From algebra Require Export upred. From program_logic Require Export resources. From algebra Require Import cofe_solver. (* The Iris program logic is parametrized by a locally contractive functor from the category of COFEs to the category of CMRAs, which is instantiated with [iProp]. The [iProp] can be used to construct impredicate CMRAs, such as the stored propositions using the agreement CMRA. *) Structure iFunctor := IFunctor { iFunctor_F :> rFunctor; iFunctor_contractive : rFunctorContractive iFunctor_F; iFunctor_empty (A : cofeT) : Empty (iFunctor_F A); iFunctor_identity (A : cofeT) : CMRAIdentity (iFunctor_F A); }. Arguments IFunctor _ {_ _ _}. Existing Instances iFunctor_contractive iFunctor_empty iFunctor_identity. Module Type iProp_solution_sig. Parameter iPreProp : language → iFunctor → cofeT. Definition iGst (Λ : language) (Σ : iFunctor) : cmraT := Σ (iPreProp Λ Σ). Definition iRes Λ Σ := res Λ (laterC (iPreProp Λ Σ)) (iGst Λ Σ). Definition iResR Λ Σ := resR Λ (laterC (iPreProp Λ Σ)) (iGst Λ Σ). Definition iWld Λ Σ := gmap positive (agree (laterC (iPreProp Λ Σ))). Definition iPst Λ := excl (state Λ). Definition iProp (Λ : language) (Σ : iFunctor) : cofeT := uPredC (iResR Λ Σ). Parameter iProp_unfold: ∀ {Λ Σ}, iProp Λ Σ -n> iPreProp Λ Σ. Parameter iProp_fold: ∀ {Λ Σ}, iPreProp Λ Σ -n> iProp Λ Σ. Parameter iProp_fold_unfold: ∀ {Λ Σ} (P : iProp Λ Σ), iProp_fold (iProp_unfold P) ≡ P. Parameter iProp_unfold_fold: ∀ {Λ Σ} (P : iPreProp Λ Σ), iProp_unfold (iProp_fold P) ≡ P. End iProp_solution_sig. Module Export iProp_solution : iProp_solution_sig. Definition iProp_result (Λ : language) (Σ : iFunctor) : solution (uPredCF (resRF Λ (▶ ∙) Σ)) := solver.result _. Definition iPreProp (Λ : language) (Σ : iFunctor) : cofeT := iProp_result Λ Σ. Definition iGst (Λ : language) (Σ : iFunctor) : cmraT := Σ (iPreProp Λ Σ). Definition iRes Λ Σ := res Λ (laterC (iPreProp Λ Σ)) (iGst Λ Σ). Definition iResR Λ Σ := resR Λ (laterC (iPreProp Λ Σ)) (iGst Λ Σ). Definition iWld Λ Σ := gmap positive (agree (laterC (iPreProp Λ Σ))). Definition iPst Λ := excl (state Λ). Definition iProp (Λ : language) (Σ : iFunctor) : cofeT := uPredC (iResR Λ Σ). Definition iProp_unfold {Λ Σ} : iProp Λ Σ -n> iPreProp Λ Σ := solution_fold (iProp_result Λ Σ). Definition iProp_fold {Λ Σ} : iPreProp Λ Σ -n> iProp Λ Σ := solution_unfold _. Lemma iProp_fold_unfold {Λ Σ} (P : iProp Λ Σ) : iProp_fold (iProp_unfold P) ≡ P. Proof. apply solution_unfold_fold. Qed. Lemma iProp_unfold_fold {Λ Σ} (P : iPreProp Λ Σ) : iProp_unfold (iProp_fold P) ≡ P. Proof. apply solution_fold_unfold. Qed. End iProp_solution. Bind Scope uPred_scope with iProp. Instance iProp_fold_inj n Λ Σ : Inj (dist n) (dist n) (@iProp_fold Λ Σ). Proof. by intros X Y H; rewrite -(iProp_unfold_fold X) H iProp_unfold_fold. Qed. Instance iProp_unfold_inj n Λ Σ : Inj (dist n) (dist n) (@iProp_unfold Λ Σ). Proof. by intros X Y H; rewrite -(iProp_fold_unfold X) H iProp_fold_unfold. Qed.