Commit 10d149b2 authored by Ralf Jung's avatar Ralf Jung

prove step_by_value :)

parent 49d7e625
......@@ -59,19 +59,37 @@ Proof.
induction v; simpl; rewrite ?IHv ?IHv1 /= ?IHv2; reflexivity.
Qed.
Section e2e. (* To get local tactics. *)
Lemma e2e e v:
e2v e = Some v -> e = v2e v.
Proof.
revert v; induction e; intros v; simpl; try discriminate.
- intros Heq. injection Heq. clear Heq. intros Heq. subst. reflexivity.
- intros Heq. injection Heq. clear Heq. intros Heq. subst. reflexivity.
- destruct (e2v e1); simpl; [|discriminate].
destruct (e2v e2); simpl; [|discriminate].
case =><-. simpl. eauto using f_equal2.
- destruct (e2v e); simpl; [|discriminate].
case =><-. simpl. eauto using f_equal.
- destruct (e2v e); simpl; [|discriminate].
case =><-. simpl. eauto using f_equal.
Ltac case0 := case =><-; simpl; eauto using f_equal, f_equal2.
Ltac case1 e1 := destruct (e2v e1); simpl; [|discriminate];
case0.
Ltac case2 e1 e2 := destruct (e2v e1); simpl; [|discriminate];
destruct (e2v e2); simpl; [|discriminate];
case0.
revert v; induction e; intros v; simpl; try discriminate; by (case2 e1 e2 || case1 e || case0).
Qed.
End e2e.
Definition eq_transport (T1 T2: Type) (Heq: T1 = T2):
T1 -> T2. (* RJ: I am *sure* this is already defined somewhere... *)
intros t1. rewrite -Heq. exact t1.
Defined.
Lemma eq_transport_id T (t: T) :
t = eq_transport T T eq_refl t.
Proof.
reflexivity.
Qed.
Lemma v2e_inj v1 v2:
v2e v1 = v2e v2 -> v1 = v2.
Proof.
revert v2; induction v1=>v2; destruct v2; simpl; try discriminate; case; eauto using f_equal, f_equal2.
- intros _. move/EqdepFacts.eq_sigT_sig_eq=>H. destruct H as (->,<-). reflexivity.
Qed.
Inductive ectx :=
......@@ -299,7 +317,7 @@ Qed.
Lemma reducible_find_redex {e K' e'} :
e = fill K' e' -> reducible e' -> find_redex e = Some (K', e').
Proof.
revert e; induction K'; intros e Hfill Hred; subst e; simpl.
revert e; induction K' => e Hfill Hred; subst e; simpl.
- (* Base case: Empty context *)
destruct Hred as (σ' & e'' & σ'' & ef & Hstep). destruct Hstep; simpl.
+ erewrite find_redex_val by eassumption. by rewrite Hv2.
......@@ -329,6 +347,24 @@ Proof.
by eapply reducible_find_redex.
Qed.
Lemma fill_not_value e K :
e2v e = None -> e2v (fill K e) = None.
Proof.
intros Hnval. induction K =>/=; try reflexivity.
- done.
- by rewrite IHK /=.
- by rewrite v2v /= IHK /=.
- by rewrite IHK /=.
- by rewrite IHK /=.
Qed.
Lemma fill_not_value2 e K v :
e2v e = None -> e2v (fill K e) = Some v -> False.
Proof.
intros Hnval Hval. erewrite fill_not_value in Hval by assumption. discriminate.
Qed.
Section step_by_value.
(* When something does a step, and another decomposition of the same
expression has a non-value e in the hole, then K is a left
sub-context of K' - in other words, e also contains the reducible
......@@ -339,6 +375,29 @@ Lemma step_by_value K K' e e' :
e2v e = None ->
exists K'', K' = comp_ctx K K''.
Proof.
intros Hfill Hred Hnval.
assert (Hfind := reducible_find_redex Hfill Hred).
Abort.
Ltac bad_fill1 Hfill := exfalso; case: Hfill => Hfill; intros; subst; eapply fill_not_value2; first eassumption;
by erewrite Hfill, ?v2v.
Ltac bad_fill2 Hfill := exfalso; case: Hfill => Hfill; intros; subst; eapply values_stuck; eassumption.
Ltac bad_red Hfill e' Hred := exfalso; destruct e'; try discriminate; [];
case: Hfill; intros; subst; destruct Hred as (σ' & e'' & σ'' & ef & Hstep);
inversion Hstep; done || (clear Hstep; subst;
eapply fill_not_value2; last (
try match goal with [ H : _ = fill _ _ |- _ ] => erewrite <-H end; simpl;
repeat match goal with [ H : e2v _ = _ |- _ ] => erewrite H; simpl end
); eassumption || done).
Ltac good Hfill IH := case: Hfill => Hfill; intros; subst;
let K'' := fresh "K''" in edestruct IH as [K'' Hcomp]; first eassumption;
exists K''; by eauto using f_equal, f_equal2, f_equal3, v2e_inj.
intros Hfill Hred Hnval.
revert K' Hfill; induction K=>K' /= Hfill; try first [
now eexists; reflexivity
| destruct K'; simpl; try discriminate; try first [
bad_red Hfill e' Hred
| bad_fill1 Hfill
| bad_fill2 Hfill
| good Hfill IHK
]
].
Qed.
End step_by_value.
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