Library iris.bi.derived_connectives
From iris.algebra Require Import monoid.
From iris.bi Require Export interface.
From iris.prelude Require Import options.
Definition bi_iff {PROP : bi} (P Q : PROP) : PROP := (P → Q) ∧ (Q → P).
Global Arguments bi_iff {_} _%I _%I : simpl never.
Global Instance: Params (@bi_iff) 1 := {}.
Infix "↔" := bi_iff : bi_scope.
Definition bi_wand_iff {PROP : bi} (P Q : PROP) : PROP :=
(P -∗ Q) ∧ (Q -∗ P).
Global Arguments bi_wand_iff {_} _%I _%I : simpl never.
Global Instance: Params (@bi_wand_iff) 1 := {}.
Infix "∗-∗" := bi_wand_iff : bi_scope.
Notation "P ∗-∗ Q" := (⊢ P ∗-∗ Q) : stdpp_scope.
Class Persistent {PROP : bi} (P : PROP) := persistent : P ⊢ <pers> P.
Global Arguments Persistent {_} _%I : simpl never.
Global Arguments persistent {_} _%I {_}.
Global Hint Mode Persistent + ! : typeclass_instances.
Global Instance: Params (@Persistent) 1 := {}.
Definition bi_affinely {PROP : bi} (P : PROP) : PROP := emp ∧ P.
Global Arguments bi_affinely {_} _%I : simpl never.
Global Instance: Params (@bi_affinely) 1 := {}.
Global Typeclasses Opaque bi_affinely.
Notation "'<affine>' P" := (bi_affinely P) : bi_scope.
Class Affine {PROP : bi} (Q : PROP) := affine : Q ⊢ emp.
Global Arguments Affine {_} _%I : simpl never.
Global Arguments affine {_} _%I {_}.
Global Hint Mode Affine + ! : typeclass_instances.
Definition bi_absorbingly {PROP : bi} (P : PROP) : PROP := True ∗ P.
Global Arguments bi_absorbingly {_} _%I : simpl never.
Global Instance: Params (@bi_absorbingly) 1 := {}.
Global Typeclasses Opaque bi_absorbingly.
Notation "'<absorb>' P" := (bi_absorbingly P) : bi_scope.
Class Absorbing {PROP : bi} (P : PROP) := absorbing : <absorb> P ⊢ P.
Global Arguments Absorbing {_} _%I : simpl never.
Global Arguments absorbing {_} _%I.
Global Hint Mode Absorbing + ! : typeclass_instances.
Definition bi_persistently_if {PROP : bi} (p : bool) (P : PROP) : PROP :=
(if p then <pers> P else P)%I.
Global Arguments bi_persistently_if {_} !_ _%I /.
Global Instance: Params (@bi_persistently_if) 2 := {}.
Global Typeclasses Opaque bi_persistently_if.
Notation "'<pers>?' p P" := (bi_persistently_if p P) : bi_scope.
Definition bi_affinely_if {PROP : bi} (p : bool) (P : PROP) : PROP :=
(if p then <affine> P else P)%I.
Global Arguments bi_affinely_if {_} !_ _%I /.
Global Instance: Params (@bi_affinely_if) 2 := {}.
Global Typeclasses Opaque bi_affinely_if.
Notation "'<affine>?' p P" := (bi_affinely_if p P) : bi_scope.
Definition bi_absorbingly_if {PROP : bi} (p : bool) (P : PROP) : PROP :=
(if p then <absorb> P else P)%I.
Global Arguments bi_absorbingly_if {_} !_ _%I /.
Global Instance: Params (@bi_absorbingly_if) 2 := {}.
Global Typeclasses Opaque bi_absorbingly_if.
Notation "'<absorb>?' p P" := (bi_absorbingly_if p P) : bi_scope.
Definition bi_intuitionistically {PROP : bi} (P : PROP) : PROP :=
(<affine> <pers> P)%I.
Global Arguments bi_intuitionistically {_} _%I : simpl never.
Global Instance: Params (@bi_intuitionistically) 1 := {}.
Global Typeclasses Opaque bi_intuitionistically.
Notation "□ P" := (bi_intuitionistically P) : bi_scope.
Definition bi_intuitionistically_if {PROP : bi} (p : bool) (P : PROP) : PROP :=
(if p then □ P else P)%I.
Global Arguments bi_intuitionistically_if {_} !_ _%I /.
Global Instance: Params (@bi_intuitionistically_if) 2 := {}.
Global Typeclasses Opaque bi_intuitionistically_if.
Notation "'□?' p P" := (bi_intuitionistically_if p P) : bi_scope.
Fixpoint bi_laterN {PROP : bi} (n : nat) (P : PROP) : PROP :=
match n with
| O ⇒ P
| S n' ⇒ ▷ ▷^n' P
end
where "▷^ n P" := (bi_laterN n P) : bi_scope.
Global Arguments bi_laterN {_} !_%nat_scope _%I.
Global Instance: Params (@bi_laterN) 2 := {}.
Notation "▷? p P" := (bi_laterN (Nat.b2n p) P) : bi_scope.
Definition bi_except_0 {PROP : bi} (P : PROP) : PROP := ▷ False ∨ P.
Global Arguments bi_except_0 {_} _%I : simpl never.
Notation "◇ P" := (bi_except_0 P) : bi_scope.
Global Instance: Params (@bi_except_0) 1 := {}.
Global Typeclasses Opaque bi_except_0.
Class Timeless {PROP : bi} (P : PROP) := timeless : ▷ P ⊢ ◇ P.
Global Arguments Timeless {_} _%I : simpl never.
Global Arguments timeless {_} _%I {_}.
Global Hint Mode Timeless + ! : typeclass_instances.
Global Instance: Params (@Timeless) 1 := {}.
From iris.bi Require Export interface.
From iris.prelude Require Import options.
Definition bi_iff {PROP : bi} (P Q : PROP) : PROP := (P → Q) ∧ (Q → P).
Global Arguments bi_iff {_} _%I _%I : simpl never.
Global Instance: Params (@bi_iff) 1 := {}.
Infix "↔" := bi_iff : bi_scope.
Definition bi_wand_iff {PROP : bi} (P Q : PROP) : PROP :=
(P -∗ Q) ∧ (Q -∗ P).
Global Arguments bi_wand_iff {_} _%I _%I : simpl never.
Global Instance: Params (@bi_wand_iff) 1 := {}.
Infix "∗-∗" := bi_wand_iff : bi_scope.
Notation "P ∗-∗ Q" := (⊢ P ∗-∗ Q) : stdpp_scope.
Class Persistent {PROP : bi} (P : PROP) := persistent : P ⊢ <pers> P.
Global Arguments Persistent {_} _%I : simpl never.
Global Arguments persistent {_} _%I {_}.
Global Hint Mode Persistent + ! : typeclass_instances.
Global Instance: Params (@Persistent) 1 := {}.
Definition bi_affinely {PROP : bi} (P : PROP) : PROP := emp ∧ P.
Global Arguments bi_affinely {_} _%I : simpl never.
Global Instance: Params (@bi_affinely) 1 := {}.
Global Typeclasses Opaque bi_affinely.
Notation "'<affine>' P" := (bi_affinely P) : bi_scope.
Class Affine {PROP : bi} (Q : PROP) := affine : Q ⊢ emp.
Global Arguments Affine {_} _%I : simpl never.
Global Arguments affine {_} _%I {_}.
Global Hint Mode Affine + ! : typeclass_instances.
Definition bi_absorbingly {PROP : bi} (P : PROP) : PROP := True ∗ P.
Global Arguments bi_absorbingly {_} _%I : simpl never.
Global Instance: Params (@bi_absorbingly) 1 := {}.
Global Typeclasses Opaque bi_absorbingly.
Notation "'<absorb>' P" := (bi_absorbingly P) : bi_scope.
Class Absorbing {PROP : bi} (P : PROP) := absorbing : <absorb> P ⊢ P.
Global Arguments Absorbing {_} _%I : simpl never.
Global Arguments absorbing {_} _%I.
Global Hint Mode Absorbing + ! : typeclass_instances.
Definition bi_persistently_if {PROP : bi} (p : bool) (P : PROP) : PROP :=
(if p then <pers> P else P)%I.
Global Arguments bi_persistently_if {_} !_ _%I /.
Global Instance: Params (@bi_persistently_if) 2 := {}.
Global Typeclasses Opaque bi_persistently_if.
Notation "'<pers>?' p P" := (bi_persistently_if p P) : bi_scope.
Definition bi_affinely_if {PROP : bi} (p : bool) (P : PROP) : PROP :=
(if p then <affine> P else P)%I.
Global Arguments bi_affinely_if {_} !_ _%I /.
Global Instance: Params (@bi_affinely_if) 2 := {}.
Global Typeclasses Opaque bi_affinely_if.
Notation "'<affine>?' p P" := (bi_affinely_if p P) : bi_scope.
Definition bi_absorbingly_if {PROP : bi} (p : bool) (P : PROP) : PROP :=
(if p then <absorb> P else P)%I.
Global Arguments bi_absorbingly_if {_} !_ _%I /.
Global Instance: Params (@bi_absorbingly_if) 2 := {}.
Global Typeclasses Opaque bi_absorbingly_if.
Notation "'<absorb>?' p P" := (bi_absorbingly_if p P) : bi_scope.
Definition bi_intuitionistically {PROP : bi} (P : PROP) : PROP :=
(<affine> <pers> P)%I.
Global Arguments bi_intuitionistically {_} _%I : simpl never.
Global Instance: Params (@bi_intuitionistically) 1 := {}.
Global Typeclasses Opaque bi_intuitionistically.
Notation "□ P" := (bi_intuitionistically P) : bi_scope.
Definition bi_intuitionistically_if {PROP : bi} (p : bool) (P : PROP) : PROP :=
(if p then □ P else P)%I.
Global Arguments bi_intuitionistically_if {_} !_ _%I /.
Global Instance: Params (@bi_intuitionistically_if) 2 := {}.
Global Typeclasses Opaque bi_intuitionistically_if.
Notation "'□?' p P" := (bi_intuitionistically_if p P) : bi_scope.
Fixpoint bi_laterN {PROP : bi} (n : nat) (P : PROP) : PROP :=
match n with
| O ⇒ P
| S n' ⇒ ▷ ▷^n' P
end
where "▷^ n P" := (bi_laterN n P) : bi_scope.
Global Arguments bi_laterN {_} !_%nat_scope _%I.
Global Instance: Params (@bi_laterN) 2 := {}.
Notation "▷? p P" := (bi_laterN (Nat.b2n p) P) : bi_scope.
Definition bi_except_0 {PROP : bi} (P : PROP) : PROP := ▷ False ∨ P.
Global Arguments bi_except_0 {_} _%I : simpl never.
Notation "◇ P" := (bi_except_0 P) : bi_scope.
Global Instance: Params (@bi_except_0) 1 := {}.
Global Typeclasses Opaque bi_except_0.
Class Timeless {PROP : bi} (P : PROP) := timeless : ▷ P ⊢ ◇ P.
Global Arguments Timeless {_} _%I : simpl never.
Global Arguments timeless {_} _%I {_}.
Global Hint Mode Timeless + ! : typeclass_instances.
Global Instance: Params (@Timeless) 1 := {}.
An optional precondition mP to Q.
TODO: We may actually consider generalizing this to a list of preconditions,
and e.g. also using it for texan triples.