@(le_to_lt_to_lt … H) /2 width=1 by next_lt/
qed.
-axiom sh_lt_dec (h): sh_lt h → sh_decidable h.
+lemma sh_lt_nexts_inv_lt (h): sh_lt h →
+ ∀s,n1,n2. (next h)^n1 s < (next h)^n2 s → n1 < n2.
+#h #Hh
+@pull_2 #n1
+elim n1 -n1
+[ #s *
+ [ #H elim (lt_refl_false … H)
+ | #n2 //
+ ]
+| #n1 #IH #s *
+ [ >iter_O #H
+ elim (lt_refl_false s)
+ /3 width=3 by nexts_lt, transitive_lt/
+ | #n2 >iter_S >iter_S <(iter_n_Sm … (next h)) <(iter_n_Sm … (next h)) #H
+ /3 width=2 by lt_S_S/
+ ]
+]
+qed-.
-axiom sh_lt_acyclic (h): sh_lt h → sh_acyclic h.
+lemma sh_lt_acyclic (h): sh_lt h → sh_acyclic h.
+#h #Hh
+@mk_sh_acyclic
+@pull_2 #n1
+elim n1 -n1
+[ #s * [ // ] #n2 >iter_O #H
+ elim (lt_refl_false s) >H in ⊢ (??%); -H
+ /2 width=1 by nexts_lt/
+| #n1 #IH #s *
+ [ >iter_O #H -IH
+ elim (lt_refl_false s) <H in ⊢ (??%); -H
+ /2 width=1 by nexts_lt/
+ | #n2 >iter_S >iter_S <(iter_n_Sm … (next h)) <(iter_n_Sm … (next h)) #H
+ /3 width=2 by eq_f/
+ ]
+]
+qed.
+
+lemma sh_lt_dec (h): sh_lt h → sh_decidable h.
+#h #Hh
+@mk_sh_decidable #s1 #s2
+elim (lt_or_ge s2 s1) #Hs
+[ @or_intror * #n #H destruct
+ @(lt_le_false … Hs) /2 width=1 by nexts_le/ (**) (* full auto too slow *)
+| @(nat_elim_le_sn … Hs) -s1 -s2 #s1 #s2 #IH #Hs12
+ elim (lt_or_eq_or_gt s2 (⫯[h]s1)) #Hs21 destruct
+ [ elim (le_to_or_lt_eq … Hs12) -Hs12 #Hs12 destruct
+ [ -IH @or_intror * #n #H destruct
+ generalize in match Hs21; -Hs21
+ <(iter_O … (next h) s1) in ⊢ (??%→?); <(iter_S … (next h)) #H
+ lapply (sh_lt_nexts_inv_lt … Hh … H) -H #H
+ <(le_n_O_to_eq n) in Hs12;
+ /2 width=2 by lt_refl_false, le_S_S_to_le/
+ | /3 width=2 by ex_intro, or_introl/
+ ]
+ | -IH @or_introl @(ex_intro … 1) // (**) (* auto fails *)
+ | lapply (transitive_lt s1 ??? Hs21) [ /2 width=1 by next_lt/ ] -Hs12 #Hs12
+ elim (IH (s2-⫯[h]s1)) -IH
+ [3: /3 width=1 by next_lt, monotonic_lt_minus_r/ ]
+ >minus_minus_m_m [2,4: /2 width=1 by lt_to_le/ ] -Hs21
+ [ * #n #H destruct
+ @or_introl @(ex_intro … (↑n)) >iter_S >iter_n_Sm //
+ | #H1 @or_intror * #n #H2 @H1 -H1 destruct
+ generalize in match Hs12; -Hs12
+ <(iter_O … (next h) s1) in ⊢ (?%?→?); #H
+ lapply (sh_lt_nexts_inv_lt … Hh … H) -H #H
+ <(S_pred … H) -H
+ @(ex_intro … (↓n)) >(iter_n_Sm … (next h)) >iter_S //
+ ]
+ ]
+]
+qed-.