1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "static_2/syntax/sort.ma".
17 (* SORT HIERARCHY ***********************************************************)
19 record is_lt (h): Prop ≝
21 next_lt: ∀s. s < ⫯[h]s (* strict monotonicity condition *)
24 (* Basic properties *********************************************************)
26 lemma nexts_le (h): is_lt h → ∀s,n. s ≤ (next h)^n s.
27 #h #Hh #s #n elim n -n [ // ] normalize #n #IH
28 lapply (next_lt … Hh ((next h)^n s)) #H
29 lapply (le_to_lt_to_lt … IH H) -IH -H /2 width=2 by lt_to_le/
32 lemma nexts_lt (h): is_lt h → ∀s,n. s < (next h)^(↑n) s.
33 #h #Hh #s #n normalize
34 lapply (nexts_le … Hh s n) #H
35 @(le_to_lt_to_lt … H) /2 width=1 by next_lt/