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 (* ********************************************************************** *)
16 (* Progetto FreeScale *)
18 (* Sviluppato da: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
19 (* Ultima modifica: 05/08/2009 *)
21 (* ********************************************************************** *)
23 include "num/quatern.ma".
24 include "num/bool_lemmas.ma".
30 ndefinition quatern_destruct_aux ≝
31 Πn1,n2:quatern.ΠP:Prop.n1 = n2 →
32 match eq_qu n1 n2 with [ true ⇒ P → P | false ⇒ P ].
34 ndefinition quatern_destruct : quatern_destruct_aux.
42 nlemma symmetric_eqqu : symmetricT quatern bool eq_qu.
50 nlemma eqqu_to_eq : ∀n1,n2.eq_qu n1 n2 = true → n1 = n2.
55 ##[ ##1,6,11,16: #H; napply refl_eq
56 ##| ##*: #H; napply (bool_destruct … H)
60 nlemma eq_to_eqqu : ∀n1,n2.n1 = n2 → eq_qu n1 n2 = true.
65 ##[ ##1,6,11,16: #H; napply refl_eq
66 ##| ##*: #H; napply (quatern_destruct … H)
70 nlemma decidable_qu : ∀x,y:quatern.decidable (x = y).
75 ##[ ##1,6,11,16: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
76 ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …);
79 napply (quatern_destruct … H)
83 nlemma neqqu_to_neq : ∀n1,n2.eq_qu n1 n2 = false → n1 ≠ n2.
88 ##[ ##1,6,11,16: #H; napply (bool_destruct … H)
89 ##| ##*: #H; #H1; napply (quatern_destruct … H1)
93 nlemma neq_to_neqqu : ∀n1,n2.n1 ≠ n2 → eq_qu n1 n2 = false.
98 ##[ ##1,6,11,16: #H; nelim (H (refl_eq …))
99 ##| ##*: #H; napply refl_eq