Commit cd8b29fe authored by Ralf Jung's avatar Ralf Jung
Browse files

explain our langauge axioms better

parent bc45284f
......@@ -4,7 +4,7 @@ From iris.algebra Require Export base.
From iris.program_logic Require Import language.
Set Default Proof Using "Type".
(* TAKE CARE: When you define an [ectxLanguage] canonical structure for your
(** TAKE CARE: When you define an [ectxLanguage] canonical structure for your
language, you need to also define a corresponding [language] canonical
structure. Use the coercion [LanguageOfEctx] as defined in the bottom of this
file for doing that. *)
......@@ -29,15 +29,23 @@ Section ectx_language_mixin.
mixin_fill_inj K : Inj (=) (=) (fill K);
mixin_fill_val K e : is_Some (to_val (fill K e)) is_Some (to_val e);
(** Given a head redex [e1'] somewhere in a term, and another decomposition
of the same term into [fill K e1] such that [e1] is not a value, then
the head redex context is [e1]'s context [K] filled with another context
[K'']. In particular, this implies [e1 = fill K'' e1'] by [fill_inj],
i.e., [e1] contains the head redex.)
This implies there can always be only one head redex, see
[head_redex_unique]. *)
mixin_step_by_val K K' e1 e1' σ1 κ e2 σ2 efs :
fill K e1 = fill K' e1'
to_val e1 = None
head_step e1' σ1 κ e2 σ2 efs
K'', K' = comp_ectx K K'';
(* If [fill K e] takes a head step, then either [e] is a value or [K] is
the empty evaluation context. In other words, if [e] is not a value then
there cannot be another redex position elsewhere in [fill K e]. *)
(** If [fill K e] takes a head step, then either [e] is a value or [K] is
the empty evaluation context. In other words, if [e] is not a value
wrapping it in a context does not add new head redex positions. *)
mixin_head_ctx_step_val K e σ1 κ e2 σ2 efs :
head_step (fill K e) σ1 κ e2 σ2 efs is_Some (to_val e) K = empty_ectx;
}.
......@@ -144,16 +152,31 @@ Section ectx_language.
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 head_prim_step e1 σ1 κ e2 σ2 efs :
head_step e1 σ1 κ e2 σ2 efs prim_step e1 σ1 κ e2 σ2 efs.
Proof. apply Ectx_step with empty_ectx; by rewrite ?fill_empty. Qed.
Lemma head_reducible_no_obs_reducible e σ :
head_reducible_no_obs e σ head_reducible e σ.
Proof. intros (?&?&?&?). eexists. eauto. Qed.
Lemma not_head_reducible e σ : ¬head_reducible e σ head_irreducible e σ.
Proof. unfold head_reducible, head_irreducible. naive_solver. Qed.
Lemma head_redex_unique K K' e e' σ :
fill K e = fill K' e'
head_reducible e σ
head_reducible e' σ
K = comp_ectx K' empty_ectx e = e'.
Proof.
intros Heq (κ & e2 & σ2 & efs & Hred) (κ' & e2' & σ2' & efs' & Hred').
edestruct (step_by_val K' K e' e) as [K'' HK]; try done.
{ exact: val_head_stuck. }
subst K. move: Heq. rewrite -fill_comp=> /fill_inj He'.
subst e'. edestruct (head_ctx_step_val _ _ _ _ _ _ _ Hred') as [Hval|HK''].
{ erewrite val_head_stuck in Hval; last done. destruct Hval. done. }
subst K''. rewrite fill_empty. done.
Qed.
Lemma head_prim_step e1 σ1 κ e2 σ2 efs :
head_step e1 σ1 κ e2 σ2 efs prim_step e1 σ1 κ e2 σ2 efs.
Proof. apply Ectx_step with empty_ectx; by rewrite ?fill_empty. Qed.
Lemma head_step_not_stuck e σ κ e' σ' efs : head_step e σ κ e' σ' efs not_stuck e σ.
Proof. rewrite /not_stuck /reducible /=. eauto 10 using head_prim_step. Qed.
......
......@@ -4,7 +4,7 @@ From iris.algebra Require Export base.
From iris.program_logic Require Import language ectx_language.
Set Default Proof Using "Type".
(* TAKE CARE: When you define an [ectxiLanguage] canonical structure for your
(** 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
......@@ -38,12 +38,20 @@ Section ectxi_language_mixin.
mixin_of_to_val e v : to_val e = Some v of_val v = e;
mixin_val_stuck e1 σ1 κ e2 σ2 efs : head_step e1 σ1 κ e2 σ2 efs to_val e1 = None;
mixin_fill_item_inj Ki : Inj (=) (=) (fill_item Ki);
mixin_fill_item_val Ki e : is_Some (to_val (fill_item Ki e)) is_Some (to_val e);
(** [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. *)
mixin_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;
(** 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);
}.
......
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