Commit b60e126a authored by Robbert Krebbers's avatar Robbert Krebbers

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.
parent b59ddcd1
......@@ -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.
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