+lemma test_char_inv :
+ ∀sig.∀P:tape sig → Prop.∀f,t,t0.P t → Rtc_true sig f t t0 → P t0.
+#sig #P #f #t #t0 #HPt * #_ //
+qed.
+
+definition test_null ≝ λalpha.test_char alpha (λ_.true).
+
+definition R_test_null_true ≝ λalpha,t1,t2.
+ current alpha t1 ≠ None ? ∧ t1 = t2.
+
+definition R_test_null_false ≝ λalpha,t1,t2.
+ current alpha t1 = None ? ∧ t1 = t2.
+
+lemma sem_test_null : ∀alpha.
+ test_null alpha ⊨ [ tc_true : R_test_null_true alpha, R_test_null_false alpha].
+#alpha #t1 cases (sem_test_char alpha (λ_.true) t1) #k * #outc * * #Hloop #Htrue
+#Hfalse %{k} %{outc} % [ %
+[ @Hloop
+| #Houtc cases (Htrue ?) [| @Houtc] * #c * #Hcurt1 #_ #Houtc1 %
+ [ >Hcurt1 % #H destruct (H) | <Houtc1 % ] ]
+| #Houtc cases (Hfalse ?) [| @Houtc] #Habsurd #Houtc %
+ [ cases (current alpha t1) in Habsurd; // #c1 #Habsurd
+ lapply (Habsurd ? (refl ??)) #H destruct (H)
+ | <Houtc % ] ]
+qed.
+