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 →
33 [ q0 ⇒ match n2 with [ q0 ⇒ P → P | _ ⇒ P ]
34 | q1 ⇒ match n2 with [ q1 ⇒ P → P | _ ⇒ P ]
35 | q2 ⇒ match n2 with [ q2 ⇒ P → P | _ ⇒ P ]
36 | q3 ⇒ match n2 with [ q3 ⇒ P → P | _ ⇒ P ]
39 ndefinition quatern_destruct : quatern_destruct_aux.
42 ##[ ##1: nelim n2; nnormalize; #H;
43 ##[ ##1: napply (λx:P.x)
44 ##| ##*: napply False_ind;
45 nchange with (match q0 with [ q0 ⇒ False | _ ⇒ True ]);
46 nrewrite > H; nnormalize; napply I
48 ##| ##2: nelim n2; nnormalize; #H;
49 ##[ ##2: napply (λx:P.x)
50 ##| ##*: napply False_ind;
51 nchange with (match q1 with [ q1 ⇒ False | _ ⇒ True ]);
52 nrewrite > H; nnormalize; napply I
54 ##| ##3: nelim n2; nnormalize; #H;
55 ##[ ##3: napply (λx:P.x)
56 ##| ##*: napply False_ind;
57 nchange with (match q2 with [ q2 ⇒ False | _ ⇒ True ]);
58 nrewrite > H; nnormalize; napply I
60 ##| ##4: nelim n2; nnormalize; #H;
61 ##[ ##4: napply (λx:P.x)
62 ##| ##*: napply False_ind;
63 nchange with (match q3 with [ q3 ⇒ False | _ ⇒ True ]);
64 nrewrite > H; nnormalize; napply I
69 nlemma symmetric_eqqu : symmetricT quatern bool eq_qu.
77 nlemma eqqu_to_eq : ∀n1,n2.eq_qu n1 n2 = true → n1 = n2.
82 ##[ ##1,6,11,16: #H; napply refl_eq
83 ##| ##*: #H; napply (bool_destruct … H)
87 nlemma eq_to_eqqu : ∀n1,n2.n1 = n2 → eq_qu n1 n2 = true.
92 ##[ ##1,6,11,16: #H; napply refl_eq
93 ##| ##*: #H; napply (quatern_destruct … H)
97 nlemma decidable_qu : ∀x,y:quatern.decidable (x = y).
102 ##[ ##1,6,11,16: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
103 ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (quatern_destruct … H)
107 nlemma neqqu_to_neq : ∀n1,n2.eq_qu n1 n2 = false → n1 ≠ n2.
112 ##[ ##1,6,11,16: #H; napply (bool_destruct … H)
113 ##| ##*: #H; #H1; napply (quatern_destruct … H1)
117 nlemma neq_to_neqqu : ∀n1,n2.n1 ≠ n2 → eq_qu n1 n2 = false.
122 ##[ ##1,6,11,16: #H; nelim (H (refl_eq …))
123 ##| ##*: #H; napply refl_eq