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 eq_to_eqqu : ∀n1,n2.n1 = n2 → eq_qu n1 n2 = true.
50 nlemma neqqu_to_neq : ∀n1,n2.eq_qu n1 n2 = false → n1 ≠ n2.
52 napply (not_to_not (n1 = n2) (eq_qu n1 n2 = true) …);
53 ##[ ##1: napply (eq_to_eqqu n1 n2)
54 ##| ##2: napply (eqfalse_to_neqtrue … H)
58 nlemma eqqu_to_eq : ∀n1,n2.eq_qu n1 n2 = true → n1 = n2.
63 ##[ ##1,6,11,16: #H; napply refl_eq
64 ##| ##*: #H; napply (bool_destruct … H)
68 nlemma neq_to_neqqu : ∀n1,n2.n1 ≠ n2 → eq_qu n1 n2 = false.
70 napply (neqtrue_to_eqfalse (eq_qu n1 n2));
71 napply (not_to_not (eq_qu n1 n2 = true) (n1 = n2) ? H);
72 napply (eqqu_to_eq n1 n2).
75 nlemma decidable_qu : ∀x,y:quatern.decidable (x = y).
77 napply (or2_elim (eq_qu x y = true) (eq_qu x y = false) ? (decidable_bexpr ?));
78 ##[ ##1: #H; napply (or2_intro1 (x = y) (x ≠ y) (eqqu_to_eq … H))
79 ##| ##2: #H; napply (or2_intro2 (x = y) (x ≠ y) (neqqu_to_neq … H))
83 nlemma symmetric_eqqu : symmetricT quatern bool eq_qu.
85 napply (or2_elim (n1 = n2) (n1 ≠ n2) ? (decidable_qu n1 n2));
86 ##[ ##1: #H; nrewrite > H; napply refl_eq
87 ##| ##2: #H; nrewrite > (neq_to_neqqu n1 n2 H);
88 napply (symmetric_eq ? (eq_qu n2 n1) false);
89 napply (neq_to_neqqu n2 n1 (symmetric_neq ? n1 n2 H))