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 : ( (∀ 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 := tc_opaque (∀ Φ : A -n> PROP, (∀ x, F Φ x -∗ Φ x) → Φ x)%I. Arguments bi_least_fixpoint : simpl never. Definition bi_greatest_fixpoint {PROP : bi} {A : ofeT} (F : (A → PROP) → (A → PROP)) (x : A) : PROP := tc_opaque (∃ Φ : A -n> PROP, (∀ x, Φ x -∗ F Φ x) ∧ Φ x)%I. Arguments bi_greatest_fixpoint : simpl never. Global Instance least_fixpoint_ne {PROP : bi} {A : ofeT} n : Proper (pointwise_relation (A → PROP) (pointwise_relation A (dist n)) ==> dist n ==> dist n) bi_least_fixpoint. Proof. solve_proper. Qed. Global Instance least_fixpoint_proper {PROP : bi} {A : ofeT} : Proper (pointwise_relation (A → PROP) (pointwise_relation A (≡)) ==> (≡) ==> (≡)) bi_least_fixpoint. Proof. solve_proper. Qed. Section least. Context {PROP : bi} {A : ofeT} (F : (A → PROP) → (A → PROP)) `{!BiMonoPred F}. Lemma least_fixpoint_unfold_2 x : F (bi_least_fixpoint F) x ⊢ bi_least_fixpoint F x. Proof. rewrite /bi_least_fixpoint /=. 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" \$! (OfeMor (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" \$! (OfeMor 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. Lemma greatest_fixpoint_ne_outer {PROP : bi} {A : ofeT} (F1 : (A → PROP) → (A → PROP)) (F2 : (A → PROP) → (A → PROP)): (∀ Φ x n, F1 Φ x ≡{n}≡ F2 Φ x) → ∀ x1 x2 n, x1 ≡{n}≡ x2 → bi_greatest_fixpoint F1 x1 ≡{n}≡ bi_greatest_fixpoint F2 x2. Proof. intros HF x1 x2 n Hx. rewrite /bi_greatest_fixpoint /=. do 3 f_equiv; last solve_proper. repeat f_equiv. apply HF. Qed. Global Instance greatest_fixpoint_ne {PROP : bi} {A : ofeT} n : Proper (pointwise_relation (A → PROP) (pointwise_relation A (dist n)) ==> dist n ==> dist n) bi_greatest_fixpoint. Proof. solve_proper. Qed. Global Instance greatest_fixpoint_proper {PROP : bi} {A : ofeT} : Proper (pointwise_relation (A → PROP) (pointwise_relation A (≡)) ==> (≡) ==> (≡)) bi_greatest_fixpoint. Proof. solve_proper. Qed. Section greatest. Context {PROP : bi} {A : ofeT} (F : (A → PROP) → (A → PROP)) `{!BiMonoPred F}. 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 (OfeMor (F (bi_greatest_fixpoint F))). iSplit; last done. iIntros "!#" (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 (OfeMor Φ). auto. Qed. End greatest.