Commit 8d840a4b authored by Ralf Jung's avatar Ralf Jung

find_redex: complicated and not useful

parent 25e9b4b1
......@@ -189,163 +189,6 @@ Proof.
clear -Hv'. intros (σ' & e'' & σ'' & ef & Hstep). destruct Hstep; simpl in *; discriminate.
Qed.
(* TODO RJ: Isn't there a shorter way to define this? Or maybe we don't need it? *)
Fixpoint find_redex (e : expr) : option (ectx * expr) :=
match e with
| Var _ => None
| Lit _ _ => None
| App e1 e2 => match find_redex e1 with
| Some (K', e') => Some (AppLCtx K' e2, e')
| None => match find_redex e2, e2v e1 with
| Some (K', e'), Some v1 => Some (AppRCtx v1 K', e')
| None, Some (LamV e1') => match e2v e2 with
| Some v2 => Some (EmptyCtx, App e1 e2)
| None => None
end
| _, _ => None (* cannot happen *)
end
end
| Lam _ => None
| Pair e1 e2 => match find_redex e1 with
| Some (K', e') => Some (PairLCtx K' e2, e')
| None => match find_redex e2, e2v e1 with
| Some (K', e'), Some v1 => Some (PairRCtx v1 K', e')
| _, _ => None
end
end
| Fst e => match find_redex e with
| Some (K', e') => Some (FstCtx K', e')
| None => match e2v e with
| Some (PairV v1 v2) => Some (EmptyCtx, Fst e)
| _ => None
end
end
| Snd e => match find_redex e with
| Some (K', e') => Some (SndCtx K', e')
| None => match e2v e with
| Some (PairV v1 v2) => Some (EmptyCtx, Snd e)
| _ => None
end
end
| InjL e => '(e', K') find_redex e; Some (InjLCtx e', K')
| InjR e => '(e', K') find_redex e; Some (InjRCtx e', K')
| Case e0 e1 e2 => match find_redex e0 with
| Some (K', e') => Some (CaseCtx K' e1 e2, e')
| None => match e2v e0 with
| Some (InjLV v0') => Some (EmptyCtx, Case e0 e1 e2)
| Some (InjRV v0') => Some (EmptyCtx, Case e0 e1 e2)
| _ => None
end
end
end.
Lemma find_redex_found e K' e' :
find_redex e = Some (K', e') -> reducible e' /\ e = fill K' e'.
Proof.
revert K' e'; induction e; intros K' e'; simpl; try discriminate.
- destruct (find_redex e1) as [[K1' e1']|].
+ intros Heq; inversion Heq. edestruct IHe1; [reflexivity|].
simpl; subst; eauto.
+ destruct (find_redex e2) as [[K2' e2']|].
* case_eq (e2v e1); [|discriminate]; intros v1 Hv1.
intros Heq; inversion Heq. edestruct IHe2; [reflexivity|].
simpl; subst; eauto using f_equal2, e2e.
* case_eq (e2v e1); [|discriminate]; intros v1 Hv1; destruct v1; try discriminate; [].
case_eq (e2v e2); [|discriminate]; intros v2 Hv2. apply e2e in Hv1. apply e2e in Hv2.
intros Heq; inversion Heq; subst. split; [|reflexivity].
do 4 eexists. eapply Beta with (σ := tt), v2v.
- destruct (find_redex e1) as [[K1' e1']|].
+ intros Heq; inversion Heq. edestruct IHe1; [reflexivity|].
simpl; subst; eauto.
+ destruct (find_redex e2) as [[K2' e2']|]; [|discriminate].
case_eq (e2v e1); [|discriminate]; intros v1 Hv1.
intros Heq; inversion Heq. edestruct IHe2; [reflexivity|].
simpl; subst; eauto using f_equal2, e2e.
- destruct (find_redex e) as [[K1' e1']|].
+ intros Heq; inversion Heq. edestruct IHe; [reflexivity|].
simpl; subst; eauto.
+ case_eq (e2v e); [|discriminate]; intros v1 Hv1; destruct v1; try discriminate; []. apply e2e in Hv1.
intros Heq; inversion Heq; subst. split; [|reflexivity].
do 4 eexists. eapply FstS with (σ := tt); fold v2e; eapply v2v. (* RJ: Why do I have to fold here? *)
- destruct (find_redex e) as [[K1' e1']|].
+ intros Heq; inversion Heq. edestruct IHe; [reflexivity|].
simpl; subst; eauto.
+ case_eq (e2v e); [|discriminate]; intros v1 Hv1; destruct v1; try discriminate; []. apply e2e in Hv1.
intros Heq; inversion Heq; subst. split; [|reflexivity].
do 4 eexists. eapply SndS with (σ := tt); fold v2e; eapply v2v. (* RJ: Why do I have to fold here? *)
- destruct (find_redex e) as [[K1' e1']|]; simpl; [|discriminate].
intros Heq; inversion Heq. edestruct IHe; [reflexivity|].
simpl; subst; eauto.
- destruct (find_redex e) as [[K1' e1']|]; simpl; [|discriminate].
intros Heq; inversion Heq. edestruct IHe; [reflexivity|].
simpl; subst; eauto.
- destruct (find_redex e) as [[K1' e1']|]; simpl.
+ intros Heq; inversion Heq. edestruct IHe; [reflexivity|].
simpl; subst; eauto.
+ case_eq (e2v e); [|discriminate]; intros v1 Hv1; destruct v1; try discriminate; [|]; apply e2e in Hv1.
* intros Heq; inversion Heq; subst. split; [|reflexivity].
do 4 eexists. eapply CaseL with (σ := tt), v2v.
* intros Heq; inversion Heq; subst. split; [|reflexivity].
do 4 eexists. eapply CaseR with (σ := tt), v2v.
Qed.
Lemma find_redex_reducible e K' e' :
find_redex e = Some (K', e') -> reducible e'.
Proof.
eapply find_redex_found.
Qed.
Lemma find_redex_fill e K' e' :
find_redex e = Some (K', e') -> e = fill K' e'.
Proof.
eapply find_redex_found.
Qed.
Lemma stuck_find_redex e :
stuck e -> find_redex e = None.
Proof.
intros Hstuck. case_eq (find_redex e); [|reflexivity]. intros [K' e'] Hfind. exfalso.
eapply Hstuck; eauto using find_redex_fill, find_redex_reducible.
Qed.
Lemma find_redex_val e v :
e2v e = Some v -> find_redex e = None.
Proof.
intros Heq. apply e2e in Heq. subst. eauto using stuck_find_redex, values_stuck.
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' => 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.
+ erewrite find_redex_val by eassumption. erewrite find_redex_val by eassumption.
by rewrite Hv1 Hv2.
+ erewrite find_redex_val by eassumption. erewrite find_redex_val by eassumption.
by rewrite Hv1 Hv2.
+ erewrite find_redex_val by eassumption. by rewrite Hv0.
+ erewrite find_redex_val by eassumption. by rewrite Hv0.
- by erewrite IHK'.
- erewrite find_redex_val by eapply v2v. by erewrite IHK'; rewrite ?v2v.
- by erewrite IHK'.
- erewrite find_redex_val by eapply v2v. by erewrite IHK'; rewrite ?v2v.
- by erewrite IHK'.
- by erewrite IHK'.
- by erewrite IHK'.
- by erewrite IHK'.
- by erewrite IHK'.
Qed.
Lemma find_redex_stuck e :
find_redex e = None -> stuck e.
Proof.
intros Hfind K' e' Hstuck Hred.
cut (find_redex e = Some (K', e')).
{ by rewrite Hfind. }
by eapply reducible_find_redex.
Qed.
Lemma fill_not_value e K :
e2v e = None -> e2v (fill K e) = None.
......
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