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/exadecim.ma".
24 include "num/bool_lemmas.ma".
30 ndefinition exadecim_destruct_aux ≝
31 Πe1,e2.ΠP:Prop.ΠH:e1 = e2.
32 match eq_ex e1 e2 with [ true ⇒ P → P | false ⇒ P ].
34 ndefinition exadecim_destruct : exadecim_destruct_aux.
42 nlemma symmetric_andex : symmetricT exadecim exadecim and_ex.
50 nlemma associative_andex1 : ∀e2,e3.(and_ex (and_ex x0 e2) e3) = (and_ex x0 (and_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
51 nlemma associative_andex2 : ∀e2,e3.(and_ex (and_ex x1 e2) e3) = (and_ex x1 (and_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
52 nlemma associative_andex3 : ∀e2,e3.(and_ex (and_ex x2 e2) e3) = (and_ex x2 (and_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
53 nlemma associative_andex4 : ∀e2,e3.(and_ex (and_ex x3 e2) e3) = (and_ex x3 (and_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
54 nlemma associative_andex5 : ∀e2,e3.(and_ex (and_ex x4 e2) e3) = (and_ex x4 (and_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
55 nlemma associative_andex6 : ∀e2,e3.(and_ex (and_ex x5 e2) e3) = (and_ex x5 (and_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
56 nlemma associative_andex7 : ∀e2,e3.(and_ex (and_ex x6 e2) e3) = (and_ex x6 (and_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
57 nlemma associative_andex8 : ∀e2,e3.(and_ex (and_ex x7 e2) e3) = (and_ex x7 (and_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
58 nlemma associative_andex9 : ∀e2,e3.(and_ex (and_ex x8 e2) e3) = (and_ex x8 (and_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
59 nlemma associative_andex10 : ∀e2,e3.(and_ex (and_ex x9 e2) e3) = (and_ex x9 (and_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
60 nlemma associative_andex11 : ∀e2,e3.(and_ex (and_ex xA e2) e3) = (and_ex xA (and_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
61 nlemma associative_andex12 : ∀e2,e3.(and_ex (and_ex xB e2) e3) = (and_ex xB (and_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
62 nlemma associative_andex13 : ∀e2,e3.(and_ex (and_ex xC e2) e3) = (and_ex xC (and_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
63 nlemma associative_andex14 : ∀e2,e3.(and_ex (and_ex xD e2) e3) = (and_ex xD (and_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
64 nlemma associative_andex15 : ∀e2,e3.(and_ex (and_ex xE e2) e3) = (and_ex xE (and_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
65 nlemma associative_andex16 : ∀e2,e3.(and_ex (and_ex xF e2) e3) = (and_ex xF (and_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
67 nlemma associative_andex : ∀e1,e2,e3.(and_ex (and_ex e1 e2) e3) = (and_ex e1 (and_ex e2 e3)).
69 ##[ ##1: napply associative_andex1 ##| ##2: napply associative_andex2
70 ##| ##3: napply associative_andex3 ##| ##4: napply associative_andex4
71 ##| ##5: napply associative_andex5 ##| ##6: napply associative_andex6
72 ##| ##7: napply associative_andex7 ##| ##8: napply associative_andex8
73 ##| ##9: napply associative_andex9 ##| ##10: napply associative_andex10
74 ##| ##11: napply associative_andex11 ##| ##12: napply associative_andex12
75 ##| ##13: napply associative_andex13 ##| ##14: napply associative_andex14
76 ##| ##15: napply associative_andex15 ##| ##16: napply associative_andex16
80 nlemma symmetric_orex : symmetricT exadecim exadecim or_ex.
88 nlemma associative_orex1 : ∀e2,e3.(or_ex (or_ex x0 e2) e3) = (or_ex x0 (or_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
89 nlemma associative_orex2 : ∀e2,e3.(or_ex (or_ex x1 e2) e3) = (or_ex x1 (or_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
90 nlemma associative_orex3 : ∀e2,e3.(or_ex (or_ex x2 e2) e3) = (or_ex x2 (or_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
91 nlemma associative_orex4 : ∀e2,e3.(or_ex (or_ex x3 e2) e3) = (or_ex x3 (or_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
92 nlemma associative_orex5 : ∀e2,e3.(or_ex (or_ex x4 e2) e3) = (or_ex x4 (or_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
93 nlemma associative_orex6 : ∀e2,e3.(or_ex (or_ex x5 e2) e3) = (or_ex x5 (or_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
94 nlemma associative_orex7 : ∀e2,e3.(or_ex (or_ex x6 e2) e3) = (or_ex x6 (or_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
95 nlemma associative_orex8 : ∀e2,e3.(or_ex (or_ex x7 e2) e3) = (or_ex x7 (or_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
96 nlemma associative_orex9 : ∀e2,e3.(or_ex (or_ex x8 e2) e3) = (or_ex x8 (or_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
97 nlemma associative_orex10 : ∀e2,e3.(or_ex (or_ex x9 e2) e3) = (or_ex x9 (or_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
98 nlemma associative_orex11 : ∀e2,e3.(or_ex (or_ex xA e2) e3) = (or_ex xA (or_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
99 nlemma associative_orex12 : ∀e2,e3.(or_ex (or_ex xB e2) e3) = (or_ex xB (or_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
100 nlemma associative_orex13 : ∀e2,e3.(or_ex (or_ex xC e2) e3) = (or_ex xC (or_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
101 nlemma associative_orex14 : ∀e2,e3.(or_ex (or_ex xD e2) e3) = (or_ex xD (or_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
102 nlemma associative_orex15 : ∀e2,e3.(or_ex (or_ex xE e2) e3) = (or_ex xE (or_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
103 nlemma associative_orex16 : ∀e2,e3.(or_ex (or_ex xF e2) e3) = (or_ex xF (or_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
105 nlemma associative_orex : ∀e1,e2,e3.(or_ex (or_ex e1 e2) e3) = (or_ex e1 (or_ex e2 e3)).
107 ##[ ##1: napply associative_orex1 ##| ##2: napply associative_orex2
108 ##| ##3: napply associative_orex3 ##| ##4: napply associative_orex4
109 ##| ##5: napply associative_orex5 ##| ##6: napply associative_orex6
110 ##| ##7: napply associative_orex7 ##| ##8: napply associative_orex8
111 ##| ##9: napply associative_orex9 ##| ##10: napply associative_orex10
112 ##| ##11: napply associative_orex11 ##| ##12: napply associative_orex12
113 ##| ##13: napply associative_orex13 ##| ##14: napply associative_orex14
114 ##| ##15: napply associative_orex15 ##| ##16: napply associative_orex16
118 nlemma symmetric_xorex : symmetricT exadecim exadecim xor_ex.
126 nlemma associative_xorex1 : ∀e2,e3.(xor_ex (xor_ex x0 e2) e3) = (xor_ex x0 (xor_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
127 nlemma associative_xorex2 : ∀e2,e3.(xor_ex (xor_ex x1 e2) e3) = (xor_ex x1 (xor_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
128 nlemma associative_xorex3 : ∀e2,e3.(xor_ex (xor_ex x2 e2) e3) = (xor_ex x2 (xor_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
129 nlemma associative_xorex4 : ∀e2,e3.(xor_ex (xor_ex x3 e2) e3) = (xor_ex x3 (xor_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
130 nlemma associative_xorex5 : ∀e2,e3.(xor_ex (xor_ex x4 e2) e3) = (xor_ex x4 (xor_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
131 nlemma associative_xorex6 : ∀e2,e3.(xor_ex (xor_ex x5 e2) e3) = (xor_ex x5 (xor_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
132 nlemma associative_xorex7 : ∀e2,e3.(xor_ex (xor_ex x6 e2) e3) = (xor_ex x6 (xor_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
133 nlemma associative_xorex8 : ∀e2,e3.(xor_ex (xor_ex x7 e2) e3) = (xor_ex x7 (xor_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
134 nlemma associative_xorex9 : ∀e2,e3.(xor_ex (xor_ex x8 e2) e3) = (xor_ex x8 (xor_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
135 nlemma associative_xorex10 : ∀e2,e3.(xor_ex (xor_ex x9 e2) e3) = (xor_ex x9 (xor_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
136 nlemma associative_xorex11 : ∀e2,e3.(xor_ex (xor_ex xA e2) e3) = (xor_ex xA (xor_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
137 nlemma associative_xorex12 : ∀e2,e3.(xor_ex (xor_ex xB e2) e3) = (xor_ex xB (xor_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
138 nlemma associative_xorex13 : ∀e2,e3.(xor_ex (xor_ex xC e2) e3) = (xor_ex xC (xor_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
139 nlemma associative_xorex14 : ∀e2,e3.(xor_ex (xor_ex xD e2) e3) = (xor_ex xD (xor_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
140 nlemma associative_xorex15 : ∀e2,e3.(xor_ex (xor_ex xE e2) e3) = (xor_ex xE (xor_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
141 nlemma associative_xorex16 : ∀e2,e3.(xor_ex (xor_ex xF e2) e3) = (xor_ex xF (xor_ex e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
143 nlemma associative_xorex : ∀e1,e2,e3.(xor_ex (xor_ex e1 e2) e3) = (xor_ex e1 (xor_ex e2 e3)).
145 ##[ ##1: napply associative_xorex1 ##| ##2: napply associative_xorex2
146 ##| ##3: napply associative_xorex3 ##| ##4: napply associative_xorex4
147 ##| ##5: napply associative_xorex5 ##| ##6: napply associative_xorex6
148 ##| ##7: napply associative_xorex7 ##| ##8: napply associative_xorex8
149 ##| ##9: napply associative_xorex9 ##| ##10: napply associative_xorex10
150 ##| ##11: napply associative_xorex11 ##| ##12: napply associative_xorex12
151 ##| ##13: napply associative_xorex13 ##| ##14: napply associative_xorex14
152 ##| ##15: napply associative_xorex15 ##| ##16: napply associative_xorex16
156 nlemma symmetric_plusex_dc_dc : ∀e1,e2,c.plus_ex_dc_dc e1 e2 c = plus_ex_dc_dc e2 e1 c.
165 nlemma plusex_dc_dc_to_dc_d : ∀e1,e2,c.fst … (plus_ex_dc_dc e1 e2 c) = plus_ex_dc_d e1 e2 c.
174 nlemma plusex_dc_dc_to_dc_c : ∀e1,e2,c.snd … (plus_ex_dc_dc e1 e2 c) = plus_ex_dc_c e1 e2 c.
183 nlemma plusex_dc_dc_to_d_dc : ∀e1,e2.plus_ex_dc_dc e1 e2 false = plus_ex_d_dc e1 e2.
191 nlemma plusex_dc_dc_to_d_d : ∀e1,e2.fst … (plus_ex_dc_dc e1 e2 false) = plus_ex_d_d e1 e2.
199 nlemma plusex_dc_dc_to_d_c : ∀e1,e2.snd … (plus_ex_dc_dc e1 e2 false) = plus_ex_d_c e1 e2.
207 nlemma symmetric_plusex_dc_d : ∀e1,e2,c.plus_ex_dc_d e1 e2 c = plus_ex_dc_d e2 e1 c.
216 nlemma symmetric_plusex_dc_c : ∀e1,e2,c.plus_ex_dc_c e1 e2 c = plus_ex_dc_c e2 e1 c.
225 nlemma symmetric_plusex_d_dc : ∀e1,e2.plus_ex_d_dc e1 e2 = plus_ex_d_dc e2 e1.
233 nlemma plusex_d_dc_to_d_d : ∀e1,e2.fst … (plus_ex_d_dc e1 e2) = plus_ex_d_d e1 e2.
241 nlemma plusex_d_dc_to_d_c : ∀e1,e2.snd … (plus_ex_d_dc e1 e2) = plus_ex_d_c e1 e2.
249 nlemma symmetric_plusex_d_d : ∀e1,e2.plus_ex_d_d e1 e2 = plus_ex_d_d e2 e1.
257 nlemma associative_plusex_d_d1 : ∀e2,e3.(plus_ex_d_d (plus_ex_d_d x0 e2) e3) = (plus_ex_d_d x0 (plus_ex_d_d e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
258 nlemma associative_plusex_d_d2 : ∀e2,e3.(plus_ex_d_d (plus_ex_d_d x1 e2) e3) = (plus_ex_d_d x1 (plus_ex_d_d e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
259 nlemma associative_plusex_d_d3 : ∀e2,e3.(plus_ex_d_d (plus_ex_d_d x2 e2) e3) = (plus_ex_d_d x2 (plus_ex_d_d e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
260 nlemma associative_plusex_d_d4 : ∀e2,e3.(plus_ex_d_d (plus_ex_d_d x3 e2) e3) = (plus_ex_d_d x3 (plus_ex_d_d e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
261 nlemma associative_plusex_d_d5 : ∀e2,e3.(plus_ex_d_d (plus_ex_d_d x4 e2) e3) = (plus_ex_d_d x4 (plus_ex_d_d e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
262 nlemma associative_plusex_d_d6 : ∀e2,e3.(plus_ex_d_d (plus_ex_d_d x5 e2) e3) = (plus_ex_d_d x5 (plus_ex_d_d e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
263 nlemma associative_plusex_d_d7 : ∀e2,e3.(plus_ex_d_d (plus_ex_d_d x6 e2) e3) = (plus_ex_d_d x6 (plus_ex_d_d e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
264 nlemma associative_plusex_d_d8 : ∀e2,e3.(plus_ex_d_d (plus_ex_d_d x7 e2) e3) = (plus_ex_d_d x7 (plus_ex_d_d e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
265 nlemma associative_plusex_d_d9 : ∀e2,e3.(plus_ex_d_d (plus_ex_d_d x8 e2) e3) = (plus_ex_d_d x8 (plus_ex_d_d e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
266 nlemma associative_plusex_d_d10 : ∀e2,e3.(plus_ex_d_d (plus_ex_d_d x9 e2) e3) = (plus_ex_d_d x9 (plus_ex_d_d e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
267 nlemma associative_plusex_d_d11 : ∀e2,e3.(plus_ex_d_d (plus_ex_d_d xA e2) e3) = (plus_ex_d_d xA (plus_ex_d_d e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
268 nlemma associative_plusex_d_d12 : ∀e2,e3.(plus_ex_d_d (plus_ex_d_d xB e2) e3) = (plus_ex_d_d xB (plus_ex_d_d e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
269 nlemma associative_plusex_d_d13 : ∀e2,e3.(plus_ex_d_d (plus_ex_d_d xC e2) e3) = (plus_ex_d_d xC (plus_ex_d_d e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
270 nlemma associative_plusex_d_d14 : ∀e2,e3.(plus_ex_d_d (plus_ex_d_d xD e2) e3) = (plus_ex_d_d xD (plus_ex_d_d e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
271 nlemma associative_plusex_d_d15 : ∀e2,e3.(plus_ex_d_d (plus_ex_d_d xE e2) e3) = (plus_ex_d_d xE (plus_ex_d_d e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
272 nlemma associative_plusex_d_d16 : ∀e2,e3.(plus_ex_d_d (plus_ex_d_d xF e2) e3) = (plus_ex_d_d xF (plus_ex_d_d e2 e3)). #e2; #e3; nelim e2; nelim e3; nnormalize; napply refl_eq. nqed.
274 nlemma associative_plusex_d_d : ∀e1,e2,e3.(plus_ex_d_d (plus_ex_d_d e1 e2) e3) = (plus_ex_d_d e1 (plus_ex_d_d e2 e3)).
276 ##[ ##1: napply associative_plusex_d_d1 ##| ##2: napply associative_plusex_d_d2
277 ##| ##3: napply associative_plusex_d_d3 ##| ##4: napply associative_plusex_d_d4
278 ##| ##5: napply associative_plusex_d_d5 ##| ##6: napply associative_plusex_d_d6
279 ##| ##7: napply associative_plusex_d_d7 ##| ##8: napply associative_plusex_d_d8
280 ##| ##9: napply associative_plusex_d_d9 ##| ##10: napply associative_plusex_d_d10
281 ##| ##11: napply associative_plusex_d_d11 ##| ##12: napply associative_plusex_d_d12
282 ##| ##13: napply associative_plusex_d_d13 ##| ##14: napply associative_plusex_d_d14
283 ##| ##15: napply associative_plusex_d_d15 ##| ##16: napply associative_plusex_d_d16
287 nlemma symmetric_plusex_d_c : ∀e1,e2.plus_ex_d_c e1 e2 = plus_ex_d_c e2 e1.
295 nlemma eq_to_eqex : ∀n1,n2.n1 = n2 → eq_ex n1 n2 = true.
303 nlemma neqex_to_neq : ∀n1,n2.eq_ex n1 n2 = false → n1 ≠ n2.
305 napply (not_to_not (n1 = n2) (eq_ex n1 n2 = true) …);
306 ##[ ##1: napply (eq_to_eqex n1 n2)
307 ##| ##2: napply (eqfalse_to_neqtrue … H)
311 nlemma eqex_to_eq : ∀n1,n2.eq_ex n1 n2 = true → n1 = n2.
316 ##[ ##1,18,35,52,69,86,103,120,137,154,171,188,205,222,239,256: #H; napply refl_eq
317 ##| ##*: #H; napply (bool_destruct … H)
321 nlemma neq_to_neqex : ∀n1,n2.n1 ≠ n2 → eq_ex n1 n2 = false.
323 napply (neqtrue_to_eqfalse (eq_ex n1 n2));
324 napply (not_to_not (eq_ex n1 n2 = true) (n1 = n2) ? H);
325 napply (eqex_to_eq n1 n2).
328 nlemma decidable_ex : ∀x,y:exadecim.decidable (x = y).
330 napply (or2_elim (eq_ex x y = true) (eq_ex x y = false) ? (decidable_bexpr ?));
331 ##[ ##1: #H; napply (or2_intro1 (x = y) (x ≠ y) (eqex_to_eq … H))
332 ##| ##2: #H; napply (or2_intro2 (x = y) (x ≠ y) (neqex_to_neq … H))
336 nlemma symmetric_eqex : symmetricT exadecim bool eq_ex.
338 napply (or2_elim (n1 = n2) (n1 ≠ n2) ? (decidable_ex n1 n2));
339 ##[ ##1: #H; nrewrite > H; napply refl_eq
340 ##| ##2: #H; nrewrite > (neq_to_neqex n1 n2 H);
341 napply (symmetric_eq ? (eq_ex n2 n1) false);
342 napply (neq_to_neqex n2 n1 (symmetric_neq ? n1 n2 H))