ectx_language.v 4.82 KB
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(** An axiomatization of evaluation-context based languages, including a proof
    that this gives rise to a "language" in the Iris sense. *)
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From iris.algebra Require Export base.
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From iris.program_logic Require Import language.
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(* We need to make thos arguments indices that we want canonical structure
   inference to use a keys. *)
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Class EctxLanguage (expr val ectx state : Type) := {
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  of_val : val  expr;
  to_val : expr  option val;
  empty_ectx : ectx;
  comp_ectx : ectx  ectx  ectx;
  fill : ectx  expr  expr;
  head_step : expr  state  expr  state  option expr  Prop;

  to_of_val v : to_val (of_val v) = Some v;
  of_to_val e v : to_val e = Some v  of_val v = e;
  val_stuck e1 σ1 e2 σ2 ef : head_step e1 σ1 e2 σ2 ef  to_val e1 = None;

  fill_empty e : fill empty_ectx e = e;
  fill_comp K1 K2 e : fill K1 (fill K2 e) = fill (comp_ectx K1 K2) e;
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  fill_inj K :> Inj (=) (=) (fill K);
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  fill_not_val K e : to_val e = None  to_val (fill K e) = None;

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  (* There are a whole lot of sensible axioms (like associativity, and left and
  right identity, we could demand for [comp_ectx] and [empty_ectx]. However,
  positivity suffices. *)
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  ectx_positive K1 K2 :
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    comp_ectx K1 K2 = empty_ectx  K1 = empty_ectx  K2 = empty_ectx;
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  step_by_val K K' e1 e1' σ1 e2 σ2 ef :
    fill K e1 = fill K' e1' 
    to_val e1 = None 
    head_step e1' σ1 e2 σ2 ef 
    exists K'', K' = comp_ectx K K'';
}.
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Arguments of_val {_ _ _ _ _} _.
Arguments to_val {_ _ _ _ _} _.
Arguments empty_ectx {_ _ _ _ _}.
Arguments comp_ectx {_ _ _ _ _} _ _.
Arguments fill {_ _ _ _ _} _ _.
Arguments head_step {_ _ _ _ _} _ _ _ _ _.

Arguments to_of_val {_ _ _ _ _} _.
Arguments of_to_val {_ _ _ _ _} _ _ _.
Arguments val_stuck {_ _ _ _ _} _ _ _ _ _ _.
Arguments fill_empty {_ _ _ _ _} _.
Arguments fill_comp {_ _ _ _ _} _ _ _.
Arguments fill_not_val {_ _ _ _ _} _ _ _.
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Arguments ectx_positive {_ _ _ _ _} _ _ _.
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Arguments step_by_val {_ _ _ _ _} _ _ _ _ _ _ _ _ _ _ _.

(* From an ectx_language, we can construct a language. *)
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Section ectx_language.
  Context {expr val ectx state} {Λ : EctxLanguage expr val ectx state}.
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  Implicit Types (e : expr) (K : ectx).

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  Definition head_reducible (e : expr) (σ : state) :=
     e' σ' ef, head_step e σ e' σ' ef.

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  Inductive prim_step (e1 : expr) (σ1 : state)
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      (e2 : expr) (σ2 : state) (ef : option expr) : Prop :=
    Ectx_step K e1' e2' :
      e1 = fill K e1'  e2 = fill K e2' 
      head_step e1' σ1 e2' σ2 ef  prim_step e1 σ1 e2 σ2 ef.
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  Lemma val_prim_stuck e1 σ1 e2 σ2 ef :
    prim_step e1 σ1 e2 σ2 ef  to_val e1 = None.
  Proof. intros [??? -> -> ?]; eauto using fill_not_val, val_stuck. Qed.

  Canonical Structure ectx_lang : language := {|
    language.expr := expr; language.val := val; language.state := state;
    language.of_val := of_val; language.to_val := to_val;
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    language.prim_step := prim_step;
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    language.to_of_val := to_of_val; language.of_to_val := of_to_val;
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    language.val_stuck := val_prim_stuck;
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  |}.

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  (* Some lemmas about this language *)
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  Lemma head_prim_step e1 σ1 e2 σ2 ef :
    head_step e1 σ1 e2 σ2 ef  prim_step e1 σ1 e2 σ2 ef.
  Proof. apply Ectx_step with empty_ectx; by rewrite ?fill_empty. Qed.

  Lemma head_prim_reducible e σ : head_reducible e σ  reducible e σ.
  Proof. intros (e'&σ'&ef&?). eexists e', σ', ef. by apply head_prim_step. Qed.
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  Lemma ectx_language_atomic e :
    ( σ e' σ' ef, head_step e σ e' σ' ef  is_Some (to_val e')) 
    ( K e', e = fill K e'  to_val e' = None  K = empty_ectx) 
    atomic e.
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  Proof.
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    intros Hatomic_step Hatomic_fill σ e' σ' ef [K e1' e2' -> -> Hstep].
    assert (K = empty_ectx) as -> by eauto 10 using val_stuck.
    rewrite fill_empty. eapply Hatomic_step. by rewrite fill_empty.
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  Qed.

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  Lemma head_reducible_prim_step e1 σ1 e2 σ2 ef :
    head_reducible e1 σ1  prim_step e1 σ1 e2 σ2 ef 
    head_step e1 σ1 e2 σ2 ef.
  Proof.
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    intros (e2''&σ2''&ef''&?) [K e1' e2' -> -> Hstep].
    destruct (step_by_val K empty_ectx e1' (fill K e1') σ1 e2'' σ2'' ef'')
      as [K' [-> _]%symmetry%ectx_positive];
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      eauto using fill_empty, fill_not_val, val_stuck.
    by rewrite !fill_empty.
  Qed.

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  (* Every evaluation context is a context. *)
  Global Instance ectx_lang_ctx K : LanguageCtx ectx_lang (fill K).
  Proof.
    split.
    - eauto using fill_not_val.
    - intros ????? [K' e1' e2' Heq1 Heq2 Hstep].
      by exists (comp_ectx K K') e1' e2'; rewrite ?Heq1 ?Heq2 ?fill_comp.
    - intros e1 σ1 e2 σ2 ? Hnval [K'' e1'' e2'' Heq1 -> Hstep].
      destruct (step_by_val K K'' e1 e1'' σ1 e2'' σ2 ef) as [K' ->]; eauto.
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      rewrite -fill_comp in Heq1; apply (inj (fill _)) in Heq1.
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      exists (fill K' e2''); rewrite -fill_comp; split; auto.
      econstructor; eauto.
  Qed.
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End ectx_language.
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Arguments ectx_lang _ {_ _ _ _}.