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 "LambdaDelta-1/wcpr0/defs.ma".
19 include "LambdaDelta-1/getl/props.ma".
22 \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h:
23 nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead
24 e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2
25 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda
26 (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))))
28 \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind
29 (\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
30 (u1: T).(\forall (k: K).((drop h O c (CHead e1 k u1)) \to (ex3_2 C T (\lambda
31 (e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2:
32 C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
33 u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
34 (u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k
35 u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead
36 e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
37 C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
38 u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
39 c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
40 T).(\forall (k: K).((drop h O c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda
41 (e2: C).(\lambda (u2: T).(drop h O c4 (CHead e2 k u2)))) (\lambda (e2:
42 C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
43 u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
44 u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
45 (e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 k u1) (CHead
46 e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead
47 c4 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
48 (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1:
49 C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u1)
50 (CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C
51 return (\lambda (_: C).C) with [(CSort _) \Rightarrow c3 | (CHead c _ _)
52 \Rightarrow c])) (CHead c3 k u1) (CHead e1 k0 u0) (drop_gen_refl (CHead c3 k
53 u1) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e:
54 C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
55 (CHead _ k1 _) \Rightarrow k1])) (CHead c3 k u1) (CHead e1 k0 u0)
56 (drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in ((let H6 \def
57 (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
58 [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c3 k u1)
59 (CHead e1 k0 u0) (drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in
60 (\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c3 e1)).(eq_ind K k (\lambda
61 (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k
62 u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
63 (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind T u1 (\lambda (t:
64 T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2)
65 (CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda
66 (_: C).(\lambda (u3: T).(pr0 t u3))))) (eq_ind C c3 (\lambda (c: C).(ex3_2 C
67 T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k
68 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_:
69 C).(\lambda (u3: T).(pr0 u1 u3))))) (ex3_2_intro C T (\lambda (e2:
70 C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k u3)))) (\lambda
71 (e2: C).(\lambda (_: T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0
72 u1 u3))) c4 u2 (drop_refl (CHead c4 k u2)) H0 H2) e1 H8) u0 H6) k0 H7))))
73 H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1:
74 C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c3 k0 u1) (CHead e1 k1
75 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 k0
76 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
77 (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) \to (\forall (e1:
78 C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c3 k0 u1) (CHead
79 e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O
80 (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
81 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b:
82 B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
83 (k0: K).((drop n O (CHead c3 (Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T
84 (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Bind b) u2) (CHead e2
85 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
86 C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0:
87 T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Bind b) u1) (CHead
88 e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c3 (CHead e1
89 k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n
90 O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
91 (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2:
92 C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Bind b) u2) (CHead e2 k0 u3))))
93 (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
94 (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
95 n O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0
96 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead
97 c4 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
98 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop
99 (Bind b) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) (\lambda (f:
100 F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
101 (k0: K).((drop n O (CHead c3 (Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T
102 (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Flat f) u2) (CHead e2
103 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
104 C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0:
105 T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Flat f) u1) (CHead
106 e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c3 (CHead
107 e1 k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop
108 (S n) O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
109 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2:
110 C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Flat f) u2) (CHead e2 k0 u3))))
111 (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
112 (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
113 (S n) O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0
114 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O
115 (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_:
116 T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1
117 (drop_drop (Flat f) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k)
118 h)))))))))) c1 c2 H))).
120 theorem wcpr0_drop_back:
121 \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h:
122 nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead
123 e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2
124 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda
125 (_: C).(\lambda (u2: T).(pr0 u2 u1)))))))))))
127 \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind
128 (\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
129 (u1: T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T
130 (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead e2 k u2)))) (\lambda
131 (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0
132 u2 u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
133 (u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k
134 u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead
135 e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
136 C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
137 u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
138 c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
139 T).(\forall (k: K).((drop h O c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda
140 (e2: C).(\lambda (u2: T).(drop h O c3 (CHead e2 k u2)))) (\lambda (e2:
141 C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2
142 u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
143 u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
144 (e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 k u2) (CHead
145 e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead
146 c3 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
147 (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1:
148 C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c4 k u2)
149 (CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C
150 return (\lambda (_: C).C) with [(CSort _) \Rightarrow c4 | (CHead c _ _)
151 \Rightarrow c])) (CHead c4 k u2) (CHead e1 k0 u0) (drop_gen_refl (CHead c4 k
152 u2) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e:
153 C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
154 (CHead _ k1 _) \Rightarrow k1])) (CHead c4 k u2) (CHead e1 k0 u0)
155 (drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in ((let H6 \def
156 (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
157 [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c4 k u2)
158 (CHead e1 k0 u0) (drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in
159 (\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c4 e1)).(eq_ind K k (\lambda
160 (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k
161 u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
162 (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind T u2 (\lambda (t:
163 T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1)
164 (CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda
165 (_: C).(\lambda (u3: T).(pr0 u3 t))))) (eq_ind C c4 (\lambda (c: C).(ex3_2 C
166 T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k
167 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_:
168 C).(\lambda (u3: T).(pr0 u3 u2))))) (ex3_2_intro C T (\lambda (e2:
169 C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k u3)))) (\lambda
170 (e2: C).(\lambda (_: T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0
171 u3 u2))) c3 u1 (drop_refl (CHead c3 k u1)) H0 H2) e1 H8) u0 H6) k0 H7))))
172 H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1:
173 C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c4 k0 u2) (CHead e1 k1
174 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k0
175 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
176 (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) \to (\forall (e1:
177 C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c4 k0 u2) (CHead
178 e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O
179 (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
180 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b:
181 B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
182 (k0: K).((drop n O (CHead c4 (Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T
183 (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Bind b) u1) (CHead e2
184 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
185 C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0:
186 T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Bind b) u2) (CHead
187 e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c4 (CHead e1
188 k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n
189 O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
190 (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2:
191 C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Bind b) u1) (CHead e2 k0 u3))))
192 (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
193 (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
194 n O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1
195 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead
196 c3 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
197 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop
198 (Bind b) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) (\lambda (f:
199 F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
200 (k0: K).((drop n O (CHead c4 (Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T
201 (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Flat f) u1) (CHead e2
202 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
203 C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0:
204 T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Flat f) u2) (CHead
205 e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c4 (CHead
206 e1 k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop
207 (S n) O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
208 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2:
209 C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Flat f) u1) (CHead e2 k0 u3))))
210 (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
211 (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
212 (S n) O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0
213 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O
214 (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_:
215 T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1
216 (drop_drop (Flat f) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k)
217 h)))))))))) c2 c1 H))).
220 \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h:
221 nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1
222 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2
223 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
224 C).(\lambda (u2: T).(pr0 u1 u2)))))))))))
226 \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind
227 (\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
228 (u1: T).(\forall (k: K).((getl h c (CHead e1 k u1)) \to (ex3_2 C T (\lambda
229 (e2: C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2:
230 C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
231 u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
232 (u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k
233 u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2
234 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
235 C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
236 u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
237 c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
238 T).(\forall (k: K).((getl h c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2:
239 C).(\lambda (u2: T).(getl h c4 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda
240 (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
241 u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
242 u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
243 (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 k u1) (CHead e1
244 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c4 k
245 u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
246 (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1:
247 C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c3 k u1)
248 (CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c3 k1 u1) (CHead e1
249 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 k1
250 u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
251 (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))))) (\lambda (b: B).(\lambda
252 (H4: (clear (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (f_equal C
253 C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
254 \Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c3
255 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let H6 \def
256 (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
257 [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0)
258 (CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let
259 H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
260 with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0
261 u0) (CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in
262 (\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c3)).(eq_ind_r K
263 (Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl
264 O (CHead c4 (Bind b) u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_:
265 T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind_r
266 T u1 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O
267 (CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda
268 (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 t u3)))))
269 (eq_ind_r C c3 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3:
270 T).(getl O (CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2:
271 C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1
272 u3))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4
273 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_:
274 T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c4 u2
275 (getl_refl b c4 u2) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f:
276 F).(\lambda (H4: (clear (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5
277 \def (H1 O e1 u0 k0 (getl_intro O c3 (CHead e1 k0 u0) c3 (drop_refl c3)
278 (clear_gen_flat f c3 (CHead e1 k0 u0) u1 H4))) in (ex3_2_ind C T (\lambda
279 (e2: C).(\lambda (u3: T).(getl O c4 (CHead e2 k0 u3)))) (\lambda (e2:
280 C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0
281 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Flat f)
282 u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
283 (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda
284 (x1: T).(\lambda (H6: (getl O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1
285 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda
286 (u3: T).(getl O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2:
287 C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0
288 u3))) x0 x1 (getl_flat c4 (CHead x0 k0 x1) O H6 f u2) H7 H8)))))) H5)))) k
289 (getl_gen_O (CHead c3 k u1) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0:
290 K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1:
291 K).((getl n (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
292 C).(\lambda (u4: T).(getl n (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda
293 (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0
294 u3 u4))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl
295 (S n) (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
296 C).(\lambda (u4: T).(getl (S n) (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda
297 (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0
298 u3 u4))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall
299 (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Bind b) u1)
300 (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n
301 (CHead c4 (Bind b) u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_:
302 T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3
303 u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
304 (getl (S n) (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (H1 n e1
305 u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T
306 (\lambda (e2: C).(\lambda (u3: T).(getl n c4 (CHead e2 k0 u3)))) (\lambda
307 (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0
308 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4
309 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
310 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0:
311 C).(\lambda (x1: T).(\lambda (H6: (getl n c4 (CHead x0 k0 x1))).(\lambda (H7:
312 (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2:
313 C).(\lambda (u3: T).(getl (S n) (CHead c4 (Bind b) u2) (CHead e2 k0 u3))))
314 (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
315 (u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Bind b) n c4 (CHead x0 k0 x1) H6 u2)
316 H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_:
317 ((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Flat
318 f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4:
319 T).(getl n (CHead c4 (Flat f) u2) (CHead e2 k0 u4)))) (\lambda (e2:
320 C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3
321 u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
322 (getl (S n) (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 (S n)
323 e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T
324 (\lambda (e2: C).(\lambda (u3: T).(getl (S n) c4 (CHead e2 k0 u3)))) (\lambda
325 (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0
326 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4
327 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
328 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0:
329 C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c4 (CHead x0 k0 x1))).(\lambda
330 (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2:
331 C).(\lambda (u3: T).(getl (S n) (CHead c4 (Flat f) u2) (CHead e2 k0 u3))))
332 (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
333 (u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat f) n c4 (CHead x0 k0 x1) H6 u2)
334 H7 H8)))))) H5))))))))) k) h)))))))))) c1 c2 H))).
336 theorem wcpr0_getl_back:
337 \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h:
338 nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1
339 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2
340 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
341 C).(\lambda (u2: T).(pr0 u2 u1)))))))))))
343 \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind
344 (\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
345 (u1: T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda
346 (e2: C).(\lambda (u2: T).(getl h c (CHead e2 k u2)))) (\lambda (e2:
347 C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2
348 u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
349 (u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k
350 u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2
351 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
352 C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
353 u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
354 c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
355 T).(\forall (k: K).((getl h c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2:
356 C).(\lambda (u2: T).(getl h c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda
357 (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2
358 u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
359 u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
360 (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 k u2) (CHead e1
361 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k
362 u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
363 (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1:
364 C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c4 k u2)
365 (CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c4 k1 u2) (CHead e1
366 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 k1
367 u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
368 (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))))) (\lambda (b: B).(\lambda
369 (H4: (clear (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (f_equal C
370 C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
371 \Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c4
372 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let H6 \def
373 (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
374 [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0)
375 (CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let
376 H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
377 with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0
378 u0) (CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in
379 (\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c4)).(eq_ind_r K
380 (Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl
381 O (CHead c3 (Bind b) u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_:
382 T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind_r
383 T u2 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O
384 (CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda
385 (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 t)))))
386 (eq_ind_r C c4 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3:
387 T).(getl O (CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2:
388 C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3
389 u2))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3
390 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_:
391 T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))) c3 u1
392 (getl_refl b c3 u1) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f:
393 F).(\lambda (H4: (clear (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5
394 \def (H1 O e1 u0 k0 (getl_intro O c4 (CHead e1 k0 u0) c4 (drop_refl c4)
395 (clear_gen_flat f c4 (CHead e1 k0 u0) u2 H4))) in (ex3_2_ind C T (\lambda
396 (e2: C).(\lambda (u3: T).(getl O c3 (CHead e2 k0 u3)))) (\lambda (e2:
397 C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3
398 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f)
399 u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
400 (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda
401 (x1: T).(\lambda (H6: (getl O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0
402 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda
403 (u3: T).(getl O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2:
404 C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3
405 u0))) x0 x1 (getl_flat c3 (CHead x0 k0 x1) O H6 f u1) H7 H8)))))) H5)))) k
406 (getl_gen_O (CHead c4 k u2) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0:
407 K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1:
408 K).((getl n (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
409 C).(\lambda (u4: T).(getl n (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda
410 (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0
411 u4 u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl
412 (S n) (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
413 C).(\lambda (u4: T).(getl (S n) (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda
414 (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0
415 u4 u3))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall
416 (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Bind b) u2)
417 (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n
418 (CHead c3 (Bind b) u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_:
419 T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4
420 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
421 (getl (S n) (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (H1 n e1
422 u0 k0 (getl_gen_S (Bind b) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T
423 (\lambda (e2: C).(\lambda (u3: T).(getl n c3 (CHead e2 k0 u3)))) (\lambda
424 (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0
425 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3
426 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
427 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0:
428 C).(\lambda (x1: T).(\lambda (H6: (getl n c3 (CHead x0 k0 x1))).(\lambda (H7:
429 (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2:
430 C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u1) (CHead e2 k0 u3))))
431 (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
432 (u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Bind b) n c3 (CHead x0 k0 x1) H6 u1)
433 H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_:
434 ((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Flat
435 f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4:
436 T).(getl n (CHead c3 (Flat f) u1) (CHead e2 k0 u4)))) (\lambda (e2:
437 C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4
438 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
439 (getl (S n) (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 (S n)
440 e1 u0 k0 (getl_gen_S (Flat f) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T
441 (\lambda (e2: C).(\lambda (u3: T).(getl (S n) c3 (CHead e2 k0 u3)))) (\lambda
442 (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0
443 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3
444 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
445 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0:
446 C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead x0 k0 x1))).(\lambda
447 (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2:
448 C).(\lambda (u3: T).(getl (S n) (CHead c3 (Flat f) u1) (CHead e2 k0 u3))))
449 (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
450 (u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Flat f) n c3 (CHead x0 k0 x1) H6 u1)
451 H7 H8)))))) H5))))))))) k) h)))))))))) c2 c1 H))).