More consistent naming for fixpoints.

- Use Φ and Ψ for predicates.
- Use _1 and _2 suffixes for the different directions of a lemma.
- Not all lemmas started with _uPred; we do not let the bigop lemmas (for instance)
  start with uPred_ either, so I just got rid of the prefix.

theories/base_logic/fix.v

@@ -7,70 +7,73 @@ Import uPred.
     the logic.  *)
 
 Definition uPred_mono_pred {M A} (F : (A → uPred M) → (A → uPred M)) :=
-  ∀ P Q, ((□ ∀ x, P x → Q x) → ∀ x, F P x → F Q x)%I.
+  ∀ Φ Ψ, ((□ ∀ x, Φ x → Ψ x) → ∀ x, F Φ x → F Ψ x)%I.
 
-Definition uPred_least_fixpoint {M A} (F : (A → uPred M) → (A → uPred M)) (x : A) : uPred M :=
-  (∀ P, □ (∀ x, F P x → P x) → P x)%I.
+Definition uPred_least_fixpoint {M A} (F : (A → uPred M) → (A → uPred M))
+    (x : A) : uPred M :=
+  (∀ Φ, □ (∀ x, F Φ x → Φ x) → Φ x)%I.
 
-Definition uPred_greatest_fixpoint {M A} (F : (A → uPred M) → (A → uPred M)) (x : A) : uPred M :=
-  (∃ P, □ (∀ x, P x → F P x) ∧ P x)%I.
+Definition uPred_greatest_fixpoint {M A} (F : (A → uPred M) → (A → uPred M))
+    (x : A) : uPred M :=
+  (∃ Φ, □ (∀ x, Φ x → F Φ x) ∧ Φ x)%I.
 
 Section least.
-  Context {M : ucmraT} {A} (F : (A → uPred M) → (A → uPred M)) (Hmono : uPred_mono_pred F).
+  Context {M : ucmraT}.
+  Context {A} (F : (A → uPred M) → (A → uPred M)) (Hmono : uPred_mono_pred F).
 
-  Lemma F_fix_implies_least_fixpoint x : F (uPred_least_fixpoint F) x ⊢ uPred_least_fixpoint F x.
+  Lemma least_fixpoint_unfold_2 x : F (uPred_least_fixpoint F) x ⊢ uPred_least_fixpoint F x.
   Proof.
-    iIntros "HF" (P) "#Hincl".
-    iApply "Hincl". iApply (Hmono _ P); last done.
+    iIntros "HF" (Φ) "#Hincl".
+    iApply "Hincl". iApply (Hmono _ Φ); last done.
     iIntros "!#" (y) "Hy". iApply "Hy". done.
   Qed.
 
-  Lemma least_fixpoint_implies_F_fix x :
+  Lemma least_fixpoint_unfold_1 x :
     uPred_least_fixpoint F x ⊢ F (uPred_least_fixpoint F) x.
   Proof.
     iIntros "HF". iApply "HF". iIntros "!#" (y) "Hy".
     iApply Hmono; last done. iIntros "!#" (z) "?".
-    by iApply F_fix_implies_least_fixpoint.
+    by iApply least_fixpoint_unfold_2.
   Qed.
 
-  Corollary uPred_least_fixpoint_unfold x :
+  Corollary least_fixpoint_unfold x :
     uPred_least_fixpoint F x ≡ F (uPred_least_fixpoint F) x.
   Proof.
-    apply (anti_symm _); auto using least_fixpoint_implies_F_fix, F_fix_implies_least_fixpoint.
+    apply (anti_symm _); auto using least_fixpoint_unfold_1, least_fixpoint_unfold_2.
   Qed.
 
-  Lemma uPred_least_fixpoint_ind (P : A → uPred M) :
-    □ (∀ y, F P y → P y) ⊢ ∀ x, uPred_least_fixpoint F x → P x.
-  Proof. iIntros "#HP" (x) "HF". iApply "HF". done. Qed.
+  Lemma least_fixpoint_ind (Φ : A → uPred M) :
+    □ (∀ y, F Φ y → Φ y) ⊢ ∀ x, uPred_least_fixpoint F x → Φ x.
+  Proof. iIntros "#HΦ" (x) "HF". iApply "HF". done. Qed.
 End least.
 
 Section greatest.
   Context {M : ucmraT} {A} (F : (A → uPred M) → (A → uPred M)) (Hmono : uPred_mono_pred F).
 
-  Lemma greatest_fixpoint_implies_F_fix x :
+  Lemma greatest_fixpoint_unfold_1 x :
     uPred_greatest_fixpoint F x ⊢ F (uPred_greatest_fixpoint F) x.
   Proof.
-    iDestruct 1 as (P) "[#Hincl HP]".
-    iApply (Hmono P (uPred_greatest_fixpoint F)).
-    - iAlways. iIntros (y) "Hy". iExists P. by iSplit.
+    iDestruct 1 as (Φ) "[#Hincl HΦ]".
+    iApply (Hmono Φ (uPred_greatest_fixpoint F)).
+    - iIntros "!#" (y) "Hy". iExists Φ. auto.
     - by iApply "Hincl".
   Qed.
 
-  Lemma F_fix_implies_greatest_fixpoint x :
+  Lemma greatest_fixpoint_unfold_2 x :
     F (uPred_greatest_fixpoint F) x ⊢ uPred_greatest_fixpoint F x.
   Proof.
     iIntros "HF". iExists (F (uPred_greatest_fixpoint F)).
-    iIntros "{$HF} !#"; iIntros (y) "Hy". iApply (Hmono with "[] Hy").
-    iAlways. iIntros (z) "?". by iApply greatest_fixpoint_implies_F_fix.
+    iIntros "{$HF} !#" (y) "Hy". iApply (Hmono with "[] Hy").
+    iIntros "!#" (z) "?". by iApply greatest_fixpoint_unfold_1.
   Qed.
 
-  Corollary uPred_greatest_fixpoint_unfold x :
+  Corollary greatest_fixpoint_unfold x :
     uPred_greatest_fixpoint F x ≡ F (uPred_greatest_fixpoint F) x.
   Proof.
-    apply (anti_symm _); auto using greatest_fixpoint_implies_F_fix, F_fix_implies_greatest_fixpoint.
+    apply (anti_symm _); auto using greatest_fixpoint_unfold_1, greatest_fixpoint_unfold_2.
   Qed.
 
-  Lemma uPred_greatest_fixpoint_coind (P : A → uPred M) :
-    □ (∀ y, P y → F P y) ⊢ ∀ x, P x → uPred_greatest_fixpoint F x.
-  Proof. iIntros "#HP" (x) "Hx". iExists P. by iIntros "{$Hx} !#". Qed.
+  Lemma greatest_fixpoint_coind (Φ : A → uPred M) :
+    □ (∀ y, Φ y → F Φ y) ⊢ ∀ x, Φ x → uPred_greatest_fixpoint F x.
+  Proof. iIntros "#HΦ" (x) "Hx". iExists Φ. by iIntros "{$Hx} !#". Qed.
 End greatest.