Commit 77c885d8 authored by Ralf Jung's avatar Ralf Jung

define a function to find the redex, and prove it correct

parent 6099138c
......@@ -57,10 +57,19 @@ Proof.
Qed.
Lemma e2e e v:
e2v e = Some v -> v2e v = e.
e2v e = Some v -> e = v2e v.
Proof.
(* TODO: First figure out how to best state this. *)
Abort.
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].
intros Heq. injection Heq. clear Heq. intros Heq. subst. simpl. eauto using f_equal2.
- destruct (e2v e); simpl; [|discriminate].
intros Heq. injection Heq. clear Heq. intros Heq. subst. simpl. eauto using f_equal.
- destruct (e2v e); simpl; [|discriminate].
intros Heq. injection Heq. clear Heq. intros Heq. subst. simpl. eauto using f_equal.
Qed.
Inductive ectx :=
| EmptyCtx
......@@ -131,16 +140,16 @@ Qed.
Definition state := unit.
Inductive prim_step : expr -> state -> expr -> state -> option expr -> Prop :=
| Beta e1 e2 v2 σ : e2v e2 = Some v2 ->
prim_step (App (Lam e1) e2) σ (e1.[e2/]) σ None
| FstS e1 v1 e2 v2 σ : e2v e1 = Some v1 -> e2v e2 = Some v2 ->
prim_step (Fst (Pair e1 e2)) σ e1 σ None
| SndS e1 v1 e2 v2 σ : e2v e1 = Some v1 -> e2v e2 = Some v2 ->
prim_step (Fst (Pair e1 e2)) σ e2 σ None
| CaseL e0 v0 e1 e2 σ : e2v e0 = Some v0 ->
prim_step (Case (InjL e0) e1 e2) σ (e1.[e0/]) σ None
| CaseR e0 v0 e1 e2 σ : e2v e0 = Some v0 ->
prim_step (Case (InjR e0) e1 e2) σ (e2.[e0/]) σ None.
| Beta e1 e2 v2 σ (Hv2 : e2v e2 = Some v2):
prim_step (App (Lam e1) e2) σ (e1.[e2/]) σ None
| FstS e1 v1 e2 v2 σ (Hv1 : e2v e1 = Some v1) (Hv2 : e2v e2 = Some v2):
prim_step (Fst (Pair e1 e2)) σ e1 σ None
| SndS e1 v1 e2 v2 σ (Hv1 : e2v e1 = Some v1) (Hv2 : e2v e2 = Some v2):
prim_step (Snd (Pair e1 e2)) σ e2 σ None
| CaseL e0 v0 e1 e2 σ (Hv0 : e2v e0 = Some v0):
prim_step (Case (InjL e0) e1 e2) σ (e1.[e0/]) σ None
| CaseR e0 v0 e1 e2 σ (Hv0 : e2v e0 = Some v0):
prim_step (Case (InjR e0) e1 e2) σ (e2.[e0/]) σ None.
Definition reducible e: Prop :=
exists σ e' σ' ef, prim_step e σ e' σ' ef.
......@@ -158,3 +167,174 @@ Proof.
{ by rewrite <-Heq, v2v. }
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'; intros 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.
(* 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
expression *)
Lemma step_by_value K K' e e' :
fill K e = fill K' e' ->
reducible e' ->
e2v e = None ->
exists K'', K' = comp_ctx K K''.
Proof.
(* TODO *)
Abort.
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