1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "Basic-1/csubst0/fwd.ma".
19 include "Basic-1/drop/fwd.ma".
21 include "Basic-1/s/props.ma".
23 theorem csubst0_drop_gt:
24 \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall
25 (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O
26 c1 e) \to (drop n O c2 e)))))))))
28 \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0)
29 \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2)
30 \to (\forall (e: C).((drop n0 O c1 e) \to (drop n0 O c2 e)))))))))) (\lambda
31 (i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda
32 (v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O
33 O c1 e)).(lt_x_O i H (drop O O c2 e)))))))))) (\lambda (n0: nat).(\lambda (H:
34 ((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall
35 (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (drop
36 n0 O c2 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda
37 (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v
38 c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))
39 (\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v
40 (CSort n1) c2)).(\lambda (e: C).(\lambda (H2: (drop (S n0) O (CSort n1)
41 e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (drop (S n0)
42 O c2 e) (\lambda (H3: (eq C e (CSort n1))).(\lambda (H4: (eq nat (S n0)
43 O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(drop
44 (S n0) O c2 c)) (let H6 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee
45 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
46 \Rightarrow True])) I O H4) in (False_ind (drop (S n0) O c2 (CSort n1)) H6))
47 e H3)))) (drop_gen_sort n1 (S n0) O e H2)))))))) (\lambda (c: C).(\lambda
48 (H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e:
49 C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (k:
50 K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H2: (csubst0 i
51 v (CHead c k t) c2)).(\lambda (e: C).(\lambda (H3: (drop (S n0) O (CHead c k
52 t) e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k
53 j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda
54 (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_:
55 C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_:
56 nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
57 v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
58 nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
59 nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
60 (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
61 nat).(csubst0 j v c c3))))) (drop (S n0) O c2 e) (\lambda (H4: (ex3_2 T nat
62 (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2:
63 T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
64 nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j:
65 nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead
66 c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (drop (S
67 n0) O c2 e) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k
68 x1))).(\lambda (H6: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t
69 x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(drop (S n0) O c0 e)) (let
70 H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0:
71 T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop
72 (S n0) O c3 e0))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda
73 (n1: nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).((drop
74 (r k0 n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1)
75 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
76 e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead c k0 x0)
77 e))))) (\lambda (b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda
78 (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to
79 (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
80 e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 c
81 e H10 x0))))) (\lambda (f: F).(\lambda (H10: (drop (r (Flat f) n0) O c
82 e)).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x1)
83 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
84 e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O)
85 (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))
86 (drop (S n0) O (CHead c (Flat f) x0) e) (\lambda (_: (eq nat x1
87 O)).(drop_drop (Flat f) n0 c e H10 x0)) (\lambda (H13: (ex2 nat (\lambda (m:
88 nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda
89 (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O
90 (CHead c (Flat f) x0) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S
91 x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e H10 x0)))) H13))
92 (lt_gen_xS x1 n0 H12)))))) k (drop_gen_drop k c e t n0 H3) H8 H9))) c2
93 H6)))))) H4)) (\lambda (H4: (ex3_2 C nat (\lambda (_: C).(\lambda (j:
94 nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
95 c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
96 c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k
97 j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
98 (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S n0) O c2 e) (\lambda
99 (x0: C).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6:
100 (eq C c2 (CHead x0 k t))).(\lambda (H7: (csubst0 x1 v c x0)).(eq_ind_r C
101 (CHead x0 k t) (\lambda (c0: C).(drop (S n0) O c0 e)) (let H8 \def (eq_ind
102 nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c
103 c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
104 e0))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda (n1:
105 nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).((drop (r k0
106 n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c
107 c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
108 e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead x0 k0 t)
109 e))))) (\lambda (b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda
110 (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to
111 (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
112 e0)))))))).(\lambda (H12: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0
113 x0 e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H10) t))))) (\lambda (f:
114 F).(\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c3:
115 C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0:
116 C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H12: (lt
117 (s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq
118 nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead x0 (Flat
119 f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7
120 e H10) t)) (\lambda (H13: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m)))
121 (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S
122 m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead x0 (Flat f) t) e)
123 (\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x
124 n0)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7 e H10) t)))) H13)) (lt_gen_xS
125 x1 n0 H12)))))) k (drop_gen_drop k c e t n0 H3) H8 H9))) c2 H6)))))) H4))
126 (\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
127 nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
128 nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
129 (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
130 nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_:
131 C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3:
132 C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
133 (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
134 C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O c2 e) (\lambda (x0:
135 T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H5: (eq nat i (s k
136 x2))).(\lambda (H6: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t
137 x0)).(\lambda (H8: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda
138 (c0: C).(drop (S n0) O c0 e)) (let H9 \def (eq_ind nat i (\lambda (n1:
139 nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall
140 (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0))))))) H1 (s k x2) H5)
141 in (let H10 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2)
142 H5) in (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall (c3:
143 C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall (e0: C).((drop
144 (S n0) O c e0) \to (drop (S n0) O c3 e0))))))) \to ((lt (s k0 x2) (S n0)) \to
145 (drop (S n0) O (CHead x1 k0 x0) e))))) (\lambda (b: B).(\lambda (H11: (drop
146 (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall (v0:
147 T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c
148 e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H13: (lt (s (Bind b) x2) (S
149 n0))).(drop_drop (Bind b) n0 x1 e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H11)
150 x0))))) (\lambda (f: F).(\lambda (H11: (drop (r (Flat f) n0) O c e)).(\lambda
151 (H12: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3)
152 \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
153 e0)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S n0))).(or_ind (eq nat x2 O)
154 (ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))
155 (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (_: (eq nat x2
156 O)).(drop_drop (Flat f) n0 x1 e (H12 x1 v H8 e H11) x0)) (\lambda (H14: (ex2
157 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m
158 n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m:
159 nat).(lt m n0)) (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (x:
160 nat).(\lambda (_: (eq nat x2 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat
161 f) n0 x1 e (H12 x1 v H8 e H11) x0)))) H14)) (lt_gen_xS x2 n0 H13)))))) k
162 (drop_gen_drop k c e t n0 H3) H9 H10))) c2 H6)))))))) H4)) (csubst0_gen_head
163 k c c2 t v i H2))))))))))) c1)))))) n).
168 theorem csubst0_drop_gt_back:
169 \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall
170 (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O
171 c2 e) \to (drop n O c1 e)))))))))
173 \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0)
174 \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2)
175 \to (\forall (e: C).((drop n0 O c2 e) \to (drop n0 O c1 e)))))))))) (\lambda
176 (i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda
177 (v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O
178 O c2 e)).(lt_x_O i H (drop O O c1 e)))))))))) (\lambda (n0: nat).(\lambda (H:
179 ((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall
180 (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (drop
181 n0 O c1 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda
182 (c1: C).(C_ind (\lambda (c: C).(\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 e)))))))
184 (\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H1: (csubst0 i
185 v (CSort n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2
186 e)).(csubst0_gen_sort c2 v i n1 H1 (drop (S n0) O (CSort n1) e))))))))
187 (\lambda (c: C).(\lambda (H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v
188 c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c
189 e)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v:
190 T).(\lambda (H2: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda
191 (H3: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j:
192 nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead
193 c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C
194 nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3:
195 C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j:
196 nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
197 C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3:
198 C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
199 (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
200 C).(\lambda (j: nat).(csubst0 j v c c3))))) (drop (S n0) O (CHead c k t) e)
201 (\lambda (H4: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k
202 j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda
203 (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_:
204 T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
205 nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
206 v t u2))) (drop (S n0) O (CHead c k t) e) (\lambda (x0: T).(\lambda (x1:
207 nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k
208 x0))).(\lambda (_: (subst0 x1 v t x0)).(let H8 \def (eq_ind C c2 (\lambda
209 (c0: C).(drop (S n0) O c0 e)) H3 (CHead c k x0) H6) in (let H9 \def (eq_ind
210 nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c
211 c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c
212 e0))))))) H1 (s k x1) H5) in (let H10 \def (eq_ind nat i (\lambda (n1:
213 nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).(((\forall
214 (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall (e0:
215 C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) \to ((lt (s k0 x1)
216 (S n0)) \to ((drop (r k0 n0) O c e) \to (drop (S n0) O (CHead c k0 t) e)))))
217 (\lambda (b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s
218 (Bind b) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop
219 (S n0) O c e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(\lambda
220 (H13: (drop (r (Bind b) n0) O c e)).(drop_drop (Bind b) n0 c e H13 t)))))
221 (\lambda (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s
222 (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop
223 (S n0) O c e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(\lambda
224 (H13: (drop (r (Flat f) n0) O c e)).(or_ind (eq nat x1 O) (ex2 nat (\lambda
225 (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O
226 (CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c
227 e H13 t)) (\lambda (H14: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m)))
228 (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S
229 m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) t) e)
230 (\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x
231 n0)).(drop_drop (Flat f) n0 c e H13 t)))) H14)) (lt_gen_xS x1 n0 H12)))))) k
232 H9 H10 (drop_gen_drop k c e x0 n0 H8)))))))))) H4)) (\lambda (H4: (ex3_2 C
233 nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3:
234 C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j:
235 nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j:
236 nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
237 c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S
238 n0) O (CHead c k t) e) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H5: (eq
239 nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead x0 k t))).(\lambda (H7:
240 (csubst0 x1 v c x0)).(let H8 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0)
241 O c0 e)) H3 (CHead x0 k t) H6) in (let H9 \def (eq_ind nat i (\lambda (n1:
242 nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall
243 (e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x1) H5)
244 in (let H10 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x1)
245 H5) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0
246 (s k0 x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S
247 n0) O c e0))))))) \to ((lt (s k0 x1) (S n0)) \to ((drop (r k0 n0) O x0 e) \to
248 (drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall
249 (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0:
250 C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H12: (lt
251 (s (Bind b) x1) (S n0))).(\lambda (H13: (drop (r (Bind b) n0) O x0
252 e)).(drop_drop (Bind b) n0 c e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H13)
253 t))))) (\lambda (f: F).(\lambda (H11: ((\forall (c3: C).(\forall (v0:
254 T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3
255 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S
256 n0))).(\lambda (H13: (drop (r (Flat f) n0) O x0 e)).(or_ind (eq nat x1 O)
257 (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))
258 (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop
259 (Flat f) n0 c e (H11 x0 v H7 e H13) t)) (\lambda (H14: (ex2 nat (\lambda (m:
260 nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda
261 (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O
262 (CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S
263 x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H11 x0 v H7 e H13)
264 t)))) H14)) (lt_gen_xS x1 n0 H12)))))) k H9 H10 (drop_gen_drop k x0 e t n0
265 H8)))))))))) H4)) (\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
266 C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3:
267 C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
268 (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
269 C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_:
270 T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2:
271 T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
272 (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
273 T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O
274 (CHead c k t) e) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2:
275 nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: (eq C c2 (CHead x1 k
276 x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: (csubst0 x2 v c
277 x1)).(let H9 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H3
278 (CHead x1 k x0) H6) in (let H10 \def (eq_ind nat i (\lambda (n1:
279 nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall
280 (e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x2) H5)
281 in (let H11 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2)
282 H5) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0
283 (s k0 x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S
284 n0) O c e0))))))) \to ((lt (s k0 x2) (S n0)) \to ((drop (r k0 n0) O x1 e) \to
285 (drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall
286 (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0:
287 C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt
288 (s (Bind b) x2) (S n0))).(\lambda (H14: (drop (r (Bind b) n0) O x1
289 e)).(drop_drop (Bind b) n0 c e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H14)
290 t))))) (\lambda (f: F).(\lambda (H12: ((\forall (c3: C).(\forall (v0:
291 T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3
292 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S
293 n0))).(\lambda (H14: (drop (r (Flat f) n0) O x1 e)).(or_ind (eq nat x2 O)
294 (ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))
295 (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x2 O)).(drop_drop
296 (Flat f) n0 c e (H12 x1 v H8 e H14) t)) (\lambda (H15: (ex2 nat (\lambda (m:
297 nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda
298 (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O
299 (CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x2 (S
300 x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H12 x1 v H8 e H14)
301 t)))) H15)) (lt_gen_xS x2 n0 H13)))))) k H10 H11 (drop_gen_drop k x1 e x0 n0
302 H9)))))))))))) H4)) (csubst0_gen_head k c c2 t v i H2))))))))))) c1)))))) n).
307 theorem csubst0_drop_lt:
308 \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall
309 (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O
310 c1 e) \to (or4 (drop n O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
311 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k:
312 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k
313 w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
314 T).(subst0 (minus i (s k n)) v u w)))))) (ex3_4 K C C T (\lambda (k:
315 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
316 u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
317 T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
318 (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (ex4_5 K C C
319 T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
320 (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda
321 (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w)))))))
322 (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
323 T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1:
324 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k
325 n)) v e1 e2))))))))))))))))
327 \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i)
328 \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2)
329 \to (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T
330 T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
331 (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
332 (w: T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
333 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w))))))
334 (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
335 T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
336 C).(\lambda (u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
337 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1
338 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
339 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
340 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0
341 O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
342 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w))))))
343 (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
344 (_: T).(csubst0 (minus i (s k n0)) v e1 e2))))))))))))))))) (\lambda (i:
345 nat).(\lambda (_: (lt O i)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v:
346 T).(\lambda (H0: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (H1: (drop O O
347 c1 e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 K C T T
348 (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c
349 (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
350 (w: T).(drop O O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
351 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w))))))
352 (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
353 T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
354 C).(\lambda (u: T).(drop O O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
355 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k O)) v e1
356 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
357 C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
358 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
359 c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
360 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w))))))
361 (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
362 (_: T).(csubst0 (minus i (s k O)) v e1 e2))))))))) (csubst0_ind (\lambda (n0:
363 nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).(or4 (drop O O c0 c)
364 (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
365 T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
366 T).(\lambda (w: T).(drop O O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda
367 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w))))))
368 (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
369 T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
370 C).(\lambda (u: T).(drop O O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
371 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n0 (s k O)) t e1
372 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
373 C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
374 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
375 c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
376 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w))))))
377 (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
378 (_: T).(csubst0 (minus n0 (s k O)) t e1 e2)))))))))))) (\lambda (k:
379 K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2:
380 T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(let H3 \def (eq_ind_r
381 nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 (minus (s k i0) (s k O))
382 (s_arith0 k i0)) in (or4_intro1 (drop O O (CHead c k u2) (CHead c k u1))
383 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
384 T).(eq C (CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
385 C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0
386 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
387 T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda
388 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k u1)
389 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
390 C).(\lambda (u: T).(drop O O (CHead c k u2) (CHead e2 k0 u)))))) (\lambda
391 (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s
392 k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda
393 (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k u1)
394 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
395 C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e2 k0
396 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
397 T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda
398 (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
399 T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C T T
400 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
401 (CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
402 C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0
403 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
404 T).(subst0 (minus (s k i0) (s k0 O)) v0 u w))))) k c u1 u2 (refl_equal C
405 (CHead c k u1)) (drop_refl (CHead c k u2)) H3)))))))))) (\lambda (k:
406 K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0:
407 T).(\lambda (H2: (csubst0 i0 v0 c3 c4)).(\lambda (H3: (or4 (drop O O c4 c3)
408 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
409 T).(eq C c3 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
410 (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0:
411 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0
412 O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
413 (_: C).(\lambda (u: T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0:
414 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0
415 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
416 T).(csubst0 (minus i0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
417 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
418 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
419 C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w)))))))
420 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
421 T).(subst0 (minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1:
422 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0
423 O)) v0 e1 e2))))))))).(\lambda (u: T).(let H4 \def (eq_ind_r nat i0 (\lambda
424 (n0: nat).(csubst0 n0 v0 c3 c4)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0))
425 in (let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(or4 (drop O O c4 c3)
426 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda
427 (_: T).(eq C c3 (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0:
428 C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda
429 (k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n0 (s
430 k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
431 C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 k0 u0)))))) (\lambda
432 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O c4 (CHead
433 e2 k0 u0)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
434 (_: T).(csubst0 (minus n0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda
435 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
436 C c3 (CHead e1 k0 u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
437 C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w)))))))
438 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda
439 (w: T).(subst0 (minus n0 (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda
440 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n0
441 (s k0 O)) v0 e1 e2))))))))) H3 (minus (s k i0) (s k O)) (s_arith0 k i0)) in
442 (or4_intro2 (drop O O (CHead c4 k u) (CHead c3 k u)) (ex3_4 K C T T (\lambda
443 (k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k
444 u) (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
445 T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e0 k0 w)))))) (\lambda
446 (k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k
447 i0) (s k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
448 C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0
449 u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0:
450 T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda
451 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O))
452 v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda
453 (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k u) (CHead e1 k0
454 u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
455 T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e2 k0 w))))))) (\lambda
456 (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
457 T).(subst0 (minus (s k i0) (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda
458 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k
459 i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda
460 (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0
461 u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0:
462 T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda
463 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O))
464 v0 e1 e2))))) k c3 c4 u (refl_equal C (CHead c3 k u)) (drop_refl (CHead c4 k
465 u)) H4)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0:
466 T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1
467 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i0 v0 c3
468 c4)).(\lambda (_: (or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k0:
469 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k0
470 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
471 T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
472 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0 O)) v0 u w))))))
473 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
474 T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
475 (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0 u)))))) (\lambda (k0:
476 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k0
477 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
478 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k0
479 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
480 T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) (\lambda (k0:
481 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
482 (minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
483 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0 O)) v0 e1
484 e2))))))))).(let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1
485 u2)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (let H6 \def (eq_ind_r
486 nat i0 (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 (minus (s k i0) (s k O))
487 (s_arith0 k i0)) in (or4_intro3 (drop O O (CHead c4 k u2) (CHead c3 k u1))
488 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
489 T).(eq C (CHead c3 k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
490 C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e0 k0
491 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
492 T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda
493 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k
494 u1) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
495 C).(\lambda (u: T).(drop O O (CHead c4 k u2) (CHead e2 k0 u)))))) (\lambda
496 (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s
497 k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda
498 (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k u1)
499 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
500 C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e2 k0
501 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
502 T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda
503 (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
504 T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex4_5_intro K C C T T
505 (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
506 (_: T).(eq C (CHead c3 k u1) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
507 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k
508 u2) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
509 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u
510 w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
511 T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2)))))) k c3 c4
512 u1 u2 (refl_equal C (CHead c3 k u1)) (drop_refl (CHead c4 k u2)) H5
513 H6)))))))))))))) i v c1 c2 H0) e (drop_gen_refl c1 e H1)))))))))) (\lambda
514 (n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt n0 i) \to (\forall (c1:
515 C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e:
516 C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T T (\lambda (k:
517 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k
518 u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
519 T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda
520 (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (ex3_4 K C C T
521 (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
522 (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda
523 (u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
524 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 e2))))))
525 (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda
526 (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda
527 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead
528 e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
529 T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (\lambda (k:
530 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
531 (minus i (s k n0)) v e1 e2)))))))))))))))))).(\lambda (i: nat).(\lambda (H:
532 (lt (S n0) i)).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
533 C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c
534 e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
535 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k:
536 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
537 e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
538 T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k:
539 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
540 u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
541 T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
542 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1
543 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
544 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
545 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
546 n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
547 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w))))))
548 (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
549 (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))))))))))) (\lambda (n1:
550 nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v (CSort n1)
551 c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CSort n1) e)).(and3_ind
552 (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop (S n0) O c2 e)
553 (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
554 T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
555 T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k:
556 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S
557 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda
558 (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda
559 (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u))))))
560 (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
561 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k:
562 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
563 (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
564 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w)))))))
565 (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
566 T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
567 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S
568 n0))) v e1 e2)))))))) (\lambda (H2: (eq C e (CSort n1))).(\lambda (H3: (eq
569 nat (S n0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c:
570 C).(or4 (drop (S n0) O c2 c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
571 C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k:
572 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
573 e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
574 T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k:
575 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k
576 u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
577 T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
578 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1
579 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
580 C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
581 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
582 n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
583 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w))))))
584 (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
585 (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2))))))))) (let H5 \def (eq_ind
586 nat (S n0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop)
587 with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind
588 (or4 (drop (S n0) O c2 (CSort n1)) (ex3_4 K C T T (\lambda (k: K).(\lambda
589 (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e0 k u))))))
590 (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
591 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u:
592 T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T
593 (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort
594 n1) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
595 C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k:
596 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k
597 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1:
598 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e1
599 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
600 T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k:
601 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
602 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
603 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1
604 e2)))))))) H5)) e H2)))) (drop_gen_sort n1 (S n0) O e H1)))))))) (\lambda (c:
605 C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to
606 (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C
607 T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
608 (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
609 (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
610 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w))))))
611 (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
612 T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
613 C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k:
614 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k
615 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1:
616 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k
617 u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
618 T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k:
619 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
620 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
621 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1
622 e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda
623 (v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda
624 (H2: (drop (S n0) O (CHead c k t) e)).(or3_ind (ex3_2 T nat (\lambda (_:
625 T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
626 nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
627 v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k
628 j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
629 (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_:
630 T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2:
631 T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
632 (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
633 T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S
634 n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
635 T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
636 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0
637 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
638 T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0:
639 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
640 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
641 T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
642 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1
643 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
644 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
645 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
646 T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
647 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
648 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
649 (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))
650 (\lambda (H3: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k
651 j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda
652 (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_:
653 T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
654 nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
655 v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda
656 (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda
657 (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2
658 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
659 T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T
660 (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
661 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
662 C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0:
663 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0
664 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
665 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0
666 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
667 T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0:
668 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
669 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
670 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0
671 (S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4:
672 (eq nat i (s k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (_:
673 (subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S
674 n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
675 T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
676 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0
677 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
678 T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0:
679 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
680 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
681 T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
682 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1
683 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
684 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
685 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
686 T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
687 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
688 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
689 (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let
690 H7 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0:
691 T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4
692 (drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1:
693 C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
694 (k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3
695 (CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
696 T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C
697 T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0
698 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
699 C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0:
700 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0
701 (S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
702 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0
703 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
704 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0:
705 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
706 (minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1:
707 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
708 (S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H4) in (let H8 \def (eq_ind nat i
709 (\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1)
710 (\lambda (n1: nat).(or4 (drop (S n0) O (CHead c k x0) e) (ex3_4 K C T T
711 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
712 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
713 T).(\lambda (w: T).(drop (S n0) O (CHead c k x0) (CHead e0 k0 w))))))
714 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
715 (minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda
716 (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda
717 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
718 (CHead c k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
719 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1
720 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
721 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
722 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
723 T).(drop (S n0) O (CHead c k x0) (CHead e2 k0 w))))))) (\lambda (k0:
724 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
725 (minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
726 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
727 (S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to
728 (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall
729 (e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
730 (\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
731 (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_:
732 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1:
733 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1)
734 (s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1:
735 C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda
736 (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
737 (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2:
738 C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2))))))
739 (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda
740 (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1:
741 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
742 n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_:
743 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0
744 u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
745 T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1
746 e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c
747 k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u:
748 T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda
749 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead
750 e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
751 T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T
752 (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
753 (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
754 C).(\lambda (u: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 k1 u))))))
755 (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
756 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1:
757 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
758 (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
759 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2
760 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
761 T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w))))))
762 (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
763 (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda
764 (b: B).(\lambda (H9: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall
765 (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0:
766 C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
767 (\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
768 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
769 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
770 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
771 x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
772 C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
773 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
774 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
775 C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1
776 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
777 C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
778 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
779 T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
780 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
781 x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
782 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
783 (S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Bind b)
784 x1))).(or4_intro0 (drop (S n0) O (CHead c (Bind b) x0) e) (ex3_4 K C T T
785 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
786 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
787 T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e0 k0 w))))))
788 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
789 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
790 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
791 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
792 T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0:
793 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
794 b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
795 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
796 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
797 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0)
798 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
799 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
800 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
801 (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
802 e2))))))) (drop_drop (Bind b) n0 c e H9 x0)))))) (\lambda (f: F).(\lambda
803 (H9: (drop (r (Flat f) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall
804 (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0)
805 O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0:
806 K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0
807 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w:
808 T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_:
809 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
810 n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
811 (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0:
812 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead
813 e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
814 T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C
815 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
816 T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
817 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3
818 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
819 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
820 n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
821 C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S
822 n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f)
823 x1))).(or4_intro0 (drop (S n0) O (CHead c (Flat f) x0) e) (ex3_4 K C T T
824 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
825 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
826 T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e0 k0 w))))))
827 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
828 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
829 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
830 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
831 T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0:
832 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
833 f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
834 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
835 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
836 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0)
837 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
838 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
839 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
840 (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
841 e2))))))) (drop_drop (Flat f) n0 c e H9 x0)))))) k (drop_gen_drop k c e t n0
842 H2) H7 H8) i H4))) c2 H5)))))) H3)) (\lambda (H3: (ex3_2 C nat (\lambda (_:
843 C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_:
844 nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
845 v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k
846 j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
847 (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O c2 e)
848 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
849 T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
850 (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0:
851 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
852 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
853 (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
854 (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0
855 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
856 T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
857 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
858 C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
859 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w)))))))
860 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
861 T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
862 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0
863 (S n0))) v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4:
864 (eq nat i (s k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6:
865 (csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop
866 (S n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
867 T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
868 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0
869 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
870 T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0:
871 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
872 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
873 T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
874 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1
875 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
876 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
877 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
878 T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
879 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
880 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
881 (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let
882 H7 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0:
883 T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4
884 (drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1:
885 C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
886 (k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3
887 (CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
888 T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C
889 T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0
890 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
891 C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0:
892 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0
893 (S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
894 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0
895 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
896 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0:
897 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
898 (minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1:
899 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
900 (S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H4) in (let H8 \def (eq_ind nat i
901 (\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1)
902 (\lambda (n1: nat).(or4 (drop (S n0) O (CHead x0 k t) e) (ex3_4 K C T T
903 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
904 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
905 T).(\lambda (w: T).(drop (S n0) O (CHead x0 k t) (CHead e0 k0 w))))))
906 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
907 (minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda
908 (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda
909 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
910 (CHead x0 k t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
911 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1
912 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
913 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
914 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
915 T).(drop (S n0) O (CHead x0 k t) (CHead e2 k0 w))))))) (\lambda (k0:
916 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
917 (minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
918 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
919 (S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to
920 (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall
921 (e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
922 (\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
923 (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_:
924 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1:
925 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1)
926 (s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1:
927 C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda
928 (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
929 (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2:
930 C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2))))))
931 (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda
932 (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1:
933 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
934 n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_:
935 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0
936 u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
937 T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1
938 e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead x0
939 k0 t) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u:
940 T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda
941 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead
942 e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
943 T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T
944 (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
945 (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
946 C).(\lambda (u: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 k1 u))))))
947 (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
948 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1:
949 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
950 (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
951 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2
952 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
953 T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w))))))
954 (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
955 (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda
956 (b: B).(\lambda (H9: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall
957 (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0:
958 C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
959 (\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
960 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
961 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
962 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
963 x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
964 C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
965 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
966 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
967 C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1
968 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
969 C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
970 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
971 T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
972 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
973 x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
974 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
975 (S n0))) v0 e1 e2))))))))))))))).(\lambda (H11: (lt (S n0) (s (Bind b)
976 x1))).(let H12 \def (IHn x1 (le_S_n (S n0) x1 H11) c x0 v H6 e H9) in
977 (or4_ind (drop n0 O x0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
978 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
979 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0
980 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
981 T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (ex3_4 K C C T (\lambda (k0:
982 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
983 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
984 T).(drop n0 O x0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
985 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1
986 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
987 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
988 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
989 T).(drop n0 O x0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
990 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0
991 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
992 (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1 e2))))))) (or4
993 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T (\lambda (k0:
994 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
995 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
996 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0:
997 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
998 x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
999 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
1000 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
1001 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1002 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
1003 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1004 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
1005 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1006 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0:
1007 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1008 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1009 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
1010 (Bind b) x1) (s k0 (S n0))) v e1 e2)))))))) (\lambda (H13: (drop n0 O x0
1011 e)).(or4_intro0 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T
1012 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1013 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1014 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w))))))
1015 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1016 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1017 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
1018 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1019 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0:
1020 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
1021 b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
1022 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1023 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1024 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
1025 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1026 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
1027 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1028 (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
1029 e2))))))) (drop_drop (Bind b) n0 x0 e H13 t))) (\lambda (H13: (ex3_4 K C T T
1030 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1031 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1032 T).(\lambda (w: T).(drop n0 O x0 (CHead e0 k0 w)))))) (\lambda (k0:
1033 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0
1034 n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k0: K).(\lambda (e0:
1035 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
1036 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0
1037 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1038 T).(subst0 (minus x1 (s k0 n0)) v u w))))) (or4 (drop (S n0) O (CHead x0
1039 (Bind b) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
1040 T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
1041 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
1042 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
1043 T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w))))))
1044 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1045 T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1046 (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0
1047 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1048 T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
1049 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1050 T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
1051 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
1052 x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1053 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1054 x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1055 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
1056 (S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4:
1057 T).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3 x2 x4))).(\lambda (H15:
1058 (drop n0 O x0 (CHead x3 x2 x5))).(\lambda (H16: (subst0 (minus x1 (s x2 n0))
1059 v x4 x5)).(eq_ind_r C (CHead x3 x2 x4) (\lambda (c0: C).(or4 (drop (S n0) O
1060 (CHead x0 (Bind b) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
1061 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda
1062 (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
1063 (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
1064 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
1065 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1066 (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0:
1067 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
1068 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1069 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
1070 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1071 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
1072 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1073 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0:
1074 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1075 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1076 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
1077 (Bind b) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O
1078 (CHead x0 (Bind b) t) (CHead x3 x2 x4)) (ex3_4 K C T T (\lambda (k0:
1079 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4)
1080 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1081 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w))))))
1082 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1083 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1084 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2
1085 x4) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1086 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u))))))
1087 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
1088 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
1089 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1090 C (CHead x3 x2 x4) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
1091 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
1092 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1093 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0
1094 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1095 C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S
1096 n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0:
1097 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4) (CHead e0 k0
1098 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
1099 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0:
1100 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1101 x1) (s k0 (S n0))) v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2 x4))
1102 (drop_drop (Bind b) n0 x0 (CHead x3 x2 x5) H15 t) (eq_ind_r nat (S (s x2 n0))
1103 (\lambda (n1: nat).(subst0 (minus (s (Bind b) x1) n1) v x4 x5)) H16 (s x2 (S
1104 n0)) (s_S x2 n0)))) e H14)))))))) H13)) (\lambda (H13: (ex3_4 K C C T
1105 (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
1106 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1107 C).(\lambda (u: T).(drop n0 O x0 (CHead e2 k0 u)))))) (\lambda (k0:
1108 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0
1109 n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1:
1110 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
1111 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2
1112 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1113 T).(csubst0 (minus x1 (s k0 n0)) v e1 e2))))) (or4 (drop (S n0) O (CHead x0
1114 (Bind b) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
1115 T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
1116 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
1117 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
1118 T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w))))))
1119 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1120 T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1121 (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0
1122 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1123 T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
1124 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1125 T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
1126 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
1127 x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1128 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1129 x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1130 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
1131 (S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4:
1132 C).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3 x2 x5))).(\lambda (H15:
1133 (drop n0 O x0 (CHead x4 x2 x5))).(\lambda (H16: (csubst0 (minus x1 (s x2 n0))
1134 v x3 x4)).(eq_ind_r C (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O
1135 (CHead x0 (Bind b) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
1136 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda
1137 (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
1138 (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
1139 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
1140 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1141 (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0:
1142 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
1143 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1144 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
1145 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1146 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
1147 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1148 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0:
1149 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1150 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1151 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
1152 (Bind b) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O
1153 (CHead x0 (Bind b) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0:
1154 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
1155 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1156 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w))))))
1157 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1158 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1159 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2
1160 x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1161 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u))))))
1162 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
1163 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
1164 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1165 C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
1166 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
1167 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1168 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0
1169 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1170 C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S
1171 n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1:
1172 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0
1173 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1174 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0:
1175 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
1176 b) x1) (s k0 (S n0))) v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2
1177 x5)) (drop_drop (Bind b) n0 x0 (CHead x4 x2 x5) H15 t) (eq_ind_r nat (S (s x2
1178 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x1) n1) v x3 x4)) H16 (s
1179 x2 (S n0)) (s_S x2 n0)))) e H14)))))))) H13)) (\lambda (H13: (ex4_5 K C C T T
1180 (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
1181 (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
1182 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2
1183 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
1184 T).(\lambda (w: T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (\lambda (k0:
1185 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
1186 (minus x1 (s k0 n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0:
1187 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1188 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1189 C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 k0 w)))))))
1190 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1191 T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1:
1192 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0
1193 n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T
1194 T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1195 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1196 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w))))))
1197 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1198 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1199 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
1200 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1201 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0:
1202 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
1203 b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
1204 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1205 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1206 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
1207 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1208 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
1209 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1210 (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
1211 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
1212 T).(\lambda (x6: T).(\lambda (H14: (eq C e (CHead x3 x2 x5))).(\lambda (H15:
1213 (drop n0 O x0 (CHead x4 x2 x6))).(\lambda (H16: (subst0 (minus x1 (s x2 n0))
1214 v x5 x6)).(\lambda (H17: (csubst0 (minus x1 (s x2 n0)) v x3 x4)).(eq_ind_r C
1215 (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind b) t)
1216 c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda
1217 (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
1218 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
1219 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
1220 T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w))))))
1221 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1222 T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1223 (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0
1224 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1225 T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
1226 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1227 T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
1228 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
1229 x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1230 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1231 x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1232 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
1233 (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x0 (Bind b) t)
1234 (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
1235 (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) (CHead e0 k0 u)))))) (\lambda
1236 (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
1237 (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
1238 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
1239 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1240 (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 u)))))) (\lambda
1241 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
1242 (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
1243 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S
1244 n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
1245 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
1246 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1247 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
1248 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1249 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
1250 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1251 (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
1252 e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1253 (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) (CHead e1 k0
1254 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
1255 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w)))))))
1256 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1257 T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0:
1258 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
1259 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) x2 x3 x4 x5 x6
1260 (refl_equal C (CHead x3 x2 x5)) (drop_drop (Bind b) n0 x0 (CHead x4 x2 x6)
1261 H15 t) (eq_ind_r nat (S (s x2 n0)) (\lambda (n1: nat).(subst0 (minus (s (Bind
1262 b) x1) n1) v x5 x6)) H16 (s x2 (S n0)) (s_S x2 n0)) (eq_ind_r nat (S (s x2
1263 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x1) n1) v x3 x4)) H17 (s
1264 x2 (S n0)) (s_S x2 n0)))) e H14)))))))))) H13)) H12)))))) (\lambda (f:
1265 F).(\lambda (H9: (drop (r (Flat f) n0) O c e)).(\lambda (H10: ((\forall (c3:
1266 C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0:
1267 C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
1268 (\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
1269 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1270 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
1271 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
1272 x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
1273 C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
1274 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
1275 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1276 C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v0 e1
1277 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1278 C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
1279 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1280 T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1281 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
1282 x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1283 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0
1284 (S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f)
1285 x1))).(let H12 \def (H10 x0 v H6 e H9) in (or4_ind (drop (S n0) O x0 e)
1286 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
1287 T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
1288 (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e0 k0 w)))))) (\lambda (k0:
1289 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 (S
1290 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1291 (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
1292 (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 k0
1293 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1294 T).(csubst0 (minus x1 (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
1295 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1296 C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1297 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w)))))))
1298 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1299 T).(subst0 (minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1300 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1
1301 (s k0 (S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e)
1302 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
1303 T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
1304 (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0
1305 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1306 T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T
1307 (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
1308 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1309 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u))))))
1310 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
1311 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
1312 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1313 C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1314 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
1315 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1316 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
1317 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1318 (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
1319 e2)))))))) (\lambda (H13: (drop (S n0) O x0 e)).(or4_intro0 (drop (S n0) O
1320 (CHead x0 (Flat f) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
1321 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
1322 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
1323 (Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1324 (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u
1325 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1326 C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_:
1327 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
1328 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1329 C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
1330 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1331 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
1332 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1333 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0:
1334 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1335 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1336 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
1337 (Flat f) x1) (s k0 (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 x0 e H13
1338 t))) (\lambda (H13: (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
1339 (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
1340 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e0 k0
1341 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1342 T).(subst0 (minus x1 (s k0 (S n0))) v u w))))))).(ex3_4_ind K C T T (\lambda
1343 (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
1344 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
1345 T).(drop (S n0) O x0 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
1346 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 (S n0))) v u
1347 w))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 K C T T (\lambda
1348 (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
1349 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
1350 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0:
1351 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
1352 x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
1353 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
1354 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
1355 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1356 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
1357 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1358 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
1359 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1360 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0:
1361 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1362 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1363 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
1364 (Flat f) x1) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3:
1365 C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3 x2
1366 x4))).(\lambda (H15: (drop (S n0) O x0 (CHead x3 x2 x5))).(\lambda (H16:
1367 (subst0 (minus x1 (s x2 (S n0))) v x4 x5)).(eq_ind_r C (CHead x3 x2 x4)
1368 (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 K C T T
1369 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
1370 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1371 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w))))))
1372 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1373 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1374 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1
1375 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1376 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0:
1377 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
1378 f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
1379 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
1380 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1381 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
1382 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1383 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
1384 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1385 (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
1386 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 x2
1387 x4)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
1388 T).(\lambda (_: T).(eq C (CHead x3 x2 x4) (CHead e0 k0 u)))))) (\lambda (k0:
1389 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
1390 (Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1391 (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u
1392 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1393 C).(\lambda (u: T).(eq C (CHead x3 x2 x4) (CHead e1 k0 u)))))) (\lambda (k0:
1394 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
1395 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1396 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
1397 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1398 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4) (CHead e1 k0
1399 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
1400 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w)))))))
1401 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1402 T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0:
1403 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
1404 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2))))))) (ex3_4_intro K C T T
1405 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
1406 (CHead x3 x2 x4) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
1407 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
1408 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
1409 T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))
1410 x2 x3 x4 x5 (refl_equal C (CHead x3 x2 x4)) (drop_drop (Flat f) n0 x0 (CHead
1411 x3 x2 x5) H15 t) H16)) e H14)))))))) H13)) (\lambda (H13: (ex3_4 K C C T
1412 (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
1413 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1414 C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 k0 u)))))) (\lambda (k0:
1415 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0
1416 (S n0))) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1:
1417 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
1418 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x0 (CHead
1419 e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1420 T).(csubst0 (minus x1 (s k0 (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead
1421 x0 (Flat f) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
1422 (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
1423 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
1424 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
1425 T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w))))))
1426 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1427 T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1428 (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0
1429 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1430 T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
1431 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1432 T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
1433 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
1434 x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1435 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
1436 x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1437 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0
1438 (S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4:
1439 C).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3 x2 x5))).(\lambda (H15:
1440 (drop (S n0) O x0 (CHead x4 x2 x5))).(\lambda (H16: (csubst0 (minus x1 (s x2
1441 (S n0))) v x3 x4)).(eq_ind_r C (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop
1442 (S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda
1443 (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u))))))
1444 (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
1445 n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda
1446 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0
1447 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
1448 C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda
1449 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
1450 (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
1451 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S
1452 n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
1453 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0
1454 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
1455 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w)))))))
1456 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1457 T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0:
1458 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
1459 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S
1460 n0) O (CHead x0 (Flat f) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0:
1461 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
1462 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1463 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w))))))
1464 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1465 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1466 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2
1467 x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1468 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u))))))
1469 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
1470 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
1471 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1472 C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
1473 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
1474 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1475 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0
1476 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1477 C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S
1478 n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1:
1479 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0
1480 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1481 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0:
1482 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
1483 f) x1) (s k0 (S n0))) v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2
1484 x5)) (drop_drop (Flat f) n0 x0 (CHead x4 x2 x5) H15 t) H16)) e H14))))))))
1485 H13)) (\lambda (H13: (ex4_5 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)))))))).(ex4_5_ind K C C T T (\lambda (k0: K).(\lambda (e1:
1493 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0
1494 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
1495 T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w))))))) (\lambda (k0:
1496 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1497 (minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
1498 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0
1499 (S n0))) v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 K
1500 C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
1501 e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1502 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w))))))
1503 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1504 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1505 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
1506 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1507 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0:
1508 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
1509 f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
1510 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1511 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1512 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
1513 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1514 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
1515 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1516 (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
1517 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
1518 T).(\lambda (x6: T).(\lambda (H14: (eq C e (CHead x3 x2 x5))).(\lambda (H15:
1519 (drop (S n0) O x0 (CHead x4 x2 x6))).(\lambda (H16: (subst0 (minus x1 (s x2
1520 (S n0))) v x5 x6)).(\lambda (H17: (csubst0 (minus x1 (s x2 (S n0))) v x3
1521 x4)).(eq_ind_r C (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead
1522 x0 (Flat f) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
1523 (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0:
1524 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
1525 (Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1526 (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u
1527 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1528 C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
1529 (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
1530 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1531 C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
1532 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1533 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
1534 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1535 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0:
1536 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1537 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1538 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
1539 (Flat f) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O
1540 (CHead x0 (Flat f) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0:
1541 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
1542 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1543 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w))))))
1544 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1545 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1546 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2
1547 x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1548 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u))))))
1549 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
1550 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
1551 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1552 C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
1553 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
1554 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1555 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0
1556 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1557 C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S
1558 n0))) v e1 e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1:
1559 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
1560 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1561 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
1562 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1563 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
1564 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1565 (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
1566 e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x3 x2 x5)) (drop_drop (Flat f)
1567 n0 x0 (CHead x4 x2 x6) H15 t) H16 H17)) e H14)))))))))) H13)) H12)))))) k
1568 (drop_gen_drop k c e t n0 H2) H7 H8) i H4))) c2 H5)))))) H3)) (\lambda (H3:
1569 (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s
1570 k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
1571 c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
1572 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
1573 c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
1574 nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
1575 nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
1576 (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
1577 nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda
1578 (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
1579 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
1580 T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
1581 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u
1582 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1583 C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_:
1584 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u))))))
1585 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
1586 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
1587 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1588 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1589 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w)))))))
1590 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1591 T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
1592 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0
1593 (S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2:
1594 nat).(\lambda (H4: (eq nat i (s k x2))).(\lambda (H5: (eq C c2 (CHead x1 k
1595 x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H7: (csubst0 x2 v c
1596 x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop (S n0) O c0 e)
1597 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
1598 T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
1599 (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 w)))))) (\lambda (k0:
1600 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
1601 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1602 (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
1603 (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 k0
1604 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1605 T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
1606 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1607 C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1608 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k0 w)))))))
1609 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1610 T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
1611 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0
1612 (S n0))) v e1 e2))))))))) (let H8 \def (eq_ind nat i (\lambda (n1:
1613 nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall
1614 (e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
1615 (\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
1616 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1617 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
1618 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S
1619 n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1620 (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0:
1621 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead
1622 e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1623 T).(csubst0 (minus n1 (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda
1624 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1625 C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1626 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w)))))))
1627 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1628 T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda
1629 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1
1630 (s k0 (S n0))) v0 e1 e2)))))))))))))) H0 (s k x2) H4) in (let H9 \def (eq_ind
1631 nat i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x2) H4) in (eq_ind_r nat (s k
1632 x2) (\lambda (n1: nat).(or4 (drop (S n0) O (CHead x1 k x0) e) (ex3_4 K C T T
1633 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1634 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1635 T).(\lambda (w: T).(drop (S n0) O (CHead x1 k x0) (CHead e0 k0 w))))))
1636 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1637 (minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda
1638 (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda
1639 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
1640 (CHead x1 k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
1641 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1
1642 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1643 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
1644 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1645 T).(drop (S n0) O (CHead x1 k x0) (CHead e2 k0 w))))))) (\lambda (k0:
1646 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1647 (minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
1648 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
1649 (S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to
1650 (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall
1651 (e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
1652 (\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
1653 (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_:
1654 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1:
1655 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2)
1656 (s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1:
1657 C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda
1658 (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
1659 (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2:
1660 C).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1 e2))))))
1661 (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda
1662 (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1:
1663 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
1664 n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_:
1665 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v0
1666 u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1667 T).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1
1668 e2)))))))))))))) \to ((lt (S n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1
1669 k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u:
1670 T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda
1671 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0)
1672 (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u:
1673 T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w)))))) (ex3_4
1674 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
1675 C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
1676 C).(\lambda (u: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 k1 u))))))
1677 (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
1678 (minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1:
1679 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1680 (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
1681 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2
1682 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
1683 T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w))))))
1684 (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
1685 (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda
1686 (b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall
1687 (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0:
1688 C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
1689 (\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
1690 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1691 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
1692 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1693 x2) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
1694 C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
1695 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
1696 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1697 C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v0 e1
1698 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1699 C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
1700 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1701 T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1702 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1703 x2) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1704 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0
1705 (S n0))) v0 e1 e2))))))))))))))).(\lambda (H12: (lt (S n0) (s (Bind b)
1706 x2))).(let H13 \def (IHn x2 (le_S_n (S n0) x2 H12) c x1 v H7 e H10) in
1707 (or4_ind (drop n0 O x1 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
1708 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
1709 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0
1710 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1711 T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (ex3_4 K C C T (\lambda (k0:
1712 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
1713 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1714 T).(drop n0 O x1 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
1715 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0 n0)) v e1
1716 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1717 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
1718 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1719 T).(drop n0 O x1 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1720 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0
1721 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1722 (_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 n0)) v e1 e2))))))) (or4
1723 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T T (\lambda (k0:
1724 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
1725 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
1726 T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0:
1727 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1728 x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
1729 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
1730 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
1731 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1732 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
1733 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1734 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
1735 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1736 T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0:
1737 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1738 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1739 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
1740 (Bind b) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (H14: (drop n0 O x1
1741 e)).(or4_intro0 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T T
1742 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1743 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1744 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w))))))
1745 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1746 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1747 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
1748 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1749 T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0:
1750 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
1751 b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
1752 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1753 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1754 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
1755 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1756 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
1757 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1758 (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
1759 e2))))))) (drop_drop (Bind b) n0 x1 e H14 x0))) (\lambda (H14: (ex3_4 K C T T
1760 (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1761 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1762 T).(\lambda (w: T).(drop n0 O x1 (CHead e0 k0 w)))))) (\lambda (k0:
1763 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0
1764 n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k0: K).(\lambda (e0:
1765 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
1766 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0
1767 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1768 T).(subst0 (minus x2 (s k0 n0)) v u w))))) (or4 (drop (S n0) O (CHead x1
1769 (Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
1770 T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
1771 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
1772 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
1773 T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w))))))
1774 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1775 T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1776 (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0
1777 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1778 T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
1779 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1780 T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
1781 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
1782 x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1783 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1784 x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1785 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0
1786 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5:
1787 T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x5))).(\lambda (H16:
1788 (drop n0 O x1 (CHead x4 x3 x6))).(\lambda (H17: (subst0 (minus x2 (s x3 n0))
1789 v x5 x6)).(eq_ind_r C (CHead x4 x3 x5) (\lambda (c0: C).(or4 (drop (S n0) O
1790 (CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
1791 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda
1792 (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
1793 (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
1794 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
1795 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1796 (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0:
1797 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
1798 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1799 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
1800 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1801 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
1802 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1803 T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0:
1804 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1805 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1806 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
1807 (Bind b) x2) (s k0 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O
1808 (CHead x1 (Bind b) x0) (CHead x4 x3 x5)) (ex3_4 K C T T (\lambda (k0:
1809 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5)
1810 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1811 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w))))))
1812 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1813 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1814 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3
1815 x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1816 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u))))))
1817 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
1818 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
1819 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1820 C (CHead x4 x3 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
1821 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
1822 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1823 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0
1824 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1825 C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S
1826 n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0:
1827 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0
1828 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
1829 T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0:
1830 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1831 x2) (s k0 (S n0))) v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x5))
1832 (drop_drop (Bind b) n0 x1 (CHead x4 x3 x6) H16 x0) (eq_ind_r nat (S (s x3
1833 n0)) (\lambda (n1: nat).(subst0 (minus (s (Bind b) x2) n1) v x5 x6)) H17 (s
1834 x3 (S n0)) (s_S x3 n0)))) e H15)))))))) H14)) (\lambda (H14: (ex3_4 K C C T
1835 (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
1836 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1837 C).(\lambda (u: T).(drop n0 O x1 (CHead e2 k0 u)))))) (\lambda (k0:
1838 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0
1839 n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1:
1840 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
1841 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2
1842 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1843 T).(csubst0 (minus x2 (s k0 n0)) v e1 e2))))) (or4 (drop (S n0) O (CHead x1
1844 (Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
1845 T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
1846 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
1847 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
1848 T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w))))))
1849 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1850 T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1851 (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0
1852 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1853 T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
1854 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1855 T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
1856 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
1857 x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1858 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1859 x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1860 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0
1861 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5:
1862 C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16:
1863 (drop n0 O x1 (CHead x5 x3 x6))).(\lambda (H17: (csubst0 (minus x2 (s x3 n0))
1864 v x4 x5)).(eq_ind_r C (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O
1865 (CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
1866 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda
1867 (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
1868 (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
1869 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
1870 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1871 (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0:
1872 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
1873 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1874 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
1875 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
1876 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
1877 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
1878 T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0:
1879 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1880 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
1881 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
1882 (Bind b) x2) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O
1883 (CHead x1 (Bind b) x0) (CHead x4 x3 x6)) (ex3_4 K C T T (\lambda (k0:
1884 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6)
1885 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1886 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w))))))
1887 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1888 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1889 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3
1890 x6) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1891 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u))))))
1892 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
1893 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
1894 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
1895 C (CHead x4 x3 x6) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
1896 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
1897 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1898 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0
1899 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
1900 C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S
1901 n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1:
1902 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
1903 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1904 T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0:
1905 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
1906 b) x2) (s k0 (S n0))) v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3
1907 x6)) (drop_drop (Bind b) n0 x1 (CHead x5 x3 x6) H16 x0) (eq_ind_r nat (S (s
1908 x3 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x2) n1) v x4 x5)) H17
1909 (s x3 (S n0)) (s_S x3 n0)))) e H15)))))))) H14)) (\lambda (H14: (ex4_5 K C C
1910 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1911 T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
1912 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead
1913 e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
1914 T).(\lambda (w: T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (\lambda (k0:
1915 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
1916 (minus x2 (s k0 n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0:
1917 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1918 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1919 C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e2 k0 w)))))))
1920 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
1921 T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1:
1922 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0
1923 n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T
1924 T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1925 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
1926 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w))))))
1927 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
1928 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
1929 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
1930 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
1931 T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0:
1932 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
1933 b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
1934 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
1935 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1936 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
1937 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1938 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
1939 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1940 (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
1941 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6:
1942 T).(\lambda (x7: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16:
1943 (drop n0 O x1 (CHead x5 x3 x7))).(\lambda (H17: (subst0 (minus x2 (s x3 n0))
1944 v x6 x7)).(\lambda (H18: (csubst0 (minus x2 (s x3 n0)) v x4 x5)).(eq_ind_r C
1945 (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0)
1946 c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda
1947 (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
1948 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
1949 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
1950 T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w))))))
1951 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1952 T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
1953 (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0
1954 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
1955 T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
1956 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
1957 T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
1958 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
1959 x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
1960 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
1961 x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
1962 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0
1963 (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Bind b) x0)
1964 (CHead x4 x3 x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
1965 (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda
1966 (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
1967 (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
1968 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
1969 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1970 (_: C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda
1971 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
1972 (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
1973 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S
1974 n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
1975 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6)
1976 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
1977 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
1978 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
1979 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
1980 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
1981 (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
1982 e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda
1983 (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
1984 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
1985 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0
1986 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
1987 T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w))))))
1988 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
1989 (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) x3 x4 x5
1990 x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Bind b) n0 x1 (CHead x5 x3
1991 x7) H16 x0) (eq_ind_r nat (S (s x3 n0)) (\lambda (n1: nat).(subst0 (minus (s
1992 (Bind b) x2) n1) v x6 x7)) H17 (s x3 (S n0)) (s_S x3 n0)) (eq_ind_r nat (S (s
1993 x3 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x2) n1) v x4 x5)) H18
1994 (s x3 (S n0)) (s_S x3 n0)))) e H15)))))))))) H14)) H13)))))) (\lambda (f:
1995 F).(\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c3:
1996 C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e0:
1997 C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
1998 (\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
1999 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2000 T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
2001 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
2002 x2) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
2003 C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
2004 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
2005 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
2006 C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v0 e1
2007 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2008 C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
2009 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2010 T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
2011 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
2012 x2) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
2013 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0
2014 (S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f)
2015 x2))).(let H13 \def (H11 x1 v H7 e H10) in (or4_ind (drop (S n0) O x1 e)
2016 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
2017 T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
2018 (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e0 k0 w)))))) (\lambda (k0:
2019 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S
2020 n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
2021 (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
2022 (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 k0
2023 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2024 T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
2025 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
2026 C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
2027 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e2 k0 w)))))))
2028 (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2029 T).(subst0 (minus x2 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
2030 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x2
2031 (s k0 (S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e)
2032 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
2033 T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
2034 (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0
2035 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2036 T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T
2037 (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
2038 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
2039 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u))))))
2040 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
2041 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
2042 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
2043 C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
2044 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
2045 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
2046 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S
2047 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
2048 (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2049 e2)))))))) (\lambda (H14: (drop (S n0) O x1 e)).(or4_intro0 (drop (S n0) O
2050 (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
2051 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
2052 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
2053 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
2054 (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u
2055 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2056 C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_:
2057 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
2058 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
2059 C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2060 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2061 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
2062 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2063 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0:
2064 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
2065 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
2066 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
2067 (Flat f) x2) (s k0 (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 x1 e H14
2068 x0))) (\lambda (H14: (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
2069 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
2070 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead
2071 e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2072 T).(subst0 (minus x2 (s k0 (S n0))) v u w))))))).(ex3_4_ind K C T T (\lambda
2073 (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
2074 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
2075 T).(drop (S n0) O x1 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
2076 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S n0))) v u
2077 w))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda
2078 (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
2079 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
2080 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0:
2081 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
2082 x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
2083 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
2084 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
2085 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
2086 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2087 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2088 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
2089 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2090 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0:
2091 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
2092 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
2093 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
2094 (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4:
2095 C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3
2096 x5))).(\lambda (H16: (drop (S n0) O x1 (CHead x4 x3 x6))).(\lambda (H17:
2097 (subst0 (minus x2 (s x3 (S n0))) v x5 x6)).(eq_ind_r C (CHead x4 x3 x5)
2098 (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K C T
2099 T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
2100 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
2101 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w))))))
2102 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
2103 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
2104 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1
2105 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
2106 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0:
2107 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
2108 f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
2109 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
2110 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
2111 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
2112 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
2113 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S
2114 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
2115 (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2116 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3
2117 x5)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
2118 T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0 u)))))) (\lambda (k0:
2119 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
2120 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
2121 (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u
2122 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2123 C).(\lambda (u: T).(eq C (CHead x4 x3 x5) (CHead e1 k0 u)))))) (\lambda (k0:
2124 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
2125 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
2126 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2127 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2128 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e1 k0
2129 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
2130 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
2131 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2132 T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
2133 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2134 (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))
2135 (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
2136 T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0 u)))))) (\lambda (k0:
2137 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
2138 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
2139 (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u
2140 w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x5)) (drop_drop (Flat f) n0 x1
2141 (CHead x4 x3 x6) H16 x0) H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 K C
2142 C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
2143 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
2144 C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 k0 u)))))) (\lambda (k0:
2145 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0
2146 (S n0))) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1:
2147 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
2148 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead
2149 e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2150 T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead
2151 x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
2152 (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
2153 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
2154 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
2155 T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
2156 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2157 T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
2158 (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
2159 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2160 T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
2161 T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2162 T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
2163 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
2164 x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
2165 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
2166 x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
2167 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0
2168 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5:
2169 C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16:
2170 (drop (S n0) O x1 (CHead x5 x3 x6))).(\lambda (H17: (csubst0 (minus x2 (s x3
2171 (S n0))) v x4 x5)).(eq_ind_r C (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop
2172 (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda
2173 (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u))))))
2174 (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
2175 n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda
2176 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0
2177 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
2178 C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda
2179 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
2180 (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
2181 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S
2182 n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
2183 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0
2184 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
2185 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
2186 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2187 T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
2188 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2189 (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))))
2190 (or4_intro2 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3 x6)) (ex3_4 K
2191 C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
2192 (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
2193 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
2194 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
2195 T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
2196 (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2197 T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_:
2198 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
2199 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
2200 C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2201 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2202 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
2203 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
2204 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
2205 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2206 T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
2207 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2208 (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))
2209 (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2210 C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0:
2211 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
2212 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
2213 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2214 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1
2215 (CHead x5 x3 x6) H16 x0) H17)) e H15)))))))) H14)) (\lambda (H14: (ex4_5 K C
2216 C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2217 T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
2218 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1
2219 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
2220 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S n0))) v u
2221 w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2222 T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1
2223 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda
2224 (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
2225 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2226 T).(drop (S n0) O x1 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
2227 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S
2228 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
2229 (_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2)))))) (or4
2230 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0:
2231 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
2232 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
2233 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0:
2234 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
2235 x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
2236 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
2237 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
2238 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
2239 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2240 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2241 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
2242 (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2243 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0:
2244 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
2245 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
2246 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
2247 (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4:
2248 C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq C e
2249 (CHead x4 x3 x6))).(\lambda (H16: (drop (S n0) O x1 (CHead x5 x3
2250 x7))).(\lambda (H17: (subst0 (minus x2 (s x3 (S n0))) v x6 x7)).(\lambda
2251 (H18: (csubst0 (minus x2 (s x3 (S n0))) v x4 x5)).(eq_ind_r C (CHead x4 x3
2252 x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K
2253 C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
2254 c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
2255 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w))))))
2256 (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
2257 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
2258 (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1
2259 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
2260 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0:
2261 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
2262 f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
2263 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
2264 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
2265 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
2266 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
2267 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S
2268 n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
2269 (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2270 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3
2271 x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
2272 T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0:
2273 K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
2274 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
2275 (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u
2276 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2277 C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0:
2278 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
2279 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
2280 (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
2281 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2282 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
2283 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
2284 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
2285 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2286 T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
2287 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2288 (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))
2289 (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
2290 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
2291 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
2292 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
2293 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2294 T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
2295 (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2296 (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) x3 x4 x5
2297 x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1 (CHead x5 x3
2298 x7) H16 x0) H17 H18)) e H15)))))))))) H14)) H13)))))) k (drop_gen_drop k c e
2299 t n0 H2) H8 H9) i H4))) c2 H5)))))))) H3)) (csubst0_gen_head k c c2 t v i
2300 H1))))))))))) c1)))))) n).
2302 Initial nodes: 39886
2305 theorem csubst0_drop_eq:
2306 \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0
2307 n v c1 c2) \to (\forall (e: C).((drop n O c1 e) \to (or4 (drop n O c2 e)
2308 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
2309 T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
2310 C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w))))))
2311 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
2312 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2313 C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
2314 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2
2315 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
2316 (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
2317 (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1
2318 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
2319 (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
2320 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
2321 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2322 T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))
2324 \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2:
2325 C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c1
2326 e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
2327 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
2328 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O
2329 c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
2330 T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
2331 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
2332 f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
2333 T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
2334 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
2335 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
2336 T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
2337 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e2
2338 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
2339 (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
2340 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
2341 e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
2342 (H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c1
2343 e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 F C T T (\lambda
2344 (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0
2345 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
2346 (w: T).(drop O O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
2347 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
2348 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c
2349 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2350 C).(\lambda (u: T).(drop O O c2 (CHead e2 (Flat f) u)))))) (\lambda (_:
2351 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
2352 (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
2353 (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f:
2354 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
2355 c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
2356 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
2357 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
2358 O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2))
2359 (\lambda (n0: nat).(or4 (drop n0 n0 c2 c1) (ex3_4 F C T T (\lambda (f:
2360 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e0 (Flat
2361 f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
2362 T).(drop n0 n0 c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
2363 C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w)))))) (ex3_4 F C C T
2364 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c1
2365 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2366 C).(\lambda (u: T).(drop n0 n0 c2 (CHead e2 (Flat f) u)))))) (\lambda (_:
2367 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1
2368 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2369 C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e1 (Flat f) u)))))))
2370 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2371 T).(drop n0 n0 c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
2372 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w))))))
2373 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2374 (_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda (H1: (csubst0
2375 y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: T).(\lambda (c:
2376 C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c0 c) (ex3_4 F C T T
2377 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c
2378 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
2379 T).(\lambda (w: T).(drop n0 n0 c0 (CHead e0 (Flat f) w)))))) (\lambda (_:
2380 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w))))))
2381 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2382 T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
2383 C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 c0 (CHead e2 (Flat f) u))))))
2384 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0
2385 t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2386 C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u)))))))
2387 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2388 T).(drop n0 n0 c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
2389 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w))))))
2390 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2391 (_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda (k: K).(K_ind (\lambda (k0:
2392 K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2:
2393 T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s k0 i) O) \to (or4
2394 (drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead c k0 u1)) (ex3_4 F C T T
2395 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead
2396 c k0 u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
2397 (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e0
2398 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2399 T).(subst0 (s k0 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
2400 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k0 u1) (CHead e1 (Flat f)
2401 u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
2402 T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e2 (Flat f) u))))))
2403 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s
2404 k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
2405 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k0 u1)
2406 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2407 C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2)
2408 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
2409 C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_:
2410 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
2411 (s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda
2412 (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1
2413 u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def (eq_ind nat
2414 (S i) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with
2415 [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or4
2416 (drop (S i) (S i) (CHead c (Bind b) u2) (CHead c (Bind b) u1)) (ex3_4 F C T T
2417 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead
2418 c (Bind b) u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
2419 C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c (Bind b) u2)
2420 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
2421 T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (ex3_4 F C C T (\lambda (f:
2422 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b)
2423 u1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2424 C).(\lambda (u: T).(drop (S i) (S i) (CHead c (Bind b) u2) (CHead e2 (Flat f)
2425 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2426 T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
2427 (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Bind
2428 b) u1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda
2429 (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c (Bind b)
2430 u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
2431 C).(\lambda (u: T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (\lambda (_:
2432 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
2433 (S i) v0 e1 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i:
2434 nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0
2435 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind
2436 nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H3) in (eq_ind_r nat O
2437 (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f) u2) (CHead c (Flat f)
2438 u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
2439 T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0 (Flat f0) u))))))
2440 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0
2441 (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
2442 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) (ex3_4 F C C T
2443 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C
2444 (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
2445 (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 (CHead c (Flat f) u2)
2446 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
2447 C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
2448 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
2449 (CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda
2450 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c
2451 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
2452 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w))))))
2453 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2454 (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro1 (drop O O (CHead c (Flat f)
2455 u2) (CHead c (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
2456 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0
2457 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
2458 (w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda
2459 (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
2460 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2461 T).(eq C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
2462 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c (Flat
2463 f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
2464 (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda
2465 (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
2466 C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0:
2467 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
2468 (CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda
2469 (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
2470 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2471 (_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0:
2472 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f)
2473 u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
2474 (_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0)
2475 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2476 T).(subst0 O v0 u w))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u1))
2477 (drop_refl (CHead c (Flat f) u2)) H4)) i H3)))))))))) k)) (\lambda (k:
2478 K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4:
2479 C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop
2480 i i c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
2481 T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f:
2482 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e0
2483 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2484 T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
2485 C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u))))))
2486 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4
2487 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
2488 C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
2489 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
2490 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2491 C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 (Flat f) w)))))))
2492 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2493 T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
2494 C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to (\forall
2495 (u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c4 k0 u)
2496 (CHead c3 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
2497 (u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e0 (Flat f) u0))))))
2498 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0
2499 i) (s k0 i) (CHead c4 k0 u) (CHead e0 (Flat f) w)))))) (\lambda (_:
2500 F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (s k0 i) v0 u0
2501 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2502 C).(\lambda (u0: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) u0))))))
2503 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0
2504 i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) (\lambda (_:
2505 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1
2506 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2507 C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f)
2508 u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
2509 T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f)
2510 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0:
2511 T).(\lambda (w: T).(subst0 (s k0 i) v0 u0 w)))))) (\lambda (_: F).(\lambda
2512 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0
2513 e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3:
2514 C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3
2515 c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T
2516 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
2517 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
2518 T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_:
2519 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
2520 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2521 T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
2522 C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u))))))
2523 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
2524 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2525 C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
2526 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2527 T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
2528 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
2529 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2530 (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat
2531 (S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat
2532 return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
2533 True])) I O H4) in (False_ind (or4 (drop (S i) (S i) (CHead c4 (Bind b) u)
2534 (CHead c3 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
2535 C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e0
2536 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
2537 (w: T).(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e0 (Flat f) w))))))
2538 (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (S
2539 i) v0 u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2540 C).(\lambda (u0: T).(eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0))))))
2541 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S i)
2542 (S i) (CHead c4 (Bind b) u) (CHead e2 (Flat f) u0)))))) (\lambda (_:
2543 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1
2544 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2545 C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e1
2546 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
2547 (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e2
2548 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
2549 (u0: T).(\lambda (w: T).(subst0 (S i) v0 u0 w)))))) (\lambda (_: F).(\lambda
2550 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1
2551 e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (c3:
2552 C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: (csubst0 i v0 c3
2553 c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T
2554 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
2555 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
2556 T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f0) w)))))) (\lambda (_:
2557 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
2558 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2559 T).(eq C c3 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
2560 C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f0) u))))))
2561 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
2562 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda
2563 (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u)))))))
2564 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2565 (w: T).(drop i i c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
2566 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
2567 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2568 (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i
2569 O)).(let H5 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4
2570 (drop n0 n0 c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda
2571 (u0: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u0)))))) (\lambda (f0:
2572 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e0
2573 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda
2574 (w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
2575 C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 (Flat f0) u0))))))
2576 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0
2577 n0 c4 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
2578 (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda
2579 (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
2580 C c3 (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
2581 (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e2 (Flat f0)
2582 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0:
2583 T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (\lambda (_: F).(\lambda (e1:
2584 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1
2585 e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0:
2586 nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r nat O (\lambda (n0:
2587 nat).(or4 (drop n0 n0 (CHead c4 (Flat f) u) (CHead c3 (Flat f) u)) (ex3_4 F C
2588 T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C
2589 (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda
2590 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c4 (Flat f) u)
2591 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0:
2592 T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0:
2593 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f)
2594 u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
2595 (e2: C).(\lambda (u0: T).(drop n0 n0 (CHead c4 (Flat f) u) (CHead e2 (Flat
2596 f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2597 T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
2598 C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Flat f)
2599 u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
2600 (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c4 (Flat f) u)
2601 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
2602 C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (\lambda (_:
2603 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
2604 n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c4 (Flat f) u) (CHead c3
2605 (Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0:
2606 T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0))))))
2607 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O
2608 (CHead c4 (Flat f) u) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
2609 (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (ex3_4 F C C
2610 T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C
2611 (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda
2612 (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u)
2613 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
2614 C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
2615 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C
2616 (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda
2617 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4
2618 (Flat f) u) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
2619 C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w))))))
2620 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2621 (_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0:
2622 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f)
2623 u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
2624 (e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0)
2625 u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2626 T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c3 (Flat f) u))
2627 (drop_refl (CHead c4 (Flat f) u)) H6)) i H4)))))))))))) k)) (\lambda (k:
2628 K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1:
2629 T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4:
2630 C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4
2631 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
2632 C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
2633 T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_:
2634 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
2635 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2636 T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
2637 C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u))))))
2638 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
2639 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2640 C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
2641 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2642 T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
2643 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
2644 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2645 (_: T).(csubst0 i v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop
2646 (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead c3 k0 u1)) (ex3_4 F C T T (\lambda
2647 (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0
2648 u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
2649 T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e0 (Flat
2650 f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2651 T).(subst0 (s k0 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
2652 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k0 u1) (CHead e1 (Flat f)
2653 u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
2654 T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e2 (Flat f) u))))))
2655 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s
2656 k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
2657 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 u1)
2658 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2659 C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2)
2660 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
2661 C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_:
2662 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
2663 (s k0 i) v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i:
2664 nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0
2665 i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3
2666 c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T
2667 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
2668 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
2669 T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_:
2670 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
2671 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2672 T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
2673 C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u))))))
2674 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
2675 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2676 C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
2677 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2678 T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
2679 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
2680 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2681 (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6
2682 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda
2683 (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5)
2684 in (False_ind (or4 (drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead c3 (Bind
2685 b) u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
2686 T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u))))))
2687 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S i)
2688 (S i) (CHead c4 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_:
2689 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (S i) v0 u w))))))
2690 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2691 T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (f:
2692 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S i) (S i) (CHead
2693 c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
2694 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C
2695 C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2696 T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))))
2697 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2698 T).(drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e2 (Flat f) w)))))))
2699 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2700 T).(subst0 (S i) v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
2701 C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2))))))))
2702 H6))))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: T).(\lambda
2703 (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c3:
2704 C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 c4)).(\lambda (H4: (((eq
2705 nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda
2706 (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u))))))
2707 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i
2708 c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
2709 T).(\lambda (w: T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f0:
2710 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat
2711 f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
2712 T).(drop i i c4 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
2713 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T
2714 T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
2715 (_: T).(eq C c3 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_:
2716 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2
2717 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
2718 (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1:
2719 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1
2720 e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6 \def (eq_ind nat i (\lambda
2721 (n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c4 c3) (ex3_4 F C T T (\lambda
2722 (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0
2723 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
2724 (w: T).(drop n0 n0 c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
2725 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) (ex3_4 F C C T
2726 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3
2727 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
2728 C).(\lambda (u: T).(drop n0 n0 c4 (CHead e2 (Flat f0) u)))))) (\lambda (_:
2729 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1
2730 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
2731 C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u)))))))
2732 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2733 (w: T).(drop n0 n0 c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda
2734 (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u
2735 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2736 T).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))))))) H4 O H5) in (let H7 \def
2737 (eq_ind nat i (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8
2738 \def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in
2739 (eq_ind_r nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c4 (Flat f) u2)
2740 (CHead c3 (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
2741 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e0
2742 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
2743 (w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) w))))))
2744 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0
2745 u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
2746 C).(\lambda (u: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u))))))
2747 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0
2748 (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda
2749 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F
2750 C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2751 T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))))
2752 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2753 (w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) w)))))))
2754 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2755 T).(subst0 n0 v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
2756 C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3
2757 (drop O O (CHead c4 (Flat f) u2) (CHead c3 (Flat f) u1)) (ex3_4 F C T T
2758 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
2759 (CHead c3 (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
2760 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2)
2761 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
2762 T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f0:
2763 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 (Flat f)
2764 u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
2765 (e2: C).(\lambda (u: T).(drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0)
2766 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2767 T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
2768 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f)
2769 u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
2770 (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2)
2771 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
2772 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_:
2773 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
2774 O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1:
2775 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f)
2776 u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
2777 (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2)
2778 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
2779 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_:
2780 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
2781 O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c3 (Flat f) u1))
2782 (drop_refl (CHead c4 (Flat f) u2)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2
2783 H1))) H) e (drop_gen_refl c1 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn:
2784 ((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to
2785 (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T
2786 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
2787 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
2788 T).(\lambda (w: T).(drop n0 O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
2789 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
2790 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2791 T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
2792 C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c2 (CHead e2 (Flat f) u))))))
2793 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
2794 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2795 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u)))))))
2796 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
2797 T).(drop n0 O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
2798 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
2799 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2800 (_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind (\lambda
2801 (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall
2802 (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 F C T T
2803 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
2804 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
2805 T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
2806 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
2807 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2808 T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
2809 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f)
2810 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2811 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
2812 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
2813 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
2814 T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
2815 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
2816 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2817 T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1:
2818 nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 (S n0) v (CSort
2819 n1) c2)).(\lambda (e: C).(\lambda (H0: (drop (S n0) O (CSort n1)
2820 e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop
2821 (S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
2822 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
2823 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
2824 e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
2825 (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
2826 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
2827 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
2828 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
2829 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
2830 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
2831 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2832 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f)
2833 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2834 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
2835 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
2836 e2)))))))) (\lambda (H1: (eq C e (CSort n1))).(\lambda (H2: (eq nat (S n0)
2837 O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(or4
2838 (drop (S n0) O c2 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
2839 (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f:
2840 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
2841 e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
2842 (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
2843 C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u))))))
2844 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
2845 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
2846 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
2847 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
2848 c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2849 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f)
2850 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2851 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
2852 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
2853 e2))))))))) (let H4 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee in
2854 nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
2855 \Rightarrow True])) I O H2) in (False_ind (or4 (drop (S n0) O c2 (CSort n1))
2856 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
2857 T).(eq C (CSort n1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
2858 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f)
2859 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
2860 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
2861 C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort n1) (CHead e1 (Flat f)
2862 u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
2863 T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
2864 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
2865 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
2866 T).(eq C (CSort n1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
2867 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
2868 e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
2869 (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
2870 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
2871 e2)))))))) H4)) e H1)))) (drop_gen_sort n1 (S n0) O e H0)))))))) (\lambda (c:
2872 C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2)
2873 \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4
2874 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
2875 C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
2876 T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
2877 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
2878 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
2879 T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
2880 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f)
2881 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2882 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
2883 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
2884 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
2885 T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
2886 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
2887 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
2888 T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))).(\lambda (k: K).(\lambda
2889 (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 (S n0) v
2890 (CHead c k t) c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CHead c k t)
2891 e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s
2892 k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2))))
2893 (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda
2894 (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda
2895 (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j:
2896 nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
2897 C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda
2898 (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2:
2899 T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
2900 T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S
2901 n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
2902 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
2903 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
2904 e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
2905 (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
2906 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
2907 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
2908 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
2909 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
2910 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
2911 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2912 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f)
2913 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2914 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
2915 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
2916 e2)))))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq
2917 nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k
2918 u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T
2919 nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2:
2920 T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
2921 nat).(subst0 j v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda
2922 (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0
2923 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
2924 (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
2925 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
2926 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
2927 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2928 C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_:
2929 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
2930 (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
2931 (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f:
2932 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
2933 n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda
2934 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
2935 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
2936 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat
2937 (S n0) (s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5:
2938 (subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S
2939 n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
2940 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
2941 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead
2942 e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
2943 (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
2944 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
2945 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
2946 O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
2947 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
2948 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
2949 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2950 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f)
2951 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
2952 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
2953 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
2954 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S
2955 n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c k0 x0) e) (ex3_4 F C T T
2956 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
2957 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
2958 T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e0 (Flat f) w))))))
2959 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
2960 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2961 C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
2962 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c
2963 k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
2964 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
2965 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
2966 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
2967 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2
2968 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
2969 (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
2970 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
2971 e2))))))))))) (\lambda (b: B).(\lambda (H6: (drop (r (Bind b) n0) O c
2972 e)).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(let H8 \def (f_equal nat
2973 nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O
2974 \Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H9 \def
2975 (eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H8) in
2976 (or4_intro0 (drop (S n0) O (CHead c (Bind b) x0) e) (ex3_4 F C T T (\lambda
2977 (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0
2978 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
2979 (w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e0 (Flat f) w))))))
2980 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
2981 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
2982 C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
2983 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c
2984 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
2985 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
2986 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
2987 T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
2988 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c
2989 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
2990 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
2991 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
2992 (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H6 x0)))))))
2993 (\lambda (f: F).(\lambda (H6: (drop (r (Flat f) n0) O c e)).(\lambda (H7: (eq
2994 nat (S n0) (s (Flat f) x1))).(let H8 \def (f_equal nat nat (\lambda (e0:
2995 nat).e0) (S n0) (s (Flat f) x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda
2996 (n1: nat).(subst0 n1 v t x0)) H5 (S n0) H8) in (or4_intro0 (drop (S n0) O
2997 (CHead c (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
2998 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
2999 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3000 n0) O (CHead c (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_:
3001 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3002 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3003 T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
3004 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) x0)
3005 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3006 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
3007 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3008 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
3009 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0)
3010 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
3011 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
3012 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
3013 O v e1 e2))))))) (drop_drop (Flat f) n0 c e H6 x0))))))) k (drop_gen_drop k c
3014 e t n0 H1) H3) c2 H4)))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_:
3015 C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_:
3016 nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
3017 v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0)
3018 (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t))))
3019 (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O
3020 c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3021 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
3022 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
3023 e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
3024 (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
3025 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
3026 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
3027 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
3028 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
3029 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3030 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3031 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f)
3032 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3033 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3034 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3035 e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0)
3036 (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1
3037 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop (S n0) O c0
3038 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda
3039 (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
3040 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f)
3041 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3042 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
3043 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
3044 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
3045 O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
3046 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
3047 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3048 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3049 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f)
3050 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3051 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3052 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3053 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S
3054 n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead x0 k0 t) e) (ex3_4 F C T T
3055 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3056 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3057 T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e0 (Flat f) w))))))
3058 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
3059 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
3060 C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
3061 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
3062 k0 t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
3063 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
3064 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3065 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3066 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2
3067 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
3068 (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3069 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3070 e2))))))))))) (\lambda (b: B).(\lambda (H6: (drop (r (Bind b) n0) O c
3071 e)).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(let H8 \def (f_equal nat
3072 nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O
3073 \Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H9 \def
3074 (eq_ind_r nat x1 (\lambda (n1: nat).(csubst0 n1 v c x0)) H5 n0 H8) in (let
3075 H10 \def (IHn c x0 v H9 e H6) in (or4_ind (drop n0 O x0 e) (ex3_4 F C T T
3076 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3077 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3078 T).(\lambda (w: T).(drop n0 O x0 (CHead e0 (Flat f) w)))))) (\lambda (_:
3079 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3080 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3081 T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
3082 C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2 (Flat f) u))))))
3083 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
3084 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
3085 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u)))))))
3086 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
3087 T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
3088 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3089 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
3090 (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Bind b) t) e)
3091 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
3092 T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
3093 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
3094 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
3095 T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
3096 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
3097 f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
3098 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda
3099 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
3100 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
3101 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u)))))))
3102 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
3103 T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda
3104 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3105 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3106 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H11:
3107 (drop n0 O x0 e)).(or4_intro0 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4
3108 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
3109 C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3110 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f)
3111 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3112 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
3113 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
3114 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
3115 O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
3116 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
3117 C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3118 T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f:
3119 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3120 n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_:
3121 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3122 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3123 T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 x0 e H11
3124 t))) (\lambda (H11: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
3125 (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
3126 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0
3127 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3128 T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0:
3129 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
3130 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O
3131 x0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
3132 T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) O (CHead x0 (Bind
3133 b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3134 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
3135 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
3136 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
3137 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3138 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
3139 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3140 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3141 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3142 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3143 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
3144 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3145 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3146 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3147 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3148 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3149 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5:
3150 T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x4))).(\lambda (H13: (drop n0 O
3151 x0 (CHead x3 (Flat x2) x5))).(\lambda (H14: (subst0 O v x4 x5)).(eq_ind_r C
3152 (CHead x3 (Flat x2) x4) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind
3153 b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3154 T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f:
3155 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
3156 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
3157 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3158 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
3159 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3160 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3161 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3162 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3163 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
3164 f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3165 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3166 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3167 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3168 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3169 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat
3170 x2) x4)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3171 T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f) u))))))
3172 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
3173 O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
3174 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3175 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
3176 x3 (Flat x2) x4) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
3177 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t)
3178 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3179 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
3180 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3181 (CHead x3 (Flat x2) x4) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda
3182 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
3183 x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
3184 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3185 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
3186 (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f:
3187 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2)
3188 x4) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3189 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f)
3190 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3191 T).(subst0 O v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x4))
3192 (drop_drop (Bind b) n0 x0 (CHead x3 (Flat x2) x5) H13 t) H14)) e H12))))))))
3193 H11)) (\lambda (H11: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda
3194 (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
3195 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2
3196 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
3197 (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda
3198 (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
3199 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O
3200 x0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3201 C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead x0
3202 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3203 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
3204 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
3205 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
3206 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3207 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
3208 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3209 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3210 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3211 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3212 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
3213 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3214 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3215 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3216 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3217 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3218 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
3219 T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H13: (drop n0 O
3220 x0 (CHead x4 (Flat x2) x5))).(\lambda (H14: (csubst0 O v x3 x4)).(eq_ind_r C
3221 (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind
3222 b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3223 T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f:
3224 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
3225 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
3226 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3227 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
3228 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3229 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3230 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3231 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3232 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
3233 f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3234 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3235 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3236 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3237 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3238 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat
3239 x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3240 T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u))))))
3241 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
3242 O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
3243 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3244 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
3245 x3 (Flat x2) x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
3246 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t)
3247 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3248 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
3249 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3250 (CHead x3 (Flat x2) x5) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda
3251 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
3252 x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
3253 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3254 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
3255 (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f:
3256 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2)
3257 x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3258 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3259 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3260 T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5))
3261 (drop_drop (Bind b) n0 x0 (CHead x4 (Flat x2) x5) H13 t) H14)) e H12))))))))
3262 H11)) (\lambda (H11: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3263 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
3264 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3265 T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) (\lambda (_:
3266 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3267 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3268 T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda
3269 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3270 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3271 C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w)))))))
3272 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3273 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3274 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0)
3275 O (CHead x0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
3276 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
3277 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
3278 O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
3279 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3280 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
3281 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3282 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3283 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3284 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3285 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
3286 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3287 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
3288 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3289 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3290 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3291 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
3292 T).(\lambda (x6: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda
3293 (H13: (drop n0 O x0 (CHead x4 (Flat x2) x6))).(\lambda (H14: (subst0 O v x5
3294 x6)).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x3 (Flat x2) x5)
3295 (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind b) t) c0) (ex3_4 F C T T
3296 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
3297 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3298 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f)
3299 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3300 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
3301 C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u))))))
3302 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
3303 O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
3304 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
3305 C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3306 T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f:
3307 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3308 n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_:
3309 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3310 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3311 T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O
3312 (CHead x0 (Bind b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f:
3313 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2)
3314 x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3315 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f)
3316 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3317 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
3318 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
3319 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
3320 (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u))))))
3321 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
3322 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
3323 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
3324 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
3325 (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat
3326 f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3327 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3328 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
3329 (ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
3330 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
3331 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
3332 (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat
3333 f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3334 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3335 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
3336 x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0
3337 x0 (CHead x4 (Flat x2) x6) H13 t) H14 H15)) e H12)))))))))) H11)) H10)))))))
3338 (\lambda (f: F).(\lambda (H6: (drop (r (Flat f) n0) O c e)).(\lambda (H7: (eq
3339 nat (S n0) (s (Flat f) x1))).(let H8 \def (f_equal nat nat (\lambda (e0:
3340 nat).e0) (S n0) (s (Flat f) x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda
3341 (n1: nat).(csubst0 n1 v c x0)) H5 (S n0) H8) in (let H10 \def (H x0 v H9 e
3342 H6) in (or4_ind (drop (S n0) O x0 e) (ex3_4 F C T T (\lambda (f0: F).(\lambda
3343 (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
3344 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3345 n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
3346 (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0:
3347 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
3348 f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
3349 T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda
3350 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
3351 C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3352 T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
3353 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3354 n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
3355 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3356 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
3357 (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e)
3358 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
3359 T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0:
3360 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
3361 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
3362 T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0:
3363 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
3364 f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
3365 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda
3366 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
3367 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
3368 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u)))))))
3369 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
3370 (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)))))))
3371 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3372 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3373 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H11:
3374 (drop (S n0) O x0 e)).(or4_intro0 (drop (S n0) O (CHead x0 (Flat f) t) e)
3375 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
3376 T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0:
3377 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
3378 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
3379 T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0:
3380 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
3381 f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
3382 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda
3383 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
3384 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
3385 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u)))))))
3386 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
3387 (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)))))))
3388 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3389 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3390 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat
3391 f) n0 x0 e H11 t))) (\lambda (H11: (ex3_4 F C T T (\lambda (f0: F).(\lambda
3392 (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
3393 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3394 n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
3395 (u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda
3396 (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0
3397 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
3398 (w: T).(drop (S n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
3399 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0)
3400 O (CHead x0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
3401 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
3402 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3403 n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
3404 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3405 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3406 T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
3407 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
3408 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3409 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
3410 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3411 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
3412 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
3413 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
3414 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
3415 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
3416 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda
3417 (x5: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x4))).(\lambda (H13: (drop
3418 (S n0) O x0 (CHead x3 (Flat x2) x5))).(\lambda (H14: (subst0 O v x4
3419 x5)).(eq_ind_r C (CHead x3 (Flat x2) x4) (\lambda (c0: C).(or4 (drop (S n0) O
3420 (CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
3421 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u))))))
3422 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3423 n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
3424 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3425 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3426 T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
3427 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
3428 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3429 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
3430 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
3431 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
3432 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
3433 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
3434 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
3435 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
3436 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3
3437 (Flat x2) x4)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
3438 T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u))))))
3439 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3440 n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
3441 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3442 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3443 T).(eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
3444 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
3445 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
3446 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
3447 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
3448 (_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u))))))) (\lambda
3449 (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
3450 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda
3451 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3452 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3453 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C
3454 T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
3455 (CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
3456 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
3457 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
3458 T).(\lambda (w: T).(subst0 O v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3
3459 (Flat x2) x4)) (drop_drop (Flat f) n0 x0 (CHead x3 (Flat x2) x5) H13 t) H14))
3460 e H12)))))))) H11)) (\lambda (H11: (ex3_4 F C C T (\lambda (f0: F).(\lambda
3461 (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
3462 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
3463 n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
3464 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C
3465 C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
3466 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
3467 C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_:
3468 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))
3469 (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0:
3470 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat
3471 f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
3472 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda
3473 (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3474 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3475 T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
3476 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
3477 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3478 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
3479 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3480 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
3481 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
3482 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
3483 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
3484 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
3485 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda
3486 (x5: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H13: (drop
3487 (S n0) O x0 (CHead x4 (Flat x2) x5))).(\lambda (H14: (csubst0 O v x3
3488 x4)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O
3489 (CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
3490 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u))))))
3491 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3492 n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
3493 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3494 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3495 T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
3496 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
3497 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3498 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
3499 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
3500 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
3501 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
3502 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
3503 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
3504 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
3505 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3
3506 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
3507 T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) u))))))
3508 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3509 n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
3510 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3511 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3512 T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
3513 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
3514 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
3515 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
3516 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
3517 (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u))))))) (\lambda
3518 (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
3519 T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda
3520 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3521 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3522 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C
3523 C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C
3524 (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
3525 (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
3526 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3527 C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead
3528 x3 (Flat x2) x5)) (drop_drop (Flat f) n0 x0 (CHead x4 (Flat x2) x5) H13 t)
3529 H14)) e H12)))))))) H11)) (\lambda (H11: (ex4_5 F C C T T (\lambda (f0:
3530 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3531 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
3532 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 (Flat f0)
3533 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3534 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3535 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3536 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda
3537 (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u)))))))
3538 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
3539 (w: T).(drop (S n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_:
3540 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3541 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3542 T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead x0
3543 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
3544 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0:
3545 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
3546 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_:
3547 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3548 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
3549 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
3550 C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0)
3551 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3552 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
3553 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat
3554 f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3555 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0)
3556 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3557 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3558 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3559 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
3560 T).(\lambda (x6: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda
3561 (H13: (drop (S n0) O x0 (CHead x4 (Flat x2) x6))).(\lambda (H14: (subst0 O v
3562 x5 x6)).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x3 (Flat x2)
3563 x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 F C
3564 T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
3565 c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
3566 T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0)
3567 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3568 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
3569 C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f0) u))))))
3570 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
3571 n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_:
3572 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
3573 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
3574 (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f0) u))))))) (\lambda (f0:
3575 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3576 n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_:
3577 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3578 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3579 T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O
3580 (CHead x0 (Flat f) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0:
3581 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2)
3582 x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
3583 (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat
3584 f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3585 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
3586 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
3587 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
3588 (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u))))))
3589 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
3590 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
3591 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
3592 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
3593 (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat
3594 f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3595 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3596 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
3597 (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
3598 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
3599 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
3600 (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat
3601 f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3602 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3603 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
3604 x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Flat f) n0
3605 x0 (CHead x4 (Flat x2) x6) H13 t) H14 H15)) e H12)))))))))) H11)) H10)))))))
3606 k (drop_gen_drop k c e t n0 H1) H3) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C
3607 nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k
3608 j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3
3609 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
3610 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
3611 c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
3612 nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda
3613 (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_:
3614 C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
3615 C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 F
3616 C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
3617 e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3618 T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
3619 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3620 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3621 T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
3622 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f)
3623 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3624 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3625 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
3626 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3627 T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
3628 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3629 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3630 T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1:
3631 C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S n0) (s k x2))).(\lambda (H4:
3632 (eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6:
3633 (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop
3634 (S n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3635 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
3636 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead
3637 e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
3638 (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
3639 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
3640 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
3641 O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
3642 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
3643 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3644 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3645 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f)
3646 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3647 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3648 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3649 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S
3650 n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1 k0 x0) e) (ex3_4 F C T T
3651 (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3652 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3653 T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e0 (Flat f) w))))))
3654 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
3655 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
3656 C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
3657 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
3658 k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
3659 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
3660 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3661 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3662 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2
3663 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
3664 (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3665 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3666 e2))))))))))) (\lambda (b: B).(\lambda (H7: (drop (r (Bind b) n0) O c
3667 e)).(\lambda (H8: (eq nat (S n0) (s (Bind b) x2))).(let H9 \def (f_equal nat
3668 nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O
3669 \Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x2) H8) in (let H10 \def
3670 (eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 n0 H9) in (let
3671 H11 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H9) in
3672 (let H12 \def (IHn c x1 v H10 e H7) in (or4_ind (drop n0 O x1 e) (ex3_4 F C T
3673 T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3674 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3675 T).(\lambda (w: T).(drop n0 O x1 (CHead e0 (Flat f) w)))))) (\lambda (_:
3676 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3677 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3678 T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
3679 C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2 (Flat f) u))))))
3680 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
3681 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
3682 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u)))))))
3683 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
3684 T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
3685 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3686 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
3687 (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Bind b) x0)
3688 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda
3689 (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
3690 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
3691 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
3692 T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
3693 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
3694 f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
3695 T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda
3696 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
3697 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
3698 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u)))))))
3699 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
3700 T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda
3701 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3702 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3703 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H13:
3704 (drop n0 O x1 e)).(or4_intro0 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4
3705 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
3706 C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3707 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f)
3708 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3709 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
3710 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
3711 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
3712 O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
3713 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
3714 C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3715 T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f:
3716 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3717 n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_:
3718 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3719 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3720 T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 x1 e H13
3721 x0))) (\lambda (H13: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
3722 (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
3723 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0
3724 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3725 T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0:
3726 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
3727 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O
3728 x1 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
3729 T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) O (CHead x1 (Bind
3730 b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3731 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
3732 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
3733 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
3734 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3735 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
3736 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3737 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3738 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3739 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3740 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
3741 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3742 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3743 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3744 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3745 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3746 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6:
3747 T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x5))).(\lambda (H15: (drop n0 O
3748 x1 (CHead x4 (Flat x3) x6))).(\lambda (H16: (subst0 O v x5 x6)).(eq_ind_r C
3749 (CHead x4 (Flat x3) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind
3750 b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3751 T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f:
3752 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
3753 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
3754 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3755 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
3756 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3757 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3758 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3759 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3760 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
3761 f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3762 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3763 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3764 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3765 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3766 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat
3767 x3) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3768 T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f) u))))))
3769 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
3770 O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
3771 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3772 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
3773 x4 (Flat x3) x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
3774 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0)
3775 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3776 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
3777 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3778 (CHead x4 (Flat x3) x5) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda
3779 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
3780 x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
3781 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3782 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
3783 (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f:
3784 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3)
3785 x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3786 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f)
3787 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3788 T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x5))
3789 (drop_drop (Bind b) n0 x1 (CHead x4 (Flat x3) x6) H15 x0) H16)) e H14))))))))
3790 H13)) (\lambda (H13: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda
3791 (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
3792 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2
3793 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
3794 (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda
3795 (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
3796 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O
3797 x1 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3798 C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead x1
3799 (Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3800 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
3801 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
3802 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
3803 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3804 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
3805 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3806 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3807 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3808 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3809 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
3810 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3811 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3812 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3813 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3814 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3815 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6:
3816 T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H15: (drop n0 O
3817 x1 (CHead x5 (Flat x3) x6))).(\lambda (H16: (csubst0 O v x4 x5)).(eq_ind_r C
3818 (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind
3819 b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3820 T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f:
3821 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
3822 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
3823 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3824 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
3825 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3826 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3827 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3828 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3829 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
3830 f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3831 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3832 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3833 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3834 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3835 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat
3836 x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
3837 T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u))))))
3838 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
3839 O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
3840 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3841 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
3842 x4 (Flat x3) x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
3843 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0)
3844 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3845 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
3846 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3847 (CHead x4 (Flat x3) x6) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda
3848 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
3849 x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
3850 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
3851 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
3852 (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f:
3853 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3)
3854 x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3855 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3856 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3857 T).(csubst0 O v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6))
3858 (drop_drop (Bind b) n0 x1 (CHead x5 (Flat x3) x6) H15 x0) H16)) e H14))))))))
3859 H13)) (\lambda (H13: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3860 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
3861 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3862 T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) (\lambda (_:
3863 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3864 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3865 T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda
3866 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
3867 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3868 C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w)))))))
3869 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3870 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
3871 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0)
3872 O (CHead x1 (Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
3873 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
3874 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
3875 O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
3876 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
3877 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
3878 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
3879 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3880 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3881 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
3882 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
3883 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
3884 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
3885 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3886 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3887 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
3888 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6:
3889 T).(\lambda (x7: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x6))).(\lambda
3890 (H15: (drop n0 O x1 (CHead x5 (Flat x3) x7))).(\lambda (H16: (subst0 O v x6
3891 x7)).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6)
3892 (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0) c0) (ex3_4 F C T
3893 T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
3894 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3895 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f)
3896 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3897 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
3898 C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u))))))
3899 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
3900 O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
3901 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
3902 C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
3903 T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f:
3904 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3905 n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_:
3906 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3907 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3908 T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O
3909 (CHead x1 (Bind b) x0) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f:
3910 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3)
3911 x6) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
3912 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f)
3913 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3914 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
3915 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1
3916 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
3917 (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u))))))
3918 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
3919 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
3920 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1
3921 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
3922 (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat
3923 f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3924 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3925 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
3926 (ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
3927 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1
3928 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
3929 (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat
3930 f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
3931 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
3932 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
3933 x3 x4 x5 x6 x7 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0
3934 x1 (CHead x5 (Flat x3) x7) H15 x0) H16 H17)) e H14)))))))))) H13))
3935 H12)))))))) (\lambda (f: F).(\lambda (H7: (drop (r (Flat f) n0) O c
3936 e)).(\lambda (H8: (eq nat (S n0) (s (Flat f) x2))).(let H9 \def (f_equal nat
3937 nat (\lambda (e0: nat).e0) (S n0) (s (Flat f) x2) H8) in (let H10 \def
3938 (eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 (S n0) H9) in
3939 (let H11 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 (S
3940 n0) H9) in (let H12 \def (H x1 v H10 e H7) in (or4_ind (drop (S n0) O x1 e)
3941 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
3942 T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0:
3943 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e0 (Flat f0)
3944 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3945 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
3946 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
3947 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
3948 n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
3949 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
3950 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
3951 (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_:
3952 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead
3953 e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
3954 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
3955 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
3956 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T
3957 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3958 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
3959 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0)
3960 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3961 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
3962 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
3963 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
3964 n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
3965 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
3966 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
3967 (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
3968 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3969 n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_:
3970 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3971 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3972 T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H13: (drop (S n0) O
3973 x1 e)).(or4_intro0 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T
3974 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
3975 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
3976 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0)
3977 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
3978 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
3979 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
3980 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
3981 n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
3982 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
3983 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
3984 (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
3985 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3986 n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_:
3987 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
3988 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
3989 T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 x1 e H13
3990 x0))) (\lambda (H13: (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
3991 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
3992 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
3993 n0) O x1 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
3994 (u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda
3995 (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0
3996 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
3997 (w: T).(drop (S n0) O x1 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
3998 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0)
3999 O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
4000 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
4001 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
4002 n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_:
4003 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
4004 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
4005 T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
4006 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
4007 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4008 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
4009 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
4010 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
4011 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
4012 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4013 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
4014 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
4015 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda
4016 (x6: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x5))).(\lambda (H15: (drop
4017 (S n0) O x1 (CHead x4 (Flat x3) x6))).(\lambda (H16: (subst0 O v x5
4018 x6)).(eq_ind_r C (CHead x4 (Flat x3) x5) (\lambda (c0: C).(or4 (drop (S n0) O
4019 (CHead x1 (Flat f) x0) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
4020 C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u))))))
4021 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
4022 n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_:
4023 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
4024 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
4025 T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
4026 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
4027 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4028 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
4029 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
4030 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
4031 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
4032 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4033 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
4034 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
4035 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead
4036 x4 (Flat x3) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda
4037 (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f0)
4038 u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
4039 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda
4040 (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
4041 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
4042 T).(eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
4043 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
4044 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
4045 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
4046 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
4047 (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u))))))) (\lambda
4048 (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
4049 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)))))))
4050 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
4051 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4052 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C
4053 T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
4054 (CHead x4 (Flat x3) x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
4055 (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
4056 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
4057 T).(\lambda (w: T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4
4058 (Flat x3) x5)) (drop_drop (Flat f) n0 x1 (CHead x4 (Flat x3) x6) H15 x0)
4059 H16)) e H14)))))))) H13)) (\lambda (H13: (ex3_4 F C C T (\lambda (f0:
4060 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
4061 f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
4062 T).(drop (S n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda
4063 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind
4064 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
4065 C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
4066 (e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 (Flat f0) u))))))
4067 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
4068 v e1 e2))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T
4069 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
4070 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
4071 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0)
4072 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
4073 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
4074 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
4075 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
4076 n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
4077 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
4078 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
4079 (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
4080 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
4081 n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_:
4082 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
4083 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4084 T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4:
4085 C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H14: (eq C e (CHead x4 (Flat
4086 x3) x6))).(\lambda (H15: (drop (S n0) O x1 (CHead x5 (Flat x3) x6))).(\lambda
4087 (H16: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0:
4088 C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 F C T T (\lambda
4089 (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0
4090 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
4091 (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w))))))
4092 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
4093 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
4094 C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0:
4095 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
4096 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
4097 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
4098 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
4099 (_: T).(eq C c0 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_:
4100 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
4101 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
4102 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
4103 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4104 (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x1 (Flat
4105 f) x0) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
4106 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0
4107 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
4108 (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w))))))
4109 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
4110 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
4111 C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))
4112 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
4113 n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
4114 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
4115 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
4116 (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0)
4117 u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
4118 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0)
4119 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
4120 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
4121 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
4122 (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
4123 C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))
4124 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
4125 n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
4126 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))
4127 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1
4128 (CHead x5 (Flat x3) x6) H15 x0) H16)) e H14)))))))) H13)) (\lambda (H13:
4129 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
4130 (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
4131 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
4132 n0) O x1 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
4133 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
4134 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4135 (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0:
4136 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
4137 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
4138 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e2 (Flat f0)
4139 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
4140 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
4141 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
4142 (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0:
4143 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat
4144 f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
4145 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda
4146 (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
4147 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
4148 T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
4149 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
4150 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4151 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
4152 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
4153 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
4154 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
4155 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4156 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
4157 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
4158 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda
4159 (x6: T).(\lambda (x7: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3)
4160 x6))).(\lambda (H15: (drop (S n0) O x1 (CHead x5 (Flat x3) x7))).(\lambda
4161 (H16: (subst0 O v x6 x7)).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C
4162 (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat
4163 f) x0) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
4164 T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0:
4165 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
4166 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_:
4167 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
4168 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
4169 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
4170 C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0)
4171 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4172 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
4173 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
4174 f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
4175 T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0)
4176 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
4177 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
4178 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
4179 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 (Flat
4180 x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
4181 T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f0) u))))))
4182 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
4183 n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_:
4184 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
4185 (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
4186 T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
4187 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
4188 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
4189 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
4190 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
4191 (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))) (\lambda
4192 (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
4193 T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)))))))
4194 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
4195 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4196 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C
4197 C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
4198 T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))))
4199 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4200 (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)))))))
4201 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
4202 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4203 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7
4204 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1 (CHead x5
4205 (Flat x3) x7) H15 x0) H16 H17)) e H14)))))))))) H13)) H12)))))))) k
4206 (drop_gen_drop k c e t n0 H1) H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2
4207 t v (S n0) H0))))))))))) c1)))) n).
4209 Initial nodes: 36162
4212 theorem csubst0_drop_eq_back:
4213 \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0
4214 n v c1 c2) \to (\forall (e: C).((drop n O c2 e) \to (or4 (drop n O c1 e)
4215 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
4216 T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
4217 C).(\lambda (u1: T).(\lambda (_: T).(drop n O c1 (CHead e0 (Flat f) u1))))))
4218 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
4219 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4220 C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
4221 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n O c1 (CHead e1
4222 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4223 (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
4224 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
4225 f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
4226 T).(\lambda (_: T).(drop n O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_:
4227 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
4228 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4229 (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))
4231 \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2:
4232 C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c2
4233 e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
4234 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
4235 (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O
4236 c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
4237 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
4238 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
4239 f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
4240 T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
4241 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
4242 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4243 (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
4244 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e1
4245 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
4246 (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda
4247 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
4248 e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
4249 (H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c2
4250 e)).(eq_ind C c2 (\lambda (c: C).(or4 (drop O O c1 c) (ex3_4 F C T T (\lambda
4251 (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c (CHead e0
4252 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
4253 (_: T).(drop O O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
4254 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
4255 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c
4256 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4257 C).(\lambda (u: T).(drop O O c1 (CHead e1 (Flat f) u)))))) (\lambda (_:
4258 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
4259 (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
4260 (_: T).(\lambda (u2: T).(eq C c (CHead e2 (Flat f) u2))))))) (\lambda (f:
4261 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O
4262 O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda
4263 (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
4264 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
4265 O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2))
4266 (\lambda (n0: nat).(or4 (drop n0 n0 c1 c2) (ex3_4 F C T T (\lambda (f:
4267 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e0 (Flat
4268 f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
4269 T).(drop n0 n0 c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
4270 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1 u2)))))) (ex3_4 F C C T
4271 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c2
4272 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4273 C).(\lambda (u: T).(drop n0 n0 c1 (CHead e1 (Flat f) u)))))) (\lambda (_:
4274 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1
4275 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4276 C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e2 (Flat f) u2)))))))
4277 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4278 (_: T).(drop n0 n0 c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda
4279 (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1
4280 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4281 T).(\lambda (_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda
4282 (H1: (csubst0 y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t:
4283 T).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c c0)
4284 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
4285 T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
4286 C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e0 (Flat f) u1))))))
4287 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0
4288 t u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4289 C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
4290 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 c (CHead e1
4291 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4292 (_: T).(csubst0 n0 t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
4293 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2
4294 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
4295 (u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e1 (Flat f) u1))))))) (\lambda
4296 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
4297 T).(subst0 n0 t u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4298 C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda
4299 (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall
4300 (u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s
4301 k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead c k0 u2))
4302 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4:
4303 T).(eq C (CHead c k0 u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda
4304 (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0
4305 u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3:
4306 T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda
4307 (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c k0 u2)
4308 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4309 C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead e1 (Flat f)
4310 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4311 T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
4312 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c k0
4313 u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda
4314 (_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0
4315 u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4316 C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda
4317 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
4318 T).(csubst0 (s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i:
4319 nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0
4320 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def
4321 (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda (_:
4322 nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in
4323 (False_ind (or4 (drop (S i) (S i) (CHead c (Bind b) u1) (CHead c (Bind b)
4324 u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
4325 (u4: T).(eq C (CHead c (Bind b) u2) (CHead e0 (Flat f) u4)))))) (\lambda (f:
4326 F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead
4327 c (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_:
4328 C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (ex3_4 F C
4329 C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C
4330 (CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda
4331 (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S i) (S i) (CHead c (Bind b)
4332 u1) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4333 C).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda
4334 (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq
4335 C (CHead c (Bind b) u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda
4336 (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i)
4337 (CHead c (Bind b) u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda
4338 (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3
4339 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4340 T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))))) H4)))))))))) (\lambda (f:
4341 F).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2:
4342 T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i
4343 O)).(let H4 \def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O
4344 H3) in (eq_ind_r nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f)
4345 u1) (CHead c (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
4346 C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0
4347 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3:
4348 T).(\lambda (_: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead e0 (Flat f0)
4349 u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4:
4350 T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
4351 C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c (Flat f) u2) (CHead e2
4352 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
4353 (u: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda
4354 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1
4355 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
4356 C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e2
4357 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
4358 C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead
4359 e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4360 C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_:
4361 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
4362 n0 v0 e1 e2))))))))) (or4_intro1 (drop O O (CHead c (Flat f) u1) (CHead c
4363 (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
4364 T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 (Flat f0) u4))))))
4365 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O
4366 (CHead c (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda
4367 (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C
4368 C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C
4369 (CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
4370 (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c (Flat f) u1)
4371 (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4372 C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
4373 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C
4374 (CHead c (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda
4375 (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c
4376 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_:
4377 C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4))))))
4378 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4379 (_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0:
4380 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f)
4381 u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
4382 (u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) (CHead e0 (Flat f0)
4383 u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4:
4384 T).(subst0 O v0 u3 u4))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u2))
4385 (drop_refl (CHead c (Flat f) u1)) H4)) i H3)))))))))) k)) (\lambda (k:
4386 K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4:
4387 C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop
4388 i i c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
4389 T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f:
4390 F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0
4391 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda
4392 (u2: T).(subst0 i v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
4393 C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u))))))
4394 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3
4395 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4396 C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
4397 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4
4398 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4399 C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f) u1)))))))
4400 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4401 (u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
4402 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to
4403 (\forall (u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead
4404 c3 k0 u) (CHead c4 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
4405 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e0 (Flat f)
4406 u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
4407 T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e0 (Flat f) u1))))))
4408 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (s
4409 k0 i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda
4410 (e2: C).(\lambda (u0: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) u0))))))
4411 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop (s k0
4412 i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) (\lambda (_:
4413 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1
4414 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4415 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f)
4416 u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
4417 T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f)
4418 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
4419 T).(\lambda (u2: T).(subst0 (s k0 i) v0 u1 u2)))))) (\lambda (_: F).(\lambda
4420 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0
4421 e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3:
4422 C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3
4423 c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T
4424 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4
4425 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
4426 T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u1)))))) (\lambda (_:
4427 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2))))))
4428 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
4429 T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
4430 C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u))))))
4431 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
4432 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4433 C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2)))))))
4434 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4435 (_: T).(drop i i c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
4436 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2))))))
4437 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4438 (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat
4439 (S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat
4440 return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
4441 True])) I O H4) in (False_ind (or4 (drop (S i) (S i) (CHead c3 (Bind b) u)
4442 (CHead c4 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
4443 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead e0
4444 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
4445 (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead e0 (Flat f) u1))))))
4446 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (S
4447 i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda
4448 (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Bind b) u) (CHead e2 (Flat f)
4449 u0)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0:
4450 T).(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0))))))
4451 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S
4452 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda
4453 (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead
4454 e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4455 C).(\lambda (u1: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u)
4456 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4457 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (S i) v0 u1 u2)))))) (\lambda
4458 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
4459 T).(csubst0 (S i) v0 e1 e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i:
4460 nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2:
4461 (csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop i i c3 c4)
4462 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
4463 (u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
4464 C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f0) u1))))))
4465 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i
4466 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
4467 C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
4468 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1
4469 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4470 (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda
4471 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2
4472 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
4473 C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0)
4474 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
4475 T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1:
4476 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1
4477 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i O)).(let H5 \def
4478 (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c3 c4)
4479 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
4480 (u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
4481 C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0 (Flat f0)
4482 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
4483 T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
4484 C).(\lambda (e2: C).(\lambda (u0: T).(eq C c4 (CHead e2 (Flat f0) u0))))))
4485 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop n0
4486 n0 c3 (CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
4487 (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda
4488 (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq
4489 C c4 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
4490 (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0)
4491 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
4492 T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1:
4493 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1
4494 e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0:
4495 nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r nat O (\lambda (n0:
4496 nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u) (CHead c4 (Flat f) u)) (ex3_4 F C
4497 T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C
4498 (CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda
4499 (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u)
4500 (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
4501 T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
4502 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f)
4503 u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
4504 (_: C).(\lambda (u0: T).(drop n0 n0 (CHead c3 (Flat f) u) (CHead e1 (Flat f0)
4505 u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4506 T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_:
4507 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Flat f)
4508 u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
4509 (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u)
4510 (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4511 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_:
4512 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
4513 n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c3 (Flat f) u) (CHead c4
4514 (Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
4515 T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2))))))
4516 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O
4517 (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda
4518 (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C
4519 C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C
4520 (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda
4521 (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u)
4522 (CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4523 C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
4524 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C
4525 (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda
4526 (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3
4527 (Flat f) u) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
4528 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2))))))
4529 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4530 (_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0:
4531 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f)
4532 u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
4533 (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0)
4534 u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4535 T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c4 (Flat f) u))
4536 (drop_refl (CHead c3 (Flat f) u)) H6)) i H4)))))))))))) k)) (\lambda (k:
4537 K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1:
4538 T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4:
4539 C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4
4540 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq
4541 C c4 (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
4542 (u3: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_:
4543 F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
4544 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
4545 T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
4546 C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u))))))
4547 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
4548 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4549 C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4)))))))
4550 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda
4551 (_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_:
4552 C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
4553 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4554 (_: T).(csubst0 i v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop
4555 (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead c4 k0 u2)) (ex3_4 F C T T (\lambda
4556 (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0
4557 u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
4558 (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e0
4559 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda
4560 (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f:
4561 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 k0 u2)
4562 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4563 C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e1 (Flat
4564 f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4565 T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
4566 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0
4567 u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda
4568 (_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0
4569 u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4570 C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda
4571 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
4572 T).(csubst0 (s k0 i) v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i:
4573 nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0
4574 i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3
4575 c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T
4576 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4
4577 (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3:
4578 T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_:
4579 F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
4580 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
4581 T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
4582 C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u))))))
4583 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
4584 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4585 C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4)))))))
4586 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda
4587 (_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_:
4588 C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
4589 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4590 (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6
4591 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda
4592 (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5)
4593 in (False_ind (or4 (drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead c4 (Bind
4594 b) u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
4595 T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e0 (Flat f) u4))))))
4596 (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i)
4597 (S i) (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_:
4598 F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3
4599 u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4600 C).(\lambda (u: T).(eq C (CHead c4 (Bind b) u2) (CHead e2 (Flat f) u))))))
4601 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S i)
4602 (S i) (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (_:
4603 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1
4604 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4605 C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e2
4606 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
4607 (u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead e1
4608 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
4609 (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (\lambda (_:
4610 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
4611 (S i) v0 e1 e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i:
4612 nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0
4613 i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3
4614 c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T
4615 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4
4616 (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3:
4617 T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f0) u3)))))) (\lambda (_:
4618 F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
4619 (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
4620 T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
4621 C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f0) u))))))
4622 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
4623 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda
4624 (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f0)
4625 u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3:
4626 T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0) u3))))))) (\lambda (_:
4627 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0
4628 i v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4629 (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat i
4630 O)).(let H6 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4
4631 (drop n0 n0 c3 c4) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda
4632 (_: T).(\lambda (u4: T).(eq C c4 (CHead e0 (Flat f0) u4)))))) (\lambda (f0:
4633 F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0
4634 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda
4635 (u4: T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda
4636 (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u))))))
4637 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0
4638 c3 (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4639 C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
4640 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4
4641 (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
4642 C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0)
4643 u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3:
4644 T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1:
4645 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1
4646 e2)))))))))) H4 O H5) in (let H7 \def (eq_ind nat i (\lambda (n0:
4647 nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8 \def (eq_ind nat i (\lambda
4648 (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in (eq_ind_r nat O (\lambda (n0:
4649 nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F
4650 C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq
4651 C (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0:
4652 F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 (CHead c3
4653 (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_:
4654 C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T
4655 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C
4656 (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
4657 (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 (CHead c3 (Flat f) u1)
4658 (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4659 C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
4660 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C
4661 (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0:
4662 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop n0
4663 n0 (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_:
4664 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0
4665 n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4666 (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3 (drop O O
4667 (CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F C T T (\lambda (f0:
4668 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f)
4669 u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
4670 (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead e0 (Flat f0)
4671 u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4:
4672 T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
4673 C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 (Flat f) u2) (CHead e2
4674 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
4675 (u: T).(drop O O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda
4676 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1
4677 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
4678 C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e2
4679 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
4680 C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead
4681 e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4682 C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_:
4683 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
4684 O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_:
4685 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f)
4686 u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
4687 (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1)
4688 (CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4689 C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_:
4690 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
4691 O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c4 (Flat f) u2))
4692 (drop_refl (CHead c3 (Flat f) u1)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2
4693 H1))) H) e (drop_gen_refl c2 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn:
4694 ((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to
4695 (\forall (e: C).((drop n0 O c2 e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T
4696 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
4697 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
4698 T).(\lambda (_: T).(drop n0 O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_:
4699 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
4700 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
4701 T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
4702 C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c1 (CHead e1 (Flat f) u))))))
4703 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
4704 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4705 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
4706 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4707 (_: T).(drop n0 O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda
4708 (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
4709 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4710 T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind
4711 (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to
4712 (\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O c e) (ex3_4 F C
4713 T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
4714 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
4715 T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
4716 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
4717 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
4718 T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
4719 C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f)
4720 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4721 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
4722 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
4723 f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
4724 T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_:
4725 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
4726 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4727 (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1:
4728 nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 (S n0) v (CSort
4729 n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 e)).(csubst0_gen_sort
4730 c2 v (S n0) n1 H (or4 (drop (S n0) O (CSort n1) e) (ex3_4 F C T T (\lambda
4731 (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0
4732 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
4733 (_: T).(drop (S n0) O (CSort n1) (CHead e0 (Flat f) u1)))))) (\lambda (_:
4734 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
4735 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
4736 T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
4737 C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat
4738 f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
4739 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
4740 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
4741 f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
4742 T).(\lambda (_: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat f) u1)))))))
4743 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4744 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
4745 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))
4746 (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S
4747 n0) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O
4748 c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
4749 (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
4750 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f)
4751 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
4752 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
4753 C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u))))))
4754 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
4755 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4756 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
4757 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
4758 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4759 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f)
4760 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
4761 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
4762 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
4763 e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda
4764 (v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t) c2)).(\lambda (e:
4765 C).(\lambda (H1: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_:
4766 T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_:
4767 nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
4768 v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k
4769 j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
4770 (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_:
4771 T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2:
4772 T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
4773 (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
4774 T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S
4775 n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
4776 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
4777 (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
4778 n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
4779 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
4780 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
4781 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4782 C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u))))))
4783 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
4784 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4785 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
4786 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4787 (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
4788 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
4789 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4790 (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H2: (ex3_2 T nat
4791 (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2:
4792 T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
4793 nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j:
4794 nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2
4795 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4
4796 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
4797 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
4798 (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
4799 n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
4800 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
4801 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
4802 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4803 C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u))))))
4804 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
4805 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4806 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
4807 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4808 (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
4809 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
4810 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
4811 (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda
4812 (x1: nat).(\lambda (H3: (eq nat (S n0) (s k x1))).(\lambda (H4: (eq C c2
4813 (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(let H6 \def (eq_ind C c2
4814 (\lambda (c0: C).(drop (S n0) O c0 e)) H1 (CHead c k x0) H4) in (K_ind
4815 (\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0 n0) O c e) \to
4816 (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda
4817 (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
4818 (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
4819 n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
4820 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
4821 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
4822 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4823 C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u))))))
4824 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
4825 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4826 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
4827 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4828 (_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) (\lambda
4829 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
4830 T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4831 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b:
4832 B).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(\lambda (H8: (drop (r
4833 (Bind b) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).(match
4834 e0 in nat return (\lambda (_: nat).nat) with [O \Rightarrow n0 | (S n1)
4835 \Rightarrow n1])) (S n0) (S x1) H7) in (let H10 \def (eq_ind_r nat x1
4836 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H9) in (or4_intro0 (drop (S n0)
4837 O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
4838 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
4839 (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
4840 n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
4841 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
4842 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
4843 T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
4844 C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead
4845 e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4846 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
4847 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
4848 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4849 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
4850 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4851 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
4852 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
4853 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H8 t))))))) (\lambda (f:
4854 F).(\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8: (drop (r
4855 (Flat f) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).e0) (S
4856 n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v
4857 t x0)) H5 (S n0) H9) in (or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e)
4858 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
4859 (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
4860 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
4861 (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
4862 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
4863 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
4864 f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
4865 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
4866 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
4867 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
4868 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2)))))))
4869 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4870 (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
4871 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4872 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
4873 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop
4874 (Flat f) n0 c e H8 t))))))) k H3 (drop_gen_drop k c e x0 n0 H6)))))))) H2))
4875 (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0)
4876 (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t))))
4877 (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat
4878 (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3:
4879 C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j:
4880 nat).(csubst0 j v c c3))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T
4881 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
4882 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
4883 T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1))))))
4884 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
4885 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4886 C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
4887 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k
4888 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
4889 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
4890 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
4891 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
4892 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1
4893 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
4894 (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda
4895 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
4896 e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0)
4897 (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1
4898 v c x0)).(let H6 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1
4899 (CHead x0 k t) H4) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to
4900 ((drop (r k0 n0) O x0 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C
4901 T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
4902 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
4903 T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1))))))
4904 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
4905 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4906 C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
4907 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
4908 k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
4909 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
4910 (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq
4911 C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda
4912 (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead
4913 e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
4914 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
4915 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
4916 O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H7: (eq nat (S n0) (s (Bind b)
4917 x1))).(\lambda (H8: (drop (r (Bind b) n0) O x0 e)).(let H9 \def (f_equal nat
4918 nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O
4919 \Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H10 \def
4920 (eq_ind_r nat x1 (\lambda (n1: nat).(csubst0 n1 v c x0)) H5 n0 H9) in (let
4921 H11 \def (IHn c x0 v H10 e H8) in (or4_ind (drop n0 O c e) (ex3_4 F C T T
4922 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
4923 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
4924 T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
4925 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
4926 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
4927 T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
4928 C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u))))))
4929 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
4930 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4931 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
4932 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4933 (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
4934 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
4935 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4936 (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
4937 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
4938 T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
4939 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
4940 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
4941 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
4942 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
4943 f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
4944 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
4945 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
4946 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4947 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
4948 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4949 (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
4950 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4951 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
4952 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
4953 (H12: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e)
4954 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
4955 T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
4956 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
4957 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
4958 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
4959 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
4960 f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
4961 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
4962 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
4963 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4964 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
4965 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4966 (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
4967 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
4968 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
4969 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop
4970 (Bind b) n0 c e H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f:
4971 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
4972 f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
4973 T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
4974 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C
4975 T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
4976 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
4977 T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
4978 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
4979 (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f:
4980 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
4981 f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
4982 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
4983 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
4984 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
4985 C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
4986 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
4987 (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
4988 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
4989 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4990 (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
4991 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
4992 (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
4993 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
4994 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
4995 (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda
4996 (x4: T).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2)
4997 x5))).(\lambda (H14: (drop n0 O c (CHead x3 (Flat x2) x4))).(\lambda (H15:
4998 (subst0 O v x4 x5)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4
4999 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f:
5000 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
5001 f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5002 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
5003 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
5004 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5005 C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
5006 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
5007 (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
5008 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
5009 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5010 (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
5011 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5012 (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
5013 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5014 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5015 (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind
5016 b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
5017 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0
5018 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
5019 (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1))))))
5020 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
5021 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5022 C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f) u))))))
5023 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
5024 O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
5025 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
5026 C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
5027 T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f)
5028 u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
5029 T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
5030 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5031 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5032 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
5033 (ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
5034 T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u2))))))
5035 (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5036 n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
5037 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
5038 x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 c
5039 (CHead x3 (Flat x2) x4) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12:
5040 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
5041 T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
5042 C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u))))))
5043 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
5044 v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda
5045 (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
5046 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1
5047 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
5048 (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
5049 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
5050 T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
5051 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
5052 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5053 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
5054 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5055 f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5056 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
5057 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
5058 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5059 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
5060 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5061 (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
5062 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5063 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
5064 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
5065 (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (H13: (eq
5066 C e (CHead x4 (Flat x2) x5))).(\lambda (H14: (drop n0 O c (CHead x3 (Flat x2)
5067 x5))).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x5)
5068 (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T
5069 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
5070 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
5071 T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f)
5072 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5073 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
5074 C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u))))))
5075 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
5076 O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
5077 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
5078 C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
5079 T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f:
5080 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5081 n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
5082 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
5083 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
5084 (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O
5085 (CHead c (Bind b) t) (CHead x4 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f:
5086 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat
5087 x2) x5) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
5088 (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat
5089 f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5090 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
5091 C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2
5092 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5093 (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u))))))
5094 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
5095 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5096 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2
5097 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5098 (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
5099 f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5100 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5101 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
5102 (ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5103 C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u))))))
5104 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
5105 O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
5106 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5
5107 (refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Bind b) n0 c (CHead x3
5108 (Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: (ex4_5 F C C T
5109 T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5110 (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
5111 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1
5112 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
5113 (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda
5114 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
5115 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5116 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
5117 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5118 (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
5119 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5120 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5121 (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
5122 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
5123 T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
5124 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
5125 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5126 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
5127 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5128 f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5129 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
5130 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
5131 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5132 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
5133 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5134 (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
5135 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5136 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
5137 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
5138 (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6:
5139 T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x6))).(\lambda (H14: (drop n0 O
5140 c (CHead x3 (Flat x2) x5))).(\lambda (H15: (subst0 O v x5 x6)).(\lambda (H16:
5141 (csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0:
5142 C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f:
5143 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
5144 f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5145 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
5146 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
5147 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5148 C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
5149 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
5150 (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
5151 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
5152 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5153 (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
5154 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5155 (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
5156 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5157 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5158 (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind
5159 b) t) (CHead x4 (Flat x2) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
5160 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0
5161 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
5162 (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1))))))
5163 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
5164 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5165 C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) u))))))
5166 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
5167 O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
5168 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
5169 C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
5170 T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f)
5171 u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
5172 T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
5173 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5174 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5175 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
5176 (ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5177 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2
5178 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5179 (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
5180 f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5181 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5182 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
5183 x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) (drop_drop (Bind b) n0
5184 c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e H13)))))))))) H12)) H11)))))))
5185 (\lambda (f: F).(\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8:
5186 (drop (r (Flat f) n0) O x0 e)).(let H9 \def (f_equal nat nat (\lambda (e0:
5187 nat).e0) (S n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1:
5188 nat).(csubst0 n1 v c x0)) H5 (S n0) H9) in (let H11 \def (H x0 v H10 e H8) in
5189 (or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
5190 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2))))))
5191 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5192 n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
5193 (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
5194 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5195 f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5196 T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
5197 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
5198 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5199 (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
5200 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead
5201 e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5202 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5203 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5204 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T
5205 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5206 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
5207 T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
5208 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5209 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
5210 C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u))))))
5211 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
5212 n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
5213 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
5214 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
5215 (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
5216 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5217 n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
5218 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
5219 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
5220 (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H12: (drop (S n0)
5221 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T
5222 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5223 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
5224 T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
5225 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5226 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
5227 C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u))))))
5228 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
5229 n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
5230 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
5231 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
5232 (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
5233 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5234 n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
5235 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
5236 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
5237 (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e
5238 H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
5239 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2))))))
5240 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5241 n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
5242 (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda
5243 (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0
5244 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
5245 T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_:
5246 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
5247 (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0:
5248 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
5249 f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5250 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
5251 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
5252 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5253 C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0:
5254 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
5255 (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
5256 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
5257 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5258 (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
5259 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5260 (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
5261 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5262 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5263 (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda
5264 (x4: T).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2)
5265 x5))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) x4))).(\lambda
5266 (H15: (subst0 O v x4 x5)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0:
5267 C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0:
5268 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
5269 f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5270 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
5271 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
5272 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5273 C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
5274 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
5275 (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
5276 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
5277 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5278 (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
5279 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5280 (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
5281 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5282 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5283 (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Flat
5284 f) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
5285 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0
5286 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
5287 T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
5288 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5289 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
5290 C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2
5291 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5292 (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
5293 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
5294 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5295 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2
5296 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
5297 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5298 (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5299 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5300 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5301 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0:
5302 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0
5303 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
5304 T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
5305 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5306 T).(subst0 O v u1 u2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5))
5307 (drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x4) H14 t) H15)) e H13))))))))
5308 H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda
5309 (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0:
5310 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead
5311 e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5312 C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0:
5313 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5314 f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5315 T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
5316 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0)
5317 O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
5318 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2))))))
5319 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5320 n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_:
5321 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5322 (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
5323 T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
5324 C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead
5325 e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5326 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
5327 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5328 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
5329 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5330 (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5331 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5332 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5333 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda
5334 (x5: T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x5))).(\lambda (H14: (drop
5335 (S n0) O c (CHead x3 (Flat x2) x5))).(\lambda (H15: (csubst0 O v x3
5336 x4)).(eq_ind_r C (CHead x4 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O
5337 (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
5338 C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2))))))
5339 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5340 n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_:
5341 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5342 (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
5343 T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
5344 C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead
5345 e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5346 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
5347 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
5348 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
5349 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5350 (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5351 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5352 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5353 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat f) t) (CHead x4
5354 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
5355 T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e0 (Flat f0)
5356 u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5357 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
5358 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
5359 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5360 C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u))))))
5361 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
5362 n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
5363 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
5364 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
5365 (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0)
5366 u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
5367 T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0)
5368 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5369 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5370 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
5371 (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5372 C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u))))))
5373 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
5374 n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
5375 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))
5376 x2 x3 x4 x5 (refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Flat f) n0 c
5377 (CHead x3 (Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12:
5378 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
5379 (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
5380 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5381 n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
5382 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5383 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5384 (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0:
5385 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5386 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
5387 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0)
5388 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5389 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5390 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
5391 (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0:
5392 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
5393 f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5394 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
5395 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
5396 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5397 C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0:
5398 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
5399 (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
5400 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
5401 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5402 (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
5403 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5404 (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
5405 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5406 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5407 (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda
5408 (x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x4
5409 (Flat x2) x6))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2)
5410 x5))).(\lambda (H15: (subst0 O v x5 x6)).(\lambda (H16: (csubst0 O v x3
5411 x4)).(eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0: C).(or4 (drop (S n0) O
5412 (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
5413 C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2))))))
5414 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5415 n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_:
5416 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5417 (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
5418 T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
5419 C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead
5420 e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5421 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
5422 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
5423 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
5424 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5425 (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5426 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5427 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5428 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x4
5429 (Flat x2) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
5430 T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 (Flat f0)
5431 u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5432 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
5433 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
5434 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5435 C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u))))))
5436 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
5437 n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
5438 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
5439 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
5440 (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0)
5441 u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
5442 T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0)
5443 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5444 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5445 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
5446 (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5447 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2
5448 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
5449 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5450 (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5451 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5452 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5453 O v e1 e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6))
5454 (drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e
5455 H13)))))))))) H12)) H11))))))) k H3 (drop_gen_drop k x0 e t n0 H6))))))))
5456 H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda
5457 (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3:
5458 C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
5459 (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
5460 C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_:
5461 T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2:
5462 T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
5463 (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
5464 T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0)
5465 O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
5466 (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f:
5467 F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5468 k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
5469 (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
5470 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5471 f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5472 T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_:
5473 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
5474 (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
5475 (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f:
5476 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5477 n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
5478 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5479 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5480 (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda
5481 (x2: nat).(\lambda (H3: (eq nat (S n0) (s k x2))).(\lambda (H4: (eq C c2
5482 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2
5483 v c x1)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1
5484 (CHead x1 k x0) H4) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x2)) \to
5485 ((drop (r k0 n0) O x1 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C
5486 T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5487 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
5488 T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1))))))
5489 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
5490 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5491 C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
5492 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
5493 k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
5494 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
5495 (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq
5496 C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda
5497 (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead
5498 e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5499 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5500 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5501 O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H8: (eq nat (S n0) (s (Bind b)
5502 x2))).(\lambda (H9: (drop (r (Bind b) n0) O x1 e)).(let H10 \def (f_equal nat
5503 nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O
5504 \Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x2) H8) in (let H11 \def
5505 (eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 n0 H10) in (let
5506 H12 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H10)
5507 in (let H13 \def (IHn c x1 v H11 e H9) in (or4_ind (drop n0 O c e) (ex3_4 F C
5508 T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5509 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
5510 T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
5511 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5512 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
5513 T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
5514 C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u))))))
5515 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
5516 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5517 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
5518 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5519 (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
5520 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5521 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5522 (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
5523 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
5524 T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
5525 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
5526 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5527 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
5528 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5529 f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5530 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
5531 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
5532 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5533 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
5534 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5535 (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
5536 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5537 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
5538 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
5539 (H14: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e)
5540 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
5541 T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
5542 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
5543 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5544 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
5545 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5546 f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5547 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
5548 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
5549 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5550 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
5551 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5552 (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
5553 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5554 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
5555 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop
5556 (Bind b) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f:
5557 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
5558 f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5559 T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
5560 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C
5561 T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5562 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
5563 T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
5564 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
5565 (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f:
5566 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
5567 f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5568 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
5569 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
5570 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5571 C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
5572 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
5573 (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
5574 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
5575 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5576 (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
5577 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5578 (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
5579 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5580 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5581 (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda
5582 (x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3)
5583 x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) x5))).(\lambda (H17:
5584 (subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4
5585 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f:
5586 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
5587 f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5588 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
5589 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
5590 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5591 C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
5592 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
5593 (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
5594 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
5595 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5596 (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
5597 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5598 (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
5599 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5600 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5601 (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind
5602 b) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
5603 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0
5604 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
5605 (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1))))))
5606 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
5607 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5608 C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u))))))
5609 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
5610 O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
5611 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
5612 C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
5613 T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f)
5614 u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
5615 T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
5616 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5617 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5618 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
5619 (ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
5620 T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2))))))
5621 (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5622 n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
5623 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
5624 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0 c
5625 (CHead x4 (Flat x3) x5) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14:
5626 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
5627 T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
5628 C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u))))))
5629 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
5630 v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda
5631 (e2: 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 n0 O c (CHead e1
5633 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
5634 (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
5635 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
5636 T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
5637 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
5638 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5639 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
5640 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5641 f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5642 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
5643 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
5644 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5645 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
5646 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5647 (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
5648 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5649 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
5650 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
5651 (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H15: (eq
5652 C e (CHead x5 (Flat x3) x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3)
5653 x6))).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6)
5654 (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T
5655 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
5656 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
5657 T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f)
5658 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5659 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
5660 C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u))))))
5661 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
5662 O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
5663 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
5664 C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
5665 T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f:
5666 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5667 n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
5668 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
5669 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
5670 (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O
5671 (CHead c (Bind b) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f:
5672 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat
5673 x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
5674 (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat
5675 f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5676 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
5677 C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
5678 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5679 (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u))))))
5680 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
5681 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5682 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
5683 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5684 (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
5685 f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5686 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5687 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
5688 (ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5689 C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f) u))))))
5690 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
5691 O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
5692 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x3 x4 x5 x6
5693 (refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop (Bind b) n0 c (CHead x4
5694 (Flat x3) x6) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: (ex4_5 F C C T
5695 T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5696 (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
5697 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1
5698 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
5699 (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda
5700 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
5701 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5702 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
5703 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5704 (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
5705 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5706 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5707 (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
5708 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
5709 T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
5710 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
5711 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5712 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
5713 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5714 f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5715 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
5716 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
5717 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5718 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
5719 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5720 (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
5721 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5722 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
5723 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
5724 (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7:
5725 T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H16: (drop n0 O
5726 c (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda (H18:
5727 (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0:
5728 C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f:
5729 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
5730 f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5731 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
5732 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
5733 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5734 C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
5735 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
5736 (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
5737 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
5738 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5739 (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
5740 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5741 (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
5742 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5743 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5744 (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind
5745 b) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
5746 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0
5747 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
5748 (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1))))))
5749 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
5750 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5751 C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f) u))))))
5752 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
5753 O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
5754 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
5755 C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
5756 T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f)
5757 u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
5758 T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
5759 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 (ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
5763 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2
5764 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5765 (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
5766 f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
5767 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
5768 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
5769 x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat x3) x7)) (drop_drop (Bind b) n0
5770 c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e H15)))))))))) H14)) H13))))))))
5771 (\lambda (f: F).(\lambda (H8: (eq nat (S n0) (s (Flat f) x2))).(\lambda (H9:
5772 (drop (r (Flat f) n0) O x1 e)).(let H10 \def (f_equal nat nat (\lambda (e0:
5773 nat).e0) (S n0) x2 H8) in (let H11 \def (eq_ind_r nat x2 (\lambda (n1:
5774 nat).(csubst0 n1 v c x1)) H6 (S n0) H10) in (let H12 \def (eq_ind_r nat x2
5775 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 (S n0) H10) in (let H13 \def (H x1
5776 v H11 e H9) in (or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0:
5777 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
5778 f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5779 T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_:
5780 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
5781 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
5782 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
5783 C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_:
5784 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
5785 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
5786 (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
5787 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5788 n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
5789 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5790 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5791 (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e)
5792 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
5793 (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
5794 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5795 (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5796 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
5797 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5798 f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5799 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
5800 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
5801 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5802 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2)))))))
5803 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5804 (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
5805 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5806 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
5807 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
5808 (H14: (drop (S n0) O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e)
5809 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
5810 (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
5811 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5812 (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5813 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
5814 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5815 f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5816 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
5817 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
5818 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5819 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2)))))))
5820 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5821 (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
5822 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5823 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
5824 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop
5825 (Flat f) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f0:
5826 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
5827 f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5828 T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_:
5829 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C
5830 T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C
5831 e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
5832 (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1))))))
5833 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
5834 u1 u2))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T
5835 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
5836 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
5837 T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
5838 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5839 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
5840 C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u))))))
5841 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
5842 n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
5843 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
5844 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
5845 (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
5846 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5847 n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
5848 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
5849 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
5850 (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda
5851 (x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4
5852 (Flat x3) x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3)
5853 x5))).(\lambda (H17: (subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6)
5854 (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T
5855 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
5856 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
5857 T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
5858 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5859 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
5860 C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u))))))
5861 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
5862 n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
5863 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
5864 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
5865 (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
5866 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5867 n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
5868 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
5869 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
5870 (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O
5871 (CHead c (Flat f) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0:
5872 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat
5873 x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
5874 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5875 (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5876 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
5877 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3)
5878 x6) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
5879 (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat
5880 f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
5881 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_:
5882 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat
5883 x3) x6) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
5884 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5885 (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
5886 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5887 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5888 (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0:
5889 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat
5890 x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
5891 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5892 (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5893 T).(\lambda (u2: T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead
5894 x4 (Flat x3) x6)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x5) H16 t)
5895 H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f0:
5896 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5897 f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5898 T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
5899 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C
5900 C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
5901 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
5902 C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_:
5903 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))
5904 (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0:
5905 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
5906 f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5907 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
5908 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
5909 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5910 C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0:
5911 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
5912 (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
5913 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
5914 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5915 (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
5916 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5917 (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
5918 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5919 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5920 (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda
5921 (x5: C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x5 (Flat x3)
5922 x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3) x6))).(\lambda
5923 (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6) (\lambda (c0:
5924 C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0:
5925 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
5926 f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
5927 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
5928 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
5929 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5930 C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
5931 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
5932 (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
5933 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
5934 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5935 (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
5936 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
5937 (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
5938 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5939 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5940 (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat
5941 f) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
5942 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e0
5943 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
5944 T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
5945 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5946 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
5947 C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
5948 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5949 (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
5950 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
5951 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5952 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
5953 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
5954 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5955 (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
5956 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
5957 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
5958 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_:
5959 C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
5960 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
5961 (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
5962 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
5963 v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop
5964 (Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17)) e H15)))))))) H14))
5965 (\lambda (H14: (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda
5966 (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0)
5967 u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
5968 T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda
5969 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
5970 T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
5971 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C
5972 C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
5973 T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
5974 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
5975 n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
5976 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
5977 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
5978 (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e)
5979 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
5980 (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
5981 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
5982 (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
5983 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
5984 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
5985 f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
5986 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
5987 (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
5988 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
5989 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2)))))))
5990 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5991 (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
5992 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
5993 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
5994 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
5995 (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7:
5996 T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H16: (drop (S
5997 n0) O c (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda
5998 (H18: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0:
5999 C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0:
6000 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
6001 f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
6002 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
6003 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
6004 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
6005 C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
6006 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
6007 (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
6008 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
6009 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
6010 (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
6011 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
6012 (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
6013 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
6014 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
6015 (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat
6016 f) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
6017 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0
6018 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
6019 T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
6020 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
6021 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
6022 C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2
6023 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
6024 (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
6025 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
6026 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
6027 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2
6028 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
6029 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
6030 (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
6031 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
6032 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
6033 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_:
6034 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat
6035 x3) x7) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
6036 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
6037 (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
6038 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
6039 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
6040 (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat
6041 x3) x7)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e
6042 H15)))))))))) H14)) H13)))))))) k H3 (drop_gen_drop k x1 e x0 n0 H7))))))))))
6043 H2)) (csubst0_gen_head k c c2 t v (S n0) H0))))))))))) c1)))) n).
6045 Initial nodes: 34765
6048 theorem csubst0_drop_lt_back:
6049 \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall
6050 (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((drop n O
6051 c2 e2) \to (or (drop n O c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n)
6052 v e1 e2)) (\lambda (e1: C).(drop n O c1 e1))))))))))))
6054 \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i)
6055 \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2)
6056 \to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C
6057 (\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O
6058 c1 e1))))))))))))) (\lambda (i: nat).(\lambda (_: (lt O i)).(\lambda (c1:
6059 C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1
6060 c2)).(\lambda (e2: C).(\lambda (H1: (drop O O c2 e2)).(eq_ind C c2 (\lambda
6061 (c: C).(or (drop O O c1 c) (ex2 C (\lambda (e1: C).(csubst0 (minus i O) v e1
6062 c)) (\lambda (e1: C).(drop O O c1 e1))))) (eq_ind nat i (\lambda (n0:
6063 nat).(or (drop O O c1 c2) (ex2 C (\lambda (e1: C).(csubst0 n0 v e1 c2))
6064 (\lambda (e1: C).(drop O O c1 e1))))) (or_intror (drop O O c1 c2) (ex2 C
6065 (\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O c1 e1)))
6066 (ex_intro2 C (\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O
6067 c1 e1)) c1 H0 (drop_refl c1))) (minus i O) (minus_n_O i)) e2 (drop_gen_refl
6068 c2 e2 H1)))))))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt
6069 n0 i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1
6070 c2) \to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C
6071 (\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O
6072 c1 e1)))))))))))))).(\lambda (i: nat).(\lambda (H: (lt (S n0) i)).(\lambda
6073 (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v
6074 c c2) \to (\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c
6075 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1:
6076 C).(drop (S n0) O c e1)))))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda
6077 (v: T).(\lambda (H0: (csubst0 i v (CSort n1) c2)).(\lambda (e2: C).(\lambda
6078 (_: (drop (S n0) O c2 e2)).(csubst0_gen_sort c2 v i n1 H0 (or (drop (S n0) O
6079 (CSort n1) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2))
6080 (\lambda (e1: C).(drop (S n0) O (CSort n1) e1))))))))))) (\lambda (c:
6081 C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to
6082 (\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c e2) (ex2 C
6083 (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop
6084 (S n0) O c e1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2:
6085 C).(\lambda (v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda
6086 (e2: C).(\lambda (H2: (drop (S n0) O c2 e2)).(or3_ind (ex3_2 T nat (\lambda
6087 (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
6088 nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
6089 v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k
6090 j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
6091 (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_:
6092 T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2:
6093 T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
6094 (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
6095 T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or (drop (S n0)
6096 O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1
6097 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (H3:
6098 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda
6099 (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2:
6100 T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_:
6101 T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
6102 nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
6103 v t u2))) (or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1:
6104 C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead
6105 c k t) e1)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq nat i (s
6106 k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t
6107 x0)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2
6108 (CHead c k x0) H5) in (let H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall
6109 (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall (e3: C).((drop (S
6110 n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0
6111 (minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))
6112 H0 (s k x1) H4) in (let H9 \def (eq_ind nat i (\lambda (n1: nat).(lt (S n0)
6113 n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) (\lambda (n1: nat).(or (drop (S
6114 n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v
6115 e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1))))) (K_ind (\lambda
6116 (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to
6117 (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C
6118 (\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1:
6119 C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 x1)) \to ((drop (r k0
6120 n0) O c e2) \to (or (drop (S n0) O (CHead c k0 t) e2) (ex2 C (\lambda (e1:
6121 C).(csubst0 (minus (s k0 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0)
6122 O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: ((\forall (c3:
6123 C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e3:
6124 C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1:
6125 C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop
6126 (S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Bind b) x1))).(\lambda
6127 (H12: (drop (r (Bind b) n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Bind
6128 b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda
6129 (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (drop_drop (Bind b) n0 c e2
6130 H12 t)))))) (\lambda (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0:
6131 T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3
6132 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s
6133 (Flat f) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c
6134 e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x1))).(\lambda (H12: (drop
6135 (r (Flat f) n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Flat f) t) e2)
6136 (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1:
6137 C).(drop (S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat f) n0 c e2 H12
6138 t)))))) k H8 H9 (drop_gen_drop k c e2 x0 n0 H7)) i H4))))))))) H3)) (\lambda
6139 (H3: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))
6140 (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3:
6141 C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_:
6142 C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_:
6143 nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
6144 v c c3))) (or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1:
6145 C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead
6146 c k t) e1)))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: (eq nat i (s
6147 k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: (csubst0 x1 v c
6148 x0)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2
6149 (CHead x0 k t) H5) in (let H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall
6150 (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall (e3: C).((drop (S
6151 n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0
6152 (minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))
6153 H0 (s k x1) H4) in (let H9 \def (eq_ind nat i (\lambda (n1: nat).(lt (S n0)
6154 n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) (\lambda (n1: nat).(or (drop (S
6155 n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v
6156 e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1))))) (K_ind (\lambda
6157 (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to
6158 (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C
6159 (\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1:
6160 C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 x1)) \to ((drop (r k0
6161 n0) O x0 e2) \to (or (drop (S n0) O (CHead c k0 t) e2) (ex2 C (\lambda (e1:
6162 C).(csubst0 (minus (s k0 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0)
6163 O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: ((\forall (c3:
6164 C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e3:
6165 C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1:
6166 C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop
6167 (S n0) O c e1))))))))))).(\lambda (H11: (lt (S n0) (s (Bind b) x1))).(\lambda
6168 (H12: (drop (r (Bind b) n0) O x0 e2)).(let H_x \def (IHn x1 (lt_S_n n0 x1
6169 H11) c x0 v H6 e2 H12) in (let H13 \def H_x in (or_ind (drop n0 O c e2) (ex2
6170 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0
6171 O c e1))) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1:
6172 C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c
6173 (Bind b) t) e1)))) (\lambda (H14: (drop n0 O c e2)).(or_introl (drop (S n0) O
6174 (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1
6175 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (drop_drop
6176 (Bind b) n0 c e2 H14 t))) (\lambda (H14: (ex2 C (\lambda (e1: C).(csubst0
6177 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O c e1)))).(ex2_ind C
6178 (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O
6179 c e1)) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1:
6180 C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c
6181 (Bind b) t) e1)))) (\lambda (x: C).(\lambda (H15: (csubst0 (minus x1 n0) v x
6182 e2)).(\lambda (H16: (drop n0 O c x)).(or_intror (drop (S n0) O (CHead c (Bind
6183 b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda
6184 (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (ex_intro2 C (\lambda (e1:
6185 C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c
6186 (Bind b) t) e1)) x H15 (drop_drop (Bind b) n0 c x H16 t)))))) H14))
6187 H13))))))) (\lambda (f: F).(\lambda (H10: ((\forall (c3: C).(\forall (v0:
6188 T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3
6189 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s
6190 (Flat f) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c
6191 e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x1))).(\lambda (H12: (drop
6192 (r (Flat f) n0) O x0 e2)).(let H_x \def (H10 x0 v H6 e2 H12) in (let H13 \def
6193 H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1: C).(csubst0 (minus
6194 x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1))) (or (drop (S n0)
6195 O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0))
6196 v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))))
6197 (\lambda (H14: (drop (S n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Flat
6198 f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2))
6199 (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat
6200 f) n0 c e2 H14 t))) (\lambda (H14: (ex2 C (\lambda (e1: C).(csubst0 (minus x1
6201 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1)))).(ex2_ind C
6202 (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop
6203 (S n0) O c e1)) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda
6204 (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O
6205 (CHead c (Flat f) t) e1)))) (\lambda (x: C).(\lambda (H15: (csubst0 (minus x1
6206 (S n0)) v x e2)).(\lambda (H16: (drop (S n0) O c x)).(or_intror (drop (S n0)
6207 O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0))
6208 v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)))
6209 (ex_intro2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda
6210 (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x H15 (drop_drop (Flat f) n0
6211 c x H16 t)))))) H14)) H13))))))) k H8 H9 (drop_gen_drop k x0 e2 t n0 H7)) i
6212 H4))))))))) H3)) (\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
6213 C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3:
6214 C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
6215 (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
6216 C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_:
6217 T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2:
6218 T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
6219 (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
6220 T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or (drop (S n0)
6221 O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1
6222 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (x0:
6223 T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H4: (eq nat i (s k
6224 x2))).(\lambda (H5: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t
6225 x0)).(\lambda (H7: (csubst0 x2 v c x1)).(let H8 \def (eq_ind C c2 (\lambda
6226 (c0: C).(drop (S n0) O c0 e2)) H2 (CHead x1 k x0) H5) in (let H9 \def (eq_ind
6227 nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c
6228 c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3)
6229 (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v0 e1 e3)) (\lambda (e1:
6230 C).(drop (S n0) O c e1)))))))))) H0 (s k x2) H4) in (let H10 \def (eq_ind nat
6231 i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x2) H4) in (eq_ind_r nat (s k x2)
6232 (\lambda (n1: nat).(or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1:
6233 C).(csubst0 (minus n1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O
6234 (CHead c k t) e1))))) (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall
6235 (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3
6236 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s
6237 k0 x2) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) \to
6238 ((lt (S n0) (s k0 x2)) \to ((drop (r k0 n0) O x1 e2) \to (or (drop (S n0) O
6239 (CHead c k0 t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s k0 x2) (S n0))
6240 v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k0 t) e1)))))))) (\lambda
6241 (b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b)
6242 x2) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0)
6243 O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v0 e1
6244 e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))).(\lambda (H12: (lt (S
6245 n0) (s (Bind b) x2))).(\lambda (H13: (drop (r (Bind b) n0) O x1 e2)).(let H_x
6246 \def (IHn x2 (lt_S_n n0 x2 H12) c x1 v H7 e2 H13) in (let H14 \def H_x in
6247 (or_ind (drop n0 O c e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1
6248 e2)) (\lambda (e1: C).(drop n0 O c e1))) (or (drop (S n0) O (CHead c (Bind b)
6249 t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1:
6250 C).(drop (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (H15: (drop n0 O c
6251 e2)).(or_introl (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1:
6252 C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c
6253 (Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 H15 t))) (\lambda (H15: (ex2 C
6254 (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O
6255 c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2))
6256 (\lambda (e1: C).(drop n0 O c e1)) (or (drop (S n0) O (CHead c (Bind b) t)
6257 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1:
6258 C).(drop (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (x: C).(\lambda (H16:
6259 (csubst0 (minus x2 n0) v x e2)).(\lambda (H17: (drop n0 O c x)).(or_intror
6260 (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0
6261 (minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t)
6262 e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda
6263 (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1)) x H16 (drop_drop (Bind b) n0
6264 c x H17 t)))))) H15)) H14))))))) (\lambda (f: F).(\lambda (H11: ((\forall
6265 (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e3:
6266 C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1:
6267 C).(csubst0 (minus (s (Flat f) x2) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop
6268 (S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x2))).(\lambda
6269 (H13: (drop (r (Flat f) n0) O x1 e2)).(let H_x \def (H11 x1 v H7 e2 H13) in
6270 (let H14 \def H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1:
6271 C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c
6272 e1))) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1:
6273 C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O
6274 (CHead c (Flat f) t) e1)))) (\lambda (H15: (drop (S n0) O c e2)).(or_introl
6275 (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0
6276 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f)
6277 t) e1))) (drop_drop (Flat f) n0 c e2 H15 t))) (\lambda (H15: (ex2 C (\lambda
6278 (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O
6279 c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2))
6280 (\lambda (e1: C).(drop (S n0) O c e1)) (or (drop (S n0) O (CHead c (Flat f)
6281 t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda
6282 (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)))) (\lambda (x: C).(\lambda
6283 (H16: (csubst0 (minus x2 (S n0)) v x e2)).(\lambda (H17: (drop (S n0) O c
6284 x)).(or_intror (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1:
6285 C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O
6286 (CHead c (Flat f) t) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus x2
6287 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x
6288 H16 (drop_drop (Flat f) n0 c x H17 t)))))) H15)) H14))))))) k H9 H10
6289 (drop_gen_drop k x1 e2 x0 n0 H8)) i H4))))))))))) H3)) (csubst0_gen_head k c
6290 c2 t v i H1))))))))))) c1)))))) n).
projects / helm.git / blob