+(* added from lambda_delta *)
+
+lemma TC_strap: ∀A. ∀R:relation A. ∀a1,a,a2.
+ R a1 a → TC … R a a2 → TC … R a1 a2.
+/3 width=3/ qed.
+
+lemma TC_reflexive: ∀A,R. reflexive A R → reflexive A (TC … R).
+/2 width=1/ qed.
+
+lemma TC_star_ind: ∀A,R. reflexive A R → ∀a1. ∀P:predicate A.
+ P a1 → (∀a,a2. TC … R a1 a → R a a2 → P a → P a2) →
+ ∀a2. TC … R a1 a2 → P a2.
+#A #R #H #a1 #P #Ha1 #IHa1 #a2 #Ha12 elim Ha12 -a2 /3 width=4/
+qed.
+
+inductive TC_dx (A:Type[0]) (R:relation A): A → A → Prop ≝
+ |inj_dx: ∀a,c. R a c → TC_dx A R a c
+ |step_dx : ∀a,b,c. R a b → TC_dx A R b c → TC_dx A R a c.
+
+lemma TC_dx_strap: ∀A. ∀R: relation A.
+ ∀a,b,c. TC_dx A R a b → R b c → TC_dx A R a c.
+#A #R #a #b #c #Hab elim Hab -a -b /3 width=3/
+qed.
+
+lemma TC_to_TC_dx: ∀A. ∀R: relation A.
+ ∀a1,a2. TC … R a1 a2 → TC_dx … R a1 a2.
+#A #R #a1 #a2 #Ha12 elim Ha12 -a2 /2 width=3/
+qed.
+
+lemma TC_dx_to_TC: ∀A. ∀R: relation A.
+ ∀a1,a2. TC_dx … R a1 a2 → TC … R a1 a2.
+#A #R #a1 #a2 #Ha12 elim Ha12 -a1 -a2 /2 width=3/
+qed.
+
+fact TC_ind_dx_aux: ∀A,R,a2. ∀P:predicate A.
+ (∀a1. R a1 a2 → P a1) →
+ (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
+ ∀a1,a. TC … R a1 a → a = a2 → P a1.
+#A #R #a2 #P #H1 #H2 #a1 #a #Ha1
+elim (TC_to_TC_dx ???? Ha1) -a1 -a
+[ #a #c #Hac #H destruct /2 width=1/
+| #a #b #c #Hab #Hbc #IH #H destruct /3 width=4/
+]
+qed-.
+
+lemma TC_ind_dx: ∀A,R,a2. ∀P:predicate A.
+ (∀a1. R a1 a2 → P a1) →
+ (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
+ ∀a1. TC … R a1 a2 → P a1.
+#A #R #a2 #P #H1 #H2 #a1 #Ha12
+@(TC_ind_dx_aux … H1 H2 … Ha12) //
+qed-.
+
+lemma TC_symmetric: ∀A,R. symmetric A R → symmetric A (TC … R).
+#A #R #HR #x #y #H @(TC_ind_dx ??????? H) -x /3 width=1/ /3 width=3/
+qed.
+
+lemma TC_star_ind_dx: ∀A,R. reflexive A R →
+ ∀a2. ∀P:predicate A. P a2 →
+ (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
+ ∀a1. TC … R a1 a2 → P a1.
+#A #R #HR #a2 #P #Ha2 #H #a1 #Ha12
+@(TC_ind_dx … P ? H … Ha12) /3 width=4/
+qed-.
+
+definition Conf3: ∀A,B. relation2 A B → relation A → Prop ≝ λA,B,S,R.
+ ∀b,a1. S a1 b → ∀a2. R a1 a2 → S a2 b.
+
+lemma TC_Conf3: ∀A,B,S,R. Conf3 A B S R → Conf3 A B S (TC … R).
+#A #B #S #R #HSR #b #a1 #Ha1 #a2 #H elim H -a2 /2 width=3/
+qed.