Commit 77c885d8 by Ralf Jung

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

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