From iris.base_logic Require Import base_logic. From iris.proofmode Require Import tactics. Set Default Proof Using "Type*". Import uPred. (** Least and greatest fixpoint of a monotone function, defined entirely inside the logic. *) Class BIMonoPred {M} {A : ofeT} (F : (A → uPred M) → (A → uPred M)) := { bi_mono_pred Φ Ψ : ((□ ∀ x, Φ x -∗ Ψ x) → ∀ x, F Φ x -∗ F Ψ x)%I; bi_mono_pred_ne Φ : NonExpansive Φ → NonExpansive (F Φ) }. Arguments bi_mono_pred {_ _ _ _} _ _. Local Existing Instance bi_mono_pred_ne. Definition uPred_least_fixpoint {M} {A : ofeT} (F : (A → uPred M) → (A → uPred M)) (x : A) : uPred M := (∀ Φ : A -n> uPredC M, □ (∀ x, F Φ x → Φ x) → Φ x)%I. Definition uPred_greatest_fixpoint {M} {A : ofeT} (F : (A → uPred M) → (A → uPred M)) (x : A) : uPred M := (∃ Φ : A -n> uPredC M, □ (∀ x, Φ x → F Φ x) ∧ Φ x)%I. Section least. Context {M} {A : ofeT} (F : (A → uPred M) → (A → uPred M)) `{!BIMonoPred F}. Global Instance least_fixpoint_ne : NonExpansive (uPred_least_fixpoint F). Proof. solve_proper. Qed. Lemma least_fixpoint_unfold_2 x : F (uPred_least_fixpoint F) x ⊢ uPred_least_fixpoint F x. Proof. iIntros "HF" (Φ) "#Hincl". iApply "Hincl". iApply (bi_mono_pred _ Φ); last done. iIntros "!#" (y) "Hy". iApply "Hy". done. Qed. Lemma least_fixpoint_unfold_1 x : uPred_least_fixpoint F x ⊢ F (uPred_least_fixpoint F) x. Proof. iIntros "HF". iApply ("HF" \$! (CofeMor (F (uPred_least_fixpoint F))) with "[#]"). iIntros "!#" (y) "Hy". iApply bi_mono_pred; last done. iIntros "!#" (z) "?". by iApply least_fixpoint_unfold_2. Qed. Corollary least_fixpoint_unfold x : uPred_least_fixpoint F x ≡ F (uPred_least_fixpoint F) x. Proof. apply (anti_symm _); auto using least_fixpoint_unfold_1, least_fixpoint_unfold_2. Qed. Lemma least_fixpoint_ind (Φ : A → uPred M) `{!NonExpansive Φ} : □ (∀ y, F Φ y → Φ y) ⊢ ∀ x, uPred_least_fixpoint F x → Φ x. Proof. iIntros "#HΦ" (x) "HF". by iApply ("HF" \$! (CofeMor Φ) with "[#]"). Qed. End least. Section greatest. Context {M} {A : ofeT} (F : (A → uPred M) → (A → uPred M)) `{!BIMonoPred F}. Global Instance greatest_fixpoint_ne : NonExpansive (uPred_greatest_fixpoint F). Proof. solve_proper. Qed. Lemma greatest_fixpoint_unfold_1 x : uPred_greatest_fixpoint F x ⊢ F (uPred_greatest_fixpoint F) x. Proof. iDestruct 1 as (Φ) "[#Hincl HΦ]". iApply (bi_mono_pred Φ (uPred_greatest_fixpoint F)). - iIntros "!#" (y) "Hy". iExists Φ. auto. - by iApply "Hincl". Qed. Lemma greatest_fixpoint_unfold_2 x : F (uPred_greatest_fixpoint F) x ⊢ uPred_greatest_fixpoint F x. Proof. iIntros "HF". iExists (CofeMor (F (uPred_greatest_fixpoint F))). iIntros "{\$HF} !#" (y) "Hy". iApply (bi_mono_pred with "[] Hy"). iIntros "!#" (z) "?". by iApply greatest_fixpoint_unfold_1. Qed. Corollary greatest_fixpoint_unfold x : uPred_greatest_fixpoint F x ≡ F (uPred_greatest_fixpoint F) x. Proof. apply (anti_symm _); auto using greatest_fixpoint_unfold_1, greatest_fixpoint_unfold_2. Qed. Lemma greatest_fixpoint_coind (Φ : A → uPred M) `{!NonExpansive Φ} : □ (∀ y, Φ y → F Φ y) ⊢ ∀ x, Φ x → uPred_greatest_fixpoint F x. Proof. iIntros "#HΦ" (x) "Hx". iExists (CofeMor Φ). by iIntros "{\$Hx} !#". Qed. End greatest.