2; eauto using dist_S. Qed. (* Later derived *) Global Instance later_mono' : Proper ((⊢) ==> (⊢)) (@uPred_later M). Proof. intros P Q; apply later_mono. Qed. Global Instance later_flip_mono' : Proper (flip (⊢) ==> flip (⊢)) (@uPred_later M). Proof. intros P Q; apply later_mono. Qed. Lemma later_True : (▷ True) ⊣⊢ True. Proof. apply (anti_symm (⊢)); auto using later_intro. Qed. Lemma later_impl P Q : ▷ (P → Q) ⊢ (▷ P → ▷ Q). Proof. apply impl_intro_l; rewrite -later_and. apply later_mono, impl_elim with P; auto. Qed. Lemma later_exist `{Inhabited A} (Φ : A → uPred M) : (▷ ∃ a, Φ a) ⊣⊢ (∃ a, ▷ Φ a). Proof. apply: anti_symm; eauto using later_exist', later_exist_1. Qed. Lemma later_wand P Q : ▷ (P -★ Q) ⊢ (▷ P -★ ▷ Q). Proof. apply wand_intro_r;rewrite -later_sep; apply later_mono,wand_elim_l. Qed. Lemma later_iff P Q : (▷ (P ↔ Q)) ⊢ (▷ P ↔ ▷ Q). Proof. by rewrite /uPred_iff later_and !later_impl. Qed. Lemma löb_strong P Q : (P ∧ ▷ Q) ⊢ Q → P ⊢ Q. Proof. intros Hlöb. apply impl_entails. rewrite -(löb (P → Q)). apply entails_impl, impl_intro_l. rewrite -{2}Hlöb. apply and_intro; first by eauto. by rewrite {1}(later_intro P) later_impl impl_elim_r. Qed. Lemma löb_strong_sep P Q : (P ★ ▷ (P -★ Q)) ⊢ Q → P ⊢ Q. Proof. move/wand_intro_l=>Hlöb. apply impl_entails. rewrite -[(_ → _)%I]always_elim. apply löb_strong. by rewrite left_id -always_wand_impl -always_later Hlöb. Qed. (* Own *) Lemma ownM_op (a1 a2 : M) : uPred_ownM (a1 ⋅ a2) ⊣⊢ (uPred_ownM a1 ★ uPred_ownM a2). Proof. unseal; split=> n x ?; split. - intros [z ?]; exists a1, (a2 ⋅ z); split; [by rewrite (assoc op)|]. split. by exists (core a1); rewrite cmra_core_r. by exists z. - intros (y1&y2&Hx&[z1 Hy1]&[z2 Hy2]); exists (z1 ⋅ z2). by rewrite (assoc op _ z1) -(comm op z1) (assoc op z1) -(assoc op _ a2) (comm op z1) -Hy1 -Hy2. Qed. Lemma always_ownM_core (a : M) : (□ uPred_ownM (core a)) ⊣⊢ uPred_ownM (core a). Proof. split=> n x; split; [by apply always_elim|unseal; intros [a' Hx]]; simpl. rewrite -(cmra_core_idemp a) Hx. apply cmra_core_preservingN, cmra_includedN_l. Qed. Lemma always_ownM (a : M) : core a ≡ a → (□ uPred_ownM a) ⊣⊢ uPred_ownM a. Proof. by intros <-; rewrite always_ownM_core. Qed. Lemma ownM_something : True ⊢ ∃ a, uPred_ownM a. Proof. unseal; split=> n x ??. by exists x; simpl. Qed. Lemma ownM_empty `{Empty M, !CMRAUnit M} : True ⊢ uPred_ownM ∅. Proof. unseal; split=> n x ??; by exists x; rewrite left_id. Qed. (* Valid *) Lemma ownM_valid (a : M) : uPred_ownM a ⊢ ✓ a. Proof. unseal; split=> n x Hv [a' ?]; cofe_subst; eauto using cmra_validN_op_l. Qed. Lemma valid_intro {A : cmraT} (a : A) : ✓ a → True ⊢ ✓ a. Proof. unseal=> ?; split=> n x ? _ /=; by apply cmra_valid_validN. Qed. Lemma valid_elim {A : cmraT} (a : A) : ¬ ✓{0} a → ✓ a ⊢ False. Proof. unseal=> Ha; split=> n x ??; apply Ha, cmra_validN_le with n; auto. Qed. Lemma always_valid {A : cmraT} (a : A) : (□ (✓ a)) ⊣⊢ (✓ a). Proof. by unseal. Qed. Lemma valid_weaken {A : cmraT} (a b : A) : ✓ (a ⋅ b) ⊢ ✓ a. Proof. unseal; split=> n x _; apply cmra_validN_op_l. Qed. (* Own and valid derived *) Lemma ownM_invalid (a : M) : ¬ ✓{0} a → uPred_ownM a ⊢ False. Proof. by intros; rewrite ownM_valid valid_elim. Qed. Global Instance ownM_mono : Proper (flip (≼) ==> (⊢)) (@uPred_ownM M). Proof. intros a b [b' ->]. rewrite ownM_op. eauto. Qed. (* Products *) Lemma prod_equivI {A B : cofeT} (x y : A * B) : (x ≡ y) ⊣⊢ (x.1 ≡ y.1 ∧ x.2 ≡ y.2). Proof. by unseal. Qed. Lemma prod_validI {A B : cmraT} (x : A * B) : (✓ x) ⊣⊢ (✓ x.1 ∧ ✓ x.2). Proof. by unseal. Qed. (* Later *) Lemma later_equivI {A : cofeT} (x y : later A) : (x ≡ y) ⊣⊢ (▷ (later_car x ≡ later_car y)). Proof. by unseal. Qed. (* Discrete *) Lemma discrete_valid {A : cmraT} `{!CMRADiscrete A} (a : A) : (✓ a) ⊣⊢ (■ ✓ a). Proof. unseal; split=> n x _. by rewrite /= -cmra_discrete_valid_iff. Qed. Lemma timeless_eq {A : cofeT} (a b : A) : Timeless a → (a ≡ b) ⊣⊢ (■ (a ≡ b)). Proof. unseal=> ?. apply (anti_symm (⊢)); split=> n x ?; by apply (timeless_iff n). Qed. (* Timeless *) Lemma timelessP_spec P : TimelessP P ↔ ∀ n x, ✓{n} x → P 0 x → P n x. Proof. split. - intros HP n x ??; induction n as [|n]; auto. move: HP; rewrite /TimelessP; unseal=> /uPred_in_entails /(_ (S n) x). by destruct 1; auto using cmra_validN_S. - move=> HP; rewrite /TimelessP; unseal; split=> -[|n] x /=; auto; left. apply HP, uPred_weaken with n x; eauto using cmra_validN_le. Qed. Global Instance const_timeless φ : TimelessP (■ φ : uPred M)%I. Proof. by apply timelessP_spec; unseal => -[|n] x. Qed. Global Instance valid_timeless {A : cmraT} `{CMRADiscrete A} (a : A) : TimelessP (✓ a : uPred M)%I. Proof. apply timelessP_spec; unseal=> n x /=. by rewrite -!cmra_discrete_valid_iff. Qed. Global Instance and_timeless P Q: TimelessP P → TimelessP Q → TimelessP (P ∧ Q). Proof. by intros ??; rewrite /TimelessP later_and or_and_r; apply and_mono. Qed. Global Instance or_timeless P Q : TimelessP P → TimelessP Q → TimelessP (P ∨ Q). Proof. intros; rewrite /TimelessP later_or (timelessP _) (timelessP Q); eauto 10. Qed. Global Instance impl_timeless P Q : TimelessP Q → TimelessP (P → Q). Proof. rewrite !timelessP_spec; unseal=> HP [|n] x ? HPQ [|n'] x' ????; auto. apply HP, HPQ, uPred_weaken with (S n') x'; eauto using cmra_validN_le. Qed. Global Instance sep_timeless P Q: TimelessP P → TimelessP Q → TimelessP (P ★ Q). Proof. intros; rewrite /TimelessP later_sep (timelessP P) (timelessP Q). apply wand_elim_l', or_elim; apply wand_intro_r; auto. apply wand_elim_r', or_elim; apply wand_intro_r; auto. rewrite ?(comm _ Q); auto. Qed. Global Instance wand_timeless P Q : TimelessP Q → TimelessP (P -★ Q). Proof. rewrite !timelessP_spec; unseal=> HP [|n] x ? HPQ [|n'] x' ???; auto. apply HP, HPQ, uPred_weaken with (S n') x'; eauto 3 using cmra_validN_le, cmra_validN_op_r. Qed. Global Instance forall_timeless {A} (Ψ : A → uPred M) : (∀ x, TimelessP (Ψ x)) → TimelessP (∀ x, Ψ x). Proof. by setoid_rewrite timelessP_spec; unseal=> HΨ n x ?? a; apply HΨ. Qed. Global Instance exist_timeless {A} (Ψ : A → uPred M) : (∀ x, TimelessP (Ψ x)) → TimelessP (∃ x, Ψ x). Proof. by setoid_rewrite timelessP_spec; unseal=> HΨ [|n] x ?; [|intros [a ?]; exists a; apply HΨ]. Qed. Global Instance always_timeless P : TimelessP P → TimelessP (□ P). Proof. intros ?; rewrite /TimelessP. by rewrite -always_const -!always_later -always_or; apply always_mono. Qed. Global Instance eq_timeless {A : cofeT} (a b : A) : Timeless a → TimelessP (a ≡ b : uPred M)%I. Proof. intro; apply timelessP_spec; unseal=> n x ??; by apply equiv_dist, timeless. Qed. Global Instance ownM_timeless (a : M) : Timeless a → TimelessP (uPred_ownM a). Proof. intro; apply timelessP_spec; unseal=> n x ??; apply cmra_included_includedN, cmra_timeless_included_l; eauto using cmra_validN_le. Qed. (* Always stable *) Global Instance const_persistent φ : Persistent (■ φ : uPred M)%I. Proof. by rewrite /Persistent always_const. Qed. Global Instance always_persistent P : Persistent (□ P). Proof. by intros; apply always_intro'. Qed. Global Instance and_persistent P Q : Persistent P → Persistent Q → Persistent (P ∧ Q). Proof. by intros; rewrite /Persistent always_and; apply and_mono. Qed. Global Instance or_persistent P Q : Persistent P → Persistent Q → Persistent (P ∨ Q). Proof. by intros; rewrite /Persistent always_or; apply or_mono. Qed. Global Instance sep_persistent P Q : Persistent P → Persistent Q → Persistent (P ★ Q). Proof. by intros; rewrite /Persistent always_sep; apply sep_mono. Qed. Global Instance forall_persistent {A} (Ψ : A → uPred M) : (∀ x, Persistent (Ψ x)) → Persistent (∀ x, Ψ x). Proof. by intros; rewrite /Persistent always_forall; apply forall_mono. Qed. Global Instance exist_persistent {A} (Ψ : A → uPred M) : (∀ x, Persistent (Ψ x)) → Persistent (∃ x, Ψ x). Proof. by intros; rewrite /Persistent always_exist; apply exist_mono. Qed. Global Instance eq_persistent {A : cofeT} (a b : A) : Persistent (a ≡ b : uPred M)%I. Proof. by intros; rewrite /Persistent always_eq. Qed. Global Instance valid_persistent {A : cmraT} (a : A) : Persistent (✓ a : uPred M)%I. Proof. by intros; rewrite /Persistent always_valid. Qed. Global Instance later_persistent P : Persistent P → Persistent (▷ P). Proof. by intros; rewrite /Persistent always_later; apply later_mono. Qed. Global Instance ownM_core_persistent (a : M) : Persistent (uPred_ownM (core a)). Proof. by rewrite /Persistent always_ownM_core. Qed. Global Instance default_persistent {A} P (Ψ : A → uPred M) (mx : option A) : Persistent P → (∀ x, Persistent (Ψ x)) → Persistent (default P mx Ψ). Proof. destruct mx; apply _. Qed. (* Derived lemmas for always stable *) Lemma always_always P `{!Persistent P} : (□ P) ⊣⊢ P. Proof. apply (anti_symm (⊢)); auto using always_elim. Qed. Lemma always_intro P Q `{!Persistent P} : P ⊢ Q → P ⊢ □ Q. Proof. rewrite -(always_always P); apply always_intro'. Qed. Lemma always_and_sep_l P Q `{!Persistent P} : (P ∧ Q) ⊣⊢ (P ★ Q). Proof. by rewrite -(always_always P) always_and_sep_l'. Qed. Lemma always_and_sep_r P Q `{!Persistent Q} : (P ∧ Q) ⊣⊢ (P ★ Q). Proof. by rewrite -(always_always Q) always_and_sep_r'. Qed. Lemma always_sep_dup P `{!Persistent P} : P ⊣⊢ (P ★ P). Proof. by rewrite -(always_always P) -always_sep_dup'. Qed. Lemma always_entails_l P Q `{!Persistent Q} : (P ⊢ Q) → P ⊢ (Q ★ P). Proof. by rewrite -(always_always Q); apply always_entails_l'. Qed. Lemma always_entails_r P Q `{!Persistent Q} : (P ⊢ Q) → P ⊢ (P ★ Q). Proof. by rewrite -(always_always Q); apply always_entails_r'. Qed. End uPred_logic. (* Hint DB for the logic *) Hint Resolve const_intro. Hint Resolve or_elim or_intro_l' or_intro_r' : I. Hint Resolve and_intro and_elim_l' and_elim_r' : I. Hint Resolve always_mono : I. Hint Resolve sep_elim_l' sep_elim_r' sep_mono : I. Hint Immediate True_intro False_elim : I. Hint Immediate iff_refl eq_refl : I. End uPred.