diff --git a/channel/heap_lang.v b/channel/heap_lang.v
index eab5262e6855d54419751f037087e0cfbaa628ba..91bd651a5d1c83005c484a4b7026277b88b16523 100644
--- a/channel/heap_lang.v
+++ b/channel/heap_lang.v
@@ -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.