Commit d18747b6 authored by Robbert Krebbers's avatar Robbert Krebbers Committed by Jacques-Henri Jourdan

Simplify box axioms.

Thanks to discussions with Ales and Amin.
parent 52c3006d
......@@ -451,21 +451,13 @@ Proof.
by unseal.
- (* P ⊢ □ emp (ADMISSIBLE) *)
intros P. unfold uPred_emp; unseal; by split=> n x ? _.
- (* emp ∧ □ P ⊢ P *)
intros P. unseal; split=> n x ? [_ ?]; simpl in *.
eauto using uPred_mono, @cmra_included_core, cmra_included_includedN.
- (* □ P ∗ Q ⊢ □ P (ADMISSIBLE) *)
intros P Q. move: (uPred_persistently P)=> P'.
unseal; split; intros n x ? (x1&x2&?&?&_); ofe_subst;
eauto using uPred_mono, cmra_includedN_l.
- (* □ P ∧ (Q ∗ R) ⊢ (□ P ∧ Q) ∗ R (ADMISSIBLE) *)
intros P Q R. unseal; split; intros n x ? [? (x1&x2&Hx&?&?)]; simpl in *.
exists (core (x1 x2) x1), x2. split_and!.
+ by rewrite -assoc cmra_core_l.
+ eapply uPred_mono; first done. rewrite -{1}cmra_core_idemp Hx.
eapply cmra_core_monoN, cmra_includedN_l.
+ eauto using uPred_mono, cmra_includedN_r.
+ done.
- (* □ P ∧ Q ⊢ (emp ∧ P) ∗ Q *)
intros P Q. unseal; split=> n x ? [??]; simpl in *.
exists (core x), x; rewrite ?cmra_core_l; auto.
Qed.
Lemma uPred_sbi_mixin (M : ucmraT) : SBIMixin
......
......@@ -872,6 +872,15 @@ Proof. intros P Q; apply persistently_mono. Qed.
Global Instance persistently_absorbing P : Absorbing ( P).
Proof. rewrite /Absorbing=> R. apply persistently_absorbing. Qed.
Lemma persistently_and_sep_assoc_1 P Q R : P (Q R) ( P Q) R.
Proof.
rewrite {1}persistently_idemp_2 persistently_and_sep_elim assoc.
apply sep_mono_l, and_intro.
- by rewrite and_elim_r absorbing.
- by rewrite and_elim_l left_id.
Qed.
Lemma persistently_and_emp_elim P : emp P P.
Proof. by rewrite comm persistently_and_sep_elim right_id and_elim_r. Qed.
Lemma persistently_elim P : P P True.
Proof.
rewrite -(right_id True%I _ ( _)%I) -{1}(left_id emp%I _ True%I).
......
......@@ -111,10 +111,8 @@ Section bi_mixin.
( a, Ψ a) a, Ψ a;
bi_mixin_persistently_emp_intro P : P emp;
bi_mixin_persistently_and_emp_elim P : emp P P;
bi_mixin_persistently_absorbing P Q : P Q P;
bi_mixin_persistently_and_sep_assoc_1 P Q R : P (Q R) ( P Q) R;
bi_mixin_persistently_and_sep_elim P Q : P Q (emp P) Q;
}.
Record SBIMixin := {
......@@ -420,12 +418,10 @@ Proof. eapply bi_mixin_persistently_exist_1, bi_bi_mixin. Qed.
Lemma persistently_emp_intro P : P emp.
Proof. eapply bi_mixin_persistently_emp_intro, bi_bi_mixin. Qed.
Lemma persistently_and_emp_elim P : emp P P.
Proof. eapply bi_mixin_persistently_and_emp_elim, bi_bi_mixin. Qed.
Lemma persistently_absorbing P Q : P Q P.
Proof. eapply (bi_mixin_persistently_absorbing bi_entails), bi_bi_mixin. Qed.
Lemma persistently_and_sep_assoc_1 P Q R : P (Q R) ( P Q) R.
Proof. eapply bi_mixin_persistently_and_sep_assoc_1, bi_bi_mixin. Qed.
Lemma persistently_and_sep_elim P Q : P Q (emp P) Q.
Proof. eapply bi_mixin_persistently_and_sep_elim, bi_bi_mixin. Qed.
End bi_laws.
Section sbi_laws.
......
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