Commit 2bc4656d by Ralf Jung

### also do least fixpoint; fix naming

parent 90ef23f7
 ... ... @@ -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. ... ...
 ... ... @@ -3,41 +3,74 @@ From iris.proofmode Require Import tactics. Set Default Proof Using "Type*". Import uPred. (** Greatest fixpoint of a monotone function, defined entirely inside the logic. TODO: Also do least fixpoint. *) (** 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 iGFix {M A} (F : (A → uPred M) → (A → uPred M)) (x : A) : uPred M := 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 iGFix. 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). iApply wand_impl_always. iIntros "#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) (x : A) : uPred_least_fixpoint F x -∗ □ (∀ y, F P y -∗ P y) -∗ P x. Proof. iIntros "HF #HP". 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 iGFix_implies_F_iGFix x : iGFix F x ⊢ F (iGFix F) x. 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 (iGFix F)). iApply (Hmono P (uPred_greatest_fixpoint F)). - iAlways. iIntros (y) "Hy". iExists P. by iSplit. - by iApply "Hincl". Qed. Lemma F_iGFix_implies_iGFix x : F (iGFix F) x ⊢ iGFix F x. Lemma F_fix_implies_greatest_fixpoint x : F (uPred_greatest_fixpoint F) x ⊢ uPred_greatest_fixpoint F x. Proof. iIntros "HF". iExists (F (iGFix F)). iIntros "HF". iExists (F (uPred_greatest_fixpoint F)). iIntros "{\$HF} !#"; iIntros (y) "Hy". iApply (Hmono with "[] Hy"). iAlways. iIntros (z). by iApply iGFix_implies_F_iGFix. iAlways. iIntros (z). by iApply greatest_fixpoint_implies_F_fix. Qed. Corollary iGFix_unfold x : iGFix F x ≡ F (iGFix F) x. Corollary uPred_greatest_fixpoint_unfold x : uPred_greatest_fixpoint F x ≡ F (uPred_greatest_fixpoint F) x. Proof. apply (anti_symm _); auto using iGFix_implies_F_iGFix, F_iGFix_implies_iGFix. apply (anti_symm _); auto using greatest_fixpoint_implies_F_fix, F_fix_implies_greatest_fixpoint. Qed. Lemma Fix_coind (P : A → uPred M) (x : A) : □ (∀ y, P y -∗ F P y) -∗ P x -∗ iGFix F x. Lemma uPred_greatest_fixpoint_coind (P : A → uPred M) (x : A) : □ (∀ y, P y -∗ F P y) -∗ P x -∗ uPred_greatest_fixpoint F x. Proof. iIntros "#HP Hx". iExists P. by iIntros "{\$Hx} !#". Qed. End iGFix. End greatest.
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