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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_destruct1 : Πe2.ΠP:Prop.ΠH:x0 = e2.match e2 with [ x0 ⇒ P → P | _ ⇒ P ].
31 #e2; #P; ncases e2; nnormalize; #H;
32 ##[ ##1: napply (λx:P.x)
33 ##| ##*: napply False_ind;
34 nchange with (match x0 with [ x0 ⇒ False | _ ⇒ True ]);
35 nrewrite > H; nnormalize; napply I
39 ndefinition exadecim_destruct2 : Πe2.ΠP:Prop.ΠH:x1 = e2.match e2 with [ x1 ⇒ P → P | _ ⇒ P ].
40 #e2; #P; ncases e2; nnormalize; #H;
41 ##[ ##2: napply (λx:P.x)
42 ##| ##*: napply False_ind;
43 nchange with (match x1 with [ x1 ⇒ False | _ ⇒ True ]);
44 nrewrite > H; nnormalize; napply I
48 ndefinition exadecim_destruct3 : Πe2.ΠP:Prop.ΠH:x2 = e2.match e2 with [ x2 ⇒ P → P | _ ⇒ P ].
49 #e2; #P; ncases e2; nnormalize; #H;
50 ##[ ##3: napply (λx:P.x)
51 ##| ##*: napply False_ind;
52 nchange with (match x2 with [ x2 ⇒ False | _ ⇒ True ]);
53 nrewrite > H; nnormalize; napply I
57 ndefinition exadecim_destruct4 : Πe2.ΠP:Prop.ΠH:x3 = e2.match e2 with [ x3 ⇒ P → P | _ ⇒ P ].
58 #e2; #P; ncases e2; nnormalize; #H;
59 ##[ ##4: napply (λx:P.x)
60 ##| ##*: napply False_ind;
61 nchange with (match x3 with [ x3 ⇒ False | _ ⇒ True ]);
62 nrewrite > H; nnormalize; napply I
66 ndefinition exadecim_destruct5 : Πe2.ΠP:Prop.ΠH:x4 = e2.match e2 with [ x4 ⇒ P → P | _ ⇒ P ].
67 #e2; #P; ncases e2; nnormalize; #H;
68 ##[ ##5: napply (λx:P.x)
69 ##| ##*: napply False_ind;
70 nchange with (match x4 with [ x4 ⇒ False | _ ⇒ True ]);
71 nrewrite > H; nnormalize; napply I
75 ndefinition exadecim_destruct6 : Πe2.ΠP:Prop.ΠH:x5 = e2.match e2 with [ x5 ⇒ P → P | _ ⇒ P ].
76 #e2; #P; ncases e2; nnormalize; #H;
77 ##[ ##6: napply (λx:P.x)
78 ##| ##*: napply False_ind;
79 nchange with (match x5 with [ x5 ⇒ False | _ ⇒ True ]);
80 nrewrite > H; nnormalize; napply I
84 ndefinition exadecim_destruct7 : Πe2.ΠP:Prop.ΠH:x6 = e2.match e2 with [ x6 ⇒ P → P | _ ⇒ P ].
85 #e2; #P; ncases e2; nnormalize; #H;
86 ##[ ##7: napply (λx:P.x)
87 ##| ##*: napply False_ind;
88 nchange with (match x6 with [ x6 ⇒ False | _ ⇒ True ]);
89 nrewrite > H; nnormalize; napply I
93 ndefinition exadecim_destruct8 : Πe2.ΠP:Prop.ΠH:x7 = e2.match e2 with [ x7 ⇒ P → P | _ ⇒ P ].
94 #e2; #P; ncases e2; nnormalize; #H;
95 ##[ ##8: napply (λx:P.x)
96 ##| ##*: napply False_ind;
97 nchange with (match x7 with [ x7 ⇒ False | _ ⇒ True ]);
98 nrewrite > H; nnormalize; napply I
102 ndefinition exadecim_destruct9 : Πe2.ΠP:Prop.ΠH:x8 = e2.match e2 with [ x8 ⇒ P → P | _ ⇒ P ].
103 #e2; #P; ncases e2; nnormalize; #H;
104 ##[ ##9: napply (λx:P.x)
105 ##| ##*: napply False_ind;
106 nchange with (match x8 with [ x8 ⇒ False | _ ⇒ True ]);
107 nrewrite > H; nnormalize; napply I
111 ndefinition exadecim_destruct10 : Πe2.ΠP:Prop.ΠH:x9 = e2.match e2 with [ x9 ⇒ P → P | _ ⇒ P ].
112 #e2; #P; ncases e2; nnormalize; #H;
113 ##[ ##10: napply (λx:P.x)
114 ##| ##*: napply False_ind;
115 nchange with (match x9 with [ x9 ⇒ False | _ ⇒ True ]);
116 nrewrite > H; nnormalize; napply I
120 ndefinition exadecim_destruct11 : Πe2.ΠP:Prop.ΠH:xA = e2.match e2 with [ xA ⇒ P → P | _ ⇒ P ].
121 #e2; #P; ncases e2; nnormalize; #H;
122 ##[ ##11: napply (λx:P.x)
123 ##| ##*: napply False_ind;
124 nchange with (match xA with [ xA ⇒ False | _ ⇒ True ]);
125 nrewrite > H; nnormalize; napply I
129 ndefinition exadecim_destruct12 : Πe2.ΠP:Prop.ΠH:xB = e2.match e2 with [ xB ⇒ P → P | _ ⇒ P ].
130 #e2; #P; ncases e2; nnormalize; #H;
131 ##[ ##12: napply (λx:P.x)
132 ##| ##*: napply False_ind;
133 nchange with (match xB with [ xB ⇒ False | _ ⇒ True ]);
134 nrewrite > H; nnormalize; napply I
138 ndefinition exadecim_destruct13 : Πe2.ΠP:Prop.ΠH:xC = e2.match e2 with [ xC ⇒ P → P | _ ⇒ P ].
139 #e2; #P; ncases e2; nnormalize; #H;
140 ##[ ##13: napply (λx:P.x)
141 ##| ##*: napply False_ind;
142 nchange with (match xC with [ xC ⇒ False | _ ⇒ True ]);
143 nrewrite > H; nnormalize; napply I
147 ndefinition exadecim_destruct14 : Πe2.ΠP:Prop.ΠH:xD = e2.match e2 with [ xD ⇒ P → P | _ ⇒ P ].
148 #e2; #P; ncases e2; nnormalize; #H;
149 ##[ ##14: napply (λx:P.x)
150 ##| ##*: napply False_ind;
151 nchange with (match xD with [ xD ⇒ False | _ ⇒ True ]);
152 nrewrite > H; nnormalize; napply I
156 ndefinition exadecim_destruct15 : Πe2.ΠP:Prop.ΠH:xE = e2.match e2 with [ xE ⇒ P → P | _ ⇒ P ].
157 #e2; #P; ncases e2; nnormalize; #H;
158 ##[ ##15: napply (λx:P.x)
159 ##| ##*: napply False_ind;
160 nchange with (match xE with [ xE ⇒ False | _ ⇒ True ]);
161 nrewrite > H; nnormalize; napply I
165 ndefinition exadecim_destruct16 : Πe2.ΠP:Prop.ΠH:xF = e2.match e2 with [ xF ⇒ P → P | _ ⇒ P ].
166 #e2; #P; ncases e2; nnormalize; #H;
167 ##[ ##16: napply (λx:P.x)
168 ##| ##*: napply False_ind;
169 nchange with (match xF with [ xF ⇒ False | _ ⇒ True ]);
170 nrewrite > H; nnormalize; napply I
174 ndefinition exadecim_destruct_aux ≝
175 Πe1,e2.ΠP:Prop.ΠH:e1 = e2.
177 [ x0 ⇒ match e2 with [ x0 ⇒ P → P | _ ⇒ P ] | x1 ⇒ match e2 with [ x1 ⇒ P → P | _ ⇒ P ]
178 | x2 ⇒ match e2 with [ x2 ⇒ P → P | _ ⇒ P ] | x3 ⇒ match e2 with [ x3 ⇒ P → P | _ ⇒ P ]
179 | x4 ⇒ match e2 with [ x4 ⇒ P → P | _ ⇒ P ] | x5 ⇒ match e2 with [ x5 ⇒ P → P | _ ⇒ P ]
180 | x6 ⇒ match e2 with [ x6 ⇒ P → P | _ ⇒ P ] | x7 ⇒ match e2 with [ x7 ⇒ P → P | _ ⇒ P ]
181 | x8 ⇒ match e2 with [ x8 ⇒ P → P | _ ⇒ P ] | x9 ⇒ match e2 with [ x9 ⇒ P → P | _ ⇒ P ]
182 | xA ⇒ match e2 with [ xA ⇒ P → P | _ ⇒ P ] | xB ⇒ match e2 with [ xB ⇒ P → P | _ ⇒ P ]
183 | xC ⇒ match e2 with [ xC ⇒ P → P | _ ⇒ P ] | xD ⇒ match e2 with [ xD ⇒ P → P | _ ⇒ P ]
184 | xE ⇒ match e2 with [ xE ⇒ P → P | _ ⇒ P ] | xF ⇒ match e2 with [ xF ⇒ P → P | _ ⇒ P ]
187 ndefinition exadecim_destruct : exadecim_destruct_aux.
189 ##[ ##1: napply exadecim_destruct1 ##| ##2: napply exadecim_destruct2
190 ##| ##3: napply exadecim_destruct3 ##| ##4: napply exadecim_destruct4
191 ##| ##5: napply exadecim_destruct5 ##| ##6: napply exadecim_destruct6
192 ##| ##7: napply exadecim_destruct7 ##| ##8: napply exadecim_destruct8
193 ##| ##9: napply exadecim_destruct9 ##| ##10: napply exadecim_destruct10
194 ##| ##11: napply exadecim_destruct11 ##| ##12: napply exadecim_destruct12
195 ##| ##13: napply exadecim_destruct13 ##| ##14: napply exadecim_destruct14
196 ##| ##15: napply exadecim_destruct15 ##| ##16: napply exadecim_destruct16
200 nlemma symmetric_eqex : symmetricT exadecim bool eq_ex.
208 nlemma symmetric_andex : symmetricT exadecim exadecim and_ex.
216 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.
217 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.
218 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.
219 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.
220 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.
221 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.
222 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.
223 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.
224 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.
225 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.
226 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.
227 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.
228 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.
229 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.
230 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.
231 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.
233 nlemma associative_andex : ∀e1,e2,e3.(and_ex (and_ex e1 e2) e3) = (and_ex e1 (and_ex e2 e3)).
235 ##[ ##1: napply associative_andex1 ##| ##2: napply associative_andex2
236 ##| ##3: napply associative_andex3 ##| ##4: napply associative_andex4
237 ##| ##5: napply associative_andex5 ##| ##6: napply associative_andex6
238 ##| ##7: napply associative_andex7 ##| ##8: napply associative_andex8
239 ##| ##9: napply associative_andex9 ##| ##10: napply associative_andex10
240 ##| ##11: napply associative_andex11 ##| ##12: napply associative_andex12
241 ##| ##13: napply associative_andex13 ##| ##14: napply associative_andex14
242 ##| ##15: napply associative_andex15 ##| ##16: napply associative_andex16
246 nlemma symmetric_orex : symmetricT exadecim exadecim or_ex.
254 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.
255 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.
256 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.
257 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.
258 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.
259 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.
260 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.
261 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.
262 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.
263 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.
264 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.
265 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.
266 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.
267 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.
268 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.
269 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.
271 nlemma associative_orex : ∀e1,e2,e3.(or_ex (or_ex e1 e2) e3) = (or_ex e1 (or_ex e2 e3)).
273 ##[ ##1: napply associative_orex1 ##| ##2: napply associative_orex2
274 ##| ##3: napply associative_orex3 ##| ##4: napply associative_orex4
275 ##| ##5: napply associative_orex5 ##| ##6: napply associative_orex6
276 ##| ##7: napply associative_orex7 ##| ##8: napply associative_orex8
277 ##| ##9: napply associative_orex9 ##| ##10: napply associative_orex10
278 ##| ##11: napply associative_orex11 ##| ##12: napply associative_orex12
279 ##| ##13: napply associative_orex13 ##| ##14: napply associative_orex14
280 ##| ##15: napply associative_orex15 ##| ##16: napply associative_orex16
284 nlemma symmetric_xorex : symmetricT exadecim exadecim xor_ex.
292 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.
293 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.
294 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.
295 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.
296 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.
297 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.
298 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.
299 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.
300 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.
301 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.
302 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.
303 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.
304 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.
305 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.
306 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.
307 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.
309 nlemma associative_xorex : ∀e1,e2,e3.(xor_ex (xor_ex e1 e2) e3) = (xor_ex e1 (xor_ex e2 e3)).
311 ##[ ##1: napply associative_xorex1 ##| ##2: napply associative_xorex2
312 ##| ##3: napply associative_xorex3 ##| ##4: napply associative_xorex4
313 ##| ##5: napply associative_xorex5 ##| ##6: napply associative_xorex6
314 ##| ##7: napply associative_xorex7 ##| ##8: napply associative_xorex8
315 ##| ##9: napply associative_xorex9 ##| ##10: napply associative_xorex10
316 ##| ##11: napply associative_xorex11 ##| ##12: napply associative_xorex12
317 ##| ##13: napply associative_xorex13 ##| ##14: napply associative_xorex14
318 ##| ##15: napply associative_xorex15 ##| ##16: napply associative_xorex16
322 nlemma symmetric_plusex_dc_dc : ∀e1,e2,c.plus_ex_dc_dc e1 e2 c = plus_ex_dc_dc e2 e1 c.
331 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.
340 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.
349 nlemma plusex_dc_dc_to_d_dc : ∀e1,e2.plus_ex_dc_dc e1 e2 false = plus_ex_d_dc e1 e2.
357 nlemma plusex_dc_dc_to_d_d : ∀e1,e2.fst … (plus_ex_dc_dc e1 e2 false) = plus_ex_d_d e1 e2.
365 nlemma plusex_dc_dc_to_d_c : ∀e1,e2.snd … (plus_ex_dc_dc e1 e2 false) = plus_ex_d_c e1 e2.
373 nlemma symmetric_plusex_dc_d : ∀e1,e2,c.plus_ex_dc_d e1 e2 c = plus_ex_dc_d e2 e1 c.
382 nlemma symmetric_plusex_dc_c : ∀e1,e2,c.plus_ex_dc_c e1 e2 c = plus_ex_dc_c e2 e1 c.
391 nlemma symmetric_plusex_d_dc : ∀e1,e2.plus_ex_d_dc e1 e2 = plus_ex_d_dc e2 e1.
399 nlemma plusex_d_dc_to_d_d : ∀e1,e2.fst … (plus_ex_d_dc e1 e2) = plus_ex_d_d e1 e2.
407 nlemma plusex_d_dc_to_d_c : ∀e1,e2.snd … (plus_ex_d_dc e1 e2) = plus_ex_d_c e1 e2.
415 nlemma symmetric_plusex_d_d : ∀e1,e2.plus_ex_d_d e1 e2 = plus_ex_d_d e2 e1.
423 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.
424 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.
425 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.
426 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.
427 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.
428 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.
429 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.
430 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.
431 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.
432 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.
433 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.
434 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.
435 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.
436 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.
437 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.
438 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.
440 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)).
442 ##[ ##1: napply associative_plusex_d_d1 ##| ##2: napply associative_plusex_d_d2
443 ##| ##3: napply associative_plusex_d_d3 ##| ##4: napply associative_plusex_d_d4
444 ##| ##5: napply associative_plusex_d_d5 ##| ##6: napply associative_plusex_d_d6
445 ##| ##7: napply associative_plusex_d_d7 ##| ##8: napply associative_plusex_d_d8
446 ##| ##9: napply associative_plusex_d_d9 ##| ##10: napply associative_plusex_d_d10
447 ##| ##11: napply associative_plusex_d_d11 ##| ##12: napply associative_plusex_d_d12
448 ##| ##13: napply associative_plusex_d_d13 ##| ##14: napply associative_plusex_d_d14
449 ##| ##15: napply associative_plusex_d_d15 ##| ##16: napply associative_plusex_d_d16
453 nlemma symmetric_plusex_d_c : ∀e1,e2.plus_ex_d_c e1 e2 = plus_ex_d_c e2 e1.
461 nlemma eqex_to_eq : ∀e1,e2:exadecim.(eq_ex e1 e2 = true) → (e1 = e2).
466 ##[ ##1,18,35,52,69,86,103,120,137,154,171,188,205,222,239,256: #H; napply refl_eq
467 ##| ##*: #H; napply (bool_destruct … H)
471 nlemma eq_to_eqex : ∀e1,e2.e1 = e2 → eq_ex e1 e2 = true.
476 ##[ ##1,18,35,52,69,86,103,120,137,154,171,188,205,222,239,256: #H; napply refl_eq
477 ##| ##*: #H; napply (exadecim_destruct … H)
481 nlemma decidable_ex : ∀x,y:exadecim.decidable (x = y).
486 ##[ ##1,18,35,52,69,86,103,120,137,154,171,188,205,222,239,256: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
487 ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (exadecim_destruct … H)
491 nlemma neqex_to_neq : ∀e1,e2:exadecim.(eq_ex e1 e2 = false) → (e1 ≠ e2).
496 ##[ ##1,18,35,52,69,86,103,120,137,154,171,188,205,222,239,256: #H; napply (bool_destruct … H)
497 ##| ##*: #H; #H1; napply (exadecim_destruct … H1)
501 nlemma neq_to_neqex : ∀e1,e2.e1 ≠ e2 → eq_ex e1 e2 = false.
506 ##[ ##1,18,35,52,69,86,103,120,137,154,171,188,205,222,239,256: #H; nelim (H (refl_eq …))
507 ##| ##*: #H; napply refl_eq