+
+(* Basic properties *********************************************************)
+
+lemma tdeq_refl: ∀h,o. reflexive … (tdeq h o).
+#h #o #T elim T -T /2 width=1 by tdeq_pair/
+* /2 width=1 by tdeq_lref, tdeq_gref/
+#s elim (deg_total h o s) /2 width=3 by tdeq_sort/
+qed.
+
+lemma tdeq_sym: ∀h,o. symmetric … (tdeq h o).
+#h #o #T1 #T2 #H elim H -T1 -T2
+/2 width=3 by tdeq_sort, tdeq_lref, tdeq_gref, tdeq_pair/
+qed-.
+
+lemma tdeq_dec: ∀h,o,T1,T2. Decidable (T1 ≛[h, o] T2).
+#h #o #T1 elim T1 -T1 [ * #s1 | #I1 #V1 #T1 #IHV #IHT ] * [1,3,5,7: * #s2 |*: #I2 #V2 #T2 ]
+[ elim (deg_total h o s1) #d1 #H1
+ elim (deg_total h o s2) #d2 #H2
+ elim (eq_nat_dec d1 d2) #Hd12 destruct /3 width=3 by tdeq_sort, or_introl/
+ @or_intror #H
+ lapply (tdeq_inv_sort_deg … H … H1 H2) -H -H1 -H2 /2 width=1 by/
+|2,3,13:
+ @or_intror #H
+ elim (tdeq_inv_sort1 … H) -H #x1 #x2 #_ #_ #H destruct
+|4,6,14:
+ @or_intror #H
+ lapply (tdeq_inv_lref1 … H) -H #H destruct
+|5:
+ elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
+ @or_intror #H
+ lapply (tdeq_inv_lref1 … H) -H #H destruct /2 width=1 by/
+|7,8,15:
+ @or_intror #H
+ lapply (tdeq_inv_gref1 … H) -H #H destruct
+|9:
+ elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
+ @or_intror #H
+ lapply (tdeq_inv_gref1 … H) -H #H destruct /2 width=1 by/
+|10,11,12:
+ @or_intror #H
+ elim (tdeq_inv_pair1 … H) -H #X1 #X2 #_ #_ #H destruct
+|16:
+ elim (eq_item2_dec I1 I2) #HI12 destruct
+ [ elim (IHV V2) -IHV #HV12
+ elim (IHT T2) -IHT #HT12
+ [ /3 width=1 by tdeq_pair, or_introl/ ]
+ ]
+ @or_intror #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+]
+qed-.
+
+(* Negated inversion lemmas *************************************************)
+
+lemma tdneq_inv_pair: ∀h,o,I1,I2,V1,V2,T1,T2.
+ (②{I1}V1.T1 ≛[h, o] ②{I2}V2.T2 → ⊥) →
+ ∨∨ I1 = I2 → ⊥
+ | (V1 ≛[h, o] V2 → ⊥)
+ | (T1 ≛[h, o] T2 → ⊥).
+#h #o #I1 #I2 #V1 #V2 #T1 #T2 #H12
+elim (eq_item2_dec I1 I2) /3 width=1 by or3_intro0/ #H destruct
+elim (tdeq_dec h o V1 V2) /3 width=1 by or3_intro1/
+elim (tdeq_dec h o T1 T2) /4 width=1 by tdeq_pair, or3_intro2/
+qed-.