Commit e128f6fb authored by Ralf Jung's avatar Ralf Jung

Merge branch 'ralf/greatest-fix' into 'master'

Implement greatest fixed point inside the logic

See merge request !60
parents ea42f994 33c3788f
......@@ -34,6 +34,7 @@ theories/base_logic/hlist.v
theories/base_logic/soundness.v
theories/base_logic/double_negation.v
theories/base_logic/deprecated.v
theories/base_logic/fix.v
theories/base_logic/lib/iprop.v
theories/base_logic/lib/own.v
theories/base_logic/lib/saved_prop.v
......
......@@ -541,6 +541,11 @@ Proof.
apply always_intro', impl_intro_r.
by rewrite always_and_sep_l' always_elim wand_elim_l.
Qed.
Lemma wand_impl_always P Q : (( P) - Q) (( P) Q).
Proof.
apply (anti_symm ()); [|by rewrite -impl_wand].
apply impl_intro_l. by rewrite always_and_sep_l' wand_elim_r.
Qed.
Lemma always_entails_l' P Q : (P Q) P Q P.
Proof. intros; rewrite -always_and_sep_l'; auto. Qed.
Lemma always_entails_r' P Q : (P Q) P P Q.
......
From iris.base_logic Require Import base_logic.
From iris.proofmode Require Import tactics.
Set Default Proof Using "Type*".
Import uPred.
(** Least and greatest fixpoint of a monotone function, defined entirely inside
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.
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_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.
Section least.
Context {M : ucmraT} {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.
Proof.
iIntros "HF" (P) "#Hincl".
iApply "Hincl". iApply (Hmono _ P); last done.
iIntros "!#" (y) "Hy". iApply "Hy". done.
Qed.
Lemma least_fixpoint_implies_F_fix 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.
Qed.
Corollary uPred_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.
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.
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 :
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.
- by iApply "Hincl".
Qed.
Lemma F_fix_implies_greatest_fixpoint x :
F (uPred_greatest_fixpoint F) x uPred_greatest_fixpoint F x.
Proof.
iIntros "HF". iExists (F (uPred_greatest_fixpoint F)).
iIntros "{$HF} !#"; iIntros (y) "Hy". iApply (Hmono with "[] Hy").
iAlways. iIntros (z) "?". by iApply greatest_fixpoint_implies_F_fix.
Qed.
Corollary uPred_greatest_fixpoint_unfold x :
uPred_greatest_fixpoint F x F (uPred_greatest_fixpoint F) x.
Proof.
apply (anti_symm _); auto using greatest_fixpoint_implies_F_fix, F_fix_implies_greatest_fixpoint.
Qed.
Lemma uPred_greatest_fixpoint_coind (P : A uPred M) :
( y, P y F P y) x, P x uPred_greatest_fixpoint F x.
Proof. iIntros "#HP" (x) "Hx". iExists P. by iIntros "{$Hx} !#". Qed.
End greatest.
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