--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/tdeq.ma".
+
+(* DEGREE-BASED EQUIVALENCE ON TERMS ****************************************)
+
+(* Main properties **********************************************************)
+
+theorem tdeq_trans: ∀h,o. Transitive … (tdeq h o).
+#h #o #T1 #T #H elim H -T1 -T
+[ #s1 #s #d #Hs1 #Hs #X #H
+ elim (tdeq_inv_sort1_deg … H … Hs) -s /2 width=3 by tdeq_sort/
+| #i1 #i #H <(tdeq_inv_lref1 … H) -H //
+| #l1 #l #H <(tdeq_inv_gref1 … H) -H //
+| #I #V1 #V #T1 #T #_ #_ #IHV #IHT #X #H
+ elim (tdeq_inv_pair1 … H) -H /3 width=1 by tdeq_pair/
+]
+qed-.
+
+theorem tdeq_canc_sn: ∀h,o. left_cancellable … (tdeq h o).
+/3 width=3 by tdeq_trans, tdeq_sym/ qed-.
+
+theorem tdeq_canc_dx: ∀h,o. right_cancellable … (tdeq h o).
+/3 width=3 by tdeq_trans, tdeq_sym/ qed-.
+
+theorem tdeq_repl: ∀h,o,T1,T2. T1 ≛[h, o] T2 →
+ ∀U1. T1 ≛[h, o] U1 → ∀U2. T2 ≛[h, o] U2 → U1 ≛[h, o] U2.
+/3 width=3 by tdeq_canc_sn, tdeq_trans/ qed-.
+
+(* Negated main properies ***************************************************)
+
+theorem tdeq_tdneq_trans: ∀h,o,T1,T. T1 ≛[h, o] T → ∀T2. (T ≛[h, o] T2 → ⊥) →
+ T1 ≛[h, o] T2 → ⊥.
+/3 width=3 by tdeq_canc_sn/ qed-.
+
+theorem tdneq_tdeq_canc_dx: ∀h,o,T1,T. (T1 ≛[h, o] T → ⊥) → ∀T2. T2 ≛[h, o] T →
+ T1 ≛[h, o] T2 → ⊥.
+/3 width=3 by tdeq_trans/ qed-.