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diff --git a/theories/lambda/operational_props.v b/theories/lambda/operational_props.v
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+From mininix Require Export lambda.operational.
+From stdpp Require Import options.
+
+Module Import lambda.
+Export lambda.
+
+(** Properties of operational semantics *)
+Lemma step_not_final e1 e2 : e1 --> e2 → ¬final e1.
+Proof. induction 1; inv 1; naive_solver. Qed.
+Lemma final_nf e : final e → nf step e.
+Proof. by intros ? [??%step_not_final]. Qed.
+
+Lemma SAppL_rtc e1 e1' e2 : e1 -->* e1' → EApp e1 e2 -->* EApp e1' e2.
+Proof. induction 1; repeat (done || econstructor). 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 lambda.