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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 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/clen/getl".
19 include "clen/defs.ma".
21 include "getl/props.ma".
23 theorem getl_ctail_clen:
24 \forall (b: B).(\forall (t: T).(\forall (c: C).(ex nat (\lambda (n:
25 nat).(getl (clen c) (CTail (Bind b) t c) (CHead (CSort n) (Bind b) t))))))
27 \lambda (b: B).(\lambda (t: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(ex
28 nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort n)
29 (Bind b) t))))) (\lambda (n: nat).(ex_intro nat (\lambda (n0: nat).(getl O
30 (CHead (CSort n) (Bind b) t) (CHead (CSort n0) (Bind b) t))) n (getl_refl b
31 (CSort n) t))) (\lambda (c0: C).(\lambda (H: (ex nat (\lambda (n: nat).(getl
32 (clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))))).(\lambda (k:
33 K).(\lambda (t0: T).(let H0 \def H in (ex_ind nat (\lambda (n: nat).(getl
34 (clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))) (ex nat
35 (\lambda (n: nat).(getl (s k (clen c0)) (CHead (CTail (Bind b) t c0) k t0)
36 (CHead (CSort n) (Bind b) t)))) (\lambda (x: nat).(\lambda (H1: (getl (clen
37 c0) (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t))).(match k in K return
38 (\lambda (k0: K).(ex nat (\lambda (n: nat).(getl (s k0 (clen c0)) (CHead
39 (CTail (Bind b) t c0) k0 t0) (CHead (CSort n) (Bind b) t))))) with [(Bind b0)
40 \Rightarrow (ex_intro nat (\lambda (n: nat).(getl (S (clen c0)) (CHead (CTail
41 (Bind b) t c0) (Bind b0) t0) (CHead (CSort n) (Bind b) t))) x (getl_head
42 (Bind b0) (clen c0) (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t) H1
43 t0)) | (Flat f) \Rightarrow (ex_intro nat (\lambda (n: nat).(getl (clen c0)
44 (CHead (CTail (Bind b) t c0) (Flat f) t0) (CHead (CSort n) (Bind b) t))) x
45 (getl_flat (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t) (clen c0) H1 f
46 t0))]))) H0)))))) c))).
48 theorem getl_gen_tail:
49 \forall (k: K).(\forall (b: B).(\forall (u1: T).(\forall (u2: T).(\forall
50 (c2: C).(\forall (c1: C).(\forall (i: nat).((getl i (CTail k u1 c1) (CHead c2
51 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
52 (\lambda (e: C).(getl i c1 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
53 nat).(eq nat i (clen c1))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
54 nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))))))))))
56 \lambda (k: K).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2: T).(\lambda
57 (c2: C).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (i: nat).((getl i
58 (CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C
59 c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4
60 nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind
61 b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort
62 n)))))))) (\lambda (n: nat).(\lambda (i: nat).(match i in nat return (\lambda
63 (n0: nat).((getl n0 (CTail k u1 (CSort n)) (CHead c2 (Bind b) u2)) \to (or
64 (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n0
65 (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0
66 (clen (CSort n)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq
67 T u1 u2)) (\lambda (n1: nat).(eq C c2 (CSort n1))))))) with [O \Rightarrow
68 (\lambda (H: (getl O (CHead (CSort n) k u1) (CHead c2 (Bind b) u2))).((match
69 k in K return (\lambda (k0: K).((clear (CHead (CSort n) k0 u1) (CHead c2
70 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k0 u1 e)))
71 (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda
72 (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K k0 (Bind b))) (\lambda (_:
73 nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))))) with [(Bind
74 b0) \Rightarrow (\lambda (H0: (clear (CHead (CSort n) (Bind b0) u1) (CHead c2
75 (Bind b) u2))).(let H1 \def (f_equal C C (\lambda (e: C).(match e in C return
76 (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow
77 c])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0) u1) (clear_gen_bind b0
78 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in ((let H2 \def (f_equal C B
79 (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
80 \Rightarrow b | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_:
81 K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead c2
82 (Bind b) u2) (CHead (CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n)
83 (CHead c2 (Bind b) u2) u1 H0)) in ((let H3 \def (f_equal C T (\lambda (e:
84 C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u2 |
85 (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind
86 b0) u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in
87 (\lambda (H4: (eq B b b0)).(\lambda (H5: (eq C c2 (CSort n))).(eq_ind_r C
88 (CSort n) (\lambda (c: C).(or (ex2 C (\lambda (e: C).(eq C c (CTail (Bind b0)
89 u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b) u2)))) (ex4 nat
90 (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b)))
91 (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c (CSort n0))))))
92 (eq_ind_r T u1 (\lambda (t: T).(or (ex2 C (\lambda (e: C).(eq C (CSort n)
93 (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b)
94 t)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind
95 b0) (Bind b))) (\lambda (_: nat).(eq T u1 t)) (\lambda (n0: nat).(eq C (CSort
96 n) (CSort n0)))))) (eq_ind_r B b0 (\lambda (b1: B).(or (ex2 C (\lambda (e:
97 C).(eq C (CSort n) (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n)
98 (CHead e (Bind b1) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda
99 (_: nat).(eq K (Bind b0) (Bind b1))) (\lambda (_: nat).(eq T u1 u1)) (\lambda
100 (n0: nat).(eq C (CSort n) (CSort n0)))))) (or_intror (ex2 C (\lambda (e:
101 C).(eq C (CSort n) (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n)
102 (CHead e (Bind b0) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda
103 (_: nat).(eq K (Bind b0) (Bind b0))) (\lambda (_: nat).(eq T u1 u1)) (\lambda
104 (n0: nat).(eq C (CSort n) (CSort n0)))) (ex4_intro nat (\lambda (_: nat).(eq
105 nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b0))) (\lambda (_: nat).(eq
106 T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0))) n (refl_equal nat
107 O) (refl_equal K (Bind b0)) (refl_equal T u1) (refl_equal C (CSort n)))) b
108 H4) u2 H3) c2 H5)))) H2)) H1))) | (Flat f) \Rightarrow (\lambda (H0: (clear
109 (CHead (CSort n) (Flat f) u1) (CHead c2 (Bind b) u2))).(clear_gen_sort (CHead
110 c2 (Bind b) u2) n (clear_gen_flat f (CSort n) (CHead c2 (Bind b) u2) u1 H0)
111 (or (ex2 C (\lambda (e: C).(eq C c2 (CTail (Flat f) u1 e))) (\lambda (e:
112 C).(getl O (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq
113 nat O O)) (\lambda (_: nat).(eq K (Flat f) (Bind b))) (\lambda (_: nat).(eq T
114 u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0)))))))]) (getl_gen_O (CHead
115 (CSort n) k u1) (CHead c2 (Bind b) u2) H))) | (S n0) \Rightarrow (\lambda (H:
116 (getl (S n0) (CHead (CSort n) k u1) (CHead c2 (Bind b) u2))).(getl_gen_sort n
117 (r k n0) (CHead c2 (Bind b) u2) (getl_gen_S k (CSort n) (CHead c2 (Bind b)
118 u2) u1 n0 H) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda
119 (e: C).(getl (S n0) (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
120 nat).(eq nat (S n0) O)) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
121 nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2 (CSort n1)))))))]))) (\lambda
122 (c: C).(\lambda (H: ((\forall (i: nat).((getl i (CTail k u1 c) (CHead c2
123 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
124 (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
125 nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
126 nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))))))).(\lambda (k0:
127 K).(\lambda (t: T).(\lambda (i: nat).(match i in nat return (\lambda (n:
128 nat).((getl n (CTail k u1 (CHead c k0 t)) (CHead c2 (Bind b) u2)) \to (or
129 (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n
130 (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n
131 (clen (CHead c k0 t)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
132 nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))))) with [O
133 \Rightarrow (\lambda (H0: (getl O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind
134 b) u2))).((match k0 in K return (\lambda (k1: K).((clear (CHead (CTail k u1
135 c) k1 t) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2
136 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c k1 t) (CHead e (Bind b)
137 u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s k1 (clen c)))) (\lambda (_:
138 nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq
139 C c2 (CSort n))))))) with [(Bind b0) \Rightarrow (\lambda (H1: (clear (CHead
140 (CTail k u1 c) (Bind b0) t) (CHead c2 (Bind b) u2))).(let H2 \def (f_equal C
141 C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
142 \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind b) u2) (CHead
143 (CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind
144 b) u2) t H1)) in ((let H3 \def (f_equal C B (\lambda (e: C).(match e in C
145 return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _)
146 \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
147 \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead
148 (CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind
149 b) u2) t H1)) in ((let H4 \def (f_equal C T (\lambda (e: C).(match e in C
150 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t)
151 \Rightarrow t])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t)
152 (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in (\lambda
153 (H5: (eq B b b0)).(\lambda (H6: (eq C c2 (CTail k u1 c))).(eq_ind T u2
154 (\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
155 (\lambda (e: C).(getl O (CHead c (Bind b0) t0) (CHead e (Bind b) u2)))) (ex4
156 nat (\lambda (_: nat).(eq nat O (s (Bind b0) (clen c)))) (\lambda (_:
157 nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq
158 C c2 (CSort n)))))) (eq_ind B b (\lambda (b1: B).(or (ex2 C (\lambda (e:
159 C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b1) u2)
160 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b1)
161 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1
162 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))) (let H7 \def (eq_ind C c2
163 (\lambda (c0: C).(\forall (i: nat).((getl i (CTail k u1 c) (CHead c0 (Bind b)
164 u2)) \to (or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e:
165 C).(getl i c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat i
166 (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1
167 u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))))) H (CTail k u1 c) H6) in
168 (eq_ind_r C (CTail k u1 c) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C
169 c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e
170 (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b) (clen c))))
171 (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda
172 (n: nat).(eq C c0 (CSort n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C
173 (CTail k u1 c) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2)
174 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b)
175 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1
176 u2)) (\lambda (n: nat).(eq C (CTail k u1 c) (CSort n)))) (ex_intro2 C
177 (\lambda (e: C).(eq C (CTail k u1 c) (CTail k u1 e))) (\lambda (e: C).(getl O
178 (CHead c (Bind b) u2) (CHead e (Bind b) u2))) c (refl_equal C (CTail k u1 c))
179 (getl_refl b c u2))) c2 H6)) b0 H5) t H4)))) H3)) H2))) | (Flat f)
180 \Rightarrow (\lambda (H1: (clear (CHead (CTail k u1 c) (Flat f) t) (CHead c2
181 (Bind b) u2))).(let H2 \def (H O (getl_intro O (CTail k u1 c) (CHead c2 (Bind
182 b) u2) (CTail k u1 c) (drop_refl (CTail k u1 c)) (clear_gen_flat f (CTail k
183 u1 c) (CHead c2 (Bind b) u2) t H1))) in (or_ind (ex2 C (\lambda (e: C).(eq C
184 c2 (CTail k u1 e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2)))) (ex4
185 nat (\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind
186 b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))
187 (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O
188 (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq
189 nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda
190 (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (H3:
191 (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O c
192 (CHead e (Bind b) u2))))).(ex2_ind C (\lambda (e: C).(eq C c2 (CTail k u1
193 e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2))) (or (ex2 C (\lambda
194 (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t)
195 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f)
196 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1
197 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (x: C).(\lambda (H4:
198 (eq C c2 (CTail k u1 x))).(\lambda (H5: (getl O c (CHead x (Bind b)
199 u2))).(eq_ind_r C (CTail k u1 x) (\lambda (c0: C).(or (ex2 C (\lambda (e:
200 C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t)
201 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f)
202 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1
203 u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))) (or_introl (ex2 C (\lambda (e:
204 C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c
205 (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s
206 (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
207 nat).(eq T u1 u2)) (\lambda (n: nat).(eq C (CTail k u1 x) (CSort n))))
208 (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda
209 (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2))) x (refl_equal C
210 (CTail k u1 x)) (getl_flat c (CHead x (Bind b) u2) O H5 f t))) c2 H4)))) H3))
211 (\lambda (H3: (ex4 nat (\lambda (_: nat).(eq nat O (clen c))) (\lambda (_:
212 nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq
213 C c2 (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq nat O (clen c)))
214 (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda
215 (n: nat).(eq C c2 (CSort n))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1
216 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2))))
217 (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_:
218 nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq
219 C c2 (CSort n))))) (\lambda (x0: nat).(\lambda (H4: (eq nat O (clen
220 c))).(\lambda (H5: (eq K k (Bind b))).(\lambda (H6: (eq T u1 u2)).(\lambda
221 (H7: (eq C c2 (CSort x0))).(eq_ind_r C (CSort x0) (\lambda (c0: C).(or (ex2 C
222 (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c
223 (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s
224 (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
225 nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))) (eq_ind T u1
226 (\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k u1 e)))
227 (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) t0)))) (ex4
228 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq
229 K k (Bind b))) (\lambda (_: nat).(eq T u1 t0)) (\lambda (n: nat).(eq C (CSort
230 x0) (CSort n)))))) (eq_ind_r K (Bind b) (\lambda (k1: K).(or (ex2 C (\lambda
231 (e: C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e: C).(getl O (CHead c
232 (Flat f) t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O (s
233 (Flat f) (clen c)))) (\lambda (_: nat).(eq K k1 (Bind b))) (\lambda (_:
234 nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort n))))))
235 (or_intror (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e)))
236 (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u1)))) (ex4
237 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq
238 K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n: nat).(eq C
239 (CSort x0) (CSort n)))) (ex4_intro nat (\lambda (_: nat).(eq nat O (s (Flat
240 f) (clen c)))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_:
241 nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort n))) x0 H4
242 (refl_equal K (Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) k H5)
243 u2 H6) c2 H7)))))) H3)) H2)))]) (getl_gen_O (CHead (CTail k u1 c) k0 t)
244 (CHead c2 (Bind b) u2) H0))) | (S n) \Rightarrow (\lambda (H0: (getl (S n)
245 (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) u2))).(let H_x \def (H (r k0
246 n) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H0)) in (let H1
247 \def H_x in (or_ind (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda
248 (e: C).(getl (r k0 n) c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
249 nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b)))
250 (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0)))) (or
251 (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n)
252 (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S
253 n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
254 nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (H2:
255 (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0
256 n) c (CHead e (Bind b) u2))))).(ex2_ind C (\lambda (e: C).(eq C c2 (CTail k
257 u1 e))) (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind b) u2))) (or (ex2 C
258 (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead
259 c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s
260 k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T
261 u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (x: C).(\lambda
262 (H3: (eq C c2 (CTail k u1 x))).(\lambda (H4: (getl (r k0 n) c (CHead x (Bind
263 b) u2))).(let H5 \def (eq_ind C c2 (\lambda (c0: C).(getl (r k0 n) (CTail k
264 u1 c) (CHead c0 (Bind b) u2))) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind
265 b) u2) t n H0) (CTail k u1 x) H3) in (eq_ind_r C (CTail k u1 x) (\lambda (c0:
266 C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl
267 (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq
268 nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
269 nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort n0)))))) (or_introl
270 (ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e:
271 C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
272 nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b)))
273 (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C (CTail k u1 x)
274 (CSort n0)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1
275 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2))) x
276 (refl_equal C (CTail k u1 x)) (getl_head k0 n c (CHead x (Bind b) u2) H4 t)))
277 c2 H3))))) H2)) (\lambda (H2: (ex4 nat (\lambda (_: nat).(eq nat (r k0 n)
278 (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1
279 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))).(ex4_ind nat (\lambda (_:
280 nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b)))
281 (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))) (or
282 (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n)
283 (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S
284 n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
285 nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (x0:
286 nat).(\lambda (H3: (eq nat (r k0 n) (clen c))).(\lambda (H4: (eq K k (Bind
287 b))).(\lambda (H5: (eq T u1 u2)).(\lambda (H6: (eq C c2 (CSort x0))).(let H7
288 \def (eq_ind C c2 (\lambda (c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0
289 (Bind b) u2))) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H0)
290 (CSort x0) H6) in (eq_ind_r C (CSort x0) (\lambda (c0: C).(or (ex2 C (\lambda
291 (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t)
292 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen
293 c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2))
294 (\lambda (n0: nat).(eq C c0 (CSort n0)))))) (let H8 \def (eq_ind_r T u2
295 (\lambda (t: T).(getl (r k0 n) (CTail k u1 c) (CHead (CSort x0) (Bind b) t)))
296 H7 u1 H5) in (eq_ind T u1 (\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C
297 (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead
298 e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c))))
299 (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 t0)) (\lambda
300 (n0: nat).(eq C (CSort x0) (CSort n0)))))) (let H9 \def (eq_ind K k (\lambda
301 (k: K).(getl (r k0 n) (CTail k u1 c) (CHead (CSort x0) (Bind b) u1))) H8
302 (Bind b) H4) in (eq_ind_r K (Bind b) (\lambda (k1: K).(or (ex2 C (\lambda (e:
303 C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0
304 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0
305 (clen c)))) (\lambda (_: nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1
306 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0)))))) (eq_ind nat (r k0 n)
307 (\lambda (n0: nat).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind
308 b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b)
309 u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 n0))) (\lambda (_:
310 nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n1:
311 nat).(eq C (CSort x0) (CSort n1)))))) (eq_ind_r nat (S n) (\lambda (n0:
312 nat).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e)))
313 (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat
314 (\lambda (_: nat).(eq nat (S n) n0)) (\lambda (_: nat).(eq K (Bind b) (Bind
315 b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n1: nat).(eq C (CSort x0)
316 (CSort n1)))))) (or_intror (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail
317 (Bind b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b)
318 u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (S n))) (\lambda (_: nat).(eq
319 K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq
320 C (CSort x0) (CSort n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat (S n) (S
321 n))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1
322 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0))) x0 (refl_equal nat (S
323 n)) (refl_equal K (Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) (s
324 k0 (r k0 n)) (s_r k0 n)) (clen c) H3) k H4)) u2 H5)) c2 H6))))))) H2))
325 H1))))])))))) c1)))))).