From iris.bi Require Export bi. From iris.proofmode Require Import tactics. Set Default Proof Using "Type*". Import bi. (** Least and greatest fixpoint of a monotone function, defined entirely inside the logic. *) Class BIMonoPred {PROP : bi} {A : ofeT} (F : (A → PROP) → (A → PROP)) := { bi_mono_pred Φ Ψ : ((bi_persistently (∀ 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 bi_least_fixpoint {PROP : bi} {A : ofeT} (F : (A → PROP) → (A → PROP)) (x : A) : PROP := (∀ Φ : A -n> PROP, bi_persistently (∀ x, F Φ x -∗ Φ x) → Φ x)%I. Definition bi_greatest_fixpoint {PROP : bi} {A : ofeT} (F : (A → PROP) → (A → PROP)) (x : A) : PROP := (∃ Φ : A -n> PROP, bi_persistently (∀ x, Φ x -∗ F Φ x) ∧ Φ x)%I. Section least. Context {PROP : bi} {A : ofeT} (F : (A → PROP) → (A → PROP)) `{!BIMonoPred F}. Global Instance least_fixpoint_ne : NonExpansive (bi_least_fixpoint F). Proof. solve_proper. Qed. Lemma least_fixpoint_unfold_2 x : F (bi_least_fixpoint F) x ⊢ bi_least_fixpoint F x. Proof. iIntros "HF" (Φ) "#Hincl". iApply "Hincl". iApply (bi_mono_pred _ Φ with "[#]"); last done. iIntros "!#" (y) "Hy". iApply ("Hy" with "[# //]"). Qed. Lemma least_fixpoint_unfold_1 x : bi_least_fixpoint F x ⊢ F (bi_least_fixpoint F) x. Proof. iIntros "HF". iApply ("HF" \$! (CofeMor (F (bi_least_fixpoint F))) with "[#]"). iIntros "!#" (y) "Hy". iApply (bi_mono_pred with "[#]"); last done. iIntros "!#" (z) "?". by iApply least_fixpoint_unfold_2. Qed. Corollary least_fixpoint_unfold x : bi_least_fixpoint F x ≡ F (bi_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 → PROP) `{!NonExpansive Φ} : □ (∀ y, F Φ y -∗ Φ y) -∗ ∀ x, bi_least_fixpoint F x -∗ Φ x. Proof. iIntros "#HΦ" (x) "HF". by iApply ("HF" \$! (CofeMor Φ) with "[#]"). Qed. Lemma least_fixpoint_strong_ind (Φ : A → PROP) `{!NonExpansive Φ} : □ (∀ y, F (λ x, Φ x ∧ bi_least_fixpoint F x) y -∗ Φ y) -∗ ∀ x, bi_least_fixpoint F x -∗ Φ x. Proof. trans (∀ x, bi_least_fixpoint F x -∗ Φ x ∧ bi_least_fixpoint F x)%I. { iIntros "#HΦ". iApply (least_fixpoint_ind with "[]"); first solve_proper. iIntros "!#" (y) "H". iSplit; first by iApply "HΦ". iApply least_fixpoint_unfold_2. iApply (bi_mono_pred with "[#] H"). by iIntros "!# * [_ ?]". } by setoid_rewrite and_elim_l. Qed. End least. Section greatest. Context {PROP : bi} {A : ofeT} (F : (A → PROP) → (A → PROP)) `{!BIMonoPred F}. Global Instance greatest_fixpoint_ne : NonExpansive (bi_greatest_fixpoint F). Proof. solve_proper. Qed. Lemma greatest_fixpoint_unfold_1 x : bi_greatest_fixpoint F x ⊢ F (bi_greatest_fixpoint F) x. Proof. iDestruct 1 as (Φ) "[#Hincl HΦ]". iApply (bi_mono_pred Φ (bi_greatest_fixpoint F) with "[#]"). - iIntros "!#" (y) "Hy". iExists Φ. auto. - by iApply "Hincl". Qed. Lemma greatest_fixpoint_unfold_2 x : F (bi_greatest_fixpoint F) x ⊢ bi_greatest_fixpoint F x. Proof. iIntros "HF". iExists (CofeMor (F (bi_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 : bi_greatest_fixpoint F x ≡ F (bi_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 → PROP) `{!NonExpansive Φ} : □ (∀ y, Φ y -∗ F Φ y) -∗ ∀ x, Φ x -∗ bi_greatest_fixpoint F x. Proof. iIntros "#HΦ" (x) "Hx". iExists (CofeMor Φ). auto. Qed. End greatest.