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 *********************)
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_n_S O i (le_O_n
2040 i)))) x6 H23)))) H22)) (\lambda (H22: (ex2 T (\lambda (t3: T).(eq T x6 (THead
2041 (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl)
2042 (s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead
2043 (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl)
2044 (s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
2045 T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
2046 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
2047 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2048 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2049 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2050 (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda
2051 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
2052 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
2053 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
2054 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2055 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2056 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2057 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2058 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2059 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead
2060 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
2061 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2062 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2063 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
2064 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2065 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda
2066 (H23: (eq T x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H24:
2067 (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x7)).(eq_ind_r T (THead (Flat
2068 Appl) (lift (S O) O x3) x7) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
2069 T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Flat Appl) u2
2070 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
2071 T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T
2072 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
2073 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2074 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
2075 x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
2076 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
2077 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
2078 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
2079 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2080 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2081 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2082 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2083 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2084 (y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead (Flat Appl)
2085 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2086 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
2087 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2088 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
2089 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2090 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2:
2091 T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
2092 O x3) x7)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
2093 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2)
2094 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2095 T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
2096 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
2097 T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead
2098 (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2099 T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
2100 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
2101 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
2102 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2103 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
2104 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
2105 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
2106 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
2107 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead (Bind b)
2108 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
2109 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
2110 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2111 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
2112 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2113 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
2114 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2115 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2116 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2117 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2118 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2119 (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))
2120 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
2121 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2122 T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
2123 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
2124 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
2125 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
2126 x0 x1 x2 x7 x3 x4 H12 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
2127 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))) (pr2_free c u1
2128 x3 H15) (pr2_free c x1 x4 H16) (pr2_delta (CHead c (Bind x0) x4) d u (S i)
2129 (getl_clear_bind x0 (CHead c (Bind x0) x4) c x4 (clear_bind x0 c x4) (CHead d
2130 (Bind Abbr) u) i H8) x2 x5 H17 x7 H24))) x6 H23)))) H22)) (\lambda (H22:
2131 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat Appl) u2
2132 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O)
2133 O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind
2134 x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T
2135 x6 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s
2136 (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3:
2137 T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (or3 (ex3_2 T T
2138 (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat
2139 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
2140 T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
2141 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
2142 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2143 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
2144 x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
2145 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
2146 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
2147 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
2148 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2149 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2150 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2151 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2152 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2153 (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl)
2154 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2155 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
2156 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2157 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
2158 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2159 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (x8:
2160 T).(\lambda (H23: (eq T x6 (THead (Flat Appl) x7 x8))).(\lambda (H24: (subst0
2161 (s (Bind x0) i) u (lift (S O) O x3) x7)).(\lambda (H25: (subst0 (s (Flat
2162 Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat Appl) x7 x8)
2163 (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
2164 (THead (Bind x0) x4 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
2165 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
2166 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2167 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2168 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2169 (t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda
2170 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
2171 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
2172 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
2173 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2174 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2175 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2176 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2177 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2178 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead
2179 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
2180 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2181 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2182 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
2183 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2184 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3:
2185 T).(eq T x7 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0)
2186 i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
2187 (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3))))
2188 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
2189 (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
2190 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
2191 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
2192 (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8))
2193 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2194 T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
2195 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
2196 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
2197 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2198 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
2199 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
2200 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
2201 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
2202 (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind b) y2 (THead
2203 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
2204 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2205 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2206 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
2207 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2208 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x9: T).(\lambda
2209 (H26: (eq T x7 (lift (S O) O x9))).(\lambda (H27: (subst0 (minus (s (Bind x0)
2210 i) (S O)) u x3 x9)).(let H28 \def (eq_ind nat (minus (s (Bind x0) i) (S O))
2211 (\lambda (n: nat).(subst0 n u x3 x9)) H27 i (s_arith1 x0 i)) in (eq_ind_r T
2212 (lift (S O) O x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda
2213 (t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat
2214 Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
2215 T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T
2216 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
2217 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2218 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
2219 x4 (THead (Flat Appl) t3 x8)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
2220 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
2221 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
2222 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
2223 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2224 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2225 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2226 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2227 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2228 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8))
2229 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
2230 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2231 T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
2232 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
2233 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
2234 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
2235 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
2236 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Flat
2237 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
2238 T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
2239 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
2240 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2241 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
2242 x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Bind Abbr) u2 t3))))))
2243 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2244 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
2245 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
2246 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2247 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2248 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2249 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2250 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2251 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
2252 O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2253 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2254 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
2255 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2256 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2257 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
2258 y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
2259 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
2260 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2261 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2262 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2263 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
2264 (Flat Appl) (lift (S O) O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl)
2265 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2266 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
2267 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2268 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
2269 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2270 (CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x8 x9 x4 H12 (refl_equal T (THead
2271 (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift
2272 (S O) O x9) x8))) (pr2_delta c d u i H8 u1 x3 H15 x9 H28) (pr2_free c x1 x4
2273 H16) (pr2_delta (CHead c (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead c
2274 (Bind x0) x4) c x4 (clear_bind x0 c x4) (CHead d (Bind Abbr) u) i H8) x2 x5
2275 H17 x8 H25))) x7 H26))))) (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O
2276 H24 (le_n_S O i (le_O_n i)))) x6 H23)))))) H22)) (subst0_gen_head (Flat Appl)
2277 u (lift (S O) O x3) x5 x6 (s (Bind x0) i) H21)) x H20)))) H19)) (\lambda
2278 (H19: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind x0)
2279 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda (_:
2280 T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O)
2281 O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
2282 (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4
2283 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat
2284 Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
2285 (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
2286 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0)
2287 x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2288 T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
2289 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
2290 T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
2291 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
2292 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
2293 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
2294 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2295 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2296 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2297 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2298 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2299 (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2300 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2301 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
2302 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2303 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2304 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
2305 y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (eq T x
2306 (THead (Bind x0) x6 x7))).(\lambda (H21: (subst0 i u x4 x6)).(\lambda (H22:
2307 (subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
2308 x7)).(eq_ind_r T (THead (Bind x0) x6 x7) (\lambda (t3: T).(or3 (ex3_2 T T
2309 (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
2310 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
2311 (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1:
2312 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
2313 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
2314 (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
2315 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
2316 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
2317 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
2318 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2319 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2320 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2321 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2322 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2323 T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
2324 O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2325 (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
2326 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2327 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2328 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
2329 y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x7 (THead (Flat
2330 Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3)
2331 u2))) (ex2 T (\lambda (t3: T).(eq T x7 (THead (Flat Appl) (lift (S O) O x3)
2332 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)))
2333 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2
2334 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O)
2335 O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind
2336 x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
2337 (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
2338 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
2339 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2340 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2341 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2342 (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda
2343 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
2344 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
2345 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
2346 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2347 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2348 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2349 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2350 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2351 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead
2352 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
2353 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2354 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2355 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
2356 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2357 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H23: (ex2 T
2358 (\lambda (u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2:
2359 T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda
2360 (u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s
2361 (Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2:
2362 T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2
2363 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
2364 T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
2365 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
2366 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2367 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
2368 x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
2369 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
2370 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
2371 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
2372 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2373 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2374 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2375 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2376 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2377 (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead (Flat Appl)
2378 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2379 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
2380 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2381 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
2382 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2383 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H24: (eq T x7
2384 (THead (Flat Appl) x8 x5))).(\lambda (H25: (subst0 (s (Bind x0) i) u (lift (S
2385 O) O x3) x8)).(eq_ind_r T (THead (Flat Appl) x8 x5) (\lambda (t3: T).(or3
2386 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 t3)
2387 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
2388 (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4
2389 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
2390 T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2391 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
2392 x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
2393 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
2394 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
2395 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
2396 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2397 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2398 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2399 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2400 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2401 (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead (Flat Appl)
2402 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2403 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
2404 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2405 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
2406 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2407 (CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x8
2408 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u
2409 x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
2410 x0) x6 (THead (Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
2411 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
2412 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2413 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2414 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2415 (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind
2416 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2417 (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
2418 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
2419 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
2420 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
2421 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2422 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2423 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2424 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
2425 (Flat Appl) x8 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2426 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2427 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
2428 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2429 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2430 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
2431 y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H26: (eq T x8 (lift (S O) O
2432 x9))).(\lambda (H27: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x9)).(let H28
2433 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u
2434 x3 x9)) H27 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x9) (\lambda (t3:
2435 T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
2436 x6 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
2437 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
2438 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2439 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2440 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2441 (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Bind
2442 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2443 (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
2444 T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
2445 u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
2446 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
2447 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2448 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2449 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2450 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
2451 (Flat Appl) t3 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2452 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2453 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
2454 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2455 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2456 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
2457 y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
2458 T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead
2459 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
2460 (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4
2461 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
2462 T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2463 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
2464 x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead (Bind Abbr) u2 t3))))))
2465 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2466 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
2467 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
2468 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2469 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2470 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2471 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2472 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2473 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
2474 O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2475 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2476 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
2477 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2478 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2479 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
2480 y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
2481 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
2482 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2483 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2484 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2485 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
2486 (Flat Appl) (lift (S O) O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl)
2487 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2488 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
2489 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2490 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
2491 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2492 (CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x9 x6 H12 (refl_equal T (THead
2493 (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift
2494 (S O) O x9) x5))) (pr2_delta c d u i H8 u1 x3 H15 x9 H28) (pr2_delta c d u i
2495 H8 x1 x4 H16 x6 H21) (pr2_free (CHead c (Bind x0) x6) x2 x5 H17))) x8
2496 H26))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H25 (le_n_S O i
2497 (le_O_n i)))) x7 H24)))) H23)) (\lambda (H23: (ex2 T (\lambda (t3: T).(eq T
2498 x7 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s
2499 (Flat Appl) (s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7
2500 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat
2501 Appl) (s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
2502 (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda
2503 (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2
2504 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
2505 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead
2506 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2507 T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2
2508 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
2509 T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
2510 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
2511 u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
2512 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
2513 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2514 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2515 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2516 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7)
2517 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
2518 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2519 T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
2520 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
2521 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
2522 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))
2523 (\lambda (x8: T).(\lambda (H24: (eq T x7 (THead (Flat Appl) (lift (S O) O x3)
2524 x8))).(\lambda (H25: (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5
2525 x8)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O x3) x8) (\lambda (t3:
2526 T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
2527 x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c
2528 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2)
2529 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2530 T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
2531 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
2532 T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
2533 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
2534 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
2535 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
2536 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2537 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2538 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2539 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2540 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2541 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead
2542 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
2543 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2544 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2545 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
2546 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2547 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T
2548 (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat
2549 Appl) (lift (S O) O x3) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
2550 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
2551 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2552 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2553 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2554 (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))
2555 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2556 T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
2557 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
2558 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
2559 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2560 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
2561 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
2562 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
2563 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
2564 (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) (THead (Bind b)
2565 y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
2566 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
2567 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2568 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
2569 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2570 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
2571 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2572 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2573 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2574 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2575 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2576 (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))
2577 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
2578 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2579 T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
2580 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
2581 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
2582 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
2583 x0 x1 x2 x8 x3 x6 H12 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
2584 (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))) (pr2_free c u1
2585 x3 H15) (pr2_delta c d u i H8 x1 x4 H16 x6 H21) (pr2_delta (CHead c (Bind x0)
2586 x6) d u (S i) (getl_clear_bind x0 (CHead c (Bind x0) x6) c x6 (clear_bind x0
2587 c x6) (CHead d (Bind Abbr) u) i H8) x2 x5 H17 x8 H25))) x7 H24)))) H23))
2588 (\lambda (H23: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead
2589 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0)
2590 i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s
2591 (Flat Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2:
2592 T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
2593 T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda
2594 (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)))
2595 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6
2596 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2597 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3))))
2598 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
2599 T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2600 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
2601 x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
2602 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
2603 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
2604 (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
2605 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2606 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2607 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2608 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2609 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2610 (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead (Flat Appl)
2611 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2612 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
2613 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2614 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
2615 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2616 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (x9:
2617 T).(\lambda (H24: (eq T x7 (THead (Flat Appl) x8 x9))).(\lambda (H25: (subst0
2618 (s (Bind x0) i) u (lift (S O) O x3) x8)).(\lambda (H26: (subst0 (s (Flat
2619 Appl) (s (Bind x0) i)) u x5 x9)).(eq_ind_r T (THead (Flat Appl) x8 x9)
2620 (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
2621 (THead (Bind x0) x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
2622 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
2623 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
2624 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2625 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2626 (t4: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda
2627 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
2628 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
2629 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
2630 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2631 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2632 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2633 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2634 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2635 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead
2636 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
2637 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2638 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2639 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
2640 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2641 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3:
2642 T).(eq T x8 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0)
2643 i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
2644 (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3))))
2645 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
2646 (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
2647 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
2648 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
2649 (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9))
2650 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2651 T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
2652 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
2653 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
2654 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2655 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
2656 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
2657 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
2658 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
2659 (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind b) y2 (THead
2660 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
2661 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2662 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
2663 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
2664 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
2665 (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x10: T).(\lambda
2666 (H27: (eq T x8 (lift (S O) O x10))).(\lambda (H28: (subst0 (minus (s (Bind
2667 x0) i) (S O)) u x3 x10)).(let H29 \def (eq_ind nat (minus (s (Bind x0) i) (S
2668 O)) (\lambda (n: nat).(subst0 n u x3 x10)) H28 i (s_arith1 x0 i)) in
2669 (eq_ind_r T (lift (S O) O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
2670 T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9))
2671 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
2672 (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4
2673 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
2674 T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2675 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
2676 x6 (THead (Flat Appl) t3 x9)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
2677 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
2678 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
2679 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
2680 T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2681 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2682 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2683 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2684 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2685 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9))
2686 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
2687 (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
2688 T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
2689 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
2690 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
2691 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
2692 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
2693 (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat
2694 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
2695 T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
2696 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
2697 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2698 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
2699 x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Bind Abbr) u2 t3))))))
2700 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2701 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
2702 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
2703 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2704 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
2705 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2706 (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
2707 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
2708 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
2709 O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2710 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2711 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
2712 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2713 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2714 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
2715 y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
2716 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
2717 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2718 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
2719 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
2720 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
2721 (Flat Appl) (lift (S O) O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl)
2722 (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
2723 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
2724 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2725 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
2726 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
2727 (CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x9 x10 x6 H12 (refl_equal T
2728 (THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl)
2729 (lift (S O) O x10) x9))) (pr2_delta c d u i H8 u1 x3 H15 x10 H29) (pr2_delta
2730 c d u i H8 x1 x4 H16 x6 H21) (pr2_delta (CHead c (Bind x0) x6) d u (S i)
2731 (getl_clear_bind x0 (CHead c (Bind x0) x6) c x6 (clear_bind x0 c x6) (CHead d
2732 (Bind Abbr) u) i H8) x2 x5 H17 x9 H26))) x8 H27))))) (subst0_gen_lift_ge u x3
2733 x8 (s (Bind x0) i) (S O) O H25 (le_n_S O i (le_O_n i)))) x7 H24)))))) H23))
2734 (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 x7 (s (Bind x0) i) H22))
2735 x H20)))))) H19)) (subst0_gen_head (Bind x0) u x4 (THead (Flat Appl) (lift (S
2736 O) O x3) x5) x i H18)) t1 H13)))))))))))))) H11)) (pr0_gen_appl u1 t1 t2
2737 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Flat Appl) u1 t1) H6)))
2738 c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c)
2739 (refl_equal T (THead (Flat Appl) u1 t1)) (refl_equal T x))))))).
2741 theorem pr2_gen_abbr:
2742 \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
2743 (THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
2744 (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_:
2745 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b:
2746 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u:
2747 T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T
2748 T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
2749 (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
2750 T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u:
2751 T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
2753 \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
2754 (H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(let H0 \def (match H in pr2 return
2755 (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t
2756 t0)).((eq C c0 c) \to ((eq T t (THead (Bind Abbr) u1 t1)) \to ((eq T t0 x)
2757 \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
2758 Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
2759 T).(\lambda (t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
2760 b) u) t1 t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead
2761 c (Bind Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2
2762 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z)))
2763 (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2))))))))
2764 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O
2765 x)))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0
2766 c)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T t2
2767 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Abbr) u1 t1)) \to ((eq
2768 T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
2769 T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
2770 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u:
2771 T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u))
2772 (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y:
2773 T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y:
2774 T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
2775 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c
2776 (Bind b) u) t1 (lift (S O) O x))))))))) (\lambda (H4: (eq T t0 (THead (Bind
2777 Abbr) u1 t1))).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (t: T).((eq T t2
2778 x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
2779 x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
2780 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u:
2781 T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u))
2782 (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y:
2783 T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y:
2784 T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
2785 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c
2786 (Bind b) u) t1 (lift (S O) O x)))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x
2787 (\lambda (t: T).((pr0 (THead (Bind Abbr) u1 t1) t) \to (or (ex3_2 T T
2788 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
2789 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
2790 (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
2791 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind
2792 Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c
2793 (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda
2794 (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall
2795 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))
2796 (\lambda (H6: (pr0 (THead (Bind Abbr) u1 t1) x)).(or_ind (ex3_2 T T (\lambda
2797 (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
2798 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0
2799 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y
2800 t3))))))) (pr0 t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
2801 (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
2802 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
2803 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u:
2804 T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T
2805 T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
2806 (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
2807 T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u:
2808 T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda (H7: (ex3_2 T T
2809 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
2810 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3:
2811 T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0
2812 O u2 y t3)))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
2813 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
2814 (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0
2815 t1 y)) (\lambda (y: T).(subst0 O u2 y t3)))))) (or (ex3_2 T T (\lambda (u2:
2816 T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
2817 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
2818 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T
2819 (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1
2820 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
2821 u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
2822 T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b:
2823 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda
2824 (x0: T).(\lambda (x1: T).(\lambda (H8: (eq T x (THead (Bind Abbr) x0
2825 x1))).(\lambda (H9: (pr0 u1 x0)).(\lambda (H_x: (or (pr0 t1 x1) (ex2 T
2826 (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O x0 y x1))))).(or_ind
2827 (pr0 t1 x1) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O x0 y
2828 x1))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
2829 Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
2830 T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
2831 b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead
2832 c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2
2833 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z)))
2834 (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))))
2835 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O
2836 x))))) (\lambda (H10: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1)
2837 (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
2838 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
2839 (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2
2840 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u:
2841 T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda
2842 (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z:
2843 T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1)
2844 z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
2845 (lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
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2851 u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
2852 T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b:
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2867 T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z x1)))) (\lambda (b:
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2878 z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
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2887 (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))))
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2891 Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
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2896 (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
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2900 x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead
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2905 T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
2906 (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
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2910 x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c
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2913 (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y:
2914 T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y:
2915 T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
2916 (Bind Abbr) u1) z t3))))))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 x1))
2917 (pr2_free c u1 x0 H9) (or3_intro1 (\forall (b: B).(\forall (u: T).(pr2 (CHead
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2919 T).(pr2 (CHead c (Bind Abbr) u) t1 x1))) (ex3_2 T T (\lambda (y: T).(\lambda
2920 (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z:
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2922 z x1)))) (ex_intro2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead
2923 c (Bind Abbr) u) t1 x1)) x0 H9 (pr2_delta (CHead c (Bind Abbr) x0) c x0 O
2924 (getl_refl Abbr c x0) t1 x2 H10 x1 H11)))))))) (pr0_subst0_back x0 x2 x1 O
2925 H11 u1 H9)) x H8)))) H_x0)) H_x)))))) H7)) (\lambda (H7: (pr0 t1 (lift (S O)
2926 O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead
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2928 (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2
2929 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u:
2930 T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda
2931 (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z:
2932 T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1)
2933 z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
2934 (lift (S O) O x)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c (Bind
2935 b) u) t1 (lift (S O) O x) H7))))) (pr0_gen_abbr u1 t1 x H6))) t2 (sym_eq T t2
2936 x H5))) t0 (sym_eq T t0 (THead (Bind Abbr) u1 t1) H4))) c0 (sym_eq C c0 c H1)
2937 H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda
2938 (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Bind Abbr) u1 t1))).(\lambda
2939 (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Bind Abbr) u1
2940 t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0
2941 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
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2943 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall
2944 (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1
2945 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T
2946 (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
2947 (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
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2950 T t0 (THead (Bind Abbr) u1 t1))).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda
2951 (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t3 t2)
2952 \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4:
2953 T).(eq T x (THead (Bind Abbr) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2
2954 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(or3 (\forall (b: B).(\forall
2955 (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))) (ex2 T (\lambda (u0: T).(pr0 u1
2956 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t4))) (ex3_2 T T
2957 (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
2958 (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
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2961 T t x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to
2962 ((pr0 (THead (Bind Abbr) u1 t1) t2) \to ((subst0 i u t2 t3) \to (or (ex3_2 T
2963 T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Abbr) u2 t4))))
2964 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
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2968 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z)))
2969 (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t4))))))))
2970 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O
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2973 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
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2977 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2
2978 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
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2983 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z
2984 t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
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2991 (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
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3018 T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
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3250 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0:
3251 T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0))
3252 (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda
3253 (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y:
3254 T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
3255 (Bind Abbr) u1) z t3))))))) x0 x3 H18 (pr2_free c u1 x0 H13) (or3_intro2
3256 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 x3))) (ex2 T
3257 (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0)
3258 t1 x3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
3259 u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
3260 T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z x3)))) (ex3_2_intro T T
3261 (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
3262 (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
3263 T).(pr2 (CHead c (Bind Abbr) u1) z x3))) x4 x1 (pr2_delta (CHead c (Bind
3264 Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 x2 H14 x4 H20) H21 (pr2_delta
3265 (CHead c (Bind Abbr) u1) d u (S i) (getl_head (Bind Abbr) i c (CHead d (Bind
3266 Abbr) u) H8 u1) x1 x1 (pr0_refl x1) x3 H19)))))))) (pr0_subst0_back x0 x2 x1
3267 O H15 u1 H13))))) H17)) (\lambda (H17: (ex3_2 T T (\lambda (u2: T).(\lambda
3268 (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
3269 T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind
3270 Abbr) i) u x1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T
3271 x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u
3272 x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1
3273 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
3274 Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
3275 T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind
3276 b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2
3277 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_:
3278 T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0
3279 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z
3280 t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
3281 (lift (S O) O x))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H18: (eq T x
3282 (THead (Bind Abbr) x3 x4))).(\lambda (H19: (subst0 i u x0 x3)).(\lambda (H20:
3283 (subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t3: T).(subst0 O u1
3284 x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2:
3285 T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
3286 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
3287 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T
3288 (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0)
3289 t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
3290 u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
3291 T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b:
3292 B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))))
3293 (\lambda (x5: T).(\lambda (H21: (subst0 O u1 x2 x5)).(\lambda (H22: (pr0 x5
3294 x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead
3295 (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
3296 (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2
3297 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
3298 (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y:
3299 T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y:
3300 T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
3301 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c
3302 (Bind b) u0) t1 (lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2:
3303 T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
3304 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
3305 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T
3306 (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0)
3307 t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
3308 u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
3309 T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))) x3 x4 H18
3310 (pr2_delta c d u i H8 u1 x0 H13 x3 H19) (or3_intro2 (\forall (b: B).(\forall
3311 (u0: T).(pr2 (CHead c (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1
3312 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 x4))) (ex3_2 T T
3313 (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
3314 (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
3315 T).(pr2 (CHead c (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y:
3316 T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y:
3317 T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
3318 (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O
3319 (getl_refl Abbr c u1) t1 x2 H14 x5 H21) H22 (pr2_delta (CHead c (Bind Abbr)
3320 u1) d u (S i) (getl_head (Bind Abbr) i c (CHead d (Bind Abbr) u) H8 u1) x1 x1
3321 (pr0_refl x1) x4 H20)))))))) (pr0_subst0_back x0 x2 x1 O H15 u1 H13)))))))
3322 H17)) (subst0_gen_head (Bind Abbr) u x0 x1 x i H16)))))) H_x0)) H_x))))))
3323 H11)) (\lambda (H11: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T
3324 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
3325 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
3326 (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
3327 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c
3328 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2
3329 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z)))
3330 (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))))
3331 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O
3332 x)))) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u
3333 (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 (lift (S O) O
3334 t2) H11 (lift (S O) O x) (subst0_lift_ge_S t2 x u i H10 O (le_O_n i)))))))
3335 (pr0_gen_abbr u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead
3336 (Bind Abbr) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0
3337 (refl_equal C c) (refl_equal T (THead (Bind Abbr) u1 t1)) (refl_equal T
3340 theorem pr2_gen_void:
3341 \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
3342 (THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
3343 (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
3344 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
3345 (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u:
3346 T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
3348 \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
3349 (H: (pr2 c (THead (Bind Void) u1 t1) x)).(let H0 \def (match H in pr2 return
3350 (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t
3351 t0)).((eq C c0 c) \to ((eq T t (THead (Bind Void) u1 t1)) \to ((eq T t0 x)
3352 \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
3353 Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
3354 T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
3355 t1 t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
3356 (lift (S O) O x)))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda
3357 (H1: (eq C c0 c)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1 t1))).(\lambda
3358 (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Void) u1
3359 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2:
3360 T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
3361 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
3362 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b:
3363 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
3364 (\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(eq_ind T (THead (Bind
3365 Void) u1 t1) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T
3366 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
3367 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
3368 (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))))))
3369 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O
3370 x)))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead
3371 (Bind Void) u1 t1) t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
3372 T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
3373 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u:
3374 T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2
3375 (CHead c (Bind b) u) t1 (lift (S O) O x))))))) (\lambda (H6: (pr0 (THead
3376 (Bind Void) u1 t1) x)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
3377 T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0
3378 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O)
3379 O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
3380 Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
3381 T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
3382 t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
3383 (lift (S O) O x))))) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
3384 T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0
3385 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind T T
3386 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
3387 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
3388 T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
3389 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
3390 (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead
3391 c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
3392 b) u) t1 (lift (S O) O x))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8:
3393 (eq T x (THead (Bind Void) x0 x1))).(\lambda (H9: (pr0 u1 x0)).(\lambda (H10:
3394 (pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1) (\lambda (t: T).(or (ex3_2
3395 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
3396 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
3397 (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))))))
3398 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O
3399 t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead
3400 (Bind Void) x0 x1) (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
3401 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
3402 (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u:
3403 T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O (THead (Bind Void) x0 x1)))))
3404 (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Void)
3405 x0 x1) (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c
3406 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u:
3407 T).(pr2 (CHead c (Bind b) u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind
3408 Void) x0 x1)) (pr2_free c u1 x0 H9) (\lambda (b: B).(\lambda (u: T).(pr2_free
3409 (CHead c (Bind b) u) t1 x1 H10))))) x H8)))))) H7)) (\lambda (H7: (pr0 t1
3410 (lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
3411 T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
3412 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u:
3413 T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2
3414 (CHead c (Bind b) u) t1 (lift (S O) O x)))) (\lambda (b: B).(\lambda (u:
3415 T).(pr2_free (CHead c (Bind b) u) t1 (lift (S O) O x) H7))))) (pr0_gen_void
3416 u1 t1 x H6))) t2 (sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Bind Void) u1
3417 t1) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2
3418 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead
3419 (Bind Void) u1 t1))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1:
3420 C).((eq T t0 (THead (Bind Void) u1 t1)) \to ((eq T t x) \to ((getl i c1
3421 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (or
3422 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2
3423 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
3424 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
3425 u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0)
3426 t1 (lift (S O) O x))))))))))) (\lambda (H6: (eq T t0 (THead (Bind Void) u1
3427 t1))).(eq_ind T (THead (Bind Void) u1 t1) (\lambda (t3: T).((eq T t x) \to
3428 ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t)
3429 \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind
3430 Void) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
3431 T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
3432 u0) t1 t4)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0)
3433 t1 (lift (S O) O x)))))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda
3434 (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (THead (Bind Void) u1
3435 t1) t2) \to ((subst0 i u t2 t3) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
3436 (t4: T).(eq T x (THead (Bind Void) u2 t4)))) (\lambda (u2: T).(\lambda (_:
3437 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
3438 (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4)))))) (\forall (b: B).(\forall (u0:
3439 T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))))))) (\lambda (H8: (getl
3440 i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (THead (Bind Void) u1 t1)
3441 t2)).(\lambda (H10: (subst0 i u t2 x)).(or_ind (ex3_2 T T (\lambda (u2:
3442 T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2:
3443 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
3444 t3)))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
3445 (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
3446 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
3447 (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0:
3448 T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (H11: (ex3_2 T
3449 T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
3450 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
3451 T).(pr0 t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
3452 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
3453 (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2:
3454 T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
3455 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
3456 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b:
3457 B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))))
3458 (\lambda (x0: T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Bind Void)
3459 x0 x1))).(\lambda (H13: (pr0 u1 x0)).(\lambda (H14: (pr0 t1 x1)).(let H15
3460 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 x)) H10 (THead (Bind Void)
3461 x0 x1) H12) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Void) u2
3462 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x
3463 (THead (Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1
3464 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void)
3465 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
3466 T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T
3467 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
3468 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
3469 (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
3470 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift
3471 (S O) O x))))) (\lambda (H16: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind
3472 Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda
3473 (u2: T).(eq T x (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0
3474 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
3475 Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
3476 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
3477 u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0)
3478 t1 (lift (S O) O x))))) (\lambda (x2: T).(\lambda (H17: (eq T x (THead (Bind
3479 Void) x2 x1))).(\lambda (H18: (subst0 i u x0 x2)).(or_introl (ex3_2 T T
3480 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
3481 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
3482 (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
3483 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift
3484 (S O) O x)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
3485 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
3486 (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead
3487 c (Bind b) u0) t1 t3))))) x2 x1 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18)
3488 (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) t1 x1
3489 H14)))))))) H16)) (\lambda (H16: (ex2 T (\lambda (t3: T).(eq T x (THead (Bind
3490 Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1
3491 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind Void) x0 t3)))
3492 (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda
3493 (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
3494 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
3495 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b:
3496 B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))))
3497 (\lambda (x2: T).(\lambda (H17: (eq T x (THead (Bind Void) x0 x2))).(\lambda
3498 (H18: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2:
3499 T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
3500 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
3501 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b:
3502 B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))
3503 (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void)
3504 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
3505 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
3506 u0) t1 t3))))) x0 x2 H17 (pr2_free c u1 x0 H13) (\lambda (b: B).(\lambda (u0:
3507 T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead
3508 d (Bind Abbr) u) H8 u0) t1 x1 H14 x2 H18)))))))) H16)) (\lambda (H16: (ex3_2
3509 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
3510 (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
3511 T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))))).(ex3_2_ind T T
3512 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
3513 (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
3514 T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T
3515 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
3516 (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
3517 (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
3518 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift
3519 (S O) O x))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H17: (eq T x
3520 (THead (Bind Void) x2 x3))).(\lambda (H18: (subst0 i u x0 x2)).(\lambda (H19:
3521 (subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2:
3522 T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
3523 T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
3524 (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b:
3525 B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))
3526 (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void)
3527 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
3528 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
3529 u0) t1 t3))))) x2 x3 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (\lambda (b:
3530 B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head
3531 (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H14 x3 H19)))))))))) H16))
3532 (subst0_gen_head (Bind Void) u x0 x1 x i H15)))))))) H11)) (\lambda (H11:
3533 (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda
3534 (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
3535 T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
3536 (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0:
3537 T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (\lambda (b: B).(\lambda
3538 (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c
3539 (CHead d (Bind Abbr) u) H8 u0) t1 (lift (S O) O t2) H11 (lift (S O) O x)
3540 (subst0_lift_ge_S t2 x u i H10 O (le_O_n i))))))) (pr0_gen_void u1 t1 t2
3541 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Bind Void) u1 t1) H6)))
3542 c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c)
3543 (refl_equal T (THead (Bind Void) u1 t1)) (refl_equal T x))))))).
3545 theorem pr2_gen_lift:
3546 \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall
3547 (d: nat).((pr2 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to
3548 (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e t1
3551 \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
3552 (d: nat).(\lambda (H: (pr2 c (lift h d t1) x)).(\lambda (e: C).(\lambda (H0:
3553 (drop h d c e)).(let H1 \def (match H in pr2 return (\lambda (c0: C).(\lambda
3554 (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t
3555 (lift h d t1)) \to ((eq T t0 x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d
3556 t2))) (\lambda (t2: T).(pr2 e t1 t2)))))))))) with [(pr2_free c0 t0 t2 H1)
3557 \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq T t0 (lift h d
3558 t1))).(\lambda (H4: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (lift
3559 h d t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t3: T).(eq T
3560 x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))))) (\lambda (H5: (eq T t0
3561 (lift h d t1))).(eq_ind T (lift h d t1) (\lambda (t: T).((eq T t2 x) \to
3562 ((pr0 t t2) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3:
3563 T).(pr2 e t1 t3)))))) (\lambda (H6: (eq T t2 x)).(eq_ind T x (\lambda (t:
3564 T).((pr0 (lift h d t1) t) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3)))
3565 (\lambda (t3: T).(pr2 e t1 t3))))) (\lambda (H7: (pr0 (lift h d t1)
3566 x)).(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr0
3567 t1 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2
3568 e t1 t3))) (\lambda (x0: T).(\lambda (H8: (eq T x (lift h d x0))).(\lambda
3569 (H9: (pr0 t1 x0)).(eq_ind_r T (lift h d x0) (\lambda (t: T).(ex2 T (\lambda
3570 (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))) (ex_intro2
3571 T (\lambda (t3: T).(eq T (lift h d x0) (lift h d t3))) (\lambda (t3: T).(pr2
3572 e t1 t3)) x0 (refl_equal T (lift h d x0)) (pr2_free e t1 x0 H9)) x H8))))
3573 (pr0_gen_lift t1 x h d H7))) t2 (sym_eq T t2 x H6))) t0 (sym_eq T t0 (lift h
3574 d t1) H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0
3575 t2 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t0
3576 (lift h d t1))).(\lambda (H6: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T
3577 t0 (lift h d t1)) \to ((eq T t x) \to ((getl i c1 (CHead d0 (Bind Abbr) u))
3578 \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (ex2 T (\lambda (t3: T).(eq T x
3579 (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))))))) (\lambda (H7: (eq T t0
3580 (lift h d t1))).(eq_ind T (lift h d t1) (\lambda (t3: T).((eq T t x) \to
3581 ((getl i c (CHead d0 (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t)
3582 \to (ex2 T (\lambda (t4: T).(eq T x (lift h d t4))) (\lambda (t4: T).(pr2 e
3583 t1 t4)))))))) (\lambda (H8: (eq T t x)).(eq_ind T x (\lambda (t3: T).((getl i
3584 c (CHead d0 (Bind Abbr) u)) \to ((pr0 (lift h d t1) t2) \to ((subst0 i u t2
3585 t3) \to (ex2 T (\lambda (t4: T).(eq T x (lift h d t4))) (\lambda (t4: T).(pr2
3586 e t1 t4))))))) (\lambda (H9: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda
3587 (H10: (pr0 (lift h d t1) t2)).(\lambda (H11: (subst0 i u t2 x)).(ex2_ind T
3588 (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2
3589 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
3590 (\lambda (x0: T).(\lambda (H12: (eq T t2 (lift h d x0))).(\lambda (H13: (pr0
3591 t1 x0)).(let H14 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 x)) H11
3592 (lift h d x0) H12) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T x (lift h d
3593 t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H15: (lt i d)).(let H16 \def
3594 (eq_ind nat d (\lambda (n: nat).(drop h n c e)) H0 (S (plus i (minus d (S
3595 i)))) (lt_plus_minus i d H15)) in (let H17 \def (eq_ind nat d (\lambda (n:
3596 nat).(subst0 i u (lift h n x0) x)) H14 (S (plus i (minus d (S i))))
3597 (lt_plus_minus i d H15)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
3598 C).(eq T u (lift h (minus d (S i)) v)))) (\lambda (v: T).(\lambda (e0:
3599 C).(getl i e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
3600 C).(drop h (minus d (S i)) d0 e0))) (ex2 T (\lambda (t3: T).(eq T x (lift h d
3601 t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (x2:
3602 C).(\lambda (H18: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H19: (getl
3603 i e (CHead x2 (Bind Abbr) x1))).(\lambda (_: (drop h (minus d (S i)) d0
3604 x2)).(let H21 \def (eq_ind T u (\lambda (t3: T).(subst0 i t3 (lift h (S (plus
3605 i (minus d (S i)))) x0) x)) H17 (lift h (minus d (S i)) x1) H18) in (ex2_ind
3606 T (\lambda (t3: T).(eq T x (lift h (S (plus i (minus d (S i)))) t3)))
3607 (\lambda (t3: T).(subst0 i x1 x0 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h
3608 d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H22: (eq
3609 T x (lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H23: (subst0 i x1 x0
3610 x3)).(let H24 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n:
3611 nat).(eq T x (lift h n x3))) H22 d (lt_plus_minus i d H15)) in (ex_intro2 T
3612 (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3
3613 H24 (pr2_delta e x2 x1 i H19 t1 x0 H13 x3 H23)))))) (subst0_gen_lift_lt x1 x0
3614 x i h (minus d (S i)) H21)))))))) (getl_drop_conf_lt Abbr c d0 u i H9 e h
3615 (minus d (S i)) H16))))) (\lambda (H15: (le d i)).(lt_le_e i (plus d h) (ex2
3616 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
3617 (\lambda (H16: (lt i (plus d h))).(subst0_gen_lift_false x0 u x h d i H15 H16
3618 H14 (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e
3619 t1 t3))))) (\lambda (H16: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq
3620 T x (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T
3621 (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
3622 (\lambda (x1: T).(\lambda (H17: (eq T x (lift h d x1))).(\lambda (H18:
3623 (subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T x (lift h d
3624 t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H17 (pr2_delta e d0 u (minus i h)
3625 (getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H9 e h d H0 H16) t1 x0 H13 x1
3626 H18))))) (subst0_gen_lift_ge u x0 x i h d H14 H16)))))))))) (pr0_gen_lift t1
3627 t2 h d H10))))) t (sym_eq T t x H8))) t0 (sym_eq T t0 (lift h d t1) H7))) c0
3628 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c) (refl_equal T
3629 (lift h d t1)) (refl_equal T x)))))))))).