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 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/fwd".
19 include "pr2/defs.ma".
23 include "getl/drop.ma".
25 include "getl/clear.ma".
28 \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to
31 \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TSort
32 n) x)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda (t:
33 T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t
34 (TSort n)) \to ((eq T t0 x) \to (eq T x (TSort n))))))))) with [(pr2_free c0
35 t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1
36 (TSort n))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t1
37 (TSort n)) \to ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T x (TSort n))))))
38 (\lambda (H4: (eq T t1 (TSort n))).(eq_ind T (TSort n) (\lambda (t: T).((eq T
39 t2 x) \to ((pr0 t t2) \to (eq T x (TSort n))))) (\lambda (H5: (eq T t2
40 x)).(eq_ind T x (\lambda (t: T).((pr0 (TSort n) t) \to (eq T x (TSort n))))
41 (\lambda (H6: (pr0 (TSort n) x)).(let H7 \def (eq_ind T x (\lambda (t:
42 T).(pr2 c (TSort n) t)) H (TSort n) (pr0_gen_sort x n H6)) in (eq_ind_r T
43 (TSort n) (\lambda (t: T).(eq T t (TSort n))) (refl_equal T (TSort n)) x
44 (pr0_gen_sort x n H6)))) t2 (sym_eq T t2 x H5))) t1 (sym_eq T t1 (TSort n)
45 H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t1 t2 H1 t
46 H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t1 (TSort
47 n))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t1 (TSort
48 n)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1 t2)
49 \to ((subst0 i u t2 t) \to (eq T x (TSort n)))))))) (\lambda (H6: (eq T t1
50 (TSort n))).(eq_ind T (TSort n) (\lambda (t0: T).((eq T t x) \to ((getl i c
51 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (eq T x
52 (TSort n))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t0: T).((getl
53 i c (CHead d (Bind Abbr) u)) \to ((pr0 (TSort n) t2) \to ((subst0 i u t2 t0)
54 \to (eq T x (TSort n)))))) (\lambda (_: (getl i c (CHead d (Bind Abbr)
55 u))).(\lambda (H9: (pr0 (TSort n) t2)).(\lambda (H10: (subst0 i u t2 x)).(let
56 H11 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 x)) H10 (TSort n)
57 (pr0_gen_sort t2 n H9)) in (subst0_gen_sort u x i n H11 (eq T x (TSort
58 n))))))) t (sym_eq T t x H7))) t1 (sym_eq T t1 (TSort n) H6))) c0 (sym_eq C
59 c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (TSort
60 n)) (refl_equal T x)))))).
63 \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to
64 (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c
65 (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S
68 \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TLRef
69 n) x)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda (t:
70 T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t
71 (TLRef n)) \to ((eq T t0 x) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda
72 (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_:
73 C).(\lambda (u: T).(eq T x (lift (S n) O u))))))))))))) with [(pr2_free c0 t1
74 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 (TLRef
75 n))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t1 (TLRef
76 n)) \to ((eq T t2 x) \to ((pr0 t1 t2) \to (or (eq T x (TLRef n)) (ex2_2 C T
77 (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda
78 (_: C).(\lambda (u: T).(eq T x (lift (S n) O u)))))))))) (\lambda (H4: (eq T
79 t1 (TLRef n))).(eq_ind T (TLRef n) (\lambda (t: T).((eq T t2 x) \to ((pr0 t
80 t2) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u:
81 T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T
82 x (lift (S n) O u))))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda
83 (t: T).((pr0 (TLRef n) t) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d:
84 C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_:
85 C).(\lambda (u: T).(eq T x (lift (S n) O u)))))))) (\lambda (H6: (pr0 (TLRef
86 n) x)).(let H7 \def (eq_ind T x (\lambda (t: T).(pr2 c (TLRef n) t)) H (TLRef
87 n) (pr0_gen_lref x n H6)) in (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T
88 t (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d
89 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t (lift (S n) O
90 u))))))) (or_introl (eq T (TLRef n) (TLRef n)) (ex2_2 C T (\lambda (d:
91 C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_:
92 C).(\lambda (u: T).(eq T (TLRef n) (lift (S n) O u))))) (refl_equal T (TLRef
93 n))) x (pr0_gen_lref x n H6)))) t2 (sym_eq T t2 x H5))) t1 (sym_eq T t1
94 (TLRef n) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0
95 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t1
96 (TLRef n))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t1
97 (TLRef n)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0
98 t1 t2) \to ((subst0 i u t2 t) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda
99 (d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_:
100 C).(\lambda (u0: T).(eq T x (lift (S n) O u0)))))))))))) (\lambda (H6: (eq T
101 t1 (TLRef n))).(eq_ind T (TLRef n) (\lambda (t0: T).((eq T t x) \to ((getl i
102 c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (or
103 (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c
104 (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift
105 (S n) O u0))))))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t0:
106 T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (TLRef n) t2) \to ((subst0 i
107 u t2 t0) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0:
108 T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0:
109 T).(eq T x (lift (S n) O u0)))))))))) (\lambda (H8: (getl i c (CHead d (Bind
110 Abbr) u))).(\lambda (H9: (pr0 (TLRef n) t2)).(\lambda (H10: (subst0 i u t2
111 x)).(let H11 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 x)) H10 (TLRef
112 n) (pr0_gen_lref t2 n H9)) in (and_ind (eq nat n i) (eq T x (lift (S n) O u))
113 (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c
114 (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift
115 (S n) O u0)))))) (\lambda (H12: (eq nat n i)).(\lambda (H13: (eq T x (lift (S
116 n) O u))).(let H14 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead
117 d (Bind Abbr) u))) H8 n H12) in (let H15 \def (eq_ind T x (\lambda (t0:
118 T).(pr2 c (TLRef n) t0)) H (lift (S n) O u) H13) in (eq_ind_r T (lift (S n) O
119 u) (\lambda (t0: T).(or (eq T t0 (TLRef n)) (ex2_2 C T (\lambda (d0:
120 C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_:
121 C).(\lambda (u0: T).(eq T t0 (lift (S n) O u0))))))) (or_intror (eq T (lift
122 (S n) O u) (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c
123 (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S
124 n) O u) (lift (S n) O u0))))) (ex2_2_intro C T (\lambda (d0: C).(\lambda (u0:
125 T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0:
126 T).(eq T (lift (S n) O u) (lift (S n) O u0)))) d u H14 (refl_equal T (lift (S
127 n) O u)))) x H13))))) (subst0_gen_lref u x i n H11)))))) t (sym_eq T t x
128 H7))) t1 (sym_eq T t1 (TLRef n) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1
129 H2))))]) in (H0 (refl_equal C c) (refl_equal T (TLRef n)) (refl_equal T
132 theorem pr2_gen_abst:
133 \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
134 (THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
135 T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2
136 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u:
137 T).(pr2 (CHead c (Bind b) u) t1 t2))))))))))
139 \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
140 (H: (pr2 c (THead (Bind Abst) u1 t1) x)).(let H0 \def (match H in pr2 return
141 (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t
142 t0)).((eq C c0 c) \to ((eq T t (THead (Bind Abst) u1 t1)) \to ((eq T t0 x)
143 \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst)
144 u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
145 T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
146 t1 t2))))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq
147 C c0 c)).(\lambda (H2: (eq T t0 (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq
148 T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Abst) u1 t1)) \to
149 ((eq T t2 x) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
150 T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
151 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u:
152 T).(pr2 (CHead c (Bind b) u) t1 t3)))))))))) (\lambda (H4: (eq T t0 (THead
153 (Bind Abst) u1 t1))).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t: T).((eq
154 T t2 x) \to ((pr0 t t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
155 T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
156 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
157 (CHead c (Bind b) u) t1 t3))))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x
158 (\lambda (t: T).((pr0 (THead (Bind Abst) u1 t1) t) \to (ex3_2 T T (\lambda
159 (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
160 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
161 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))))) (\lambda (H6:
162 (pr0 (THead (Bind Abst) u1 t1) x)).(ex3_2_ind T T (\lambda (u2: T).(\lambda
163 (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
164 T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T
165 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3))))
166 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
167 (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))))))
168 (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind Abst) x0
169 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def
170 (eq_ind T x (\lambda (t: T).(pr2 c (THead (Bind Abst) u1 t1) t)) H (THead
171 (Bind Abst) x0 x1) H7) in (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t:
172 T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2
173 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
174 T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
175 t1 t3))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead
176 (Bind Abst) x0 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
177 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
178 (u: T).(pr2 (CHead c (Bind b) u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind
179 Abst) x0 x1)) (pr2_free c u1 x0 H8) (\lambda (b: B).(\lambda (u: T).(pr2_free
180 (CHead c (Bind b) u) t1 x1 H9)))) x H7))))))) (pr0_gen_abst u1 t1 x H6))) t2
181 (sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Bind Abst) u1 t1) H4))) c0
182 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2)
183 \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Bind
184 Abst) u1 t1))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T
185 t0 (THead (Bind Abst) u1 t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind
186 Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (ex3_2 T T (\lambda
187 (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
188 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
189 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))))))))) (\lambda
190 (H6: (eq T t0 (THead (Bind Abst) u1 t1))).(eq_ind T (THead (Bind Abst) u1 t1)
191 (\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to
192 ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (ex3_2 T T (\lambda (u2: T).(\lambda
193 (t4: T).(eq T x (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
194 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
195 (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))))))))))) (\lambda (H7: (eq T t
196 x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to
197 ((pr0 (THead (Bind Abst) u1 t1) t2) \to ((subst0 i u t2 t3) \to (ex3_2 T T
198 (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Abst) u2 t4))))
199 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
200 (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
201 t4)))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9:
202 (pr0 (THead (Bind Abst) u1 t1) t2)).(\lambda (H10: (subst0 i u t2
203 x)).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
204 Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
205 T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
206 T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
207 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0:
208 T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x0: T).(\lambda (x1:
209 T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H12: (pr0 u1
210 x0)).(\lambda (H13: (pr0 t1 x1)).(let H14 \def (eq_ind T t2 (\lambda (t3:
211 T).(subst0 i u t3 x)) H10 (THead (Bind Abst) x0 x1) H11) in (or3_ind (ex2 T
212 (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0
213 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Abst) x0 t3)))
214 (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3))) (ex3_2 T T (\lambda
215 (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
216 T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
217 T).(subst0 (s (Bind Abst) i) u x1 t3)))) (ex3_2 T T (\lambda (u2: T).(\lambda
218 (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
219 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
220 (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (H15: (ex2 T (\lambda
221 (u2: T).(eq T x (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0
222 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 x1)))
223 (\lambda (u2: T).(subst0 i u x0 u2)) (ex3_2 T T (\lambda (u2: T).(\lambda
224 (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
225 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
226 (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: T).(\lambda
227 (H16: (eq T x (THead (Bind Abst) x2 x1))).(\lambda (H17: (subst0 i u x0
228 x2)).(let H18 \def (eq_ind T x (\lambda (t3: T).(pr2 c (THead (Bind Abst) u1
229 t1) t3)) H (THead (Bind Abst) x2 x1) H16) in (eq_ind_r T (THead (Bind Abst)
230 x2 x1) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3
231 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
232 (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead
233 c (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3:
234 T).(eq T (THead (Bind Abst) x2 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2:
235 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
236 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x1
237 (refl_equal T (THead (Bind Abst) x2 x1)) (pr2_delta c d u i H8 u1 x0 H12 x2
238 H17) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) t1 x1
239 H13)))) x H16))))) H15)) (\lambda (H15: (ex2 T (\lambda (t3: T).(eq T x
240 (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1
241 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind Abst) x0 t3)))
242 (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)) (ex3_2 T T (\lambda (u2:
243 T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
244 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
245 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2:
246 T).(\lambda (H16: (eq T x (THead (Bind Abst) x0 x2))).(\lambda (H17: (subst0
247 (s (Bind Abst) i) u x1 x2)).(let H18 \def (eq_ind T x (\lambda (t3: T).(pr2 c
248 (THead (Bind Abst) u1 t1) t3)) H (THead (Bind Abst) x0 x2) H16) in (eq_ind_r
249 T (THead (Bind Abst) x0 x2) (\lambda (t3: T).(ex3_2 T T (\lambda (u2:
250 T).(\lambda (t4: T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2:
251 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall
252 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))))))) (ex3_2_intro
253 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x2) (THead
254 (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
255 (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead
256 c (Bind b) u0) t1 t3))))) x0 x2 (refl_equal T (THead (Bind Abst) x0 x2))
257 (pr2_free c u1 x0 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c
258 (Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0)
259 t1 x1 H13 x2 H17)))) x H16))))) H15)) (\lambda (H15: (ex3_2 T T (\lambda (u2:
260 T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
261 T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
262 T).(subst0 (s (Bind Abst) i) u x1 t3))))).(ex3_2_ind T T (\lambda (u2:
263 T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
264 T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
265 T).(subst0 (s (Bind Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda
266 (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
267 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
268 (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: T).(\lambda (x3:
269 T).(\lambda (H16: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (H17: (subst0
270 i u x0 x2)).(\lambda (H18: (subst0 (s (Bind Abst) i) u x1 x3)).(let H19 \def
271 (eq_ind T x (\lambda (t3: T).(pr2 c (THead (Bind Abst) u1 t1) t3)) H (THead
272 (Bind Abst) x2 x3) H16) in (eq_ind_r T (THead (Bind Abst) x2 x3) (\lambda
273 (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
274 Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
275 T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
276 u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
277 (THead (Bind Abst) x2 x3) (THead (Bind Abst) u2 t3)))) (\lambda (u2:
278 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
279 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x3
280 (refl_equal T (THead (Bind Abst) x2 x3)) (pr2_delta c d u i H8 u1 x0 H12 x2
281 H17) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S
282 i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H13 x3
283 H18)))) x H16))))))) H15)) (subst0_gen_head (Bind Abst) u x0 x1 x i
284 H14)))))))) (pr0_gen_abst u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T
285 t0 (THead (Bind Abst) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))])
286 in (H0 (refl_equal C c) (refl_equal T (THead (Bind Abst) u1 t1)) (refl_equal
289 theorem pr2_gen_cast:
290 \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
291 (THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
292 (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_:
293 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c
296 \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
297 (H: (pr2 c (THead (Flat Cast) u1 t1) x)).(let H0 \def (match H in pr2 return
298 (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t
299 t0)).((eq C c0 c) \to ((eq T t (THead (Flat Cast) u1 t1)) \to ((eq T t0 x)
300 \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat
301 Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
302 T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c t1 x))))))))) with [(pr2_free c0
303 t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0
304 (THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda
305 (_: C).((eq T t0 (THead (Flat Cast) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2)
306 \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
307 Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
308 T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)))))) (\lambda (H4: (eq T t0
309 (THead (Flat Cast) u1 t1))).(eq_ind T (THead (Flat Cast) u1 t1) (\lambda (t:
310 T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
311 (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
312 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c
313 t1 x))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead
314 (Flat Cast) u1 t1) t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
315 T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
316 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x))))
317 (\lambda (H6: (pr0 (THead (Flat Cast) u1 t1) x)).(or_ind (ex3_2 T T (\lambda
318 (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2:
319 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
320 t3)))) (pr0 t1 x) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
321 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
322 (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (H7:
323 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2
324 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
325 T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
326 (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
327 T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or (ex3_2 T
328 T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3))))
329 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
330 (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x0: T).(\lambda (x1:
331 T).(\lambda (H8: (eq T x (THead (Flat Cast) x0 x1))).(\lambda (H9: (pr0 u1
332 x0)).(\lambda (H10: (pr0 t1 x1)).(let H11 \def (eq_ind T x (\lambda (t:
333 T).(pr2 c (THead (Flat Cast) u1 t1) t)) H (THead (Flat Cast) x0 x1) H8) in
334 (eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda
335 (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2:
336 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
337 t3)))) (pr2 c t1 t))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
338 T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda (u2:
339 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
340 t3)))) (pr2 c t1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T (\lambda (u2:
341 T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2
342 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
343 T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x1 (refl_equal T (THead (Flat Cast) x0
344 x1)) (pr2_free c u1 x0 H9) (pr2_free c t1 x1 H10))) x H8))))))) H7)) (\lambda
345 (H7: (pr0 t1 x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
346 T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
347 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)
348 (pr2_free c t1 x H7))) (pr0_gen_cast u1 t1 x H6))) t2 (sym_eq T t2 x H5))) t0
349 (sym_eq T t0 (THead (Flat Cast) u1 t1) H4))) c0 (sym_eq C c0 c H1) H2 H3
350 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq
351 C c0 c)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(\lambda (H5: (eq
352 T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Flat Cast) u1 t1)) \to
353 ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to
354 ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
355 x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
356 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x))))))))
357 (\lambda (H6: (eq T t0 (THead (Flat Cast) u1 t1))).(eq_ind T (THead (Flat
358 Cast) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr)
359 u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2:
360 T).(\lambda (t4: T).(eq T x (THead (Flat Cast) u2 t4)))) (\lambda (u2:
361 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1
362 t4)))) (pr2 c t1 x))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t3:
363 T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (THead (Flat Cast) u1 t1)
364 t2) \to ((subst0 i u t2 t3) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4:
365 T).(eq T x (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2
366 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (pr2 c t1
367 x)))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0
368 (THead (Flat Cast) u1 t1) t2)).(\lambda (H10: (subst0 i u t2 x)).(or_ind
369 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2
370 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
371 T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda (u2:
372 T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2:
373 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
374 t3)))) (pr2 c t1 x)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
375 T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
376 T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind
377 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3))))
378 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
379 T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
380 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
381 (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x0:
382 T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Flat Cast) x0
383 x1))).(\lambda (H13: (pr0 u1 x0)).(\lambda (H14: (pr0 t1 x1)).(let H15 \def
384 (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 x)) H10 (THead (Flat Cast) x0
385 x1) H12) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Flat Cast) u2
386 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x
387 (THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1
388 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast)
389 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
390 T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))) (or (ex3_2 T T
391 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3))))
392 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
393 (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (H16: (ex2 T (\lambda (u2:
394 T).(eq T x (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0
395 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Flat Cast) u2 x1)))
396 (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
397 (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
398 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c
399 t1 x)) (\lambda (x2: T).(\lambda (H17: (eq T x (THead (Flat Cast) x2
400 x1))).(\lambda (H18: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2:
401 T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2:
402 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
403 t3)))) (pr2 c t1 x) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
404 x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
405 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x1 H17 (pr2_delta c
406 d u i H8 u1 x0 H13 x2 H18) (pr2_free c t1 x1 H14)))))) H16)) (\lambda (H16:
407 (ex2 T (\lambda (t3: T).(eq T x (THead (Flat Cast) x0 t3))) (\lambda (t3:
408 T).(subst0 (s (Flat Cast) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x
409 (THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1
410 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
411 Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
412 T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x2: T).(\lambda
413 (H17: (eq T x (THead (Flat Cast) x0 x2))).(\lambda (H18: (subst0 (s (Flat
414 Cast) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
415 T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
416 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)
417 (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast)
418 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
419 T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x2 H17 (pr2_free c u1 x0 H13)
420 (pr2_delta c d u i H8 t1 x1 H14 x2 H18)))))) H16)) (\lambda (H16: (ex3_2 T T
421 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3))))
422 (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
423 T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))))).(ex3_2_ind T T
424 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3))))
425 (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
426 T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (or (ex3_2 T T
427 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3))))
428 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
429 (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x2: T).(\lambda (x3:
430 T).(\lambda (H17: (eq T x (THead (Flat Cast) x2 x3))).(\lambda (H18: (subst0
431 i u x0 x2)).(\lambda (H19: (subst0 (s (Flat Cast) i) u x1 x3)).(or_introl
432 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2
433 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
434 T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) (ex3_2_intro T T (\lambda
435 (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2:
436 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
437 t3))) x2 x3 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (pr2_delta c d u i H8
438 t1 x1 H14 x3 H19)))))))) H16)) (subst0_gen_head (Flat Cast) u x0 x1 x i
439 H15)))))))) H11)) (\lambda (H11: (pr0 t1 t2)).(or_intror (ex3_2 T T (\lambda
440 (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2:
441 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
442 t3)))) (pr2 c t1 x) (pr2_delta c d u i H8 t1 t2 H11 x H10))) (pr0_gen_cast u1
443 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Flat Cast) u1 t1)
444 H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c)
445 (refl_equal T (THead (Flat Cast) u1 t1)) (refl_equal T x))))))).
447 theorem pr2_gen_csort:
448 \forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2)
451 \lambda (t1: T).(\lambda (t2: T).(\lambda (n: nat).(\lambda (H: (pr2 (CSort
452 n) t1 t2)).(let H0 \def (match H in pr2 return (\lambda (c: C).(\lambda (t:
453 T).(\lambda (t0: T).(\lambda (_: (pr2 c t t0)).((eq C c (CSort n)) \to ((eq T
454 t t1) \to ((eq T t0 t2) \to (pr0 t1 t2)))))))) with [(pr2_free c t0 t3 H0)
455 \Rightarrow (\lambda (H1: (eq C c (CSort n))).(\lambda (H2: (eq T t0
456 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CSort n) (\lambda (_: C).((eq T
457 t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr0 t1 t2))))) (\lambda (H4:
458 (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to
459 (pr0 t1 t2)))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0
460 t1 t) \to (pr0 t1 t2))) (\lambda (H6: (pr0 t1 t2)).H6) t3 (sym_eq T t3 t2
461 H5))) t0 (sym_eq T t0 t1 H4))) c (sym_eq C c (CSort n) H1) H2 H3 H0)))) |
462 (pr2_delta c d u i H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c (CSort
463 n))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CSort
464 n) (\lambda (c0: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c0 (CHead d
465 (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr0 t1 t2)))))))
466 (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to
467 ((getl i (CSort n) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u
468 t3 t) \to (pr0 t1 t2)))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda
469 (t4: T).((getl i (CSort n) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to
470 ((subst0 i u t3 t4) \to (pr0 t1 t2))))) (\lambda (H8: (getl i (CSort n)
471 (CHead d (Bind Abbr) u))).(\lambda (_: (pr0 t1 t3)).(\lambda (_: (subst0 i u
472 t3 t2)).(getl_gen_sort n i (CHead d (Bind Abbr) u) H8 (pr0 t1 t2))))) t
473 (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c (sym_eq C c (CSort n) H3) H4
474 H5 H0 H1 H2))))]) in (H0 (refl_equal C (CSort n)) (refl_equal T t1)
475 (refl_equal T t2)))))).
477 theorem pr2_gen_appl:
478 \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
479 (THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
480 (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
481 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T
482 T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
483 t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
484 (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_:
485 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
486 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
487 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T
488 T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
489 (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
490 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
491 T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
492 (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead
493 (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
494 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
495 (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
496 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
497 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
498 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))))
500 \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
501 (H: (pr2 c (THead (Flat Appl) u1 t1) x)).(let H0 \def (match H in pr2 return
502 (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t
503 t0)).((eq C c0 c) \to ((eq T t (THead (Flat Appl) u1 t1)) \to ((eq T t0 x)
504 \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat
505 Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
506 T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
507 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
508 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2:
509 T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
510 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
511 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
512 (u: T).(pr2 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda
513 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
514 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
515 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
516 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
517 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
518 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
519 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
520 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
521 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
522 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
523 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))))))) with [(pr2_free c0
524 t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0
525 (THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda
526 (_: C).((eq T t0 (THead (Flat Appl) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2)
527 \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
528 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
529 T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
530 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
531 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
532 T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
533 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
534 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
535 (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
536 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
537 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
538 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
539 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
540 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
541 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
542 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
543 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
544 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
545 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
546 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))))) (\lambda (H4: (eq T t0
547 (THead (Flat Appl) u1 t1))).(eq_ind T (THead (Flat Appl) u1 t1) (\lambda (t:
548 T).((eq T t2 x) \to ((pr0 t t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
549 (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
550 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T
551 T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
552 t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
553 (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
554 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
555 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
556 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T
557 T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
558 (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
559 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
560 T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
561 (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead
562 (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
563 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
564 (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
565 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
566 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
567 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))) (\lambda
568 (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead (Flat Appl) u1 t1)
569 t) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
570 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
571 T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
572 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
573 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
574 T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
575 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
576 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
577 (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
578 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
579 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
580 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
581 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
582 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
583 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
584 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
585 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
586 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
587 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
588 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))) (\lambda (H6: (pr0 (THead
589 (Flat Appl) u1 t1) x)).(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
590 T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0
591 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T
592 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
593 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
594 T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
595 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
596 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))
597 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
598 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
599 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
600 (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
601 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x
602 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda
603 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
604 T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
605 T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
606 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
607 T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T
608 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3))))
609 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
610 (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
611 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
612 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x
613 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
614 T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
615 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
616 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
617 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
618 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
619 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
620 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
621 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead
622 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
623 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
624 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
625 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
626 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
627 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H7: (ex3_2 T T
628 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3))))
629 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
630 T).(pr0 t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
631 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
632 (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2:
633 T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
634 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
635 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
636 T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
637 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
638 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
639 (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
640 T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
641 z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
642 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
643 (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
644 (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
645 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
646 (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
647 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
648 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
649 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
650 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
651 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
652 y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq T x
653 (THead (Flat Appl) x0 x1))).(\lambda (H9: (pr0 u1 x0)).(\lambda (H10: (pr0 t1
654 x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T
655 (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3))))
656 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
657 (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
658 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
659 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t
660 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
661 T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
662 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
663 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
664 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
665 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
666 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
667 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
668 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
669 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
670 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
671 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
672 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
673 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
674 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T
675 (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead
676 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
677 (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda
678 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
679 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
680 T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead (Bind Abbr) u2
681 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
682 T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
683 T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
684 z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
685 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
686 (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
687 (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
688 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
689 (y2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Bind b) y2 (THead (Flat Appl)
690 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
691 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
692 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
693 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
694 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
695 (CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2:
696 T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2
697 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
698 T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x1 (refl_equal T (THead (Flat Appl) x0
699 x1)) (pr2_free c u1 x0 H9) (pr2_free c t1 x1 H10))) x H8)))))) H7)) (\lambda
700 (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
701 T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
702 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
703 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
704 (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
705 T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T (\lambda (y1:
706 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
707 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
708 (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
709 T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
710 (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T
711 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3))))
712 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
713 (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
714 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
715 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x
716 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
717 T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
718 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
719 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
720 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
721 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
722 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
723 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
724 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead
725 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
726 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
727 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
728 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
729 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
730 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda
731 (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t1 (THead (Bind
732 Abst) x0 x1))).(\lambda (H9: (eq T x (THead (Bind Abbr) x2 x3))).(\lambda
733 (H10: (pr0 u1 x2)).(\lambda (H11: (pr0 x1 x3)).(eq_ind_r T (THead (Bind Abbr)
734 x2 x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
735 T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
736 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T
737 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
738 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
739 T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
740 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
741 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
742 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T
743 T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
744 (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
745 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
746 T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
747 (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead
748 (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
749 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
750 (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
751 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
752 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
753 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (eq_ind_r
754 T (THead (Bind Abst) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2:
755 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2
756 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
757 T).(\lambda (t3: T).(pr2 c t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
758 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1
759 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
760 T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda
761 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
762 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
763 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T
764 T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
765 (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
766 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
767 T t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
768 (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
769 Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
770 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
771 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
772 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
773 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
774 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
775 y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
776 T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
777 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
778 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
779 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
780 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
781 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2
782 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
783 T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
784 T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
785 z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
786 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
787 (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
788 (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1
789 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
790 T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind Abbr) x2 x3) (THead
791 (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
792 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
793 (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
794 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
795 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
796 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro
797 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
798 T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
799 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr)
800 x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
801 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
802 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
803 (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))) x0 x1 x2 x3 (refl_equal T
804 (THead (Bind Abst) x0 x1)) (refl_equal T (THead (Bind Abbr) x2 x3)) (pr2_free
805 c u1 x2 H10) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c (Bind b) u)
806 x1 x3 H11))))) t1 H8) x H9))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T T
807 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
808 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
809 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
810 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
811 T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b)
812 v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda
813 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0
814 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
815 T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
816 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
817 (t3: T).(pr0 z1 t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda
818 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not
819 (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1:
820 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b)
821 y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
822 T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead (Flat
823 Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
824 (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2)))))))
825 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
826 (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_:
827 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
828 t3))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead
829 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
830 (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda
831 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
832 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
833 T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
834 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
835 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
836 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T
837 T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
838 (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
839 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
840 T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
841 (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead
842 (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
843 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
844 (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
845 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
846 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
847 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
848 (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
849 T).(\lambda (x5: T).(\lambda (H8: (not (eq B x0 Abst))).(\lambda (H9: (eq T
850 t1 (THead (Bind x0) x1 x2))).(\lambda (H10: (eq T x (THead (Bind x0) x4
851 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda (H11: (pr0 u1
852 x3)).(\lambda (H12: (pr0 x1 x4)).(\lambda (H13: (pr0 x2 x5)).(eq_ind_r T
853 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t:
854 T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat
855 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
856 T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
857 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
858 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
859 T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
860 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
861 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
862 (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
863 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
864 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
865 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
866 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
867 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b)
868 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
869 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
870 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
871 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
872 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
873 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (eq_ind_r T (THead (Bind
874 x0) x1 x2) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
875 T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead
876 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
877 (\lambda (_: T).(\lambda (t3: T).(pr2 c t t3)))) (ex4_4 T T T T (\lambda (y1:
878 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind
879 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
880 (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))
881 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
882 T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
883 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
884 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
885 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
886 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
887 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind
888 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
889 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
890 (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl)
891 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
892 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
893 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
894 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
895 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
896 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2:
897 T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
898 O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
899 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2)
900 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
901 T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
902 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
903 T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead
904 (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
905 T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
906 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
907 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
908 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
909 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
910 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
911 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
912 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
913 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b)
914 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
915 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
916 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
917 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
918 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
919 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
920 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
921 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
922 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
923 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
924 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
925 (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))
926 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
927 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
928 T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
929 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
930 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
931 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
932 x0 x1 x2 x5 x3 x4 H8 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
933 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))) (pr2_free c u1
934 x3 H11) (pr2_free c x1 x4 H12) (pr2_free (CHead c (Bind x0) x4) x2 x5 H13)))
935 t1 H9) x H10))))))))))))) H7)) (pr0_gen_appl u1 t1 x H6))) t2 (sym_eq T t2 x
936 H5))) t0 (sym_eq T t0 (THead (Flat Appl) u1 t1) H4))) c0 (sym_eq C c0 c H1)
937 H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda
938 (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(\lambda
939 (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Flat Appl) u1
940 t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0
941 t2) \to ((subst0 i u t2 t) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
942 T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
943 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T
944 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
945 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
946 T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
947 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
948 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
949 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
950 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
951 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
952 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
953 (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
954 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x
955 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
956 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
957 T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
958 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
959 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
960 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
961 z2))))))))))))))) (\lambda (H6: (eq T t0 (THead (Flat Appl) u1 t1))).(eq_ind
962 T (THead (Flat Appl) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c
963 (CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or3
964 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2
965 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
966 T).(\lambda (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
967 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
968 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
969 T).(eq T x (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
970 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
971 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
972 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
973 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
974 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
975 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
976 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
977 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
978 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
979 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
980 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
981 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
982 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
983 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))))) (\lambda (H7: (eq T t
984 x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to
985 ((pr0 (THead (Flat Appl) u1 t1) t2) \to ((subst0 i u t2 t3) \to (or3 (ex3_2 T
986 T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2 t4))))
987 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
988 (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
989 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
990 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T x
991 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
992 T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
993 T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
994 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
995 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
996 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
997 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
998 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
999 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead
1000 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1001 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
1002 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1003 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
1004 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1005 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))))) (\lambda (H8: (getl i c
1006 (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (THead (Flat Appl) u1 t1)
1007 t2)).(\lambda (H10: (subst0 i u t2 x)).(or3_ind (ex3_2 T T (\lambda (u2:
1008 T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1009 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
1010 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1011 T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1012 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
1013 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1014 (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1015 T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b:
1016 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1017 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
1018 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
1019 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1020 (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead
1021 (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
1022 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
1023 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1024 T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
1025 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1026 (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
1027 T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
1028 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T
1029 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
1030 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1031 T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
1032 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
1033 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
1034 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
1035 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1036 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1037 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1038 (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
1039 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x
1040 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
1041 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1042 T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
1043 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
1044 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
1045 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))
1046 (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
1047 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
1048 (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind T T (\lambda (u2:
1049 T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1050 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
1051 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
1052 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
1053 T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1054 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
1055 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
1056 T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
1057 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
1058 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
1059 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
1060 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1061 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1062 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
1063 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1064 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
1065 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
1066 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
1067 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1068 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
1069 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1070 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda
1071 (x1: T).(\lambda (H12: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H13:
1072 (pr0 u1 x0)).(\lambda (H14: (pr0 t1 x1)).(let H15 \def (eq_ind T t2 (\lambda
1073 (t3: T).(subst0 i u t3 x)) H10 (THead (Flat Appl) x0 x1) H12) in (or3_ind
1074 (ex2 T (\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1))) (\lambda (u2:
1075 T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Flat Appl) x0
1076 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (ex3_2 T T
1077 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3))))
1078 (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
1079 T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3)))) (or3 (ex3_2 T T
1080 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3))))
1081 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
1082 (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
1083 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
1084 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x
1085 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1086 T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
1087 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
1088 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
1089 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1090 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
1091 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
1092 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1093 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead
1094 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1095 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
1096 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1097 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
1098 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1099 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H16: (ex2 T
1100 (\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0
1101 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1)))
1102 (\lambda (u2: T).(subst0 i u x0 u2)) (or3 (ex3_2 T T (\lambda (u2:
1103 T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1104 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
1105 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1106 T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1107 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
1108 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1109 (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1110 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1111 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1112 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1113 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1114 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
1115 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
1116 (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift
1117 (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
1118 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
1119 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1120 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
1121 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
1122 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (H17: (eq T x
1123 (THead (Flat Appl) x2 x1))).(\lambda (H18: (subst0 i u x0 x2)).(eq_ind_r T
1124 (THead (Flat Appl) x2 x1) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
1125 T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
1126 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1
1127 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1128 T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1129 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
1130 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1131 (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1132 T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1133 u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1134 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1135 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1136 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
1137 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
1138 (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift
1139 (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
1140 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
1141 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1142 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
1143 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
1144 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2:
1145 T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2
1146 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
1147 T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1148 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
1149 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
1150 T).(eq T (THead (Flat Appl) x2 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda
1151 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
1152 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
1153 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
1154 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1155 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1156 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1157 (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
1158 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
1159 (THead (Flat Appl) x2 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O
1160 u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1161 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
1162 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1163 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1164 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
1165 y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
1166 (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1167 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
1168 t3))) x2 x1 (refl_equal T (THead (Flat Appl) x2 x1)) (pr2_delta c d u i H8 u1
1169 x0 H13 x2 H18) (pr2_free c t1 x1 H14))) x H17)))) H16)) (\lambda (H16: (ex2 T
1170 (\lambda (t3: T).(eq T x (THead (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0
1171 (s (Flat Appl) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead
1172 (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))
1173 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl)
1174 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
1175 T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1176 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
1177 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
1178 T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
1179 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
1180 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
1181 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
1182 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1183 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1184 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
1185 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1186 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
1187 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
1188 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
1189 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1190 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
1191 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1192 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda
1193 (H17: (eq T x (THead (Flat Appl) x0 x2))).(\lambda (H18: (subst0 (s (Flat
1194 Appl) i) u x1 x2)).(eq_ind_r T (THead (Flat Appl) x0 x2) (\lambda (t3:
1195 T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat
1196 Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
1197 T).(\lambda (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1198 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
1199 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
1200 T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
1201 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
1202 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
1203 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
1204 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1205 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1206 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
1207 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1208 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
1209 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
1210 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1211 (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
1212 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
1213 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
1214 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))
1215 (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat
1216 Appl) x0 x2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
1217 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T
1218 T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
1219 t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
1220 (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind Abbr)
1221 u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
1222 T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1223 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1224 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1225 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1226 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1227 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
1228 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
1229 (u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind b) y2
1230 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1231 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
1232 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1233 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
1234 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1235 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda
1236 (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Flat Appl)
1237 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
1238 T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x2 (refl_equal T (THead (Flat Appl) x0
1239 x2)) (pr2_free c u1 x0 H13) (pr2_delta c d u i H8 t1 x1 H14 x2 H18))) x
1240 H17)))) H16)) (\lambda (H16: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
1241 T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u
1242 x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1
1243 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
1244 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2)))
1245 (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (or3
1246 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2
1247 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
1248 T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1249 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
1250 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
1251 T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
1252 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
1253 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
1254 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
1255 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1256 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1257 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
1258 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1259 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
1260 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
1261 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
1262 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1263 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
1264 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1265 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda
1266 (x3: T).(\lambda (H17: (eq T x (THead (Flat Appl) x2 x3))).(\lambda (H18:
1267 (subst0 i u x0 x2)).(\lambda (H19: (subst0 (s (Flat Appl) i) u x1
1268 x3)).(eq_ind_r T (THead (Flat Appl) x2 x3) (\lambda (t3: T).(or3 (ex3_2 T T
1269 (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
1270 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
1271 (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
1272 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
1273 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3
1274 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1275 T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
1276 T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
1277 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
1278 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1279 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
1280 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
1281 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1282 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead
1283 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1284 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
1285 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1286 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
1287 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1288 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T
1289 (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead
1290 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
1291 (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda
1292 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
1293 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1294 T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind Abbr) u2
1295 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
1296 T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1297 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1298 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1299 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1300 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1301 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
1302 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
1303 (u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind b) y2
1304 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1305 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
1306 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1307 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
1308 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1309 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda
1310 (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Flat Appl)
1311 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
1312 T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x3 (refl_equal T (THead (Flat Appl) x2
1313 x3)) (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (pr2_delta c d u i H8 t1 x1 H14
1314 x3 H19))) x H17)))))) H16)) (subst0_gen_head (Flat Appl) u x0 x1 x i
1315 H15)))))))) H11)) (\lambda (H11: (ex4_4 T T T T (\lambda (y1: T).(\lambda
1316 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
1317 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
1318 T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
1319 T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
1320 (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T
1321 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
1322 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1323 T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
1324 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
1325 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3
1326 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2
1327 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
1328 T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1329 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
1330 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
1331 T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
1332 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
1333 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
1334 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
1335 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1336 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1337 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
1338 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1339 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
1340 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
1341 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
1342 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1343 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
1344 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1345 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda
1346 (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq T t1 (THead
1347 (Bind Abst) x0 x1))).(\lambda (H13: (eq T t2 (THead (Bind Abbr) x2
1348 x3))).(\lambda (H14: (pr0 u1 x2)).(\lambda (H15: (pr0 x1 x3)).(let H16 \def
1349 (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 x)) H10 (THead (Bind Abbr) x2
1350 x3) H13) in (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(or3
1351 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2
1352 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
1353 T).(\lambda (t4: T).(pr2 c t3 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1354 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind Abst) y1
1355 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
1356 T).(eq T x (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
1357 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
1358 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
1359 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
1360 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1361 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1362 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3
1363 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1364 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
1365 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
1366 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
1367 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1368 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
1369 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1370 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda
1371 (u2: T).(eq T x (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2
1372 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Abbr) x2 t3))) (\lambda
1373 (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (ex3_2 T T (\lambda (u2:
1374 T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
1375 T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t3:
1376 T).(subst0 (s (Bind Abbr) i) u x3 t3)))) (or3 (ex3_2 T T (\lambda (u2:
1377 T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1378 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
1379 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1380 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
1381 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1382 T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
1383 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
1384 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
1385 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
1386 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1387 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1388 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1389 (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
1390 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
1391 T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
1392 O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1393 (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
1394 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1395 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1396 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
1397 y2) z1 z2))))))))) (\lambda (H17: (ex2 T (\lambda (u2: T).(eq T x (THead
1398 (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2)))).(ex2_ind T
1399 (\lambda (u2: T).(eq T x (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0
1400 i u x2 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead
1401 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
1402 (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3))))
1403 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
1404 T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda
1405 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
1406 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1407 (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1408 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1409 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1410 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1411 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1412 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
1413 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1414 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
1415 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
1416 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
1417 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1418 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
1419 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1420 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda
1421 (H18: (eq T x (THead (Bind Abbr) x4 x3))).(\lambda (H19: (subst0 i u x2
1422 x4)).(eq_ind_r T (THead (Bind Abbr) x4 x3) (\lambda (t3: T).(or3 (ex3_2 T T
1423 (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
1424 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
1425 (t4: T).(pr2 c (THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1:
1426 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst)
1427 x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
1428 T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4))))))
1429 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
1430 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4:
1431 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4))))))))
1432 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1433 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1434 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1435 (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
1436 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
1437 T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
1438 O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1439 (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
1440 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1441 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1442 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
1443 y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
1444 T).(eq T (THead (Bind Abbr) x4 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1445 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
1446 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1447 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
1448 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1449 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2
1450 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
1451 T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1452 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1453 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1454 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1455 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1456 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
1457 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1458 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
1459 Abbr) x4 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1460 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1461 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
1462 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1463 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1464 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
1465 y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
1466 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
1467 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1468 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2
1469 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
1470 T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1471 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1472 u0) z1 t3))))))) x0 x1 x4 x3 (refl_equal T (THead (Bind Abst) x0 x1))
1473 (refl_equal T (THead (Bind Abbr) x4 x3)) (pr2_delta c d u i H8 u1 x2 H14 x4
1474 H19) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) x1 x3
1475 H15))))) x H18)))) H17)) (\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x
1476 (THead (Bind Abbr) x2 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3
1477 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind Abbr) x2 t3)))
1478 (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3)) (or3 (ex3_2 T T (\lambda
1479 (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1480 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
1481 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1482 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
1483 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1484 T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
1485 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
1486 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
1487 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
1488 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1489 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1490 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1491 (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
1492 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
1493 T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
1494 O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1495 (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
1496 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1497 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1498 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
1499 y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (H18: (eq T x (THead (Bind Abbr)
1500 x2 x4))).(\lambda (H19: (subst0 (s (Bind Abbr) i) u x3 x4)).(eq_ind_r T
1501 (THead (Bind Abbr) x2 x4) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
1502 T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
1503 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
1504 (THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1505 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
1506 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1507 T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
1508 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
1509 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
1510 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
1511 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1512 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1513 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1514 (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
1515 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
1516 T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
1517 O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1518 (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
1519 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1520 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1521 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
1522 y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
1523 T).(eq T (THead (Bind Abbr) x2 x4) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1524 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
1525 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1526 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
1527 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1528 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2
1529 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
1530 T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1531 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1532 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1533 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1534 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1535 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
1536 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1537 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
1538 Abbr) x2 x4) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1539 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1540 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
1541 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1542 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1543 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
1544 y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
1545 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
1546 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1547 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2
1548 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
1549 T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1550 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1551 u0) z1 t3))))))) x0 x1 x2 x4 (refl_equal T (THead (Bind Abst) x0 x1))
1552 (refl_equal T (THead (Bind Abbr) x2 x4)) (pr2_free c u1 x2 H14) (\lambda (b:
1553 B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i)
1554 (getl_clear_bind b (CHead c (Bind b) u0) c u0 (clear_bind b c u0) (CHead d
1555 (Bind Abbr) u) i H8) x1 x3 H15 x4 H19))))) x H18)))) H17)) (\lambda (H17:
1556 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2
1557 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_:
1558 T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))))).(ex3_2_ind T T
1559 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
1560 (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_:
1561 T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (or3 (ex3_2 T T
1562 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3))))
1563 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
1564 (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1:
1565 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst)
1566 x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
1567 T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))))
1568 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
1569 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
1570 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
1571 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1572 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1573 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1574 (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
1575 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
1576 T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
1577 O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1578 (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
1579 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1580 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1581 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
1582 y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H18: (eq T x
1583 (THead (Bind Abbr) x4 x5))).(\lambda (H19: (subst0 i u x2 x4)).(\lambda (H20:
1584 (subst0 (s (Bind Abbr) i) u x3 x5)).(eq_ind_r T (THead (Bind Abbr) x4 x5)
1585 (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3
1586 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
1587 (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) x0 x1) t4))))
1588 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
1589 T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda
1590 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
1591 (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1592 T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
1593 T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
1594 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
1595 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1596 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
1597 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
1598 Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
1599 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
1600 t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
1601 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1602 T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
1603 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
1604 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
1605 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
1606 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
1607 (THead (Bind Abbr) x4 x5) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1608 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
1609 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1610 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
1611 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1612 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2
1613 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
1614 T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1615 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1616 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1617 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1618 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1619 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
1620 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1621 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
1622 Abbr) x4 x5) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1623 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1624 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
1625 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1626 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1627 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
1628 y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
1629 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
1630 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1631 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2
1632 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
1633 T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1634 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1635 u0) z1 t3))))))) x0 x1 x4 x5 (refl_equal T (THead (Bind Abst) x0 x1))
1636 (refl_equal T (THead (Bind Abbr) x4 x5)) (pr2_delta c d u i H8 u1 x2 H14 x4
1637 H19) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S
1638 i) (getl_clear_bind b (CHead c (Bind b) u0) c u0 (clear_bind b c u0) (CHead d
1639 (Bind Abbr) u) i H8) x1 x3 H15 x5 H20))))) x H18)))))) H17)) (subst0_gen_head
1640 (Bind Abbr) u x2 x3 x i H16)) t1 H12)))))))))) H11)) (\lambda (H11: (ex6_6 B
1641 T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1642 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1643 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1644 (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
1645 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T
1646 t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3)))))))))
1647 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
1648 T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
1649 T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
1650 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1651 T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))).(ex6_6_ind B T T T T
1652 T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1653 (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
1654 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
1655 T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
1656 (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead
1657 (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
1658 B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
1659 (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
1660 T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
1661 (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
1662 T).(\lambda (t3: T).(pr0 z1 t3))))))) (or3 (ex3_2 T T (\lambda (u2:
1663 T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1664 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
1665 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1666 T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1667 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
1668 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1669 (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1670 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1671 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1672 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1673 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1674 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
1675 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
1676 (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift
1677 (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
1678 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
1679 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1680 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
1681 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
1682 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: B).(\lambda (x1:
1683 T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
1684 T).(\lambda (H12: (not (eq B x0 Abst))).(\lambda (H13: (eq T t1 (THead (Bind
1685 x0) x1 x2))).(\lambda (H14: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl)
1686 (lift (S O) O x3) x5)))).(\lambda (H15: (pr0 u1 x3)).(\lambda (H16: (pr0 x1
1687 x4)).(\lambda (H17: (pr0 x2 x5)).(let H18 \def (eq_ind T t2 (\lambda (t3:
1688 T).(subst0 i u t3 x)) H10 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
1689 O x3) x5)) H14) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(or3
1690 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2
1691 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
1692 T).(\lambda (t4: T).(pr2 c t3 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1693 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind Abst) y1
1694 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
1695 T).(eq T x (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
1696 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
1697 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
1698 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
1699 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1700 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1701 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3
1702 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1703 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
1704 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
1705 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
1706 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1707 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
1708 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1709 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda
1710 (u2: T).(eq T x (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3)
1711 x5)))) (\lambda (u2: T).(subst0 i u x4 u2))) (ex2 T (\lambda (t3: T).(eq T x
1712 (THead (Bind x0) x4 t3))) (\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead
1713 (Flat Appl) (lift (S O) O x3) x5) t3))) (ex3_2 T T (\lambda (u2: T).(\lambda
1714 (t3: T).(eq T x (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_:
1715 T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0)
1716 i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3)))) (or3 (ex3_2 T T (\lambda
1717 (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1718 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
1719 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
1720 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1721 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1722 (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
1723 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
1724 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
1725 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
1726 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1727 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1728 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
1729 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
1730 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
1731 (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1732 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1733 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
1734 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1735 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1736 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
1737 y2) z1 z2))))))))) (\lambda (H19: (ex2 T (\lambda (u2: T).(eq T x (THead
1738 (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2:
1739 T).(subst0 i u x4 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Bind x0)
1740 u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: T).(subst0 i u
1741 x4 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead
1742 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
1743 (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4
1744 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
1745 T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1746 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
1747 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1748 (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1749 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1750 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1751 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1752 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1753 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1754 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1755 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead
1756 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1757 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
1758 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1759 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
1760 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1761 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda
1762 (H20: (eq T x (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3)
1763 x5)))).(\lambda (H21: (subst0 i u x4 x6)).(eq_ind_r T (THead (Bind x0) x6
1764 (THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(or3 (ex3_2 T T
1765 (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
1766 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
1767 (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1:
1768 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
1769 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
1770 (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
1771 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
1772 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
1773 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
1774 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1775 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1776 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1777 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
1778 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
1779 T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
1780 O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1781 (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
1782 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1783 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1784 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
1785 y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
1786 T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead
1787 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
1788 (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4
1789 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
1790 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 x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3))))))
1793 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
1794 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
1795 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (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) x6 (THead (Flat Appl) (lift (S O)
1802 O x3) 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 c u1 u2))))))) (\lambda (_:
1805 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1806 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1807 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (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) x6 (THead
1814 (Flat Appl) (lift (S O) O x3) 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 c u1 u2)))))))
1817 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1818 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
1819 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
1820 (CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x3 x6 H12 (refl_equal T (THead
1821 (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift
1822 (S O) O x3) x5))) (pr2_free c u1 x3 H15) (pr2_delta c d u i H8 x1 x4 H16 x6
1823 H21) (pr2_free (CHead c (Bind x0) x6) x2 x5 H17))) x H20)))) H19)) (\lambda
1824 (H19: (ex2 T (\lambda (t3: T).(eq T x (THead (Bind x0) x4 t3))) (\lambda (t3:
1825 T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
1826 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind x0) x4 t3))) (\lambda
1827 (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
1828 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
1829 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
1830 T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
1831 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
1832 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1833 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
1834 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1835 (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1836 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1837 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1838 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1839 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1840 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1841 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1842 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead
1843 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1844 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
1845 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1846 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
1847 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1848 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda
1849 (H20: (eq T x (THead (Bind x0) x4 x6))).(\lambda (H21: (subst0 (s (Bind x0)
1850 i) u (THead (Flat Appl) (lift (S O) O x3) x5) x6)).(eq_ind_r T (THead (Bind
1851 x0) x4 x6) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4:
1852 T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_:
1853 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0)
1854 x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1855 T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
1856 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
1857 T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
1858 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
1859 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
1860 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
1861 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1862 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1863 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
1864 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
1865 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
1866 (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1867 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1868 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
1869 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1870 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1871 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
1872 y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x6 (THead (Flat
1873 Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3)
1874 u2))) (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S O) O x3)
1875 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)))
1876 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat Appl) u2
1877 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O)
1878 O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind
1879 x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
1880 (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1881 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
1882 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
1883 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1884 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1885 (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda
1886 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
1887 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
1888 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
1889 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1890 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1891 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1892 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
1893 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
1894 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead
1895 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1896 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
1897 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1898 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
1899 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1900 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H22: (ex2 T
1901 (\lambda (u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2:
1902 T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda
1903 (u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s
1904 (Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2:
1905 T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2
1906 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
1907 T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
1908 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
1909 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1910 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
1911 x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
1912 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
1913 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
1914 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
1915 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1916 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1917 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
1918 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
1919 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
1920 (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl)
1921 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
1922 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
1923 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1924 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
1925 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
1926 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (H23: (eq T x6
1927 (THead (Flat Appl) x7 x5))).(\lambda (H24: (subst0 (s (Bind x0) i) u (lift (S
1928 O) O x3) x7)).(eq_ind_r T (THead (Flat Appl) x7 x5) (\lambda (t3: T).(or3
1929 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3)
1930 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
1931 (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4
1932 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
1933 T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1934 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
1935 x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
1936 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
1937 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
1938 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
1939 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1940 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1941 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
1942 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
1943 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
1944 (y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead (Flat Appl)
1945 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
1946 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
1947 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1948 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
1949 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
1950 (CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x7
1951 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u
1952 x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
1953 x0) x4 (THead (Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1954 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
1955 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
1956 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1957 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1958 (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind
1959 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1960 (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1961 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1962 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1963 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1964 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1965 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1966 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1967 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
1968 (Flat Appl) x7 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1969 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1970 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
1971 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1972 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1973 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
1974 y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H25: (eq T x7 (lift (S O) O
1975 x8))).(\lambda (H26: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x8)).(let H27
1976 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u
1977 x3 x8)) H26 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x8) (\lambda (t3:
1978 T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
1979 x4 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
1980 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
1981 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
1982 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1983 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1984 (t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Bind
1985 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1986 (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1987 T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1988 u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1989 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1990 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1991 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
1992 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1993 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
1994 (Flat Appl) t3 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1995 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1996 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
1997 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1998 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1999 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
2000 y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
2001 T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead
2002 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
2003 (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4
2004 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
2005 T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2006 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
2007 x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead (Bind Abbr) u2 t3))))))
2008 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2009 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
2010 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
2011 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2012 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2013 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2014 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2015 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2016 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
2017 O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2018 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2019 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
2020 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2021 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2022 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
2023 y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
2024 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
2025 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2026 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2027 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2028 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
2029 (Flat Appl) (lift (S O) O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl)
2030 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2031 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
2032 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2033 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
2034 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2035 (CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x8 x4 H12 (refl_equal T (THead
2036 (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift
2037 (S O) O x8) x5))) (pr2_delta c d u i H8 u1 x3 H15 x8 H27) (pr2_free c x1 x4
2038 H16) (pr2_free (CHead c (Bind x0) x4) x2 x5 H17))) x7 H25)))))
2039 (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O H24 (le_S_n (S O) (S i)
2040 (lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm O i (le_O_n i))))))) x6
2041 H23)))) H22)) (\lambda (H22: (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat
2042 Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s
2043 (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead (Flat
2044 Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s
2045 (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
2046 T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
2047 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
2048 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2049 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2050 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2051 (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda
2052 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
2053 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
2054 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
2055 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2056 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2057 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2058 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2059 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2060 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead
2061 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
2062 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2063 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2064 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
2065 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2066 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda
2067 (H23: (eq T x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H24:
2068 (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x7)).(eq_ind_r T (THead (Flat
2069 Appl) (lift (S O) O x3) x7) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
2070 T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Flat Appl) u2
2071 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
2072 T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T
2073 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
2074 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2075 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
2076 x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
2077 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
2078 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
2079 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
2080 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2081 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2082 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2083 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2084 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2085 (y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead (Flat Appl)
2086 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2087 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
2088 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2089 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
2090 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2091 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2:
2092 T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
2093 O x3) x7)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
2094 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2)
2095 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2096 T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
2097 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
2098 T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead
2099 (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2100 T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
2101 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
2102 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
2103 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2104 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
2105 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
2106 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
2107 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
2108 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead (Bind b)
2109 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
2110 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
2111 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2112 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
2113 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2114 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
2115 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2116 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2117 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2118 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2119 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2120 (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))
2121 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
2122 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2123 T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
2124 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
2125 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
2126 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
2127 x0 x1 x2 x7 x3 x4 H12 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
2128 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))) (pr2_free c u1
2129 x3 H15) (pr2_free c x1 x4 H16) (pr2_delta (CHead c (Bind x0) x4) d u (S i)
2130 (getl_clear_bind x0 (CHead c (Bind x0) x4) c x4 (clear_bind x0 c x4) (CHead d
2131 (Bind Abbr) u) i H8) x2 x5 H17 x7 H24))) x6 H23)))) H22)) (\lambda (H22:
2132 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat Appl) u2
2133 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O)
2134 O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind
2135 x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T
2136 x6 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s
2137 (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3:
2138 T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (or3 (ex3_2 T T
2139 (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat
2140 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
2141 T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
2142 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
2143 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2144 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
2145 x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
2146 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
2147 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
2148 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
2149 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2150 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2151 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2152 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2153 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2154 (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl)
2155 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2156 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
2157 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2158 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
2159 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2160 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (x8:
2161 T).(\lambda (H23: (eq T x6 (THead (Flat Appl) x7 x8))).(\lambda (H24: (subst0
2162 (s (Bind x0) i) u (lift (S O) O x3) x7)).(\lambda (H25: (subst0 (s (Flat
2163 Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat Appl) x7 x8)
2164 (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
2165 (THead (Bind x0) x4 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
2166 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
2167 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2168 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2169 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2170 (t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda
2171 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
2172 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
2173 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
2174 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2175 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2176 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2177 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2178 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2179 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead
2180 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
2181 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2182 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2183 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
2184 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2185 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3:
2186 T).(eq T x7 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0)
2187 i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
2188 (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3))))
2189 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
2190 (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
2191 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
2192 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
2193 (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8))
2194 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2195 T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
2196 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
2197 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
2198 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2199 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
2200 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
2201 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
2202 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
2203 (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind b) y2 (THead
2204 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
2205 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2206 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2207 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
2208 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2209 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x9: T).(\lambda
2210 (H26: (eq T x7 (lift (S O) O x9))).(\lambda (H27: (subst0 (minus (s (Bind x0)
2211 i) (S O)) u x3 x9)).(let H28 \def (eq_ind nat (minus (s (Bind x0) i) (S O))
2212 (\lambda (n: nat).(subst0 n u x3 x9)) H27 i (s_arith1 x0 i)) in (eq_ind_r T
2213 (lift (S O) O x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda
2214 (t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat
2215 Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
2216 T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T
2217 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
2218 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2219 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
2220 x4 (THead (Flat Appl) t3 x8)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
2221 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
2222 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
2223 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
2224 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2225 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2226 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2227 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2228 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2229 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8))
2230 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
2231 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2232 T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
2233 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
2234 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
2235 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
2236 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
2237 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Flat
2238 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
2239 T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
2240 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
2241 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2242 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
2243 x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Bind Abbr) u2 t3))))))
2244 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2245 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
2246 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
2247 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2248 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2249 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2250 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2251 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2252 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
2253 O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2254 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2255 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
2256 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2257 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2258 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
2259 y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
2260 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
2261 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2262 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2263 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2264 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
2265 (Flat Appl) (lift (S O) O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl)
2266 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2267 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
2268 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2269 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
2270 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2271 (CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x8 x9 x4 H12 (refl_equal T (THead
2272 (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift
2273 (S O) O x9) x8))) (pr2_delta c d u i H8 u1 x3 H15 x9 H28) (pr2_free c x1 x4
2274 H16) (pr2_delta (CHead c (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead c
2275 (Bind x0) x4) c x4 (clear_bind x0 c x4) (CHead d (Bind Abbr) u) i H8) x2 x5
2276 H17 x8 H25))) x7 H26))))) (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O
2277 H24 (le_S_n (S O) (S i) (lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm
2278 O i (le_O_n i))))))) x6 H23)))))) H22)) (subst0_gen_head (Flat Appl) u (lift
2279 (S O) O x3) x5 x6 (s (Bind x0) i) H21)) x H20)))) H19)) (\lambda (H19: (ex3_2
2280 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind x0) u2 t3))))
2281 (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda (_:
2282 T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O)
2283 O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
2284 (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4
2285 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat
2286 Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
2287 (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
2288 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0)
2289 x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2290 T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
2291 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
2292 T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
2293 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
2294 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
2295 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
2296 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2297 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2298 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2299 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2300 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2301 (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2302 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2303 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
2304 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2305 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2306 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
2307 y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (eq T x
2308 (THead (Bind x0) x6 x7))).(\lambda (H21: (subst0 i u x4 x6)).(\lambda (H22:
2309 (subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
2310 x7)).(eq_ind_r T (THead (Bind x0) x6 x7) (\lambda (t3: T).(or3 (ex3_2 T T
2311 (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
2312 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
2313 (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1:
2314 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
2315 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
2316 (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
2317 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
2318 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
2319 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
2320 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2321 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2322 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2323 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2324 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2325 T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
2326 O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2327 (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
2328 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2329 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2330 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
2331 y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x7 (THead (Flat
2332 Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3)
2333 u2))) (ex2 T (\lambda (t3: T).(eq T x7 (THead (Flat Appl) (lift (S O) O x3)
2334 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)))
2335 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2
2336 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O)
2337 O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind
2338 x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
2339 (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
2340 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
2341 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2342 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2343 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2344 (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda
2345 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
2346 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
2347 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
2348 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2349 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2350 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2351 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2352 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2353 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead
2354 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
2355 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2356 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2357 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
2358 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2359 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H23: (ex2 T
2360 (\lambda (u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2:
2361 T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda
2362 (u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s
2363 (Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2:
2364 T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2
2365 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
2366 T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
2367 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
2368 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2369 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
2370 x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
2371 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
2372 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
2373 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
2374 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2375 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2376 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2377 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2378 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2379 (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead (Flat Appl)
2380 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2381 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
2382 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2383 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
2384 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2385 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H24: (eq T x7
2386 (THead (Flat Appl) x8 x5))).(\lambda (H25: (subst0 (s (Bind x0) i) u (lift (S
2387 O) O x3) x8)).(eq_ind_r T (THead (Flat Appl) x8 x5) (\lambda (t3: T).(or3
2388 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 t3)
2389 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
2390 (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4
2391 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
2392 T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2393 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
2394 x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
2395 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
2396 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
2397 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
2398 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2399 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2400 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2401 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2402 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2403 (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead (Flat Appl)
2404 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2405 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
2406 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2407 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
2408 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2409 (CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x8
2410 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u
2411 x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
2412 x0) x6 (THead (Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
2413 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
2414 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2415 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2416 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2417 (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind
2418 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2419 (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
2420 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
2421 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
2422 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
2423 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2424 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2425 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2426 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
2427 (Flat Appl) x8 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2428 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2429 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
2430 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2431 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2432 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
2433 y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H26: (eq T x8 (lift (S O) O
2434 x9))).(\lambda (H27: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x9)).(let H28
2435 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u
2436 x3 x9)) H27 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x9) (\lambda (t3:
2437 T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
2438 x6 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
2439 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
2440 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2441 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2442 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2443 (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Bind
2444 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2445 (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
2446 T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
2447 u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
2448 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
2449 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2450 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2451 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2452 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
2453 (Flat Appl) t3 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2454 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2455 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
2456 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2457 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2458 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
2459 y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
2460 T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead
2461 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
2462 (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4
2463 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
2464 T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2465 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
2466 x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead (Bind Abbr) u2 t3))))))
2467 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2468 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
2469 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
2470 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2471 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2472 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2473 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2474 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2475 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
2476 O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2477 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2478 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
2479 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2480 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2481 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
2482 y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
2483 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
2484 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2485 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2486 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2487 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
2488 (Flat Appl) (lift (S O) O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl)
2489 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2490 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
2491 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2492 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
2493 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2494 (CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x9 x6 H12 (refl_equal T (THead
2495 (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift
2496 (S O) O x9) x5))) (pr2_delta c d u i H8 u1 x3 H15 x9 H28) (pr2_delta c d u i
2497 H8 x1 x4 H16 x6 H21) (pr2_free (CHead c (Bind x0) x6) x2 x5 H17))) x8
2498 H26))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H25 (le_S_n (S
2499 O) (S i) (lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm O i (le_O_n
2500 i))))))) x7 H24)))) H23)) (\lambda (H23: (ex2 T (\lambda (t3: T).(eq T x7
2501 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat
2502 Appl) (s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7
2503 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat
2504 Appl) (s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
2505 (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda
2506 (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2
2507 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
2508 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead
2509 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2510 T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2
2511 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
2512 T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
2513 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
2514 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
2515 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
2516 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2517 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2518 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2519 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7)
2520 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
2521 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2522 T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
2523 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
2524 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
2525 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))
2526 (\lambda (x8: T).(\lambda (H24: (eq T x7 (THead (Flat Appl) (lift (S O) O x3)
2527 x8))).(\lambda (H25: (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5
2528 x8)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O x3) x8) (\lambda (t3:
2529 T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
2530 x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c
2531 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2)
2532 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2533 T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
2534 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
2535 T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
2536 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
2537 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
2538 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
2539 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2540 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2541 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2542 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2543 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2544 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead
2545 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
2546 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2547 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2548 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
2549 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2550 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T
2551 (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat
2552 Appl) (lift (S O) O x3) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
2553 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
2554 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2555 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2556 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2557 (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))
2558 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2559 T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
2560 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
2561 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
2562 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2563 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
2564 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
2565 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
2566 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
2567 (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) (THead (Bind b)
2568 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
2569 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
2570 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2571 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
2572 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2573 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
2574 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2575 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2576 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2577 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2578 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2579 (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))
2580 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
2581 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2582 T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
2583 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
2584 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
2585 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
2586 x0 x1 x2 x8 x3 x6 H12 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
2587 (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))) (pr2_free c u1
2588 x3 H15) (pr2_delta c d u i H8 x1 x4 H16 x6 H21) (pr2_delta (CHead c (Bind x0)
2589 x6) d u (S i) (getl_clear_bind x0 (CHead c (Bind x0) x6) c x6 (clear_bind x0
2590 c x6) (CHead d (Bind Abbr) u) i H8) x2 x5 H17 x8 H25))) x7 H24)))) H23))
2591 (\lambda (H23: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead
2592 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0)
2593 i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s
2594 (Flat Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2:
2595 T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
2596 T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda
2597 (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)))
2598 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6
2599 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2600 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3))))
2601 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
2602 T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2603 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
2604 x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
2605 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
2606 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
2607 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
2608 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2609 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2610 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2611 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2612 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2613 (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead (Flat Appl)
2614 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2615 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
2616 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2617 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
2618 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2619 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (x9:
2620 T).(\lambda (H24: (eq T x7 (THead (Flat Appl) x8 x9))).(\lambda (H25: (subst0
2621 (s (Bind x0) i) u (lift (S O) O x3) x8)).(\lambda (H26: (subst0 (s (Flat
2622 Appl) (s (Bind x0) i)) u x5 x9)).(eq_ind_r T (THead (Flat Appl) x8 x9)
2623 (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
2624 (THead (Bind x0) x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
2625 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
2626 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2627 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2628 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2629 (t4: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda
2630 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
2631 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
2632 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
2633 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2634 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2635 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2636 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2637 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2638 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead
2639 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
2640 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2641 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2642 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
2643 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2644 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3:
2645 T).(eq T x8 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0)
2646 i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
2647 (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3))))
2648 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
2649 (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
2650 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
2651 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
2652 (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9))
2653 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2654 T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
2655 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
2656 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
2657 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2658 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
2659 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
2660 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
2661 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
2662 (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind b) y2 (THead
2663 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
2664 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2665 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2666 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
2667 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2668 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x10: T).(\lambda
2669 (H27: (eq T x8 (lift (S O) O x10))).(\lambda (H28: (subst0 (minus (s (Bind
2670 x0) i) (S O)) u x3 x10)).(let H29 \def (eq_ind nat (minus (s (Bind x0) i) (S
2671 O)) (\lambda (n: nat).(subst0 n u x3 x10)) H28 i (s_arith1 x0 i)) in
2672 (eq_ind_r T (lift (S O) O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
2673 T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9))
2674 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
2675 (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4
2676 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
2677 T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2678 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
2679 x6 (THead (Flat Appl) t3 x9)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
2680 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
2681 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
2682 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
2683 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2684 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2685 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2686 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2687 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2688 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9))
2689 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
2690 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2691 T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
2692 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
2693 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
2694 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
2695 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
2696 (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat
2697 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
2698 T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
2699 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
2700 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2701 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
2702 x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Bind Abbr) u2 t3))))))
2703 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2704 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
2705 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
2706 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2707 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2708 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2709 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2710 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2711 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
2712 O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2713 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2714 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
2715 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2716 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2717 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
2718 y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
2719 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
2720 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2721 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2722 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2723 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
2724 (Flat Appl) (lift (S O) O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl)
2725 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2726 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
2727 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2728 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
2729 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2730 (CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x9 x10 x6 H12 (refl_equal T
2731 (THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl)
2732 (lift (S O) O x10) x9))) (pr2_delta c d u i H8 u1 x3 H15 x10 H29) (pr2_delta
2733 c d u i H8 x1 x4 H16 x6 H21) (pr2_delta (CHead c (Bind x0) x6) d u (S i)
2734 (getl_clear_bind x0 (CHead c (Bind x0) x6) c x6 (clear_bind x0 c x6) (CHead d
2735 (Bind Abbr) u) i H8) x2 x5 H17 x9 H26))) x8 H27))))) (subst0_gen_lift_ge u x3
2736 x8 (s (Bind x0) i) (S O) O H25 (le_S_n (S O) (S i) (lt_le_S (S O) (S (S i))
2737 (lt_n_S O (S i) (le_lt_n_Sm O i (le_O_n i))))))) x7 H24)))))) H23))
2738 (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 x7 (s (Bind x0) i) H22))
2739 x H20)))))) H19)) (subst0_gen_head (Bind x0) u x4 (THead (Flat Appl) (lift (S
2740 O) O x3) x5) x i H18)) t1 H13)))))))))))))) H11)) (pr0_gen_appl u1 t1 t2
2741 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Flat Appl) u1 t1) H6)))
2742 c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c)
2743 (refl_equal T (THead (Flat Appl) u1 t1)) (refl_equal T x))))))).
2745 theorem pr2_gen_abbr:
2746 \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
2747 (THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
2748 (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_:
2749 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b:
2750 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u:
2751 T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T
2752 T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
2753 (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
2754 T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u:
2755 T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
2757 \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
2758 (H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(let H0 \def (match H in pr2 return
2759 (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t
2760 t0)).((eq C c0 c) \to ((eq T t (THead (Bind Abbr) u1 t1)) \to ((eq T t0 x)
2761 \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
2762 Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
2763 T).(\lambda (t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
2764 b) u) t1 t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead
2765 c (Bind Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2
2766 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z)))
2767 (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2))))))))
2768 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O
2769 x)))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0
2770 c)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T t2
2771 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Abbr) u1 t1)) \to ((eq
2772 T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
2773 T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
2774 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u:
2775 T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u))
2776 (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y:
2777 T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y:
2778 T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
2779 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c
2780 (Bind b) u) t1 (lift (S O) O x))))))))) (\lambda (H4: (eq T t0 (THead (Bind
2781 Abbr) u1 t1))).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (t: T).((eq T t2
2782 x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
2783 x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2784 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u:
2785 T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u))
2786 (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y:
2787 T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y:
2788 T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
2789 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c
2790 (Bind b) u) t1 (lift (S O) O x)))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x
2791 (\lambda (t: T).((pr0 (THead (Bind Abbr) u1 t1) t) \to (or (ex3_2 T T
2792 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
2793 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
2794 (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
2795 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind
2796 Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c
2797 (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda
2798 (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall
2799 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))
2800 (\lambda (H6: (pr0 (THead (Bind Abbr) u1 t1) x)).(or_ind (ex3_2 T T (\lambda
2801 (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
2802 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0
2803 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y
2804 t3))))))) (pr0 t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
2805 (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
2806 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
2807 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u:
2808 T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T
2809 T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
2810 (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
2811 T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u:
2812 T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda (H7: (ex3_2 T T
2813 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
2814 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3:
2815 T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0
2816 O u2 y t3)))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
2817 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
2818 (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0
2819 t1 y)) (\lambda (y: T).(subst0 O u2 y t3)))))) (or (ex3_2 T T (\lambda (u2:
2820 T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
2821 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
2822 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T
2823 (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1
2824 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
2825 u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
2826 T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b:
2827 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda
2828 (x0: T).(\lambda (x1: T).(\lambda (H8: (eq T x (THead (Bind Abbr) x0
2829 x1))).(\lambda (H9: (pr0 u1 x0)).(\lambda (H_x: (or (pr0 t1 x1) (ex2 T
2830 (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O x0 y x1))))).(or_ind
2831 (pr0 t1 x1) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O x0 y
2832 x1))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
2833 Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
2834 T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
2835 b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead
2836 c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2
2837 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z)))
2838 (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))))
2839 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O
2840 x))))) (\lambda (H10: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1)
2841 (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
2842 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
2843 (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2
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2846 (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z:
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2848 z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
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2854 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
2855 u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
2856 T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b:
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2859 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
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2862 (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1
2863 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
2864 u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
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2869 x1))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
2870 u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
2871 T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z x1)))) (\lambda (b:
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2880 (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z:
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2882 z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
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2889 c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2
2890 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z)))
2891 (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))))
2892 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O
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2894 x1)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
2895 Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
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2899 T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
2900 (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
2901 T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u:
2902 T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O (THead (Bind Abbr) x0 x1))))))
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2904 x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead
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2909 T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
2910 (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
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2913 (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr)
2914 x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c
2915 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u:
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2917 (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y:
2918 T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y:
2919 T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
2920 (Bind Abbr) u1) z t3))))))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 x1))
2921 (pr2_free c u1 x0 H9) (or3_intro1 (\forall (b: B).(\forall (u: T).(pr2 (CHead
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2923 T).(pr2 (CHead c (Bind Abbr) u) t1 x1))) (ex3_2 T T (\lambda (y: T).(\lambda
2924 (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z:
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2926 z x1)))) (ex_intro2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead
2927 c (Bind Abbr) u) t1 x1)) x0 H9 (pr2_delta (CHead c (Bind Abbr) x0) c x0 O
2928 (getl_refl Abbr c x0) t1 x2 H10 x1 H11)))))))) (pr0_subst0_back x0 x2 x1 O
2929 H11 u1 H9)) x H8)))) H_x0)) H_x)))))) H7)) (\lambda (H7: (pr0 t1 (lift (S O)
2930 O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead
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2932 (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2
2933 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u:
2934 T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda
2935 (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z:
2936 T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1)
2937 z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
2938 (lift (S O) O x)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c (Bind
2939 b) u) t1 (lift (S O) O x) H7))))) (pr0_gen_abbr u1 t1 x H6))) t2 (sym_eq T t2
2940 x H5))) t0 (sym_eq T t0 (THead (Bind Abbr) u1 t1) H4))) c0 (sym_eq C c0 c H1)
2941 H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda
2942 (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Bind Abbr) u1 t1))).(\lambda
2943 (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Bind Abbr) u1
2944 t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0
2945 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
2946 T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
2947 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall
2948 (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1
2949 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T
2950 (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
2951 (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
2952 T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0:
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2954 T t0 (THead (Bind Abbr) u1 t1))).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda
2955 (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t3 t2)
2956 \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4:
2957 T).(eq T x (THead (Bind Abbr) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2
2958 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(or3 (\forall (b: B).(\forall
2959 (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))) (ex2 T (\lambda (u0: T).(pr0 u1
2960 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t4))) (ex3_2 T T
2961 (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
2962 (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
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2965 T t x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to
2966 ((pr0 (THead (Bind Abbr) u1 t1) t2) \to ((subst0 i u t2 t3) \to (or (ex3_2 T
2967 T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Abbr) u2 t4))))
2968 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
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2972 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z)))
2973 (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t4))))))))
2974 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O
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2976 (pr0 (THead (Bind Abbr) u1 t1) t2)).(\lambda (H10: (subst0 i u t2 x)).(or_ind
2977 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
2978 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
2979 T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
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2981 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2
2982 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
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2985 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_:
2986 T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0
2987 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z
2988 t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
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2995 (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
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2997 T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
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2999 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T
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3018 (\lambda (t3: T).(subst0 i u t3 x)) H10 (THead (Bind Abbr) x0 x1) H12) in
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3022 T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
3023 (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
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3025 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
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3248 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_:
3249 T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0
3250 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z
3251 t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
3252 (lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
3253 x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
3254 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0:
3255 T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0))
3256 (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda
3257 (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y:
3258 T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
3259 (Bind Abbr) u1) z t3))))))) x0 x3 H18 (pr2_free c u1 x0 H13) (or3_intro2
3260 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 x3))) (ex2 T
3261 (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0)
3262 t1 x3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
3263 u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
3264 T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z x3)))) (ex3_2_intro T T
3265 (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
3266 (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
3267 T).(pr2 (CHead c (Bind Abbr) u1) z x3))) x4 x1 (pr2_delta (CHead c (Bind
3268 Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 x2 H14 x4 H20) H21 (pr2_delta
3269 (CHead c (Bind Abbr) u1) d u (S i) (getl_head (Bind Abbr) i c (CHead d (Bind
3270 Abbr) u) H8 u1) x1 x1 (pr0_refl x1) x3 H19)))))))) (pr0_subst0_back x0 x2 x1
3271 O H15 u1 H13))))) H17)) (\lambda (H17: (ex3_2 T T (\lambda (u2: T).(\lambda
3272 (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
3273 T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind
3274 Abbr) i) u x1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T
3275 x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u
3276 x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1
3277 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
3278 Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
3279 T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind
3280 b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2
3281 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_:
3282 T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0
3283 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z
3284 t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
3285 (lift (S O) O x))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H18: (eq T x
3286 (THead (Bind Abbr) x3 x4))).(\lambda (H19: (subst0 i u x0 x3)).(\lambda (H20:
3287 (subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t3: T).(subst0 O u1
3288 x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2:
3289 T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
3290 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
3291 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T
3292 (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0)
3293 t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
3294 u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
3295 T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b:
3296 B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))))
3297 (\lambda (x5: T).(\lambda (H21: (subst0 O u1 x2 x5)).(\lambda (H22: (pr0 x5
3298 x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead
3299 (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
3300 (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2
3301 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
3302 (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y:
3303 T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y:
3304 T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
3305 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c
3306 (Bind b) u0) t1 (lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2:
3307 T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
3308 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
3309 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T
3310 (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0)
3311 t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
3312 u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
3313 T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))) x3 x4 H18
3314 (pr2_delta c d u i H8 u1 x0 H13 x3 H19) (or3_intro2 (\forall (b: B).(\forall
3315 (u0: T).(pr2 (CHead c (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1
3316 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 x4))) (ex3_2 T T
3317 (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
3318 (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
3319 T).(pr2 (CHead c (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y:
3320 T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y:
3321 T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
3322 (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O
3323 (getl_refl Abbr c u1) t1 x2 H14 x5 H21) H22 (pr2_delta (CHead c (Bind Abbr)
3324 u1) d u (S i) (getl_head (Bind Abbr) i c (CHead d (Bind Abbr) u) H8 u1) x1 x1
3325 (pr0_refl x1) x4 H20)))))))) (pr0_subst0_back x0 x2 x1 O H15 u1 H13)))))))
3326 H17)) (subst0_gen_head (Bind Abbr) u x0 x1 x i H16)))))) H_x0)) H_x))))))
3327 H11)) (\lambda (H11: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T
3328 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
3329 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
3330 (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
3331 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c
3332 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2
3333 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z)))
3334 (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))))
3335 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O
3336 x)))) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u
3337 (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 (lift (S O) O
3338 t2) H11 (lift (S O) O x) (subst0_lift_ge_S t2 x u i H10 O (le_O_n i)))))))
3339 (pr0_gen_abbr u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead
3340 (Bind Abbr) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0
3341 (refl_equal C c) (refl_equal T (THead (Bind Abbr) u1 t1)) (refl_equal T
3344 theorem pr2_gen_void:
3345 \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
3346 (THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
3347 (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
3348 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
3349 (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u:
3350 T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
3352 \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
3353 (H: (pr2 c (THead (Bind Void) u1 t1) x)).(let H0 \def (match H in pr2 return
3354 (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t
3355 t0)).((eq C c0 c) \to ((eq T t (THead (Bind Void) u1 t1)) \to ((eq T t0 x)
3356 \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
3357 Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
3358 T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
3359 t1 t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
3360 (lift (S O) O x)))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda
3361 (H1: (eq C c0 c)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1 t1))).(\lambda
3362 (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Void) u1
3363 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2:
3364 T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
3365 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
3366 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b:
3367 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
3368 (\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(eq_ind T (THead (Bind
3369 Void) u1 t1) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T
3370 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
3371 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
3372 (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))))))
3373 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O
3374 x)))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead
3375 (Bind Void) u1 t1) t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
3376 T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
3377 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u:
3378 T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2
3379 (CHead c (Bind b) u) t1 (lift (S O) O x))))))) (\lambda (H6: (pr0 (THead
3380 (Bind Void) u1 t1) x)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
3381 T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0
3382 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O)
3383 O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
3384 Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
3385 T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
3386 t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
3387 (lift (S O) O x))))) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
3388 T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0
3389 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind T T
3390 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
3391 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
3392 T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
3393 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
3394 (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead
3395 c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
3396 b) u) t1 (lift (S O) O x))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8:
3397 (eq T x (THead (Bind Void) x0 x1))).(\lambda (H9: (pr0 u1 x0)).(\lambda (H10:
3398 (pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1) (\lambda (t: T).(or (ex3_2
3399 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
3400 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
3401 (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))))))
3402 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O
3403 t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead
3404 (Bind Void) x0 x1) (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
3405 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
3406 (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u:
3407 T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O (THead (Bind Void) x0 x1)))))
3408 (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Void)
3409 x0 x1) (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c
3410 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u:
3411 T).(pr2 (CHead c (Bind b) u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind
3412 Void) x0 x1)) (pr2_free c u1 x0 H9) (\lambda (b: B).(\lambda (u: T).(pr2_free
3413 (CHead c (Bind b) u) t1 x1 H10))))) x H8)))))) H7)) (\lambda (H7: (pr0 t1
3414 (lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
3415 T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
3416 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u:
3417 T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2
3418 (CHead c (Bind b) u) t1 (lift (S O) O x)))) (\lambda (b: B).(\lambda (u:
3419 T).(pr2_free (CHead c (Bind b) u) t1 (lift (S O) O x) H7))))) (pr0_gen_void
3420 u1 t1 x H6))) t2 (sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Bind Void) u1
3421 t1) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2
3422 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead
3423 (Bind Void) u1 t1))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1:
3424 C).((eq T t0 (THead (Bind Void) u1 t1)) \to ((eq T t x) \to ((getl i c1
3425 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (or
3426 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2
3427 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
3428 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
3429 u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0)
3430 t1 (lift (S O) O x))))))))))) (\lambda (H6: (eq T t0 (THead (Bind Void) u1
3431 t1))).(eq_ind T (THead (Bind Void) u1 t1) (\lambda (t3: T).((eq T t x) \to
3432 ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t)
3433 \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind
3434 Void) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
3435 T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
3436 u0) t1 t4)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0)
3437 t1 (lift (S O) O x)))))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda
3438 (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (THead (Bind Void) u1
3439 t1) t2) \to ((subst0 i u t2 t3) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
3440 (t4: T).(eq T x (THead (Bind Void) u2 t4)))) (\lambda (u2: T).(\lambda (_:
3441 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
3442 (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4)))))) (\forall (b: B).(\forall (u0:
3443 T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))))))) (\lambda (H8: (getl
3444 i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (THead (Bind Void) u1 t1)
3445 t2)).(\lambda (H10: (subst0 i u t2 x)).(or_ind (ex3_2 T T (\lambda (u2:
3446 T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2:
3447 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
3448 t3)))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
3449 (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
3450 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
3451 (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0:
3452 T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (H11: (ex3_2 T
3453 T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
3454 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
3455 T).(pr0 t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
3456 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
3457 (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2:
3458 T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
3459 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
3460 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b:
3461 B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))))
3462 (\lambda (x0: T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Bind Void)
3463 x0 x1))).(\lambda (H13: (pr0 u1 x0)).(\lambda (H14: (pr0 t1 x1)).(let H15
3464 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 x)) H10 (THead (Bind Void)
3465 x0 x1) H12) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Void) u2
3466 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x
3467 (THead (Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1
3468 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void)
3469 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
3470 T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T
3471 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
3472 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
3473 (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
3474 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift
3475 (S O) O x))))) (\lambda (H16: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind
3476 Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda
3477 (u2: T).(eq T x (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0
3478 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
3479 Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
3480 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
3481 u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0)
3482 t1 (lift (S O) O x))))) (\lambda (x2: T).(\lambda (H17: (eq T x (THead (Bind
3483 Void) x2 x1))).(\lambda (H18: (subst0 i u x0 x2)).(or_introl (ex3_2 T T
3484 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
3485 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
3486 (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
3487 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift
3488 (S O) O x)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
3489 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
3490 (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead
3491 c (Bind b) u0) t1 t3))))) x2 x1 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18)
3492 (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) t1 x1
3493 H14)))))))) H16)) (\lambda (H16: (ex2 T (\lambda (t3: T).(eq T x (THead (Bind
3494 Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1
3495 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind Void) x0 t3)))
3496 (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda
3497 (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
3498 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
3499 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b:
3500 B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))))
3501 (\lambda (x2: T).(\lambda (H17: (eq T x (THead (Bind Void) x0 x2))).(\lambda
3502 (H18: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2:
3503 T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
3504 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
3505 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b:
3506 B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))
3507 (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void)
3508 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
3509 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
3510 u0) t1 t3))))) x0 x2 H17 (pr2_free c u1 x0 H13) (\lambda (b: B).(\lambda (u0:
3511 T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead
3512 d (Bind Abbr) u) H8 u0) t1 x1 H14 x2 H18)))))))) H16)) (\lambda (H16: (ex3_2
3513 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
3514 (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
3515 T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))))).(ex3_2_ind T T
3516 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
3517 (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
3518 T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T
3519 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
3520 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
3521 (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
3522 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift
3523 (S O) O x))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H17: (eq T x
3524 (THead (Bind Void) x2 x3))).(\lambda (H18: (subst0 i u x0 x2)).(\lambda (H19:
3525 (subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2:
3526 T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
3527 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
3528 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b:
3529 B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))
3530 (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void)
3531 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
3532 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
3533 u0) t1 t3))))) x2 x3 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (\lambda (b:
3534 B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head
3535 (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H14 x3 H19)))))))))) H16))
3536 (subst0_gen_head (Bind Void) u x0 x1 x i H15)))))))) H11)) (\lambda (H11:
3537 (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda
3538 (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
3539 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
3540 (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0:
3541 T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (\lambda (b: B).(\lambda
3542 (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c
3543 (CHead d (Bind Abbr) u) H8 u0) t1 (lift (S O) O t2) H11 (lift (S O) O x)
3544 (subst0_lift_ge_S t2 x u i H10 O (le_O_n i))))))) (pr0_gen_void u1 t1 t2
3545 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Bind Void) u1 t1) H6)))
3546 c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c)
3547 (refl_equal T (THead (Bind Void) u1 t1)) (refl_equal T x))))))).
3549 theorem pr2_gen_lift:
3550 \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall
3551 (d: nat).((pr2 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to
3552 (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e t1
3555 \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
3556 (d: nat).(\lambda (H: (pr2 c (lift h d t1) x)).(\lambda (e: C).(\lambda (H0:
3557 (drop h d c e)).(let H1 \def (match H in pr2 return (\lambda (c0: C).(\lambda
3558 (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t
3559 (lift h d t1)) \to ((eq T t0 x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d
3560 t2))) (\lambda (t2: T).(pr2 e t1 t2)))))))))) with [(pr2_free c0 t0 t2 H1)
3561 \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq T t0 (lift h d
3562 t1))).(\lambda (H4: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (lift
3563 h d t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t3: T).(eq T
3564 x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))))) (\lambda (H5: (eq T t0
3565 (lift h d t1))).(eq_ind T (lift h d t1) (\lambda (t: T).((eq T t2 x) \to
3566 ((pr0 t t2) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3:
3567 T).(pr2 e t1 t3)))))) (\lambda (H6: (eq T t2 x)).(eq_ind T x (\lambda (t:
3568 T).((pr0 (lift h d t1) t) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3)))
3569 (\lambda (t3: T).(pr2 e t1 t3))))) (\lambda (H7: (pr0 (lift h d t1)
3570 x)).(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr0
3571 t1 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2
3572 e t1 t3))) (\lambda (x0: T).(\lambda (H8: (eq T x (lift h d x0))).(\lambda
3573 (H9: (pr0 t1 x0)).(eq_ind_r T (lift h d x0) (\lambda (t: T).(ex2 T (\lambda
3574 (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))) (ex_intro2
3575 T (\lambda (t3: T).(eq T (lift h d x0) (lift h d t3))) (\lambda (t3: T).(pr2
3576 e t1 t3)) x0 (refl_equal T (lift h d x0)) (pr2_free e t1 x0 H9)) x H8))))
3577 (pr0_gen_lift t1 x h d H7))) t2 (sym_eq T t2 x H6))) t0 (sym_eq T t0 (lift h
3578 d t1) H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0
3579 t2 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t0
3580 (lift h d t1))).(\lambda (H6: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T
3581 t0 (lift h d t1)) \to ((eq T t x) \to ((getl i c1 (CHead d0 (Bind Abbr) u))
3582 \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (ex2 T (\lambda (t3: T).(eq T x
3583 (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))))))) (\lambda (H7: (eq T t0
3584 (lift h d t1))).(eq_ind T (lift h d t1) (\lambda (t3: T).((eq T t x) \to
3585 ((getl i c (CHead d0 (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t)
3586 \to (ex2 T (\lambda (t4: T).(eq T x (lift h d t4))) (\lambda (t4: T).(pr2 e
3587 t1 t4)))))))) (\lambda (H8: (eq T t x)).(eq_ind T x (\lambda (t3: T).((getl i
3588 c (CHead d0 (Bind Abbr) u)) \to ((pr0 (lift h d t1) t2) \to ((subst0 i u t2
3589 t3) \to (ex2 T (\lambda (t4: T).(eq T x (lift h d t4))) (\lambda (t4: T).(pr2
3590 e t1 t4))))))) (\lambda (H9: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda
3591 (H10: (pr0 (lift h d t1) t2)).(\lambda (H11: (subst0 i u t2 x)).(ex2_ind T
3592 (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2
3593 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
3594 (\lambda (x0: T).(\lambda (H12: (eq T t2 (lift h d x0))).(\lambda (H13: (pr0
3595 t1 x0)).(let H14 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 x)) H11
3596 (lift h d x0) H12) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T x (lift h d
3597 t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H15: (lt i d)).(let H16 \def
3598 (eq_ind nat d (\lambda (n: nat).(drop h n c e)) H0 (S (plus i (minus d (S
3599 i)))) (lt_plus_minus i d H15)) in (let H17 \def (eq_ind nat d (\lambda (n:
3600 nat).(subst0 i u (lift h n x0) x)) H14 (S (plus i (minus d (S i))))
3601 (lt_plus_minus i d H15)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
3602 C).(eq T u (lift h (minus d (S i)) v)))) (\lambda (v: T).(\lambda (e0:
3603 C).(getl i e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
3604 C).(drop h (minus d (S i)) d0 e0))) (ex2 T (\lambda (t3: T).(eq T x (lift h d
3605 t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (x2:
3606 C).(\lambda (H18: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H19: (getl
3607 i e (CHead x2 (Bind Abbr) x1))).(\lambda (_: (drop h (minus d (S i)) d0
3608 x2)).(let H21 \def (eq_ind T u (\lambda (t3: T).(subst0 i t3 (lift h (S (plus
3609 i (minus d (S i)))) x0) x)) H17 (lift h (minus d (S i)) x1) H18) in (ex2_ind
3610 T (\lambda (t3: T).(eq T x (lift h (S (plus i (minus d (S i)))) t3)))
3611 (\lambda (t3: T).(subst0 i x1 x0 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h
3612 d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H22: (eq
3613 T x (lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H23: (subst0 i x1 x0
3614 x3)).(let H24 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n:
3615 nat).(eq T x (lift h n x3))) H22 d (lt_plus_minus i d H15)) in (ex_intro2 T
3616 (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3
3617 H24 (pr2_delta e x2 x1 i H19 t1 x0 H13 x3 H23)))))) (subst0_gen_lift_lt x1 x0
3618 x i h (minus d (S i)) H21)))))))) (getl_drop_conf_lt Abbr c d0 u i H9 e h
3619 (minus d (S i)) H16))))) (\lambda (H15: (le d i)).(lt_le_e i (plus d h) (ex2
3620 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
3621 (\lambda (H16: (lt i (plus d h))).(subst0_gen_lift_false x0 u x h d i H15 H16
3622 H14 (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e
3623 t1 t3))))) (\lambda (H16: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq
3624 T x (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T
3625 (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
3626 (\lambda (x1: T).(\lambda (H17: (eq T x (lift h d x1))).(\lambda (H18:
3627 (subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T x (lift h d
3628 t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H17 (pr2_delta e d0 u (minus i h)
3629 (getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H9 e h d H0 H16) t1 x0 H13 x1
3630 H18))))) (subst0_gen_lift_ge u x0 x i h d H14 H16)))))))))) (pr0_gen_lift t1
3631 t2 h d H10))))) t (sym_eq T t x H8))) t0 (sym_eq T t0 (lift h d t1) H7))) c0
3632 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c) (refl_equal T
3633 (lift h d t1)) (refl_equal T x)))))))))).