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1 (**************************************************************************)
2 (*       ___                                                              *)
3 (*      ||M||                                                             *)
4 (*      ||A||       A project by Andrea Asperti                           *)
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9 (*      \   /                                                             *)
10 (*       \ /        This file is distributed under the terms of the       *)
11 (*        v         GNU General Public License Version 2                  *)
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13 (**************************************************************************)
14
15 include "static_2/syntax/sh_lt.ma".
16 include "static_2/syntax/sd_d.ma".
17
18 (* SORT DEGREE **************************************************************)
19
20 (* Properties with sh_lt ****************************************************)
21
22 lemma deg_SO_gt (h): sh_lt h →
23       ∀s1,s2. s1 < s2 → deg_SO h s1 s2 𝟎.
24 #h #Hh #s1 #s2 #Hs12 @deg_SO_zero
25 #n @(nat_ind_succ … n) -n
26 [ <sh_nexts_zero #H destruct
27   elim (nlt_inv_refl … Hs12)
28 | #n #_ <sh_nexts_succ #H
29   lapply (sh_next_lt h Hh (⇡*[h,n]s2)) >H -H #H
30   lapply (nlt_trans … H … Hs12) -s1 #H1
31   /3 width=5 by nlt_ge_false, sh_nexts_le/ (* full auto too slow *)
32 ]
33 qed.