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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 *)
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15 (* This file was automatically generated: do not edit *********************)
17 include "basic_1/csubst0/fwd.ma".
19 include "basic_1/drop/fwd.ma".
21 lemma csubst0_drop_gt:
22 \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall
23 (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O
24 c1 e) \to (drop n O c2 e)))))))))
26 \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0)
27 \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2)
28 \to (\forall (e: C).((drop n0 O c1 e) \to (drop n0 O c2 e)))))))))) (\lambda
29 (i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda
30 (v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O
31 O c1 e)).(lt_x_O i H (drop O O c2 e)))))))))) (\lambda (n0: nat).(\lambda (H:
32 ((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall
33 (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (drop
34 n0 O c2 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda
35 (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v
36 c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))
37 (\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v
38 (CSort n1) c2)).(\lambda (e: C).(\lambda (H2: (drop (S n0) O (CSort n1)
39 e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (drop (S n0)
40 O c2 e) (\lambda (H3: (eq C e (CSort n1))).(\lambda (H4: (eq nat (S n0)
41 O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(drop
42 (S n0) O c2 c)) (let H6 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee
43 with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind
44 (drop (S n0) O c2 (CSort n1)) H6)) e H3)))) (drop_gen_sort n1 (S n0) O e
45 H2)))))))) (\lambda (c: C).(\lambda (H1: ((\forall (c2: C).(\forall (v:
46 T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S
47 n0) O c2 e)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda
48 (v: T).(\lambda (H2: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda
49 (H3: (drop (S n0) O (CHead c k t) e)).(let H4 \def (csubst0_gen_head k c c2 t
50 v i H2) in (or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i
51 (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2))))
52 (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda
53 (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_:
54 nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
55 v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
56 nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
57 nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
58 (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
59 nat).(csubst0 j v c c3))))) (drop (S n0) O c2 e) (\lambda (H5: (ex3_2 T nat
60 (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2:
61 T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
62 nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j:
63 nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead
64 c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (drop (S
65 n0) O c2 e) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H6: (eq nat i (s k
66 x1))).(\lambda (H7: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t
67 x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(drop (S n0) O c0 e)) (let
68 H9 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0:
69 T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop
70 (S n0) O c3 e0))))))) H1 (s k x1) H6) in (let H10 \def (eq_ind nat i (\lambda
71 (n1: nat).(lt n1 (S n0))) H0 (s k x1) H6) in (K_ind (\lambda (k0: K).((drop
72 (r k0 n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1)
73 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
74 e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead c k0 x0)
75 e))))) (\lambda (b: B).(\lambda (H11: (drop (r (Bind b) n0) O c e)).(\lambda
76 (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to
77 (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
78 e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 c
79 e H11 x0))))) (\lambda (f: F).(\lambda (H11: (drop (r (Flat f) n0) O c
80 e)).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x1)
81 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
82 e0)))))))).(\lambda (H13: (lt (s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O)
83 (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))
84 (drop (S n0) O (CHead c (Flat f) x0) e) (\lambda (_: (eq nat x1
85 O)).(drop_drop (Flat f) n0 c e H11 x0)) (\lambda (H14: (ex2 nat (\lambda (m:
86 nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda
87 (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O
88 (CHead c (Flat f) x0) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S
89 x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e H11 x0)))) H14))
90 (lt_gen_xS x1 n0 H13)))))) k (drop_gen_drop k c e t n0 H3) H9 H10))) c2
91 H7)))))) H5)) (\lambda (H5: (ex3_2 C nat (\lambda (_: C).(\lambda (j:
92 nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
93 c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
94 c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k
95 j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
96 (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S n0) O c2 e) (\lambda
97 (x0: C).(\lambda (x1: nat).(\lambda (H6: (eq nat i (s k x1))).(\lambda (H7:
98 (eq C c2 (CHead x0 k t))).(\lambda (H8: (csubst0 x1 v c x0)).(eq_ind_r C
99 (CHead x0 k t) (\lambda (c0: C).(drop (S n0) O c0 e)) (let H9 \def (eq_ind
100 nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c
101 c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
102 e0))))))) H1 (s k x1) H6) in (let H10 \def (eq_ind nat i (\lambda (n1:
103 nat).(lt n1 (S n0))) H0 (s k x1) H6) in (K_ind (\lambda (k0: K).((drop (r k0
104 n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c
105 c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
106 e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead x0 k0 t)
107 e))))) (\lambda (b: B).(\lambda (H11: (drop (r (Bind b) n0) O c e)).(\lambda
108 (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to
109 (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
110 e0)))))))).(\lambda (H13: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0
111 x0 e (H x1 (lt_S_n x1 n0 H13) c x0 v H8 e H11) t))))) (\lambda (f:
112 F).(\lambda (H11: (drop (r (Flat f) n0) O c e)).(\lambda (H12: ((\forall (c3:
113 C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0:
114 C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H13: (lt
115 (s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq
116 nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead x0 (Flat
117 f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 x0 e (H12 x0 v H8
118 e H11) t)) (\lambda (H14: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m)))
119 (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S
120 m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead x0 (Flat f) t) e)
121 (\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x
122 n0)).(drop_drop (Flat f) n0 x0 e (H12 x0 v H8 e H11) t)))) H14)) (lt_gen_xS
123 x1 n0 H13)))))) k (drop_gen_drop k c e t n0 H3) H9 H10))) c2 H7)))))) H5))
124 (\lambda (H5: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
125 nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
126 nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
127 (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
128 nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_:
129 C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3:
130 C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
131 (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
132 C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O c2 e) (\lambda (x0:
133 T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H6: (eq nat i (s k
134 x2))).(\lambda (H7: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t
135 x0)).(\lambda (H9: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda
136 (c0: C).(drop (S n0) O c0 e)) (let H10 \def (eq_ind nat i (\lambda (n1:
137 nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall
138 (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0))))))) H1 (s k x2) H6)
139 in (let H11 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2)
140 H6) in (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall (c3:
141 C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall (e0: C).((drop
142 (S n0) O c e0) \to (drop (S n0) O c3 e0))))))) \to ((lt (s k0 x2) (S n0)) \to
143 (drop (S n0) O (CHead x1 k0 x0) e))))) (\lambda (b: B).(\lambda (H12: (drop
144 (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall (v0:
145 T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c
146 e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H14: (lt (s (Bind b) x2) (S
147 n0))).(drop_drop (Bind b) n0 x1 e (H x2 (lt_S_n x2 n0 H14) c x1 v H9 e H12)
148 x0))))) (\lambda (f: F).(\lambda (H12: (drop (r (Flat f) n0) O c e)).(\lambda
149 (H13: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3)
150 \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
151 e0)))))))).(\lambda (H14: (lt (s (Flat f) x2) (S n0))).(or_ind (eq nat x2 O)
152 (ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))
153 (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (_: (eq nat x2
154 O)).(drop_drop (Flat f) n0 x1 e (H13 x1 v H9 e H12) x0)) (\lambda (H15: (ex2
155 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m
156 n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m:
157 nat).(lt m n0)) (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (x:
158 nat).(\lambda (_: (eq nat x2 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat
159 f) n0 x1 e (H13 x1 v H9 e H12) x0)))) H15)) (lt_gen_xS x2 n0 H14)))))) k
160 (drop_gen_drop k c e t n0 H3) H10 H11))) c2 H7)))))))) H5)) H4)))))))))))
163 lemma csubst0_drop_gt_back:
164 \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall
165 (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O
166 c2 e) \to (drop n O c1 e)))))))))
168 \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0)
169 \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2)
170 \to (\forall (e: C).((drop n0 O c2 e) \to (drop n0 O c1 e)))))))))) (\lambda
171 (i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda
172 (v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O
173 O c2 e)).(lt_x_O i H (drop O O c1 e)))))))))) (\lambda (n0: nat).(\lambda (H:
174 ((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall
175 (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (drop
176 n0 O c1 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda
177 (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v
178 c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))
179 (\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H1: (csubst0 i
180 v (CSort n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2
181 e)).(csubst0_gen_sort c2 v i n1 H1 (drop (S n0) O (CSort n1) e))))))))
182 (\lambda (c: C).(\lambda (H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v
183 c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c
184 e)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v:
185 T).(\lambda (H2: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda
186 (H3: (drop (S n0) O c2 e)).(let H4 \def (csubst0_gen_head k c c2 t v i H2) in
187 (or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j))))
188 (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2:
189 T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_:
190 C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_:
191 nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
192 v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
193 nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
194 nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
195 (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
196 nat).(csubst0 j v c c3))))) (drop (S n0) O (CHead c k t) e) (\lambda (H5:
197 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda
198 (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2:
199 T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_:
200 T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
201 nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
202 v t u2))) (drop (S n0) O (CHead c k t) e) (\lambda (x0: T).(\lambda (x1:
203 nat).(\lambda (H6: (eq nat i (s k x1))).(\lambda (H7: (eq C c2 (CHead c k
204 x0))).(\lambda (_: (subst0 x1 v t x0)).(let H9 \def (eq_ind C c2 (\lambda
205 (c0: C).(drop (S n0) O c0 e)) H3 (CHead c k x0) H7) in (let H10 \def (eq_ind
206 nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c
207 c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c
208 e0))))))) H1 (s k x1) H6) in (let H11 \def (eq_ind nat i (\lambda (n1:
209 nat).(lt n1 (S n0))) H0 (s k x1) H6) in (K_ind (\lambda (k0: K).(((\forall
210 (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall (e0:
211 C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) \to ((lt (s k0 x1)
212 (S n0)) \to ((drop (r k0 n0) O c e) \to (drop (S n0) O (CHead c k0 t) e)))))
213 (\lambda (b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s
214 (Bind b) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop
215 (S n0) O c e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(\lambda
216 (H14: (drop (r (Bind b) n0) O c e)).(drop_drop (Bind b) n0 c e H14 t)))))
217 (\lambda (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s
218 (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop
219 (S n0) O c e0)))))))).(\lambda (H13: (lt (s (Flat f) x1) (S n0))).(\lambda
220 (H14: (drop (r (Flat f) n0) O c e)).(or_ind (eq nat x1 O) (ex2 nat (\lambda
221 (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O
222 (CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c
223 e H14 t)) (\lambda (H15: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m)))
224 (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S
225 m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) t) e)
226 (\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x
227 n0)).(drop_drop (Flat f) n0 c e H14 t)))) H15)) (lt_gen_xS x1 n0 H13)))))) k
228 H10 H11 (drop_gen_drop k c e x0 n0 H9)))))))))) H5)) (\lambda (H5: (ex3_2 C
229 nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3:
230 C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j:
231 nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j:
232 nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
233 c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S
234 n0) O (CHead c k t) e) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H6: (eq
235 nat i (s k x1))).(\lambda (H7: (eq C c2 (CHead x0 k t))).(\lambda (H8:
236 (csubst0 x1 v c x0)).(let H9 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0)
237 O c0 e)) H3 (CHead x0 k t) H7) in (let H10 \def (eq_ind nat i (\lambda (n1:
238 nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall
239 (e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x1) H6)
240 in (let H11 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x1)
241 H6) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0
242 (s k0 x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S
243 n0) O c e0))))))) \to ((lt (s k0 x1) (S n0)) \to ((drop (r k0 n0) O x0 e) \to
244 (drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall
245 (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0:
246 C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt
247 (s (Bind b) x1) (S n0))).(\lambda (H14: (drop (r (Bind b) n0) O x0
248 e)).(drop_drop (Bind b) n0 c e (H x1 (lt_S_n x1 n0 H13) c x0 v H8 e H14)
249 t))))) (\lambda (f: F).(\lambda (H12: ((\forall (c3: C).(\forall (v0:
250 T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3
251 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt (s (Flat f) x1) (S
252 n0))).(\lambda (H14: (drop (r (Flat f) n0) O x0 e)).(or_ind (eq nat x1 O)
253 (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))
254 (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop
255 (Flat f) n0 c e (H12 x0 v H8 e H14) t)) (\lambda (H15: (ex2 nat (\lambda (m:
256 nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda
257 (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O
258 (CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S
259 x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H12 x0 v H8 e H14)
260 t)))) H15)) (lt_gen_xS x1 n0 H13)))))) k H10 H11 (drop_gen_drop k x0 e t n0
261 H9)))))))))) H5)) (\lambda (H5: (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
262 C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3:
263 C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
264 (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
265 C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_:
266 T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2:
267 T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
268 (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
269 T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O
270 (CHead c k t) e) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2:
271 nat).(\lambda (H6: (eq nat i (s k x2))).(\lambda (H7: (eq C c2 (CHead x1 k
272 x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H9: (csubst0 x2 v c
273 x1)).(let H10 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H3
274 (CHead x1 k x0) H7) in (let H11 \def (eq_ind nat i (\lambda (n1:
275 nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall
276 (e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x2) H6)
277 in (let H12 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2)
278 H6) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0
279 (s k0 x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S
280 n0) O c e0))))))) \to ((lt (s k0 x2) (S n0)) \to ((drop (r k0 n0) O x1 e) \to
281 (drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall
282 (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0:
283 C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H14: (lt
284 (s (Bind b) x2) (S n0))).(\lambda (H15: (drop (r (Bind b) n0) O x1
285 e)).(drop_drop (Bind b) n0 c e (H x2 (lt_S_n x2 n0 H14) c x1 v H9 e H15)
286 t))))) (\lambda (f: F).(\lambda (H13: ((\forall (c3: C).(\forall (v0:
287 T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3
288 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H14: (lt (s (Flat f) x2) (S
289 n0))).(\lambda (H15: (drop (r (Flat f) n0) O x1 e)).(or_ind (eq nat x2 O)
290 (ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))
291 (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x2 O)).(drop_drop
292 (Flat f) n0 c e (H13 x1 v H9 e H15) t)) (\lambda (H16: (ex2 nat (\lambda (m:
293 nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda
294 (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O
295 (CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x2 (S
296 x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H13 x1 v H9 e H15)
297 t)))) H16)) (lt_gen_xS x2 n0 H14)))))) k H11 H12 (drop_gen_drop k x1 e x0 n0
298 H10)))))))))))) H5)) H4))))))))))) c1)))))) n).
300 lemma csubst0_drop_lt:
301 \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall
302 (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O
303 c1 e) \to (or4 (drop n O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
304 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k:
305 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k
306 w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
307 T).(subst0 (minus i (s k n)) v u w)))))) (ex3_4 K C C T (\lambda (k:
308 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
309 u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
310 T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
311 (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (ex4_5 K C C
312 T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
313 (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda
314 (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w)))))))
315 (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
316 T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1:
317 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k
318 n)) v e1 e2))))))))))))))))
320 \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i)
321 \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2)
322 \to (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T
323 T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
324 (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
325 (w: T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
326 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w))))))
327 (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
328 T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
329 C).(\lambda (u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
330 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1
331 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
332 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
333 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0
334 O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
335 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w))))))
336 (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
337 (_: T).(csubst0 (minus i (s k n0)) v e1 e2))))))))))))))))) (\lambda (i:
338 nat).(\lambda (_: (lt O i)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v:
339 T).(\lambda (H0: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (H1: (drop O O
340 c1 e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 K C T T
341 (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c
342 (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
343 (w: T).(drop O O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
344 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w))))))
345 (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
346 T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
347 C).(\lambda (u: T).(drop O O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
348 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k O)) v e1
349 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
350 C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
351 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
352 c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
353 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w))))))
354 (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
355 (_: T).(csubst0 (minus i (s k O)) v e1 e2))))))))) (csubst0_ind (\lambda (n0:
356 nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).(or4 (drop O O c0 c)
357 (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
358 T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
359 T).(\lambda (w: T).(drop O O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda
360 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w))))))
361 (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
362 T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
363 C).(\lambda (u: T).(drop O O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
364 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n0 (s k O)) t e1
365 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
366 C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
367 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
368 c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
369 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w))))))
370 (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
371 (_: T).(csubst0 (minus n0 (s k O)) t e1 e2)))))))))))) (\lambda (k:
372 K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2:
373 T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(let H3 \def (eq_ind_r
374 nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 (minus (s k i0) (s k O))
375 (s_arith0 k i0)) in (or4_intro1 (drop O O (CHead c k u2) (CHead c k u1))
376 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
377 T).(eq C (CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
378 C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0
379 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
380 T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda
381 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k u1)
382 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
383 C).(\lambda (u: T).(drop O O (CHead c k u2) (CHead e2 k0 u)))))) (\lambda
384 (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s
385 k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda
386 (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k u1)
387 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
388 C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e2 k0
389 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
390 T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda
391 (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
392 T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C T T
393 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
394 (CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
395 C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0
396 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
397 T).(subst0 (minus (s k i0) (s k0 O)) v0 u w))))) k c u1 u2 (refl_equal C
398 (CHead c k u1)) (drop_refl (CHead c k u2)) H3)))))))))) (\lambda (k:
399 K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0:
400 T).(\lambda (H2: (csubst0 i0 v0 c3 c4)).(\lambda (H3: (or4 (drop O O c4 c3)
401 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
402 T).(eq C c3 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
403 (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0:
404 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0
405 O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
406 (_: C).(\lambda (u: T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0:
407 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0
408 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
409 T).(csubst0 (minus i0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
410 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
411 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
412 C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w)))))))
413 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
414 T).(subst0 (minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1:
415 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0
416 O)) v0 e1 e2))))))))).(\lambda (u: T).(let H4 \def (eq_ind_r nat i0 (\lambda
417 (n0: nat).(csubst0 n0 v0 c3 c4)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0))
418 in (let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(or4 (drop O O c4 c3)
419 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda
420 (_: T).(eq C c3 (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0:
421 C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda
422 (k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n0 (s
423 k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
424 C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 k0 u0)))))) (\lambda
425 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O c4 (CHead
426 e2 k0 u0)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
427 (_: T).(csubst0 (minus n0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda
428 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
429 C c3 (CHead e1 k0 u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
430 C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w)))))))
431 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda
432 (w: T).(subst0 (minus n0 (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda
433 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n0
434 (s k0 O)) v0 e1 e2))))))))) H3 (minus (s k i0) (s k O)) (s_arith0 k i0)) in
435 (or4_intro2 (drop O O (CHead c4 k u) (CHead c3 k u)) (ex3_4 K C T T (\lambda
436 (k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k
437 u) (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
438 T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e0 k0 w)))))) (\lambda
439 (k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k
440 i0) (s k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
441 C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0
442 u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0:
443 T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda
444 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O))
445 v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda
446 (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k u) (CHead e1 k0
447 u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
448 T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e2 k0 w))))))) (\lambda
449 (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
450 T).(subst0 (minus (s k i0) (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda
451 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k
452 i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda
453 (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0
454 u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0:
455 T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda
456 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O))
457 v0 e1 e2))))) k c3 c4 u (refl_equal C (CHead c3 k u)) (drop_refl (CHead c4 k
458 u)) H4)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0:
459 T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1
460 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i0 v0 c3
461 c4)).(\lambda (_: (or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k0:
462 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k0
463 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
464 T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
465 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0 O)) v0 u w))))))
466 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
467 T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
468 (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0 u)))))) (\lambda (k0:
469 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k0
470 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
471 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k0
472 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
473 T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) (\lambda (k0:
474 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
475 (minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
476 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0 O)) v0 e1
477 e2))))))))).(let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1
478 u2)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (let H6 \def (eq_ind_r
479 nat i0 (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 (minus (s k i0) (s k O))
480 (s_arith0 k i0)) in (or4_intro3 (drop O O (CHead c4 k u2) (CHead c3 k u1))
481 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
482 T).(eq C (CHead c3 k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
483 C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e0 k0
484 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
485 T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda
486 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k
487 u1) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
488 C).(\lambda (u: T).(drop O O (CHead c4 k u2) (CHead e2 k0 u)))))) (\lambda
489 (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s
490 k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda
491 (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k u1)
492 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
493 C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e2 k0
494 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
495 T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda
496 (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
497 T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex4_5_intro K C C T T
498 (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
499 (_: T).(eq C (CHead c3 k u1) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
500 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k
501 u2) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
502 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u
503 w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
504 T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2)))))) k c3 c4
505 u1 u2 (refl_equal C (CHead c3 k u1)) (drop_refl (CHead c4 k u2)) H5
506 H6)))))))))))))) i v c1 c2 H0) e (drop_gen_refl c1 e H1)))))))))) (\lambda
507 (n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt n0 i) \to (\forall (c1:
508 C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e:
509 C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T T (\lambda (k:
510 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k
511 u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
512 T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda
513 (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (ex3_4 K C C T
514 (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
515 (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda
516 (u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
517 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 e2))))))
518 (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda
519 (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda
520 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead
521 e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
522 T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (\lambda (k:
523 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
524 (minus i (s k n0)) v e1 e2)))))))))))))))))).(\lambda (i: nat).(\lambda (H:
525 (lt (S n0) i)).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
526 C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c
527 e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
528 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k:
529 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
530 e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
531 T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k:
532 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
533 u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
534 T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
535 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1
536 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
537 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
538 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
539 n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
540 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w))))))
541 (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
542 (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))))))))))) (\lambda (n1:
543 nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v (CSort n1)
544 c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CSort n1) e)).(and3_ind
545 (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop (S n0) O c2 e)
546 (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
547 T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
548 T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k:
549 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S
550 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda
551 (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda
552 (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u))))))
553 (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
554 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k:
555 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
556 (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
557 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w)))))))
558 (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
559 T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
560 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S
561 n0))) v e1 e2)))))))) (\lambda (H2: (eq C e (CSort n1))).(\lambda (H3: (eq
562 nat (S n0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c:
563 C).(or4 (drop (S n0) O c2 c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
564 C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k:
565 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
566 e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
567 T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k:
568 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k
569 u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
570 T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
571 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1
572 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
573 C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
574 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
575 n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
576 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w))))))
577 (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
578 (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2))))))))) (let H5 \def (eq_ind
579 nat (S n0) (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _)
580 \Rightarrow True])) I O H3) in (False_ind (or4 (drop (S n0) O c2 (CSort n1))
581 (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
582 T).(eq C (CSort n1) (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0:
583 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w))))))
584 (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
585 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda
586 (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort n1) (CHead e1 k u))))))
587 (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
588 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2:
589 C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C
590 T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
591 (_: T).(eq C (CSort n1) (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_:
592 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
593 e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
594 T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k:
595 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
596 (minus i (s k (S n0))) v e1 e2)))))))) H5)) e H2)))) (drop_gen_sort n1 (S n0)
597 O e H1)))))))) (\lambda (c: C).(\lambda (H0: ((\forall (c2: C).(\forall (v:
598 T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4
599 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda
600 (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda
601 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w))))))
602 (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
603 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda
604 (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda
605 (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2
606 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2:
607 C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C
608 T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
609 (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda
610 (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k
611 w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
612 T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k:
613 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
614 (minus i (s k (S n0))) v e1 e2))))))))))))))).(\lambda (k: K).(\lambda (t:
615 T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H1: (csubst0 i v (CHead c k t)
616 c2)).(\lambda (e: C).(\lambda (H2: (drop (S n0) O (CHead c k t) e)).(let H3
617 \def (csubst0_gen_head k c c2 t v i H1) in (or3_ind (ex3_2 T nat (\lambda (_:
618 T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
619 nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
620 v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k
621 j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
622 (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_:
623 T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2:
624 T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
625 (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
626 T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S
627 n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
628 T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
629 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0
630 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
631 T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0:
632 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
633 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
634 T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
635 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1
636 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
637 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
638 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
639 T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
640 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
641 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
642 (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))
643 (\lambda (H4: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k
644 j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda
645 (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_:
646 T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
647 nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
648 v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda
649 (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda
650 (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2
651 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
652 T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T
653 (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
654 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
655 C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0:
656 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0
657 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
658 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0
659 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
660 T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0:
661 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
662 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
663 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0
664 (S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5:
665 (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k x0))).(\lambda (_:
666 (subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S
667 n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
668 T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
669 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0
670 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
671 T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0:
672 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
673 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
674 T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
675 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1
676 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
677 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
678 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
679 T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
680 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
681 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
682 (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let
683 H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0:
684 T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4
685 (drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1:
686 C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
687 (k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3
688 (CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
689 T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C
690 T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0
691 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
692 C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0:
693 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0
694 (S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
695 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0
696 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
697 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0:
698 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
699 (minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1:
700 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
701 (S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H5) in (let H9 \def (eq_ind nat i
702 (\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H5) in (eq_ind_r nat (s k x1)
703 (\lambda (n1: nat).(or4 (drop (S n0) O (CHead c k x0) e) (ex3_4 K C T T
704 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
705 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
706 T).(\lambda (w: T).(drop (S n0) O (CHead c k x0) (CHead e0 k0 w))))))
707 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
708 (minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda
709 (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda
710 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
711 (CHead c k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
712 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1
713 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
714 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
715 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
716 T).(drop (S n0) O (CHead c k x0) (CHead e2 k0 w))))))) (\lambda (k0:
717 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
718 (minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
719 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
720 (S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to
721 (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall
722 (e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
723 (\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
724 (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_:
725 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1:
726 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1)
727 (s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1:
728 C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda
729 (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
730 (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2:
731 C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2))))))
732 (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda
733 (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1:
734 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
735 n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_:
736 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0
737 u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
738 T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1
739 e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c
740 k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u:
741 T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda
742 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead
743 e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
744 T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T
745 (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
746 (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
747 C).(\lambda (u: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 k1 u))))))
748 (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
749 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1:
750 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
751 (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
752 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2
753 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
754 T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w))))))
755 (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
756 (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda
757 (b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall
758 (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0:
759 C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
760 (\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
761 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
762 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
763 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
764 x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
765 C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
766 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
767 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
768 C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1
769 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
770 C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
771 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
772 T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
773 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
774 x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
775 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
776 (S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Bind b)
777 x1))).(or4_intro0 (drop (S n0) O (CHead c (Bind b) x0) e) (ex3_4 K C T T
778 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
779 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
780 T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e0 k0 w))))))
781 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
782 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
783 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
784 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
785 T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0:
786 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
787 b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
788 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
789 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
790 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0)
791 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
792 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
793 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
794 (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
795 e2))))))) (drop_drop (Bind b) n0 c e H10 x0)))))) (\lambda (f: F).(\lambda
796 (H10: (drop (r (Flat f) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall
797 (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0)
798 O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0:
799 K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0
800 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w:
801 T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_:
802 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
803 n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
804 (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0:
805 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead
806 e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
807 T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C
808 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
809 T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
810 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3
811 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
812 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
813 n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
814 C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S
815 n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f)
816 x1))).(or4_intro0 (drop (S n0) O (CHead c (Flat f) x0) e) (ex3_4 K C T T
817 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
818 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
819 T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e0 k0 w))))))
820 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
821 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
822 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
823 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
824 T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0:
825 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
826 f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
827 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
828 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
829 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0)
830 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
831 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
832 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
833 (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
834 e2))))))) (drop_drop (Flat f) n0 c e H10 x0)))))) k (drop_gen_drop k c e t n0
835 H2) H8 H9) i H5))) c2 H6)))))) H4)) (\lambda (H4: (ex3_2 C nat (\lambda (_:
836 C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_:
837 nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
838 v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k
839 j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
840 (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O c2 e)
841 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
842 T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
843 (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0:
844 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
845 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
846 (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
847 (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0
848 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
849 T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
850 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
851 C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
852 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w)))))))
853 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
854 T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
855 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0
856 (S n0))) v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H5:
857 (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead x0 k t))).(\lambda (H7:
858 (csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop
859 (S n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
860 T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
861 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0
862 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
863 T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0:
864 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
865 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
866 T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
867 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1
868 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
869 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
870 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
871 T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
872 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
873 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
874 (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let
875 H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0:
876 T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4
877 (drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1:
878 C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
879 (k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3
880 (CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
881 T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C
882 T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0
883 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
884 C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0:
885 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0
886 (S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
887 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0
888 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
889 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0:
890 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
891 (minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1:
892 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
893 (S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H5) in (let H9 \def (eq_ind nat i
894 (\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H5) in (eq_ind_r nat (s k x1)
895 (\lambda (n1: nat).(or4 (drop (S n0) O (CHead x0 k t) e) (ex3_4 K C T T
896 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
897 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
898 T).(\lambda (w: T).(drop (S n0) O (CHead x0 k t) (CHead e0 k0 w))))))
899 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
900 (minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda
901 (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda
902 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
903 (CHead x0 k t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
904 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1
905 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
906 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
907 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
908 T).(drop (S n0) O (CHead x0 k t) (CHead e2 k0 w))))))) (\lambda (k0:
909 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
910 (minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
911 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
912 (S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to
913 (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall
914 (e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
915 (\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
916 (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_:
917 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1:
918 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1)
919 (s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1:
920 C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda
921 (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
922 (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2:
923 C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2))))))
924 (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda
925 (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1:
926 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
927 n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_:
928 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0
929 u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
930 T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1
931 e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead x0
932 k0 t) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u:
933 T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda
934 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead
935 e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
936 T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T
937 (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
938 (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
939 C).(\lambda (u: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 k1 u))))))
940 (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
941 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1:
942 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
943 (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
944 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2
945 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
946 T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w))))))
947 (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
948 (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda
949 (b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall
950 (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0:
951 C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
952 (\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
953 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
954 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
955 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
956 x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
957 C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
958 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
959 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
960 C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1
961 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
962 C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
963 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
964 T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
965 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
966 x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
967 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
968 (S n0))) v0 e1 e2))))))))))))))).(\lambda (H12: (lt (S n0) (s (Bind b)
969 x1))).(let H13 \def (IHn x1 (le_S_n (S n0) x1 H12) c x0 v H7 e H10) in
970 (or4_ind (drop n0 O x0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
971 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
972 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0
973 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
974 T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (ex3_4 K C C T (\lambda (k0:
975 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
976 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
977 T).(drop n0 O x0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
978 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1
979 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
980 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
981 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
982 T).(drop n0 O x0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
983 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0
984 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
985 (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1 e2))))))) (or4
986 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T (\lambda (k0:
987 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
988 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
989 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0:
990 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
991 x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
992 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
993 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
994 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
995 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
996 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
997 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
998 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
999 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0:
1000 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1001 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1002 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
1003 (Bind b) x1) (s k0 (S n0))) v e1 e2)))))))) (\lambda (H14: (drop n0 O x0
1004 e)).(or4_intro0 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T
1005 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1006 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1007 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w))))))
1008 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1009 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1010 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
1011 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1012 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0:
1013 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
1014 b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
1015 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1016 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1017 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
1018 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1019 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
1020 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1021 (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
1022 e2))))))) (drop_drop (Bind b) n0 x0 e H14 t))) (\lambda (H14: (ex3_4 K C T T
1023 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1024 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1025 T).(\lambda (w: T).(drop n0 O x0 (CHead e0 k0 w)))))) (\lambda (k0:
1026 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0
1027 n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k0: K).(\lambda (e0:
1028 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
1029 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0
1030 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1031 T).(subst0 (minus x1 (s k0 n0)) v u w))))) (or4 (drop (S n0) O (CHead x0
1032 (Bind b) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
1033 T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
1034 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
1035 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
1036 T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w))))))
1037 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1038 T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1039 (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0
1040 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1041 T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
1042 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1043 T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
1044 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
1045 x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1046 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1047 x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1048 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
1049 (S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4:
1050 T).(\lambda (x5: T).(\lambda (H15: (eq C e (CHead x3 x2 x4))).(\lambda (H16:
1051 (drop n0 O x0 (CHead x3 x2 x5))).(\lambda (H17: (subst0 (minus x1 (s x2 n0))
1052 v x4 x5)).(eq_ind_r C (CHead x3 x2 x4) (\lambda (c0: C).(or4 (drop (S n0) O
1053 (CHead x0 (Bind b) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
1054 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda
1055 (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
1056 (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
1057 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
1058 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1059 (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0:
1060 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
1061 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1062 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
1063 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1064 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
1065 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1066 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0:
1067 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1068 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1069 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
1070 (Bind b) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O
1071 (CHead x0 (Bind b) t) (CHead x3 x2 x4)) (ex3_4 K C T T (\lambda (k0:
1072 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4)
1073 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1074 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w))))))
1075 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1076 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1077 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2
1078 x4) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1079 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u))))))
1080 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
1081 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
1082 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1083 C (CHead x3 x2 x4) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
1084 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
1085 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1086 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0
1087 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1088 C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S
1089 n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0:
1090 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4) (CHead e0 k0
1091 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
1092 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0:
1093 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1094 x1) (s k0 (S n0))) v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2 x4))
1095 (drop_drop (Bind b) n0 x0 (CHead x3 x2 x5) H16 t) (eq_ind_r nat (S (s x2 n0))
1096 (\lambda (n1: nat).(subst0 (minus (s (Bind b) x1) n1) v x4 x5)) H17 (s x2 (S
1097 n0)) (s_S x2 n0)))) e H15)))))))) H14)) (\lambda (H14: (ex3_4 K C C T
1098 (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
1099 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1100 C).(\lambda (u: T).(drop n0 O x0 (CHead e2 k0 u)))))) (\lambda (k0:
1101 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0
1102 n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1:
1103 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
1104 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2
1105 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1106 T).(csubst0 (minus x1 (s k0 n0)) v e1 e2))))) (or4 (drop (S n0) O (CHead x0
1107 (Bind b) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
1108 T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
1109 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
1110 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
1111 T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w))))))
1112 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1113 T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1114 (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0
1115 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1116 T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
1117 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1118 T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
1119 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
1120 x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1121 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1122 x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1123 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
1124 (S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4:
1125 C).(\lambda (x5: T).(\lambda (H15: (eq C e (CHead x3 x2 x5))).(\lambda (H16:
1126 (drop n0 O x0 (CHead x4 x2 x5))).(\lambda (H17: (csubst0 (minus x1 (s x2 n0))
1127 v x3 x4)).(eq_ind_r C (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O
1128 (CHead x0 (Bind b) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
1129 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda
1130 (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
1131 (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
1132 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
1133 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1134 (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0:
1135 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
1136 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1137 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
1138 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1139 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
1140 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1141 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0:
1142 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1143 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1144 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
1145 (Bind b) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O
1146 (CHead x0 (Bind b) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0:
1147 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
1148 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1149 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w))))))
1150 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1151 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1152 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2
1153 x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1154 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u))))))
1155 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
1156 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
1157 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1158 C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
1159 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
1160 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1161 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0
1162 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1163 C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S
1164 n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1:
1165 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0
1166 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1167 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0:
1168 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
1169 b) x1) (s k0 (S n0))) v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2
1170 x5)) (drop_drop (Bind b) n0 x0 (CHead x4 x2 x5) H16 t) (eq_ind_r nat (S (s x2
1171 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x1) n1) v x3 x4)) H17 (s
1172 x2 (S n0)) (s_S x2 n0)))) e H15)))))))) H14)) (\lambda (H14: (ex4_5 K C C T T
1173 (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
1174 (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
1175 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2
1176 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
1177 T).(\lambda (w: T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (\lambda (k0:
1178 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
1179 (minus x1 (s k0 n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0:
1180 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1181 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1182 C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 k0 w)))))))
1183 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1184 T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1:
1185 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0
1186 n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T
1187 T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1188 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1189 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w))))))
1190 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1191 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1192 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
1193 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1194 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0:
1195 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
1196 b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
1197 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1198 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1199 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
1200 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1201 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
1202 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1203 (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
1204 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
1205 T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x3 x2 x5))).(\lambda (H16:
1206 (drop n0 O x0 (CHead x4 x2 x6))).(\lambda (H17: (subst0 (minus x1 (s x2 n0))
1207 v x5 x6)).(\lambda (H18: (csubst0 (minus x1 (s x2 n0)) v x3 x4)).(eq_ind_r C
1208 (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind b) t)
1209 c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda
1210 (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
1211 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
1212 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
1213 T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w))))))
1214 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1215 T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1216 (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0
1217 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1218 T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
1219 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1220 T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
1221 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
1222 x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1223 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1224 x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1225 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
1226 (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x0 (Bind b) t)
1227 (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
1228 (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) (CHead e0 k0 u)))))) (\lambda
1229 (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
1230 (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
1231 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
1232 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1233 (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 u)))))) (\lambda
1234 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
1235 (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
1236 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S
1237 n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
1238 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
1239 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1240 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
1241 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1242 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
1243 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1244 (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
1245 e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1246 (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) (CHead e1 k0
1247 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
1248 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w)))))))
1249 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1250 T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0:
1251 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
1252 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) x2 x3 x4 x5 x6
1253 (refl_equal C (CHead x3 x2 x5)) (drop_drop (Bind b) n0 x0 (CHead x4 x2 x6)
1254 H16 t) (eq_ind_r nat (S (s x2 n0)) (\lambda (n1: nat).(subst0 (minus (s (Bind
1255 b) x1) n1) v x5 x6)) H17 (s x2 (S n0)) (s_S x2 n0)) (eq_ind_r nat (S (s x2
1256 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x1) n1) v x3 x4)) H18 (s
1257 x2 (S n0)) (s_S x2 n0)))) e H15)))))))))) H14)) H13)))))) (\lambda (f:
1258 F).(\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c3:
1259 C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0:
1260 C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
1261 (\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
1262 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1263 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
1264 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
1265 x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
1266 C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
1267 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
1268 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1269 C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v0 e1
1270 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1271 C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
1272 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1273 T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1274 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
1275 x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1276 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0
1277 (S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f)
1278 x1))).(let H13 \def (H11 x0 v H7 e H10) in (or4_ind (drop (S n0) O x0 e)
1279 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
1280 T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
1281 (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e0 k0 w)))))) (\lambda (k0:
1282 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 (S
1283 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1284 (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
1285 (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 k0
1286 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1287 T).(csubst0 (minus x1 (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
1288 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1289 C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1290 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w)))))))
1291 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1292 T).(subst0 (minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1293 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1
1294 (s k0 (S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e)
1295 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
1296 T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
1297 (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0
1298 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1299 T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T
1300 (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
1301 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1302 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u))))))
1303 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
1304 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
1305 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1306 C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1307 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
1308 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1309 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
1310 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1311 (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
1312 e2)))))))) (\lambda (H14: (drop (S n0) O x0 e)).(or4_intro0 (drop (S n0) O
1313 (CHead x0 (Flat f) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
1314 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
1315 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
1316 (Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1317 (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u
1318 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1319 C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_:
1320 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
1321 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1322 C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
1323 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1324 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
1325 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1326 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0:
1327 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1328 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1329 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
1330 (Flat f) x1) (s k0 (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 x0 e H14
1331 t))) (\lambda (H14: (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
1332 (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
1333 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e0 k0
1334 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1335 T).(subst0 (minus x1 (s k0 (S n0))) v u w))))))).(ex3_4_ind K C T T (\lambda
1336 (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
1337 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
1338 T).(drop (S n0) O x0 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
1339 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 (S n0))) v u
1340 w))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 K C T T (\lambda
1341 (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
1342 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
1343 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0:
1344 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
1345 x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
1346 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
1347 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
1348 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1349 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
1350 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1351 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
1352 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1353 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0:
1354 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1355 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1356 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
1357 (Flat f) x1) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3:
1358 C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H15: (eq C e (CHead x3 x2
1359 x4))).(\lambda (H16: (drop (S n0) O x0 (CHead x3 x2 x5))).(\lambda (H17:
1360 (subst0 (minus x1 (s x2 (S n0))) v x4 x5)).(eq_ind_r C (CHead x3 x2 x4)
1361 (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 K C T T
1362 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
1363 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1364 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w))))))
1365 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1366 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1367 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1
1368 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1369 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0:
1370 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
1371 f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
1372 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
1373 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1374 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
1375 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1376 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
1377 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1378 (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
1379 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 x2
1380 x4)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
1381 T).(\lambda (_: T).(eq C (CHead x3 x2 x4) (CHead e0 k0 u)))))) (\lambda (k0:
1382 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
1383 (Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1384 (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u
1385 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1386 C).(\lambda (u: T).(eq C (CHead x3 x2 x4) (CHead e1 k0 u)))))) (\lambda (k0:
1387 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
1388 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1389 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
1390 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1391 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4) (CHead e1 k0
1392 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
1393 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w)))))))
1394 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1395 T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0:
1396 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
1397 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2))))))) (ex3_4_intro K C T T
1398 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
1399 (CHead x3 x2 x4) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
1400 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
1401 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
1402 T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))
1403 x2 x3 x4 x5 (refl_equal C (CHead x3 x2 x4)) (drop_drop (Flat f) n0 x0 (CHead
1404 x3 x2 x5) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 K C C T
1405 (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
1406 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1407 C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 k0 u)))))) (\lambda (k0:
1408 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0
1409 (S n0))) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1:
1410 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
1411 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x0 (CHead
1412 e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1413 T).(csubst0 (minus x1 (s k0 (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead
1414 x0 (Flat f) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
1415 (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
1416 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
1417 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
1418 T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w))))))
1419 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1420 T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1421 (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0
1422 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1423 T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
1424 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1425 T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
1426 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
1427 x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1428 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
1429 x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1430 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0
1431 (S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4:
1432 C).(\lambda (x5: T).(\lambda (H15: (eq C e (CHead x3 x2 x5))).(\lambda (H16:
1433 (drop (S n0) O x0 (CHead x4 x2 x5))).(\lambda (H17: (csubst0 (minus x1 (s x2
1434 (S n0))) v x3 x4)).(eq_ind_r C (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop
1435 (S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda
1436 (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u))))))
1437 (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
1438 n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda
1439 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0
1440 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
1441 C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda
1442 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
1443 (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
1444 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S
1445 n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
1446 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0
1447 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
1448 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w)))))))
1449 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1450 T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0:
1451 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
1452 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S
1453 n0) O (CHead x0 (Flat f) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0:
1454 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
1455 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1456 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w))))))
1457 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1458 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1459 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2
1460 x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1461 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u))))))
1462 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
1463 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
1464 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1465 C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
1466 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
1467 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1468 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0
1469 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1470 C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S
1471 n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1:
1472 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0
1473 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1474 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0:
1475 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
1476 f) x1) (s k0 (S n0))) v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2
1477 x5)) (drop_drop (Flat f) n0 x0 (CHead x4 x2 x5) H16 t) H17)) e H15))))))))
1478 H14)) (\lambda (H14: (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
1479 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0
1480 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
1481 T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w))))))) (\lambda (k0:
1482 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1483 (minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
1484 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0
1485 (S n0))) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: K).(\lambda (e1:
1486 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0
1487 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
1488 T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w))))))) (\lambda (k0:
1489 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1490 (minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
1491 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0
1492 (S n0))) v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 K
1493 C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
1494 e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1495 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w))))))
1496 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1497 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1498 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
1499 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1500 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0:
1501 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
1502 f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
1503 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1504 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1505 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
1506 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1507 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
1508 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1509 (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
1510 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
1511 T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x3 x2 x5))).(\lambda (H16:
1512 (drop (S n0) O x0 (CHead x4 x2 x6))).(\lambda (H17: (subst0 (minus x1 (s x2
1513 (S n0))) v x5 x6)).(\lambda (H18: (csubst0 (minus x1 (s x2 (S n0))) v x3
1514 x4)).(eq_ind_r C (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead
1515 x0 (Flat f) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
1516 (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0:
1517 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
1518 (Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1519 (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u
1520 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1521 C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
1522 (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
1523 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1524 C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
1525 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1526 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
1527 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1528 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0:
1529 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1530 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1531 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
1532 (Flat f) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O
1533 (CHead x0 (Flat f) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0:
1534 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
1535 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1536 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w))))))
1537 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1538 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1539 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2
1540 x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1541 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u))))))
1542 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
1543 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
1544 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1545 C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
1546 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
1547 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1548 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0
1549 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1550 C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S
1551 n0))) v e1 e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1:
1552 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
1553 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1554 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
1555 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1556 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
1557 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1558 (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
1559 e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x3 x2 x5)) (drop_drop (Flat f)
1560 n0 x0 (CHead x4 x2 x6) H16 t) H17 H18)) e H15)))))))))) H14)) H13)))))) k
1561 (drop_gen_drop k c e t n0 H2) H8 H9) i H5))) c2 H6)))))) H4)) (\lambda (H4:
1562 (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s
1563 k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
1564 c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
1565 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
1566 c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
1567 nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
1568 nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
1569 (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
1570 nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda
1571 (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
1572 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
1573 T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
1574 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u
1575 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1576 C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_:
1577 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u))))))
1578 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
1579 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
1580 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1581 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1582 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w)))))))
1583 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1584 T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
1585 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0
1586 (S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2:
1587 nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: (eq C c2 (CHead x1 k
1588 x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: (csubst0 x2 v c
1589 x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop (S n0) O c0 e)
1590 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
1591 T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
1592 (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 w)))))) (\lambda (k0:
1593 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
1594 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1595 (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
1596 (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 k0
1597 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1598 T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
1599 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1600 C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1601 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k0 w)))))))
1602 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1603 T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
1604 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0
1605 (S n0))) v e1 e2))))))))) (let H9 \def (eq_ind nat i (\lambda (n1:
1606 nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall
1607 (e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
1608 (\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
1609 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1610 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
1611 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S
1612 n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1613 (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0:
1614 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead
1615 e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1616 T).(csubst0 (minus n1 (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda
1617 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1618 C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1619 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w)))))))
1620 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1621 T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda
1622 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1
1623 (s k0 (S n0))) v0 e1 e2)))))))))))))) H0 (s k x2) H5) in (let H10 \def
1624 (eq_ind nat i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x2) H5) in (eq_ind_r
1625 nat (s k x2) (\lambda (n1: nat).(or4 (drop (S n0) O (CHead x1 k x0) e) (ex3_4
1626 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
1627 C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1628 T).(\lambda (w: T).(drop (S n0) O (CHead x1 k x0) (CHead e0 k0 w))))))
1629 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1630 (minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda
1631 (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda
1632 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
1633 (CHead x1 k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
1634 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1
1635 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1636 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
1637 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1638 T).(drop (S n0) O (CHead x1 k x0) (CHead e2 k0 w))))))) (\lambda (k0:
1639 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1640 (minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
1641 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
1642 (S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to
1643 (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall
1644 (e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
1645 (\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
1646 (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_:
1647 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1:
1648 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2)
1649 (s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1:
1650 C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda
1651 (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
1652 (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2:
1653 C).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1 e2))))))
1654 (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda
1655 (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1:
1656 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
1657 n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_:
1658 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v0
1659 u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1660 T).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1
1661 e2)))))))))))))) \to ((lt (S n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1
1662 k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u:
1663 T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda
1664 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0)
1665 (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u:
1666 T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w)))))) (ex3_4
1667 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
1668 C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
1669 C).(\lambda (u: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 k1 u))))))
1670 (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
1671 (minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1:
1672 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1673 (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
1674 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2
1675 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
1676 T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w))))))
1677 (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
1678 (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda
1679 (b: B).(\lambda (H11: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall
1680 (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0:
1681 C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
1682 (\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
1683 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1684 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
1685 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1686 x2) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
1687 C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
1688 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
1689 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1690 C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v0 e1
1691 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1692 C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
1693 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1694 T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1695 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1696 x2) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1697 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0
1698 (S n0))) v0 e1 e2))))))))))))))).(\lambda (H13: (lt (S n0) (s (Bind b)
1699 x2))).(let H14 \def (IHn x2 (le_S_n (S n0) x2 H13) c x1 v H8 e H11) in
1700 (or4_ind (drop n0 O x1 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
1701 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
1702 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0
1703 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1704 T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (ex3_4 K C C T (\lambda (k0:
1705 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
1706 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1707 T).(drop n0 O x1 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
1708 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0 n0)) v e1
1709 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1710 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
1711 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1712 T).(drop n0 O x1 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1713 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0
1714 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1715 (_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 n0)) v e1 e2))))))) (or4
1716 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T T (\lambda (k0:
1717 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
1718 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
1719 T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0:
1720 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1721 x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
1722 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
1723 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
1724 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1725 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
1726 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1727 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
1728 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1729 T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0:
1730 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1731 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1732 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
1733 (Bind b) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (H15: (drop n0 O x1
1734 e)).(or4_intro0 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T T
1735 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1736 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1737 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w))))))
1738 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1739 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1740 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
1741 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1742 T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0:
1743 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
1744 b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
1745 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1746 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1747 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
1748 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1749 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
1750 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1751 (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
1752 e2))))))) (drop_drop (Bind b) n0 x1 e H15 x0))) (\lambda (H15: (ex3_4 K C T T
1753 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1754 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1755 T).(\lambda (w: T).(drop n0 O x1 (CHead e0 k0 w)))))) (\lambda (k0:
1756 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0
1757 n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k0: K).(\lambda (e0:
1758 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
1759 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0
1760 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1761 T).(subst0 (minus x2 (s k0 n0)) v u w))))) (or4 (drop (S n0) O (CHead x1
1762 (Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
1763 T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
1764 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
1765 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
1766 T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w))))))
1767 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1768 T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1769 (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0
1770 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1771 T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
1772 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1773 T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
1774 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
1775 x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1776 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1777 x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1778 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0
1779 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5:
1780 T).(\lambda (x6: T).(\lambda (H16: (eq C e (CHead x4 x3 x5))).(\lambda (H17:
1781 (drop n0 O x1 (CHead x4 x3 x6))).(\lambda (H18: (subst0 (minus x2 (s x3 n0))
1782 v x5 x6)).(eq_ind_r C (CHead x4 x3 x5) (\lambda (c0: C).(or4 (drop (S n0) O
1783 (CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
1784 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda
1785 (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
1786 (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
1787 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
1788 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1789 (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0:
1790 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
1791 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1792 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
1793 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1794 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
1795 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1796 T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0:
1797 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1798 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1799 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
1800 (Bind b) x2) (s k0 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O
1801 (CHead x1 (Bind b) x0) (CHead x4 x3 x5)) (ex3_4 K C T T (\lambda (k0:
1802 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5)
1803 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1804 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w))))))
1805 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1806 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1807 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3
1808 x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1809 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u))))))
1810 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
1811 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
1812 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1813 C (CHead x4 x3 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
1814 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
1815 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1816 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0
1817 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1818 C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S
1819 n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0:
1820 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0
1821 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
1822 T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0:
1823 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1824 x2) (s k0 (S n0))) v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x5))
1825 (drop_drop (Bind b) n0 x1 (CHead x4 x3 x6) H17 x0) (eq_ind_r nat (S (s x3
1826 n0)) (\lambda (n1: nat).(subst0 (minus (s (Bind b) x2) n1) v x5 x6)) H18 (s
1827 x3 (S n0)) (s_S x3 n0)))) e H16)))))))) H15)) (\lambda (H15: (ex3_4 K C C T
1828 (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
1829 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1830 C).(\lambda (u: T).(drop n0 O x1 (CHead e2 k0 u)))))) (\lambda (k0:
1831 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0
1832 n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1:
1833 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
1834 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2
1835 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1836 T).(csubst0 (minus x2 (s k0 n0)) v e1 e2))))) (or4 (drop (S n0) O (CHead x1
1837 (Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
1838 T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
1839 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
1840 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
1841 T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w))))))
1842 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1843 T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1844 (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0
1845 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1846 T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
1847 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1848 T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
1849 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
1850 x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1851 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1852 x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1853 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0
1854 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5:
1855 C).(\lambda (x6: T).(\lambda (H16: (eq C e (CHead x4 x3 x6))).(\lambda (H17:
1856 (drop n0 O x1 (CHead x5 x3 x6))).(\lambda (H18: (csubst0 (minus x2 (s x3 n0))
1857 v x4 x5)).(eq_ind_r C (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O
1858 (CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
1859 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda
1860 (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
1861 (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
1862 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
1863 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1864 (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0:
1865 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
1866 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1867 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
1868 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1869 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
1870 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1871 T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0:
1872 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1873 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1874 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
1875 (Bind b) x2) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O
1876 (CHead x1 (Bind b) x0) (CHead x4 x3 x6)) (ex3_4 K C T T (\lambda (k0:
1877 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6)
1878 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1879 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w))))))
1880 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1881 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1882 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3
1883 x6) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1884 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u))))))
1885 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
1886 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
1887 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1888 C (CHead x4 x3 x6) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
1889 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
1890 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1891 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0
1892 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1893 C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S
1894 n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1:
1895 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
1896 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1897 T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0:
1898 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
1899 b) x2) (s k0 (S n0))) v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3
1900 x6)) (drop_drop (Bind b) n0 x1 (CHead x5 x3 x6) H17 x0) (eq_ind_r nat (S (s
1901 x3 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x2) n1) v x4 x5)) H18
1902 (s x3 (S n0)) (s_S x3 n0)))) e H16)))))))) H15)) (\lambda (H15: (ex4_5 K C C
1903 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1904 T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
1905 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead
1906 e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
1907 T).(\lambda (w: T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (\lambda (k0:
1908 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
1909 (minus x2 (s k0 n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0:
1910 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1911 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1912 C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e2 k0 w)))))))
1913 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1914 T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1:
1915 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0
1916 n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T
1917 T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1918 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1919 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w))))))
1920 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1921 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1922 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
1923 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1924 T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0:
1925 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
1926 b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
1927 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1928 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1929 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
1930 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1931 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
1932 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1933 (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
1934 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6:
1935 T).(\lambda (x7: T).(\lambda (H16: (eq C e (CHead x4 x3 x6))).(\lambda (H17:
1936 (drop n0 O x1 (CHead x5 x3 x7))).(\lambda (H18: (subst0 (minus x2 (s x3 n0))
1937 v x6 x7)).(\lambda (H19: (csubst0 (minus x2 (s x3 n0)) v x4 x5)).(eq_ind_r C
1938 (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0)
1939 c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda
1940 (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
1941 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
1942 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
1943 T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w))))))
1944 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1945 T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1946 (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0
1947 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1948 T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
1949 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1950 T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
1951 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
1952 x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1953 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1954 x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1955 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0
1956 (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Bind b) x0)
1957 (CHead x4 x3 x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
1958 (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda
1959 (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
1960 (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
1961 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
1962 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1963 (_: C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda
1964 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
1965 (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
1966 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S
1967 n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
1968 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6)
1969 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1970 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
1971 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1972 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
1973 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1974 (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
1975 e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1976 (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
1977 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
1978 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0
1979 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
1980 T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w))))))
1981 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
1982 (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) x3 x4 x5
1983 x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Bind b) n0 x1 (CHead x5 x3
1984 x7) H17 x0) (eq_ind_r nat (S (s x3 n0)) (\lambda (n1: nat).(subst0 (minus (s
1985 (Bind b) x2) n1) v x6 x7)) H18 (s x3 (S n0)) (s_S x3 n0)) (eq_ind_r nat (S (s
1986 x3 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x2) n1) v x4 x5)) H19
1987 (s x3 (S n0)) (s_S x3 n0)))) e H16)))))))))) H15)) H14)))))) (\lambda (f:
1988 F).(\lambda (H11: (drop (r (Flat f) n0) O c e)).(\lambda (H12: ((\forall (c3:
1989 C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e0:
1990 C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
1991 (\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
1992 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1993 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
1994 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
1995 x2) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
1996 C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
1997 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
1998 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1999 C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v0 e1
2000 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2001 C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
2002 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2003 T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
2004 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
2005 x2) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
2006 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0
2007 (S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f)
2008 x2))).(let H14 \def (H12 x1 v H8 e H11) in (or4_ind (drop (S n0) O x1 e)
2009 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
2010 T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
2011 (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e0 k0 w)))))) (\lambda (k0:
2012 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S
2013 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
2014 (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
2015 (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 k0
2016 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2017 T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
2018 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
2019 C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
2020 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e2 k0 w)))))))
2021 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2022 T).(subst0 (minus x2 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
2023 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x2
2024 (s k0 (S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e)
2025 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
2026 T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
2027 (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0
2028 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2029 T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T
2030 (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
2031 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
2032 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u))))))
2033 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
2034 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
2035 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
2036 C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
2037 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
2038 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
2039 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S
2040 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
2041 (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2042 e2)))))))) (\lambda (H15: (drop (S n0) O x1 e)).(or4_intro0 (drop (S n0) O
2043 (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
2044 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
2045 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
2046 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
2047 (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u
2048 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2049 C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_:
2050 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
2051 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
2052 C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2053 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2054 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
2055 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2056 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0:
2057 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
2058 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
2059 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
2060 (Flat f) x2) (s k0 (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 x1 e H15
2061 x0))) (\lambda (H15: (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
2062 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
2063 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead
2064 e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2065 T).(subst0 (minus x2 (s k0 (S n0))) v u w))))))).(ex3_4_ind K C T T (\lambda
2066 (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
2067 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
2068 T).(drop (S n0) O x1 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
2069 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S n0))) v u
2070 w))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda
2071 (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
2072 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
2073 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0:
2074 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
2075 x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
2076 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
2077 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
2078 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
2079 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2080 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2081 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
2082 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2083 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0:
2084 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
2085 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
2086 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
2087 (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4:
2088 C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H16: (eq C e (CHead x4 x3
2089 x5))).(\lambda (H17: (drop (S n0) O x1 (CHead x4 x3 x6))).(\lambda (H18:
2090 (subst0 (minus x2 (s x3 (S n0))) v x5 x6)).(eq_ind_r C (CHead x4 x3 x5)
2091 (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K C T
2092 T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
2093 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
2094 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w))))))
2095 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
2096 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
2097 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1
2098 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
2099 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0:
2100 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
2101 f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
2102 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
2103 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
2104 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
2105 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
2106 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S
2107 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
2108 (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2109 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3
2110 x5)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
2111 T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0 u)))))) (\lambda (k0:
2112 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
2113 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
2114 (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u
2115 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2116 C).(\lambda (u: T).(eq C (CHead x4 x3 x5) (CHead e1 k0 u)))))) (\lambda (k0:
2117 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
2118 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
2119 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2120 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2121 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e1 k0
2122 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
2123 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
2124 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2125 T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
2126 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2127 (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))
2128 (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
2129 T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0 u)))))) (\lambda (k0:
2130 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
2131 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
2132 (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u
2133 w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x5)) (drop_drop (Flat f) n0 x1
2134 (CHead x4 x3 x6) H17 x0) H18)) e H16)))))))) H15)) (\lambda (H15: (ex3_4 K C
2135 C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
2136 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
2137 C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 k0 u)))))) (\lambda (k0:
2138 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0
2139 (S n0))) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1:
2140 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
2141 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead
2142 e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2143 T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead
2144 x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
2145 (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
2146 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
2147 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
2148 T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
2149 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2150 T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
2151 (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
2152 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2153 T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
2154 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2155 T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
2156 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
2157 x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
2158 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
2159 x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
2160 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0
2161 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5:
2162 C).(\lambda (x6: T).(\lambda (H16: (eq C e (CHead x4 x3 x6))).(\lambda (H17:
2163 (drop (S n0) O x1 (CHead x5 x3 x6))).(\lambda (H18: (csubst0 (minus x2 (s x3
2164 (S n0))) v x4 x5)).(eq_ind_r C (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop
2165 (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda
2166 (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u))))))
2167 (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
2168 n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda
2169 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0
2170 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
2171 C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda
2172 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
2173 (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
2174 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S
2175 n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
2176 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0
2177 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
2178 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
2179 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2180 T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
2181 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2182 (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))))
2183 (or4_intro2 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3 x6)) (ex3_4 K
2184 C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
2185 (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
2186 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
2187 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
2188 T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
2189 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2190 T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_:
2191 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
2192 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
2193 C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2194 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2195 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
2196 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
2197 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
2198 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2199 T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
2200 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2201 (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))
2202 (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2203 C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0:
2204 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
2205 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
2206 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2207 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1
2208 (CHead x5 x3 x6) H17 x0) H18)) e H16)))))))) H15)) (\lambda (H15: (ex4_5 K C
2209 C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2210 T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
2211 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1
2212 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
2213 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S n0))) v u
2214 w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2215 T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1
2216 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda
2217 (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
2218 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2219 T).(drop (S n0) O x1 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
2220 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S
2221 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
2222 (_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2)))))) (or4
2223 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0:
2224 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
2225 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
2226 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0:
2227 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
2228 x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
2229 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
2230 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
2231 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
2232 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2233 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2234 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
2235 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2236 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0:
2237 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
2238 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
2239 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
2240 (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4:
2241 C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H16: (eq C e
2242 (CHead x4 x3 x6))).(\lambda (H17: (drop (S n0) O x1 (CHead x5 x3
2243 x7))).(\lambda (H18: (subst0 (minus x2 (s x3 (S n0))) v x6 x7)).(\lambda
2244 (H19: (csubst0 (minus x2 (s x3 (S n0))) v x4 x5)).(eq_ind_r C (CHead x4 x3
2245 x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K
2246 C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
2247 c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
2248 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w))))))
2249 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
2250 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
2251 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1
2252 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
2253 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0:
2254 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
2255 f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
2256 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
2257 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
2258 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
2259 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
2260 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S
2261 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
2262 (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2263 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3
2264 x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
2265 T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0:
2266 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
2267 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
2268 (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u
2269 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2270 C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0:
2271 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
2272 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
2273 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2274 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2275 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
2276 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
2277 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
2278 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2279 T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
2280 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2281 (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))
2282 (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2283 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
2284 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
2285 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
2286 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2287 T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
2288 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2289 (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) x3 x4 x5
2290 x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1 (CHead x5 x3
2291 x7) H17 x0) H18 H19)) e H16)))))))))) H15)) H14)))))) k (drop_gen_drop k c e
2292 t n0 H2) H9 H10) i H5))) c2 H6)))))))) H4)) H3))))))))))) c1)))))) n).
2294 lemma csubst0_drop_eq:
2295 \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0
2296 n v c1 c2) \to (\forall (e: C).((drop n O c1 e) \to (or4 (drop n O c2 e)
2297 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
2298 T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
2299 C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w))))))
2300 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
2301 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2302 C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
2303 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2
2304 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
2305 (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
2306 (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1
2307 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
2308 (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
2309 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
2310 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2311 T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))
2313 \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2:
2314 C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c1
2315 e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
2316 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
2317 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O
2318 c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
2319 T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
2320 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
2321 f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
2322 T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
2323 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
2324 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
2325 T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
2326 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e2
2327 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
2328 (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
2329 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
2330 e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
2331 (H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c1
2332 e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 F C T T (\lambda
2333 (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0
2334 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
2335 (w: T).(drop O O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
2336 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
2337 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c
2338 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2339 C).(\lambda (u: T).(drop O O c2 (CHead e2 (Flat f) u)))))) (\lambda (_:
2340 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
2341 (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
2342 (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f:
2343 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
2344 c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
2345 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
2346 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
2347 O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2))
2348 (\lambda (n0: nat).(or4 (drop n0 n0 c2 c1) (ex3_4 F C T T (\lambda (f:
2349 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e0 (Flat
2350 f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
2351 T).(drop n0 n0 c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
2352 C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w)))))) (ex3_4 F C C T
2353 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c1
2354 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2355 C).(\lambda (u: T).(drop n0 n0 c2 (CHead e2 (Flat f) u)))))) (\lambda (_:
2356 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1
2357 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2358 C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e1 (Flat f) u)))))))
2359 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2360 T).(drop n0 n0 c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
2361 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w))))))
2362 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2363 (_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda (H1: (csubst0
2364 y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: T).(\lambda (c:
2365 C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c0 c) (ex3_4 F C T T
2366 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c
2367 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
2368 T).(\lambda (w: T).(drop n0 n0 c0 (CHead e0 (Flat f) w)))))) (\lambda (_:
2369 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w))))))
2370 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2371 T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
2372 C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 c0 (CHead e2 (Flat f) u))))))
2373 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0
2374 t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2375 C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u)))))))
2376 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2377 T).(drop n0 n0 c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
2378 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w))))))
2379 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2380 (_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda (k: K).(K_ind (\lambda (k0:
2381 K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2:
2382 T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s k0 i) O) \to (or4
2383 (drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead c k0 u1)) (ex3_4 F C T T
2384 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead
2385 c k0 u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
2386 (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e0
2387 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2388 T).(subst0 (s k0 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
2389 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k0 u1) (CHead e1 (Flat f)
2390 u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
2391 T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e2 (Flat f) u))))))
2392 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s
2393 k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
2394 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k0 u1)
2395 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2396 C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2)
2397 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
2398 C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_:
2399 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
2400 (s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda
2401 (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1
2402 u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def (eq_ind nat
2403 (S i) (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _)
2404 \Rightarrow True])) I O H3) in (False_ind (or4 (drop (S i) (S i) (CHead c
2405 (Bind b) u2) (CHead c (Bind b) u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda
2406 (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Bind b) u1) (CHead e0
2407 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
2408 (w: T).(drop (S i) (S i) (CHead c (Bind b) u2) (CHead e0 (Flat f) w))))))
2409 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (S i)
2410 v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2411 C).(\lambda (u: T).(eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u))))))
2412 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S i)
2413 (S i) (CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_:
2414 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1
2415 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2416 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Bind b) u1) (CHead e1
2417 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
2418 (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c (Bind b) u2) (CHead e2
2419 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
2420 (u: T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (\lambda (_: F).(\lambda
2421 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1
2422 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0:
2423 T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1
2424 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind nat i
2425 (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H3) in (eq_ind_r nat O (\lambda
2426 (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f) u2) (CHead c (Flat f) u1))
2427 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
2428 T).(eq C (CHead c (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0:
2429 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c
2430 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_:
2431 C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) (ex3_4 F C C T
2432 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C
2433 (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
2434 (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 (CHead c (Flat f) u2)
2435 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
2436 C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
2437 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
2438 (CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda
2439 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c
2440 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
2441 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w))))))
2442 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2443 (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro1 (drop O O (CHead c (Flat f)
2444 u2) (CHead c (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
2445 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0
2446 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
2447 (w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda
2448 (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
2449 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2450 T).(eq C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
2451 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c (Flat
2452 f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
2453 (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda
2454 (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
2455 C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0:
2456 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
2457 (CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda
2458 (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
2459 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2460 (_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0:
2461 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f)
2462 u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
2463 (_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0)
2464 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2465 T).(subst0 O v0 u w))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u1))
2466 (drop_refl (CHead c (Flat f) u2)) H4)) i H3)))))))))) k)) (\lambda (k:
2467 K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4:
2468 C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop
2469 i i c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
2470 T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f:
2471 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e0
2472 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2473 T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
2474 C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u))))))
2475 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4
2476 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
2477 C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
2478 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
2479 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2480 C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 (Flat f) w)))))))
2481 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2482 T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
2483 C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to (\forall
2484 (u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c4 k0 u)
2485 (CHead c3 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
2486 (u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e0 (Flat f) u0))))))
2487 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0
2488 i) (s k0 i) (CHead c4 k0 u) (CHead e0 (Flat f) w)))))) (\lambda (_:
2489 F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (s k0 i) v0 u0
2490 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2491 C).(\lambda (u0: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) u0))))))
2492 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0
2493 i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) (\lambda (_:
2494 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1
2495 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2496 C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f)
2497 u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
2498 T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f)
2499 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0:
2500 T).(\lambda (w: T).(subst0 (s k0 i) v0 u0 w)))))) (\lambda (_: F).(\lambda
2501 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0
2502 e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3:
2503 C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3
2504 c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T
2505 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
2506 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
2507 T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_:
2508 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
2509 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2510 T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
2511 C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u))))))
2512 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
2513 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2514 C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
2515 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2516 T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
2517 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
2518 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2519 (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat
2520 (S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee with [O
2521 \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (or4
2522 (drop (S i) (S i) (CHead c4 (Bind b) u) (CHead c3 (Bind b) u)) (ex3_4 F C T T
2523 (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C
2524 (CHead c3 (Bind b) u) (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda
2525 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c4 (Bind b)
2526 u) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0:
2527 T).(\lambda (w: T).(subst0 (S i) v0 u0 w)))))) (ex3_4 F C C T (\lambda (f:
2528 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Bind b)
2529 u) (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2530 C).(\lambda (u0: T).(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e2 (Flat
2531 f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2532 T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
2533 (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3
2534 (Bind b) u) (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_:
2535 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead
2536 c4 (Bind b) u) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
2537 C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (S i) v0 u0
2538 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2539 T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))))) H5))))))))))) (\lambda (f:
2540 F).(\lambda (i: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0:
2541 T).(\lambda (H2: (csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4
2542 (drop i i c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda
2543 (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u)))))) (\lambda (f0:
2544 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e0
2545 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
2546 (w: T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
2547 C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f0) u))))))
2548 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i i
2549 c4 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
2550 C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
2551 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
2552 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
2553 C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 (Flat f0) w)))))))
2554 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2555 T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
2556 C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda
2557 (u: T).(\lambda (H4: (eq nat i O)).(let H5 \def (eq_ind nat i (\lambda (n0:
2558 nat).((eq nat n0 O) \to (or4 (drop n0 n0 c4 c3) (ex3_4 F C T T (\lambda (f0:
2559 F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat
2560 f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
2561 T).(drop n0 n0 c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_:
2562 C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C T
2563 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c3
2564 (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
2565 C).(\lambda (u0: T).(drop n0 n0 c4 (CHead e2 (Flat f0) u0)))))) (\lambda (_:
2566 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1
2567 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
2568 C).(\lambda (u0: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u0)))))))
2569 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2570 (w: T).(drop n0 n0 c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda
2571 (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0
2572 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2573 T).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))))))) H3 O H4) in (let H6 \def
2574 (eq_ind nat i (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r
2575 nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c4 (Flat f) u) (CHead c3
2576 (Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0:
2577 T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0))))))
2578 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0
2579 (CHead c4 (Flat f) u) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
2580 (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C
2581 T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C
2582 (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda
2583 (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 n0 (CHead c4 (Flat f) u)
2584 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
2585 C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
2586 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C
2587 (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda
2588 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c4
2589 (Flat f) u) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
2590 C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0 w))))))
2591 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2592 (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c4 (Flat f)
2593 u) (CHead c3 (Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
2594 C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e0
2595 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
2596 (w: T).(drop O O (CHead c4 (Flat f) u) (CHead e0 (Flat f0) w)))))) (\lambda
2597 (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0
2598 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
2599 C).(\lambda (u0: T).(eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0))))))
2600 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O
2601 (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda
2602 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C
2603 C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0:
2604 T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))))
2605 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2606 (w: T).(drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0) w))))))) (\lambda
2607 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
2608 T).(subst0 O v0 u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
2609 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F
2610 C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq
2611 C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0:
2612 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4
2613 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1:
2614 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2))))) f c3 c4 u
2615 (refl_equal C (CHead c3 (Flat f) u)) (drop_refl (CHead c4 (Flat f) u)) H6)) i
2616 H4)))))))))))) k)) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i:
2617 nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: T).((subst0 i v0 u1 u2)
2618 \to (\forall (c3: C).(\forall (c4: C).((csubst0 i v0 c3 c4) \to ((((eq nat i
2619 O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
2620 C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u))))))
2621 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4
2622 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
2623 T).(\lambda (w: T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f:
2624 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat
2625 f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
2626 T).(drop i i c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
2627 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T
2628 T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
2629 (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
2630 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2
2631 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
2632 (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1:
2633 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1
2634 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead
2635 c4 k0 u2) (CHead c3 k0 u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
2636 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 u1) (CHead e0 (Flat f)
2637 u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
2638 T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e0 (Flat f) w))))))
2639 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0
2640 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2641 C).(\lambda (u: T).(eq C (CHead c3 k0 u1) (CHead e1 (Flat f) u)))))) (\lambda
2642 (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0 i) (s k0
2643 i) (CHead c4 k0 u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
2644 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F
2645 C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2646 T).(\lambda (_: T).(eq C (CHead c3 k0 u1) (CHead e1 (Flat f) u)))))))
2647 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2648 T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e2 (Flat f) w)))))))
2649 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2650 T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
2651 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0 e1
2652 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (v0:
2653 T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1
2654 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3
2655 c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T
2656 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
2657 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
2658 T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_:
2659 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
2660 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2661 T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
2662 C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u))))))
2663 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
2664 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2665 C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
2666 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2667 T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
2668 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
2669 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2670 (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6
2671 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee with [O \Rightarrow False
2672 | (S _) \Rightarrow True])) I O H5) in (False_ind (or4 (drop (S i) (S i)
2673 (CHead c4 (Bind b) u2) (CHead c3 (Bind b) u1)) (ex3_4 F C T T (\lambda (f:
2674 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Bind b)
2675 u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
2676 T).(\lambda (w: T).(drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e0 (Flat
2677 f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2678 T).(subst0 (S i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
2679 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 (Bind b) u1) (CHead e1
2680 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
2681 (u: T).(drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e2 (Flat f) u))))))
2682 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S
2683 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda
2684 (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e1
2685 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
2686 (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e2
2687 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
2688 (u: T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (\lambda (_: F).(\lambda
2689 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1
2690 e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0:
2691 T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1
2692 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3
2693 c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T
2694 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
2695 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
2696 T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f0) w)))))) (\lambda (_:
2697 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
2698 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2699 T).(eq C c3 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
2700 C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f0) u))))))
2701 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
2702 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda
2703 (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u)))))))
2704 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2705 (w: T).(drop i i c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
2706 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
2707 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2708 (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6
2709 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c4
2710 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda
2711 (_: T).(eq C c3 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0:
2712 C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e0 (Flat f0) w))))))
2713 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0
2714 u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
2715 C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f0) u)))))) (\lambda (f0:
2716 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 c4 (CHead e2
2717 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
2718 (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda
2719 (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1
2720 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
2721 (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e2 (Flat f0) w))))))) (\lambda
2722 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2723 T).(subst0 n0 v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
2724 C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))))))) H4 O H5) in
2725 (let H7 \def (eq_ind nat i (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 O H5)
2726 in (let H8 \def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O
2727 H5) in (eq_ind_r nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c4 (Flat f)
2728 u2) (CHead c3 (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
2729 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e0
2730 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
2731 (w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) w))))))
2732 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0
2733 u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
2734 C).(\lambda (u: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u))))))
2735 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0
2736 (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda
2737 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F
2738 C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2739 T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))))
2740 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2741 (w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) w)))))))
2742 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2743 T).(subst0 n0 v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
2744 C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3
2745 (drop O O (CHead c4 (Flat f) u2) (CHead c3 (Flat f) u1)) (ex3_4 F C T T
2746 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
2747 (CHead c3 (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
2748 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2)
2749 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
2750 T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f0:
2751 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 (Flat f)
2752 u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
2753 (e2: C).(\lambda (u: T).(drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0)
2754 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2755 T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
2756 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f)
2757 u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
2758 (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2)
2759 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
2760 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_:
2761 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
2762 O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1:
2763 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f)
2764 u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
2765 (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2)
2766 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
2767 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_:
2768 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
2769 O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c3 (Flat f) u1))
2770 (drop_refl (CHead c4 (Flat f) u2)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2
2771 H1))) H) e (drop_gen_refl c1 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn:
2772 ((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to
2773 (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T
2774 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
2775 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
2776 T).(\lambda (w: T).(drop n0 O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
2777 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
2778 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2779 T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
2780 C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c2 (CHead e2 (Flat f) u))))))
2781 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
2782 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2783 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u)))))))
2784 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2785 T).(drop n0 O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
2786 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
2787 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2788 (_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind (\lambda
2789 (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall
2790 (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 F C T T
2791 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
2792 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
2793 T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
2794 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
2795 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2796 T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
2797 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f)
2798 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2799 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
2800 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
2801 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
2802 T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
2803 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
2804 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2805 T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1:
2806 nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 (S n0) v (CSort
2807 n1) c2)).(\lambda (e: C).(\lambda (H0: (drop (S n0) O (CSort n1)
2808 e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop
2809 (S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
2810 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
2811 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
2812 e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
2813 (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
2814 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
2815 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
2816 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
2817 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
2818 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
2819 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2820 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f)
2821 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2822 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
2823 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
2824 e2)))))))) (\lambda (H1: (eq C e (CSort n1))).(\lambda (H2: (eq nat (S n0)
2825 O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(or4
2826 (drop (S n0) O c2 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
2827 (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f:
2828 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
2829 e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
2830 (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
2831 C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u))))))
2832 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
2833 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
2834 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
2835 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
2836 c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2837 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f)
2838 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2839 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
2840 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
2841 e2))))))))) (let H4 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee with
2842 [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind (or4
2843 (drop (S n0) O c2 (CSort n1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
2844 C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e0 (Flat f)
2845 u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
2846 T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
2847 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
2848 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort
2849 n1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2850 C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_:
2851 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
2852 (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
2853 (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e1 (Flat f) u))))))) (\lambda
2854 (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop
2855 (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
2856 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
2857 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2858 (_: T).(csubst0 O v e1 e2)))))))) H4)) e H1)))) (drop_gen_sort n1 (S n0) O e
2859 H0)))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v:
2860 T).((csubst0 (S n0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4
2861 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
2862 (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
2863 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
2864 e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
2865 (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
2866 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
2867 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
2868 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
2869 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
2870 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
2871 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2872 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f)
2873 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2874 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
2875 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
2876 e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda
2877 (v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t) c2)).(\lambda (e:
2878 C).(\lambda (H1: (drop (S n0) O (CHead c k t) e)).(let H2 \def
2879 (csubst0_gen_head k c c2 t v (S n0) H0) in (or3_ind (ex3_2 T nat (\lambda (_:
2880 T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_:
2881 nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
2882 v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k
2883 j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
2884 (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_:
2885 T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2:
2886 T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
2887 (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
2888 T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S
2889 n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
2890 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
2891 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
2892 e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
2893 (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
2894 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
2895 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
2896 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
2897 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
2898 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
2899 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2900 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f)
2901 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2902 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
2903 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
2904 e2)))))))) (\lambda (H3: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq
2905 nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k
2906 u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T
2907 nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2:
2908 T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
2909 nat).(subst0 j v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda
2910 (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0
2911 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
2912 (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
2913 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
2914 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
2915 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2916 C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_:
2917 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
2918 (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
2919 (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f:
2920 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
2921 n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda
2922 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
2923 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
2924 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq nat
2925 (S n0) (s k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (H6:
2926 (subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S
2927 n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
2928 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
2929 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead
2930 e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
2931 (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
2932 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
2933 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
2934 O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
2935 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
2936 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
2937 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2938 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f)
2939 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2940 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
2941 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
2942 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S
2943 n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c k0 x0) e) (ex3_4 F C T T
2944 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
2945 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
2946 T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e0 (Flat f) w))))))
2947 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
2948 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2949 C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
2950 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c
2951 k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
2952 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
2953 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
2954 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2955 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2
2956 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
2957 (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
2958 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
2959 e2))))))))))) (\lambda (b: B).(\lambda (H7: (drop (r (Bind b) n0) O c
2960 e)).(\lambda (H8: (eq nat (S n0) (s (Bind b) x1))).(let H9 \def (f_equal nat
2961 nat (\lambda (e0: nat).(match e0 with [O \Rightarrow n0 | (S n1) \Rightarrow
2962 n1])) (S n0) (S x1) H8) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1:
2963 nat).(subst0 n1 v t x0)) H6 n0 H9) in (or4_intro0 (drop (S n0) O (CHead c
2964 (Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
2965 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
2966 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c
2967 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
2968 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
2969 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
2970 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2971 C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e2 (Flat f)
2972 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2973 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
2974 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
2975 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
2976 T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e2 (Flat f)
2977 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2978 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
2979 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
2980 (drop_drop (Bind b) n0 c e H7 x0))))))) (\lambda (f: F).(\lambda (H7: (drop
2981 (r (Flat f) n0) O c e)).(\lambda (H8: (eq nat (S n0) (s (Flat f) x1))).(let
2982 H9 \def (f_equal nat nat (\lambda (e0: nat).e0) (S n0) (s (Flat f) x1) H8) in
2983 (let H10 \def (eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v t x0)) H6 (S
2984 n0) H9) in (or4_intro0 (drop (S n0) O (CHead c (Flat f) x0) e) (ex3_4 F C T T
2985 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
2986 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
2987 T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e0 (Flat f0)
2988 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2989 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
2990 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
2991 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
2992 n0) O (CHead c (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
2993 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
2994 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
2995 (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
2996 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
2997 n0) O (CHead c (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_:
2998 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
2999 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3000 T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e H7
3001 x0))))))) k (drop_gen_drop k c e t n0 H1) H4) c2 H5)))))) H3)) (\lambda (H3:
3002 (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))
3003 (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3:
3004 C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_:
3005 C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_:
3006 nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
3007 v c c3))) (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda
3008 (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
3009 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
3010 O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
3011 T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
3012 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
3013 f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
3014 T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
3015 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
3016 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
3017 T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
3018 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
3019 e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
3020 (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3021 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3022 e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: (eq nat (S n0)
3023 (s k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: (csubst0 x1
3024 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop (S n0) O c0
3025 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda
3026 (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
3027 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f)
3028 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3029 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
3030 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
3031 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
3032 O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
3033 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
3034 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3035 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3036 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f)
3037 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3038 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3039 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3040 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S
3041 n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead x0 k0 t) e) (ex3_4 F C T T
3042 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3043 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3044 T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e0 (Flat f) w))))))
3045 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
3046 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
3047 C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
3048 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
3049 k0 t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
3050 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
3051 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3052 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3053 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2
3054 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
3055 (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3056 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3057 e2))))))))))) (\lambda (b: B).(\lambda (H7: (drop (r (Bind b) n0) O c
3058 e)).(\lambda (H8: (eq nat (S n0) (s (Bind b) x1))).(let H9 \def (f_equal nat
3059 nat (\lambda (e0: nat).(match e0 with [O \Rightarrow n0 | (S n1) \Rightarrow
3060 n1])) (S n0) (S x1) H8) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1:
3061 nat).(csubst0 n1 v c x0)) H6 n0 H9) in (let H11 \def (IHn c x0 v H10 e H7) in
3062 (or4_ind (drop n0 O x0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
3063 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
3064 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O
3065 x0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
3066 T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
3067 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
3068 f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
3069 T).(drop n0 O x0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
3070 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
3071 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
3072 T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
3073 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2
3074 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
3075 (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3076 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
3077 (or4 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f:
3078 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat
3079 f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
3080 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda
3081 (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3082 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3083 T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
3084 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t)
3085 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3086 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
3087 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3088 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3089 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
3090 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
3091 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
3092 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
3093 O v e1 e2)))))))) (\lambda (H12: (drop n0 O x0 e)).(or4_intro0 (drop (S n0) O
3094 (CHead x0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
3095 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
3096 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
3097 O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
3098 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3099 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
3100 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3101 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3102 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3103 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3104 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
3105 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3106 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3107 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3108 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3109 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
3110 (drop_drop (Bind b) n0 x0 e H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda
3111 (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0
3112 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
3113 (w: T).(drop n0 O x0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
3114 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T
3115 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3116 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3117 T).(\lambda (w: T).(drop n0 O x0 (CHead e0 (Flat f) w)))))) (\lambda (_:
3118 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4
3119 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f:
3120 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat
3121 f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
3122 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda
3123 (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3124 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3125 T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
3126 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t)
3127 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3128 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
3129 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3130 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3131 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
3132 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
3133 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
3134 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
3135 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda
3136 (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x4))).(\lambda (H14: (drop
3137 n0 O x0 (CHead x3 (Flat x2) x5))).(\lambda (H15: (subst0 O v x4
3138 x5)).(eq_ind_r C (CHead x3 (Flat x2) x4) (\lambda (c0: C).(or4 (drop (S n0) O
3139 (CHead x0 (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
3140 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u))))))
3141 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
3142 O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
3143 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3144 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
3145 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3146 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3147 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3148 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3149 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
3150 f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3151 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3152 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3153 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3154 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3155 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat
3156 x2) x4)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3157 T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f) u))))))
3158 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
3159 O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
3160 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3161 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
3162 x3 (Flat x2) x4) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
3163 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t)
3164 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3165 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
3166 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3167 (CHead x3 (Flat x2) x4) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda
3168 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
3169 x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
3170 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3171 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
3172 (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f:
3173 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2)
3174 x4) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3175 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f)
3176 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3177 T).(subst0 O v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x4))
3178 (drop_drop (Bind b) n0 x0 (CHead x3 (Flat x2) x5) H14 t) H15)) e H13))))))))
3179 H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda
3180 (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
3181 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2
3182 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
3183 (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda
3184 (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
3185 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O
3186 x0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3187 C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead x0
3188 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3189 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
3190 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
3191 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
3192 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3193 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
3194 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3195 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3196 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3197 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3198 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
3199 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3200 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3201 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3202 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3203 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3204 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
3205 T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H14: (drop n0 O
3206 x0 (CHead x4 (Flat x2) x5))).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C
3207 (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind
3208 b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3209 T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f:
3210 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
3211 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
3212 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3213 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
3214 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3215 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3216 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3217 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3218 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
3219 f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3220 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3221 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3222 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3223 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3224 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat
3225 x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3226 T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u))))))
3227 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
3228 O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
3229 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3230 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
3231 x3 (Flat x2) x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
3232 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t)
3233 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3234 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
3235 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3236 (CHead x3 (Flat x2) x5) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda
3237 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
3238 x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
3239 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3240 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
3241 (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f:
3242 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2)
3243 x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3244 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3245 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3246 T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5))
3247 (drop_drop (Bind b) n0 x0 (CHead x4 (Flat x2) x5) H14 t) H15)) e H13))))))))
3248 H12)) (\lambda (H12: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3249 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
3250 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3251 T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) (\lambda (_:
3252 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3253 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3254 T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda
3255 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3256 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3257 C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w)))))))
3258 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3259 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3260 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0)
3261 O (CHead x0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
3262 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
3263 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
3264 O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
3265 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3266 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
3267 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3268 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3269 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3270 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3271 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
3272 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3273 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3274 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3275 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3276 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3277 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
3278 T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x5))).(\lambda
3279 (H14: (drop n0 O x0 (CHead x4 (Flat x2) x6))).(\lambda (H15: (subst0 O v x5
3280 x6)).(\lambda (H16: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x3 (Flat x2) x5)
3281 (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind b) t) c0) (ex3_4 F C T T
3282 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
3283 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3284 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f)
3285 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3286 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
3287 C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u))))))
3288 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
3289 O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
3290 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
3291 C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3292 T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f:
3293 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3294 n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_:
3295 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3296 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3297 T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O
3298 (CHead x0 (Bind b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f:
3299 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2)
3300 x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3301 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f)
3302 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3303 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
3304 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
3305 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
3306 (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u))))))
3307 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
3308 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
3309 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
3310 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
3311 (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat
3312 f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3313 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3314 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
3315 (ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
3316 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
3317 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
3318 (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat
3319 f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3320 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3321 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
3322 x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0
3323 x0 (CHead x4 (Flat x2) x6) H14 t) H15 H16)) e H13)))))))))) H12)) H11)))))))
3324 (\lambda (f: F).(\lambda (H7: (drop (r (Flat f) n0) O c e)).(\lambda (H8: (eq
3325 nat (S n0) (s (Flat f) x1))).(let H9 \def (f_equal nat nat (\lambda (e0:
3326 nat).e0) (S n0) (s (Flat f) x1) H8) in (let H10 \def (eq_ind_r nat x1
3327 (\lambda (n1: nat).(csubst0 n1 v c x0)) H6 (S n0) H9) in (let H11 \def (H x0
3328 v H10 e H7) in (or4_ind (drop (S n0) O x0 e) (ex3_4 F C T T (\lambda (f0:
3329 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat
3330 f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
3331 T).(drop (S n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_:
3332 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3333 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
3334 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
3335 C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_:
3336 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
3337 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
3338 (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
3339 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3340 n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
3341 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3342 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
3343 (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e)
3344 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
3345 T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0:
3346 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
3347 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
3348 T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0:
3349 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
3350 f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
3351 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda
3352 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
3353 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
3354 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u)))))))
3355 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
3356 (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)))))))
3357 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3358 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3359 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H12:
3360 (drop (S n0) O x0 e)).(or4_intro0 (drop (S n0) O (CHead x0 (Flat f) t) e)
3361 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
3362 T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0:
3363 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
3364 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
3365 T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0:
3366 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
3367 f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
3368 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda
3369 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
3370 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
3371 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u)))))))
3372 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
3373 (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)))))))
3374 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3375 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3376 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat
3377 f) n0 x0 e H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f0: F).(\lambda
3378 (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
3379 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3380 n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
3381 (u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda
3382 (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0
3383 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
3384 (w: T).(drop (S n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
3385 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0)
3386 O (CHead x0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
3387 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
3388 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3389 n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
3390 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3391 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3392 T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
3393 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
3394 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3395 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
3396 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3397 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
3398 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
3399 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
3400 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
3401 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
3402 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda
3403 (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x4))).(\lambda (H14: (drop
3404 (S n0) O x0 (CHead x3 (Flat x2) x5))).(\lambda (H15: (subst0 O v x4
3405 x5)).(eq_ind_r C (CHead x3 (Flat x2) x4) (\lambda (c0: C).(or4 (drop (S n0) O
3406 (CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
3407 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u))))))
3408 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3409 n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
3410 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3411 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3412 T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
3413 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
3414 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3415 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
3416 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
3417 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
3418 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
3419 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
3420 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
3421 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
3422 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3
3423 (Flat x2) x4)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
3424 T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u))))))
3425 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3426 n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
3427 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3428 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3429 T).(eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
3430 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
3431 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
3432 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
3433 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
3434 (_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u))))))) (\lambda
3435 (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
3436 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda
3437 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3438 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3439 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C
3440 T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
3441 (CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
3442 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
3443 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
3444 T).(\lambda (w: T).(subst0 O v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3
3445 (Flat x2) x4)) (drop_drop (Flat f) n0 x0 (CHead x3 (Flat x2) x5) H14 t) H15))
3446 e H13)))))))) H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f0: F).(\lambda
3447 (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
3448 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
3449 n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
3450 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C
3451 C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
3452 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
3453 C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_:
3454 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))
3455 (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0:
3456 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat
3457 f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
3458 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda
3459 (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3460 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3461 T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
3462 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
3463 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3464 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
3465 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3466 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
3467 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
3468 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
3469 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
3470 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
3471 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda
3472 (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H14: (drop
3473 (S n0) O x0 (CHead x4 (Flat x2) x5))).(\lambda (H15: (csubst0 O v x3
3474 x4)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O
3475 (CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
3476 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u))))))
3477 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3478 n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
3479 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3480 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3481 T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
3482 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
3483 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3484 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
3485 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
3486 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
3487 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
3488 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
3489 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
3490 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
3491 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3
3492 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
3493 T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) u))))))
3494 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3495 n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
3496 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3497 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3498 T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
3499 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
3500 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
3501 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
3502 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
3503 (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u))))))) (\lambda
3504 (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
3505 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda
3506 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3507 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3508 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C
3509 C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C
3510 (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
3511 (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
3512 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3513 C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead
3514 x3 (Flat x2) x5)) (drop_drop (Flat f) n0 x0 (CHead x4 (Flat x2) x5) H14 t)
3515 H15)) e H13)))))))) H12)) (\lambda (H12: (ex4_5 F C C T T (\lambda (f0:
3516 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3517 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
3518 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 (Flat f0)
3519 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3520 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3521 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3522 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda
3523 (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u)))))))
3524 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
3525 (w: T).(drop (S n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_:
3526 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3527 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3528 T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead x0
3529 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
3530 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0:
3531 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
3532 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_:
3533 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3534 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
3535 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
3536 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0)
3537 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3538 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
3539 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat
3540 f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3541 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0)
3542 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3543 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3544 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3545 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
3546 T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x5))).(\lambda
3547 (H14: (drop (S n0) O x0 (CHead x4 (Flat x2) x6))).(\lambda (H15: (subst0 O v
3548 x5 x6)).(\lambda (H16: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x3 (Flat x2)
3549 x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 F C
3550 T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
3551 c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
3552 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0)
3553 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3554 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
3555 C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f0) u))))))
3556 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
3557 n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_:
3558 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
3559 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
3560 (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f0) u))))))) (\lambda (f0:
3561 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3562 n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_:
3563 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3564 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3565 T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O
3566 (CHead x0 (Flat f) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0:
3567 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2)
3568 x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
3569 (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat
3570 f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3571 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
3572 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
3573 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
3574 (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u))))))
3575 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
3576 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
3577 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
3578 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
3579 (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat
3580 f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3581 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3582 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
3583 (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
3584 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
3585 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
3586 (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat
3587 f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3588 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3589 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
3590 x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Flat f) n0
3591 x0 (CHead x4 (Flat x2) x6) H14 t) H15 H16)) e H13)))))))))) H12)) H11)))))))
3592 k (drop_gen_drop k c e t n0 H1) H4) c2 H5)))))) H3)) (\lambda (H3: (ex4_3 T C
3593 nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k
3594 j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3
3595 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
3596 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
3597 c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
3598 nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda
3599 (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_:
3600 C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
3601 C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 F
3602 C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
3603 e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3604 T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
3605 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3606 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3607 T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
3608 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f)
3609 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3610 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3611 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
3612 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3613 T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
3614 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3615 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3616 T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1:
3617 C).(\lambda (x2: nat).(\lambda (H4: (eq nat (S n0) (s k x2))).(\lambda (H5:
3618 (eq C c2 (CHead x1 k x0))).(\lambda (H6: (subst0 x2 v t x0)).(\lambda (H7:
3619 (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop
3620 (S n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3621 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
3622 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead
3623 e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
3624 (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
3625 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
3626 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
3627 O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
3628 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
3629 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3630 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3631 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f)
3632 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3633 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3634 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3635 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S
3636 n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1 k0 x0) e) (ex3_4 F C T T
3637 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3638 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3639 T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e0 (Flat f) w))))))
3640 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
3641 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
3642 C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
3643 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
3644 k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
3645 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
3646 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3647 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3648 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2
3649 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
3650 (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3651 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3652 e2))))))))))) (\lambda (b: B).(\lambda (H8: (drop (r (Bind b) n0) O c
3653 e)).(\lambda (H9: (eq nat (S n0) (s (Bind b) x2))).(let H10 \def (f_equal nat
3654 nat (\lambda (e0: nat).(match e0 with [O \Rightarrow n0 | (S n1) \Rightarrow
3655 n1])) (S n0) (S x2) H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n1:
3656 nat).(csubst0 n1 v c x1)) H7 n0 H10) in (let H12 \def (eq_ind_r nat x2
3657 (\lambda (n1: nat).(subst0 n1 v t x0)) H6 n0 H10) in (let H13 \def (IHn c x1
3658 v H11 e H8) in (or4_ind (drop n0 O x1 e) (ex3_4 F C T T (\lambda (f:
3659 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat
3660 f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
3661 T).(drop n0 O x1 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
3662 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3663 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
3664 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3665 C).(\lambda (u: T).(drop n0 O x1 (CHead e2 (Flat f) u)))))) (\lambda (_:
3666 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
3667 (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
3668 (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f:
3669 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0
3670 O x1 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
3671 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
3672 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
3673 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 F C T T
3674 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3675 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3676 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f)
3677 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3678 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
3679 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
3680 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
3681 O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
3682 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
3683 C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3684 T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f:
3685 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3686 n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_:
3687 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3688 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3689 T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H14: (drop n0 O x1
3690 e)).(or4_intro0 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 F C T T
3691 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3692 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3693 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f)
3694 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3695 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
3696 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
3697 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
3698 O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
3699 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
3700 C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3701 T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f:
3702 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3703 n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_:
3704 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3705 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3706 T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 x1 e H14
3707 x0))) (\lambda (H14: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
3708 (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
3709 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0
3710 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3711 T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0:
3712 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
3713 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O
3714 x1 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
3715 T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) O (CHead x1 (Bind
3716 b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3717 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
3718 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
3719 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
3720 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3721 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
3722 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3723 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3724 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3725 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3726 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
3727 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3728 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3729 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3730 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3731 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3732 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6:
3733 T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) x5))).(\lambda (H16: (drop n0 O
3734 x1 (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x5 x6)).(eq_ind_r C
3735 (CHead x4 (Flat x3) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind
3736 b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3737 T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f:
3738 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
3739 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
3740 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3741 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
3742 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3743 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3744 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3745 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3746 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
3747 f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3748 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3749 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3750 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3751 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3752 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat
3753 x3) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3754 T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f) u))))))
3755 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
3756 O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
3757 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3758 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
3759 x4 (Flat x3) x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
3760 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0)
3761 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3762 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
3763 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3764 (CHead x4 (Flat x3) x5) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda
3765 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
3766 x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
3767 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3768 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
3769 (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f:
3770 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3)
3771 x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3772 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f)
3773 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3774 T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x5))
3775 (drop_drop (Bind b) n0 x1 (CHead x4 (Flat x3) x6) H16 x0) H17)) e H15))))))))
3776 H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda
3777 (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
3778 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2
3779 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
3780 (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda
3781 (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
3782 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O
3783 x1 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3784 C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead x1
3785 (Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3786 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
3787 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
3788 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
3789 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3790 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
3791 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3792 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3793 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3794 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3795 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
3796 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3797 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3798 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3799 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3800 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3801 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6:
3802 T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H16: (drop n0 O
3803 x1 (CHead x5 (Flat x3) x6))).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C
3804 (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind
3805 b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3806 T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f:
3807 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
3808 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
3809 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3810 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
3811 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3812 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3813 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3814 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3815 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
3816 f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3817 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3818 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3819 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3820 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3821 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat
3822 x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3823 T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u))))))
3824 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
3825 O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
3826 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3827 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
3828 x4 (Flat x3) x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
3829 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0)
3830 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3831 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
3832 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3833 (CHead x4 (Flat x3) x6) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda
3834 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
3835 x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
3836 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3837 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
3838 (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f:
3839 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3)
3840 x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3841 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3842 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3843 T).(csubst0 O v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6))
3844 (drop_drop (Bind b) n0 x1 (CHead x5 (Flat x3) x6) H16 x0) H17)) e H15))))))))
3845 H14)) (\lambda (H14: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3846 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
3847 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3848 T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) (\lambda (_:
3849 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3850 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3851 T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda
3852 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3853 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3854 C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w)))))))
3855 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3856 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3857 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0)
3858 O (CHead x1 (Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
3859 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
3860 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
3861 O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
3862 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3863 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
3864 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3865 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3866 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3867 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3868 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
3869 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3870 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3871 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3872 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3873 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3874 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6:
3875 T).(\lambda (x7: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) x6))).(\lambda
3876 (H16: (drop n0 O x1 (CHead x5 (Flat x3) x7))).(\lambda (H17: (subst0 O v x6
3877 x7)).(\lambda (H18: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6)
3878 (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0) c0) (ex3_4 F C T
3879 T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
3880 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3881 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f)
3882 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3883 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
3884 C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u))))))
3885 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
3886 O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
3887 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
3888 C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3889 T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f:
3890 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3891 n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_:
3892 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3893 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3894 T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O
3895 (CHead x1 (Bind b) x0) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f:
3896 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3)
3897 x6) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3898 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f)
3899 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3900 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
3901 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1
3902 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
3903 (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u))))))
3904 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
3905 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
3906 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1
3907 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
3908 (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat
3909 f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3910 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3911 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
3912 (ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
3913 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1
3914 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
3915 (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat
3916 f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3917 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3918 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
3919 x3 x4 x5 x6 x7 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0
3920 x1 (CHead x5 (Flat x3) x7) H16 x0) H17 H18)) e H15)))))))))) H14))
3921 H13)))))))) (\lambda (f: F).(\lambda (H8: (drop (r (Flat f) n0) O c
3922 e)).(\lambda (H9: (eq nat (S n0) (s (Flat f) x2))).(let H10 \def (f_equal nat
3923 nat (\lambda (e0: nat).e0) (S n0) (s (Flat f) x2) H9) in (let H11 \def
3924 (eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H7 (S n0) H10) in
3925 (let H12 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H6 (S
3926 n0) H10) in (let H13 \def (H x1 v H11 e H8) in (or4_ind (drop (S n0) O x1 e)
3927 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
3928 T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0:
3929 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e0 (Flat f0)
3930 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3931 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
3932 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
3933 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
3934 n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
3935 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
3936 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
3937 (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_:
3938 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead
3939 e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
3940 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
3941 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
3942 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T
3943 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3944 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
3945 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0)
3946 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3947 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
3948 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
3949 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
3950 n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
3951 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
3952 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
3953 (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
3954 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3955 n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_:
3956 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3957 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3958 T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H14: (drop (S n0) O
3959 x1 e)).(or4_intro0 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T
3960 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3961 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
3962 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0)
3963 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3964 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
3965 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
3966 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
3967 n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
3968 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
3969 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
3970 (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
3971 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3972 n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_:
3973 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3974 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3975 T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 x1 e H14
3976 x0))) (\lambda (H14: (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
3977 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
3978 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3979 n0) O x1 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
3980 (u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda
3981 (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0
3982 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
3983 (w: T).(drop (S n0) O x1 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
3984 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0)
3985 O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
3986 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
3987 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3988 n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_:
3989 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3990 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3991 T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
3992 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
3993 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3994 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
3995 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3996 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
3997 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
3998 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
3999 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
4000 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
4001 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda
4002 (x6: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) x5))).(\lambda (H16: (drop
4003 (S n0) O x1 (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x5
4004 x6)).(eq_ind_r C (CHead x4 (Flat x3) x5) (\lambda (c0: C).(or4 (drop (S n0) O
4005 (CHead x1 (Flat f) x0) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
4006 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u))))))
4007 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
4008 n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_:
4009 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
4010 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
4011 T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
4012 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
4013 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4014 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
4015 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
4016 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
4017 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
4018 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4019 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
4020 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
4021 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead
4022 x4 (Flat x3) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda
4023 (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f0)
4024 u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
4025 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda
4026 (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
4027 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
4028 T).(eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
4029 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
4030 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
4031 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
4032 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
4033 (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u))))))) (\lambda
4034 (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
4035 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)))))))
4036 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
4037 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4038 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C
4039 T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
4040 (CHead x4 (Flat x3) x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
4041 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
4042 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
4043 T).(\lambda (w: T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4
4044 (Flat x3) x5)) (drop_drop (Flat f) n0 x1 (CHead x4 (Flat x3) x6) H16 x0)
4045 H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f0:
4046 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
4047 f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
4048 T).(drop (S n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda
4049 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind
4050 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
4051 C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
4052 (e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 (Flat f0) u))))))
4053 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
4054 v e1 e2))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T
4055 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
4056 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
4057 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0)
4058 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
4059 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
4060 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
4061 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
4062 n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
4063 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
4064 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
4065 (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
4066 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
4067 n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_:
4068 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
4069 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4070 T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4:
4071 C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 (Flat
4072 x3) x6))).(\lambda (H16: (drop (S n0) O x1 (CHead x5 (Flat x3) x6))).(\lambda
4073 (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0:
4074 C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 F C T T (\lambda
4075 (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0
4076 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
4077 (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w))))))
4078 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
4079 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
4080 C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0:
4081 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
4082 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
4083 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
4084 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
4085 (_: T).(eq C c0 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_:
4086 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
4087 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
4088 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
4089 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4090 (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x1 (Flat
4091 f) x0) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
4092 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0
4093 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
4094 (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w))))))
4095 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
4096 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
4097 C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))
4098 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
4099 n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
4100 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
4101 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
4102 (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0)
4103 u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
4104 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0)
4105 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
4106 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
4107 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
4108 (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
4109 C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))
4110 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
4111 n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
4112 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))
4113 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1
4114 (CHead x5 (Flat x3) x6) H16 x0) H17)) e H15)))))))) H14)) (\lambda (H14:
4115 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
4116 (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
4117 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
4118 n0) O x1 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
4119 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
4120 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4121 (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0:
4122 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
4123 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
4124 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e2 (Flat f0)
4125 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
4126 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
4127 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
4128 (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0:
4129 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat
4130 f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
4131 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda
4132 (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
4133 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
4134 T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
4135 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
4136 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4137 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
4138 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
4139 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
4140 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
4141 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4142 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
4143 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
4144 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda
4145 (x6: T).(\lambda (x7: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3)
4146 x6))).(\lambda (H16: (drop (S n0) O x1 (CHead x5 (Flat x3) x7))).(\lambda
4147 (H17: (subst0 O v x6 x7)).(\lambda (H18: (csubst0 O v x4 x5)).(eq_ind_r C
4148 (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat
4149 f) x0) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
4150 T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0:
4151 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
4152 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_:
4153 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
4154 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
4155 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
4156 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0)
4157 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4158 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
4159 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
4160 f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
4161 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0)
4162 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
4163 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
4164 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
4165 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 (Flat
4166 x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
4167 T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f0) u))))))
4168 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
4169 n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_:
4170 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
4171 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
4172 T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
4173 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
4174 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
4175 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
4176 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
4177 (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))) (\lambda
4178 (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
4179 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)))))))
4180 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
4181 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4182 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C
4183 C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
4184 T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))))
4185 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4186 (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)))))))
4187 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
4188 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4189 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7
4190 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1 (CHead x5
4191 (Flat x3) x7) H16 x0) H17 H18)) e H15)))))))))) H14)) H13)))))))) k
4192 (drop_gen_drop k c e t n0 H1) H4) c2 H5)))))))) H3)) H2))))))))))) c1)))) n).
4194 lemma csubst0_drop_eq_back:
4195 \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0
4196 n v c1 c2) \to (\forall (e: C).((drop n O c2 e) \to (or4 (drop n O c1 e)
4197 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
4198 T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
4199 C).(\lambda (u1: T).(\lambda (_: T).(drop n O c1 (CHead e0 (Flat f) u1))))))
4200 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
4201 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4202 C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
4203 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n O c1 (CHead e1
4204 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4205 (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
4206 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
4207 f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
4208 T).(\lambda (_: T).(drop n O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_:
4209 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
4210 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4211 (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))
4213 \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2:
4214 C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c2
4215 e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
4216 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
4217 (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O
4218 c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
4219 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
4220 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
4221 f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
4222 T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
4223 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
4224 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4225 (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
4226 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e1
4227 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
4228 (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda
4229 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
4230 e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
4231 (H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c2
4232 e)).(eq_ind C c2 (\lambda (c: C).(or4 (drop O O c1 c) (ex3_4 F C T T (\lambda
4233 (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c (CHead e0
4234 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
4235 (_: T).(drop O O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
4236 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
4237 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c
4238 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4239 C).(\lambda (u: T).(drop O O c1 (CHead e1 (Flat f) u)))))) (\lambda (_:
4240 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
4241 (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
4242 (_: T).(\lambda (u2: T).(eq C c (CHead e2 (Flat f) u2))))))) (\lambda (f:
4243 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O
4244 O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda
4245 (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
4246 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
4247 O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2))
4248 (\lambda (n0: nat).(or4 (drop n0 n0 c1 c2) (ex3_4 F C T T (\lambda (f:
4249 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e0 (Flat
4250 f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
4251 T).(drop n0 n0 c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
4252 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1 u2)))))) (ex3_4 F C C T
4253 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c2
4254 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4255 C).(\lambda (u: T).(drop n0 n0 c1 (CHead e1 (Flat f) u)))))) (\lambda (_:
4256 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1
4257 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4258 C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e2 (Flat f) u2)))))))
4259 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4260 (_: T).(drop n0 n0 c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda
4261 (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1
4262 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4263 T).(\lambda (_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda
4264 (H1: (csubst0 y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t:
4265 T).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c c0)
4266 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
4267 T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
4268 C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e0 (Flat f) u1))))))
4269 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0
4270 t u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4271 C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
4272 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 c (CHead e1
4273 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4274 (_: T).(csubst0 n0 t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
4275 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2
4276 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
4277 (u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e1 (Flat f) u1))))))) (\lambda
4278 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
4279 T).(subst0 n0 t u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4280 C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda
4281 (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall
4282 (u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s
4283 k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead c k0 u2))
4284 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4:
4285 T).(eq C (CHead c k0 u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda
4286 (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0
4287 u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3:
4288 T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda
4289 (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c k0 u2)
4290 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4291 C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead e1 (Flat f)
4292 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4293 T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
4294 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c k0
4295 u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda
4296 (_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0
4297 u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4298 C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda
4299 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
4300 T).(csubst0 (s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i:
4301 nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0
4302 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def
4303 (eq_ind nat (S i) (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S
4304 _) \Rightarrow True])) I O H3) in (False_ind (or4 (drop (S i) (S i) (CHead c
4305 (Bind b) u1) (CHead c (Bind b) u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda
4306 (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Bind b) u2) (CHead
4307 e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3:
4308 T).(\lambda (_: T).(drop (S i) (S i) (CHead c (Bind b) u1) (CHead e0 (Flat f)
4309 u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4:
4310 T).(subst0 (S i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
4311 C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c (Bind b) u2) (CHead e2
4312 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
4313 (u: T).(drop (S i) (S i) (CHead c (Bind b) u1) (CHead e1 (Flat f) u))))))
4314 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S
4315 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda
4316 (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Bind b) u2) (CHead
4317 e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4318 C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead c (Bind b) u1)
4319 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4320 C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (\lambda
4321 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
4322 T).(csubst0 (S i) v0 e1 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i:
4323 nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0
4324 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind
4325 nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H3) in (eq_ind_r nat O
4326 (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f) u1) (CHead c (Flat f)
4327 u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
4328 T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 (Flat f0) u4))))))
4329 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0
4330 n0 (CHead c (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_:
4331 F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3
4332 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
4333 C).(\lambda (u: T).(eq C (CHead c (Flat f) u2) (CHead e2 (Flat f0) u))))))
4334 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0
4335 (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda
4336 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F
4337 C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
4338 T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e2 (Flat f0) u4)))))))
4339 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda
4340 (_: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead e1 (Flat f0) u3)))))))
4341 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda
4342 (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
4343 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))))))
4344 (or4_intro1 (drop O O (CHead c (Flat f) u1) (CHead c (Flat f) u2)) (ex3_4 F C
4345 T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C
4346 (CHead c (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda
4347 (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1)
4348 (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3:
4349 T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0:
4350 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c (Flat f)
4351 u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
4352 (_: C).(\lambda (u: T).(drop O O (CHead c (Flat f) u1) (CHead e1 (Flat f0)
4353 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4354 T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_:
4355 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f)
4356 u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
4357 (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1)
4358 (CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4359 C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_:
4360 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
4361 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0:
4362 C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0
4363 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3:
4364 T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) (CHead e0 (Flat f0)
4365 u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4:
4366 T).(subst0 O v0 u3 u4))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u2))
4367 (drop_refl (CHead c (Flat f) u1)) H4)) i H3)))))))))) k)) (\lambda (k:
4368 K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4:
4369 C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop
4370 i i c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
4371 T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f:
4372 F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0
4373 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda
4374 (u2: T).(subst0 i v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
4375 C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u))))))
4376 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3
4377 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4378 C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
4379 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4
4380 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4381 C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f) u1)))))))
4382 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4383 (u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
4384 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to
4385 (\forall (u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead
4386 c3 k0 u) (CHead c4 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
4387 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e0 (Flat f)
4388 u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
4389 T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e0 (Flat f) u1))))))
4390 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (s
4391 k0 i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda
4392 (e2: C).(\lambda (u0: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) u0))))))
4393 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop (s k0
4394 i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) (\lambda (_:
4395 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1
4396 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4397 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f)
4398 u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
4399 T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f)
4400 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
4401 T).(\lambda (u2: T).(subst0 (s k0 i) v0 u1 u2)))))) (\lambda (_: F).(\lambda
4402 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0
4403 e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3:
4404 C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3
4405 c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T
4406 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4
4407 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
4408 T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u1)))))) (\lambda (_:
4409 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2))))))
4410 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
4411 T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
4412 C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u))))))
4413 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
4414 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4415 C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2)))))))
4416 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4417 (_: T).(drop i i c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
4418 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2))))))
4419 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4420 (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat
4421 (S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee with [O
4422 \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (or4
4423 (drop (S i) (S i) (CHead c3 (Bind b) u) (CHead c4 (Bind b) u)) (ex3_4 F C T T
4424 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C
4425 (CHead c4 (Bind b) u) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda
4426 (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b)
4427 u) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
4428 T).(\lambda (u2: T).(subst0 (S i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f:
4429 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Bind b)
4430 u) (CHead e2 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4431 C).(\lambda (u0: T).(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead e1 (Flat
4432 f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4433 T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
4434 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4
4435 (Bind b) u) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
4436 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S i) (S i) (CHead
4437 c3 (Bind b) u) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
4438 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (S i) v0 u1
4439 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4440 T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))))) H5))))))))))) (\lambda (f:
4441 F).(\lambda (i: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0:
4442 T).(\lambda (H2: (csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4
4443 (drop i i c3 c4) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda
4444 (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0:
4445 F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0
4446 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda
4447 (u2: T).(subst0 i v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
4448 C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u))))))
4449 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i
4450 c3 (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4451 C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
4452 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4
4453 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
4454 C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0)
4455 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
4456 T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1:
4457 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1
4458 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i O)).(let H5 \def
4459 (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c3 c4)
4460 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
4461 (u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
4462 C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0 (Flat f0)
4463 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
4464 T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
4465 C).(\lambda (e2: C).(\lambda (u0: T).(eq C c4 (CHead e2 (Flat f0) u0))))))
4466 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop n0
4467 n0 c3 (CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
4468 (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda
4469 (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq
4470 C c4 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
4471 (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0)
4472 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
4473 T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1:
4474 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1
4475 e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0:
4476 nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r nat O (\lambda (n0:
4477 nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u) (CHead c4 (Flat f) u)) (ex3_4 F C
4478 T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C
4479 (CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda
4480 (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u)
4481 (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
4482 T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
4483 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f)
4484 u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
4485 (_: C).(\lambda (u0: T).(drop n0 n0 (CHead c3 (Flat f) u) (CHead e1 (Flat f0)
4486 u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4487 T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_:
4488 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Flat f)
4489 u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
4490 (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u)
4491 (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4492 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_:
4493 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
4494 n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c3 (Flat f) u) (CHead c4
4495 (Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
4496 T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2))))))
4497 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O
4498 (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda
4499 (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C
4500 C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C
4501 (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda
4502 (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u)
4503 (CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4504 C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
4505 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C
4506 (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda
4507 (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3
4508 (Flat f) u) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
4509 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2))))))
4510 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4511 (_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0:
4512 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f)
4513 u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
4514 (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0)
4515 u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4516 T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c4 (Flat f) u))
4517 (drop_refl (CHead c3 (Flat f) u)) H6)) i H4)))))))))))) k)) (\lambda (k:
4518 K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1:
4519 T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4:
4520 C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4
4521 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq
4522 C c4 (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
4523 (u3: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_:
4524 F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
4525 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
4526 T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
4527 C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u))))))
4528 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
4529 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4530 C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4)))))))
4531 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda
4532 (_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_:
4533 C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
4534 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4535 (_: T).(csubst0 i v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop
4536 (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead c4 k0 u2)) (ex3_4 F C T T (\lambda
4537 (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0
4538 u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
4539 (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e0
4540 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda
4541 (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f:
4542 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 k0 u2)
4543 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4544 C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e1 (Flat
4545 f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4546 T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
4547 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0
4548 u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda
4549 (_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0
4550 u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4551 C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda
4552 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
4553 T).(csubst0 (s k0 i) v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i:
4554 nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0
4555 i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3
4556 c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T
4557 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4
4558 (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3:
4559 T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_:
4560 F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
4561 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
4562 T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
4563 C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u))))))
4564 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
4565 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4566 C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4)))))))
4567 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda
4568 (_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_:
4569 C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
4570 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4571 (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6
4572 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee with [O \Rightarrow False
4573 | (S _) \Rightarrow True])) I O H5) in (False_ind (or4 (drop (S i) (S i)
4574 (CHead c3 (Bind b) u1) (CHead c4 (Bind b) u2)) (ex3_4 F C T T (\lambda (f:
4575 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Bind b)
4576 u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
4577 (u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead e0
4578 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda
4579 (u4: T).(subst0 (S i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda
4580 (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 (Bind b) u2) (CHead
4581 e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
4582 (u: T).(drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u))))))
4583 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S
4584 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda
4585 (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead
4586 e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4587 C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u1)
4588 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4589 C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (\lambda
4590 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
4591 T).(csubst0 (S i) v0 e1 e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda
4592 (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2:
4593 (subst0 i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0
4594 i v0 c3 c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F
4595 C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq
4596 C c4 (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
4597 (u3: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f0) u3)))))) (\lambda
4598 (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3
4599 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
4600 C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
4601 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1
4602 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4603 (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda
4604 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2
4605 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
4606 C).(\lambda (u3: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0)
4607 u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3:
4608 T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1:
4609 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1
4610 e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6 \def (eq_ind nat i (\lambda
4611 (n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c3 c4) (ex3_4 F C T T (\lambda
4612 (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e0
4613 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3:
4614 T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0 (Flat f0) u3)))))) (\lambda (_:
4615 F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3
4616 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
4617 C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
4618 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 c3 (CHead e1
4619 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4620 (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda
4621 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2
4622 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
4623 C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0)
4624 u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3:
4625 T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1:
4626 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1
4627 e2)))))))))) H4 O H5) in (let H7 \def (eq_ind nat i (\lambda (n0:
4628 nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8 \def (eq_ind nat i (\lambda
4629 (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in (eq_ind_r nat O (\lambda (n0:
4630 nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F
4631 C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq
4632 C (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0:
4633 F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 (CHead c3
4634 (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_:
4635 C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T
4636 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C
4637 (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
4638 (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 (CHead c3 (Flat f) u1)
4639 (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4640 C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
4641 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C
4642 (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0:
4643 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop n0
4644 n0 (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_:
4645 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0
4646 n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4647 (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3 (drop O O
4648 (CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F C T T (\lambda (f0:
4649 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f)
4650 u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
4651 (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead e0 (Flat f0)
4652 u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4:
4653 T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
4654 C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 (Flat f) u2) (CHead e2
4655 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
4656 (u: T).(drop O O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda
4657 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1
4658 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
4659 C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e2
4660 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
4661 C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead
4662 e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4663 C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_:
4664 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
4665 O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_:
4666 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f)
4667 u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
4668 (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1)
4669 (CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4670 C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_:
4671 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
4672 O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c4 (Flat f) u2))
4673 (drop_refl (CHead c3 (Flat f) u1)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2
4674 H1))) H) e (drop_gen_refl c2 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn:
4675 ((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to
4676 (\forall (e: C).((drop n0 O c2 e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T
4677 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
4678 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
4679 T).(\lambda (_: T).(drop n0 O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_:
4680 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
4681 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
4682 T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
4683 C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c1 (CHead e1 (Flat f) u))))))
4684 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
4685 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4686 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
4687 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4688 (_: T).(drop n0 O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda
4689 (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
4690 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4691 T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind
4692 (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to
4693 (\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O c e) (ex3_4 F C
4694 T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
4695 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
4696 T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
4697 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
4698 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
4699 T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
4700 C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f)
4701 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4702 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
4703 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
4704 f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
4705 T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_:
4706 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
4707 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4708 (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1:
4709 nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 (S n0) v (CSort
4710 n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 e)).(csubst0_gen_sort
4711 c2 v (S n0) n1 H (or4 (drop (S n0) O (CSort n1) e) (ex3_4 F C T T (\lambda
4712 (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0
4713 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
4714 (_: T).(drop (S n0) O (CSort n1) (CHead e0 (Flat f) u1)))))) (\lambda (_:
4715 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
4716 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
4717 T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
4718 C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat
4719 f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4720 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
4721 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
4722 f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
4723 T).(\lambda (_: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat f) u1)))))))
4724 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4725 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
4726 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))
4727 (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S
4728 n0) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O
4729 c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
4730 (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
4731 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f)
4732 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
4733 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
4734 C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u))))))
4735 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
4736 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4737 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
4738 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
4739 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4740 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f)
4741 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
4742 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
4743 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
4744 e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda
4745 (v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t) c2)).(\lambda (e:
4746 C).(\lambda (H1: (drop (S n0) O c2 e)).(let H2 \def (csubst0_gen_head k c c2
4747 t v (S n0) H0) in (or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq
4748 nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k
4749 u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat
4750 (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3:
4751 C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j:
4752 nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
4753 C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda
4754 (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2:
4755 T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
4756 T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S
4757 n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
4758 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
4759 (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
4760 n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
4761 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
4762 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
4763 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4764 C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u))))))
4765 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
4766 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4767 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
4768 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4769 (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
4770 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
4771 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4772 (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H3: (ex3_2 T nat
4773 (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2:
4774 T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
4775 nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j:
4776 nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2
4777 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4
4778 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
4779 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
4780 (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
4781 n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
4782 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
4783 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
4784 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4785 C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u))))))
4786 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
4787 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4788 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
4789 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4790 (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
4791 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
4792 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4793 (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda
4794 (x1: nat).(\lambda (H4: (eq nat (S n0) (s k x1))).(\lambda (H5: (eq C c2
4795 (CHead c k x0))).(\lambda (H6: (subst0 x1 v t x0)).(let H7 \def (eq_ind C c2
4796 (\lambda (c0: C).(drop (S n0) O c0 e)) H1 (CHead c k x0) H5) in (K_ind
4797 (\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0 n0) O c e) \to
4798 (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda
4799 (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
4800 (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
4801 n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
4802 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
4803 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
4804 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4805 C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u))))))
4806 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
4807 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4808 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
4809 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4810 (_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) (\lambda
4811 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
4812 T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4813 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b:
4814 B).(\lambda (H8: (eq nat (S n0) (s (Bind b) x1))).(\lambda (H9: (drop (r
4815 (Bind b) n0) O c e)).(let H10 \def (f_equal nat nat (\lambda (e0: nat).(match
4816 e0 with [O \Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H8) in
4817 (let H11 \def (eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v t x0)) H6 n0
4818 H10) in (or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T
4819 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
4820 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
4821 T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f)
4822 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
4823 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
4824 C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u))))))
4825 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
4826 O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
4827 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
4828 C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
4829 T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f:
4830 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
4831 n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
4832 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
4833 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4834 (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e
4835 H9 t))))))) (\lambda (f: F).(\lambda (H8: (eq nat (S n0) (s (Flat f)
4836 x1))).(\lambda (H9: (drop (r (Flat f) n0) O c e)).(let H10 \def (f_equal nat
4837 nat (\lambda (e0: nat).e0) (S n0) (s (Flat f) x1) H8) in (let H11 \def
4838 (eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v t x0)) H6 (S n0) H10) in
4839 (or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda
4840 (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0
4841 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
4842 T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
4843 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
4844 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
4845 C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u))))))
4846 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
4847 n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
4848 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
4849 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
4850 (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
4851 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
4852 n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
4853 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
4854 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4855 (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e
4856 H9 t))))))) k H4 (drop_gen_drop k c e x0 n0 H7)))))))) H3)) (\lambda (H3:
4857 (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))
4858 (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3:
4859 C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_:
4860 C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_:
4861 nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
4862 v c c3))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f:
4863 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
4864 f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
4865 T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
4866 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
4867 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
4868 T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
4869 C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1
4870 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4871 (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
4872 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
4873 f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
4874 T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1)))))))
4875 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4876 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
4877 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
4878 (x0: C).(\lambda (x1: nat).(\lambda (H4: (eq nat (S n0) (s k x1))).(\lambda
4879 (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: (csubst0 x1 v c x0)).(let H7
4880 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1 (CHead x0 k t)
4881 H5) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0
4882 n0) O x0 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda
4883 (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0
4884 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
4885 (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
4886 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
4887 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
4888 T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
4889 C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1
4890 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4891 (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
4892 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
4893 f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
4894 T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1)))))))
4895 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4896 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
4897 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))
4898 (\lambda (b: B).(\lambda (H8: (eq nat (S n0) (s (Bind b) x1))).(\lambda (H9:
4899 (drop (r (Bind b) n0) O x0 e)).(let H10 \def (f_equal nat nat (\lambda (e0:
4900 nat).(match e0 with [O \Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S
4901 x1) H8) in (let H11 \def (eq_ind_r nat x1 (\lambda (n1: nat).(csubst0 n1 v c
4902 x0)) H6 n0 H10) in (let H12 \def (IHn c x0 v H11 e H9) in (or4_ind (drop n0 O
4903 c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
4904 (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
4905 C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1))))))
4906 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
4907 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4908 C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
4909 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1
4910 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4911 (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
4912 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
4913 f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
4914 T).(\lambda (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_:
4915 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
4916 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4917 (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c
4918 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
4919 T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f:
4920 F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
4921 (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
4922 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
4923 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
4924 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4925 C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
4926 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4927 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
4928 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
4929 f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
4930 T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
4931 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
4932 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
4933 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
4934 e2)))))))) (\lambda (H13: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead
4935 c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
4936 T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f:
4937 F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
4938 (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
4939 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
4940 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
4941 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4942 C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
4943 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4944 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
4945 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
4946 f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
4947 T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
4948 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
4949 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
4950 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
4951 (drop_drop (Bind b) n0 c e H13 t))) (\lambda (H13: (ex3_4 F C T T (\lambda
4952 (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0
4953 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
4954 (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
4955 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C
4956 T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
4957 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
4958 T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
4959 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
4960 (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f:
4961 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
4962 f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
4963 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
4964 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
4965 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4966 C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
4967 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
4968 (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
4969 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
4970 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4971 (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
4972 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
4973 (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
4974 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
4975 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4976 (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda
4977 (x4: T).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3 (Flat x2)
4978 x5))).(\lambda (H15: (drop n0 O c (CHead x3 (Flat x2) x4))).(\lambda (H16:
4979 (subst0 O v x4 x5)).(let H17 \def (eq_ind C e (\lambda (c0: C).(drop n0 O x0
4980 c0)) H9 (CHead x3 (Flat x2) x5) H14) in (eq_ind_r C (CHead x3 (Flat x2) x5)
4981 (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T
4982 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
4983 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
4984 T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f)
4985 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
4986 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
4987 C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u))))))
4988 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
4989 O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
4990 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
4991 C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
4992 T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f:
4993 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
4994 n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
4995 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
4996 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4997 (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O
4998 (CHead c (Bind b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f:
4999 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat
5000 x2) x5) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
5001 (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat
5002 f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5003 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
5004 C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2
5005 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5006 (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u))))))
5007 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
5008 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5009 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2
5010 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5011 (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
5012 f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5013 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5014 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
5015 (ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
5016 T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u2))))))
5017 (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5018 n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
5019 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
5020 x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 c
5021 (CHead x3 (Flat x2) x4) H15 t) H16)) e H14))))))))) H13)) (\lambda (H13:
5022 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
5023 T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
5024 C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u))))))
5025 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
5026 v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda
5027 (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
5028 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1
5029 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
5030 (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
5031 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
5032 T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
5033 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
5034 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5035 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
5036 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5037 f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5038 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
5039 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
5040 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5041 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
5042 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5043 (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
5044 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5045 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
5046 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
5047 (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (H14: (eq
5048 C e (CHead x4 (Flat x2) x5))).(\lambda (H15: (drop n0 O c (CHead x3 (Flat x2)
5049 x5))).(\lambda (H16: (csubst0 O v x3 x4)).(let H17 \def (eq_ind C e (\lambda
5050 (c0: C).(drop n0 O x0 c0)) H9 (CHead x4 (Flat x2) x5) H14) in (eq_ind_r C
5051 (CHead x4 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind
5052 b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
5053 T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f:
5054 F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5055 (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
5056 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
5057 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0
5058 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
5059 C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
5060 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
5061 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
5062 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat
5063 f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
5064 T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
5065 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5066 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5067 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
5068 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Bind b) t) (CHead x4 (Flat
5069 x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
5070 T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e0 (Flat f) u2))))))
5071 (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5072 n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
5073 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5074 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
5075 T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u)))))) (\lambda (f:
5076 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
5077 (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
5078 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
5079 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5080 (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u2))))))) (\lambda
5081 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_:
5082 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda
5083 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5084 T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5085 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C
5086 C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C
5087 (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda
5088 (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t)
5089 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5090 C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead
5091 x4 (Flat x2) x5)) (drop_drop (Bind b) n0 c (CHead x3 (Flat x2) x5) H15 t)
5092 H16)) e H14))))))))) H13)) (\lambda (H13: (ex4_5 F C C T T (\lambda (f:
5093 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5094 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
5095 C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 (Flat f) u1)))))))
5096 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5097 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
5098 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind
5099 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
5100 T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f:
5101 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0
5102 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5103 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5104 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5105 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T
5106 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5107 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
5108 T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f)
5109 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5110 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
5111 C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u))))))
5112 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
5113 O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
5114 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
5115 C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
5116 T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f:
5117 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5118 n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
5119 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
5120 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
5121 (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda
5122 (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H14: (eq
5123 C e (CHead x4 (Flat x2) x6))).(\lambda (H15: (drop n0 O c (CHead x3 (Flat x2)
5124 x5))).(\lambda (H16: (subst0 O v x5 x6)).(\lambda (H17: (csubst0 O v x3
5125 x4)).(let H18 \def (eq_ind C e (\lambda (c0: C).(drop n0 O x0 c0)) H9 (CHead
5126 x4 (Flat x2) x6) H14) in (eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0:
5127 C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f:
5128 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
5129 f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5130 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
5131 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
5132 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5133 C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
5134 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
5135 (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
5136 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
5137 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5138 (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
5139 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5140 (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
5141 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5142 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5143 (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind
5144 b) t) (CHead x4 (Flat x2) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
5145 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0
5146 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
5147 (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1))))))
5148 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
5149 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5150 C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) u))))))
5151 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
5152 O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
5153 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
5154 C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
5155 T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f)
5156 u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
5157 T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
5158 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5159 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5160 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
5161 (ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5162 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2
5163 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5164 (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
5165 f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5166 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5167 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
5168 x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) (drop_drop (Bind b) n0
5169 c (CHead x3 (Flat x2) x5) H15 t) H16 H17)) e H14))))))))))) H13)) H12)))))))
5170 (\lambda (f: F).(\lambda (H8: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H9:
5171 (drop (r (Flat f) n0) O x0 e)).(let H10 \def (f_equal nat nat (\lambda (e0:
5172 nat).e0) (S n0) (s (Flat f) x1) H8) in (let H11 \def (eq_ind_r nat x1
5173 (\lambda (n1: nat).(csubst0 n1 v c x0)) H6 (S n0) H10) in (let H12 \def (H x0
5174 v H11 e H9) in (or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0:
5175 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
5176 f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5177 T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_:
5178 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
5179 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
5180 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
5181 C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_:
5182 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
5183 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
5184 (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
5185 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5186 n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
5187 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5188 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5189 (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e)
5190 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
5191 (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
5192 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5193 (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5194 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
5195 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5196 f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5197 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
5198 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
5199 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5200 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2)))))))
5201 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5202 (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
5203 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5204 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
5205 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
5206 (H13: (drop (S n0) O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e)
5207 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
5208 (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
5209 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5210 (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5211 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
5212 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5213 f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5214 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
5215 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
5216 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5217 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2)))))))
5218 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5219 (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
5220 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5221 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
5222 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop
5223 (Flat f) n0 c e H13 t))) (\lambda (H13: (ex3_4 F C T T (\lambda (f0:
5224 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
5225 f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5226 T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_:
5227 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C
5228 T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C
5229 e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
5230 (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1))))))
5231 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
5232 u1 u2))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T
5233 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5234 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
5235 T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
5236 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5237 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
5238 C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u))))))
5239 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
5240 n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
5241 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
5242 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
5243 (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
5244 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5245 n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
5246 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
5247 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
5248 (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda
5249 (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3
5250 (Flat x2) x5))).(\lambda (H15: (drop (S n0) O c (CHead x3 (Flat x2)
5251 x4))).(\lambda (H16: (subst0 O v x4 x5)).(let H17 \def (eq_ind C e (\lambda
5252 (c0: C).(drop (S n0) O x0 c0)) H9 (CHead x3 (Flat x2) x5) H14) in (eq_ind_r C
5253 (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat
5254 f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
5255 T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0:
5256 F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5257 (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_:
5258 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
5259 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0
5260 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
5261 C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0)
5262 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
5263 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_:
5264 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat
5265 f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
5266 T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0)
5267 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5268 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5269 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
5270 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Flat f) t) (CHead x3 (Flat
5271 x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
5272 T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0)
5273 u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5274 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
5275 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
5276 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5277 C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f0) u))))))
5278 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
5279 n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
5280 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
5281 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
5282 (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f0)
5283 u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
5284 T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0)
5285 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5286 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5287 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
5288 (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
5289 T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0)
5290 u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5291 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
5292 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
5293 u2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Flat
5294 f) n0 c (CHead x3 (Flat x2) x4) H15 t) H16)) e H14))))))))) H13)) (\lambda
5295 (H13: (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5296 C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0:
5297 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead
5298 e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5299 C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0:
5300 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5301 f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5302 T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
5303 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0)
5304 O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
5305 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2))))))
5306 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5307 n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_:
5308 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5309 (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
5310 T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
5311 C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead
5312 e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5313 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
5314 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5315 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
5316 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5317 (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5318 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5319 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5320 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda
5321 (x5: T).(\lambda (H14: (eq C e (CHead x4 (Flat x2) x5))).(\lambda (H15: (drop
5322 (S n0) O c (CHead x3 (Flat x2) x5))).(\lambda (H16: (csubst0 O v x3 x4)).(let
5323 H17 \def (eq_ind C e (\lambda (c0: C).(drop (S n0) O x0 c0)) H9 (CHead x4
5324 (Flat x2) x5) H14) in (eq_ind_r C (CHead x4 (Flat x2) x5) (\lambda (c0:
5325 C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0:
5326 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
5327 f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5328 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
5329 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
5330 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5331 C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
5332 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
5333 (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
5334 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
5335 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5336 (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
5337 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5338 (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
5339 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5340 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5341 (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat
5342 f) t) (CHead x4 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
5343 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e0
5344 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
5345 T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
5346 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5347 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
5348 C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2
5349 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5350 (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
5351 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
5352 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5353 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2
5354 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
5355 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5356 (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5357 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5358 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5359 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_:
5360 C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2
5361 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5362 (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
5363 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
5364 v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop
5365 (Flat f) n0 c (CHead x3 (Flat x2) x5) H15 t) H16)) e H14))))))))) H13))
5366 (\lambda (H13: (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda
5367 (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0)
5368 u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
5369 T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda
5370 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5371 T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5372 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C
5373 C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
5374 T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
5375 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5376 n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
5377 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5378 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5379 (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e)
5380 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
5381 (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
5382 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5383 (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5384 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
5385 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5386 f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5387 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
5388 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
5389 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5390 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2)))))))
5391 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5392 (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
5393 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5394 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
5395 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
5396 (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6:
5397 T).(\lambda (H14: (eq C e (CHead x4 (Flat x2) x6))).(\lambda (H15: (drop (S
5398 n0) O c (CHead x3 (Flat x2) x5))).(\lambda (H16: (subst0 O v x5 x6)).(\lambda
5399 (H17: (csubst0 O v x3 x4)).(let H18 \def (eq_ind C e (\lambda (c0: C).(drop
5400 (S n0) O x0 c0)) H9 (CHead x4 (Flat x2) x6) H14) in (eq_ind_r C (CHead x4
5401 (Flat x2) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0)
5402 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
5403 (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
5404 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5405 (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5406 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
5407 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat
5408 f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5409 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
5410 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
5411 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5412 C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2)))))))
5413 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5414 (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
5415 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5416 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
5417 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))
5418 (or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 (Flat x2) x6))
5419 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
5420 (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 (Flat f0) u2)))))) (\lambda
5421 (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O
5422 (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda
5423 (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C
5424 T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C
5425 (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
5426 (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t)
5427 (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5428 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
5429 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C
5430 (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
5431 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5432 n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
5433 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
5434 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
5435 (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T
5436 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5437 (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u2))))))) (\lambda
5438 (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_:
5439 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda
5440 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5441 T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5442 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x2 x3 x4 x5 x6
5443 (refl_equal C (CHead x4 (Flat x2) x6)) (drop_drop (Flat f) n0 c (CHead x3
5444 (Flat x2) x5) H15 t) H16 H17)) e H14))))))))))) H13)) H12))))))) k H4
5445 (drop_gen_drop k x0 e t n0 H7)))))))) H3)) (\lambda (H3: (ex4_3 T C nat
5446 (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))))
5447 (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k
5448 u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
5449 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
5450 c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
5451 nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda
5452 (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_:
5453 C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
5454 C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O (CHead c k t)
5455 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
5456 (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
5457 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e0
5458 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda
5459 (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
5460 C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u))))))
5461 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
5462 O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
5463 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
5464 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5465 (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
5466 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k
5467 t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5468 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5469 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5470 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2:
5471 nat).(\lambda (H4: (eq nat (S n0) (s k x2))).(\lambda (H5: (eq C c2 (CHead x1
5472 k x0))).(\lambda (H6: (subst0 x2 v t x0)).(\lambda (H7: (csubst0 x2 v c
5473 x1)).(let H8 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1
5474 (CHead x1 k x0) H5) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x2)) \to
5475 ((drop (r k0 n0) O x1 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C
5476 T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5477 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
5478 T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1))))))
5479 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
5480 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5481 C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
5482 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
5483 k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
5484 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
5485 (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq
5486 C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda
5487 (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead
5488 e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5489 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5490 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5491 O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H9: (eq nat (S n0) (s (Bind b)
5492 x2))).(\lambda (H10: (drop (r (Bind b) n0) O x1 e)).(let H11 \def (f_equal
5493 nat nat (\lambda (e0: nat).(match e0 with [O \Rightarrow n0 | (S n1)
5494 \Rightarrow n1])) (S n0) (S x2) H9) in (let H12 \def (eq_ind_r nat x2
5495 (\lambda (n1: nat).(csubst0 n1 v c x1)) H7 n0 H11) in (let H13 \def (eq_ind_r
5496 nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H6 n0 H11) in (let H14 \def
5497 (IHn c x1 v H12 e H10) in (or4_ind (drop n0 O c e) (ex3_4 F C T T (\lambda
5498 (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0
5499 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
5500 (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
5501 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
5502 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
5503 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
5504 C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_:
5505 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
5506 (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
5507 (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f:
5508 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0
5509 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5510 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5511 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5512 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T
5513 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5514 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
5515 T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f)
5516 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5517 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
5518 C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u))))))
5519 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
5520 O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
5521 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
5522 C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
5523 T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f:
5524 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5525 n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
5526 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
5527 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
5528 (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H15: (drop n0 O c
5529 e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T
5530 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5531 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
5532 T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f)
5533 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5534 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
5535 C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u))))))
5536 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
5537 O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
5538 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
5539 C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
5540 T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f:
5541 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5542 n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
5543 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
5544 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
5545 (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e
5546 H15 t))) (\lambda (H15: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
5547 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
5548 (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O
5549 c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5550 T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda (f:
5551 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
5552 f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5553 T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
5554 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) (or4 (drop (S n0)
5555 O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
5556 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
5557 (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5558 n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
5559 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5560 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
5561 T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
5562 C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead
5563 e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5564 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
5565 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5566 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
5567 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
5568 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5569 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5570 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5571 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda
5572 (x6: T).(\lambda (H16: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H17: (drop
5573 n0 O c (CHead x4 (Flat x3) x5))).(\lambda (H18: (subst0 O v x5 x6)).(let H19
5574 \def (eq_ind C e (\lambda (c0: C).(drop n0 O x1 c0)) H10 (CHead x4 (Flat x3)
5575 x6) H16) in (eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop
5576 (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda
5577 (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f) u2))))))
5578 (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5579 n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
5580 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5581 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
5582 T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
5583 C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead
5584 e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5585 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
5586 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
5587 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
5588 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
5589 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5590 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5591 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5592 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind b) t) (CHead x4
5593 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
5594 T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2))))))
5595 (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5596 n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
5597 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5598 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
5599 T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u)))))) (\lambda (f:
5600 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
5601 (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
5602 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
5603 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5604 (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u2))))))) (\lambda
5605 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_:
5606 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda
5607 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5608 T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5609 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C
5610 T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C
5611 (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda
5612 (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
5613 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5614 T).(\lambda (u2: T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead
5615 x4 (Flat x3) x6)) (drop_drop (Bind b) n0 c (CHead x4 (Flat x3) x5) H17 t)
5616 H18)) e H16))))))))) H15)) (\lambda (H15: (ex3_4 F C C T (\lambda (f:
5617 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5618 f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5619 T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
5620 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C
5621 C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
5622 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
5623 C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_:
5624 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))
5625 (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f:
5626 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
5627 f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5628 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
5629 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
5630 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5631 C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
5632 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
5633 (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
5634 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
5635 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5636 (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
5637 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5638 (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
5639 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5640 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5641 (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda
5642 (x5: C).(\lambda (x6: T).(\lambda (H16: (eq C e (CHead x5 (Flat x3)
5643 x6))).(\lambda (H17: (drop n0 O c (CHead x4 (Flat x3) x6))).(\lambda (H18:
5644 (csubst0 O v x4 x5)).(let H19 \def (eq_ind C e (\lambda (c0: C).(drop n0 O x1
5645 c0)) H10 (CHead x5 (Flat x3) x6) H16) in (eq_ind_r C (CHead x5 (Flat x3) x6)
5646 (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T
5647 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
5648 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
5649 T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f)
5650 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5651 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
5652 C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u))))))
5653 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
5654 O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
5655 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
5656 C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
5657 T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f:
5658 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5659 n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
5660 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
5661 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
5662 (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O
5663 (CHead c (Bind b) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f:
5664 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat
5665 x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
5666 (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat
5667 f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5668 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
5669 C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
5670 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5671 (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u))))))
5672 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
5673 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5674 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
5675 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5676 (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
5677 f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5678 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5679 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
5680 (ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5681 C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f) u))))))
5682 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
5683 O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
5684 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x3 x4 x5 x6
5685 (refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop (Bind b) n0 c (CHead x4
5686 (Flat x3) x6) H17 t) H18)) e H16))))))))) H15)) (\lambda (H15: (ex4_5 F C C T
5687 T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5688 (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
5689 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1
5690 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
5691 (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda
5692 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
5693 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5694 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
5695 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5696 (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
5697 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5698 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5699 (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
5700 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
5701 T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
5702 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
5703 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5704 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
5705 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5706 f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5707 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
5708 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
5709 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5710 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
5711 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5712 (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
5713 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5714 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
5715 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
5716 (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7:
5717 T).(\lambda (H16: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H17: (drop n0 O
5718 c (CHead x4 (Flat x3) x6))).(\lambda (H18: (subst0 O v x6 x7)).(\lambda (H19:
5719 (csubst0 O v x4 x5)).(let H20 \def (eq_ind C e (\lambda (c0: C).(drop n0 O x1
5720 c0)) H10 (CHead x5 (Flat x3) x7) H16) in (eq_ind_r C (CHead x5 (Flat x3) x7)
5721 (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T
5722 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
5723 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
5724 T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f)
5725 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5726 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
5727 C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u))))))
5728 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
5729 O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
5730 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
5731 C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
5732 T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f:
5733 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5734 n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
5735 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
5736 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
5737 (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O
5738 (CHead c (Bind b) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f:
5739 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat
5740 x3) x7) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
5741 (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat
5742 f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5743 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
5744 C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2
5745 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5746 (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u))))))
5747 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
5748 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5749 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2
5750 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5751 (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
5752 f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5753 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5754 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
5755 (ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5756 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2
5757 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5758 (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
5759 f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5760 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5761 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
5762 x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat x3) x7)) (drop_drop (Bind b) n0
5763 c (CHead x4 (Flat x3) x6) H17 t) H18 H19)) e H16))))))))))) H15)) H14))))))))
5764 (\lambda (f: F).(\lambda (H9: (eq nat (S n0) (s (Flat f) x2))).(\lambda (H10:
5765 (drop (r (Flat f) n0) O x1 e)).(let H11 \def (f_equal nat nat (\lambda (e0:
5766 nat).e0) (S n0) (s (Flat f) x2) H9) in (let H12 \def (eq_ind_r nat x2
5767 (\lambda (n1: nat).(csubst0 n1 v c x1)) H7 (S n0) H11) in (let H13 \def
5768 (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H6 (S n0) H11) in
5769 (let H14 \def (H x1 v H12 e H10) in (or4_ind (drop (S n0) O c e) (ex3_4 F C T
5770 T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5771 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
5772 T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_:
5773 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5774 (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
5775 T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
5776 C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0)
5777 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
5778 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_:
5779 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
5780 f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
5781 T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda
5782 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5783 T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5784 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S
5785 n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
5786 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2))))))
5787 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5788 n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_:
5789 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5790 (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
5791 T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
5792 C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead
5793 e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5794 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
5795 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5796 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
5797 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5798 (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5799 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5800 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5801 O v e1 e2)))))))) (\lambda (H15: (drop (S n0) O c e)).(or4_intro0 (drop (S
5802 n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
5803 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2))))))
5804 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5805 n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_:
5806 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5807 (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
5808 T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
5809 C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead
5810 e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5811 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
5812 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5813 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
5814 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5815 (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5816 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5817 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5818 O v e1 e2))))))) (drop_drop (Flat f) n0 c e H15 t))) (\lambda (H15: (ex3_4 F
5819 C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq
5820 C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
5821 (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1))))))
5822 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
5823 u1 u2))))))).(ex3_4_ind F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda
5824 (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0:
5825 F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead
5826 e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5827 T).(\lambda (u2: T).(subst0 O v u1 u2))))) (or4 (drop (S n0) O (CHead c (Flat
5828 f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
5829 T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0:
5830 F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5831 (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_:
5832 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
5833 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
5834 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
5835 C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0)
5836 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
5837 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_:
5838 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
5839 f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
5840 T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0)
5841 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5842 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5843 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
5844 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6:
5845 T).(\lambda (H16: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H17: (drop (S
5846 n0) O c (CHead x4 (Flat x3) x5))).(\lambda (H18: (subst0 O v x5 x6)).(let H19
5847 \def (eq_ind C e (\lambda (c0: C).(drop (S n0) O x1 c0)) H10 (CHead x4 (Flat
5848 x3) x6) H16) in (eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4
5849 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0:
5850 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
5851 f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5852 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
5853 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
5854 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5855 C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
5856 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
5857 (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
5858 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
5859 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5860 (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
5861 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5862 (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
5863 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5864 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5865 (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Flat
5866 f) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
5867 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0
5868 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
5869 T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
5870 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5871 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
5872 C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2
5873 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5874 (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
5875 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
5876 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5877 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2
5878 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
5879 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5880 (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5881 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5882 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5883 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0:
5884 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0
5885 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
5886 T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
5887 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5888 T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6))
5889 (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x5) H17 t) H18)) e H16)))))))))
5890 H15)) (\lambda (H15: (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda
5891 (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0:
5892 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead
5893 e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5894 C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0:
5895 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5896 f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5897 T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
5898 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0)
5899 O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
5900 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2))))))
5901 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5902 n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_:
5903 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5904 (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
5905 T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
5906 C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead
5907 e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5908 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
5909 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5910 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
5911 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5912 (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5913 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5914 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5915 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda
5916 (x6: T).(\lambda (H16: (eq C e (CHead x5 (Flat x3) x6))).(\lambda (H17: (drop
5917 (S n0) O c (CHead x4 (Flat x3) x6))).(\lambda (H18: (csubst0 O v x4 x5)).(let
5918 H19 \def (eq_ind C e (\lambda (c0: C).(drop (S n0) O x1 c0)) H10 (CHead x5
5919 (Flat x3) x6) H16) in (eq_ind_r C (CHead x5 (Flat x3) x6) (\lambda (c0:
5920 C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0:
5921 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
5922 f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5923 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
5924 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
5925 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5926 C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
5927 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
5928 (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
5929 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
5930 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5931 (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
5932 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5933 (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
5934 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5935 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5936 (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat
5937 f) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
5938 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e0
5939 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
5940 T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
5941 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5942 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
5943 C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
5944 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5945 (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
5946 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
5947 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5948 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
5949 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
5950 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5951 (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5952 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5953 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5954 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_:
5955 C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
5956 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5957 (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
5958 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
5959 v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop
5960 (Flat f) n0 c (CHead x4 (Flat x3) x6) H17 t) H18)) e H16))))))))) H15))
5961 (\lambda (H15: (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda
5962 (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0)
5963 u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
5964 T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda
5965 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5966 T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5967 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C
5968 C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
5969 T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
5970 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5971 n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
5972 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5973 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5974 (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e)
5975 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
5976 (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
5977 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5978 (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5979 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
5980 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5981 f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5982 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
5983 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
5984 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5985 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2)))))))
5986 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5987 (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
5988 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5989 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
5990 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
5991 (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7:
5992 T).(\lambda (H16: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H17: (drop (S
5993 n0) O c (CHead x4 (Flat x3) x6))).(\lambda (H18: (subst0 O v x6 x7)).(\lambda
5994 (H19: (csubst0 O v x4 x5)).(let H20 \def (eq_ind C e (\lambda (c0: C).(drop
5995 (S n0) O x1 c0)) H10 (CHead x5 (Flat x3) x7) H16) in (eq_ind_r C (CHead x5
5996 (Flat x3) x7) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0)
5997 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
5998 (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
5999 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
6000 (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
6001 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
6002 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat
6003 f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
6004 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
6005 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
6006 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
6007 C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2)))))))
6008 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
6009 (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
6010 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
6011 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
6012 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))
6013 (or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x5 (Flat x3) x7))
6014 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
6015 (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0 (Flat f0) u2)))))) (\lambda
6016 (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O
6017 (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda
6018 (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C
6019 T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C
6020 (CHead x5 (Flat x3) x7) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
6021 (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t)
6022 (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
6023 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
6024 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C
6025 (CHead x5 (Flat x3) x7) (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
6026 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
6027 n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
6028 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
6029 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
6030 (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T
6031 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
6032 (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f0) u2))))))) (\lambda
6033 (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_:
6034 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda
6035 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
6036 T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
6037 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7
6038 (refl_equal C (CHead x5 (Flat x3) x7)) (drop_drop (Flat f) n0 c (CHead x4
6039 (Flat x3) x6) H17 t) H18 H19)) e H16))))))))))) H15)) H14)))))))) k H4
6040 (drop_gen_drop k x1 e x0 n0 H8)))))))))) H3)) H2))))))))))) c1)))) n).
6042 lemma csubst0_drop_lt_back:
6043 \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall
6044 (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((drop n O
6045 c2 e2) \to (or (drop n O c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n)
6046 v e1 e2)) (\lambda (e1: C).(drop n O c1 e1))))))))))))
6048 \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i)
6049 \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2)
6050 \to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C
6051 (\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O
6052 c1 e1))))))))))))) (\lambda (i: nat).(\lambda (_: (lt O i)).(\lambda (c1:
6053 C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1
6054 c2)).(\lambda (e2: C).(\lambda (H1: (drop O O c2 e2)).(eq_ind C c2 (\lambda
6055 (c: C).(or (drop O O c1 c) (ex2 C (\lambda (e1: C).(csubst0 (minus i O) v e1
6056 c)) (\lambda (e1: C).(drop O O c1 e1))))) (eq_ind nat i (\lambda (n0:
6057 nat).(or (drop O O c1 c2) (ex2 C (\lambda (e1: C).(csubst0 n0 v e1 c2))
6058 (\lambda (e1: C).(drop O O c1 e1))))) (or_intror (drop O O c1 c2) (ex2 C
6059 (\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O c1 e1)))
6060 (ex_intro2 C (\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O
6061 c1 e1)) c1 H0 (drop_refl c1))) (minus i O) (minus_n_O i)) e2 (drop_gen_refl
6062 c2 e2 H1)))))))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt
6063 n0 i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1
6064 c2) \to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C
6065 (\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O
6066 c1 e1)))))))))))))).(\lambda (i: nat).(\lambda (H: (lt (S n0) i)).(\lambda
6067 (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v
6068 c c2) \to (\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c
6069 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1:
6070 C).(drop (S n0) O c e1)))))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda
6071 (v: T).(\lambda (H0: (csubst0 i v (CSort n1) c2)).(\lambda (e2: C).(\lambda
6072 (_: (drop (S n0) O c2 e2)).(csubst0_gen_sort c2 v i n1 H0 (or (drop (S n0) O
6073 (CSort n1) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2))
6074 (\lambda (e1: C).(drop (S n0) O (CSort n1) e1))))))))))) (\lambda (c:
6075 C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to
6076 (\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c e2) (ex2 C
6077 (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop
6078 (S n0) O c e1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2:
6079 C).(\lambda (v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda
6080 (e2: C).(\lambda (H2: (drop (S n0) O c2 e2)).(let H3 \def (csubst0_gen_head k
6081 c c2 t v i H1) in (or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq
6082 nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k
6083 u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat
6084 (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3:
6085 C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j:
6086 nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
6087 C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3:
6088 C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
6089 (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
6090 C).(\lambda (j: nat).(csubst0 j v c c3))))) (or (drop (S n0) O (CHead c k t)
6091 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1:
6092 C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (H4: (ex3_2 T nat (\lambda
6093 (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
6094 nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
6095 v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k
6096 j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda
6097 (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or (drop (S n0) O (CHead c k
6098 t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda
6099 (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (x0: T).(\lambda (x1:
6100 nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k
6101 x0))).(\lambda (_: (subst0 x1 v t x0)).(let H8 \def (eq_ind C c2 (\lambda
6102 (c0: C).(drop (S n0) O c0 e2)) H2 (CHead c k x0) H6) in (let H9 \def (eq_ind
6103 nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c
6104 c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3)
6105 (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v0 e1 e3)) (\lambda (e1:
6106 C).(drop (S n0) O c e1)))))))))) H0 (s k x1) H5) in (let H10 \def (eq_ind nat
6107 i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H5) in (eq_ind_r nat (s k x1)
6108 (\lambda (n1: nat).(or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1:
6109 C).(csubst0 (minus n1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O
6110 (CHead c k t) e1))))) (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall
6111 (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3
6112 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s
6113 k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) \to
6114 ((lt (S n0) (s k0 x1)) \to ((drop (r k0 n0) O c e2) \to (or (drop (S n0) O
6115 (CHead c k0 t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0))
6116 v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k0 t) e1)))))))) (\lambda
6117 (b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b)
6118 x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0)
6119 O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1
6120 e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))).(\lambda (_: (lt (S n0)
6121 (s (Bind b) x1))).(\lambda (H13: (drop (r (Bind b) n0) O c e2)).(or_introl
6122 (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0
6123 (minus (s (Bind b) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O
6124 (CHead c (Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 H13 t)))))) (\lambda
6125 (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f)
6126 x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0)
6127 O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Flat f) x1) (S n0)) v0 e1
6128 e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))).(\lambda (_: (lt (S n0)
6129 (s (Flat f) x1))).(\lambda (H13: (drop (r (Flat f) n0) O c e2)).(or_introl
6130 (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0
6131 (minus (s (Flat f) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O
6132 (CHead c (Flat f) t) e1))) (drop_drop (Flat f) n0 c e2 H13 t)))))) k H9 H10
6133 (drop_gen_drop k c e2 x0 n0 H8)) i H5))))))))) H4)) (\lambda (H4: (ex3_2 C
6134 nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3:
6135 C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j:
6136 nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j:
6137 nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
6138 c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or (drop
6139 (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0))
6140 v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (x0:
6141 C).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C
6142 c2 (CHead x0 k t))).(\lambda (H7: (csubst0 x1 v c x0)).(let H8 \def (eq_ind C
6143 c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2 (CHead x0 k t) H6) in (let H9
6144 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0:
6145 T).((csubst0 n1 v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or
6146 (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v0 e1
6147 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) H0 (s k x1) H5) in (let
6148 H10 \def (eq_ind nat i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H5) in
6149 (eq_ind_r nat (s k x1) (\lambda (n1: nat).(or (drop (S n0) O (CHead c k t)
6150 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v e1 e2)) (\lambda
6151 (e1: C).(drop (S n0) O (CHead c k t) e1))))) (K_ind (\lambda (k0:
6152 K).(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to
6153 (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C
6154 (\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1:
6155 C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 x1)) \to ((drop (r k0
6156 n0) O x0 e2) \to (or (drop (S n0) O (CHead c k0 t) e2) (ex2 C (\lambda (e1:
6157 C).(csubst0 (minus (s k0 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0)
6158 O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: ((\forall (c3:
6159 C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e3:
6160 C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1:
6161 C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop
6162 (S n0) O c e1))))))))))).(\lambda (H12: (lt (S n0) (s (Bind b) x1))).(\lambda
6163 (H13: (drop (r (Bind b) n0) O x0 e2)).(let H_x \def (IHn x1 (lt_S_n n0 x1
6164 H12) c x0 v H7 e2 H13) in (let H14 \def H_x in (or_ind (drop n0 O c e2) (ex2
6165 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0
6166 O c e1))) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1:
6167 C).(csubst0 (minus (s (Bind b) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop
6168 (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (H15: (drop n0 O c
6169 e2)).(or_introl (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1:
6170 C).(csubst0 (minus (s (Bind b) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop
6171 (S n0) O (CHead c (Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 H15 t)))
6172 (\lambda (H15: (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2))
6173 (\lambda (e1: C).(drop n0 O c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0
6174 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O c e1)) (or (drop (S n0) O
6175 (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b)
6176 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t)
6177 e1)))) (\lambda (x: C).(\lambda (H16: (csubst0 (minus x1 n0) v x
6178 e2)).(\lambda (H17: (drop n0 O c x)).(or_intror (drop (S n0) O (CHead c (Bind
6179 b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x1) (S n0)) v
6180 e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (ex_intro2
6181 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x1) (S n0)) v e1 e2)) (\lambda
6182 (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1)) x H16 (drop_drop (Bind b) n0
6183 c x H17 t)))))) H15)) H14))))))) (\lambda (f: F).(\lambda (H11: ((\forall
6184 (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e3:
6185 C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1:
6186 C).(csubst0 (minus (s (Flat f) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop
6187 (S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x1))).(\lambda
6188 (H13: (drop (r (Flat f) n0) O x0 e2)).(let H_x \def (H11 x0 v H7 e2 H13) in
6189 (let H14 \def H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1:
6190 C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c
6191 e1))) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1:
6192 C).(csubst0 (minus (s (Flat f) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop
6193 (S n0) O (CHead c (Flat f) t) e1)))) (\lambda (H15: (drop (S n0) O c
6194 e2)).(or_introl (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1:
6195 C).(csubst0 (minus (s (Flat f) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop
6196 (S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat f) n0 c e2 H15 t)))
6197 (\lambda (H15: (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2))
6198 (\lambda (e1: C).(drop (S n0) O c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0
6199 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1)) (or (drop
6200 (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s
6201 (Flat f) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat
6202 f) t) e1)))) (\lambda (x: C).(\lambda (H16: (csubst0 (minus x1 (S n0)) v x
6203 e2)).(\lambda (H17: (drop (S n0) O c x)).(or_intror (drop (S n0) O (CHead c
6204 (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Flat f) x1) (S
6205 n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)))
6206 (ex_intro2 C (\lambda (e1: C).(csubst0 (minus (s (Flat f) x1) (S n0)) v e1
6207 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x H16
6208 (drop_drop (Flat f) n0 c x H17 t)))))) H15)) H14))))))) k H9 H10
6209 (drop_gen_drop k x0 e2 t n0 H8)) i H5))))))))) H4)) (\lambda (H4: (ex4_3 T C
6210 nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))))
6211 (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k
6212 u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
6213 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
6214 c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
6215 nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
6216 nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
6217 (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
6218 nat).(csubst0 j v c c3)))) (or (drop (S n0) O (CHead c k t) e2) (ex2 C
6219 (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop
6220 (S n0) O (CHead c k t) e1)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2:
6221 nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: (eq C c2 (CHead x1 k
6222 x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: (csubst0 x2 v c
6223 x1)).(let H9 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2
6224 (CHead x1 k x0) H6) in (let H10 \def (eq_ind nat i (\lambda (n1:
6225 nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall
6226 (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda
6227 (e1: C).(csubst0 (minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O
6228 c e1)))))))))) H0 (s k x2) H5) in (let H11 \def (eq_ind nat i (\lambda (n1:
6229 nat).(lt (S n0) n1)) H (s k x2) H5) in (eq_ind_r nat (s k x2) (\lambda (n1:
6230 nat).(or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0
6231 (minus n1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t)
6232 e1))))) (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0
6233 (s k0 x2) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop
6234 (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s k0 x2) (S n0)) v0
6235 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0
6236 x2)) \to ((drop (r k0 n0) O x1 e2) \to (or (drop (S n0) O (CHead c k0 t) e2)
6237 (ex2 C (\lambda (e1: C).(csubst0 (minus (s k0 x2) (S n0)) v e1 e2)) (\lambda
6238 (e1: C).(drop (S n0) O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_:
6239 ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to
6240 (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C
6241 (\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v0 e1 e3)) (\lambda
6242 (e1: C).(drop (S n0) O c e1))))))))))).(\lambda (H13: (lt (S n0) (s (Bind b)
6243 x2))).(\lambda (H14: (drop (r (Bind b) n0) O x1 e2)).(let H_x \def (IHn x2
6244 (lt_S_n n0 x2 H13) c x1 v H8 e2 H14) in (let H15 \def H_x in (or_ind (drop n0
6245 O c e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda
6246 (e1: C).(drop n0 O c e1))) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C
6247 (\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v e1 e2)) (\lambda
6248 (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (H16: (drop n0 O
6249 c e2)).(or_introl (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda
6250 (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v e1 e2)) (\lambda (e1:
6251 C).(drop (S n0) O (CHead c (Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 H16
6252 t))) (\lambda (H16: (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2))
6253 (\lambda (e1: C).(drop n0 O c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0
6254 (minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O c e1)) (or (drop (S n0) O
6255 (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b)
6256 x2) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t)
6257 e1)))) (\lambda (x: C).(\lambda (H17: (csubst0 (minus x2 n0) v x
6258 e2)).(\lambda (H18: (drop n0 O c x)).(or_intror (drop (S n0) O (CHead c (Bind
6259 b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v
6260 e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (ex_intro2
6261 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v e1 e2)) (\lambda
6262 (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1)) x H17 (drop_drop (Bind b) n0
6263 c x H18 t)))))) H16)) H15))))))) (\lambda (f: F).(\lambda (H12: ((\forall
6264 (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e3:
6265 C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1:
6266 C).(csubst0 (minus (s (Flat f) x2) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop
6267 (S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x2))).(\lambda
6268 (H14: (drop (r (Flat f) n0) O x1 e2)).(let H_x \def (H12 x1 v H8 e2 H14) in
6269 (let H15 \def H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1:
6270 C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c
6271 e1))) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1:
6272 C).(csubst0 (minus (s (Flat f) x2) (S n0)) v e1 e2)) (\lambda (e1: C).(drop
6273 (S n0) O (CHead c (Flat f) t) e1)))) (\lambda (H16: (drop (S n0) O c
6274 e2)).(or_introl (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1:
6275 C).(csubst0 (minus (s (Flat f) x2) (S n0)) v e1 e2)) (\lambda (e1: C).(drop
6276 (S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat f) n0 c e2 H16 t)))
6277 (\lambda (H16: (ex2 C (\lambda (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2))
6278 (\lambda (e1: C).(drop (S n0) O c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0
6279 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1)) (or (drop
6280 (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s
6281 (Flat f) x2) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat
6282 f) t) e1)))) (\lambda (x: C).(\lambda (H17: (csubst0 (minus x2 (S n0)) v x
6283 e2)).(\lambda (H18: (drop (S n0) O c x)).(or_intror (drop (S n0) O (CHead c
6284 (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Flat f) x2) (S
6285 n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)))
6286 (ex_intro2 C (\lambda (e1: C).(csubst0 (minus (s (Flat f) x2) (S n0)) v e1
6287 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x H17
6288 (drop_drop (Flat f) n0 c x H18 t)))))) H16)) H15))))))) k H10 H11
6289 (drop_gen_drop k x1 e2 x0 n0 H9)) i H5))))))))))) H4)) H3))))))))))) c1))))))
projects / helm.git / blob