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 (* This file was automatically generated: do not edit *********************)
17 include "Basic-1/lift/defs.ma".
20 \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort
23 \lambda (n: nat).(\lambda (_: nat).(\lambda (_: nat).(refl_equal T (TSort
30 \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((lt n d) \to (eq T
31 (lift h d (TLRef n)) (TLRef n)))))
33 \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (lt n
34 d)).(eq_ind bool true (\lambda (b: bool).(eq T (TLRef (match b with [true
35 \Rightarrow n | false \Rightarrow (plus n h)])) (TLRef n))) (refl_equal T
36 (TLRef n)) (blt n d) (sym_eq bool (blt n d) true (lt_blt d n H)))))).
42 \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T
43 (lift h d (TLRef n)) (TLRef (plus n h))))))
45 \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (le d
46 n)).(eq_ind bool false (\lambda (b: bool).(eq T (TLRef (match b with [true
47 \Rightarrow n | false \Rightarrow (plus n h)])) (TLRef (plus n h))))
48 (refl_equal T (TLRef (plus n h))) (blt n d) (sym_eq bool (blt n d) false
55 \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
56 (d: nat).(eq T (lift h d (THead k u t)) (THead k (lift h d u) (lift h (s k d)
59 \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
60 (d: nat).(refl_equal T (THead k (lift h d u) (lift h (s k d) t))))))).
66 \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
67 (d: nat).(eq T (lift h d (THead (Bind b) u t)) (THead (Bind b) (lift h d u)
68 (lift h (S d) t)))))))
70 \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
71 (d: nat).(refl_equal T (THead (Bind b) (lift h d u) (lift h (S d) t))))))).
77 \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
78 (d: nat).(eq T (lift h d (THead (Flat f) u t)) (THead (Flat f) (lift h d u)
81 \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
82 (d: nat).(refl_equal T (THead (Flat f) (lift h d u) (lift h d t))))))).
87 theorem lift_gen_sort:
88 \forall (h: nat).(\forall (d: nat).(\forall (n: nat).(\forall (t: T).((eq T
89 (TSort n) (lift h d t)) \to (eq T t (TSort n))))))
91 \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (t: T).(T_ind
92 (\lambda (t0: T).((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n))))
93 (\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d (TSort
94 n0)))).(sym_eq T (TSort n) (TSort n0) H))) (\lambda (n0: nat).(\lambda (H:
95 (eq T (TSort n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TSort
96 n)) (\lambda (_: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0))
97 (\lambda (t0: T).(eq T (TSort n) t0)) H (TLRef n0) (lift_lref_lt n0 h d (let
98 H1 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda
99 (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
100 (THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind
101 (lt n0 d) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match
102 ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True |
103 (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n0)
104 H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2)))) (\lambda (_: (le d
105 n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T
106 (TSort n) t0)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d (let H1 \def
107 (eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_:
108 T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
109 (THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind
110 (le d n0) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match
111 ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True |
112 (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef
113 (plus n0 h)) H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2))))))) (\lambda
114 (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TSort n) (lift h d t0)) \to (eq
115 T t0 (TSort n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TSort n) (lift h d
116 t1)) \to (eq T t1 (TSort n))))).(\lambda (H1: (eq T (TSort n) (lift h d
117 (THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda
118 (t2: T).(eq T (TSort n) t2)) H1 (THead k (lift h d t0) (lift h (s k d) t1))
119 (lift_head k t0 t1 h d)) in (let H3 \def (eq_ind T (TSort n) (\lambda (ee:
120 T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
121 True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
122 (THead k (lift h d t0) (lift h (s k d) t1)) H2) in (False_ind (eq T (THead k
123 t0 t1) (TSort n)) H3))))))))) t)))).
128 theorem lift_gen_lref:
129 \forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T
130 (TLRef i) (lift h d t)) \to (or (land (lt i d) (eq T t (TLRef i))) (land (le
131 (plus d h) i) (eq T t (TLRef (minus i h)))))))))
133 \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(\forall (h:
134 nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (or (land (lt i d)
135 (eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i
136 h)))))))))) (\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda
137 (i: nat).(\lambda (H: (eq T (TLRef i) (lift h d (TSort n)))).(let H0 \def
138 (eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TSort
139 n) (lift_sort n h d)) in (let H1 \def (eq_ind T (TLRef i) (\lambda (ee:
140 T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
141 False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
142 (TSort n) H0) in (False_ind (or (land (lt i d) (eq T (TSort n) (TLRef i)))
143 (land (le (plus d h) i) (eq T (TSort n) (TLRef (minus i h))))) H1))))))))
144 (\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i:
145 nat).(\lambda (H: (eq T (TLRef i) (lift h d (TLRef n)))).(lt_le_e n d (or
146 (land (lt i d) (eq T (TLRef n) (TLRef i))) (land (le (plus d h) i) (eq T
147 (TLRef n) (TLRef (minus i h))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind
148 T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TLRef n)
149 (lift_lref_lt n h d H0)) in (let H2 \def (f_equal T nat (\lambda (e:
150 T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i |
151 (TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef
152 n) H1) in (eq_ind_r nat n (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef
153 n) (TLRef n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0
154 h)))))) (or_introl (land (lt n d) (eq T (TLRef n) (TLRef n))) (land (le (plus
155 d h) n) (eq T (TLRef n) (TLRef (minus n h)))) (conj (lt n d) (eq T (TLRef n)
156 (TLRef n)) H0 (refl_equal T (TLRef n)))) i H2)))) (\lambda (H0: (le d
157 n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef
158 i) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def
159 (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
160 [(TSort _) \Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _)
161 \Rightarrow i])) (TLRef i) (TLRef (plus n h)) H1) in (eq_ind_r nat (plus n h)
162 (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef n0))) (land (le
163 (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) (eq_ind_r nat n
164 (\lambda (n0: nat).(or (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n
165 h)))) (land (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n0)))))
166 (or_intror (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land
167 (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n))) (conj (le (plus d h)
168 (plus n h)) (eq T (TLRef n) (TLRef n)) (le_plus_plus d n h h H0 (le_n h))
169 (refl_equal T (TLRef n)))) (minus (plus n h) h) (minus_plus_r n h)) i
170 H2)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (d:
171 nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to
172 (or (land (lt i d) (eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0
173 (TLRef (minus i h))))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (d:
174 nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t1)) \to
175 (or (land (lt i d) (eq T t1 (TLRef i))) (land (le (plus d h) i) (eq T t1
176 (TLRef (minus i h))))))))))).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i:
177 nat).(\lambda (H1: (eq T (TLRef i) (lift h d (THead k t0 t1)))).(let H2 \def
178 (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TLRef i) t2)) H1
179 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let
180 H3 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda
181 (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
182 (THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d)
183 t1)) H2) in (False_ind (or (land (lt i d) (eq T (THead k t0 t1) (TLRef i)))
184 (land (le (plus d h) i) (eq T (THead k t0 t1) (TLRef (minus i h)))))
190 theorem lift_gen_lref_lt:
191 \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((lt n d) \to (\forall
192 (t: T).((eq T (TLRef n) (lift h d t)) \to (eq T t (TLRef n)))))))
194 \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt n
195 d)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef n) (lift h d t))).(let H_x
196 \def (lift_gen_lref t d h n H0) in (let H1 \def H_x in (or_ind (land (lt n d)
197 (eq T t (TLRef n))) (land (le (plus d h) n) (eq T t (TLRef (minus n h)))) (eq
198 T t (TLRef n)) (\lambda (H2: (land (lt n d) (eq T t (TLRef n)))).(land_ind
199 (lt n d) (eq T t (TLRef n)) (eq T t (TLRef n)) (\lambda (_: (lt n
200 d)).(\lambda (H4: (eq T t (TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0:
201 T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) t H4))) H2)) (\lambda (H2:
202 (land (le (plus d h) n) (eq T t (TLRef (minus n h))))).(land_ind (le (plus d
203 h) n) (eq T t (TLRef (minus n h))) (eq T t (TLRef n)) (\lambda (H3: (le (plus
204 d h) n)).(\lambda (H4: (eq T t (TLRef (minus n h)))).(eq_ind_r T (TLRef
205 (minus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false (plus d h) n (eq
206 T (TLRef (minus n h)) (TLRef n)) H3 (lt_le_S n (plus d h) (le_plus_trans (S
207 n) d h H))) t H4))) H2)) H1)))))))).
212 theorem lift_gen_lref_false:
213 \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n
214 (plus d h)) \to (\forall (t: T).((eq T (TLRef n) (lift h d t)) \to (\forall
217 \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d
218 n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(\lambda (H1: (eq T
219 (TLRef n) (lift h d t))).(\lambda (P: Prop).(let H_x \def (lift_gen_lref t d
220 h n H1) in (let H2 \def H_x in (or_ind (land (lt n d) (eq T t (TLRef n)))
221 (land (le (plus d h) n) (eq T t (TLRef (minus n h)))) P (\lambda (H3: (land
222 (lt n d) (eq T t (TLRef n)))).(land_ind (lt n d) (eq T t (TLRef n)) P
223 (\lambda (H4: (lt n d)).(\lambda (_: (eq T t (TLRef n))).(le_false d n P H
224 H4))) H3)) (\lambda (H3: (land (le (plus d h) n) (eq T t (TLRef (minus n
225 h))))).(land_ind (le (plus d h) n) (eq T t (TLRef (minus n h))) P (\lambda
226 (H4: (le (plus d h) n)).(\lambda (_: (eq T t (TLRef (minus n h)))).(le_false
227 (plus d h) n P H4 H0))) H3)) H2)))))))))).
232 theorem lift_gen_lref_ge:
233 \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall
234 (t: T).((eq T (TLRef (plus n h)) (lift h d t)) \to (eq T t (TLRef n)))))))
236 \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d
237 n)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d
238 t))).(let H_x \def (lift_gen_lref t d h (plus n h) H0) in (let H1 \def H_x in
239 (or_ind (land (lt (plus n h) d) (eq T t (TLRef (plus n h)))) (land (le (plus
240 d h) (plus n h)) (eq T t (TLRef (minus (plus n h) h)))) (eq T t (TLRef n))
241 (\lambda (H2: (land (lt (plus n h) d) (eq T t (TLRef (plus n h))))).(land_ind
242 (lt (plus n h) d) (eq T t (TLRef (plus n h))) (eq T t (TLRef n)) (\lambda
243 (H3: (lt (plus n h) d)).(\lambda (H4: (eq T t (TLRef (plus n h)))).(eq_ind_r
244 T (TLRef (plus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false d n (eq
245 T (TLRef (plus n h)) (TLRef n)) H (lt_le_S n d (simpl_lt_plus_r h n d
246 (lt_le_trans (plus n h) d (plus d h) H3 (le_plus_l d h))))) t H4))) H2))
247 (\lambda (H2: (land (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n
248 h) h))))).(land_ind (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n
249 h) h))) (eq T t (TLRef n)) (\lambda (_: (le (plus d h) (plus n h))).(\lambda
250 (H4: (eq T t (TLRef (minus (plus n h) h)))).(eq_ind_r T (TLRef (minus (plus n
251 h) h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (f_equal nat T TLRef (minus
252 (plus n h) h) n (minus_plus_r n h)) t H4))) H2)) H1)))))))).
257 theorem lift_gen_head:
258 \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
259 nat).(\forall (d: nat).((eq T (THead k u t) (lift h d x)) \to (ex3_2 T T
260 (\lambda (y: T).(\lambda (z: T).(eq T x (THead k y z)))) (\lambda (y:
261 T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
262 T).(eq T t (lift h (s k d) z)))))))))))
264 \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind
265 (\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k u t)
266 (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead
267 k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
268 (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))))) (\lambda (n:
269 nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t)
270 (lift h d (TSort n)))).(let H0 \def (eq_ind T (lift h d (TSort n)) (\lambda
271 (t0: T).(eq T (THead k u t) t0)) H (TSort n) (lift_sort n h d)) in (let H1
272 \def (eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda
273 (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
274 | (THead _ _ _) \Rightarrow True])) I (TSort n) H0) in (False_ind (ex3_2 T T
275 (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead k y z)))) (\lambda (y:
276 T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
277 T).(eq T t (lift h (s k d) z))))) H1))))))) (\lambda (n: nat).(\lambda (h:
278 nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) (lift h d (TLRef
279 n)))).(lt_le_e n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n)
280 (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y))))
281 (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) (\lambda (H0:
282 (lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T
283 (THead k u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def
284 (eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda (_:
285 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
286 (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T
287 (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y:
288 T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
289 T).(eq T t (lift h (s k d) z))))) H2)))) (\lambda (H0: (le d n)).(let H1 \def
290 (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead k u t) t0)) H
291 (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (eq_ind T (THead
292 k u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
293 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
294 \Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind (ex3_2 T T
295 (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y:
296 T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
297 T).(eq T t (lift h (s k d) z))))) H2))))))))) (\lambda (k0: K).(\lambda (t0:
298 T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t)
299 (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead
300 k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
301 (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (t1:
302 T).(\lambda (H0: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t)
303 (lift h d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead
304 k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
305 (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (h:
306 nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k u t) (lift h d (THead k0
307 t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (t2:
308 T).(eq T (THead k u t) t2)) H1 (THead k0 (lift h d t0) (lift h (s k0 d) t1))
309 (lift_head k0 t0 t1 h d)) in (let H3 \def (f_equal T K (\lambda (e: T).(match
310 e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
311 \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u t) (THead k0
312 (lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H4 \def (f_equal T T
313 (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
314 \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2]))
315 (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H5
316 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
317 with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t2)
318 \Rightarrow t2])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1))
319 H2) in (\lambda (H6: (eq T u (lift h d t0))).(\lambda (H7: (eq K k k0)).(let
320 H8 \def (eq_ind_r K k0 (\lambda (k1: K).(eq T t (lift h (s k1 d) t1))) H5 k
321 H7) in (eq_ind K k (\lambda (k1: K).(ex3_2 T T (\lambda (y: T).(\lambda (z:
322 T).(eq T (THead k1 t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_:
323 T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s
324 k d) z)))))) (let H9 \def (eq_ind T t (\lambda (t2: T).(\forall (h0:
325 nat).(\forall (d0: nat).((eq T (THead k u t2) (lift h0 d0 t1)) \to (ex3_2 T T
326 (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z)))) (\lambda (y:
327 T).(\lambda (_: T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z:
328 T).(eq T t2 (lift h0 (s k d0) z))))))))) H0 (lift h (s k d) t1) H8) in (let
329 H10 \def (eq_ind T t (\lambda (t2: T).(\forall (h0: nat).(\forall (d0:
330 nat).((eq T (THead k u t2) (lift h0 d0 t0)) \to (ex3_2 T T (\lambda (y:
331 T).(\lambda (z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_:
332 T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift
333 h0 (s k d0) z))))))))) H (lift h (s k d) t1) H8) in (eq_ind_r T (lift h (s k
334 d) t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T
335 (THead k t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u
336 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (s k d)
337 z)))))) (let H11 \def (eq_ind T u (\lambda (t2: T).(\forall (h0:
338 nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0
339 t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead k y z))))
340 (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_:
341 T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H10
342 (lift h d t0) H6) in (let H12 \def (eq_ind T u (\lambda (t2: T).(\forall (h0:
343 nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0
344 t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z))))
345 (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_:
346 T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H9
347 (lift h d t0) H6) in (eq_ind_r T (lift h d t0) (\lambda (t2: T).(ex3_2 T T
348 (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead k y z))))
349 (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_:
350 T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d) z))))))
351 (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead
352 k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y))))
353 (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d)
354 z)))) t0 t1 (refl_equal T (THead k t0 t1)) (refl_equal T (lift h d t0))
355 (refl_equal T (lift h (s k d) t1))) u H6))) t H8))) k0 H7))))) H4))
361 theorem lift_gen_bind:
362 \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
363 nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d x)) \to (ex3_2 T
364 T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda
365 (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
366 T).(eq T t (lift h (S d) z)))))))))))
368 \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h:
369 nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u t) (lift h d
370 x))).(let H_x \def (lift_gen_head (Bind b) u t x h d H) in (let H0 \def H_x
371 in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y
372 z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
373 T).(\lambda (z: T).(eq T t (lift h (S d) z)))) (ex3_2 T T (\lambda (y:
374 T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda
375 (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift
376 h (S d) z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead
377 (Bind b) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t
378 (lift h (S d) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0:
379 T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Bind b) y
380 z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
381 T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) (eq_ind_r T (lift h (S d)
382 x1) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead
383 (Bind b) x0 x1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T
384 u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h (S d)
385 z)))))) (eq_ind_r T (lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y:
386 T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind b) y z))))
387 (\lambda (y: T).(\lambda (_: T).(eq T t0 (lift h d y)))) (\lambda (_:
388 T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) z)))))) (ex3_2_intro
389 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind
390 b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d x0) (lift h d
391 y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d)
392 z)))) x0 x1 (refl_equal T (THead (Bind b) x0 x1)) (refl_equal T (lift h d
393 x0)) (refl_equal T (lift h (S d) x1))) u H2) t H3) x H1)))))) H0))))))))).
398 theorem lift_gen_flat:
399 \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
400 nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d x)) \to (ex3_2 T
401 T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda
402 (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
403 T).(eq T t (lift h d z)))))))))))
405 \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h:
406 nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d
407 x))).(let H_x \def (lift_gen_head (Flat f) u t x h d H) in (let H0 \def H_x
408 in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y
409 z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
410 T).(\lambda (z: T).(eq T t (lift h d z)))) (ex3_2 T T (\lambda (y:
411 T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda
412 (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift
413 h d z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead
414 (Flat f) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t
415 (lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t0: T).(ex3_2 T
416 T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) y z)))) (\lambda
417 (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
418 T).(eq T t (lift h d z)))))) (eq_ind_r T (lift h d x1) (\lambda (t0:
419 T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) x0 x1)
420 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
421 y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h d z)))))) (eq_ind_r T
422 (lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq
423 T (THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_:
424 T).(eq T t0 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d
425 x1) (lift h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T
426 (THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_:
427 T).(eq T (lift h d x0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T
428 (lift h d x1) (lift h d z)))) x0 x1 (refl_equal T (THead (Flat f) x0 x1))
429 (refl_equal T (lift h d x0)) (refl_equal T (lift h d x1))) u H2) t H3) x
430 H1)))))) H0))))))))).