Library iris.program_logic.ectxi_language

An axiomatization of languages based on evaluation context items, including a proof that these are instances of general ectx-based languages.
From iris.algebra Require Export base.
From iris.program_logic Require Import language ectx_language.
From iris Require Import options.

TAKE CARE: When you define an ectxiLanguage canonical structure for your language, you need to also define a corresponding language and ectxLanguage canonical structure for canonical structure inference to work properly. You should use the coercion EctxLanguageOfEctxi and LanguageOfEctx for that, and not ectxi_lang and ectxi_lang_ectx, otherwise the canonical projections will not point to the right terms.
A full concrete example of setting up your language can be found in heap_lang. Below you can find the relevant parts:
Module heap_lang.
Lemma heap_lang_mixin : EctxiLanguageMixin of_val to_val fill_item head_step. Proof. Qed. End heap_lang.
Canonical Structure heap_ectxi_lang := EctxiLanguage heap_lang.heap_lang_mixin. Canonical Structure heap_ectx_lang := EctxLanguageOfEctxi heap_ectxi_lang. Canonical Structure heap_lang := LanguageOfEctx heap_ectx_lang.
fill_item is always injective on the expression for a fixed context.
    mixin_fill_item_inj Ki : Inj (=) (=) (fill_item Ki);
    
fill_item with (potentially different) non-value expressions is injective on the context.
If fill_item Ki e takes a head step, then e is a value (unlike for ectx_language, an empty context is impossible here). In other words, if e is not a value then wrapping it in a context does not add new head redex positions.
    mixin_head_ctx_step_val Ki e σ1 κ e2 σ2 efs :
      head_step (fill_item Ki e) σ1 κ e2 σ2 efs is_Some (to_val e);
  }.
End ectxi_language_mixin.

Structure ectxiLanguage := EctxiLanguage {
  expr : Type;
  val : Type;
  ectx_item : Type;
  state : Type;
  observation : Type;

  of_val : val expr;
  to_val : expr option val;
  fill_item : ectx_item expr expr;
  head_step : expr state list observation expr state list expr Prop;

  ectxi_language_mixin :
    EctxiLanguageMixin of_val to_val fill_item head_step
}.

Bind Scope expr_scope with expr.
Bind Scope val_scope with val.

Arguments EctxiLanguage {_ _ _ _ _ _ _ _ _} _.
Arguments of_val {_} _.
Arguments to_val {_} _.
Arguments fill_item {_} _ _.
Arguments head_step {_} _ _ _ _ _ _.

Section ectxi_language.
  Context {Λ : ectxiLanguage}.
  Implicit Types (e : expr Λ) (Ki : ectx_item Λ).
  Notation ectx := (list (ectx_item Λ)).

  Global Instance fill_item_inj Ki : Inj (=) (=) (fill_item Ki).
  Proof. apply ectxi_language_mixin. Qed.
  Lemma fill_item_val Ki e : is_Some (to_val (fill_item Ki e)) is_Some (to_val e).
  Proof. apply ectxi_language_mixin. Qed.
  Lemma fill_item_no_val_inj Ki1 Ki2 e1 e2 :
    to_val e1 = None to_val e2 = None
    fill_item Ki1 e1 = fill_item Ki2 e2 Ki1 = Ki2.
  Proof. apply ectxi_language_mixin. Qed.
  Lemma head_ctx_step_val Ki e σ1 κ e2 σ2 efs :
    head_step (fill_item Ki e) σ1 κ e2 σ2 efs is_Some (to_val e).
  Proof. apply ectxi_language_mixin. Qed.

  Definition fill (K : ectx) (e : expr Λ) : expr Λ := foldl (flip fill_item) e K.

  Lemma fill_app (K1 K2 : ectx) e : fill (K1 ++ K2) e = fill K2 (fill K1 e).
  Proof. apply foldl_app. Qed.

  Definition ectxi_lang_ectx_mixin :
    EctxLanguageMixin of_val to_val [] (flip (++)) fill head_step.
  Proof.
    assert (fill_val : K e, is_Some (to_val (fill K e)) is_Some (to_val e)).
    { intros K. induction K as [|Ki K IH]=> e //=. by intros ?%IH%fill_item_val. }
    assert (fill_not_val : K e, to_val e = None to_val (fill K e) = None).
    { intros K e. rewrite !eq_None_not_Some. eauto. }
    split.
    - apply ectxi_language_mixin.
    - apply ectxi_language_mixin.
    - apply ectxi_language_mixin.
    - done.
    - intros K1 K2 e. by rewrite /fill /= foldl_app.
    - intros K; induction K as [|Ki K IH]; rewrite /Inj; naive_solver.
    - done.
    - intros K K' e1 κ e1' σ1 e2 σ2 efs Hfill Hred Hstep; revert K' Hfill.
      induction K as [|Ki K IH] using rev_ind⇒ /= K' Hfill; eauto using app_nil_r.
      destruct K' as [|Ki' K' _] using @rev_ind; simplify_eq/=.
      { rewrite fill_app in Hstep. apply head_ctx_step_val in Hstep.
        apply fill_val in Hstep. by apply not_eq_None_Some in Hstep. }
      rewrite !fill_app /= in Hfill.
      assert (Ki = Ki') as →.
      { eapply fill_item_no_val_inj, Hfill; eauto using val_head_stuck.
        apply fill_not_val. revert Hstep. apply ectxi_language_mixin. }
      simplify_eq. destruct (IH K') as [K'' ->]; auto.
       K''. by rewrite assoc.
    - intros K e1 σ1 κ e2 σ2 efs.
      destruct K as [|Ki K _] using rev_ind; simpl; first by auto.
      rewrite fill_app /=.
      intros ?%head_ctx_step_val; eauto using fill_val.
  Qed.

  Canonical Structure ectxi_lang_ectx := EctxLanguage ectxi_lang_ectx_mixin.
  Canonical Structure ectxi_lang := LanguageOfEctx ectxi_lang_ectx.

  Lemma fill_not_val K e : to_val e = None to_val (fill K e) = None.
  Proof. rewrite !eq_None_not_Some. eauto using fill_val. Qed.

  Lemma ectxi_language_sub_redexes_are_values e :
    ( Ki e', e = fill_item Ki e' is_Some (to_val e'))
    sub_redexes_are_values e.
  Proof.
    intros Hsub K e' →. destruct K as [|Ki K _] using @rev_ind⇒ //=.
    intros []%eq_None_not_Some. eapply fill_val, Hsub. by rewrite /= fill_app.
  Qed.

  Global Instance ectxi_lang_ctx_item Ki : LanguageCtx (fill_item Ki).
  Proof. change (LanguageCtx (fill [Ki])). apply _. Qed.
End ectxi_language.

Arguments ectxi_lang_ectx : clear implicits.
Arguments ectxi_lang : clear implicits.
Coercion ectxi_lang_ectx : ectxiLanguage >-> ectxLanguage.
Coercion ectxi_lang : ectxiLanguage >-> language.

Definition EctxLanguageOfEctxi (Λ : ectxiLanguage) : ectxLanguage :=
  let '@EctxiLanguage E V C St K of_val to_val fill head mix := Λ in
  @EctxLanguage E V (list C) St K of_val to_val _ _ _ _
    (@ectxi_lang_ectx_mixin (@EctxiLanguage E V C St K of_val to_val fill head mix)).