Port fixpoints to BIs.

_CoqProject

@@ -31,13 +31,13 @@ theories/bi/bi.v
 theories/bi/tactics.v
 theories/bi/fractional.v
 theories/bi/counter_examples.v
+theories/bi/fixpoint.v
 theories/base_logic/upred.v
 theories/base_logic/derived.v
 theories/base_logic/base_logic.v
 theories/base_logic/soundness.v
 theories/base_logic/double_negation.v
 theories/base_logic/deprecated.v
-theories/base_logic/fixpoint.v
 theories/base_logic/proofmode.v
 theories/base_logic/proofmode_classes.v
 theories/base_logic/lib/iprop.v

theories/bi/fixpoint.v

+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 :=
+  (∀ Φ : A -n> PROP, □ (∀ 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, □ (∀ 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.