+
+lemma tpr_thom: ∀T1,T2. T1 ⇒ T2 → T1 ≈ T1 → T1 ≈ T2.
+#T1 #T2 #H elim H -T1 -T2 //
+[ #I #V1 #V2 #T1 #T2 #_ #_ #_ #IHT12 #H
+ elim (thom_inv_flat1 … H) -H #W2 #U2 #HT1U2 #HT1 #_ #H1 #H2 destruct
+ lapply (IHT12 HT1U2) -IHT12 -HT1U2 #HUT2
+ lapply (simple_thom_repl_dx … HUT2 HT1) /2 width=1/
+| #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #H
+ elim (thom_inv_flat1 … H) -H #W2 #U2 #_ #H
+ elim (simple_inv_bind … H)
+| #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #_ #_ #H
+ elim (thom_inv_bind1 … H) -H #W2 #U2 #H destruct //
+| #V2 #V1 #V #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H
+ elim (thom_inv_flat1 … H) -H #U1 #U2 #_ #H
+ elim (simple_inv_bind … H)
+| #V #T #T1 #T2 #_ #_ #_ #H
+ elim (thom_inv_bind1 … H) -H #W2 #U2 #H destruct
+| #V #T1 #T2 #_ #_ #H
+ elim (thom_inv_flat1 … H) -H #W2 #U2 #_ #_ #_ #H destruct
+]
+qed.
+
+lemma twhnf_thom: ∀T. T ≈ T → 𝕎ℍℕ[T].
+/2 width=1/ qed.