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 set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/nf2".
19 include "ty3/arity.ma".
23 include "nf2/arity.ma".
25 definition ty3_nf2_inv_abst_premise:
26 C \to (T \to (T \to Prop))
28 \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\forall (d: C).(\forall (wi:
29 T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi)) \to (\forall (vs:
30 TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi)) (THead (Bind Abst) w
31 u)) \to False)))))))).
33 theorem ty3_nf2_inv_abst_premise_csort:
34 \forall (w: T).(\forall (u: T).(\forall (m: nat).(ty3_nf2_inv_abst_premise
37 \lambda (w: T).(\lambda (u: T).(\lambda (m: nat).(\lambda (d: C).(\lambda
38 (wi: T).(\lambda (i: nat).(\lambda (H: (getl i (CSort m) (CHead d (Bind Abst)
39 wi))).(\lambda (vs: TList).(\lambda (_: (pc3 (CSort m) (THeads (Flat Appl) vs
40 (lift (S i) O wi)) (THead (Bind Abst) w u))).(getl_gen_sort m i (CHead d
41 (Bind Abst) wi) H False))))))))).
43 theorem ty3_nf2_inv_all:
44 \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
45 u) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T
46 t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w)))
47 (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w) u0)))) (ex nat
48 (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws:
49 TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
50 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
51 TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))
53 \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
54 (ty3 g c t u)).(\lambda (H0: (nf2 c t)).(let H_x \def (ty3_arity g c t u H)
55 in (let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda
56 (a1: A).(arity g c u (asucc g a1))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda
57 (u0: T).(eq T t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
58 T).(nf2 c w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w)
59 u0)))) (ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat
60 (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef
61 i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
62 TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x: A).(\lambda (H2:
63 (arity g c t x)).(\lambda (_: (arity g c u (asucc g x))).(arity_nf2_inv_all g
64 c t x H2 H0)))) H1)))))))).
66 theorem ty3_nf2_inv_sort:
67 \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (m: nat).((ty3 g c t
68 (TSort m)) \to ((nf2 c t) \to (or (ex2 nat (\lambda (n: nat).(eq T t (TSort
69 n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws:
70 TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
71 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
72 TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))
74 \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (m: nat).(\lambda
75 (H: (ty3 g c t (TSort m))).(\lambda (H0: (nf2 c t)).(let H_x \def
76 (ty3_nf2_inv_all g c t (TSort m) H H0) in (let H1 \def H_x in (or3_ind (ex3_2
77 T T (\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind Abst) w u))))
78 (\lambda (w: T).(\lambda (_: T).(nf2 c w))) (\lambda (w: T).(\lambda (u:
79 T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t
80 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t
81 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
82 nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))
83 (or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat
84 m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
85 t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
86 nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef
87 i)))))) (\lambda (H2: (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t
88 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w)))
89 (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w)
90 u))))).(ex3_2_ind T T (\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind
91 Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) (\lambda (w:
92 T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u))) (or (ex2 nat (\lambda
93 (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2
94 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl)
95 ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws)))
96 (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0:
97 T).(\lambda (x1: T).(\lambda (H3: (eq T t (THead (Bind Abst) x0
98 x1))).(\lambda (_: (nf2 c x0)).(\lambda (_: (nf2 (CHead c (Bind Abst) x0)
99 x1)).(let H6 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H
100 (THead (Bind Abst) x0 x1) H3) in (eq_ind_r T (THead (Bind Abst) x0 x1)
101 (\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda
102 (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws:
103 TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
104 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
105 TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (ex4_3_ind T T T (\lambda
106 (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2)
107 (TSort m))))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c x0
108 t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind
109 Abst) x0) x1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t1: T).(ty3 g
110 (CHead c (Bind Abst) x0) t2 t1)))) (or (ex2 nat (\lambda (n: nat).(eq T
111 (THead (Bind Abst) x0 x1) (TSort n))) (\lambda (n: nat).(eq nat m (next g
112 n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead
113 (Bind Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
114 TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i:
115 nat).(nf2 c (TLRef i)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
116 T).(\lambda (H7: (pc3 c (THead (Bind Abst) x0 x2) (TSort m))).(\lambda (_:
117 (ty3 g c x0 x3)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x0) x1
118 x2)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x0) x2 x4)).(pc3_gen_sort_abst
119 c x0 x2 m (pc3_s c (TSort m) (THead (Bind Abst) x0 x2) H7) (or (ex2 nat
120 (\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n))) (\lambda (n:
121 nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda
122 (i: nat).(eq T (THead (Bind Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i)))))
123 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
124 TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))))) (ty3_gen_bind g Abst c
125 x0 x1 (TSort m) H6)) t H3))))))) H2)) (\lambda (H2: (ex nat (\lambda (n:
126 nat).(eq T t (TSort n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n)))
127 (or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat
128 m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
129 t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
130 nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef
131 i)))))) (\lambda (x: nat).(\lambda (H3: (eq T t (TSort x))).(let H4 \def
132 (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (TSort x) H3) in
133 (eq_ind_r T (TSort x) (\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T
134 t0 (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat
135 (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef
136 i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
137 TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (eq_ind nat (next g x)
138 (\lambda (n: nat).(or (ex2 nat (\lambda (n0: nat).(eq T (TSort x) (TSort
139 n0))) (\lambda (n0: nat).(eq nat n (next g n0)))) (ex3_2 TList nat (\lambda
140 (ws: TList).(\lambda (i: nat).(eq T (TSort x) (THeads (Flat Appl) ws (TLRef
141 i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
142 TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_introl (ex2 nat (\lambda
143 (n: nat).(eq T (TSort x) (TSort n))) (\lambda (n: nat).(eq nat (next g x)
144 (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
145 (TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda
146 (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef
147 i))))) (ex_intro2 nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) (\lambda
148 (n: nat).(eq nat (next g x) (next g n))) x (refl_equal T (TSort x))
149 (refl_equal nat (next g x)))) m (pc3_gen_sort c (next g x) m (ty3_gen_sort g
150 c (TSort m) x H4))) t H3)))) H2)) (\lambda (H2: (ex3_2 TList nat (\lambda
151 (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
152 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
153 TList).(\lambda (i: nat).(nf2 c (TLRef i)))))).(ex3_2_ind TList nat (\lambda
154 (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
155 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
156 TList).(\lambda (i: nat).(nf2 c (TLRef i)))) (or (ex2 nat (\lambda (n:
157 nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2
158 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl)
159 ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws)))
160 (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0:
161 TList).(\lambda (x1: nat).(\lambda (H3: (eq T t (THeads (Flat Appl) x0 (TLRef
162 x1)))).(\lambda (H4: (nfs2 c x0)).(\lambda (H5: (nf2 c (TLRef x1))).(let H6
163 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (THeads (Flat
164 Appl) x0 (TLRef x1)) H3) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1))
165 (\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda
166 (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws:
167 TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
168 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
169 TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_intror (ex2 nat (\lambda
170 (n: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (TSort n))) (\lambda (n:
171 nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda
172 (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws
173 (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda
174 (_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex3_2_intro TList nat
175 (\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef
176 x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
177 nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))
178 x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H4 H5)) t H3)))))))
181 theorem ty3_nf2_gen__ty3_nf2_inv_abst_aux:
182 \forall (c: C).(\forall (w1: T).(\forall (u1: T).((ty3_nf2_inv_abst_premise
183 c w1 u1) \to (\forall (t: T).(\forall (w2: T).(\forall (u2: T).((pc3 c (THead
184 (Flat Appl) t (THead (Bind Abst) w2 u2)) (THead (Bind Abst) w1 u1)) \to
185 (ty3_nf2_inv_abst_premise c w2 u2))))))))
187 \lambda (c: C).(\lambda (w1: T).(\lambda (u1: T).(\lambda (H: ((\forall (d:
188 C).(\forall (wi: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi))
189 \to (\forall (vs: TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi))
190 (THead (Bind Abst) w1 u1)) \to False)))))))).(\lambda (t: T).(\lambda (w2:
191 T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Flat Appl) t (THead (Bind
192 Abst) w2 u2)) (THead (Bind Abst) w1 u1))).(\lambda (d: C).(\lambda (wi:
193 T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d (Bind Abst)
194 wi))).(\lambda (vs: TList).(\lambda (H2: (pc3 c (THeads (Flat Appl) vs (lift
195 (S i) O wi)) (THead (Bind Abst) w2 u2))).(H d wi i H1 (TCons t vs) (pc3_t
196 (THead (Flat Appl) t (THead (Bind Abst) w2 u2)) c (THead (Flat Appl) t
197 (THeads (Flat Appl) vs (lift (S i) O wi))) (pc3_thin_dx c (THeads (Flat Appl)
198 vs (lift (S i) O wi)) (THead (Bind Abst) w2 u2) H2 t Appl) (THead (Bind Abst)
199 w1 u1) H0))))))))))))))).
201 theorem ty3_nf2_inv_abst:
202 \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u:
203 T).((ty3 g c t (THead (Bind Abst) w u)) \to ((nf2 c t) \to ((nf2 c w) \to
204 ((ty3_nf2_inv_abst_premise c w u) \to (ex4_2 T T (\lambda (v: T).(\lambda (_:
205 T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g
206 c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v
207 u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w)
210 \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u:
211 T).(\lambda (H: (ty3 g c t (THead (Bind Abst) w u))).(\lambda (H0: (nf2 c
212 t)).(\lambda (H1: (nf2 c w)).(\lambda (H2: (ty3_nf2_inv_abst_premise c w
213 u)).(let H_x \def (ty3_nf2_inv_all g c t (THead (Bind Abst) w u) H H0) in
214 (let H3 \def H_x in (or3_ind (ex3_2 T T (\lambda (w0: T).(\lambda (u0: T).(eq
215 T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c
216 w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0))))
217 (ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws:
218 TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
219 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
220 TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex4_2 T T (\lambda (v:
221 T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_:
222 T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
223 (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
224 (Bind Abst) w) v)))) (\lambda (H4: (ex3_2 T T (\lambda (w0: T).(\lambda (u0:
225 T).(eq T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2
226 c w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0)
227 u0))))).(ex3_2_ind T T (\lambda (w0: T).(\lambda (u0: T).(eq T t (THead (Bind
228 Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c w0))) (\lambda (w0:
229 T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0))) (ex4_2 T T (\lambda
230 (v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_:
231 T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
232 (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
233 (Bind Abst) w) v)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t
234 (THead (Bind Abst) x0 x1))).(\lambda (H6: (nf2 c x0)).(\lambda (H7: (nf2
235 (CHead c (Bind Abst) x0) x1)).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3
236 g c t0 (THead (Bind Abst) w u))) H (THead (Bind Abst) x0 x1) H5) in (eq_ind_r
237 T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(ex4_2 T T (\lambda (v:
238 T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_:
239 T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
240 (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
241 (Bind Abst) w) v))))) (ex_ind T (\lambda (t0: T).(ty3 g c (THead (Bind Abst)
242 w u) t0)) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst)
243 x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w
244 w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u)))
245 (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda
246 (x: T).(\lambda (H9: (ty3 g c (THead (Bind Abst) w u) x)).(ex4_3_ind T T T
247 (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) w
248 t2) x)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c w t0))))
249 (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst)
250 w) u t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead c
251 (Bind Abst) w) t2 t1)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T
252 (THead (Bind Abst) x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda
253 (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c
254 (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind
255 Abst) w) v)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda
256 (_: (pc3 c (THead (Bind Abst) w x2) x)).(\lambda (H11: (ty3 g c w
257 x3)).(\lambda (H12: (ty3 g (CHead c (Bind Abst) w) u x2)).(\lambda (_: (ty3 g
258 (CHead c (Bind Abst) w) x2 x4)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda
259 (_: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (THead (Bind Abst) w
260 u))))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c x0 t0))))
261 (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst)
262 x0) x1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead
263 c (Bind Abst) x0) t2 t1)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T
264 (THead (Bind Abst) x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda
265 (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c
266 (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind
267 Abst) w) v)))) (\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda
268 (H14: (pc3 c (THead (Bind Abst) x0 x5) (THead (Bind Abst) w u))).(\lambda (_:
269 (ty3 g c x0 x6)).(\lambda (H16: (ty3 g (CHead c (Bind Abst) x0) x1
270 x5)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x0) x5 x7)).(and_ind (pc3 c x0
271 w) (\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) x5 u))) (ex4_2
272 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
273 (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0)))
274 (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u)))
275 (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda
276 (H18: (pc3 c x0 w)).(\lambda (H19: ((\forall (b: B).(\forall (u0: T).(pc3
277 (CHead c (Bind b) u0) x5 u))))).(let H_y \def (pc3_nf2 c x0 w H18 H6 H1) in
278 (let H20 \def (eq_ind T x0 (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) t0)
279 x1 x5)) H16 w H_y) in (let H21 \def (eq_ind T x0 (\lambda (t0: T).(nf2 (CHead
280 c (Bind Abst) t0) x1)) H7 w H_y) in (eq_ind_r T w (\lambda (t0: T).(ex4_2 T T
281 (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) t0 x1) (THead (Bind
282 Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v:
283 T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v:
284 T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))) (ex4_2_intro T T
285 (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) w x1) (THead (Bind
286 Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v:
287 T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v:
288 T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))) x1 x3 (refl_equal T
289 (THead (Bind Abst) w x1)) H11 (ty3_conv g (CHead c (Bind Abst) w) u x2 H12 x1
290 x5 H20 (H19 Abst w)) H21) x0 H_y)))))) (pc3_gen_abst c x0 w x5 u H14)))))))))
291 (ty3_gen_bind g Abst c x0 x1 (THead (Bind Abst) w u) H8)))))))))
292 (ty3_gen_bind g Abst c w u x H9)))) (ty3_correct g c (THead (Bind Abst) x0
293 x1) (THead (Bind Abst) w u) H8)) t H5))))))) H4)) (\lambda (H4: (ex nat
294 (\lambda (n: nat).(eq T t (TSort n))))).(ex_ind nat (\lambda (n: nat).(eq T t
295 (TSort n))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T t (THead (Bind
296 Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v:
297 T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v:
298 T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (x:
299 nat).(\lambda (H5: (eq T t (TSort x))).(let H6 \def (eq_ind T t (\lambda (t0:
300 T).(ty3 g c t0 (THead (Bind Abst) w u))) H (TSort x) H5) in (eq_ind_r T
301 (TSort x) (\lambda (t0: T).(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T
302 t0 (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w
303 w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u)))
304 (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))))
305 (pc3_gen_sort_abst c w u (next g x) (ty3_gen_sort g c (THead (Bind Abst) w u)
306 x H6) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TSort x) (THead (Bind
307 Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v:
308 T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v:
309 T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))) t H5)))) H4)) (\lambda
310 (H4: (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads
311 (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c
312 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))).(ex3_2_ind
313 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl)
314 ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws)))
315 (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))) (ex4_2 T T (\lambda
316 (v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_:
317 T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
318 (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
319 (Bind Abst) w) v)))) (\lambda (x0: TList).(\lambda (x1: nat).(\lambda (H5:
320 (eq T t (THeads (Flat Appl) x0 (TLRef x1)))).(\lambda (_: (nfs2 c
321 x0)).(\lambda (H7: (nf2 c (TLRef x1))).(let H8 \def (eq_ind T t (\lambda (t0:
322 T).(ty3 g c t0 (THead (Bind Abst) w u))) H (THeads (Flat Appl) x0 (TLRef x1))
323 H5) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda (t0: T).(ex4_2
324 T T (\lambda (v: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v))))
325 (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda
326 (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_:
327 T).(nf2 (CHead c (Bind Abst) w) v))))) (let H9 \def H2 in ((let H10 \def H8
328 in (unintro T u (\lambda (t0: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1))
329 (THead (Bind Abst) w t0)) \to ((ty3_nf2_inv_abst_premise c w t0) \to (ex4_2 T
330 T (\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1))
331 (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0)))
332 (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v t0)))
333 (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))))))
334 (unintro T w (\lambda (t0: T).(\forall (x: T).((ty3 g c (THeads (Flat Appl)
335 x0 (TLRef x1)) (THead (Bind Abst) t0 x)) \to ((ty3_nf2_inv_abst_premise c t0
336 x) \to (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl)
337 x0 (TLRef x1)) (THead (Bind Abst) t0 v)))) (\lambda (_: T).(\lambda (w0:
338 T).(ty3 g c t0 w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind
339 Abst) t0) v x))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst)
340 t0) v)))))))) (TList_ind (\lambda (t0: TList).(\forall (x: T).(\forall (x2:
341 T).((ty3 g c (THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x x2)) \to
342 ((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda
343 (_: T).(eq T (THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x v))))
344 (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda
345 (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_:
346 T).(nf2 (CHead c (Bind Abst) x) v))))))))) (\lambda (x: T).(\lambda (x2:
347 T).(\lambda (H11: (ty3 g c (TLRef x1) (THead (Bind Abst) x x2))).(\lambda
348 (H12: (ty3_nf2_inv_abst_premise c x x2)).(or_ind (ex3_3 C T T (\lambda (_:
349 C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind
350 Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c
351 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0:
352 T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda
353 (_: T).(pc3 c (lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e:
354 C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0)))))
355 (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex4_2
356 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind Abst) x
357 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v:
358 T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v:
359 T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (H13: (ex3_3 C
360 T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O
361 t0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
362 (_: T).(getl x1 c (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
363 T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T (\lambda (_:
364 C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind
365 Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c
366 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0:
367 T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef
368 x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x
369 w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2)))
370 (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda
371 (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c (lift (S x1) O
372 x5) (THead (Bind Abst) x x2))).(\lambda (H15: (getl x1 c (CHead x3 (Bind
373 Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(nf2_gen_lref c x3 x4 x1 H15 H7
374 (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind
375 Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v:
376 T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v:
377 T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v))))))))))) H13)) (\lambda
378 (H13: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c
379 (lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0:
380 T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e:
381 C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T
382 (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c (lift (S x1) O u0)
383 (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
384 T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0:
385 T).(\lambda (t0: T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda
386 (_: T).(eq T (TLRef x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda
387 (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c
388 (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind
389 Abst) x) v)))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda
390 (H14: (pc3 c (lift (S x1) O x4) (THead (Bind Abst) x x2))).(\lambda (H15:
391 (getl x1 c (CHead x3 (Bind Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let
392 H_x0 \def (H12 x3 x4 x1 H15 TNil H14) in (let H17 \def H_x0 in (False_ind
393 (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind
394 Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v:
395 T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v:
396 T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) H17))))))))) H13))
397 (ty3_gen_lref g c (THead (Bind Abst) x x2) x1 H11)))))) (\lambda (t0:
398 T).(\lambda (t1: TList).(\lambda (H11: ((\forall (x: T).(\forall (x2:
399 T).((ty3 g c (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2)) \to
400 ((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda
401 (_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x v))))
402 (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda
403 (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_:
404 T).(nf2 (CHead c (Bind Abst) x) v)))))))))).(\lambda (x: T).(\lambda (x2:
405 T).(\lambda (H12: (ty3 g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
406 (TLRef x1))) (THead (Bind Abst) x x2))).(\lambda (H13:
407 (ty3_nf2_inv_abst_premise c x x2)).(ex3_2_ind T T (\lambda (u0: T).(\lambda
408 (t2: T).(pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) u0 t2)) (THead (Bind
409 Abst) x x2)))) (\lambda (u0: T).(\lambda (t2: T).(ty3 g c (THeads (Flat Appl)
410 t1 (TLRef x1)) (THead (Bind Abst) u0 t2)))) (\lambda (u0: T).(\lambda (_:
411 T).(ty3 g c t0 u0))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead
412 (Flat Appl) t0 (THeads (Flat Appl) t1 (TLRef x1))) (THead (Bind Abst) x v))))
413 (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda
414 (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_:
415 T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (x3: T).(\lambda (x4:
416 T).(\lambda (H14: (pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4))
417 (THead (Bind Abst) x x2))).(\lambda (H15: (ty3 g c (THeads (Flat Appl) t1
418 (TLRef x1)) (THead (Bind Abst) x3 x4))).(\lambda (_: (ty3 g c t0 x3)).(let
419 H_y \def (ty3_nf2_gen__ty3_nf2_inv_abst_aux c x x2 H13 t0 x3 x4 H14) in (let
420 H_x0 \def (H11 x3 x4 H15 H_y) in (let H17 \def H_x0 in (ex4_2_ind T T
421 (\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1))
422 (THead (Bind Abst) x3 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x3
423 w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x3) v x4)))
424 (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x3) v))) (ex4_2 T T
425 (\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat
426 Appl) t1 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda
427 (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c
428 (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind
429 Abst) x) v)))) (\lambda (x5: T).(\lambda (x6: T).(\lambda (H18: (eq T (THeads
430 (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x3 x5))).(\lambda (_: (ty3 g c
431 x3 x6)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x3) x5 x4)).(\lambda (_:
432 (nf2 (CHead c (Bind Abst) x3) x5)).(TList_ind (\lambda (t2: TList).((eq T
433 (THeads (Flat Appl) t2 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T
434 (\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat
435 Appl) t2 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda
436 (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c
437 (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind
438 Abst) x) v)))))) (\lambda (H22: (eq T (THeads (Flat Appl) TNil (TLRef x1))
439 (THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (TLRef x1) (\lambda (ee:
440 T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
441 False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
442 (THead (Bind Abst) x3 x5) H22) in (False_ind (ex4_2 T T (\lambda (v:
443 T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) TNil
444 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3
445 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v
446 x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v))))
447 H23))) (\lambda (t2: T).(\lambda (t3: TList).(\lambda (_: (((eq T (THeads
448 (Flat Appl) t3 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T (\lambda
449 (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) t3
450 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3
451 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v
452 x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x)
453 v))))))).(\lambda (H22: (eq T (THeads (Flat Appl) (TCons t2 t3) (TLRef x1))
454 (THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (THead (Flat Appl) t2
455 (THeads (Flat Appl) t3 (TLRef x1))) (\lambda (ee: T).(match ee in T return
456 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
457 \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
458 (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
459 True])])) I (THead (Bind Abst) x3 x5) H22) in (False_ind (ex4_2 T T (\lambda
460 (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) (TCons
461 t2 t3) (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0:
462 T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind
463 Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x)
464 v)))) H23)))))) t1 H18))))))) H17))))))))) (ty3_gen_appl g c t0 (THeads (Flat
465 Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2) H12))))))))) x0)) H10)) H9)) t
466 H5))))))) H4)) H3))))))))))).