modalities.v 8.08 KB
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From iris.bi Require Export bi.
From stdpp Require Import namespaces.
Set Default Proof Using "Type".
Import bi.

(** The `iModIntro` tactic is not tied the Iris modalities, but can be
instantiated with a variety of modalities.

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For the purpose of MoSeL, a modality is a mapping of propositions
`M : PROP → PROP` (where `PROP` is a type of bi-assertions) that is
monotone and distributes over finite products. Specifically, the following
rules have to be satisfied.

      P ⊢ Q        emp ⊢ M emp
    ----------
    M P ⊢ M Q      M P ∗ M Q ⊢ M (P ∗ Q)

Together those conditions allow one to introduce the modality in the
goal, while stripping away the modalities in the context.

Additionally, upon introducing a modality one can perform a number of
associated actions on the intuitionistic and spatial contexts.
Such an action can be one of the following:
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- 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 transform each hypotheses using a type class
  `C : PROP → PROP → Prop`, where the first parameter is the input and the
  second parameter is the output. Hypotheses that cannot be transformed (i.e.
  for which no instance of `C` can be found) will be cleared.
- Introduction will clear the context.
- Introduction will keep the context as-if.

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Formally, these actions correspond to the inductive type [modality_action].
For each of those actions you have to prove that the transformation is valid.

To instantiate the modality you have to define: 1) a mixin `modality_mixin`,
2) a record `modality`, 3) a `FromModal` type class instance from `classes.v`.

For examples consult `modality_id` at the end of this file, or the instances
in the `modality_instances.v` file.

Note that in MoSeL modality can map the propositions between two different BI-algebras.
For instance, the <affine> modality maps propositions of an arbitrary BI-algebra into
the sub-BI-algebra of affine propositions.
*)
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Inductive modality_action (PROP1 : bi) : bi  Type :=
  | MIEnvIsEmpty {PROP2 : bi} : modality_action PROP1 PROP2
  | MIEnvForall (C : PROP1  Prop) : modality_action PROP1 PROP1
  | MIEnvTransform {PROP2 : bi} (C : PROP2  PROP1  Prop) : modality_action PROP1 PROP2
  | MIEnvClear {PROP2} : modality_action PROP1 PROP2
  | MIEnvId : modality_action PROP1 PROP1.
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Arguments MIEnvIsEmpty {_ _}.
Arguments MIEnvForall {_} _.
Arguments MIEnvTransform {_ _} _.
Arguments MIEnvClear {_ _}.
Arguments MIEnvId {_}.

Notation MIEnvFilter C := (MIEnvTransform (TCDiag C)).

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Definition modality_intuitionistic_action_spec {PROP1 PROP2}
    (s : modality_action PROP1 PROP2) : (PROP1  PROP2)  Prop :=
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  match s with
  | MIEnvIsEmpty => λ M, True
  | MIEnvForall C => λ M,
     ( P, C P   P  M ( P)) 
     ( P Q, M P  M Q  M (P  Q))
  | MIEnvTransform C => λ M,
     ( P Q, C P Q   P  M ( Q)) 
     ( P Q, M P  M Q  M (P  Q))
  | MIEnvClear => λ M, True
  | MIEnvId => λ M,  P,  P  M ( P)
  end.

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Definition modality_spatial_action_spec {PROP1 PROP2}
    (s : modality_action PROP1 PROP2) : (PROP1  PROP2)  Prop :=
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  match s with
  | MIEnvIsEmpty => λ M, True
  | MIEnvForall C => λ M,  P, C P  P  M P
  | MIEnvTransform C => λ M,  P Q, C P Q  P  M Q
  | MIEnvClear => λ M,  P, Absorbing (M P)
  | MIEnvId => λ M,  P, P  M P
  end.

(* A modality is then a record packing together the modality with the laws it
should satisfy to justify the given actions for both contexts: *)
Record modality_mixin {PROP1 PROP2 : bi} (M : PROP1  PROP2)
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    (iaction saction : modality_action PROP1 PROP2) := {
  modality_mixin_intuitionistic : modality_intuitionistic_action_spec iaction M;
  modality_mixin_spatial : modality_spatial_action_spec saction M;
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  modality_mixin_emp : emp  M emp;
  modality_mixin_mono P Q : (P  Q)  M P  M Q;
  modality_mixin_sep P Q : M P  M Q  M (P  Q)
}.

Record modality (PROP1 PROP2 : bi) := Modality {
  modality_car :> PROP1  PROP2;
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  modality_intuitionistic_action : modality_action PROP1 PROP2;
  modality_spatial_action : modality_action PROP1 PROP2;
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  modality_mixin_of :
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    modality_mixin modality_car modality_intuitionistic_action modality_spatial_action
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}.
Arguments Modality {_ _} _ {_ _} _.
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Arguments modality_intuitionistic_action {_ _} _.
Arguments modality_spatial_action {_ _} _.
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Section modality.
  Context {PROP1 PROP2} (M : modality PROP1 PROP2).

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  Lemma modality_intuitionistic_transform C P Q :
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    modality_intuitionistic_action M = MIEnvTransform C  C P Q   P  M ( Q).
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  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma modality_and_transform C P Q :
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    modality_intuitionistic_action M = MIEnvTransform C  M P  M Q  M (P  Q).
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  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma modality_spatial_transform C P Q :
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    modality_spatial_action M = MIEnvTransform C  C P Q  P  M Q.
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  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma modality_spatial_clear P :
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    modality_spatial_action M = MIEnvClear  Absorbing (M P).
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  Proof. destruct M as [??? []]; naive_solver. Qed.

  Lemma modality_emp : emp  M emp.
  Proof. eapply modality_mixin_emp, modality_mixin_of. Qed.
  Lemma modality_mono P Q : (P  Q)  M P  M Q.
  Proof. eapply modality_mixin_mono, modality_mixin_of. Qed.
  Lemma modality_sep P Q : M P  M Q  M (P  Q).
  Proof. eapply modality_mixin_sep, modality_mixin_of. Qed.
  Global Instance modality_mono' : Proper (() ==> ()) M.
  Proof. intros P Q. apply modality_mono. Qed.
  Global Instance modality_flip_mono' : Proper (flip () ==> flip ()) M.
  Proof. intros P Q. apply modality_mono. Qed.
  Global Instance modality_proper : Proper (() ==> ()) M.
  Proof. intros P Q. rewrite !equiv_spec=> -[??]; eauto using modality_mono. Qed.
End modality.

Section modality1.
  Context {PROP} (M : modality PROP PROP).

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  Lemma modality_intuitionistic_forall C P :
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    modality_intuitionistic_action M = MIEnvForall C  C P   P  M ( P).
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  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma modality_and_forall C P Q :
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    modality_intuitionistic_action M = MIEnvForall C  M P  M Q  M (P  Q).
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  Proof. destruct M as [??? []]; naive_solver. Qed.
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  Lemma modality_intuitionistic_id P :
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    modality_intuitionistic_action M = MIEnvId   P  M ( P).
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  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma modality_spatial_forall C P :
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    modality_spatial_action M = MIEnvForall C  C P  P  M P.
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  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma modality_spatial_id P :
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    modality_spatial_action M = MIEnvId  P  M P.
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  Proof. destruct M as [??? []]; naive_solver. Qed.

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  Lemma modality_intuitionistic_forall_big_and C Ps :
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    modality_intuitionistic_action M = MIEnvForall C 
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    Forall C Ps   [] Ps  M ( [] Ps).
  Proof.
    induction 2 as [|P Ps ? _ IH]; simpl.
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    - by rewrite intuitionistically_True_emp -modality_emp.
    - rewrite intuitionistically_and -modality_and_forall // -IH.
      by rewrite {1}(modality_intuitionistic_forall _ P).
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  Qed.
  Lemma modality_spatial_forall_big_sep C Ps :
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    modality_spatial_action M = MIEnvForall C 
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    Forall C Ps  [] Ps  M ([] Ps).
  Proof.
    induction 2 as [|P Ps ? _ IH]; simpl.
    - by rewrite -modality_emp.
    - by rewrite -modality_sep -IH {1}(modality_spatial_forall _ P).
  Qed.
End modality1.

(** The identity modality [modality_id] can be used in combination with
[FromModal modality_id] to support introduction for modalities that enjoy
[P ⊢ M P]. This is done by defining an instance [FromModal modality_id (M P) P],
which will instruct [iModIntro] to introduce the modality without modifying the
proof mode context. Examples of such modalities are [bupd], [fupd], [except_0],
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[monPred_subjectively] and [bi_absorbingly]. *)
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Lemma modality_id_mixin {PROP : bi} : modality_mixin (@id PROP) MIEnvId MIEnvId.
Proof. split; simpl; eauto. Qed.
Definition modality_id {PROP : bi} := Modality (@id PROP) modality_id_mixin.