From mininix Require Export dynlang.operational.
From stdpp Require Import options.

Module Import dynlang.
Export dynlang.

(** Properties of operational semantics *)
Lemma step_not_final e1 e2 : e1 --> e2 → ¬final e1.
Proof. induction 1; simpl; repeat case_match; naive_solver. Qed.
Lemma final_nf e : final e → nf step e.
Proof. by intros ? [??%step_not_final]. Qed.

Lemma SAbsL_rtc ex1 ex1' e : ex1 -->* ex1' → EAbs ex1 e -->* EAbs ex1' e.
Proof. induction 1; econstructor; eauto using step. Qed.
Lemma SAppL_rtc e1 e1' e2 : e1 -->* e1' → EApp e1 e2 -->* EApp e1' e2.
Proof. induction 1; econstructor; eauto using step. Qed.
Lemma SId_rtc ds e1 e1' : e1 -->* e1' → EId ds e1 -->* EId ds e1'.
Proof. induction 1; econstructor; eauto using step. Qed.

Ltac inv_step := repeat
  match goal with H : ?e --> _ |- _ => assert_fails (is_var e); inv H end.

Lemma step_det e d1 d2 :
  e --> d1 →
  e --> d2 →
  d1 = d2.
Proof.
  intros Hred1. revert d2.
  induction Hred1; intros d2 Hred2; inv Hred2; try by inv_step;
    f_equal; by apply IHHred1.
Qed.

End dynlang.