classes.v 23.6 KB
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From iris.bi Require Export bi.
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From stdpp Require Import namespaces.
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Set Default Proof Using "Type".
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Import bi.

Class FromAssumption {PROP : bi} (p : bool) (P Q : PROP) :=
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  from_assumption : ?p P  Q.
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Arguments FromAssumption {_} _ _%I _%I : simpl never.
Arguments from_assumption {_} _ _%I _%I {_}.
(* No need to restrict Hint Mode, we have a default instance that will always
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be used in case of evars *)
Hint Mode FromAssumption + + - - : typeclass_instances.
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Class IntoPure {PROP : bi} (P : PROP) (φ : Prop) :=
  into_pure : P  ⌜φ⌝.
Arguments IntoPure {_} _%I _%type_scope : simpl never.
Arguments into_pure {_} _%I _%type_scope {_}.
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Hint Mode IntoPure + ! - : typeclass_instances.

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(* [IntoPureT] is a variant of [IntoPure] with the argument in [Type] to avoid
some shortcoming of unification in Coq's type class search. An example where we
use this workaround is to repair the following instance:

  Global Instance into_exist_and_pure P Q (φ : Prop) :
    IntoPure P φ → IntoExist (P ∧ Q) (λ _ : φ, Q).

Coq is unable to use this instance: [class_apply] -- which is used by type class
search -- fails with the error that it cannot unify [Prop] and [Type]. This is
probably caused because [class_apply] uses an ancient unification algorith. The
[refine] tactic -- which uses a better unification algorithm -- succeeds to
apply the above instance.

Since we do not want to define [Hint Extern] declarations using [refine] for
any instance like [into_exist_and_pure], we factor this out in the class
[IntoPureT]. This way, we only have to declare a [Hint Extern] using [refine]
once, and use [IntoPureT] in any instance like [into_exist_and_pure].

TODO: Report this as a Coq bug, or wait for https://github.com/coq/coq/pull/991
to be finished and merged someday. *)
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Class IntoPureT {PROP : bi} (P : PROP) (φ : Type) :=
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  into_pureT :  ψ : Prop, φ = ψ  IntoPure P ψ.
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Lemma into_pureT_hint {PROP : bi} (P : PROP) (φ : Prop) : IntoPure P φ  IntoPureT P φ.
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Proof. by exists φ. Qed.
Hint Extern 0 (IntoPureT _ _) =>
  notypeclasses refine (into_pureT_hint _ _ _) : typeclass_instances.

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(** [FromPure] is used when introducing a pure assertion. It is used
    by iPure, the "[%]" specialization pattern, and the [with "[%]"]
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    pattern when using [iAssert].
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    The [a] Boolean asserts whether we introduce the pure assertion in
    an affine way or in an absorbing way. When [FromPure true P φ] is
    derived, then [FromPure false P φ] can always be derived too. We
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    use [true] for specialization patterns and [false] for the
    [iPureIntro] tactic.
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    This Boolean is not needed for [IntoPure], because in the case of
    [IntoPure], we can have the same behavior by asking that [P] be
    [Affine]. *)
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Class FromPure {PROP : bi} (a : bool) (P : PROP) (φ : Prop) :=
  from_pure : bi_affinely_if a ⌜φ⌝  P.
Arguments FromPure {_} _ _%I _%type_scope : simpl never.
Arguments from_pure {_} _ _%I _%type_scope {_}.
Hint Mode FromPure + + ! - : typeclass_instances.

Class FromPureT {PROP : bi} (a : bool) (P : PROP) (φ : Type) :=
  from_pureT :  ψ : Prop, φ = ψ  FromPure a P ψ.
Lemma from_pureT_hint {PROP : bi} (a : bool) (P : PROP) (φ : Prop) :
  FromPure a P φ  FromPureT a P φ.
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Proof. by exists φ. Qed.
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Hint Extern 0 (FromPureT _ _ _) =>
  notypeclasses refine (from_pureT_hint _ _ _ _) : typeclass_instances.
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Class IntoInternalEq {PROP : sbi} {A : ofeT} (P : PROP) (x y : A) :=
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  into_internal_eq : P  x  y.
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Arguments IntoInternalEq {_ _} _%I _%type_scope _%type_scope : simpl never.
Arguments into_internal_eq {_ _} _%I _%type_scope _%type_scope {_}.
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Hint Mode IntoInternalEq + - ! - - : typeclass_instances.

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Class IntoPersistent {PROP : bi} (p : bool) (P Q : PROP) :=
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  into_persistent : bi_persistently_if p P  bi_persistently Q.
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Arguments IntoPersistent {_} _ _%I _%I : simpl never.
Arguments into_persistent {_} _ _%I _%I {_}.
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Hint Mode IntoPersistent + + ! - : typeclass_instances.
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(* The `iAlways` tactic is not tied to `persistently` and `affinely`, but can be
instantiated with a variety of comonadic (always-style) modalities.

In order to plug in an always-style modality, one has to decide for both the
persistent and spatial what action should be performed upon introducing the
modality:

- Introduction is only allowed when the context is empty.
- Introduction is only allowed when all hypotheses satisfy some predicate
  `C : PROP → Prop` (where `C` should be a type class).
- Introduction will only keep the hypotheses that satisfy some predicate
  `C : PROP → Prop` (where `C` should be a type class).
- Introduction will clear the context.
- Introduction will keep the context as-if.

Formally, these actions correspond to the following inductive type: *)
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Inductive always_intro_spec (PROP : bi) :=
  | AIEnvIsEmpty
  | AIEnvForall (C : PROP  Prop)
  | AIEnvFilter (C : PROP  Prop)
  | AIEnvClear
  | AIEnvId.
Arguments AIEnvIsEmpty {_}.
Arguments AIEnvForall {_} _.
Arguments AIEnvFilter {_} _.
Arguments AIEnvClear {_}.
Arguments AIEnvId {_}.

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(* An always-style modality is then a record packing together the modality with
the laws it should satisfy to justify the given actions for both contexts: *)
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Record always_modality_mixin {PROP : bi} (M : PROP  PROP)
    (pspec sspec : always_intro_spec PROP) := {
  always_modality_mixin_persistent :
    match pspec with
    | AIEnvIsEmpty => True
    | AIEnvForall C | AIEnvFilter C =>  P, C P   P  M ( P)
    | AIEnvClear => True
    | AIEnvId =>  P,  P  M ( P)
    end;
  always_modality_mixin_spatial :
    match sspec with
    | AIEnvIsEmpty => True
    | AIEnvForall C =>  P, C P  P  M P
    | AIEnvFilter C => ( P, C P  P  M P)  ( P, Absorbing (M P))
    | AIEnvClear =>  P, Absorbing (M P)
    | AIEnvId => False
    end;
  always_modality_mixin_emp : emp  M emp;
  always_modality_mixin_mono P Q : (P  Q)  M P  M Q;
  always_modality_mixin_and P Q : M P  M Q  M (P  Q);
  always_modality_mixin_sep P Q : M P  M Q  M (P  Q)
}.

Record always_modality (PROP : bi) := AlwaysModality {
  always_modality_car :> PROP  PROP;
  always_modality_persistent_spec : always_intro_spec PROP;
  always_modality_spatial_spec : always_intro_spec PROP;
  always_modality_mixin_of : always_modality_mixin
    always_modality_car
    always_modality_persistent_spec always_modality_spatial_spec
}.
Arguments AlwaysModality {_} _ {_ _} _.
Arguments always_modality_persistent_spec {_} _.
Arguments always_modality_spatial_spec {_} _.

Section always_modality.
  Context {PROP} (M : always_modality PROP).

  Lemma always_modality_persistent_forall C P :
    always_modality_persistent_spec M = AIEnvForall C  C P   P  M ( P).
  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma always_modality_persistent_filter C P :
    always_modality_persistent_spec M = AIEnvFilter C  C P   P  M ( P).
  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma always_modality_persistent_id P :
    always_modality_persistent_spec M = AIEnvId   P  M ( P).
  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma always_modality_spatial_forall C P :
    always_modality_spatial_spec M = AIEnvForall C  C P  P  M P.
  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma always_modality_spatial_filter C P :
    always_modality_spatial_spec M = AIEnvFilter C  C P  P  M P.
  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma always_modality_spatial_filter_absorbing C P :
    always_modality_spatial_spec M = AIEnvFilter C  Absorbing (M P).
  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma always_modality_spatial_clear P :
    always_modality_spatial_spec M = AIEnvClear  Absorbing (M P).
  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma always_modality_spatial_id :
    always_modality_spatial_spec M  AIEnvId.
  Proof. destruct M as [??? []]; naive_solver. Qed.

  Lemma always_modality_emp : emp  M emp.
  Proof. eapply always_modality_mixin_emp, always_modality_mixin_of. Qed.
  Lemma always_modality_mono P Q : (P  Q)  M P  M Q.
  Proof. eapply always_modality_mixin_mono, always_modality_mixin_of. Qed.
  Lemma always_modality_and P Q : M P  M Q  M (P  Q).
  Proof. eapply always_modality_mixin_and, always_modality_mixin_of. Qed.
  Lemma always_modality_sep P Q : M P  M Q  M (P  Q).
  Proof. eapply always_modality_mixin_sep, always_modality_mixin_of. Qed.
  Global Instance always_modality_mono' : Proper (() ==> ()) M.
  Proof. intros P Q. apply always_modality_mono. Qed.
  Global Instance always_modality_flip_mono' : Proper (flip () ==> flip ()) M.
  Proof. intros P Q. apply always_modality_mono. Qed.
  Global Instance always_modality_proper : Proper (() ==> ()) M.
  Proof. intros P Q. rewrite !equiv_spec=> -[??]; eauto using always_modality_mono. Qed.

  Lemma always_modality_persistent_forall_big_and C Ps :
    always_modality_persistent_spec M = AIEnvForall C 
    Forall C Ps   [] Ps  M ( [] Ps).
  Proof.
    induction 2 as [|P Ps ? _ IH]; simpl.
    - by rewrite persistently_pure affinely_True_emp affinely_emp -always_modality_emp.
    - rewrite affinely_persistently_and -always_modality_and -IH.
      by rewrite {1}(always_modality_persistent_forall _ P).
  Qed.
  Lemma always_modality_spatial_forall_big_sep C Ps :
    always_modality_spatial_spec M = AIEnvForall C 
    Forall C Ps  [] Ps  M ([] Ps).
  Proof.
    induction 2 as [|P Ps ? _ IH]; simpl.
    - by rewrite -always_modality_emp.
    - by rewrite -always_modality_sep -IH {1}(always_modality_spatial_forall _ P).
  Qed.
End always_modality.

Class FromAlways {PROP : bi} (M : always_modality PROP) (P Q : PROP) :=
  from_always : M Q  P.
Arguments FromAlways {_} _ _%I _%I : simpl never.
Arguments from_always {_} _ _%I _%I {_}.
Hint Mode FromAlways + - ! - : typeclass_instances.
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Class FromAffinely {PROP : bi} (P Q : PROP) :=
  from_affinely : bi_affinely Q  P.
Arguments FromAffinely {_} _%I _%type_scope : simpl never.
Arguments from_affinely {_} _%I _%type_scope {_}.
Hint Mode FromAffinely + ! - : typeclass_instances.
Hint Mode FromAffinely + - ! : typeclass_instances.
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Class IntoAbsorbingly {PROP : bi} (P Q : PROP) :=
  into_absorbingly : P  bi_absorbingly Q.
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Arguments IntoAbsorbingly {_} _%I _%I.
Arguments into_absorbingly {_} _%I _%I {_}.
Hint Mode IntoAbsorbingly + ! -  : typeclass_instances.
Hint Mode IntoAbsorbingly + - ! : typeclass_instances.
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(*
Converting an assumption [R] into a wand [P -∗ Q] is done in three stages:

- Strip modalities and universal quantifiers of [R] until an arrow or a wand
  has been obtained.
- Balance modalities in the arguments [P] and [Q] to match the goal (which used
  for [iApply]) or the premise (when used with [iSpecialize] and a specific
  hypothesis).
- Instantiate the premise of the wand or implication.
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*)
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Class IntoWand {PROP : bi} (p q : bool) (R P Q : PROP) :=
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  into_wand : ?p R  ?q P - Q.
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Arguments IntoWand {_} _ _ _%I _%I _%I : simpl never.
Arguments into_wand {_} _ _ _%I _%I _%I {_}.
Hint Mode IntoWand + + + ! - - : typeclass_instances.

Class IntoWand' {PROP : bi} (p q : bool) (R P Q : PROP) :=
  into_wand' : IntoWand p q R P Q.
Arguments IntoWand' {_} _ _ _%I _%I _%I : simpl never.
Hint Mode IntoWand' + + + ! ! - : typeclass_instances.
Hint Mode IntoWand' + + + ! - ! : typeclass_instances.

Instance into_wand_wand' {PROP : bi} p q (P Q P' Q' : PROP) :
  IntoWand' p q (P - Q) P' Q'  IntoWand p q (P - Q) P' Q' | 100.
Proof. done. Qed.
Instance into_wand_impl' {PROP : bi} p q (P Q P' Q' : PROP) :
  IntoWand' p q (P  Q) P' Q'  IntoWand p q (P  Q) P' Q' | 100.
Proof. done. Qed.
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Class FromWand {PROP : bi} (P Q1 Q2 : PROP) := from_wand : (Q1 - Q2)  P.
Arguments FromWand {_} _%I _%I _%I : simpl never.
Arguments from_wand {_} _%I _%I _%I {_}.
Hint Mode FromWand + ! - - : typeclass_instances.

Class FromImpl {PROP : bi} (P Q1 Q2 : PROP) := from_impl : (Q1  Q2)  P.
Arguments FromImpl {_} _%I _%I _%I : simpl never.
Arguments from_impl {_} _%I _%I _%I {_}.
Hint Mode FromImpl + ! - - : typeclass_instances.

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Class FromSep {PROP : bi} (P Q1 Q2 : PROP) := from_sep : Q1  Q2  P.
Arguments FromSep {_} _%I _%I _%I : simpl never.
Arguments from_sep {_} _%I _%I _%I {_}.
Hint Mode FromSep + ! - - : typeclass_instances.
Hint Mode FromSep + - ! ! : typeclass_instances. (* For iCombine *)

Class FromAnd {PROP : bi} (P Q1 Q2 : PROP) := from_and : Q1  Q2  P.
Arguments FromAnd {_} _%I _%I _%I : simpl never.
Arguments from_and {_} _%I _%I _%I {_}.
Hint Mode FromAnd + ! - - : typeclass_instances.
Hint Mode FromAnd + - ! ! : typeclass_instances. (* For iCombine *)

Class IntoAnd {PROP : bi} (p : bool) (P Q1 Q2 : PROP) :=
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  into_and : ?p P  ?p (Q1  Q2).
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Arguments IntoAnd {_} _ _%I _%I _%I : simpl never.
Arguments into_and {_} _ _%I _%I _%I {_}.
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Hint Mode IntoAnd + + ! - - : typeclass_instances.
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Class IntoSep {PROP : bi} (P Q1 Q2 : PROP) :=
  into_sep : P  Q1  Q2.
Arguments IntoSep {_} _%I _%I _%I : simpl never.
Arguments into_sep {_} _%I _%I _%I {_}.
Hint Mode IntoSep + ! - - : typeclass_instances.
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Class FromOr {PROP : bi} (P Q1 Q2 : PROP) := from_or : Q1  Q2  P.
Arguments FromOr {_} _%I _%I _%I : simpl never.
Arguments from_or {_} _%I _%I _%I {_}.
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Hint Mode FromOr + ! - - : typeclass_instances.
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Class IntoOr {PROP : bi} (P Q1 Q2 : PROP) := into_or : P  Q1  Q2.
Arguments IntoOr {_} _%I _%I _%I : simpl never.
Arguments into_or {_} _%I _%I _%I {_}.
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Hint Mode IntoOr + ! - - : typeclass_instances.
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Class FromExist {PROP : bi} {A} (P : PROP) (Φ : A  PROP) :=
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  from_exist : ( x, Φ x)  P.
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Arguments FromExist {_ _} _%I _%I : simpl never.
Arguments from_exist {_ _} _%I _%I {_}.
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Hint Mode FromExist + - ! - : typeclass_instances.
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Class IntoExist {PROP : bi} {A} (P : PROP) (Φ : A  PROP) :=
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  into_exist : P   x, Φ x.
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Arguments IntoExist {_ _} _%I _%I : simpl never.
Arguments into_exist {_ _} _%I _%I {_}.
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Hint Mode IntoExist + - ! - : typeclass_instances.
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Class IntoForall {PROP : bi} {A} (P : PROP) (Φ : A  PROP) :=
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  into_forall : P   x, Φ x.
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Arguments IntoForall {_ _} _%I _%I : simpl never.
Arguments into_forall {_ _} _%I _%I {_}.
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Hint Mode IntoForall + - ! - : typeclass_instances.

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Class FromForall {PROP : bi} {A} (P : PROP) (Φ : A  PROP) :=
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  from_forall : ( x, Φ x)  P.
Arguments from_forall {_ _} _ _ {_}.
Hint Mode FromForall + - ! - : typeclass_instances.

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Class IsExcept0 {PROP : sbi} (Q : PROP) := is_except_0 :  Q  Q.
Arguments IsExcept0 {_} _%I : simpl never.
Arguments is_except_0 {_} _%I {_}.
Hint Mode IsExcept0 + ! : typeclass_instances.

Class FromModal {PROP : bi} (P Q : PROP) := from_modal : Q  P.
Arguments FromModal {_} _%I _%I : simpl never.
Arguments from_modal {_} _%I _%I {_}.
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Hint Mode FromModal + ! - : typeclass_instances.
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Class ElimModal {PROP : bi} (φ : Prop) (P P' : PROP) (Q Q' : PROP) :=
  elim_modal : φ  P  (P' - Q')  Q.
Arguments ElimModal {_} _ _%I _%I _%I _%I : simpl never.
Arguments elim_modal {_} _ _%I _%I _%I _%I {_}.
Hint Mode ElimModal + - ! - ! - : typeclass_instances.
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(* Used by the specialization pattern [ > ] in [iSpecialize] and [iAssert] to
add a modality to the goal corresponding to a premise/asserted proposition. *)
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Class AddModal {PROP : bi} (P P' : PROP) (Q : PROP) :=
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  add_modal : P  (P' - Q)  Q.
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Arguments AddModal {_} _%I _%I _%I : simpl never.
Arguments add_modal {_} _%I _%I _%I {_}.
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Hint Mode AddModal + - ! ! : typeclass_instances.

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Lemma add_modal_id {PROP : bi} (P Q : PROP) : AddModal P P Q.
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Proof. by rewrite /AddModal wand_elim_r. Qed.
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Class IsCons {A} (l : list A) (x : A) (k : list A) := is_cons : l = x :: k.
Class IsApp {A} (l k1 k2 : list A) := is_app : l = k1 ++ k2.
Global Hint Mode IsCons + ! - - : typeclass_instances.
Global Hint Mode IsApp + ! - - : typeclass_instances.

Instance is_cons_cons {A} (x : A) (l : list A) : IsCons (x :: l) x l.
Proof. done. Qed.
Instance is_app_app {A} (l1 l2 : list A) : IsApp (l1 ++ l2) l1 l2.
Proof. done. Qed.
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Class Frame {PROP : bi} (p : bool) (R P Q : PROP) := frame : ?p R  Q  P.
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Arguments Frame {_} _ _%I _%I _%I.
Arguments frame {_ _} _%I _%I _%I {_}.
Hint Mode Frame + + ! ! - : typeclass_instances.

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(* The boolean [progress] indicates whether actual framing has been performed.
If it is [false], then the default instance [maybe_frame_default] below has been
used. *)
Class MaybeFrame {PROP : bi} (p : bool) (R P Q : PROP) (progress : bool) :=
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  maybe_frame : ?p R  Q  P.
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Arguments MaybeFrame {_} _ _%I _%I _%I _.
Arguments maybe_frame {_} _ _%I _%I _%I _ {_}.
Hint Mode MaybeFrame + + ! ! - - : typeclass_instances.
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Instance maybe_frame_frame {PROP : bi} p (R P Q : PROP) :
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  Frame p R P Q  MaybeFrame p R P Q true.
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Proof. done. Qed.
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Instance maybe_frame_default_persistent {PROP : bi} (R P : PROP) :
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  MaybeFrame true R P P false | 100.
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Proof. intros. rewrite /MaybeFrame /=. by rewrite sep_elim_r. Qed.
Instance maybe_frame_default {PROP : bi} (R P : PROP) :
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  TCOr (Affine R) (Absorbing P)  MaybeFrame false R P P false | 100.
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Proof. intros. rewrite /MaybeFrame /=. apply: sep_elim_r. Qed.

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Class IntoExcept0 {PROP : sbi} (P Q : PROP) := into_except_0 : P   Q.
Arguments IntoExcept0 {_} _%I _%I : simpl never.
Arguments into_except_0 {_} _%I _%I {_}.
Hint Mode IntoExcept0 + ! - : typeclass_instances.
Hint Mode IntoExcept0 + - ! : typeclass_instances.

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(* The class [MaybeIntoLaterN] has only two instances:
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- The default instance [MaybeIntoLaterN n P P], i.e. [▷^n P -∗ P]
- The instance [IntoLaterN n P Q → MaybeIntoLaterN n P Q], where [IntoLaterN]
  is identical to [MaybeIntoLaterN], but is supposed to make progress, i.e. it
  should actually strip a later.
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The point of using the auxilary class [IntoLaterN] is to ensure that the
default instance is not applied deeply inside a term, which may result in too
many definitions being unfolded (see issue #55).
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For binary connectives we have the following instances:

<<
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IntoLaterN n P P'       MaybeIntoLaterN n Q Q'
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------------------------------------------
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     IntoLaterN n (P /\ Q) (P' /\ Q')
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      IntoLaterN n Q Q'
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-------------------------------
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IntoLaterN n (P /\ Q) (P /\ Q')
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>>
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The Boolean [only_head] indicates whether laters should only be stripped in
head position or also below other logical connectives. For [iNext] it should
strip laters below other logical connectives, but this should not happen while
framing, e.g. the following should succeed:

<<
Lemma test_iFrame_later_1 P Q : P ∗ ▷ Q -∗ ▷ (P ∗ ▷ Q).
Proof. iIntros "H". iFrame "H". Qed.
>>
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*)
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Class MaybeIntoLaterN {PROP : sbi} (only_head : bool) (n : nat) (P Q : PROP) :=
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  maybe_into_laterN : P  ^n Q.
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Arguments MaybeIntoLaterN {_} _ _%nat_scope _%I _%I.
Arguments maybe_into_laterN {_} _ _%nat_scope _%I _%I {_}.
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Hint Mode MaybeIntoLaterN + + + - - : typeclass_instances.
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Class IntoLaterN {PROP : sbi} (only_head : bool) (n : nat) (P Q : PROP) :=
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  into_laterN :> MaybeIntoLaterN only_head n P Q.
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Arguments IntoLaterN {_} _ _%nat_scope _%I _%I.
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Hint Mode IntoLaterN + + + ! - : typeclass_instances.
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Instance maybe_into_laterN_default {PROP : sbi} only_head n (P : PROP) :
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  MaybeIntoLaterN only_head n P P | 1000.
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Proof. apply laterN_intro. Qed.
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(* In the case both parameters are evars and n=0, we have to stop the
   search and unify both evars immediately instead of looping using
   other instances. *)
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Instance maybe_into_laterN_default_0 {PROP : sbi} only_head (P : PROP) :
  MaybeIntoLaterN only_head 0 P P | 0.
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Proof. apply _. Qed.
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Class FromLaterN {PROP : sbi} (n : nat) (P Q : PROP) := from_laterN : ^n Q  P.
Arguments FromLaterN {_} _%nat_scope _%I _%I.
Arguments from_laterN {_} _%nat_scope _%I _%I {_}.
Hint Mode FromLaterN + - ! - : typeclass_instances.

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Class AsValid {PROP : bi} (φ : Prop) (P : PROP) := as_valid : φ  P.
Arguments AsValid {_} _%type _%I.

Class AsValid0 {PROP : bi} (φ : Prop) (P : PROP) :=
  as_valid_here : AsValid φ P.
Arguments AsValid0 {_} _%type _%I.
Existing Instance as_valid_here | 0.

Lemma as_valid_1 (φ : Prop) {PROP : bi} (P : PROP) `{!AsValid φ P} : φ  P.
Proof. by apply as_valid. Qed.
Lemma as_valid_2 (φ : Prop) {PROP : bi} (P : PROP) `{!AsValid φ P} : P  φ.
Proof. by apply as_valid. Qed.

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(* Input: [P]; Outputs: [N],
   Extracts the namespace associated with an invariant assertion. Used for [iInv]. *)
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Class IntoInv {PROP : bi} (P: PROP) (N: namespace).
Arguments IntoInv {_} _%I _.
Hint Mode IntoInv + ! - : typeclass_instances.

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(* Input: [Pinv]
   Arguments:
   - [Pinv] is an invariant assertion
   - [Pin] is an additional assertion needed for opening an invariant
   - [Pout] is the assertion obtained by opening the invariant
   - [Q] is a goal on which iInv may be invoked
   - [Q'] is the transformed goal that must be proved after opening the invariant.
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   There are similarities to the definition of ElimModal, however we
   want to be general enough to support uses in settings where there
   is not a clearly associated instance of ElimModal of the right form
   (e.g. to handle Iris 2.0 usage of iInv).
*)
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Class ElimInv {PROP : bi} (φ : Prop) (N : namespace) (Pinv Pin Pout Q Q' : PROP) :=
  elim_inv : φ  Pinv  Pin  (Pout - Q')  Q.
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Arguments ElimInv {_} _ _ _  _%I _%I _%I _%I : simpl never.
Arguments elim_inv {_} _ _ _%I _%I _%I _%I _%I _%I.
Hint Mode ElimInv + - - ! - - - - : typeclass_instances.

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(* We make sure that tactics that perform actions on *specific* hypotheses or
parts of the goal look through the [tc_opaque] connective, which is used to make
definitions opaque for type class search. For example, when using `iDestruct`,
an explicit hypothesis is affected, and as such, we should look through opaque
definitions. However, when using `iFrame` or `iNext`, arbitrary hypotheses or
parts of the goal are affected, and as such, type class opacity should be
respected.

This means that there are [tc_opaque] instances for all proofmode type classes
with the exception of:

- [FromAssumption] used by [iAssumption]
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- [Frame] and [MaybeFrame] used by [iFrame]
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- [MaybeIntoLaterN] and [FromLaterN] used by [iNext]
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- [IntoPersistent] used by [iPersistent]
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*)
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Instance into_pure_tc_opaque {PROP : bi} (P : PROP) φ :
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  IntoPure P φ  IntoPure (tc_opaque P) φ := id.
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Instance from_pure_tc_opaque {PROP : bi} (a : bool) (P : PROP) φ :
  FromPure a P φ  FromPure a (tc_opaque P) φ := id.
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Instance from_laterN_tc_opaque {PROP : sbi} n (P Q : PROP) :
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  FromLaterN n P Q  FromLaterN n (tc_opaque P) Q := id.
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Instance from_wand_tc_opaque {PROP : bi} (P Q1 Q2 : PROP) :
  FromWand P Q1 Q2  FromWand (tc_opaque P) Q1 Q2 := id.
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Instance into_wand_tc_opaque {PROP : bi} p q (R P Q : PROP) :
  IntoWand p q R P Q  IntoWand p q (tc_opaque R) P Q := id.
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(* Higher precedence than [from_and_sep] so that [iCombine] does not loop. *)
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Instance from_and_tc_opaque {PROP : bi} (P Q1 Q2 : PROP) :
  FromAnd P Q1 Q2  FromAnd (tc_opaque P) Q1 Q2 | 102 := id.
Instance into_and_tc_opaque {PROP : bi} p (P Q1 Q2 : PROP) :
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  IntoAnd p P Q1 Q2  IntoAnd p (tc_opaque P) Q1 Q2 := id.
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Instance from_or_tc_opaque {PROP : bi} (P Q1 Q2 : PROP) :
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  FromOr P Q1 Q2  FromOr (tc_opaque P) Q1 Q2 := id.
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Instance into_or_tc_opaque {PROP : bi} (P Q1 Q2 : PROP) :
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  IntoOr P Q1 Q2  IntoOr (tc_opaque P) Q1 Q2 := id.
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Instance from_exist_tc_opaque {PROP : bi} {A} (P : PROP) (Φ : A  PROP) :
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  FromExist P Φ  FromExist (tc_opaque P) Φ := id.
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Instance into_exist_tc_opaque {PROP : bi} {A} (P : PROP) (Φ : A  PROP) :
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  IntoExist P Φ  IntoExist (tc_opaque P) Φ := id.
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Instance into_forall_tc_opaque {PROP : bi} {A} (P : PROP) (Φ : A  PROP) :
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  IntoForall P Φ  IntoForall (tc_opaque P) Φ := id.
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Instance from_modal_tc_opaque {PROP : bi} (P Q : PROP) :
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  FromModal P Q  FromModal (tc_opaque P) Q := id.
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Instance elim_modal_tc_opaque {PROP : bi} φ (P P' Q Q' : PROP) :
  ElimModal φ P P' Q Q'  ElimModal φ (tc_opaque P) P' Q Q' := id.