1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "LambdaDelta-1/pr2/defs.ma".
19 include "LambdaDelta-1/pr0/fwd.ma".
21 include "LambdaDelta-1/getl/drop.ma".
23 include "LambdaDelta-1/getl/clear.ma".
26 \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to
29 \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TSort
30 n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr2 c t x)) (\lambda (t:
31 T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda
32 (_: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0
33 t))))) (\lambda (_: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0
34 t1 t2)).(\lambda (H2: (eq T t1 (TSort n))).(let H3 \def (eq_ind T t1 (\lambda
35 (t: T).(pr0 t t2)) H1 (TSort n) H2) in (eq_ind_r T (TSort n) (\lambda (t:
36 T).(eq T t2 t)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T t (TSort n)))
37 (refl_equal T (TSort n)) t2 (pr0_gen_sort t2 n H3)) t1 H2))))))) (\lambda
38 (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl
39 i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda
40 (H2: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda
41 (H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0
42 t2)) H2 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t t0))
43 (let H6 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 t)) H3 (TSort n)
44 (pr0_gen_sort t2 n H5)) in (subst0_gen_sort u t i n H6 (eq T t (TSort n))))
45 t1 H4))))))))))))) c y x H0))) H)))).
48 \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to
49 (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c
50 (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S
53 \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TLRef
54 n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr2 c t x)) (\lambda (t:
55 T).(or (eq T x t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead
56 d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O
57 u))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0:
58 C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (eq T t0 t)
59 (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr)
60 u)))) (\lambda (_: C).(\lambda (u: T).(eq T t0 (lift (S n) O u))))))))))
61 (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1
62 t2)).(\lambda (H2: (eq T t1 (TLRef n))).(let H3 \def (eq_ind T t1 (\lambda
63 (t: T).(pr0 t t2)) H1 (TLRef n) H2) in (eq_ind_r T (TLRef n) (\lambda (t:
64 T).(or (eq T t2 t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0
65 (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S
66 n) O u))))))) (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t (TLRef n))
67 (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr)
68 u)))) (\lambda (_: C).(\lambda (u: T).(eq T t (lift (S n) O u)))))))
69 (or_introl (eq T (TLRef n) (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u:
70 T).(getl n c0 (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq
71 T (TLRef n) (lift (S n) O u))))) (refl_equal T (TLRef n))) t2 (pr0_gen_lref
72 t2 n H3)) t1 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
73 T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr)
74 u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H2: (pr0 t1 t2)).(\lambda
75 (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t1 (TLRef
76 n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0 t2)) H2 (TLRef n) H4)
77 in (eq_ind_r T (TLRef n) (\lambda (t0: T).(or (eq T t t0) (ex2_2 C T (\lambda
78 (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (_:
79 C).(\lambda (u0: T).(eq T t (lift (S n) O u0))))))) (let H6 \def (eq_ind T t2
80 (\lambda (t0: T).(subst0 i u t0 t)) H3 (TLRef n) (pr0_gen_lref t2 n H5)) in
81 (land_ind (eq nat n i) (eq T t (lift (S n) O u)) (or (eq T t (TLRef n))
82 (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr)
83 u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t (lift (S n) O u0))))))
84 (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t (lift (S n) O
85 u))).(eq_ind_r T (lift (S n) O u) (\lambda (t0: T).(or (eq T t0 (TLRef n))
86 (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr)
87 u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t0 (lift (S n) O u0))))))) (let
88 H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind Abbr)
89 u))) H1 n H7) in (or_intror (eq T (lift (S n) O u) (TLRef n)) (ex2_2 C T
90 (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0))))
91 (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0)))))
92 (ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind
93 Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S
94 n) O u0)))) d u H9 (refl_equal T (lift (S n) O u))))) t H8)))
95 (subst0_gen_lref u t i n H6))) t1 H4))))))))))))) c y x H0))) H)))).
98 \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
99 (THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
100 T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2
101 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u:
102 T).(pr2 (CHead c (Bind b) u) t1 t2))))))))))
104 \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
105 (H: (pr2 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1
106 t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2:
107 T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2:
108 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
109 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y:
110 T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda (t:
111 T).(\lambda (t0: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
112 (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2))))
113 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
114 (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
115 t2)))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1:
116 (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abst) u1 t1))).(let H3 \def
117 (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abst) u1 t1) H2) in
118 (ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst)
119 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
120 T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
121 T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
122 T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
123 (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3)))))) (\lambda (x0: T).(\lambda (x1:
124 T).(\lambda (H4: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr0 u1
125 x0)).(\lambda (H6: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Abst) x0 x1)
126 (\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
127 (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
128 (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead
129 c0 (Bind b) u) t1 t3))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3:
130 T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2:
131 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
132 (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))))) x0 x1
133 (refl_equal T (THead (Bind Abst) x0 x1)) (pr2_free c0 u1 x0 H5) (\lambda (b:
134 B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 H6)))) t2 H4))))))
135 (pr0_gen_abst u1 t1 t2 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
136 (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr)
137 u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda
138 (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Bind
139 Abst) u1 t1))).(let H5 \def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2
140 (THead (Bind Abst) u1 t1) H4) in (ex3_2_ind T T (\lambda (u2: T).(\lambda
141 (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
142 T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T
143 (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3))))
144 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
145 (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
146 t3)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t2 (THead
147 (Bind Abst) x0 x1))).(\lambda (H7: (pr0 u1 x0)).(\lambda (H8: (pr0 t1
148 x1)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead
149 (Bind Abst) x0 x1) H6) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead
150 (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda
151 (t3: T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind
152 Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
153 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0
154 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3))))
155 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2
156 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
157 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
158 u0) t1 t3)))))) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind
159 Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda
160 (u2: T).(eq T t (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0
161 u2)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst)
162 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
163 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
164 u0) t1 t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x2
165 x1))).(\lambda (H12: (subst0 i u x0 x2)).(eq_ind_r T (THead (Bind Abst) x2
166 x1) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3
167 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
168 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
169 (CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2:
170 T).(\lambda (t3: T).(eq T (THead (Bind Abst) x2 x1) (THead (Bind Abst) u2
171 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
172 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
173 u0) t1 t3))))) x2 x1 (refl_equal T (THead (Bind Abst) x2 x1)) (pr2_delta c0 d
174 u i H1 u1 x0 H7 x2 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0
175 (Bind b) u0) t1 x1 H8)))) t H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3:
176 T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind
177 Abst) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abst)
178 x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)) (ex3_2 T T
179 (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3))))
180 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
181 (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
182 t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x0
183 x2))).(\lambda (H12: (subst0 (s (Bind Abst) i) u x1 x2)).(eq_ind_r T (THead
184 (Bind Abst) x0 x2) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4:
185 T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
186 T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
187 (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda
188 (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x2) (THead (Bind Abst)
189 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
190 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
191 u0) t1 t3))))) x0 x2 (refl_equal T (THead (Bind Abst) x0 x2)) (pr2_free c0 u1
192 x0 H7) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u
193 (S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x2
194 H12)))) t H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda
195 (t3: T).(eq T t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
196 T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind
197 Abst) i) u x1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T
198 t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u
199 x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1
200 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst)
201 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
202 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
203 u0) t1 t3)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T t
204 (THead (Bind Abst) x2 x3))).(\lambda (H12: (subst0 i u x0 x2)).(\lambda (H13:
205 (subst0 (s (Bind Abst) i) u x1 x3)).(eq_ind_r T (THead (Bind Abst) x2 x3)
206 (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
207 (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
208 (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead
209 c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3:
210 T).(eq T (THead (Bind Abst) x2 x3) (THead (Bind Abst) u2 t3)))) (\lambda (u2:
211 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
212 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))))) x2 x3
213 (refl_equal T (THead (Bind Abst) x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H7 x2
214 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u
215 (S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x3
216 H13)))) t H11)))))) H10)) (subst0_gen_head (Bind Abst) u x0 x1 t i H9))))))))
217 (pr0_gen_abst u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
219 theorem pr2_gen_cast:
220 \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
221 (THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
222 (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_:
223 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c
226 \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
227 (H: (pr2 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1
228 t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
229 T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2:
230 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1
231 t2)))) (pr2 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind
232 (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Cast)
233 u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead
234 (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
235 (\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (pr2 c0 t1 t0))))))
236 (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (pr0 t0
237 t2)).(\lambda (H2: (eq T t0 (THead (Flat Cast) u1 t1))).(let H3 \def (eq_ind
238 T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Cast) u1 t1) H2) in (or_ind
239 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2
240 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
241 T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda (u2:
242 T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
243 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
244 t1 t3)))) (pr2 c0 t1 t2)) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda
245 (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
246 T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind
247 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3))))
248 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
249 T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
250 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
251 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t2))
252 (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead (Flat Cast)
253 x0 x1))).(\lambda (H6: (pr0 u1 x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T
254 (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2:
255 T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2:
256 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
257 t1 t3)))) (pr2 c0 t1 t))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda
258 (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda
259 (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3:
260 T).(pr2 c0 t1 t3)))) (pr2 c0 t1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T
261 (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead
262 (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
263 (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x1 (refl_equal T (THead
264 (Flat Cast) x0 x1)) (pr2_free c0 u1 x0 H6) (pr2_free c0 t1 x1 H7))) t2
265 H5)))))) H4)) (\lambda (H4: (pr0 t1 t2)).(or_intror (ex3_2 T T (\lambda (u2:
266 T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
267 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
268 t1 t3)))) (pr2 c0 t1 t2) (pr2_free c0 t1 t2 H4))) (pr0_gen_cast u1 t1 t2
269 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
270 nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0:
271 T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3:
272 (subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(let H5
273 \def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Cast) u1 t1)
274 H4) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
275 (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
276 (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda
277 (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2:
278 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
279 t1 t3)))) (pr2 c0 t1 t)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda
280 (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
281 T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind
282 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3))))
283 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
284 T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
285 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
286 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t))
287 (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t2 (THead (Flat Cast)
288 x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def
289 (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Flat Cast) x0 x1)
290 H7) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1)))
291 (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead
292 (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))
293 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2
294 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
295 T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))) (or (ex3_2 T T
296 (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3))))
297 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
298 (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (H11: (ex2 T (\lambda (u2:
299 T).(eq T t (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0
300 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1)))
301 (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
302 (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
303 T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2
304 c0 t1 t)) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2
305 x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2:
306 T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2:
307 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
308 t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3:
309 T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
310 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x1 H12
311 (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9)))))) H11))
312 (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Cast) x0 t3)))
313 (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))).(ex2_ind T (\lambda
314 (t3: T).(eq T t (THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat
315 Cast) i) u x1 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
316 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
317 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t))
318 (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x0 x2))).(\lambda
319 (H13: (subst0 (s (Flat Cast) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2:
320 T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2:
321 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
322 t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3:
323 T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
324 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 H12
325 (pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13)))))) H11))
326 (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
327 (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2)))
328 (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1
329 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat
330 Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2)))
331 (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (or
332 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2
333 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
334 T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (x2:
335 T).(\lambda (x3: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2
336 x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat
337 Cast) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
338 T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
339 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)
340 (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast)
341 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
342 T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0
343 H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1 H9 x3 H14)))))))) H11))
344 (subst0_gen_head (Flat Cast) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0
345 t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
346 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
347 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)
348 (pr2_delta c0 d u i H1 t1 t2 H6 t H3))) (pr0_gen_cast u1 t1 t2
349 H5)))))))))))))) c y x H0))) H))))).
351 theorem pr2_gen_csort:
352 \forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2)
355 \lambda (t1: T).(\lambda (t2: T).(\lambda (n: nat).(\lambda (H: (pr2 (CSort
356 n) t1 t2)).(insert_eq C (CSort n) (\lambda (c: C).(pr2 c t1 t2)) (\lambda (_:
357 C).(pr0 t1 t2)) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind
358 (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CSort n)) \to (pr0
359 t t0))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
360 (pr0 t3 t4)).(\lambda (_: (eq C c (CSort n))).H1))))) (\lambda (c:
361 C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c
362 (CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0
363 t3 t4)).(\lambda (t: T).(\lambda (_: (subst0 i u t4 t)).(\lambda (H4: (eq C c
364 (CSort n))).(let H5 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead d
365 (Bind Abbr) u))) H1 (CSort n) H4) in (getl_gen_sort n i (CHead d (Bind Abbr)
366 u) H5 (pr0 t3 t)))))))))))))) y t1 t2 H0))) H)))).
368 theorem pr2_gen_appl:
369 \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
370 (THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
371 (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
372 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T
373 T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
374 t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
375 (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_:
376 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
377 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
378 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T
379 T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
380 (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
381 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
382 T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
383 (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead
384 (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
385 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
386 (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
387 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
388 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
389 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))))
391 \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
392 (H: (pr2 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1
393 t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda
394 (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2:
395 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1
396 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
397 T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
398 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
399 Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
400 (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
401 T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
402 z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
403 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
404 (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
405 (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
406 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
407 (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
408 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
409 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
410 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
411 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
412 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
413 y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind
414 (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl)
415 u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead
416 (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
417 (\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (ex4_4 T T T T (\lambda
418 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
419 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
420 T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_:
421 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
422 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
423 (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t2)))))))) (ex6_6 B T T
424 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
425 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
426 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
427 (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
428 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
429 t0 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
430 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
431 T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
432 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
433 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
434 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
435 z2))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda
436 (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Flat Appl) u1 t1))).(let H3
437 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Appl) u1 t1)
438 H2) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
439 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
440 (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
441 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
442 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
443 (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
444 T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
445 (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
446 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
447 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
448 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
449 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
450 T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
451 b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
452 B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
453 (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
454 T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
455 (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
456 T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2:
457 T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
458 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
459 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
460 T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
461 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
462 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
463 (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
464 T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
465 u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
466 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
467 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
468 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
469 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
470 (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift
471 (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
472 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
473 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
474 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
475 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
476 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H4: (ex3_2 T T (\lambda (u2:
477 T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
478 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
479 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
480 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
481 (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2:
482 T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
483 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
484 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
485 T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
486 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
487 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
488 (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
489 T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
490 u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
491 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
492 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
493 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
494 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
495 (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift
496 (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
497 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
498 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
499 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
500 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
501 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1:
502 T).(\lambda (H5: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H6: (pr0 u1
503 x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Flat Appl) x0 x1)
504 (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
505 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
506 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T
507 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
508 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
509 T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
510 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
511 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
512 (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T
513 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
514 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
515 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
516 (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
517 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t
518 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
519 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
520 T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
521 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
522 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
523 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
524 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
525 (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
526 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
527 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
528 T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
529 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl)
530 x0 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
531 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
532 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
533 (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
534 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
535 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
536 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
537 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
538 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Flat
539 Appl) x0 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
540 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
541 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
542 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
543 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
544 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
545 y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
546 (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
547 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
548 t1 t3))) x0 x1 (refl_equal T (THead (Flat Appl) x0 x1)) (pr2_free c0 u1 x0
549 H6) (pr2_free c0 t1 x1 H7))) t2 H5)))))) H4)) (\lambda (H4: (ex4_4 T T T T
550 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
551 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
552 T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
553 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
554 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
555 t3))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
556 T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
557 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
558 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
559 (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
560 T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
561 (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
562 T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4
563 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
564 T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
565 (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
566 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
567 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
568 (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T
569 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
570 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
571 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
572 (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
573 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
574 t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
575 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
576 T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
577 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
578 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
579 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
580 z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
581 T).(\lambda (H5: (eq T t1 (THead (Bind Abst) x0 x1))).(\lambda (H6: (eq T t2
582 (THead (Bind Abbr) x2 x3))).(\lambda (H7: (pr0 u1 x2)).(\lambda (H8: (pr0 x1
583 x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(or3 (ex3_2 T T
584 (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3))))
585 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
586 (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
587 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
588 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t
589 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
590 T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
591 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
592 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
593 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
594 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
595 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
596 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
597 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
598 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
599 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
600 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
601 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
602 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
603 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (eq_ind_r T (THead (Bind
604 Abst) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
605 T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
606 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
607 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
608 T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_:
609 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr)
610 x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
611 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
612 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
613 (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
614 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
615 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
616 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t
617 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
618 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
619 Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
620 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
621 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
622 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
623 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
624 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
625 y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
626 T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
627 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
628 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
629 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
630 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
631 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2
632 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
633 T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
634 T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
635 u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
636 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
637 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
638 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
639 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
640 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
641 Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
642 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
643 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
644 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
645 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
646 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
647 y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
648 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
649 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
650 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2
651 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
652 T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
653 T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
654 u) z1 t3))))))) x0 x1 x2 x3 (refl_equal T (THead (Bind Abst) x0 x1))
655 (refl_equal T (THead (Bind Abbr) x2 x3)) (pr2_free c0 u1 x2 H7) (\lambda (b:
656 B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) x1 x3 H8))))) t1 H5) t2
657 H6))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda
658 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not
659 (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1:
660 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b)
661 y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
662 T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat
663 Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
664 (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2)))))))
665 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
666 (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_:
667 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
668 t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
669 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
670 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
671 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
672 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
673 (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift
674 (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
675 T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2)))))))
676 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
677 (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_:
678 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
679 t3))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
680 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
681 (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda
682 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
683 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
684 T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
685 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
686 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
687 (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T
688 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
689 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
690 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
691 (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
692 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
693 t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
694 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
695 T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
696 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
697 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
698 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
699 z2))))))))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
700 T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not (eq B x0
701 Abst))).(\lambda (H6: (eq T t1 (THead (Bind x0) x1 x2))).(\lambda (H7: (eq T
702 t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda
703 (H8: (pr0 u1 x3)).(\lambda (H9: (pr0 x1 x4)).(\lambda (H10: (pr0 x2
704 x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3)
705 x5)) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
706 t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
707 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T
708 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
709 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
710 T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
711 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
712 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
713 (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T
714 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
715 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
716 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
717 (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
718 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t
719 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
720 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
721 T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
722 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
723 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
724 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
725 z2)))))))))) (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t: T).(or3 (ex3_2
726 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat
727 Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
728 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
729 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
730 T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_:
731 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
732 x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3))))))
733 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
734 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
735 T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3))))))))
736 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
737 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
738 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
739 (_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
740 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
741 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b)
742 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
743 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
744 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
745 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
746 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
747 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T
748 (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat
749 Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
750 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
751 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
752 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
753 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
754 (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))
755 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
756 T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
757 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
758 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
759 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
760 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
761 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
762 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
763 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
764 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b)
765 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
766 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
767 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
768 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
769 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
770 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
771 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
772 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
773 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
774 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
775 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
776 (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))
777 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
778 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
779 T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
780 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
781 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
782 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))
783 x0 x1 x2 x5 x3 x4 H5 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
784 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))) (pr2_free c0
785 u1 x3 H8) (pr2_free c0 x1 x4 H9) (pr2_free (CHead c0 (Bind x0) x4) x2 x5
786 H10))) t1 H6) t2 H7))))))))))))) H4)) (pr0_gen_appl u1 t1 t2 H3))))))))
787 (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
788 (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2:
789 T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2
790 t)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(let H5 \def (eq_ind T
791 t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Appl) u1 t1) H4) in (or3_ind
792 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2
793 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
794 T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
795 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
796 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
797 T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
798 T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
799 (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
800 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
801 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
802 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
803 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
804 T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
805 b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
806 B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
807 (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
808 T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
809 (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
810 T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2:
811 T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
812 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
813 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
814 T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
815 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
816 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
817 (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
818 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
819 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
820 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
821 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
822 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
823 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
824 (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
825 (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
826 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
827 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
828 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
829 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
830 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H6: (ex3_2 T T (\lambda (u2:
831 T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
832 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
833 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
834 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
835 (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2:
836 T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
837 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
838 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
839 T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
840 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
841 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
842 (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
843 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
844 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
845 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
846 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
847 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
848 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
849 (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
850 (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
851 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
852 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
853 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
854 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
855 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1:
856 T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr0 u1
857 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3:
858 T).(subst0 i u t3 t)) H3 (THead (Flat Appl) x0 x1) H7) in (or3_ind (ex2 T
859 (\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0
860 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3)))
861 (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (ex3_2 T T (\lambda
862 (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
863 T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
864 T).(subst0 (s (Flat Appl) i) u x1 t3)))) (or3 (ex3_2 T T (\lambda (u2:
865 T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
866 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
867 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
868 T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
869 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
870 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
871 (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
872 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
873 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
874 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
875 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
876 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
877 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
878 (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
879 (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
880 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
881 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
882 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
883 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
884 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H11: (ex2 T (\lambda (u2:
885 T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 i u x0
886 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1)))
887 (\lambda (u2: T).(subst0 i u x0 u2)) (or3 (ex3_2 T T (\lambda (u2:
888 T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
889 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
890 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
891 T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
892 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
893 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
894 (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
895 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
896 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
897 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
898 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
899 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
900 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
901 (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
902 (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
903 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
904 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
905 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
906 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
907 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (H12: (eq T t
908 (THead (Flat Appl) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(eq_ind_r T
909 (THead (Flat Appl) x2 x1) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
910 T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
911 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
912 t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
913 T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
914 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
915 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
916 (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
917 T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
918 u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
919 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
920 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
921 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
922 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
923 (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift
924 (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
925 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
926 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
927 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
928 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
929 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2:
930 T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2
931 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
932 T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
933 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
934 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
935 (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Bind Abbr) u2 t3))))))
936 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
937 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
938 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
939 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
940 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
941 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
942 (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
943 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
944 (THead (Flat Appl) x2 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O
945 u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
946 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
947 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
948 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
949 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
950 y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
951 (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
952 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
953 t1 t3))) x2 x1 (refl_equal T (THead (Flat Appl) x2 x1)) (pr2_delta c0 d u i
954 H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9))) t H12)))) H11)) (\lambda (H11:
955 (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3))) (\lambda (t3:
956 T).(subst0 (s (Flat Appl) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t
957 (THead (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1
958 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat
959 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
960 (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
961 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
962 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
963 (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
964 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
965 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
966 (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
967 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
968 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
969 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
970 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
971 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b)
972 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
973 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
974 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
975 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
976 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
977 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda
978 (H12: (eq T t (THead (Flat Appl) x0 x2))).(\lambda (H13: (subst0 (s (Flat
979 Appl) i) u x1 x2)).(eq_ind_r T (THead (Flat Appl) x0 x2) (\lambda (t3:
980 T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat
981 Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
982 (_: T).(\lambda (t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1:
983 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
984 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
985 (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
986 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
987 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
988 (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
989 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
990 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
991 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
992 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
993 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
994 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
995 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
996 (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
997 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2)))))))
998 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
999 (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))
1000 (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat
1001 Appl) x0 x2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
1002 T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4
1003 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
1004 T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
1005 (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind Abbr)
1006 u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
1007 T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1008 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
1009 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1010 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1011 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1012 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
1013 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
1014 (u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind b) y2
1015 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1016 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
1017 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1018 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
1019 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1020 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda
1021 (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Flat Appl)
1022 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
1023 T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 (refl_equal T (THead (Flat Appl)
1024 x0 x2)) (pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13))) t
1025 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
1026 T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u
1027 x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1
1028 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat
1029 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2)))
1030 (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (or3
1031 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2
1032 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
1033 T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
1034 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
1035 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1036 (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
1037 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
1038 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
1039 (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
1040 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1041 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1042 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
1043 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1044 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b)
1045 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
1046 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
1047 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1048 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
1049 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1050 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda
1051 (x3: T).(\lambda (H12: (eq T t (THead (Flat Appl) x2 x3))).(\lambda (H13:
1052 (subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat Appl) i) u x1
1053 x3)).(eq_ind_r T (THead (Flat Appl) x2 x3) (\lambda (t3: T).(or3 (ex3_2 T T
1054 (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
1055 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
1056 (t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
1057 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
1058 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3
1059 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1060 T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
1061 T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
1062 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
1063 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1064 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
1065 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
1066 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1067 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead
1068 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1069 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
1070 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1071 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
1072 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1073 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T
1074 (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead
1075 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
1076 (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda
1077 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
1078 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1079 T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind Abbr) u2
1080 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
1081 T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1082 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
1083 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1084 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1085 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1086 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
1087 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
1088 (u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind b) y2
1089 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1090 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
1091 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1092 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
1093 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1094 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda
1095 (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Flat Appl)
1096 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
1097 T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 (refl_equal T (THead (Flat Appl)
1098 x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1
1099 H9 x3 H14))) t H12)))))) H11)) (subst0_gen_head (Flat Appl) u x0 x1 t i
1100 H10)))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
1101 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
1102 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2
1103 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1104 T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
1105 (_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T (\lambda (y1:
1106 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
1107 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1108 (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
1109 T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
1110 (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T
1111 (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3))))
1112 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
1113 (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
1114 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
1115 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t
1116 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1117 T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
1118 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
1119 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
1120 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1121 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
1122 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
1123 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1124 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
1125 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1126 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
1127 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1128 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
1129 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1130 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda
1131 (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T t1 (THead (Bind
1132 Abst) x0 x1))).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda
1133 (H9: (pr0 u1 x2)).(\lambda (H10: (pr0 x1 x3)).(let H11 \def (eq_ind T t2
1134 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Bind Abbr) x2 x3) H8) in
1135 (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(or3 (ex3_2 T T
1136 (\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2 t4))))
1137 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
1138 (t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
1139 T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind Abst) y1 z1))))))
1140 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t
1141 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1142 T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
1143 T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
1144 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
1145 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1146 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
1147 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind
1148 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1149 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
1150 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1151 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
1152 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1153 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
1154 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1155 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda
1156 (u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2
1157 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3))) (\lambda
1158 (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (ex3_2 T T (\lambda (u2:
1159 T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
1160 T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t3:
1161 T).(subst0 (s (Bind Abbr) i) u x3 t3)))) (or3 (ex3_2 T T (\lambda (u2:
1162 T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1163 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
1164 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1165 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
1166 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1167 T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
1168 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
1169 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
1170 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
1171 T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1172 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1173 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1174 (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
1175 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
1176 T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
1177 O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1178 (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
1179 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1180 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1181 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
1182 y2) z1 z2))))))))) (\lambda (H12: (ex2 T (\lambda (u2: T).(eq T t (THead
1183 (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2)))).(ex2_ind T
1184 (\lambda (u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0
1185 i u x2 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
1186 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
1187 (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3))))
1188 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
1189 T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda
1190 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
1191 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1192 (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1193 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
1194 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1195 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1196 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1197 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
1198 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1199 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b)
1200 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
1201 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
1202 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1203 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
1204 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1205 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda
1206 (H13: (eq T t (THead (Bind Abbr) x4 x3))).(\lambda (H14: (subst0 i u x2
1207 x4)).(eq_ind_r T (THead (Bind Abbr) x4 x3) (\lambda (t3: T).(or3 (ex3_2 T T
1208 (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
1209 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
1210 (t4: T).(pr2 c0 (THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1:
1211 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst)
1212 x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
1213 T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4))))))
1214 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
1215 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4:
1216 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4))))))))
1217 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1218 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1219 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1220 (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
1221 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
1222 T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
1223 O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1224 (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
1225 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1226 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1227 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
1228 y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
1229 T).(eq T (THead (Bind Abbr) x4 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1230 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
1231 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1232 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
1233 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1234 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2
1235 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
1236 T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1237 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
1238 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1239 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1240 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1241 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
1242 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1243 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
1244 Abbr) x4 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1245 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1246 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
1247 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1248 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1249 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
1250 y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
1251 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
1252 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1253 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2
1254 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
1255 T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1256 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
1257 u0) z1 t3))))))) x0 x1 x4 x3 (refl_equal T (THead (Bind Abst) x0 x1))
1258 (refl_equal T (THead (Bind Abbr) x4 x3)) (pr2_delta c0 d u i H1 u1 x2 H9 x4
1259 H14) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) x1 x3
1260 H10))))) t H13)))) H12)) (\lambda (H12: (ex2 T (\lambda (t3: T).(eq T t
1261 (THead (Bind Abbr) x2 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3
1262 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3)))
1263 (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3)) (or3 (ex3_2 T T (\lambda
1264 (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1265 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
1266 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1267 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
1268 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1269 T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
1270 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
1271 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
1272 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
1273 T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1274 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1275 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1276 (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
1277 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
1278 T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
1279 O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1280 (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
1281 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1282 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1283 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
1284 y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (H13: (eq T t (THead (Bind Abbr)
1285 x2 x4))).(\lambda (H14: (subst0 (s (Bind Abbr) i) u x3 x4)).(eq_ind_r T
1286 (THead (Bind Abbr) x2 x4) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
1287 T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
1288 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
1289 (THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1290 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
1291 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1292 T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
1293 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
1294 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
1295 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
1296 T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1297 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1298 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1299 (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
1300 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
1301 T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
1302 O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1303 (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
1304 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1305 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1306 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
1307 y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
1308 T).(eq T (THead (Bind Abbr) x2 x4) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1309 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
1310 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1311 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
1312 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1313 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2
1314 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
1315 T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1316 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
1317 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1318 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1319 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1320 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
1321 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1322 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
1323 Abbr) x2 x4) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1324 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1325 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
1326 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1327 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1328 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
1329 y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
1330 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
1331 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1332 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2
1333 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
1334 T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1335 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
1336 u0) z1 t3))))))) x0 x1 x2 x4 (refl_equal T (THead (Bind Abst) x0 x1))
1337 (refl_equal T (THead (Bind Abbr) x2 x4)) (pr2_free c0 u1 x2 H9) (\lambda (b:
1338 B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i)
1339 (getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0) (CHead d
1340 (Bind Abbr) u) i H1) x1 x3 H10 x4 H14))))) t H13)))) H12)) (\lambda (H12:
1341 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2
1342 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_:
1343 T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))))).(ex3_2_ind T T
1344 (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
1345 (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_:
1346 T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (or3 (ex3_2 T T
1347 (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3))))
1348 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
1349 (t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1:
1350 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst)
1351 x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
1352 T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))))
1353 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
1354 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
1355 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
1356 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1357 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1358 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1359 (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
1360 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
1361 T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
1362 O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1363 (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
1364 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1365 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1366 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
1367 y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H13: (eq T t
1368 (THead (Bind Abbr) x4 x5))).(\lambda (H14: (subst0 i u x2 x4)).(\lambda (H15:
1369 (subst0 (s (Bind Abbr) i) u x3 x5)).(eq_ind_r T (THead (Bind Abbr) x4 x5)
1370 (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3
1371 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
1372 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind Abst) x0 x1)
1373 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1374 T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1
1375 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
1376 T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
1377 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
1378 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
1379 (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
1380 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1381 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1382 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
1383 (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b:
1384 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
1385 (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1386 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1387 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
1388 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1389 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1390 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
1391 y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
1392 T).(eq T (THead (Bind Abbr) x4 x5) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1393 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
1394 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1395 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
1396 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1397 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2
1398 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
1399 T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1400 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
1401 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1402 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1403 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1404 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
1405 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1406 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
1407 Abbr) x4 x5) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1408 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1409 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
1410 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1411 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1412 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
1413 y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
1414 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
1415 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1416 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2
1417 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
1418 T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1419 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
1420 u0) z1 t3))))))) x0 x1 x4 x5 (refl_equal T (THead (Bind Abst) x0 x1))
1421 (refl_equal T (THead (Bind Abbr) x4 x5)) (pr2_delta c0 d u i H1 u1 x2 H9 x4
1422 H14) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u
1423 (S i) (getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0)
1424 (CHead d (Bind Abbr) u) i H1) x1 x3 H10 x5 H15))))) t H13)))))) H12))
1425 (subst0_gen_head (Bind Abbr) u x2 x3 t i H11)) t1 H7)))))))))) H6)) (\lambda
1426 (H6: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1427 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
1428 (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
1429 (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
1430 B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda
1431 (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2)
1432 t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1433 T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda
1434 (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0
1435 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1436 T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))).(ex6_6_ind B T T T T
1437 T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1438 (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
1439 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
1440 T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
1441 (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead
1442 (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
1443 B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
1444 (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
1445 T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
1446 (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
1447 T).(\lambda (t3: T).(pr0 z1 t3))))))) (or3 (ex3_2 T T (\lambda (u2:
1448 T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1449 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
1450 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1451 T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1452 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
1453 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1454 (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1455 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
1456 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1457 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1458 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1459 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
1460 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
1461 (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
1462 (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
1463 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
1464 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1465 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
1466 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
1467 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: B).(\lambda (x1:
1468 T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
1469 T).(\lambda (H7: (not (eq B x0 Abst))).(\lambda (H8: (eq T t1 (THead (Bind
1470 x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl)
1471 (lift (S O) O x3) x5)))).(\lambda (H10: (pr0 u1 x3)).(\lambda (H11: (pr0 x1
1472 x4)).(\lambda (H12: (pr0 x2 x5)).(let H13 \def (eq_ind T t2 (\lambda (t3:
1473 T).(subst0 i u t3 t)) H3 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O
1474 x3) x5)) H9) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(or3
1475 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2
1476 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
1477 T).(\lambda (t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1:
1478 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind
1479 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1480 (t4: T).(eq T t (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
1481 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
1482 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
1483 (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
1484 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1485 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1486 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3
1487 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1488 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b)
1489 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
1490 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
1491 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1492 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
1493 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1494 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda
1495 (u2: T).(eq T t (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3)
1496 x5)))) (\lambda (u2: T).(subst0 i u x4 u2))) (ex2 T (\lambda (t3: T).(eq T t
1497 (THead (Bind x0) x4 t3))) (\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead
1498 (Flat Appl) (lift (S O) O x3) x5) t3))) (ex3_2 T T (\lambda (u2: T).(\lambda
1499 (t3: T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_:
1500 T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0)
1501 i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3)))) (or3 (ex3_2 T T (\lambda
1502 (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1503 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
1504 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
1505 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1506 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1507 (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
1508 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
1509 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
1510 (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
1511 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1512 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1513 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
1514 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
1515 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
1516 (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1517 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1518 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
1519 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1520 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1521 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
1522 y2) z1 z2))))))))) (\lambda (H14: (ex2 T (\lambda (u2: T).(eq T t (THead
1523 (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2:
1524 T).(subst0 i u x4 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Bind x0)
1525 u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: T).(subst0 i u
1526 x4 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
1527 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
1528 (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3))))
1529 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
1530 T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1531 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
1532 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1533 (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1534 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
1535 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1536 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1537 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1538 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1539 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1540 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
1541 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1542 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
1543 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1544 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
1545 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1546 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda
1547 (H15: (eq T t (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3)
1548 x5)))).(\lambda (H16: (subst0 i u x4 x6)).(eq_ind_r T (THead (Bind x0) x6
1549 (THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(or3 (ex3_2 T T
1550 (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
1551 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
1552 (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1:
1553 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
1554 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
1555 (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
1556 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
1557 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
1558 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
1559 T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1560 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1561 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1562 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
1563 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
1564 T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
1565 O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1566 (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
1567 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1568 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1569 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
1570 y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
1571 T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead
1572 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
1573 (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3))))
1574 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
1575 T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1576 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
1577 x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3))))))
1578 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
1579 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
1580 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
1581 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1582 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1583 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1584 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
1585 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
1586 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
1587 O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1588 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1589 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
1590 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1591 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1592 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
1593 y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
1594 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
1595 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1596 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1597 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1598 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
1599 (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl)
1600 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
1601 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
1602 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1603 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
1604 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
1605 (CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x3 x6 H7 (refl_equal T (THead
1606 (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift
1607 (S O) O x3) x5))) (pr2_free c0 u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6
1608 H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) t H15)))) H14)) (\lambda
1609 (H14: (ex2 T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda (t3:
1610 T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
1611 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda
1612 (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
1613 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat
1614 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
1615 (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
1616 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
1617 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1618 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
1619 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1620 (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1621 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
1622 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1623 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1624 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1625 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1626 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1627 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
1628 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1629 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
1630 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1631 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
1632 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1633 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda
1634 (H15: (eq T t (THead (Bind x0) x4 x6))).(\lambda (H16: (subst0 (s (Bind x0)
1635 i) u (THead (Flat Appl) (lift (S O) O x3) x5) x6)).(eq_ind_r T (THead (Bind
1636 x0) x4 x6) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4:
1637 T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_:
1638 T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0)
1639 x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1640 T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
1641 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
1642 T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
1643 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
1644 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
1645 (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
1646 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1647 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1648 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
1649 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
1650 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
1651 (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1652 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1653 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
1654 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1655 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1656 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
1657 y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x6 (THead (Flat
1658 Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3)
1659 u2))) (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S O) O x3)
1660 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)))
1661 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat Appl) u2
1662 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O)
1663 O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind
1664 x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
1665 (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1666 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
1667 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
1668 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1669 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1670 (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda
1671 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
1672 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
1673 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
1674 T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1675 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1676 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1677 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
1678 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
1679 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead
1680 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1681 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
1682 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1683 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
1684 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1685 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H17: (ex2 T
1686 (\lambda (u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2:
1687 T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda
1688 (u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s
1689 (Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2:
1690 T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2
1691 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
1692 T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
1693 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
1694 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1695 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
1696 x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
1697 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
1698 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
1699 (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
1700 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1701 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1702 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
1703 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
1704 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
1705 (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl)
1706 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
1707 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
1708 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1709 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
1710 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
1711 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (H18: (eq T
1712 x6 (THead (Flat Appl) x7 x5))).(\lambda (H19: (subst0 (s (Bind x0) i) u (lift
1713 (S O) O x3) x7)).(eq_ind_r T (THead (Flat Appl) x7 x5) (\lambda (t3: T).(or3
1714 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3)
1715 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
1716 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4))))
1717 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
1718 T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1719 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
1720 x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
1721 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
1722 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
1723 (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
1724 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1725 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1726 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
1727 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
1728 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
1729 (y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead (Flat Appl)
1730 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
1731 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
1732 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1733 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
1734 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
1735 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x7
1736 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u
1737 x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
1738 x0) x4 (THead (Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1739 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
1740 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
1741 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1742 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1743 (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind
1744 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1745 (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1746 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
1747 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1748 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1749 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1750 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1751 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1752 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
1753 (Flat Appl) x7 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1754 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1755 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
1756 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1757 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1758 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
1759 y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H20: (eq T x7 (lift (S O) O
1760 x8))).(\lambda (H21: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x8)).(let H22
1761 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u
1762 x3 x8)) H21 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x8) (\lambda (t3:
1763 T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
1764 x4 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
1765 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
1766 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
1767 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1768 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1769 (t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Bind
1770 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1771 (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1772 T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
1773 u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1774 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1775 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1776 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1777 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1778 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
1779 (Flat Appl) t3 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1780 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1781 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
1782 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1783 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1784 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
1785 y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
1786 T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead
1787 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
1788 (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3))))
1789 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
1790 T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1791 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
1792 x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead (Bind Abbr) u2 t3))))))
1793 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
1794 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
1795 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
1796 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1797 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1798 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1799 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
1800 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
1801 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
1802 O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1803 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1804 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
1805 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1806 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1807 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
1808 y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
1809 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
1810 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1811 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1812 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1813 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
1814 (Flat Appl) (lift (S O) O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl)
1815 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
1816 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
1817 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1818 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
1819 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
1820 (CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x8 x4 H7 (refl_equal T (THead
1821 (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift
1822 (S O) O x8) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x8 H22) (pr2_free c0 x1 x4
1823 H11) (pr2_free (CHead c0 (Bind x0) x4) x2 x5 H12))) x7 H20)))))
1824 (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O H19 (le_n_S O i (le_O_n
1825 i)))) x6 H18)))) H17)) (\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x6 (THead
1826 (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl)
1827 (s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead
1828 (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl)
1829 (s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
1830 T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1831 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
1832 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
1833 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1834 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1835 (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda
1836 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
1837 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
1838 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
1839 T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1840 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1841 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1842 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
1843 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
1844 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead
1845 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1846 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
1847 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1848 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
1849 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1850 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda
1851 (H18: (eq T x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H19:
1852 (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x7)).(eq_ind_r T (THead (Flat
1853 Appl) (lift (S O) O x3) x7) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
1854 T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Flat Appl) u2
1855 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
1856 T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T
1857 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
1858 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1859 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
1860 x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
1861 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
1862 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
1863 (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
1864 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1865 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1866 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
1867 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
1868 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
1869 (y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead (Flat Appl)
1870 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
1871 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
1872 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1873 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
1874 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
1875 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2:
1876 T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
1877 O x3) x7)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
1878 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2)
1879 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1880 T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
1881 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
1882 T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead
1883 (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1884 T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
1885 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
1886 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
1887 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1888 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
1889 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
1890 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
1891 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
1892 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead (Bind b)
1893 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
1894 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
1895 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1896 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
1897 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1898 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
1899 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1900 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1901 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
1902 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
1903 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
1904 (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))
1905 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
1906 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1907 T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
1908 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
1909 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
1910 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))
1911 x0 x1 x2 x7 x3 x4 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
1912 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))) (pr2_free c0
1913 u1 x3 H10) (pr2_free c0 x1 x4 H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S
1914 i) (getl_clear_bind x0 (CHead c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4)
1915 (CHead d (Bind Abbr) u) i H1) x2 x5 H12 x7 H19))) x6 H18)))) H17)) (\lambda
1916 (H17: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat
1917 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u
1918 (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat
1919 Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
1920 (t3: T).(eq T x6 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
1921 T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda
1922 (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (or3 (ex3_2 T T
1923 (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat
1924 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
1925 (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
1926 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
1927 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1928 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
1929 x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
1930 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
1931 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
1932 (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
1933 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1934 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1935 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
1936 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
1937 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
1938 (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl)
1939 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
1940 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
1941 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1942 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
1943 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
1944 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (x8:
1945 T).(\lambda (H18: (eq T x6 (THead (Flat Appl) x7 x8))).(\lambda (H19: (subst0
1946 (s (Bind x0) i) u (lift (S O) O x3) x7)).(\lambda (H20: (subst0 (s (Flat
1947 Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat Appl) x7 x8)
1948 (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
1949 (THead (Bind x0) x4 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
1950 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
1951 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
1952 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1953 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1954 (t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda
1955 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
1956 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
1957 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
1958 T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1959 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1960 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1961 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
1962 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
1963 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead
1964 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1965 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
1966 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1967 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
1968 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1969 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3:
1970 T).(eq T x7 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0)
1971 i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
1972 (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3))))
1973 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
1974 (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
1975 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
1976 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
1977 (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8))
1978 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1979 T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
1980 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
1981 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
1982 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1983 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
1984 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
1985 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
1986 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
1987 (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind b) y2 (THead
1988 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1989 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
1990 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1991 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
1992 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1993 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x9: T).(\lambda
1994 (H21: (eq T x7 (lift (S O) O x9))).(\lambda (H22: (subst0 (minus (s (Bind x0)
1995 i) (S O)) u x3 x9)).(let H23 \def (eq_ind nat (minus (s (Bind x0) i) (S O))
1996 (\lambda (n: nat).(subst0 n u x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T
1997 (lift (S O) O x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda
1998 (t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat
1999 Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
2000 (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T
2001 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
2002 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2003 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
2004 x4 (THead (Flat Appl) t3 x8)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
2005 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
2006 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
2007 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
2008 T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2009 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2010 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2011 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2012 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2013 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8))
2014 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
2015 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2016 T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
2017 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
2018 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
2019 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
2020 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
2021 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Flat
2022 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
2023 (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
2024 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
2025 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2026 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
2027 x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Bind Abbr) u2 t3))))))
2028 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
2029 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
2030 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
2031 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2032 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2033 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2034 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2035 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2036 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
2037 O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2038 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2039 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
2040 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2041 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2042 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
2043 y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
2044 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
2045 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2046 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2047 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2048 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
2049 (Flat Appl) (lift (S O) O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl)
2050 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2051 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
2052 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2053 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
2054 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2055 (CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x8 x9 x4 H7 (refl_equal T (THead
2056 (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift
2057 (S O) O x9) x8))) (pr2_delta c0 d u i H1 u1 x3 H10 x9 H23) (pr2_free c0 x1 x4
2058 H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead
2059 c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4) (CHead d (Bind Abbr) u) i H1) x2
2060 x5 H12 x8 H20))) x7 H21))))) (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S
2061 O) O H19 (le_n_S O i (le_O_n i)))) x6 H18)))))) H17)) (subst0_gen_head (Flat
2062 Appl) u (lift (S O) O x3) x5 x6 (s (Bind x0) i) H16)) t H15)))) H14))
2063 (\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
2064 (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2)))
2065 (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl)
2066 (lift (S O) O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3:
2067 T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_:
2068 T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0)
2069 i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda
2070 (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
2071 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
2072 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2073 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2074 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2075 (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
2076 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
2077 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
2078 (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
2079 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2080 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2081 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2082 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2083 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2084 (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2085 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2086 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
2087 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2088 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2089 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
2090 y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq T t
2091 (THead (Bind x0) x6 x7))).(\lambda (H16: (subst0 i u x4 x6)).(\lambda (H17:
2092 (subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
2093 x7)).(eq_ind_r T (THead (Bind x0) x6 x7) (\lambda (t3: T).(or3 (ex3_2 T T
2094 (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
2095 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
2096 (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1:
2097 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
2098 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
2099 (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
2100 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
2101 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
2102 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
2103 T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2104 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2105 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2106 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2107 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2108 T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
2109 O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2110 (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
2111 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2112 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2113 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
2114 y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x7 (THead (Flat
2115 Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3)
2116 u2))) (ex2 T (\lambda (t3: T).(eq T x7 (THead (Flat Appl) (lift (S O) O x3)
2117 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)))
2118 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2
2119 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O)
2120 O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind
2121 x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
2122 (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
2123 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
2124 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2125 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2126 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2127 (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda
2128 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
2129 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
2130 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
2131 T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2132 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2133 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2134 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2135 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2136 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead
2137 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
2138 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
2139 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2140 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
2141 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2142 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H18: (ex2 T
2143 (\lambda (u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2:
2144 T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda
2145 (u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s
2146 (Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2:
2147 T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2
2148 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
2149 T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
2150 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
2151 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2152 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
2153 x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
2154 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
2155 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
2156 (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
2157 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2158 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2159 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2160 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2161 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2162 (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead (Flat Appl)
2163 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2164 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
2165 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2166 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
2167 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2168 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H19: (eq T
2169 x7 (THead (Flat Appl) x8 x5))).(\lambda (H20: (subst0 (s (Bind x0) i) u (lift
2170 (S O) O x3) x8)).(eq_ind_r T (THead (Flat Appl) x8 x5) (\lambda (t3: T).(or3
2171 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 t3)
2172 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
2173 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4))))
2174 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
2175 T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2176 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
2177 x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
2178 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
2179 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
2180 (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
2181 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2182 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2183 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2184 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2185 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2186 (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead (Flat Appl)
2187 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2188 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
2189 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2190 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
2191 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2192 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x8
2193 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u
2194 x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
2195 x0) x6 (THead (Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
2196 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
2197 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2198 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2199 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2200 (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind
2201 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2202 (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
2203 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
2204 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
2205 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
2206 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2207 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2208 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2209 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
2210 (Flat Appl) x8 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2211 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2212 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
2213 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2214 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2215 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
2216 y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H21: (eq T x8 (lift (S O) O
2217 x9))).(\lambda (H22: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x9)).(let H23
2218 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u
2219 x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x9) (\lambda (t3:
2220 T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
2221 x6 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
2222 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
2223 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2224 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2225 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2226 (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Bind
2227 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2228 (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
2229 T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
2230 u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
2231 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
2232 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2233 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2234 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2235 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
2236 (Flat Appl) t3 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2237 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2238 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
2239 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2240 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2241 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
2242 y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
2243 T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead
2244 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
2245 (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3))))
2246 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
2247 T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2248 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
2249 x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead (Bind Abbr) u2 t3))))))
2250 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
2251 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
2252 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
2253 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2254 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2255 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2256 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2257 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2258 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
2259 O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2260 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2261 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
2262 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2263 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2264 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
2265 y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
2266 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
2267 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2268 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2269 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2270 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
2271 (Flat Appl) (lift (S O) O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl)
2272 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2273 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
2274 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2275 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
2276 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2277 (CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x9 x6 H7 (refl_equal T (THead
2278 (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift
2279 (S O) O x9) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x9 H23) (pr2_delta c0 d u
2280 i H1 x1 x4 H11 x6 H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) x8
2281 H21))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i
2282 (le_O_n i)))) x7 H19)))) H18)) (\lambda (H18: (ex2 T (\lambda (t3: T).(eq T
2283 x7 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s
2284 (Flat Appl) (s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7
2285 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat
2286 Appl) (s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
2287 (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda
2288 (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3:
2289 T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
2290 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
2291 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
2292 (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2
2293 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
2294 T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
2295 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
2296 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
2297 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
2298 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2299 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2300 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2301 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7)
2302 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
2303 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2304 T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
2305 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
2306 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
2307 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
2308 z2))))))))) (\lambda (x8: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) (lift
2309 (S O) O x3) x8))).(\lambda (H20: (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5
2310 x8)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O x3) x8) (\lambda (t3:
2311 T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
2312 x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0
2313 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2)
2314 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2315 T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
2316 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
2317 T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
2318 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
2319 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
2320 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
2321 T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2322 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2323 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2324 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2325 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2326 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead
2327 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
2328 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
2329 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2330 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
2331 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2332 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T
2333 (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat
2334 Appl) (lift (S O) O x3) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
2335 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
2336 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2337 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2338 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2339 (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))
2340 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2341 T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
2342 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
2343 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
2344 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2345 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
2346 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
2347 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
2348 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
2349 (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) (THead (Bind b)
2350 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
2351 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
2352 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2353 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
2354 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2355 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
2356 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2357 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2358 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2359 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2360 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2361 (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))
2362 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
2363 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2364 T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
2365 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
2366 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
2367 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))
2368 x0 x1 x2 x8 x3 x6 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
2369 (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))) (pr2_free c0
2370 u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0
2371 (Bind x0) x6) d u (S i) (getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6
2372 (clear_bind x0 c0 x6) (CHead d (Bind Abbr) u) i H1) x2 x5 H12 x8 H20))) x7
2373 H19)))) H18)) (\lambda (H18: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
2374 T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s
2375 (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3:
2376 T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T
2377 (\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 t3))))
2378 (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3)
2379 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0)
2380 i)) u x5 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead
2381 (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
2382 T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0)
2383 x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2384 T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
2385 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
2386 T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
2387 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
2388 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
2389 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
2390 T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2391 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2392 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2393 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2394 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2395 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead
2396 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
2397 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
2398 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2399 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
2400 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2401 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda
2402 (x9: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) x8 x9))).(\lambda (H20:
2403 (subst0 (s (Bind x0) i) u (lift (S O) O x3) x8)).(\lambda (H21: (subst0 (s
2404 (Flat Appl) (s (Bind x0) i)) u x5 x9)).(eq_ind_r T (THead (Flat Appl) x8 x9)
2405 (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
2406 (THead (Bind x0) x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
2407 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
2408 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2409 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2410 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2411 (t4: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda
2412 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
2413 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
2414 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
2415 T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2416 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2417 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2418 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2419 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2420 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead
2421 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
2422 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
2423 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2424 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
2425 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2426 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3:
2427 T).(eq T x8 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0)
2428 i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
2429 (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3))))
2430 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
2431 (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
2432 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
2433 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
2434 (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9))
2435 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2436 T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
2437 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
2438 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
2439 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2440 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
2441 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
2442 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
2443 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
2444 (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind b) y2 (THead
2445 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
2446 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
2447 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2448 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
2449 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2450 (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x10: T).(\lambda
2451 (H22: (eq T x8 (lift (S O) O x10))).(\lambda (H23: (subst0 (minus (s (Bind
2452 x0) i) (S O)) u x3 x10)).(let H24 \def (eq_ind nat (minus (s (Bind x0) i) (S
2453 O)) (\lambda (n: nat).(subst0 n u x3 x10)) H23 i (s_arith1 x0 i)) in
2454 (eq_ind_r T (lift (S O) O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
2455 T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9))
2456 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
2457 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4))))
2458 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
2459 T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2460 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
2461 x6 (THead (Flat Appl) t3 x9)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
2462 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
2463 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
2464 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
2465 T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2466 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2467 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2468 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2469 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2470 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9))
2471 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
2472 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2473 T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
2474 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
2475 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
2476 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
2477 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
2478 (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat
2479 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
2480 (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
2481 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
2482 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2483 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
2484 x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Bind Abbr) u2 t3))))))
2485 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
2486 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
2487 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
2488 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2489 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2490 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2491 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2492 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2493 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
2494 O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2495 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2496 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
2497 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2498 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2499 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
2500 y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
2501 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
2502 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2503 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2504 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2505 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
2506 (Flat Appl) (lift (S O) O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl)
2507 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2508 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
2509 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2510 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
2511 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2512 (CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x9 x10 x6 H7 (refl_equal T
2513 (THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl)
2514 (lift (S O) O x10) x9))) (pr2_delta c0 d u i H1 u1 x3 H10 x10 H24) (pr2_delta
2515 c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 (Bind x0) x6) d u (S i)
2516 (getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 (clear_bind x0 c0 x6)
2517 (CHead d (Bind Abbr) u) i H1) x2 x5 H12 x9 H21))) x8 H22)))))
2518 (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i (le_O_n
2519 i)))) x7 H19)))))) H18)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5
2520 x7 (s (Bind x0) i) H17)) t H15)))))) H14)) (subst0_gen_head (Bind x0) u x4
2521 (THead (Flat Appl) (lift (S O) O x3) x5) t i H13)) t1 H8)))))))))))))) H6))
2522 (pr0_gen_appl u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
2524 theorem pr2_gen_abbr:
2525 \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
2526 (THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
2527 (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_:
2528 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b:
2529 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u:
2530 T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T
2531 T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
2532 (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
2533 T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u:
2534 T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
2536 \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
2537 (H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1
2538 t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
2539 T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
2540 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3
2541 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T
2542 (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1
2543 t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
2544 u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
2545 T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b:
2546 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))
2547 (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda
2548 (t: T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T
2549 T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2))))
2550 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
2551 (t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
2552 t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind
2553 Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0
2554 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z)))
2555 (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t2))))))))
2556 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O
2557 t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1:
2558 (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(let H3 \def
2559 (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abbr) u1 t1) H2) in
2560 (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
2561 Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
2562 T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
2563 (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2)) (or
2564 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
2565 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
2566 T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind
2567 b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead
2568 c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
2569 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
2570 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z
2571 t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
2572 (lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
2573 T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
2574 T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T
2575 (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0
2576 t3)))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
2577 (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
2578 (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
2579 (\lambda (y0: T).(subst0 O u2 y0 t3)))))) (or (ex3_2 T T (\lambda (u2:
2580 T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
2581 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
2582 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T
2583 (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1
2584 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr)
2585 u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
2586 T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
2587 B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2)))))
2588 (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead (Bind Abbr)
2589 x0 x1))).(\lambda (H6: (pr0 u1 x0)).(\lambda (H_x: (or (pr0 t1 x1) (ex2 T
2590 (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0
2591 x1))))).(or_ind (pr0 t1 x1) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda
2592 (y0: T).(subst0 O x0 y0 x1))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
2593 T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
2594 T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
2595 B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u:
2596 T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2
2597 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0)))
2598 (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
2599 T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u:
2600 T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (H7: (pr0 t1
2601 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or (ex3_2 T T
2602 (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
2603 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
2604 (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
2605 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind
2606 Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0
2607 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z)))
2608 (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))))
2609 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O
2610 t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead
2611 (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
2612 T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
2613 B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u:
2614 T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2
2615 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0)))
2616 (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
2617 T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u:
2618 T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O (THead (Bind Abbr) x0 x1)))))
2619 (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr)
2620 x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0
2621 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u:
2622 T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u))
2623 (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda
2624 (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
2625 T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
2626 c0 (Bind Abbr) u1) z t3))))))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 x1))
2627 (pr2_free c0 u1 x0 H6) (or3_intro0 (\forall (b: B).(\forall (u: T).(pr2
2628 (CHead c0 (Bind b) u) t1 x1))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda
2629 (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 x1))) (ex3_2 T T (\lambda (y0:
2630 T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
2631 T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
2632 c0 (Bind Abbr) u1) z x1)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead
2633 c0 (Bind b) u) t1 x1 H7)))))) t2 H5)) (\lambda (H_x0: (ex2 T (\lambda (y0:
2634 T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0 x1)))).(ex2_ind T (\lambda
2635 (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0 x1)) (or (ex3_2 T T
2636 (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))
2637 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
2638 (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
2639 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind
2640 Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0
2641 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z)))
2642 (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))))
2643 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O
2644 t2))))) (\lambda (x2: T).(\lambda (H7: (pr0 t1 x2)).(\lambda (H8: (subst0 O
2645 x0 x2 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or (ex3_2 T
2646 T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
2647 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
2648 (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
2649 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind
2650 Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0
2651 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z)))
2652 (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))))
2653 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O
2654 t)))))) (ex2_ind T (\lambda (t: T).(subst0 O u1 x2 t)) (\lambda (t: T).(pr0 t
2655 x1)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
2656 Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
2657 T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
2658 B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u:
2659 T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2
2660 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0)))
2661 (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
2662 T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u:
2663 T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O (THead (Bind Abbr) x0 x1))))))
2664 (\lambda (x3: T).(\lambda (_: (subst0 O u1 x2 x3)).(\lambda (_: (pr0 x3
2665 x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead
2666 (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
2667 T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
2668 B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u:
2669 T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2
2670 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0)))
2671 (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
2672 T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u:
2673 T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O (THead (Bind Abbr) x0 x1)))))
2674 (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr)
2675 x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0
2676 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u:
2677 T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u))
2678 (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda
2679 (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
2680 T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
2681 c0 (Bind Abbr) u1) z t3))))))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 x1))
2682 (pr2_free c0 u1 x0 H6) (or3_intro1 (\forall (b: B).(\forall (u: T).(pr2
2683 (CHead c0 (Bind b) u) t1 x1))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda
2684 (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 x1))) (ex3_2 T T (\lambda (y0:
2685 T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
2686 T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
2687 c0 (Bind Abbr) u1) z x1)))) (ex_intro2 T (\lambda (u: T).(pr0 u1 u)) (\lambda
2688 (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 x1)) x0 H6 (pr2_delta (CHead c0 (Bind
2689 Abbr) x0) c0 x0 O (getl_refl Abbr c0 x0) t1 x2 H7 x1 H8))))))))
2690 (pr0_subst0_back x0 x2 x1 O H8 u1 H6)) t2 H5)))) H_x0)) H_x)))))) H4))
2691 (\lambda (H4: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2:
2692 T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
2693 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
2694 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T
2695 (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1
2696 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr)
2697 u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
2698 T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
2699 B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2))))
2700 (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 (lift (S
2701 O) O t2) H4))))) (pr0_gen_abbr u1 t1 t2 H3)))))))) (\lambda (c0: C).(\lambda
2702 (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d
2703 (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0
2704 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0
2705 (THead (Bind Abbr) u1 t1))).(let H5 \def (eq_ind T t0 (\lambda (t3: T).(pr0
2706 t3 t2)) H2 (THead (Bind Abbr) u1 t1) H4) in (or_ind (ex3_2 T T (\lambda (u2:
2707 T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
2708 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0
2709 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0
2710 t3))))))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
2711 (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
2712 T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
2713 B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0:
2714 T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3)))
2715 (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1
2716 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda
2717 (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall
2718 (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (H6:
2719 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
2720 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
2721 T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
2722 (\lambda (y0: T).(subst0 O u2 y0 t3)))))))).(ex3_2_ind T T (\lambda (u2:
2723 T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
2724 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0
2725 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0
2726 t3)))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
2727 (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
2728 (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2
2729 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
2730 (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0:
2731 T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
2732 T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
2733 c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead
2734 c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x0: T).(\lambda (x1:
2735 T).(\lambda (H7: (eq T t2 (THead (Bind Abbr) x0 x1))).(\lambda (H8: (pr0 u1
2736 x0)).(\lambda (H_x: (or (pr0 t1 x1) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
2737 (\lambda (y0: T).(subst0 O x0 y0 x1))))).(or_ind (pr0 t1 x1) (ex2 T (\lambda
2738 (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0 x1))) (or (ex3_2 T T
2739 (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
2740 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
2741 (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
2742 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0
2743 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
2744 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
2745 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z
2746 t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
2747 (lift (S O) O t))))) (\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2
2748 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Bind Abbr) x0 x1) H7) in
2749 (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead (Bind Abbr) u2 x1))) (\lambda
2750 (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind
2751 Abbr) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))) (ex3_2 T
2752 T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
2753 (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
2754 T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3)))) (or (ex3_2 T T
2755 (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
2756 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
2757 (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
2758 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0
2759 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
2760 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
2761 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z
2762 t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
2763 (lift (S O) O t))))) (\lambda (H11: (ex2 T (\lambda (u2: T).(eq T t (THead
2764 (Bind Abbr) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T
2765 (\lambda (u2: T).(eq T t (THead (Bind Abbr) u2 x1))) (\lambda (u2: T).(subst0
2766 i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
2767 (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
2768 (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2
2769 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
2770 (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0:
2771 T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
2772 T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
2773 c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead
2774 c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x2: T).(\lambda (H12: (eq T
2775 t (THead (Bind Abbr) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl
2776 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2
2777 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
2778 T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0
2779 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0:
2780 T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0:
2781 T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
2782 T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
2783 c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead
2784 c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2:
2785 T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
2786 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
2787 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
2788 (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
2789 t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
2790 Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
2791 T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x2 x1 H12
2792 (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (or3_intro0 (\forall (b: B).(\forall
2793 (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 x1))) (ex2 T (\lambda (u0: T).(pr0 u1
2794 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x1))) (ex3_2 T T
2795 (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0)))
2796 (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
2797 T).(pr2 (CHead c0 (Bind Abbr) u1) z x1)))) (\lambda (b: B).(\lambda (u0:
2798 T).(pr2_free (CHead c0 (Bind b) u0) t1 x1 H9))))))))) H11)) (\lambda (H11:
2799 (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x0 t3))) (\lambda (t3:
2800 T).(subst0 (s (Bind Abbr) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t
2801 (THead (Bind Abbr) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1
2802 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
2803 Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
2804 (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0
2805 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0:
2806 T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0:
2807 T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
2808 T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
2809 c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead
2810 c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x2: T).(\lambda (H12: (eq T
2811 t (THead (Bind Abbr) x0 x2))).(\lambda (H13: (subst0 (s (Bind Abbr) i) u x1
2812 x2)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
2813 (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
2814 (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2
2815 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
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2821 T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
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2828 (pr2_free c0 u1 x0 H8) (or3_intro0 (\forall (b: B).(\forall (u0: T).(pr2
2829 (CHead c0 (Bind b) u0) t1 x2))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
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2834 (CHead c0 (Bind b) u0) d u (S i) (getl_head (Bind b) i c0 (CHead d (Bind
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2836 (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
2837 (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
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2847 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
2848 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z
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2866 T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
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2873 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z x3))))
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2875 (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x3
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2881 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0:
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2888 t1 x2)).(\lambda (H10: (subst0 O x0 x2 x1)).(let H11 \def (eq_ind T t2
2889 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Bind Abbr) x0 x1) H7) in
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2893 T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
2894 (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
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2901 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
2902 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z
2903 t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
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2909 (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2
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2937 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
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2941 T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
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2947 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
2948 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z x1))))
2949 (ex_intro2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0
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2958 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
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2964 (\lambda (x3: T).(\lambda (H13: (eq T t (THead (Bind Abbr) x0 x3))).(\lambda
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2969 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
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2978 (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2
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2985 T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
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2987 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
2988 (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
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2992 (pr2_free c0 u1 x0 H8) (or3_intro2 (\forall (b: B).(\forall (u0: T).(pr2
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2998 T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z:
2999 T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr)
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3004 (H12: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr)
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3007 (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
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3010 (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
3011 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
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3013 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0
3014 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
3015 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
3016 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z
3017 t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
3018 (lift (S O) O t))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H13: (eq T t
3019 (THead (Bind Abbr) x3 x4))).(\lambda (H14: (subst0 i u x0 x3)).(\lambda (H15:
3020 (subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t3: T).(subst0 O u1
3021 x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2:
3022 T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
3023 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
3024 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
3025 (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
3026 t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
3027 Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
3028 T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
3029 B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
3030 (\lambda (x5: T).(\lambda (H16: (subst0 O u1 x2 x5)).(\lambda (H17: (pr0 x5
3031 x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
3032 (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
3033 (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2
3034 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
3035 (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0:
3036 T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
3037 T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
3038 c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead
3039 c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2:
3040 T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
3041 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
3042 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
3043 (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
3044 t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
3045 Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
3046 T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x3 x4 H13
3047 (pr2_delta c0 d u i H1 u1 x0 H8 x3 H14) (or3_intro2 (\forall (b: B).(\forall
3048 (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1
3049 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x4))) (ex3_2 T T
3050 (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0)))
3051 (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
3052 T).(pr2 (CHead c0 (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y0:
3053 T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
3054 T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
3055 c0 (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O
3056 (getl_refl Abbr c0 u1) t1 x2 H9 x5 H16) H17 (pr2_delta (CHead c0 (Bind Abbr)
3057 u1) d u (S i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1
3058 x1 (pr0_refl x1) x4 H15)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8)))))))
3059 H12)) (subst0_gen_head (Bind Abbr) u x0 x1 t i H11)))))) H_x0)) H_x))))))
3060 H6)) (\lambda (H6: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda
3061 (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
3062 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
3063 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
3064 (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
3065 t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
3066 Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
3067 T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
3068 B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
3069 (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i)
3070 (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2)
3071 H6 (lift (S O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i)))))))
3072 (pr0_gen_abbr u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
3074 theorem pr2_gen_void:
3075 \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
3076 (THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
3077 (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
3078 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
3079 (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u:
3080 T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
3082 \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
3083 (H: (pr2 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1
3084 t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
3085 T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2:
3086 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
3087 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b:
3088 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))
3089 (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda
3090 (t: T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T
3091 T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2))))
3092 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
3093 (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
3094 t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift
3095 (S O) O t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2:
3096 T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1
3097 t1))).(let H3 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind
3098 Void) u1 t1) H2) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
3099 T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1
3100 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O
3101 t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
3102 Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
3103 (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind
3104 b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u)
3105 t1 (lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda
3106 (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
3107 T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind
3108 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
3109 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
3110 T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
3111 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
3112 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
3113 (CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead
3114 c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (x0: T).(\lambda (x1:
3115 T).(\lambda (H5: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H6: (pr0 u1
3116 x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1)
3117 (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
3118 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
3119 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
3120 (CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead
3121 c0 (Bind b) u) t1 (lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2:
3122 T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2
3123 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
3124 T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
3125 u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
3126 (lift (S O) O (THead (Bind Void) x0 x1))))) (ex3_2_intro T T (\lambda (u2:
3127 T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2
3128 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
3129 T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
3130 u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) (pr2_free c0 u1
3131 x0 H6) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1
3132 H7))))) t2 H5)))))) H4)) (\lambda (H4: (pr0 t1 (lift (S O) O t2))).(or_intror
3133 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2
3134 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
3135 T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
3136 u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
3137 (lift (S O) O t2)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0
3138 (Bind b) u) t1 (lift (S O) O t2) H4))))) (pr0_gen_void u1 t1 t2 H3))))))))
3139 (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
3140 (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2:
3141 T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2
3142 t)).(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(let H5 \def (eq_ind T
3143 t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Bind Void) u1 t1) H4) in (or_ind
3144 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2
3145 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
3146 T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T
3147 (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
3148 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
3149 (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
3150 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
3151 (lift (S O) O t))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
3152 T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
3153 T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind
3154 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
3155 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
3156 T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
3157 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
3158 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
3159 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2
3160 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x0: T).(\lambda (x1:
3161 T).(\lambda (H7: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H8: (pr0 u1
3162 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3:
3163 T).(subst0 i u t3 t)) H3 (THead (Bind Void) x0 x1) H7) in (or3_ind (ex2 T
3164 (\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0
3165 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3)))
3166 (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (ex3_2 T T (\lambda
3167 (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
3168 T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
3169 T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T (\lambda (u2:
3170 T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
3171 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
3172 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
3173 B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
3174 (\lambda (H11: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1)))
3175 (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t
3176 (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T
3177 T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
3178 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
3179 (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
3180 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
3181 (lift (S O) O t))))) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind
3182 Void) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T
3183 (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
3184 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
3185 (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
3186 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
3187 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
3188 t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
3189 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
3190 (CHead c0 (Bind b) u0) t1 t3))))) x2 x1 H12 (pr2_delta c0 d u i H1 u1 x0 H8
3191 x2 H13) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) t1
3192 x1 H9)))))))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead
3193 (Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1
3194 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3)))
3195 (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda
3196 (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
3197 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
3198 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
3199 B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
3200 (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind Void) x0 x2))).(\lambda
3201 (H13: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2:
3202 T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
3203 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
3204 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
3205 B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
3206 (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void)
3207 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
3208 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
3209 u0) t1 t3))))) x0 x2 H12 (pr2_free c0 u1 x0 H8) (\lambda (b: B).(\lambda (u0:
3210 T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head (Bind b) i c0
3211 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x2 H13)))))))) H11)) (\lambda (H11:
3212 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2
3213 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
3214 T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))))).(ex3_2_ind T T
3215 (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
3216 (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
3217 T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T
3218 (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
3219 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
3220 (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
3221 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
3222 (lift (S O) O t))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq T t
3223 (THead (Bind Void) x2 x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14:
3224 (subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2:
3225 T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
3226 T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
3227 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
3228 B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
3229 (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void)
3230 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
3231 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
3232 u0) t1 t3))))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (\lambda (b:
3233 B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head
3234 (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x3 H14)))))))))) H11))
3235 (subst0_gen_head (Bind Void) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0
3236 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
3237 T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
3238 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0:
3239 T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0:
3240 T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) (\lambda (b:
3241 B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head
3242 (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) H6 (lift (S
3243 O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) (pr0_gen_void u1 t1
3244 t2 H5)))))))))))))) c y x H0))) H))))).
3246 theorem pr2_gen_lift:
3247 \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall
3248 (d: nat).((pr2 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to
3249 (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e t1
3252 \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
3253 (d: nat).(\lambda (H: (pr2 c (lift h d t1) x)).(insert_eq T (lift h d t1)
3254 (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e)
3255 \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e
3256 t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0:
3257 C).(\lambda (t: T).(\lambda (t0: T).((eq T t (lift h d t1)) \to (\forall (e:
3258 C).((drop h d c0 e) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h d t2)))
3259 (\lambda (t2: T).(pr2 e t1 t2))))))))) (\lambda (c0: C).(\lambda (t0:
3260 T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (lift h
3261 d t1))).(\lambda (e: C).(\lambda (_: (drop h d c0 e)).(let H4 \def (eq_ind T
3262 t0 (\lambda (t: T).(pr0 t t2)) H1 (lift h d t1) H2) in (ex2_ind T (\lambda
3263 (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 T
3264 (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
3265 (\lambda (x0: T).(\lambda (H5: (eq T t2 (lift h d x0))).(\lambda (H6: (pr0 t1
3266 x0)).(eq_ind_r T (lift h d x0) (\lambda (t: T).(ex2 T (\lambda (t3: T).(eq T
3267 t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))) (ex_intro2 T (\lambda
3268 (t3: T).(eq T (lift h d x0) (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))
3269 x0 (refl_equal T (lift h d x0)) (pr2_free e t1 x0 H6)) t2 H5))))
3270 (pr0_gen_lift t1 t2 h d H4)))))))))) (\lambda (c0: C).(\lambda (d0:
3271 C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d0 (Bind
3272 Abbr) u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0
3273 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0
3274 (lift h d t1))).(\lambda (e: C).(\lambda (H5: (drop h d c0 e)).(let H6 \def
3275 (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (lift h d t1) H4) in (ex2_ind T
3276 (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2
3277 T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
3278 (\lambda (x0: T).(\lambda (H7: (eq T t2 (lift h d x0))).(\lambda (H8: (pr0 t1
3279 x0)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (lift h
3280 d x0) H7) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T t (lift h d t3)))
3281 (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H10: (lt i d)).(let H11 \def
3282 (eq_ind nat d (\lambda (n: nat).(subst0 i u (lift h n x0) t)) H9 (S (plus i
3283 (minus d (S i)))) (lt_plus_minus i d H10)) in (let H12 \def (eq_ind nat d
3284 (\lambda (n: nat).(drop h n c0 e)) H5 (S (plus i (minus d (S i))))
3285 (lt_plus_minus i d H10)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
3286 C).(eq T u (lift h (minus d (S i)) v)))) (\lambda (v: T).(\lambda (e0:
3287 C).(getl i e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
3288 C).(drop h (minus d (S i)) d0 e0))) (ex2 T (\lambda (t3: T).(eq T t (lift h d
3289 t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (x2:
3290 C).(\lambda (H13: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H14: (getl
3291 i e (CHead x2 (Bind Abbr) x1))).(\lambda (_: (drop h (minus d (S i)) d0
3292 x2)).(let H16 \def (eq_ind T u (\lambda (t3: T).(subst0 i t3 (lift h (S (plus
3293 i (minus d (S i)))) x0) t)) H11 (lift h (minus d (S i)) x1) H13) in (ex2_ind
3294 T (\lambda (t3: T).(eq T t (lift h (S (plus i (minus d (S i)))) t3)))
3295 (\lambda (t3: T).(subst0 i x1 x0 t3)) (ex2 T (\lambda (t3: T).(eq T t (lift h
3296 d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H17: (eq
3297 T t (lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H18: (subst0 i x1 x0
3298 x3)).(let H19 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n:
3299 nat).(eq T t (lift h n x3))) H17 d (lt_plus_minus i d H10)) in (ex_intro2 T
3300 (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3
3301 H19 (pr2_delta e x2 x1 i H14 t1 x0 H8 x3 H18)))))) (subst0_gen_lift_lt x1 x0
3302 t i h (minus d (S i)) H16)))))))) (getl_drop_conf_lt Abbr c0 d0 u i H1 e h
3303 (minus d (S i)) H12))))) (\lambda (H10: (le d i)).(lt_le_e i (plus d h) (ex2
3304 T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
3305 (\lambda (H11: (lt i (plus d h))).(subst0_gen_lift_false x0 u t h d i H10 H11
3306 H9 (ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1
3307 t3))))) (\lambda (H11: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq T t
3308 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T
3309 (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
3310 (\lambda (x1: T).(\lambda (H12: (eq T t (lift h d x1))).(\lambda (H13:
3311 (subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T t (lift h d
3312 t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H12 (pr2_delta e d0 u (minus i h)
3313 (getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c0 H1 e h d H5 H11) t1 x0 H8 x1
3314 H13))))) (subst0_gen_lift_ge u x0 t i h d H9 H11)))))))))) (pr0_gen_lift t1
3315 t2 h d H6)))))))))))))))) c y x H0))) H)))))).
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