+
+lemma cpm_tdeq_inv_atom_sn (n) (h) (I) (G) (L):
+ ∀X. ⦃G,L⦄ ⊢ ⓪{I} ➡[n,h] X → ⓪{I} ≛ X →
+ ∨∨ ∧∧ X = ⓪{I} & n = 0
+ | ∃∃s. X = ⋆(⫯[h]s) & I = Sort s & n = 1.
+#n #h * #s #G #L #X #H1 #H2
+[ elim (cpm_inv_sort1 … H1) -H1
+ cases n -n [| #n ] #H #Hn destruct -H2
+ [ /3 width=1 by or_introl, conj/
+ | <(le_n_O_to_eq n) [| /2 width=3 by le_S_S_to_le/ ] -n
+ /3 width=3 by ex3_intro, or_intror/
+ ]
+| elim (cpm_tdeq_inv_lref_sn … H1 H2) -H1 -H2 /3 width=1 by or_introl, conj/
+| elim (cpm_inv_gref1 … H1) -H1 -H2 /3 width=1 by or_introl, conj/
+]
+qed-.