1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "pr0/props.ma".
20 \forall (t1: T).(\forall (t2: T).(\forall (P: ((T \to (T \to
21 Prop)))).((((pr0 t1 t2) \to (\forall (t: T).((eq T t t1) \to ((eq T t t2) \to
22 (P t1 t2)))))) \to ((((pr0 t1 t2) \to (\forall (u1: T).(\forall (u2:
23 T).(\forall (t0: T).(\forall (t3: T).(\forall (k: K).((eq T (THead k u1 t0)
24 t1) \to ((eq T (THead k u2 t3) t2) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to (P
25 t1 t2)))))))))))) \to ((((pr0 t1 t2) \to (\forall (u: T).(\forall (v1:
26 T).(\forall (v2: T).(\forall (t0: T).(\forall (t3: T).((eq T (THead (Flat
27 Appl) v1 (THead (Bind Abst) u t0)) t1) \to ((eq T (THead (Bind Abbr) v2 t3)
28 t2) \to ((pr0 v1 v2) \to ((pr0 t0 t3) \to (P t1 t2)))))))))))) \to ((((pr0 t1
29 t2) \to (\forall (b: B).(\forall (v1: T).(\forall (v2: T).(\forall (u1:
30 T).(\forall (u2: T).(\forall (t0: T).(\forall (t3: T).((eq T (THead (Flat
31 Appl) v1 (THead (Bind b) u1 t0)) t1) \to ((eq T (THead (Bind b) u2 (THead
32 (Flat Appl) (lift (S O) O v2) t3)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1
33 v2) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to (P t1 t2)))))))))))))))) \to
34 ((((pr0 t1 t2) \to (\forall (u1: T).(\forall (u2: T).(\forall (t0:
35 T).(\forall (t3: T).(\forall (w: T).((eq T (THead (Bind Abbr) u1 t0) t1) \to
36 ((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to
37 ((subst0 O u2 t3 w) \to (P t1 t2))))))))))))) \to ((((pr0 t1 t2) \to (\forall
38 (b: B).(\forall (t0: T).(\forall (t3: T).(\forall (u: T).((eq T (THead (Bind
39 b) u (lift (S O) O t0)) t1) \to ((eq T t3 t2) \to ((not (eq B b Abst)) \to
40 ((pr0 t0 t3) \to (P t1 t2))))))))))) \to ((((pr0 t1 t2) \to (\forall (t0:
41 T).(\forall (t3: T).(\forall (u: T).((eq T (THead (Flat Cast) u t0) t1) \to
42 ((eq T t3 t2) \to ((pr0 t0 t3) \to (P t1 t2))))))))) \to ((pr0 t1 t2) \to (P
45 \lambda (t1: T).(\lambda (t2: T).(\lambda (P: ((T \to (T \to
46 Prop)))).(\lambda (H: (((pr0 t1 t2) \to (\forall (t: T).((eq T t t1) \to ((eq
47 T t t2) \to (P t1 t2))))))).(\lambda (H0: (((pr0 t1 t2) \to (\forall (u1:
48 T).(\forall (u2: T).(\forall (t0: T).(\forall (t3: T).(\forall (k: K).((eq T
49 (THead k u1 t0) t1) \to ((eq T (THead k u2 t3) t2) \to ((pr0 u1 u2) \to ((pr0
50 t0 t3) \to (P t1 t2))))))))))))).(\lambda (H1: (((pr0 t1 t2) \to (\forall (u:
51 T).(\forall (v1: T).(\forall (v2: T).(\forall (t0: T).(\forall (t3: T).((eq T
52 (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t1) \to ((eq T (THead (Bind
53 Abbr) v2 t3) t2) \to ((pr0 v1 v2) \to ((pr0 t0 t3) \to (P t1
54 t2))))))))))))).(\lambda (H2: (((pr0 t1 t2) \to (\forall (b: B).(\forall (v1:
55 T).(\forall (v2: T).(\forall (u1: T).(\forall (u2: T).(\forall (t0:
56 T).(\forall (t3: T).((eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t1)
57 \to ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t2)
58 \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t0 t3)
59 \to (P t1 t2))))))))))))))))).(\lambda (H3: (((pr0 t1 t2) \to (\forall (u1:
60 T).(\forall (u2: T).(\forall (t0: T).(\forall (t3: T).(\forall (w: T).((eq T
61 (THead (Bind Abbr) u1 t0) t1) \to ((eq T (THead (Bind Abbr) u2 w) t2) \to
62 ((pr0 u1 u2) \to ((pr0 t0 t3) \to ((subst0 O u2 t3 w) \to (P t1
63 t2)))))))))))))).(\lambda (H4: (((pr0 t1 t2) \to (\forall (b: B).(\forall
64 (t0: T).(\forall (t3: T).(\forall (u: T).((eq T (THead (Bind b) u (lift (S O)
65 O t0)) t1) \to ((eq T t3 t2) \to ((not (eq B b Abst)) \to ((pr0 t0 t3) \to (P
66 t1 t2)))))))))))).(\lambda (H5: (((pr0 t1 t2) \to (\forall (t0: T).(\forall
67 (t3: T).(\forall (u: T).((eq T (THead (Flat Cast) u t0) t1) \to ((eq T t3 t2)
68 \to ((pr0 t0 t3) \to (P t1 t2)))))))))).(\lambda (H6: (pr0 t1 t2)).(let H7
69 \def (match H6 in pr0 return (\lambda (t: T).(\lambda (t0: T).(\lambda (_:
70 (pr0 t t0)).((eq T t t1) \to ((eq T t0 t2) \to (P t1 t2)))))) with [(pr0_refl
71 t) \Rightarrow (\lambda (H7: (eq T t t1)).(\lambda (H8: (eq T t t2)).(H H6 t
72 H7 H8))) | (pr0_comp u1 u2 H7 t0 t3 H8 k) \Rightarrow (\lambda (H9: (eq T
73 (THead k u1 t0) t1)).(\lambda (H10: (eq T (THead k u2 t3) t2)).(H0 H6 u1 u2
74 t0 t3 k H9 H10 H7 H8))) | (pr0_beta u v1 v2 H7 t0 t3 H8) \Rightarrow (\lambda
75 (H9: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t1)).(\lambda
76 (H10: (eq T (THead (Bind Abbr) v2 t3) t2)).(H1 H6 u v1 v2 t0 t3 H9 H10 H7
77 H8))) | (pr0_upsilon b H7 v1 v2 H8 u1 u2 H9 t0 t3 H10) \Rightarrow (\lambda
78 (H11: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t1)).(\lambda (H12:
79 (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t2)).(H2
80 H6 b v1 v2 u1 u2 t0 t3 H11 H12 H7 H8 H9 H10))) | (pr0_delta u1 u2 H7 t0 t3 H8
81 w H9) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) u1 t0)
82 t1)).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(H3 H6 u1 u2 t0 t3 w
83 H10 H11 H7 H8 H9))) | (pr0_zeta b H7 t0 t3 H8 u) \Rightarrow (\lambda (H9:
84 (eq T (THead (Bind b) u (lift (S O) O t0)) t1)).(\lambda (H10: (eq T t3
85 t2)).(H4 H6 b t0 t3 u H9 H10 H7 H8))) | (pr0_epsilon t0 t3 H7 u) \Rightarrow
86 (\lambda (H8: (eq T (THead (Flat Cast) u t0) t1)).(\lambda (H9: (eq T t3
87 t2)).(H5 H6 t0 t3 u H8 H9 H7)))]) in (H7 (refl_equal T t1) (refl_equal T
91 \forall (x: T).(\forall (n: nat).((pr0 (TSort n) x) \to (eq T x (TSort n))))
93 \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TSort n)
94 x)).(pr0_inv_coq (TSort n) x (\lambda (t: T).(\lambda (t0: T).(eq T t0 t)))
95 (\lambda (H0: (pr0 (TSort n) x)).(\lambda (t: T).(\lambda (H1: (eq T t (TSort
96 n))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T t (\lambda (t0: T).(eq
97 T t0 (TSort n))) H1 x H2) in (let H4 \def (eq_ind T x (\lambda (t0: T).(pr0
98 (TSort n) t0)) H0 (TSort n) H3) in (let H5 \def (eq_ind T x (\lambda (t0:
99 T).(pr0 (TSort n) t0)) H (TSort n) H3) in (eq_ind_r T (TSort n) (\lambda (t0:
100 T).(eq T t0 (TSort n))) (refl_equal T (TSort n)) x H3)))))))) (\lambda (H0:
101 (pr0 (TSort n) x)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t0:
102 T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T (THead k u1 t0)
103 (TSort n))).(\lambda (H3: (eq T (THead k u2 t3) x)).(\lambda (_: (pr0 u1
104 u2)).(\lambda (_: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x (\lambda (t:
105 T).(pr0 (TSort n) t)) H0 (THead k u2 t3) H3) in (let H6 \def (eq_ind_r T x
106 (\lambda (t: T).(pr0 (TSort n) t)) H (THead k u2 t3) H3) in (eq_ind T (THead
107 k u2 t3) (\lambda (t: T).(eq T t (TSort n))) (let H7 \def (eq_ind T (THead k
108 u1 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
109 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
110 \Rightarrow True])) I (TSort n) H2) in (False_ind (eq T (THead k u2 t3)
111 (TSort n)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (TSort n) x)).(\lambda
112 (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t3:
113 T).(\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (TSort
114 n))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t3) x)).(\lambda (_: (pr0 v1
115 v2)).(\lambda (_: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x (\lambda (t:
116 T).(pr0 (TSort n) t)) H0 (THead (Bind Abbr) v2 t3) H3) in (let H6 \def
117 (eq_ind_r T x (\lambda (t: T).(pr0 (TSort n) t)) H (THead (Bind Abbr) v2 t3)
118 H3) in (eq_ind T (THead (Bind Abbr) v2 t3) (\lambda (t: T).(eq T t (TSort
119 n))) (let H7 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0))
120 (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
121 \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
122 True])) I (TSort n) H2) in (False_ind (eq T (THead (Bind Abbr) v2 t3) (TSort
123 n)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (TSort n) x)).(\lambda (b:
124 B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (u1: T).(\lambda (u2:
125 T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H4: (eq T (THead (Flat Appl)
126 v1 (THead (Bind b) u1 t0)) (TSort n))).(\lambda (H5: (eq T (THead (Bind b) u2
127 (THead (Flat Appl) (lift (S O) O v2) t3)) x)).(\lambda (_: (not (eq B b
128 Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0
129 t0 t3)).(let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TSort n) t)) H0
130 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (let H8
131 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TSort n) t)) H (THead (Bind b) u2
132 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (eq_ind T (THead (Bind b) u2
133 (THead (Flat Appl) (lift (S O) O v2) t3)) (\lambda (t: T).(eq T t (TSort n)))
134 (let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda
135 (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
136 \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
137 True])) I (TSort n) H4) in (False_ind (eq T (THead (Bind b) u2 (THead (Flat
138 Appl) (lift (S O) O v2) t3)) (TSort n)) H9)) x H5))))))))))))))))) (\lambda
139 (H0: (pr0 (TSort n) x)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t0:
140 T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind Abbr) u1
141 t0) (TSort n))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).(\lambda (_:
142 (pr0 u1 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda (_: (subst0 O u2 t3 w)).(let
143 H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TSort n) t)) H0 (THead (Bind
144 Abbr) u2 w) H4) in (let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TSort n)
145 t)) H (THead (Bind Abbr) u2 w) H4) in (eq_ind T (THead (Bind Abbr) u2 w)
146 (\lambda (t: T).(eq T t (TSort n))) (let H8 \def (eq_ind T (THead (Bind Abbr)
147 u1 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
148 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
149 \Rightarrow True])) I (TSort n) H3) in (False_ind (eq T (THead (Bind Abbr) u2
150 w) (TSort n)) H8)) x H4)))))))))))))) (\lambda (_: (pr0 (TSort n)
151 x)).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (u:
152 T).(\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (TSort
153 n))).(\lambda (H3: (eq T t3 x)).(\lambda (_: (not (eq B b Abst))).(\lambda
154 (H4: (pr0 t0 t3)).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H4 x
155 H3) in (let H6 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda
156 (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
157 \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
158 True])) I (TSort n) H2) in (False_ind (eq T x (TSort n)) H6))))))))))))
159 (\lambda (_: (pr0 (TSort n) x)).(\lambda (t0: T).(\lambda (t3: T).(\lambda
160 (u: T).(\lambda (H1: (eq T (THead (Flat Cast) u t0) (TSort n))).(\lambda (H2:
161 (eq T t3 x)).(\lambda (H3: (pr0 t0 t3)).(let H4 \def (eq_ind T t3 (\lambda
162 (t: T).(pr0 t0 t)) H3 x H2) in (let H5 \def (eq_ind T (THead (Flat Cast) u
163 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
164 _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
165 \Rightarrow True])) I (TSort n) H1) in (False_ind (eq T x (TSort n))
168 theorem pr0_gen_lref:
169 \forall (x: T).(\forall (n: nat).((pr0 (TLRef n) x) \to (eq T x (TLRef n))))
171 \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TLRef n)
172 x)).(pr0_inv_coq (TLRef n) x (\lambda (t: T).(\lambda (t0: T).(eq T t0 t)))
173 (\lambda (H0: (pr0 (TLRef n) x)).(\lambda (t: T).(\lambda (H1: (eq T t (TLRef
174 n))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T t (\lambda (t0: T).(eq
175 T t0 (TLRef n))) H1 x H2) in (let H4 \def (eq_ind T x (\lambda (t0: T).(pr0
176 (TLRef n) t0)) H0 (TLRef n) H3) in (let H5 \def (eq_ind T x (\lambda (t0:
177 T).(pr0 (TLRef n) t0)) H (TLRef n) H3) in (eq_ind_r T (TLRef n) (\lambda (t0:
178 T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) x H3)))))))) (\lambda (H0:
179 (pr0 (TLRef n) x)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t0:
180 T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T (THead k u1 t0)
181 (TLRef n))).(\lambda (H3: (eq T (THead k u2 t3) x)).(\lambda (_: (pr0 u1
182 u2)).(\lambda (_: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x (\lambda (t:
183 T).(pr0 (TLRef n) t)) H0 (THead k u2 t3) H3) in (let H6 \def (eq_ind_r T x
184 (\lambda (t: T).(pr0 (TLRef n) t)) H (THead k u2 t3) H3) in (eq_ind T (THead
185 k u2 t3) (\lambda (t: T).(eq T t (TLRef n))) (let H7 \def (eq_ind T (THead k
186 u1 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
187 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
188 \Rightarrow True])) I (TLRef n) H2) in (False_ind (eq T (THead k u2 t3)
189 (TLRef n)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (TLRef n) x)).(\lambda
190 (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t3:
191 T).(\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (TLRef
192 n))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t3) x)).(\lambda (_: (pr0 v1
193 v2)).(\lambda (_: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x (\lambda (t:
194 T).(pr0 (TLRef n) t)) H0 (THead (Bind Abbr) v2 t3) H3) in (let H6 \def
195 (eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n) t)) H (THead (Bind Abbr) v2 t3)
196 H3) in (eq_ind T (THead (Bind Abbr) v2 t3) (\lambda (t: T).(eq T t (TLRef
197 n))) (let H7 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0))
198 (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
199 \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
200 True])) I (TLRef n) H2) in (False_ind (eq T (THead (Bind Abbr) v2 t3) (TLRef
201 n)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (TLRef n) x)).(\lambda (b:
202 B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (u1: T).(\lambda (u2:
203 T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H4: (eq T (THead (Flat Appl)
204 v1 (THead (Bind b) u1 t0)) (TLRef n))).(\lambda (H5: (eq T (THead (Bind b) u2
205 (THead (Flat Appl) (lift (S O) O v2) t3)) x)).(\lambda (_: (not (eq B b
206 Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0
207 t0 t3)).(let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n) t)) H0
208 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (let H8
209 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n) t)) H (THead (Bind b) u2
210 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (eq_ind T (THead (Bind b) u2
211 (THead (Flat Appl) (lift (S O) O v2) t3)) (\lambda (t: T).(eq T t (TLRef n)))
212 (let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda
213 (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
214 \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
215 True])) I (TLRef n) H4) in (False_ind (eq T (THead (Bind b) u2 (THead (Flat
216 Appl) (lift (S O) O v2) t3)) (TLRef n)) H9)) x H5))))))))))))))))) (\lambda
217 (H0: (pr0 (TLRef n) x)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t0:
218 T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind Abbr) u1
219 t0) (TLRef n))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).(\lambda (_:
220 (pr0 u1 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda (_: (subst0 O u2 t3 w)).(let
221 H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n) t)) H0 (THead (Bind
222 Abbr) u2 w) H4) in (let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n)
223 t)) H (THead (Bind Abbr) u2 w) H4) in (eq_ind T (THead (Bind Abbr) u2 w)
224 (\lambda (t: T).(eq T t (TLRef n))) (let H8 \def (eq_ind T (THead (Bind Abbr)
225 u1 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
226 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
227 \Rightarrow True])) I (TLRef n) H3) in (False_ind (eq T (THead (Bind Abbr) u2
228 w) (TLRef n)) H8)) x H4)))))))))))))) (\lambda (_: (pr0 (TLRef n)
229 x)).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (u:
230 T).(\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (TLRef
231 n))).(\lambda (H3: (eq T t3 x)).(\lambda (_: (not (eq B b Abst))).(\lambda
232 (H4: (pr0 t0 t3)).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H4 x
233 H3) in (let H6 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda
234 (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
235 \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
236 True])) I (TLRef n) H2) in (False_ind (eq T x (TLRef n)) H6))))))))))))
237 (\lambda (_: (pr0 (TLRef n) x)).(\lambda (t0: T).(\lambda (t3: T).(\lambda
238 (u: T).(\lambda (H1: (eq T (THead (Flat Cast) u t0) (TLRef n))).(\lambda (H2:
239 (eq T t3 x)).(\lambda (H3: (pr0 t0 t3)).(let H4 \def (eq_ind T t3 (\lambda
240 (t: T).(pr0 t0 t)) H3 x H2) in (let H5 \def (eq_ind T (THead (Flat Cast) u
241 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
242 _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
243 \Rightarrow True])) I (TLRef n) H1) in (False_ind (eq T x (TLRef n))
246 theorem pr0_gen_abst:
247 \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abst) u1
248 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
249 Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
250 T).(\lambda (t2: T).(pr0 t1 t2)))))))
252 \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
253 (Bind Abst) u1 t1) x)).(pr0_inv_coq (THead (Bind Abst) u1 t1) x (\lambda (_:
254 T).(\lambda (t0: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0
255 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
256 (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))))) (\lambda (H0: (pr0 (THead
257 (Bind Abst) u1 t1) x)).(\lambda (t: T).(\lambda (H1: (eq T t (THead (Bind
258 Abst) u1 t1))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T t (\lambda
259 (t0: T).(eq T t0 (THead (Bind Abst) u1 t1))) H1 x H2) in (let H4 \def (eq_ind
260 T x (\lambda (t0: T).(pr0 (THead (Bind Abst) u1 t1) t0)) H0 (THead (Bind
261 Abst) u1 t1) H3) in (let H5 \def (eq_ind T x (\lambda (t0: T).(pr0 (THead
262 (Bind Abst) u1 t1) t0)) H (THead (Bind Abst) u1 t1) H3) in (eq_ind_r T (THead
263 (Bind Abst) u1 t1) (\lambda (t0: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t2:
264 T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_:
265 T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))))
266 (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abst)
267 u1 t1) (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1
268 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T
269 (THead (Bind Abst) u1 t1)) (pr0_refl u1) (pr0_refl t1)) x H3)))))))) (\lambda
270 (H0: (pr0 (THead (Bind Abst) u1 t1) x)).(\lambda (u0: T).(\lambda (u2:
271 T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T
272 (THead k u0 t0) (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T (THead k u2
273 t3) x)).(\lambda (H1: (pr0 u0 u2)).(\lambda (H4: (pr0 t0 t3)).(let H5 \def
274 (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H0 (THead k
275 u2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind
276 Abst) u1 t1) t)) H (THead k u2 t3) H3) in (eq_ind T (THead k u2 t3) (\lambda
277 (t: T).(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Bind
278 Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
279 T).(\lambda (t2: T).(pr0 t1 t2))))) (let H7 \def (f_equal T K (\lambda (e:
280 T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k |
281 (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0)
282 (THead (Bind Abst) u1 t1) H2) in ((let H8 \def (f_equal T T (\lambda (e:
283 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 |
284 (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0)
285 (THead (Bind Abst) u1 t1) H2) in ((let H9 \def (f_equal T T (\lambda (e:
286 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
287 (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0)
288 (THead (Bind Abst) u1 t1) H2) in (\lambda (H10: (eq T u0 u1)).(\lambda (H11:
289 (eq K k (Bind Abst))).(let H12 \def (eq_ind K k (\lambda (k0: K).(pr0 (THead
290 (Bind Abst) u1 t1) (THead k0 u2 t3))) H6 (Bind Abst) H11) in (let H13 \def
291 (eq_ind K k (\lambda (k0: K).(pr0 (THead (Bind Abst) u1 t1) (THead k0 u2
292 t3))) H5 (Bind Abst) H11) in (eq_ind_r K (Bind Abst) (\lambda (k0: K).(ex3_2
293 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k0 u2 t3) (THead (Bind
294 Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
295 T).(\lambda (t2: T).(pr0 t1 t2))))) (let H14 \def (eq_ind T t0 (\lambda (t:
296 T).(pr0 t t3)) H4 t1 H9) in (let H15 \def (eq_ind T u0 (\lambda (t: T).(pr0 t
297 u2)) H1 u1 H10) in (ex3_2_intro T T (\lambda (u3: T).(\lambda (t2: T).(eq T
298 (THead (Bind Abst) u2 t3) (THead (Bind Abst) u3 t2)))) (\lambda (u3:
299 T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
300 t2))) u2 t3 (refl_equal T (THead (Bind Abst) u2 t3)) H15 H14))) k H11))))))
301 H8)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (THead (Bind Abst) u1 t1)
302 x)).(\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0:
303 T).(\lambda (t3: T).(\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind
304 Abst) u t0)) (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T (THead (Bind
305 Abbr) v2 t3) x)).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t0 t3)).(let H5
306 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H0
307 (THead (Bind Abbr) v2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t:
308 T).(pr0 (THead (Bind Abst) u1 t1) t)) H (THead (Bind Abbr) v2 t3) H3) in
309 (eq_ind T (THead (Bind Abbr) v2 t3) (\lambda (t: T).(ex3_2 T T (\lambda (u2:
310 T).(\lambda (t2: T).(eq T t (THead (Bind Abst) u2 t2)))) (\lambda (u2:
311 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
312 t2))))) (let H7 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u
313 t0)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
314 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
315 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
316 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1
317 t1) H2) in (False_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T
318 (THead (Bind Abbr) v2 t3) (THead (Bind Abst) u2 t2)))) (\lambda (u2:
319 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
320 t2)))) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (THead (Bind Abst) u1 t1)
321 x)).(\lambda (b: B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (u0:
322 T).(\lambda (u2: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H4: (eq T
323 (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abst) u1
324 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
325 v2) t3)) x)).(\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1
326 v2)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0 t0 t3)).(let H7 \def
327 (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H0 (THead
328 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (let H8 \def
329 (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H (THead
330 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (eq_ind T (THead
331 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (\lambda (t: T).(ex3_2
332 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Bind Abst) u3 t2))))
333 (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
334 T).(pr0 t1 t2))))) (let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind
335 b) u0 t0)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
336 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
337 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
338 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1
339 t1) H4) in (False_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T
340 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Bind
341 Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
342 T).(\lambda (t2: T).(pr0 t1 t2)))) H9)) x H5))))))))))))))))) (\lambda (H0:
343 (pr0 (THead (Bind Abst) u1 t1) x)).(\lambda (u0: T).(\lambda (u2: T).(\lambda
344 (t0: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind
345 Abbr) u0 t0) (THead (Bind Abst) u1 t1))).(\lambda (H4: (eq T (THead (Bind
346 Abbr) u2 w) x)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda
347 (_: (subst0 O u2 t3 w)).(let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0
348 (THead (Bind Abst) u1 t1) t)) H0 (THead (Bind Abbr) u2 w) H4) in (let H7 \def
349 (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H (THead
350 (Bind Abbr) u2 w) H4) in (eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t:
351 T).(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Bind Abst) u3
352 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
353 T).(\lambda (t2: T).(pr0 t1 t2))))) (let H8 \def (eq_ind T (THead (Bind Abbr)
354 u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
355 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
356 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
357 \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
358 True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
359 \Rightarrow False])])) I (THead (Bind Abst) u1 t1) H3) in (False_ind (ex3_2 T
360 T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u2 w) (THead
361 (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
362 (_: T).(\lambda (t2: T).(pr0 t1 t2)))) H8)) x H4)))))))))))))) (\lambda (H0:
363 (pr0 (THead (Bind Abst) u1 t1) x)).(\lambda (b: B).(\lambda (t0: T).(\lambda
364 (t3: T).(\lambda (u: T).(\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O
365 t0)) (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T t3 x)).(\lambda (H1:
366 (not (eq B b Abst))).(\lambda (H4: (pr0 t0 t3)).(let H5 \def (eq_ind T t3
367 (\lambda (t: T).(pr0 t0 t)) H4 x H3) in (let H6 \def (f_equal T B (\lambda
368 (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b
369 | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
370 (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
371 b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H2) in
372 ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
373 T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _)
374 \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1
375 t1) H2) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T return
376 (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
377 \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow
378 (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true
379 \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t2) \Rightarrow
380 (THead k (lref_map f d u0) (lref_map f (s k d) t2))]) in lref_map) (\lambda
381 (x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map
382 (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n)
383 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
384 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t2)
385 \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t2))]) in
386 lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t)
387 \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1
388 t1) H2) in (\lambda (_: (eq T u u1)).(\lambda (H10: (eq B b Abst)).(let H11
389 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abst H10) in (let
390 H12 \def (eq_ind_r T t1 (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t) x)) H0
391 (lift (S O) O t0) H8) in (let H13 \def (eq_ind_r T t1 (\lambda (t: T).(pr0
392 (THead (Bind Abst) u1 t) x)) H (lift (S O) O t0) H8) in (eq_ind T (lift (S O)
393 O t0) (\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x
394 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
395 (\lambda (_: T).(\lambda (t2: T).(pr0 t t2))))) (let H14 \def (match (H11
396 (refl_equal B Abst)) in False return (\lambda (_: False).(ex3_2 T T (\lambda
397 (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2:
398 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 (lift
399 (S O) O t0) t2))))) with []) in H14) t1 H8))))))) H7)) H6))))))))))))
400 (\lambda (_: (pr0 (THead (Bind Abst) u1 t1) x)).(\lambda (t0: T).(\lambda
401 (t3: T).(\lambda (u: T).(\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead
402 (Bind Abst) u1 t1))).(\lambda (H2: (eq T t3 x)).(\lambda (H3: (pr0 t0
403 t3)).(let H4 \def (eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H3 x H2) in (let
404 H5 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (ee: T).(match ee in T
405 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
406 \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
407 (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
408 True])])) I (THead (Bind Abst) u1 t1) H1) in (False_ind (ex3_2 T T (\lambda
409 (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2:
410 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
411 t2)))) H5)))))))))) H)))).
413 theorem pr0_gen_appl:
414 \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Appl) u1
415 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
416 (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
417 (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1:
418 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
419 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
420 (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
421 T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
422 (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
423 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
424 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
425 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
426 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
427 T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b)
428 v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda
429 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0
430 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
431 T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
432 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
433 (t2: T).(pr0 z1 t2))))))))))))
435 \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
436 (Flat Appl) u1 t1) x)).(pr0_inv_coq (THead (Flat Appl) u1 t1) x (\lambda (_:
437 T).(\lambda (t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T
438 t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1
439 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda
440 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
441 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
442 T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_:
443 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
444 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))
445 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
446 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
447 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
448 (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
449 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T
450 t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2)))))))))
451 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
452 T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
453 T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
454 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
455 T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))))))) (\lambda (H0: (pr0
456 (THead (Flat Appl) u1 t1) x)).(\lambda (t: T).(\lambda (H1: (eq T t (THead
457 (Flat Appl) u1 t1))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T t
458 (\lambda (t0: T).(eq T t0 (THead (Flat Appl) u1 t1))) H1 x H2) in (let H4
459 \def (eq_ind T x (\lambda (t0: T).(pr0 (THead (Flat Appl) u1 t1) t0)) H0
460 (THead (Flat Appl) u1 t1) H3) in (let H5 \def (eq_ind T x (\lambda (t0:
461 T).(pr0 (THead (Flat Appl) u1 t1) t0)) H (THead (Flat Appl) u1 t1) H3) in
462 (eq_ind_r T (THead (Flat Appl) u1 t1) (\lambda (t0: T).(or3 (ex3_2 T T
463 (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Appl) u2 t2))))
464 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
465 T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
466 (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
467 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind
468 Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
469 (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
470 T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b:
471 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
472 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
473 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
474 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
475 (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead (Bind b) v2 (THead
476 (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_:
477 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
478 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
479 T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
480 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
481 (t2: T).(pr0 z1 t2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda
482 (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Flat Appl) u2 t2)))) (\lambda
483 (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0
484 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
485 T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
486 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl)
487 u1 t1) (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
488 T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
489 (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
490 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
491 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
492 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
493 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
494 T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T (THead (Flat
495 Appl) u1 t1) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2)
496 t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
497 T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda
498 (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0
499 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
500 T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))) (ex3_2_intro T T
501 (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead
502 (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
503 (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Flat Appl)
504 u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H3)))))))) (\lambda (H0: (pr0 (THead
505 (Flat Appl) u1 t1) x)).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0:
506 T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T (THead k u0 t0)
507 (THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T (THead k u2 t3) x)).(\lambda
508 (H1: (pr0 u0 u2)).(\lambda (H4: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x
509 (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t1) t)) H0 (THead k u2 t3) H3) in
510 (let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t1) t))
511 H (THead k u2 t3) H3) in (eq_ind T (THead k u2 t3) (\lambda (t: T).(or3
512 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Flat Appl) u3
513 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
514 T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
515 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
516 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2:
517 T).(eq T t (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_:
518 T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
519 (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
520 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
521 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
522 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
523 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
524 T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T t (THead (Bind b)
525 v2 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda
526 (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0
527 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
528 T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
529 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
530 (t2: T).(pr0 z1 t2)))))))))) (let H7 \def (f_equal T K (\lambda (e: T).(match
531 e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
532 \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Flat
533 Appl) u1 t1) H2) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T
534 return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
535 \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat
536 Appl) u1 t1) H2) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e in T
537 return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
538 \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat
539 Appl) u1 t1) H2) in (\lambda (H10: (eq T u0 u1)).(\lambda (H11: (eq K k (Flat
540 Appl))).(let H12 \def (eq_ind K k (\lambda (k0: K).(pr0 (THead (Flat Appl) u1
541 t1) (THead k0 u2 t3))) H6 (Flat Appl) H11) in (let H13 \def (eq_ind K k
542 (\lambda (k0: K).(pr0 (THead (Flat Appl) u1 t1) (THead k0 u2 t3))) H5 (Flat
543 Appl) H11) in (eq_ind_r K (Flat Appl) (\lambda (k0: K).(or3 (ex3_2 T T
544 (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k0 u2 t3) (THead (Flat Appl)
545 u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
546 T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
547 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
548 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2:
549 T).(eq T (THead k0 u2 t3) (THead (Bind Abbr) u3 t2)))))) (\lambda (_:
550 T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda
551 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))
552 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
553 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
554 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
555 (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
556 T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T
557 (THead k0 u2 t3) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3)
558 t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3:
559 T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda
560 (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0
561 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
562 T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (let H14 \def
563 (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H4 t1 H9) in (let H15 \def (eq_ind T
564 u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H10) in (or3_intro0 (ex3_2 T T (\lambda
565 (u3: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u2 t3) (THead (Flat Appl)
566 u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
567 T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
568 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
569 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2:
570 T).(eq T (THead (Flat Appl) u2 t3) (THead (Bind Abbr) u3 t2)))))) (\lambda
571 (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))))
572 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1
573 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
574 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
575 (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
576 (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
577 B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda
578 (t2: T).(eq T (THead (Flat Appl) u2 t3) (THead (Bind b) v2 (THead (Flat Appl)
579 (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
580 T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3)))))))
581 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
582 (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_:
583 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1
584 t2)))))))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead
585 (Flat Appl) u2 t3) (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_:
586 T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u2 t3
587 (refl_equal T (THead (Flat Appl) u2 t3)) H15 H14)))) k H11)))))) H8)) H7)) x
588 H3))))))))))))) (\lambda (H0: (pr0 (THead (Flat Appl) u1 t1) x)).(\lambda (u:
589 T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t3:
590 T).(\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead
591 (Flat Appl) u1 t1))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t3)
592 x)).(\lambda (H1: (pr0 v1 v2)).(\lambda (H4: (pr0 t0 t3)).(let H5 \def
593 (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t1) t)) H0 (THead
594 (Bind Abbr) v2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0
595 (THead (Flat Appl) u1 t1) t)) H (THead (Bind Abbr) v2 t3) H3) in (eq_ind T
596 (THead (Bind Abbr) v2 t3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2:
597 T).(\lambda (t2: T).(eq T t (THead (Flat Appl) u2 t2)))) (\lambda (u2:
598 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
599 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
600 T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
601 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T t (THead (Bind
602 Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
603 (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
604 T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b:
605 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
606 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
607 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
608 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
609 (u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T t (THead (Bind b) v3 (THead
610 (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_:
611 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
612 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
613 T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
614 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
615 (t2: T).(pr0 z1 t2)))))))))) (let H7 \def (f_equal T T (\lambda (e: T).(match
616 e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _)
617 \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead
618 (Bind Abst) u t0)) (THead (Flat Appl) u1 t1) H2) in ((let H8 \def (f_equal T
619 T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
620 \Rightarrow (THead (Bind Abst) u t0) | (TLRef _) \Rightarrow (THead (Bind
621 Abst) u t0) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead
622 (Bind Abst) u t0)) (THead (Flat Appl) u1 t1) H2) in (\lambda (H9: (eq T v1
623 u1)).(let H10 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H1 u1 H9) in (let
624 H11 \def (eq_ind_r T t1 (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t) (THead
625 (Bind Abbr) v2 t3))) H6 (THead (Bind Abst) u t0) H8) in (let H12 \def
626 (eq_ind_r T t1 (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t) (THead (Bind
627 Abbr) v2 t3))) H5 (THead (Bind Abst) u t0) H8) in (eq_ind T (THead (Bind
628 Abst) u t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
629 T).(eq T (THead (Bind Abbr) v2 t3) (THead (Flat Appl) u2 t2)))) (\lambda (u2:
630 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t
631 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
632 T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_:
633 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr)
634 v2 t3) (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
635 T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
636 (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
637 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
638 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
639 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t
640 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
641 T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T (THead (Bind
642 Abbr) v2 t3) (THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2)
643 t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
644 T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda
645 (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0
646 y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
647 T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (or3_intro1 (ex3_2 T
648 T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) v2 t3) (THead
649 (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
650 (_: T).(\lambda (t2: T).(pr0 (THead (Bind Abst) u t0) t2)))) (ex4_4 T T T T
651 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
652 (THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
653 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr)
654 v2 t3) (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
655 T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
656 (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
657 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
658 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
659 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
660 (THead (Bind Abst) u t0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
661 B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda
662 (t2: T).(eq T (THead (Bind Abbr) v2 t3) (THead (Bind b) v3 (THead (Flat Appl)
663 (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
664 T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2)))))))
665 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
666 (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_:
667 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1
668 t2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
669 (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) y1
670 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2:
671 T).(eq T (THead (Bind Abbr) v2 t3) (THead (Bind Abbr) u2 t2)))))) (\lambda
672 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))))
673 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1
674 t2))))) u t0 v2 t3 (refl_equal T (THead (Bind Abst) u t0)) (refl_equal T
675 (THead (Bind Abbr) v2 t3)) H10 H4)) t1 H8)))))) H7)) x H3)))))))))))))
676 (\lambda (H0: (pr0 (THead (Flat Appl) u1 t1) x)).(\lambda (b: B).(\lambda
677 (v1: T).(\lambda (v2: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0:
678 T).(\lambda (t3: T).(\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b)
679 u0 t0)) (THead (Flat Appl) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2
680 (THead (Flat Appl) (lift (S O) O v2) t3)) x)).(\lambda (H1: (not (eq B b
681 Abst))).(\lambda (H2: (pr0 v1 v2)).(\lambda (H3: (pr0 u0 u2)).(\lambda (H6:
682 (pr0 t0 t3)).(let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat
683 Appl) u1 t1) t)) H0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
684 t3)) H5) in (let H8 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat
685 Appl) u1 t1) t)) H (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
686 t3)) H5) in (eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
687 t3)) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T
688 t (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
689 (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1:
690 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
691 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
692 (t2: T).(eq T t (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_:
693 T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
694 (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
695 (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
696 T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
697 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
698 (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
699 T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t2: T).(eq T t (THead (Bind
700 b0) v3 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_:
701 B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
702 (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
703 T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
704 (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
705 T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (let H9 \def (f_equal T T (\lambda
706 (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1
707 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat
708 Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1) H4) in ((let H10
709 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
710 with [(TSort _) \Rightarrow (THead (Bind b) u0 t0) | (TLRef _) \Rightarrow
711 (THead (Bind b) u0 t0) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1
712 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1) H4) in (\lambda (H11: (eq T
713 v1 u1)).(let H12 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H2 u1 H11) in
714 (let H13 \def (eq_ind_r T t1 (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t)
715 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)))) H8 (THead
716 (Bind b) u0 t0) H10) in (let H14 \def (eq_ind_r T t1 (\lambda (t: T).(pr0
717 (THead (Flat Appl) u1 t) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
718 v2) t3)))) H7 (THead (Bind b) u0 t0) H10) in (eq_ind T (THead (Bind b) u0 t0)
719 (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T
720 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Flat
721 Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
722 T).(\lambda (t2: T).(pr0 t t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
723 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1
724 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2:
725 T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead
726 (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
727 T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
728 (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b0:
729 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
730 (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
731 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind
732 b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
733 (u3: T).(\lambda (v3: T).(\lambda (t2: T).(eq T (THead (Bind b) u2 (THead
734 (Flat Appl) (lift (S O) O v2) t3)) (THead (Bind b0) v3 (THead (Flat Appl)
735 (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
736 T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3)))))))
737 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
738 (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_:
739 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1
740 t2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T
741 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Flat
742 Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
743 T).(\lambda (t2: T).(pr0 (THead (Bind b) u0 t0) t2)))) (ex4_4 T T T T
744 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
745 (THead (Bind b) u0 t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
746 T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind b) u2
747 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Bind Abbr) u3 t2))))))
748 (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1
749 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2:
750 T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_:
751 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
752 b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda
753 (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead
754 (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
755 T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t2: T).(eq T (THead (Bind b)
756 u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Bind b0) v3 (THead (Flat
757 Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
758 (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3)))))))
759 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
760 (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_:
761 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1
762 t2)))))))) (ex6_6_intro B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda
763 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0
764 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
765 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind
766 b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
767 (u3: T).(\lambda (v3: T).(\lambda (t2: T).(eq T (THead (Bind b) u2 (THead
768 (Flat Appl) (lift (S O) O v2) t3)) (THead (Bind b0) v3 (THead (Flat Appl)
769 (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
770 T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3)))))))
771 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
772 (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_:
773 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1
774 t2))))))) b u0 t0 v2 u2 t3 H1 (refl_equal T (THead (Bind b) u0 t0))
775 (refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)))
776 H12 H3 H6)) t1 H10)))))) H9)) x H5))))))))))))))))) (\lambda (H0: (pr0 (THead
777 (Flat Appl) u1 t1) x)).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0:
778 T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind Abbr) u0
779 t0) (THead (Flat Appl) u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w)
780 x)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda (_: (subst0
781 O u2 t3 w)).(let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat
782 Appl) u1 t1) t)) H0 (THead (Bind Abbr) u2 w) H4) in (let H7 \def (eq_ind_r T
783 x (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t1) t)) H (THead (Bind Abbr) u2
784 w) H4) in (eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).(or3 (ex3_2 T T
785 (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Flat Appl) u3 t2))))
786 (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
787 T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
788 (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
789 T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Bind
790 Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
791 (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
792 T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b:
793 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
794 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
795 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
796 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
797 (u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T t (THead (Bind b) v2 (THead
798 (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_:
799 T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
800 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
801 T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
802 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
803 (t2: T).(pr0 z1 t2)))))))))) (let H8 \def (eq_ind T (THead (Bind Abbr) u0 t0)
804 (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
805 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
806 (match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True |
807 (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind
808 (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u2
809 w) (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
810 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda
811 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
812 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
813 T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3
814 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_:
815 T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
816 (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
817 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
818 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
819 T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
820 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
821 (v2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind b) v2
822 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_:
823 T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
824 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
825 T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
826 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
827 (t2: T).(pr0 z1 t2))))))))) H8)) x H4)))))))))))))) (\lambda (_: (pr0 (THead
828 (Flat Appl) u1 t1) x)).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3:
829 T).(\lambda (u: T).(\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0))
830 (THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T t3 x)).(\lambda (_: (not (eq
831 B b Abst))).(\lambda (H4: (pr0 t0 t3)).(let H5 \def (eq_ind T t3 (\lambda (t:
832 T).(pr0 t0 t)) H4 x H3) in (let H6 \def (eq_ind T (THead (Bind b) u (lift (S
833 O) O t0)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
834 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
835 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
836 \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1
837 t1) H2) in (False_ind (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T
838 x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
839 (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1:
840 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
841 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
842 (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
843 T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
844 (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
845 (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
846 T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
847 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
848 (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
849 T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind
850 b0) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_:
851 B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
852 (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
853 T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
854 (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
855 T).(\lambda (t2: T).(pr0 z1 t2))))))))) H6)))))))))))) (\lambda (_: (pr0
856 (THead (Flat Appl) u1 t1) x)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (u:
857 T).(\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Flat Appl) u1
858 t1))).(\lambda (H2: (eq T t3 x)).(\lambda (H3: (pr0 t0 t3)).(let H4 \def
859 (eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H3 x H2) in (let H5 \def (eq_ind T
860 (THead (Flat Cast) u t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
861 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
862 (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
863 [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return
864 (\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow
865 True])])])) I (THead (Flat Appl) u1 t1) H1) in (False_ind (or3 (ex3_2 T T
866 (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2))))
867 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
868 T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
869 (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
870 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
871 Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
872 (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
873 T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b:
874 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
875 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
876 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
877 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
878 (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) v2 (THead
879 (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_:
880 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
881 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
882 T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
883 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
884 (t2: T).(pr0 z1 t2))))))))) H5)))))))))) H)))).
886 theorem pr0_gen_cast:
887 \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Cast) u1
888 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
889 (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
890 (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x)))))
892 \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
893 (Flat Cast) u1 t1) x)).(pr0_inv_coq (THead (Flat Cast) u1 t1) x (\lambda (_:
894 T).(\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0
895 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
896 (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t0)))) (\lambda (H0:
897 (pr0 (THead (Flat Cast) u1 t1) x)).(\lambda (t: T).(\lambda (H1: (eq T t
898 (THead (Flat Cast) u1 t1))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T
899 t (\lambda (t0: T).(eq T t0 (THead (Flat Cast) u1 t1))) H1 x H2) in (let H4
900 \def (eq_ind T x (\lambda (t0: T).(pr0 (THead (Flat Cast) u1 t1) t0)) H0
901 (THead (Flat Cast) u1 t1) H3) in (let H5 \def (eq_ind T x (\lambda (t0:
902 T).(pr0 (THead (Flat Cast) u1 t1) t0)) H (THead (Flat Cast) u1 t1) H3) in
903 (eq_ind_r T (THead (Flat Cast) u1 t1) (\lambda (t0: T).(or (ex3_2 T T
904 (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2))))
905 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
906 T).(pr0 t1 t2)))) (pr0 t1 t0))) (or_introl (ex3_2 T T (\lambda (u2:
907 T).(\lambda (t2: T).(eq T (THead (Flat Cast) u1 t1) (THead (Flat Cast) u2
908 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
909 T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Flat Cast) u1 t1))
910 (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast)
911 u1 t1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1
912 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T
913 (THead (Flat Cast) u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H3))))))))
914 (\lambda (H0: (pr0 (THead (Flat Cast) u1 t1) x)).(\lambda (u0: T).(\lambda
915 (u2: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T
916 (THead k u0 t0) (THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T (THead k u2
917 t3) x)).(\lambda (H1: (pr0 u0 u2)).(\lambda (H4: (pr0 t0 t3)).(let H5 \def
918 (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Cast) u1 t1) t)) H0 (THead k
919 u2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat
920 Cast) u1 t1) t)) H (THead k u2 t3) H3) in (eq_ind T (THead k u2 t3) (\lambda
921 (t: T).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Flat
922 Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
923 T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t))) (let H7 \def (f_equal T K
924 (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
925 \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
926 (THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in ((let H8 \def (f_equal T T
927 (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
928 \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
929 (THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in ((let H9 \def (f_equal T T
930 (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
931 \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
932 (THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in (\lambda (H10: (eq T u0
933 u1)).(\lambda (H11: (eq K k (Flat Cast))).(let H12 \def (eq_ind K k (\lambda
934 (k0: K).(pr0 (THead (Flat Cast) u1 t1) (THead k0 u2 t3))) H6 (Flat Cast) H11)
935 in (let H13 \def (eq_ind K k (\lambda (k0: K).(pr0 (THead (Flat Cast) u1 t1)
936 (THead k0 u2 t3))) H5 (Flat Cast) H11) in (eq_ind_r K (Flat Cast) (\lambda
937 (k0: K).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k0 u2
938 t3) (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
939 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead k0 u2
940 t3)))) (let H14 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H4 t1 H9) in
941 (let H15 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H10) in
942 (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Flat
943 Cast) u2 t3) (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_:
944 T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1
945 (THead (Flat Cast) u2 t3)) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t2:
946 T).(eq T (THead (Flat Cast) u2 t3) (THead (Flat Cast) u3 t2)))) (\lambda (u3:
947 T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
948 t2))) u2 t3 (refl_equal T (THead (Flat Cast) u2 t3)) H15 H14)))) k H11))))))
949 H8)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (THead (Flat Cast) u1 t1)
950 x)).(\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0:
951 T).(\lambda (t3: T).(\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind
952 Abst) u t0)) (THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T (THead (Bind
953 Abbr) v2 t3) x)).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t0 t3)).(let H5
954 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Cast) u1 t1) t)) H0
955 (THead (Bind Abbr) v2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t:
956 T).(pr0 (THead (Flat Cast) u1 t1) t)) H (THead (Bind Abbr) v2 t3) H3) in
957 (eq_ind T (THead (Bind Abbr) v2 t3) (\lambda (t: T).(or (ex3_2 T T (\lambda
958 (u2: T).(\lambda (t2: T).(eq T t (THead (Flat Cast) u2 t2)))) (\lambda (u2:
959 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
960 t2)))) (pr0 t1 t))) (let H7 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind
961 Abst) u t0)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
962 with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
963 _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
964 \Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_:
965 F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead
966 (Flat Cast) u1 t1) H2) in (False_ind (or (ex3_2 T T (\lambda (u2: T).(\lambda
967 (t2: T).(eq T (THead (Bind Abbr) v2 t3) (THead (Flat Cast) u2 t2)))) (\lambda
968 (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0
969 t1 t2)))) (pr0 t1 (THead (Bind Abbr) v2 t3))) H7)) x H3))))))))))))) (\lambda
970 (H0: (pr0 (THead (Flat Cast) u1 t1) x)).(\lambda (b: B).(\lambda (v1:
971 T).(\lambda (v2: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0:
972 T).(\lambda (t3: T).(\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b)
973 u0 t0)) (THead (Flat Cast) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2
974 (THead (Flat Appl) (lift (S O) O v2) t3)) x)).(\lambda (_: (not (eq B b
975 Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0
976 t0 t3)).(let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Cast) u1
977 t1) t)) H0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5)
978 in (let H8 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Cast) u1 t1)
979 t)) H (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in
980 (eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3))
981 (\lambda (t: T).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t
982 (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
983 (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t))) (let H9 \def
984 (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee:
985 T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
986 False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
987 return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f)
988 \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow
989 True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H4) in
990 (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead
991 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Flat Cast) u3
992 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
993 T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Bind b) u2 (THead (Flat
994 Appl) (lift (S O) O v2) t3)))) H9)) x H5))))))))))))))))) (\lambda (H0: (pr0
995 (THead (Flat Cast) u1 t1) x)).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0:
996 T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind Abbr) u0
997 t0) (THead (Flat Cast) u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w)
998 x)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda (_: (subst0
999 O u2 t3 w)).(let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat
1000 Cast) u1 t1) t)) H0 (THead (Bind Abbr) u2 w) H4) in (let H7 \def (eq_ind_r T
1001 x (\lambda (t: T).(pr0 (THead (Flat Cast) u1 t1) t)) H (THead (Bind Abbr) u2
1002 w) H4) in (eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).(or (ex3_2 T T
1003 (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Flat Cast) u3 t2))))
1004 (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
1005 T).(pr0 t1 t2)))) (pr0 t1 t))) (let H8 \def (eq_ind T (THead (Bind Abbr) u0
1006 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
1007 _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
1008 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
1009 \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1
1010 t1) H3) in (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T
1011 (THead (Bind Abbr) u2 w) (THead (Flat Cast) u3 t2)))) (\lambda (u3:
1012 T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
1013 t2)))) (pr0 t1 (THead (Bind Abbr) u2 w))) H8)) x H4)))))))))))))) (\lambda
1014 (_: (pr0 (THead (Flat Cast) u1 t1) x)).(\lambda (b: B).(\lambda (t0:
1015 T).(\lambda (t3: T).(\lambda (u: T).(\lambda (H2: (eq T (THead (Bind b) u
1016 (lift (S O) O t0)) (THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T t3
1017 x)).(\lambda (_: (not (eq B b Abst))).(\lambda (H4: (pr0 t0 t3)).(let H5 \def
1018 (eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H4 x H3) in (let H6 \def (eq_ind T
1019 (THead (Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match ee in T return
1020 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
1021 \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
1022 (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
1023 False])])) I (THead (Flat Cast) u1 t1) H2) in (False_ind (or (ex3_2 T T
1024 (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2))))
1025 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
1026 T).(pr0 t1 t2)))) (pr0 t1 x)) H6)))))))))))) (\lambda (_: (pr0 (THead (Flat
1027 Cast) u1 t1) x)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (u: T).(\lambda
1028 (H1: (eq T (THead (Flat Cast) u t0) (THead (Flat Cast) u1 t1))).(\lambda (H2:
1029 (eq T t3 x)).(\lambda (H3: (pr0 t0 t3)).(let H4 \def (eq_ind T t3 (\lambda
1030 (t: T).(pr0 t0 t)) H3 x H2) in (let H5 \def (f_equal T T (\lambda (e:
1031 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u |
1032 (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u
1033 t0) (THead (Flat Cast) u1 t1) H1) in ((let H6 \def (f_equal T T (\lambda (e:
1034 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
1035 (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast)
1036 u t0) (THead (Flat Cast) u1 t1) H1) in (\lambda (_: (eq T u u1)).(let H8 \def
1037 (eq_ind T t0 (\lambda (t: T).(pr0 t x)) H4 t1 H6) in (or_intror (ex3_2 T T
1038 (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2))))
1039 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
1040 T).(pr0 t1 t2)))) (pr0 t1 x) H8)))) H5)))))))))) H)))).
1042 theorem pr0_gen_abbr:
1043 \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abbr) u1
1044 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
1045 (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
1046 (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y))
1047 (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x))))))
1049 \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
1050 (Bind Abbr) u1 t1) x)).(let H0 \def (match H in pr0 return (\lambda (t:
1051 T).(\lambda (t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Bind Abbr) u1
1052 t1)) \to ((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
1053 T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0
1054 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda
1055 (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S
1056 O) O x)))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead
1057 (Bind Abbr) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Bind Abbr)
1058 u1 t1) (\lambda (t0: T).((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2:
1059 T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
1060 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0
1061 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y
1062 t2))))))) (pr0 t1 (lift (S O) O x))))) (\lambda (H2: (eq T (THead (Bind Abbr)
1063 u1 t1) x)).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (t0: T).(or (ex3_2 T
1064 T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2))))
1065 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2:
1066 T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0
1067 O u2 y t2))))))) (pr0 t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda
1068 (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u1 t1) (THead (Bind Abbr)
1069 u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
1070 T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y))
1071 (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O (THead (Bind
1072 Abbr) u1 t1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T
1073 (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
1074 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0
1075 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y
1076 t2)))))) u1 t1 (refl_equal T (THead (Bind Abbr) u1 t1)) (pr0_refl u1)
1077 (or_introl (pr0 t1 t1) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y:
1078 T).(subst0 O u1 y t1))) (pr0_refl t1)))) x H2)) t (sym_eq T t (THead (Bind
1079 Abbr) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda
1080 (H2: (eq T (THead k u0 t0) (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T
1081 (THead k u2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e in T
1082 return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
1083 \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind
1084 Abbr) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T
1085 return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
1086 \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind
1087 Abbr) u1 t1) H2) in ((let H6 \def (f_equal T K (\lambda (e: T).(match e in T
1088 return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
1089 \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind
1090 Abbr) u1 t1) H2) in (eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u0 u1) \to
1091 ((eq T t0 t1) \to ((eq T (THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0
1092 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind
1093 Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
1094 T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
1095 (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x)))))))))
1096 (\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to
1097 ((eq T (THead (Bind Abbr) u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to (or
1098 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3
1099 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
1100 T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
1101 (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x))))))))
1102 (\lambda (H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Bind
1103 Abbr) u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda
1104 (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3:
1105 T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0
1106 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y
1107 t3))))))) (pr0 t1 (lift (S O) O x))))))) (\lambda (H9: (eq T (THead (Bind
1108 Abbr) u2 t2) x)).(eq_ind T (THead (Bind Abbr) u2 t2) (\lambda (t: T).((pr0 u1
1109 u2) \to ((pr0 t1 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq
1110 T t (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
1111 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y:
1112 T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O
1113 t)))))) (\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(or_introl
1114 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 t2)
1115 (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
1116 (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0
1117 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O (THead
1118 (Bind Abbr) u2 t2))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T
1119 (THead (Bind Abbr) u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3:
1120 T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0
1121 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y
1122 t3)))))) u2 t2 (refl_equal T (THead (Bind Abbr) u2 t2)) H10 (or_introl (pr0
1123 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y
1124 t2))) H11))))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k
1125 (sym_eq K k (Bind Abbr) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0
1126 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind
1127 Abst) u t0)) (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T (THead (Bind
1128 Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind
1129 Abst) u t0)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
1130 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
1131 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
1132 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1
1133 t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2)
1134 \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
1135 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
1136 (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0
1137 t1 y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x))))))
1138 H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow
1139 (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead
1140 (Bind Abbr) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat
1141 Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl)
1142 v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e in T return (\lambda (_:
1143 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
1144 (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
1145 [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
1146 Abbr) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl)
1147 (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0
1148 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3:
1149 T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0
1150 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda
1151 (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S
1152 O) O x)))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2)
1153 \Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr)
1154 u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def
1155 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
1156 [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
1157 \Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H3) in
1158 ((let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
1159 T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t
1160 _) \Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H3)
1161 in (eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Abbr)
1162 u2 w) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or
1163 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3
1164 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
1165 T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
1166 (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x)))))))))
1167 (\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Bind
1168 Abbr) u2 w) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to ((subst0 O u2 t2 w) \to
1169 (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr)
1170 u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
1171 T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
1172 (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x))))))))
1173 (\lambda (H8: (eq T (THead (Bind Abbr) u2 w) x)).(eq_ind T (THead (Bind Abbr)
1174 u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to ((subst0 O u2 t2 w)
1175 \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Bind
1176 Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
1177 T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
1178 (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O t)))))))
1179 (\lambda (H9: (pr0 u1 u2)).(\lambda (H10: (pr0 t1 t2)).(\lambda (H11: (subst0
1180 O u2 t2 w)).(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T
1181 (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 t3)))) (\lambda (u3:
1182 T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0
1183 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y
1184 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 w))) (ex3_2_intro T T
1185 (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind
1186 Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
1187 T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
1188 (\lambda (y: T).(subst0 O u3 y t3)))))) u2 w (refl_equal T (THead (Bind Abbr)
1189 u2 w)) H9 (or_intror (pr0 t1 w) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda
1190 (y: T).(subst0 O u2 y w))) (ex_intro2 T (\lambda (y: T).(pr0 t1 y)) (\lambda
1191 (y: T).(subst0 O u2 y w)) t2 H10 H11))))))) x H8)) t0 (sym_eq T t0 t1 H7)))
1192 u0 (sym_eq T u0 u1 H6))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u)
1193 \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead
1194 (Bind Abbr) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (f_equal T T
1195 (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
1196 \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
1197 \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
1198 (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
1199 | (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1200 t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (TLRef _)
1201 \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
1202 \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
1203 (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
1204 | (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1205 t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t)
1206 \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1
1207 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T return
1208 (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
1209 (THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead
1210 (Bind Abbr) u1 t1) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e
1211 in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _)
1212 \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_:
1213 K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead
1214 (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H2) in (eq_ind B Abbr
1215 (\lambda (b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2
1216 x) \to ((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda
1217 (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
1218 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0
1219 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y
1220 t3))))))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u u1)).(eq_ind
1221 T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not
1222 (eq B Abbr Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2:
1223 T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
1224 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0
1225 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y
1226 t3))))))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T (lift (S O) O
1227 t0) t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not
1228 (eq B Abbr Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2:
1229 T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
1230 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t
1231 t3) (ex2 T (\lambda (y: T).(pr0 t y)) (\lambda (y: T).(subst0 O u2 y
1232 t3))))))) (pr0 t (lift (S O) O x))))))) (\lambda (H9: (eq T t2 x)).(eq_ind T
1233 x (\lambda (t: T).((not (eq B Abbr Abst)) \to ((pr0 t0 t) \to (or (ex3_2 T T
1234 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
1235 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3:
1236 T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y: T).(pr0 (lift (S O) O
1237 t0) y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 (lift (S O) O t0) (lift
1238 (S O) O x)))))) (\lambda (_: (not (eq B Abbr Abst))).(\lambda (H11: (pr0 t0
1239 x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead
1240 (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
1241 (u2: T).(\lambda (t3: T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y:
1242 T).(pr0 (lift (S O) O t0) y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0
1243 (lift (S O) O t0) (lift (S O) O x)) (pr0_lift t0 x H11 (S O) O)))) t2 (sym_eq
1244 T t2 x H9))) t1 H8)) u (sym_eq T u u1 H7))) b (sym_eq B b Abbr H6))) H5))
1245 H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T
1246 (THead (Flat Cast) u t0) (THead (Bind Abbr) u1 t1))).(\lambda (H2: (eq T t2
1247 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e
1248 in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
1249 _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
1250 (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
1251 True])])) I (THead (Bind Abbr) u1 t1) H1) in (False_ind ((eq T t2 x) \to
1252 ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
1253 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
1254 (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0
1255 t1 y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x)))))
1256 H3)) H2 H0)))]) in (H0 (refl_equal T (THead (Bind Abbr) u1 t1)) (refl_equal T
1259 theorem pr0_gen_void:
1260 \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Void) u1
1261 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
1262 (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
1263 (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x))))))
1265 \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
1266 (Bind Void) u1 t1) x)).(let H0 \def (match H in pr0 return (\lambda (t:
1267 T).(\lambda (t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Bind Void) u1
1268 t1)) \to ((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
1269 T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0
1270 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O)
1271 O x)))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind
1272 Void) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Bind Void) u1 t1)
1273 (\lambda (t0: T).((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
1274 (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
1275 T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1
1276 (lift (S O) O x))))) (\lambda (H2: (eq T (THead (Bind Void) u1 t1)
1277 x)).(eq_ind T (THead (Bind Void) u1 t1) (\lambda (t0: T).(or (ex3_2 T T
1278 (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2))))
1279 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
1280 T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda
1281 (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) u1 t1) (THead (Bind Void)
1282 u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
1283 T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O (THead (Bind Void)
1284 u1 t1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead
1285 (Bind Void) u1 t1) (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
1286 T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1
1287 (refl_equal T (THead (Bind Void) u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H2))
1288 t (sym_eq T t (THead (Bind Void) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2
1289 H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Bind Void) u1
1290 t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (f_equal T T
1291 (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
1292 \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
1293 (THead k u0 t0) (THead (Bind Void) u1 t1) H2) in ((let H5 \def (f_equal T T
1294 (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
1295 \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
1296 (THead k u0 t0) (THead (Bind Void) u1 t1) H2) in ((let H6 \def (f_equal T K
1297 (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
1298 \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
1299 (THead k u0 t0) (THead (Bind Void) u1 t1) H2) in (eq_ind K (Bind Void)
1300 (\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2)
1301 x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3:
1302 T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3:
1303 T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
1304 t3)))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T
1305 u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Void) u2 t2) x) \to
1306 ((pr0 t u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda
1307 (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_:
1308 T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1
1309 (lift (S O) O x)))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t:
1310 T).((eq T (THead (Bind Void) u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to
1311 (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Void)
1312 u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
1313 T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))))) (\lambda
1314 (H9: (eq T (THead (Bind Void) u2 t2) x)).(eq_ind T (THead (Bind Void) u2 t2)
1315 (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to (or (ex3_2 T T (\lambda
1316 (u3: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u3 t3)))) (\lambda (u3:
1317 T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
1318 t3)))) (pr0 t1 (lift (S O) O t)))))) (\lambda (H10: (pr0 u1 u2)).(\lambda
1319 (H11: (pr0 t1 t2)).(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3:
1320 T).(eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 t3)))) (\lambda (u3:
1321 T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
1322 t3)))) (pr0 t1 (lift (S O) O (THead (Bind Void) u2 t2))) (ex3_2_intro T T
1323 (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Void) u2 t2) (THead
1324 (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
1325 (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Bind Void)
1326 u2 t2)) H10 H11)))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7)))
1327 k (sym_eq K k (Bind Void) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0
1328 t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind
1329 Abst) u t0)) (THead (Bind Void) u1 t1))).(\lambda (H3: (eq T (THead (Bind
1330 Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind
1331 Abst) u t0)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
1332 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
1333 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
1334 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Void) u1
1335 t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2)
1336 \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
1337 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
1338 (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))))
1339 H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow
1340 (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead
1341 (Bind Void) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat
1342 Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl)
1343 v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e in T return (\lambda (_:
1344 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
1345 (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
1346 [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
1347 Void) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl)
1348 (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0
1349 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3:
1350 T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0
1351 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O)
1352 O x)))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2)
1353 \Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Void)
1354 u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def
1355 (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (e: T).(match e in T return
1356 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
1357 \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
1358 (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
1359 B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void
1360 \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind Void) u1
1361 t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to
1362 ((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T (\lambda (u3:
1363 T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3:
1364 T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
1365 t3)))) (pr0 t1 (lift (S O) O x))))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0
1366 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0))
1367 (THead (Bind Void) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (f_equal
1368 T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
1369 \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
1370 \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
1371 (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
1372 | (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1373 t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (TLRef _)
1374 \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
1375 \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
1376 (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
1377 | (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1378 t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t)
1379 \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1
1380 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T return
1381 (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
1382 (THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead
1383 (Bind Void) u1 t1) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e
1384 in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _)
1385 \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_:
1386 K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead
1387 (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 t1) H2) in (eq_ind B Void
1388 (\lambda (b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2
1389 x) \to ((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda
1390 (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
1391 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
1392 t3)))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T
1393 u1 (\lambda (_: T).((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not
1394 (eq B Void Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2:
1395 T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
1396 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
1397 t3)))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T (lift (S O) O t0)
1398 t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not (eq B
1399 Void Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
1400 (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
1401 T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (pr0 t (lift
1402 (S O) O x))))))) (\lambda (H9: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((not
1403 (eq B Void Abst)) \to ((pr0 t0 t) \to (or (ex3_2 T T (\lambda (u2:
1404 T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
1405 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift
1406 (S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O x)))))) (\lambda (_:
1407 (not (eq B Void Abst))).(\lambda (H11: (pr0 t0 x)).(or_intror (ex3_2 T T
1408 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
1409 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
1410 T).(pr0 (lift (S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O x))
1411 (pr0_lift t0 x H11 (S O) O)))) t2 (sym_eq T t2 x H9))) t1 H8)) u (sym_eq T u
1412 u1 H7))) b (sym_eq B b Void H6))) H5)) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2
1413 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Bind
1414 Void) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead
1415 (Flat Cast) u t0) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop)
1416 with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
1417 _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
1418 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Void) u1
1419 t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T
1420 (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
1421 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
1422 T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))) H3)) H2 H0)))]) in (H0
1423 (refl_equal T (THead (Bind Void) u1 t1)) (refl_equal T x)))))).
1425 theorem pr0_gen_lift:
1426 \forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr0
1427 (lift h d t1) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
1428 (t2: T).(pr0 t1 t2)))))))
1430 \lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda
1431 (H: (pr0 (lift h d t1) x)).(insert_eq T (lift h d t1) (\lambda (t: T).(pr0 t
1432 x)) (\lambda (_: T).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
1433 (t2: T).(pr0 t1 t2)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(unintro nat
1434 d (\lambda (n: nat).((eq T y (lift h n t1)) \to (ex2 T (\lambda (t2: T).(eq T
1435 x (lift h n t2))) (\lambda (t2: T).(pr0 t1 t2))))) (unintro T t1 (\lambda (t:
1436 T).(\forall (x0: nat).((eq T y (lift h x0 t)) \to (ex2 T (\lambda (t2: T).(eq
1437 T x (lift h x0 t2))) (\lambda (t2: T).(pr0 t t2)))))) (pr0_ind (\lambda (t:
1438 T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1
1439 x0)) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h x1 t2))) (\lambda (t2:
1440 T).(pr0 x0 t2)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1:
1441 nat).(\lambda (H1: (eq T t (lift h x1 x0))).(ex_intro2 T (\lambda (t2: T).(eq
1442 T t (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)) x0 H1 (pr0_refl x0))))))
1443 (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2:
1444 ((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T
1445 (\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0
1446 t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2
1447 t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1
1448 x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4:
1449 T).(pr0 x0 t4)))))))).(\lambda (k: K).(\lambda (x0: T).(\lambda (x1:
1450 nat).(\lambda (H5: (eq T (THead k u1 t2) (lift h x1 x0))).(K_ind (\lambda
1451 (k0: K).((eq T (THead k0 u1 t2) (lift h x1 x0)) \to (ex2 T (\lambda (t4:
1452 T).(eq T (THead k0 u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))
1453 (\lambda (b: B).(\lambda (H6: (eq T (THead (Bind b) u1 t2) (lift h x1
1454 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind
1455 b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0))))
1456 (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda
1457 (t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0
1458 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead
1459 (Bind b) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9: (eq T
1460 t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda (t:
1461 T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4)))
1462 (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h
1463 (S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T
1464 (THead (Bind b) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b)
1465 x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3 (lift h (S x1)
1466 x4))).(\lambda (H10: (pr0 x3 x4)).(eq_ind_r T (lift h (S x1) x4) (\lambda (t:
1467 T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t) (lift h x1 t4)))
1468 (\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4)))) (ex2_ind T (\lambda (t4:
1469 T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda
1470 (t4: T).(eq T (THead (Bind b) u2 (lift h (S x1) x4)) (lift h x1 t4)))
1471 (\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda
1472 (H_x0: (eq T u2 (lift h x1 x5))).(\lambda (H11: (pr0 x2 x5)).(eq_ind_r T
1473 (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b)
1474 t (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b)
1475 x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1
1476 x5) (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind
1477 b) x2 x3) t4)) (THead (Bind b) x5 x4) (sym_eq T (lift h x1 (THead (Bind b) x5
1478 x4)) (THead (Bind b) (lift h x1 x5) (lift h (S x1) x4)) (lift_bind b x5 x4 h
1479 x1)) (pr0_comp x2 x5 H11 x3 x4 H10 (Bind b))) u2 H_x0)))) (H2 x2 x1 H8)) t3
1480 H_x)))) (H4 x3 (S x1) H9)) x0 H7)))))) (lift_gen_bind b u1 t2 x0 h x1 H6))))
1481 (\lambda (f: F).(\lambda (H6: (eq T (THead (Flat f) u1 t2) (lift h x1
1482 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat
1483 f) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0))))
1484 (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T (\lambda
1485 (t4: T).(eq T (THead (Flat f) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0
1486 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead
1487 (Flat f) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9: (eq T
1488 t2 (lift h x1 x3))).(eq_ind_r T (THead (Flat f) x2 x3) (\lambda (t: T).(ex2 T
1489 (\lambda (t4: T).(eq T (THead (Flat f) u2 t3) (lift h x1 t4))) (\lambda (t4:
1490 T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4)))
1491 (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Flat f)
1492 u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2 x3) t4)))
1493 (\lambda (x4: T).(\lambda (H_x: (eq T t3 (lift h x1 x4))).(\lambda (H10: (pr0
1494 x3 x4)).(eq_ind_r T (lift h x1 x4) (\lambda (t: T).(ex2 T (\lambda (t4:
1495 T).(eq T (THead (Flat f) u2 t) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead
1496 (Flat f) x2 x3) t4)))) (ex2_ind T (\lambda (t4: T).(eq T u2 (lift h x1 t4)))
1497 (\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Flat f)
1498 u2 (lift h x1 x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2
1499 x3) t4))) (\lambda (x5: T).(\lambda (H_x0: (eq T u2 (lift h x1 x5))).(\lambda
1500 (H11: (pr0 x2 x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda
1501 (t4: T).(eq T (THead (Flat f) t (lift h x1 x4)) (lift h x1 t4))) (\lambda
1502 (t4: T).(pr0 (THead (Flat f) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq
1503 T (THead (Flat f) (lift h x1 x5) (lift h x1 x4)) (lift h x1 t4))) (\lambda
1504 (t4: T).(pr0 (THead (Flat f) x2 x3) t4)) (THead (Flat f) x5 x4) (sym_eq T
1505 (lift h x1 (THead (Flat f) x5 x4)) (THead (Flat f) (lift h x1 x5) (lift h x1
1506 x4)) (lift_flat f x5 x4 h x1)) (pr0_comp x2 x5 H11 x3 x4 H10 (Flat f))) u2
1507 H_x0)))) (H2 x2 x1 H8)) t3 H_x)))) (H4 x3 x1 H9)) x0 H7)))))) (lift_gen_flat
1508 f u1 t2 x0 h x1 H6)))) k H5))))))))))))) (\lambda (u: T).(\lambda (v1:
1509 T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (H2: ((\forall (x0:
1510 T).(\forall (x1: nat).((eq T v1 (lift h x1 x0)) \to (ex2 T (\lambda (t2:
1511 T).(eq T v2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda
1512 (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall
1513 (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4:
1514 T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda
1515 (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead
1516 (Bind Abst) u t2)) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda
1517 (z: T).(eq T x0 (THead (Flat Appl) y0 z)))) (\lambda (y0: T).(\lambda (_:
1518 T).(eq T v1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (THead
1519 (Bind Abst) u t2) (lift h x1 z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind
1520 Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2:
1521 T).(\lambda (x3: T).(\lambda (H6: (eq T x0 (THead (Flat Appl) x2
1522 x3))).(\lambda (H7: (eq T v1 (lift h x1 x2))).(\lambda (H8: (eq T (THead
1523 (Bind Abst) u t2) (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl) x2 x3)
1524 (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift
1525 h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (ex3_2_ind T T (\lambda (y0:
1526 T).(\lambda (z: T).(eq T x3 (THead (Bind Abst) y0 z)))) (\lambda (y0:
1527 T).(\lambda (_: T).(eq T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z:
1528 T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind
1529 Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 x3)
1530 t4))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H9: (eq T x3 (THead (Bind
1531 Abst) x4 x5))).(\lambda (_: (eq T u (lift h x1 x4))).(\lambda (H11: (eq T t2
1532 (lift h (S x1) x5))).(eq_ind_r T (THead (Bind Abst) x4 x5) (\lambda (t:
1533 T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4)))
1534 (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 t) t4)))) (ex2_ind T (\lambda
1535 (t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x5 t4)) (ex2 T
1536 (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4))) (\lambda
1537 (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4))) (\lambda
1538 (x6: T).(\lambda (H_x: (eq T t3 (lift h (S x1) x6))).(\lambda (H12: (pr0 x5
1539 x6)).(eq_ind_r T (lift h (S x1) x6) (\lambda (t: T).(ex2 T (\lambda (t4:
1540 T).(eq T (THead (Bind Abbr) v2 t) (lift h x1 t4))) (\lambda (t4: T).(pr0
1541 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4)))) (ex2_ind T (\lambda
1542 (t4: T).(eq T v2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T
1543 (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 (lift h (S x1) x6)) (lift h x1
1544 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5))
1545 t4))) (\lambda (x7: T).(\lambda (H_x0: (eq T v2 (lift h x1 x7))).(\lambda
1546 (H13: (pr0 x2 x7)).(eq_ind_r T (lift h x1 x7) (\lambda (t: T).(ex2 T (\lambda
1547 (t4: T).(eq T (THead (Bind Abbr) t (lift h (S x1) x6)) (lift h x1 t4)))
1548 (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4))))
1549 (ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x7) (lift h
1550 (S x1) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2
1551 (THead (Bind Abst) x4 x5)) t4)) (THead (Bind Abbr) x7 x6) (sym_eq T (lift h
1552 x1 (THead (Bind Abbr) x7 x6)) (THead (Bind Abbr) (lift h x1 x7) (lift h (S
1553 x1) x6)) (lift_bind Abbr x7 x6 h x1)) (pr0_beta x4 x2 x7 H13 x5 x6 H12)) v2
1554 H_x0)))) (H2 x2 x1 H7)) t3 H_x)))) (H4 x5 (S x1) H11)) x3 H9))))))
1555 (lift_gen_bind Abst u t2 x3 h x1 H8)) x0 H6)))))) (lift_gen_flat Appl v1
1556 (THead (Bind Abst) u t2) x0 h x1 H5)))))))))))))) (\lambda (b: B).(\lambda
1557 (H1: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0
1558 v1 v2)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: nat).((eq T v1 (lift h
1559 x1 x0)) \to (ex2 T (\lambda (t2: T).(eq T v2 (lift h x1 t2))) (\lambda (t2:
1560 T).(pr0 x0 t2)))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1
1561 u2)).(\lambda (H5: ((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1
1562 x0)) \to (ex2 T (\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2:
1563 T).(pr0 x0 t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2
1564 t3)).(\lambda (H7: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1
1565 x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4:
1566 T).(pr0 x0 t4)))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H8: (eq T
1567 (THead (Flat Appl) v1 (THead (Bind b) u1 t2)) (lift h x1 x0))).(ex3_2_ind T T
1568 (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z))))
1569 (\lambda (y0: T).(\lambda (_: T).(eq T v1 (lift h x1 y0)))) (\lambda (_:
1570 T).(\lambda (z: T).(eq T (THead (Bind b) u1 t2) (lift h x1 z)))) (ex2 T
1571 (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
1572 v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2:
1573 T).(\lambda (x3: T).(\lambda (H9: (eq T x0 (THead (Flat Appl) x2
1574 x3))).(\lambda (H10: (eq T v1 (lift h x1 x2))).(\lambda (H11: (eq T (THead
1575 (Bind b) u1 t2) (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl) x2 x3)
1576 (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat
1577 Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4))))
1578 (ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x3 (THead (Bind b) y0
1579 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) (\lambda
1580 (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda (t4:
1581 T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h
1582 x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 x3) t4))) (\lambda (x4:
1583 T).(\lambda (x5: T).(\lambda (H12: (eq T x3 (THead (Bind b) x4 x5))).(\lambda
1584 (H13: (eq T u1 (lift h x1 x4))).(\lambda (H14: (eq T t2 (lift h (S x1)
1585 x5))).(eq_ind_r T (THead (Bind b) x4 x5) (\lambda (t: T).(ex2 T (\lambda (t4:
1586 T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h
1587 x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 t) t4)))) (ex2_ind T
1588 (\lambda (t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x5 t4))
1589 (ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S
1590 O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2
1591 (THead (Bind b) x4 x5)) t4))) (\lambda (x6: T).(\lambda (H_x: (eq T t3 (lift
1592 h (S x1) x6))).(\lambda (H15: (pr0 x5 x6)).(eq_ind_r T (lift h (S x1) x6)
1593 (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat
1594 Appl) (lift (S O) O v2) t)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead
1595 (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex2_ind T (\lambda (t4: T).(eq
1596 T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x4 t4)) (ex2 T (\lambda (t4:
1597 T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift h (S
1598 x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead
1599 (Bind b) x4 x5)) t4))) (\lambda (x7: T).(\lambda (H_x0: (eq T u2 (lift h x1
1600 x7))).(\lambda (H16: (pr0 x4 x7)).(eq_ind_r T (lift h x1 x7) (\lambda (t:
1601 T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) t (THead (Flat Appl) (lift
1602 (S O) O v2) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0
1603 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex2_ind T (\lambda (t4:
1604 T).(eq T v2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda
1605 (t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) (lift (S O) O
1606 v2) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat
1607 Appl) x2 (THead (Bind b) x4 x5)) t4))) (\lambda (x8: T).(\lambda (H_x1: (eq T
1608 v2 (lift h x1 x8))).(\lambda (H17: (pr0 x2 x8)).(eq_ind_r T (lift h x1 x8)
1609 (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7)
1610 (THead (Flat Appl) (lift (S O) O t) (lift h (S x1) x6))) (lift h x1 t4)))
1611 (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4))))
1612 (eq_ind T (lift h (plus (S O) x1) (lift (S O) O x8)) (\lambda (t: T).(ex2 T
1613 (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) t
1614 (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat
1615 Appl) x2 (THead (Bind b) x4 x5)) t4)))) (eq_ind T (lift h (S x1) (THead (Flat
1616 Appl) (lift (S O) O x8) x6)) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T
1617 (THead (Bind b) (lift h x1 x7) t) (lift h x1 t4))) (\lambda (t4: T).(pr0
1618 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex_intro2 T (\lambda
1619 (t4: T).(eq T (THead (Bind b) (lift h x1 x7) (lift h (S x1) (THead (Flat
1620 Appl) (lift (S O) O x8) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead
1621 (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)) (THead (Bind b) x7 (THead (Flat
1622 Appl) (lift (S O) O x8) x6)) (sym_eq T (lift h x1 (THead (Bind b) x7 (THead
1623 (Flat Appl) (lift (S O) O x8) x6))) (THead (Bind b) (lift h x1 x7) (lift h (S
1624 x1) (THead (Flat Appl) (lift (S O) O x8) x6))) (lift_bind b x7 (THead (Flat
1625 Appl) (lift (S O) O x8) x6) h x1)) (pr0_upsilon b H1 x2 x8 H17 x4 x7 H16 x5
1626 x6 H15)) (THead (Flat Appl) (lift h (S x1) (lift (S O) O x8)) (lift h (S x1)
1627 x6)) (lift_flat Appl (lift (S O) O x8) x6 h (S x1))) (lift (S O) O (lift h x1
1628 x8)) (lift_d x8 h (S O) x1 O (le_O_n x1))) v2 H_x1)))) (H3 x2 x1 H10)) u2
1629 H_x0)))) (H5 x4 x1 H13)) t3 H_x)))) (H7 x5 (S x1) H14)) x3 H12))))))
1630 (lift_gen_bind b u1 t2 x3 h x1 H11)) x0 H9)))))) (lift_gen_flat Appl v1
1631 (THead (Bind b) u1 t2) x0 h x1 H8))))))))))))))))))) (\lambda (u1:
1632 T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: ((\forall (x0:
1633 T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T (\lambda (t2:
1634 T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda
1635 (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall
1636 (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4:
1637 T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (w:
1638 T).(\lambda (H5: (subst0 O u2 t3 w)).(\lambda (x0: T).(\lambda (x1:
1639 nat).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t2) (lift h x1
1640 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind
1641 Abbr) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0))))
1642 (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda
1643 (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0
1644 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead
1645 (Bind Abbr) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9:
1646 (eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda
1647 (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1
1648 t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3
1649 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4:
1650 T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0
1651 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3
1652 (lift h (S x1) x4))).(\lambda (H10: (pr0 x3 x4)).(let H11 \def (eq_ind T t3
1653 (\lambda (t: T).(subst0 O u2 t w)) H5 (lift h (S x1) x4) H_x) in (ex2_ind T
1654 (\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2
1655 T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda
1656 (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x5: T).(\lambda (H_x0:
1657 (eq T u2 (lift h x1 x5))).(\lambda (H12: (pr0 x2 x5)).(eq_ind_r T (lift h x1
1658 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) t w)
1659 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (let
1660 H13 \def (eq_ind T u2 (\lambda (t: T).(subst0 O t (lift h (S x1) x4) w)) H11
1661 (lift h x1 x5) H_x0) in (let H14 \def (refl_equal nat (S (plus O x1))) in
1662 (let H15 \def (eq_ind nat (S x1) (\lambda (n: nat).(subst0 O (lift h x1 x5)
1663 (lift h n x4) w)) H13 (S (plus O x1)) H14) in (ex2_ind T (\lambda (t4: T).(eq
1664 T w (lift h (S (plus O x1)) t4))) (\lambda (t4: T).(subst0 O x5 x4 t4)) (ex2
1665 T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) w) (lift h x1
1666 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x6:
1667 T).(\lambda (H16: (eq T w (lift h (S (plus O x1)) x6))).(\lambda (H17:
1668 (subst0 O x5 x4 x6)).(eq_ind_r T (lift h (S (plus O x1)) x6) (\lambda (t:
1669 T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) t) (lift h
1670 x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (ex_intro2 T
1671 (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O
1672 x1)) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3)
1673 t4)) (THead (Bind Abbr) x5 x6) (sym_eq T (lift h x1 (THead (Bind Abbr) x5
1674 x6)) (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O x1)) x6))
1675 (lift_bind Abbr x5 x6 h (plus O x1))) (pr0_delta x2 x5 H12 x3 x4 H10 x6 H17))
1676 w H16)))) (subst0_gen_lift_lt x5 x4 w O h x1 H15))))) u2 H_x0)))) (H2 x2 x1
1677 H8)))))) (H4 x3 (S x1) H9)) x0 H7)))))) (lift_gen_bind Abbr u1 t2 x0 h x1
1678 H6))))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda
1679 (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H3: ((\forall
1680 (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4:
1681 T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u:
1682 T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq T (THead (Bind b) u
1683 (lift (S O) O t2)) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda
1684 (z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq
1685 T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift (S O) O t2)
1686 (lift h (S x1) z)))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4)))
1687 (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
1688 (H5: (eq T x0 (THead (Bind b) x2 x3))).(\lambda (_: (eq T u (lift h x1
1689 x2))).(\lambda (H7: (eq T (lift (S O) O t2) (lift h (S x1) x3))).(eq_ind_r T
1690 (THead (Bind b) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift
1691 h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (let H8 \def (eq_ind_r nat (plus (S
1692 O) x1) (\lambda (n: nat).(eq nat (S x1) n)) (refl_equal nat (plus (S O) x1))
1693 (plus x1 (S O)) (plus_comm x1 (S O))) in (let H9 \def (eq_ind nat (S x1)
1694 (\lambda (n: nat).(eq T (lift (S O) O t2) (lift h n x3))) H7 (plus x1 (S O))
1695 H8) in (ex2_ind T (\lambda (t4: T).(eq T x3 (lift (S O) O t4))) (\lambda (t4:
1696 T).(eq T t2 (lift h x1 t4))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1
1697 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x4:
1698 T).(\lambda (H10: (eq T x3 (lift (S O) O x4))).(\lambda (H11: (eq T t2 (lift
1699 h x1 x4))).(eq_ind_r T (lift (S O) O x4) (\lambda (t: T).(ex2 T (\lambda (t4:
1700 T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 t)
1701 t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4:
1702 T).(pr0 x4 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda
1703 (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4))) (\lambda (x5:
1704 T).(\lambda (H_x: (eq T t3 (lift h x1 x5))).(\lambda (H12: (pr0 x4
1705 x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T
1706 t (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O
1707 x4)) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (lift h x1 x5) (lift h x1
1708 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4)) x5
1709 (refl_equal T (lift h x1 x5)) (pr0_zeta b H1 x4 x5 H12 x2)) t3 H_x)))) (H3 x4
1710 x1 H11)) x3 H10)))) (lift_gen_lift t2 x3 (S O) h O x1 (le_O_n x1) H9)))) x0
1711 H5)))))) (lift_gen_bind b u (lift (S O) O t2) x0 h x1 H4)))))))))))) (\lambda
1712 (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H2: ((\forall
1713 (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4:
1714 T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u:
1715 T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq T (THead (Flat Cast)
1716 u t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T
1717 x0 (THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift
1718 h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T
1719 (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))
1720 (\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (eq T x0 (THead (Flat Cast)
1721 x2 x3))).(\lambda (_: (eq T u (lift h x1 x2))).(\lambda (H6: (eq T t2 (lift h
1722 x1 x3))).(eq_ind_r T (THead (Flat Cast) x2 x3) (\lambda (t: T).(ex2 T
1723 (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4))))
1724 (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0
1725 x3 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4:
1726 T).(pr0 (THead (Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T
1727 t3 (lift h x1 x4))).(\lambda (H7: (pr0 x3 x4)).(eq_ind_r T (lift h x1 x4)
1728 (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t (lift h x1 t4))) (\lambda
1729 (t4: T).(pr0 (THead (Flat Cast) x2 x3) t4)))) (ex_intro2 T (\lambda (t4:
1730 T).(eq T (lift h x1 x4) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat
1731 Cast) x2 x3) t4)) x4 (refl_equal T (lift h x1 x4)) (pr0_epsilon x3 x4 H7 x2))
1732 t3 H_x)))) (H2 x3 x1 H6)) x0 H4)))))) (lift_gen_flat Cast u t2 x0 h x1
1733 H3)))))))))) y x H0))))) H))))).
projects / helm.git / blob