1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "Basic-1/ty3/props.ma".
19 include "Basic-1/pc3/subst1.ma".
21 include "Basic-1/getl/getl.ma".
23 theorem ty3_gen_cabbr:
24 \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
25 t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c
26 (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to
27 (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda
28 (_: T).(subst1 d u t1 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
29 T).(subst1 d u t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
30 g a y1 y2))))))))))))))))
32 \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
33 (H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda
34 (t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead
35 e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a:
36 C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
37 T).(subst1 d u t (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
38 T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
39 g a y1 y2))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t:
40 T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e: C).(\forall (u:
41 T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0:
42 C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T
43 T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t3 (lift (S O) d y1))))
44 (\lambda (_: T).(\lambda (y2: T).(subst1 d u t (lift (S O) d y2)))) (\lambda
45 (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (u:
46 T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: ((\forall (e:
47 C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0))
48 \to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d
49 a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S
50 O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d
51 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
52 y2)))))))))))))).(\lambda (H4: (pc3 c0 t4 t3)).(\lambda (e: C).(\lambda (u0:
53 T).(\lambda (d: nat).(\lambda (H5: (getl d c0 (CHead e (Bind Abbr)
54 u0))).(\lambda (a0: C).(\lambda (H6: (csubst1 d u0 c0 a0)).(\lambda (a:
55 C).(\lambda (H7: (drop (S O) d a0 a)).(let H8 \def (H3 e u0 d H5 a0 H6 a H7)
56 in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O)
57 d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d
58 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T
59 (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda
60 (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d y2)))) (\lambda (y1:
61 T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1:
62 T).(\lambda (H9: (subst1 d u0 u (lift (S O) d x0))).(\lambda (H10: (subst1 d
63 u0 t4 (lift (S O) d x1))).(\lambda (H11: (ty3 g a x0 x1)).(let H12 \def (H1 e
64 u0 d H5 a0 H6 a H7) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_:
65 T).(subst1 d u0 t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
66 T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
67 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift
68 (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d
69 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2:
70 T).(\lambda (x3: T).(\lambda (H13: (subst1 d u0 t3 (lift (S O) d
71 x2))).(\lambda (_: (subst1 d u0 t (lift (S O) d x3))).(\lambda (H15: (ty3 g a
72 x2 x3)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u
73 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift
74 (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 H9
75 H13 (ty3_conv g a x2 x3 H15 x0 x1 H11 (pc3_gen_cabbr c0 t4 t3 H4 e u0 d H5 a0
76 H6 a H7 x1 H10 x2 H13)))))))) H12))))))) H8)))))))))))))))))))) (\lambda (c0:
77 C).(\lambda (m: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (d:
78 nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0:
79 C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O)
80 d a0 a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (TSort
81 m) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u (TSort
82 (next g m)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
83 y1 y2))) (TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t:
84 T).(subst1 d u (TSort m) t)) (subst1_refl d u (TSort m)) (lift (S O) d (TSort
85 m)) (lift_sort m (S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t:
86 T).(subst1 d u (TSort (next g m)) t)) (subst1_refl d u (TSort (next g m)))
87 (lift (S O) d (TSort (next g m))) (lift_sort (next g m) (S O) d)) (ty3_sort g
88 a m)))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda
89 (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t:
90 T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: C).(\forall (u0:
91 T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Abbr) u0)) \to (\forall (a0:
92 C).((csubst1 d0 u0 d a0) \to (\forall (a: C).((drop (S O) d0 a0 a) \to (ex3_2
93 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 u (lift (S O) d0 y1))))
94 (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 t (lift (S O) d0 y2))))
95 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e:
96 C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e
97 (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0
98 a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(lt_eq_gt_e n d0
99 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S
100 O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t)
101 (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
102 (\lambda (H6: (lt n d0)).(let H7 \def (eq_ind nat (minus d0 n) (\lambda (n0:
103 nat).(getl n0 (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0)))
104 (getl_conf_le d0 (CHead e (Bind Abbr) u0) c0 H3 (CHead d (Bind Abbr) u) n H0
105 (le_S_n n d0 (le_S (S n) d0 H6))) (S (minus d0 (S n))) (minus_x_Sy d0 n H6))
106 in (ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 n) u0 (CHead d (Bind Abbr)
107 u) e2)) (\lambda (e2: C).(getl n a0 e2)) (ex3_2 T T (\lambda (y1: T).(\lambda
108 (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
109 (y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
110 T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x: C).(\lambda (H8: (csubst1
111 (minus d0 n) u0 (CHead d (Bind Abbr) u) x)).(\lambda (H9: (getl n a0 x)).(let
112 H10 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(csubst1 n0 u0 (CHead d
113 (Bind Abbr) u) x)) H8 (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) in (let H11
114 \def (csubst1_gen_head (Bind Abbr) d x u u0 (minus d0 (S n)) H10) in
115 (ex3_2_ind T C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 (Bind
116 Abbr) u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 (minus d0 (S n)) u0 u
117 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 (minus d0 (S n)) u0 d c2)))
118 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S
119 O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t)
120 (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
121 (\lambda (x0: T).(\lambda (x1: C).(\lambda (H12: (eq C x (CHead x1 (Bind
122 Abbr) x0))).(\lambda (H13: (subst1 (minus d0 (S n)) u0 u x0)).(\lambda (H14:
123 (csubst1 (minus d0 (S n)) u0 d x1)).(let H15 \def (eq_ind C x (\lambda (c1:
124 C).(getl n a0 c1)) H9 (CHead x1 (Bind Abbr) x0) H12) in (let H16 \def (eq_ind
125 nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 (S (plus n (minus d0 (S
126 n)))) (lt_plus_minus n d0 H6)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
127 C).(eq T x0 (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0:
128 C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
129 C).(drop (S O) (minus d0 (S n)) x1 e0))) (ex3_2 T T (\lambda (y1: T).(\lambda
130 (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
131 (y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
132 T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3:
133 C).(\lambda (H17: (eq T x0 (lift (S O) (minus d0 (S n)) x2))).(\lambda (H18:
134 (getl n a (CHead x3 (Bind Abbr) x2))).(\lambda (H19: (drop (S O) (minus d0 (S
135 n)) x1 x3)).(let H20 \def (eq_ind T x0 (\lambda (t0: T).(subst1 (minus d0 (S
136 n)) u0 u t0)) H13 (lift (S O) (minus d0 (S n)) x2) H17) in (let H21 \def (H2
137 e u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Abbr) u0) u
138 (minus d0 (S n)) H7) x1 H14 x3 H19) in (ex3_2_ind T T (\lambda (y1:
139 T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n))
140 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (minus d0 (S n)) u0 t (lift
141 (S O) (minus d0 (S n)) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x3 y1
142 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n)
143 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S
144 n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
145 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H22: (subst1 (minus d0 (S
146 n)) u0 u (lift (S O) (minus d0 (S n)) x4))).(\lambda (H23: (subst1 (minus d0
147 (S n)) u0 t (lift (S O) (minus d0 (S n)) x5))).(\lambda (H24: (ty3 g x3 x4
148 x5)).(let H25 \def (eq_ind T x4 (\lambda (t0: T).(ty3 g x3 t0 x5)) H24 x2
149 (subst1_confluence_lift u x4 u0 (minus d0 (S n)) H22 x2 H20)) in (eq_ind_r
150 nat (plus (minus d0 (S n)) (S n)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
151 T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
152 T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) d0 y2))))
153 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (S
154 n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda
155 (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
156 (y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) (lift (S O)
157 n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro
158 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0
159 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n))
160 u0 (lift (S n) O t) (lift (S O) (plus (S n) (minus d0 (S n))) y2)))) (\lambda
161 (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x5)
162 (eq_ind_r T (TLRef n) (\lambda (t0: T).(subst1 d0 u0 (TLRef n) t0))
163 (subst1_refl d0 u0 (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O)
164 d0 H6)) (eq_ind_r T (lift (S n) O (lift (S O) (minus d0 (S n)) x5)) (\lambda
165 (t0: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) t0))
166 (subst1_lift_ge t (lift (S O) (minus d0 (S n)) x5) u0 (minus d0 (S n)) (S n)
167 H23 O (le_O_n (minus d0 (S n)))) (lift (S O) (plus (S n) (minus d0 (S n)))
168 (lift (S n) O x5)) (lift_d x5 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus
169 d0 (S n))))) (ty3_abbr g n a x3 x2 H18 x5 H25)) d0 (le_plus_minus (S n) d0
170 H6)) d0 (le_plus_minus_sym (S n) d0 H6)))))))) H21)))))))) (getl_drop_conf_lt
171 Abbr a0 x1 x0 n H15 a (S O) (minus d0 (S n)) H16))))))))) H11))))))
172 (csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0)))) (\lambda
173 (H6: (eq nat n d0)).(let H7 \def (eq_ind_r nat d0 (\lambda (n0: nat).(drop (S
174 O) n0 a0 a)) H5 n H6) in (let H8 \def (eq_ind_r nat d0 (\lambda (n0:
175 nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let H9 \def (eq_ind_r nat d0
176 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr) u0))) H3 n H6) in (eq_ind
177 nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1
178 n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2:
179 T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1:
180 T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C (CHead d
181 (Bind Abbr) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Abbr) u0)
182 (getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in
183 (let H11 \def (f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda
184 (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1]))
185 (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind
186 Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in ((let H12 \def (f_equal C T
187 (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _)
188 \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u)
189 (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e
190 (Bind Abbr) u0) H9)) in (\lambda (H13: (eq C d e)).(let H14 \def (eq_ind_r T
191 u0 (\lambda (t0: T).(getl n c0 (CHead e (Bind Abbr) t0))) H10 u H12) in (let
192 H15 \def (eq_ind_r T u0 (\lambda (t0: T).(csubst1 n t0 c0 a0)) H8 u H12) in
193 (eq_ind T u (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
194 T).(subst1 n t0 (TLRef n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2:
195 T).(subst1 n t0 (lift (S n) O t) (lift (S O) n y2)))) (\lambda (y1:
196 T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (eq_ind_r C e (\lambda
197 (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u))) H14 d H13) in (ex3_2_intro T T
198 (\lambda (y1: T).(\lambda (_: T).(subst1 n u (TLRef n) (lift (S O) n y1))))
199 (\lambda (_: T).(\lambda (y2: T).(subst1 n u (lift (S n) O t) (lift (S O) n
200 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (lift n O u) (lift
201 n O t) (subst1_single n u (TLRef n) (lift (S O) n (lift n O u)) (eq_ind_r T
202 (lift (plus (S O) n) O u) (\lambda (t0: T).(subst0 n u (TLRef n) t0))
203 (subst0_lref u n) (lift (S O) n (lift n O u)) (lift_free u n (S O) O n (le_n
204 (plus O n)) (le_O_n n)))) (eq_ind_r T (lift (plus (S O) n) O t) (\lambda (t0:
205 T).(subst1 n u (lift (S n) O t) t0)) (subst1_refl n u (lift (S n) O t)) (lift
206 (S O) n (lift n O t)) (lift_free t n (S O) O n (le_n (plus O n)) (le_O_n n)))
207 (ty3_lift g d u t H1 a O n (getl_conf_ge_drop Abbr a0 d u n (csubst1_getl_ge
208 n n (le_n n) c0 a0 u H15 (CHead d (Bind Abbr) u) H16) a H7)))) u0 H12)))))
209 H11))) d0 H6))))) (\lambda (H6: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n
210 (S O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
211 T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
212 (y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
213 T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat (plus (S O) (minus n (S
214 O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1
215 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
216 T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
217 T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S O))
218 (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1
219 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
220 T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
221 T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1:
222 T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) (lift
223 (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O
224 t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
225 (TLRef (minus n (S O))) (lift n O t) (eq_ind_r T (TLRef (plus (minus n (S O))
226 (S O))) (\lambda (t0: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O)))
227 t0)) (subst1_refl d0 u0 (TLRef (plus (minus n (S O)) (S O)))) (lift (S O) d0
228 (TLRef (minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus
229 d0 n H6))) (eq_ind_r T (lift (plus (S O) n) O t) (\lambda (t0: T).(subst1 d0
230 u0 (lift (S n) O t) t0)) (subst1_refl d0 u0 (lift (S n) O t)) (lift (S O) d0
231 (lift n O t)) (lift_free t n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0)
232 (plus O n) H6)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0:
233 nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t))) (ty3_abbr g (minus n (S
234 O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr) u) a0 (csubst1_getl_ge d0
235 n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0) a
236 (S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6
237 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0
238 n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_sym (S O) (minus n (S
239 O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S
240 O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0)
241 H6))))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
242 C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst)
243 u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e:
244 C).(\forall (u0: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Abbr) u0))
245 \to (\forall (a0: C).((csubst1 d0 u0 d a0) \to (\forall (a: C).((drop (S O)
246 d0 a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 u
247 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 t (lift
248 (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
249 y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda
250 (H3: (getl d0 c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4:
251 (csubst1 d0 u0 c0 a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0
252 a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0
253 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1
254 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2:
255 T).(ty3 g a y1 y2)))) (\lambda (H6: (lt n d0)).(let H7 \def (eq_ind nat
256 (minus d0 n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abst) u) (CHead e
257 (Bind Abbr) u0))) (getl_conf_le d0 (CHead e (Bind Abbr) u0) c0 H3 (CHead d
258 (Bind Abst) u) n H0 (le_S_n n d0 (le_S (S n) d0 H6))) (S (minus d0 (S n)))
259 (minus_x_Sy d0 n H6)) in (ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 n) u0
260 (CHead d (Bind Abst) u) e2)) (\lambda (e2: C).(getl n a0 e2)) (ex3_2 T T
261 (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0
262 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift
263 (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda
264 (x: C).(\lambda (H8: (csubst1 (minus d0 n) u0 (CHead d (Bind Abst) u)
265 x)).(\lambda (H9: (getl n a0 x)).(let H10 \def (eq_ind nat (minus d0 n)
266 (\lambda (n0: nat).(csubst1 n0 u0 (CHead d (Bind Abst) u) x)) H8 (S (minus d0
267 (S n))) (minus_x_Sy d0 n H6)) in (let H11 \def (csubst1_gen_head (Bind Abst)
268 d x u u0 (minus d0 (S n)) H10) in (ex3_2_ind T C (\lambda (u2: T).(\lambda
269 (c2: C).(eq C x (CHead c2 (Bind Abst) u2)))) (\lambda (u2: T).(\lambda (_:
270 C).(subst1 (minus d0 (S n)) u0 u u2))) (\lambda (_: T).(\lambda (c2:
271 C).(csubst1 (minus d0 (S n)) u0 d c2))) (ex3_2 T T (\lambda (y1: T).(\lambda
272 (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
273 (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1:
274 T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1:
275 C).(\lambda (H12: (eq C x (CHead x1 (Bind Abst) x0))).(\lambda (H13: (subst1
276 (minus d0 (S n)) u0 u x0)).(\lambda (H14: (csubst1 (minus d0 (S n)) u0 d
277 x1)).(let H15 \def (eq_ind C x (\lambda (c1: C).(getl n a0 c1)) H9 (CHead x1
278 (Bind Abst) x0) H12) in (let H16 \def (eq_ind nat d0 (\lambda (n0: nat).(drop
279 (S O) n0 a0 a)) H5 (S (plus n (minus d0 (S n)))) (lt_plus_minus n d0 H6)) in
280 (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T x0 (lift (S O) (minus d0
281 (S n)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl n a (CHead e0 (Bind Abst)
282 v)))) (\lambda (_: T).(\lambda (e0: C).(drop (S O) (minus d0 (S n)) x1 e0)))
283 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S
284 O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u)
285 (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
286 (\lambda (x2: T).(\lambda (x3: C).(\lambda (H17: (eq T x0 (lift (S O) (minus
287 d0 (S n)) x2))).(\lambda (H18: (getl n a (CHead x3 (Bind Abst) x2))).(\lambda
288 (H19: (drop (S O) (minus d0 (S n)) x1 x3)).(let H20 \def (eq_ind T x0
289 (\lambda (t0: T).(subst1 (minus d0 (S n)) u0 u t0)) H13 (lift (S O) (minus d0
290 (S n)) x2) H17) in (let H21 \def (H2 e u0 (minus d0 (S n)) (getl_gen_S (Bind
291 Abst) d (CHead e (Bind Abbr) u0) u (minus d0 (S n)) H7) x1 H14 x3 H19) in
292 (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u
293 (lift (S O) (minus d0 (S n)) y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1
294 (minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n)) y2)))) (\lambda (y1:
295 T).(\lambda (y2: T).(ty3 g x3 y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda
296 (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
297 (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1:
298 T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5:
299 T).(\lambda (H22: (subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n))
300 x4))).(\lambda (_: (subst1 (minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n))
301 x5))).(\lambda (H24: (ty3 g x3 x4 x5)).(let H25 \def (eq_ind T x4 (\lambda
302 (t0: T).(ty3 g x3 t0 x5)) H24 x2 (subst1_confluence_lift u x4 u0 (minus d0 (S
303 n)) H22 x2 H20)) in (eq_ind_r nat (plus (minus d0 (S n)) (S n)) (\lambda (n0:
304 nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n)
305 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n0 u0 (lift (S
306 n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
307 y2))))) (eq_ind_r nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2
308 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0
309 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n))
310 u0 (lift (S n) O u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2:
311 T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_:
312 T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
313 (y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O u) (lift (S O)
314 (plus (S n) (minus d0 (S n))) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g
315 a y1 y2))) (TLRef n) (lift (S n) O x2) (eq_ind_r T (TLRef n) (\lambda (t0:
316 T).(subst1 d0 u0 (TLRef n) t0)) (subst1_refl d0 u0 (TLRef n)) (lift (S O) d0
317 (TLRef n)) (lift_lref_lt n (S O) d0 H6)) (eq_ind_r T (lift (S n) O (lift (S
318 O) (minus d0 (S n)) x2)) (\lambda (t0: T).(subst1 (plus (minus d0 (S n)) (S
319 n)) u0 (lift (S n) O u) t0)) (subst1_lift_ge u (lift (S O) (minus d0 (S n))
320 x2) u0 (minus d0 (S n)) (S n) H20 O (le_O_n (minus d0 (S n)))) (lift (S O)
321 (plus (S n) (minus d0 (S n))) (lift (S n) O x2)) (lift_d x2 (S O) (S n)
322 (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) (ty3_abst g n a x3 x2 H18 x5
323 H25)) d0 (le_plus_minus (S n) d0 H6)) d0 (le_plus_minus_sym (S n) d0
324 H6)))))))) H21)))))))) (getl_drop_conf_lt Abst a0 x1 x0 n H15 a (S O) (minus
325 d0 (S n)) H16))))))))) H11)))))) (csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead
326 d (Bind Abst) u) H0)))) (\lambda (H6: (eq nat n d0)).(let H7 \def (eq_ind_r
327 nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 n H6) in (let H8 \def
328 (eq_ind_r nat d0 (\lambda (n0: nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let
329 H9 \def (eq_ind_r nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr)
330 u0))) H3 n H6) in (eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
331 T).(\lambda (_: T).(subst1 n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_:
332 T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O u) (lift (S O) n0 y2))))
333 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C
334 (CHead d (Bind Abst) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind
335 Abbr) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0)
336 H9)) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee:
337 C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
338 False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop)
339 with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with
340 [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) |
341 (Flat _) \Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c0
342 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind
343 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 n u0 (TLRef n) (lift (S
344 O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n u0 (lift (S n) O u)
345 (lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
346 H11))) d0 H6))))) (\lambda (H6: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n
347 (S O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
348 T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
349 (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1:
350 T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat (plus (S O) (minus n (S
351 O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1
352 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
353 T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1:
354 T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S O))
355 (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1
356 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
357 T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1:
358 T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1:
359 T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) (lift
360 (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O
361 u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
362 (TLRef (minus n (S O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O))
363 (S O))) (\lambda (t0: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O)))
364 t0)) (subst1_refl d0 u0 (TLRef (plus (minus n (S O)) (S O)))) (lift (S O) d0
365 (TLRef (minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus
366 d0 n H6))) (eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t0: T).(subst1 d0
367 u0 (lift (S n) O u) t0)) (subst1_refl d0 u0 (lift (S n) O u)) (lift (S O) d0
368 (lift n O u)) (lift_free u n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0)
369 (plus O n) H6)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0:
370 nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O u))) (ty3_abst g (minus n (S
371 O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abst) u) a0 (csubst1_getl_ge d0
372 n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abst) u) H0) a
373 (S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6
374 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0
375 n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_sym (S O) (minus n (S
376 O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S
377 O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0)
378 H6))))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t:
379 T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: ((\forall (e: C).(\forall (u0:
380 T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0:
381 C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2
382 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) d y1))))
383 (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda
384 (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (b:
385 B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b)
386 u) t3 t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d:
387 nat).((getl d (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0)) \to (\forall
388 (a0: C).((csubst1 d u0 (CHead c0 (Bind b) u) a0) \to (\forall (a: C).((drop
389 (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 t3
390 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift
391 (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
392 y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda
393 (H4: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H5:
394 (csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S O) d a0 a)).(let
395 H7 \def (H1 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda
396 (_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
397 T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
398 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead
399 (Bind b) u t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1
400 d u0 (THead (Bind b) u t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
401 (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8:
402 (subst1 d u0 u (lift (S O) d x0))).(\lambda (_: (subst1 d u0 t (lift (S O) d
403 x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H3 e u0 (S d) (getl_head
404 (Bind b) d c0 (CHead e (Bind Abbr) u0) H4 u) (CHead a0 (Bind b) (lift (S O) d
405 x0)) (csubst1_bind b d u0 u (lift (S O) d x0) H8 c0 a0 H5) (CHead a (Bind b)
406 x0) (drop_skip_bind (S O) d a0 a H6 b x0)) in (ex3_2_ind T T (\lambda (y1:
407 T).(\lambda (_: T).(subst1 (S d) u0 t3 (lift (S O) (S d) y1)))) (\lambda (_:
408 T).(\lambda (y2: T).(subst1 (S d) u0 t4 (lift (S O) (S d) y2)))) (\lambda
409 (y1: T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T
410 (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S
411 O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u
412 t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
413 (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 (S d) u0 t3 (lift (S
414 O) (S d) x2))).(\lambda (H13: (subst1 (S d) u0 t4 (lift (S O) (S d)
415 x3))).(\lambda (H14: (ty3 g (CHead a (Bind b) x0) x2 x3)).(ex3_2_intro T T
416 (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S
417 O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u
418 t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
419 (THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (eq_ind_r T (THead (Bind b)
420 (lift (S O) d x0) (lift (S O) (S d) x2)) (\lambda (t0: T).(subst1 d u0 (THead
421 (Bind b) u t3) t0)) (subst1_head u0 u (lift (S O) d x0) d H8 (Bind b) t3
422 (lift (S O) (S d) x2) H12) (lift (S O) d (THead (Bind b) x0 x2)) (lift_bind b
423 x0 x2 (S O) d)) (eq_ind_r T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S
424 d) x3)) (\lambda (t0: T).(subst1 d u0 (THead (Bind b) u t4) t0)) (subst1_head
425 u0 u (lift (S O) d x0) d H8 (Bind b) t4 (lift (S O) (S d) x3) H13) (lift (S
426 O) d (THead (Bind b) x0 x3)) (lift_bind b x0 x3 (S O) d)) (ty3_bind g a x0 x1
427 H10 b x2 x3 H14))))))) H11))))))) H7)))))))))))))))))))) (\lambda (c0:
428 C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1:
429 ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e
430 (Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a:
431 C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
432 T).(subst1 d u0 w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
433 T).(subst1 d u0 u (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
434 g a y1 y2)))))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g
435 c0 v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0:
436 T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0:
437 C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2
438 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 v (lift (S O) d y1))))
439 (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind Abst) u t) (lift
440 (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
441 y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda
442 (H4: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H5:
443 (csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S O) d a0 a)).(let
444 H7 \def (H3 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda
445 (_: T).(subst1 d u0 v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
446 T).(subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d y2)))) (\lambda (y1:
447 T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda
448 (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_:
449 T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u
450 t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
451 (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (subst1 d u0 v (lift (S O) d
452 x0))).(\lambda (H9: (subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d
453 x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H1 e u0 d H4 a0 H5 a H6)
454 in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 w (lift (S O)
455 d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 u (lift (S O) d y2))))
456 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1:
457 T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1))))
458 (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead
459 (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
460 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 d u0 w
461 (lift (S O) d x2))).(\lambda (H13: (subst1 d u0 u (lift (S O) d
462 x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H_x \def (subst1_gen_head (Bind
463 Abst) u0 u t (lift (S O) d x1) d H9) in (let H15 \def H_x in (ex3_2_ind T T
464 (\lambda (u2: T).(\lambda (t3: T).(eq T (lift (S O) d x1) (THead (Bind Abst)
465 u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 d u0 u u2))) (\lambda (_:
466 T).(\lambda (t3: T).(subst1 (S d) u0 t t3))) (ex3_2 T T (\lambda (y1:
467 T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1))))
468 (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead
469 (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
470 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H16: (eq T (lift (S
471 O) d x1) (THead (Bind Abst) x4 x5))).(\lambda (H17: (subst1 d u0 u
472 x4)).(\lambda (H18: (subst1 (S d) u0 t x5)).(let H19 \def (sym_eq T (lift (S
473 O) d x1) (THead (Bind Abst) x4 x5) H16) in (ex3_2_ind T T (\lambda (y:
474 T).(\lambda (z: T).(eq T x1 (THead (Bind Abst) y z)))) (\lambda (y:
475 T).(\lambda (_: T).(eq T x4 (lift (S O) d y)))) (\lambda (_: T).(\lambda (z:
476 T).(eq T x5 (lift (S O) (S d) z)))) (ex3_2 T T (\lambda (y1: T).(\lambda (_:
477 T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_:
478 T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u
479 t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
480 (\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (eq T x1 (THead (Bind Abst)
481 x6 x7))).(\lambda (H21: (eq T x4 (lift (S O) d x6))).(\lambda (H22: (eq T x5
482 (lift (S O) (S d) x7))).(let H23 \def (eq_ind T x5 (\lambda (t0: T).(subst1
483 (S d) u0 t t0)) H18 (lift (S O) (S d) x7) H22) in (let H24 \def (eq_ind T x4
484 (\lambda (t0: T).(subst1 d u0 u t0)) H17 (lift (S O) d x6) H21) in (let H25
485 \def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H10 (THead (Bind Abst) x6
486 x7) H20) in (let H26 \def (eq_ind T x6 (\lambda (t0: T).(ty3 g a x0 (THead
487 (Bind Abst) t0 x7))) H25 x3 (subst1_confluence_lift u x6 u0 d H24 x3 H13)) in
488 (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat
489 Appl) w v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0
490 (THead (Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda
491 (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x2 x0) (THead
492 (Flat Appl) x2 (THead (Bind Abst) x3 x7)) (eq_ind_r T (THead (Flat Appl)
493 (lift (S O) d x2) (lift (S O) d x0)) (\lambda (t0: T).(subst1 d u0 (THead
494 (Flat Appl) w v) t0)) (subst1_head u0 w (lift (S O) d x2) d H12 (Flat Appl) v
495 (lift (S O) d x0) H8) (lift (S O) d (THead (Flat Appl) x2 x0)) (lift_flat
496 Appl x2 x0 (S O) d)) (eq_ind_r T (THead (Flat Appl) (lift (S O) d x2) (lift
497 (S O) d (THead (Bind Abst) x3 x7))) (\lambda (t0: T).(subst1 d u0 (THead
498 (Flat Appl) w (THead (Bind Abst) u t)) t0)) (subst1_head u0 w (lift (S O) d
499 x2) d H12 (Flat Appl) (THead (Bind Abst) u t) (lift (S O) d (THead (Bind
500 Abst) x3 x7)) (eq_ind_r T (THead (Bind Abst) (lift (S O) d x3) (lift (S O) (S
501 d) x7)) (\lambda (t0: T).(subst1 (s (Flat Appl) d) u0 (THead (Bind Abst) u t)
502 t0)) (subst1_head u0 u (lift (S O) d x3) (s (Flat Appl) d) H13 (Bind Abst) t
503 (lift (S O) (S d) x7) H23) (lift (S O) d (THead (Bind Abst) x3 x7))
504 (lift_bind Abst x3 x7 (S O) d))) (lift (S O) d (THead (Flat Appl) x2 (THead
505 (Bind Abst) x3 x7))) (lift_flat Appl x2 (THead (Bind Abst) x3 x7) (S O) d))
506 (ty3_appl g a x2 x3 H14 x0 x7 H26))))))))))) (lift_gen_bind Abst x4 x5 x1 (S
507 O) d H19)))))))) H15)))))))) H11))))))) H7))))))))))))))))))) (\lambda (c0:
508 C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda
509 (H1: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e
510 (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a:
511 C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
512 T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
513 T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
514 g a y1 y2)))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4
515 t0)).(\lambda (H3: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl
516 d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to
517 (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda
518 (_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
519 T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
520 g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d:
521 nat).(\lambda (H4: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0:
522 C).(\lambda (H5: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S
523 O) d a0 a)).(let H7 \def (H3 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda
524 (y1: T).(\lambda (_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_:
525 T).(\lambda (y2: T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1:
526 T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda
527 (_: T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda
528 (_: T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d
529 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0:
530 T).(\lambda (x1: T).(\lambda (H8: (subst1 d u t4 (lift (S O) d x0))).(\lambda
531 (H9: (subst1 d u t0 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let
532 H11 \def (H1 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda
533 (_: T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
534 T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
535 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (THead
536 (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
537 T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d y2)))) (\lambda (y1:
538 T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3:
539 T).(\lambda (H12: (subst1 d u t3 (lift (S O) d x2))).(\lambda (H13: (subst1 d
540 u t4 (lift (S O) d x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H15 \def
541 (eq_ind T x3 (\lambda (t: T).(ty3 g a x2 t)) H14 x0 (subst1_confluence_lift
542 t4 x3 u d H13 x0 H8)) in (ex3_2_intro T T (\lambda (y1: T).(\lambda (_:
543 T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_:
544 T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d
545 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast)
546 x0 x2) (THead (Flat Cast) x1 x0) (eq_ind_r T (THead (Flat Cast) (lift (S O) d
547 x0) (lift (S O) d x2)) (\lambda (t: T).(subst1 d u (THead (Flat Cast) t4 t3)
548 t)) (subst1_head u t4 (lift (S O) d x0) d H8 (Flat Cast) t3 (lift (S O) d x2)
549 H12) (lift (S O) d (THead (Flat Cast) x0 x2)) (lift_flat Cast x0 x2 (S O) d))
550 (eq_ind_r T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (\lambda
551 (t: T).(subst1 d u (THead (Flat Cast) t0 t4) t)) (subst1_head u t0 (lift (S
552 O) d x1) d H9 (Flat Cast) t4 (lift (S O) d x0) H8) (lift (S O) d (THead (Flat
553 Cast) x1 x0)) (lift_flat Cast x1 x0 (S O) d)) (ty3_cast g a x2 x0 H15 x1
554 H10)))))))) H11))))))) H7)))))))))))))))))) c t1 t2 H))))).
559 theorem ty3_gen_cvoid:
560 \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
561 t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c
562 (CHead e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c a) \to (ex3_2 T
563 T (\lambda (y1: T).(\lambda (_: T).(eq T t1 (lift (S O) d y1)))) (\lambda (_:
564 T).(\lambda (y2: T).(eq T t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
565 (y2: T).(ty3 g a y1 y2))))))))))))))
567 \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
568 (H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda
569 (t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead
570 e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T
571 (\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) (\lambda (_:
572 T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
573 (y2: T).(ty3 g a y1 y2))))))))))))) (\lambda (c0: C).(\lambda (t3:
574 T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e:
575 C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to
576 (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda
577 (_: T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t
578 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
579 y2)))))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 u
580 t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl
581 d c0 (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to
582 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1))))
583 (\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1:
584 T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (H4: (pc3 c0 t4
585 t3)).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H5: (getl d
586 c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H6: (drop (S O) d c0
587 a)).(let H7 \def (H3 e u0 d H5 a H6) in (ex3_2_ind T T (\lambda (y1:
588 T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
589 T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
590 y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d
591 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2))))
592 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0:
593 T).(\lambda (x1: T).(\lambda (H8: (eq T u (lift (S O) d x0))).(\lambda (H9:
594 (eq T t4 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def
595 (eq_ind T t4 (\lambda (t0: T).(pc3 c0 t0 t3)) H4 (lift (S O) d x1) H9) in
596 (let H12 \def (eq_ind T t4 (\lambda (t0: T).(ty3 g c0 u t0)) H2 (lift (S O) d
597 x1) H9) in (let H13 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S
598 O) d x1))) H12 (lift (S O) d x0) H8) in (eq_ind_r T (lift (S O) d x0)
599 (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift
600 (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2))))
601 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H1 e u0
602 d H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift
603 (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d y2))))
604 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1:
605 T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1)))) (\lambda (_:
606 T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
607 (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15:
608 (eq T t3 (lift (S O) d x2))).(\lambda (H16: (eq T t (lift (S O) d
609 x3))).(\lambda (H17: (ty3 g a x2 x3)).(let H18 \def (eq_ind T t (\lambda (t0:
610 T).(ty3 g c0 t3 t0)) H0 (lift (S O) d x3) H16) in (let H19 \def (eq_ind T t3
611 (\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x3))) H18 (lift (S O) d x2) H15)
612 in (let H20 \def (eq_ind T t3 (\lambda (t0: T).(pc3 c0 (lift (S O) d x1) t0))
613 H11 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda (t0:
614 T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift
615 (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2))))
616 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T
617 (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1))))
618 (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2))))
619 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 (refl_equal T (lift
620 (S O) d x0)) (refl_equal T (lift (S O) d x2)) (ty3_conv g a x2 x3 H17 x0 x1
621 H10 (pc3_gen_lift c0 x1 x2 (S O) d H20 a H6))) t3 H15))))))))) H14)) u
622 H8))))))))) H7)))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda
623 (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (_: (getl d c0 (CHead e
624 (Bind Void) u))).(\lambda (a: C).(\lambda (_: (drop (S O) d c0
625 a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TSort m) (lift
626 (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (TSort (next g m))
627 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
628 (TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t: T).(eq T
629 (TSort m) t)) (refl_equal T (TSort m)) (lift (S O) d (TSort m)) (lift_sort m
630 (S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(eq T (TSort (next g
631 m)) t)) (refl_equal T (TSort (next g m))) (lift (S O) d (TSort (next g m)))
632 (lift_sort (next g m) (S O) d)) (ty3_sort g a m)))))))))) (\lambda (n:
633 nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n
634 c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u
635 t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: T).(\forall (d0: nat).((getl
636 d0 d (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d0 d a) \to
637 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d0 y1))))
638 (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1:
639 T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0:
640 T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void)
641 u0))).(\lambda (a: C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0
642 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0
643 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0
644 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt
645 n d0)).(let H6 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0
646 (CHead d (Bind Abbr) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e
647 (Bind Void) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 (le_S_n n d0 (le_S (S n)
648 d0 H5))) (S (minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind
649 nat d0 (\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 (S (plus n (minus d0 (S
650 n)))) (lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
651 C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0:
652 C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
653 C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda (y1: T).(\lambda
654 (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
655 T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda
656 (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H8:
657 (eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9: (getl n a (CHead x1
658 (Bind Abbr) x0))).(\lambda (H10: (drop (S O) (minus d0 (S n)) d x1)).(let H11
659 \def (eq_ind T u (\lambda (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall
660 (d1: nat).((getl d1 d (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop
661 (S O) d1 d a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift
662 (S O) d1 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d1
663 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift
664 (S O) (minus d0 (S n)) x0) H8) in (let H12 \def (eq_ind T u (\lambda (t0:
665 T).(ty3 g d t0 t)) H1 (lift (S O) (minus d0 (S n)) x0) H8) in (let H13 \def
666 (H11 e u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Void) u0)
667 u (minus d0 (S n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda
668 (_: T).(eq T (lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n))
669 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n))
670 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T
671 (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1))))
672 (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2))))
673 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2:
674 T).(\lambda (x3: T).(\lambda (H14: (eq T (lift (S O) (minus d0 (S n)) x0)
675 (lift (S O) (minus d0 (S n)) x2))).(\lambda (H15: (eq T t (lift (S O) (minus
676 d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let H17 \def (eq_ind T t
677 (\lambda (t0: T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t0)) H12 (lift (S
678 O) (minus d0 (S n)) x3) H15) in (eq_ind_r T (lift (S O) (minus d0 (S n)) x3)
679 (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n)
680 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O
681 t0) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
682 y2))))) (let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(ty3 g x1 t0 x3)) H16
683 x0 (lift_inj x0 x2 (S O) (minus d0 (S n)) H14)) in (eq_ind T (lift (S O)
684 (plus (S n) (minus d0 (S n))) (lift (S n) O x3)) (\lambda (t0: T).(ex3_2 T T
685 (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1))))
686 (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1:
687 T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat d0 (\lambda (n0:
688 nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O)
689 d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) n0 (lift (S n) O
690 x3)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
691 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n)
692 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d0
693 (lift (S n) O x3)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2:
694 T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x3) (eq_ind_r T (TLRef n)
695 (\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d0
696 (TLRef n)) (lift_lref_lt n (S O) d0 H5)) (refl_equal T (lift (S O) d0 (lift
697 (S n) O x3))) (ty3_abbr g n a x1 x0 H9 x3 H18)) (plus (S n) (minus d0 (S n)))
698 (le_plus_minus (S n) d0 H5)) (lift (S n) O (lift (S O) (minus d0 (S n)) x3))
699 (lift_d x3 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n)))))) t
700 H15))))))) H13))))))))) (getl_drop_conf_lt Abbr c0 d u n H0 a (S O) (minus d0
701 (S n)) H7))))) (\lambda (H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0
702 (\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r
703 nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in
704 (eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
705 T).(eq T (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq
706 T (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2:
707 T).(ty3 g a y1 y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abbr) u)
708 (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0
709 (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def
710 (eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return
711 (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
712 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
713 \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
714 True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
715 \Rightarrow False])])) I (CHead e (Bind Void) u0) (getl_mono c0 (CHead d
716 (Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (False_ind (ex3_2 T T
717 (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) n y1))))
718 (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) n y2))))
719 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) H9))) d0 H5)))) (\lambda
720 (H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S O)))) (\lambda (n0:
721 nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O)
722 d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O)
723 d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat
724 (plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
725 T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_:
726 T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda
727 (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S
728 O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq
729 T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T
730 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
731 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef
732 (plus (minus n (S O)) (S O))) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
733 (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
734 T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S O))) (lift n O t)
735 (eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda (t0: T).(eq T
736 (TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef (plus (minus n
737 (S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) (lift_lref_ge (minus
738 n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T (lift (plus (S O) n) O
739 t) (\lambda (t0: T).(eq T (lift (S n) O t) t0)) (refl_equal T (lift (S n) O
740 t)) (lift (S O) d0 (lift n O t)) (lift_free t n (S O) O d0 (le_S_n d0 (plus O
741 n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S
742 O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t)))
743 (ty3_abbr g (minus n (S O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr)
744 u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le
745 n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n
746 (le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n (S O))) (plus_sym
747 (S O) (minus n (S O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus
748 O (minus n (S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0)
749 H5))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
750 C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst)
751 u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e:
752 C).(\forall (u0: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Void) u0))
753 \to (\forall (a: C).((drop (S O) d0 d a) \to (ex3_2 T T (\lambda (y1:
754 T).(\lambda (_: T).(eq T u (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
755 (y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
756 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d0:
757 nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void) u0))).(\lambda (a:
758 C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda
759 (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
760 T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda
761 (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt n d0)).(let H6
762 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 (CHead d (Bind
763 Abst) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e (Bind Void) u0)
764 c0 H3 (CHead d (Bind Abst) u) n H0 (le_S_n n d0 (le_S (S n) d0 H5))) (S
765 (minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind nat d0
766 (\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 (S (plus n (minus d0 (S n))))
767 (lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
768 C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0:
769 C).(getl n a (CHead e0 (Bind Abst) v)))) (\lambda (_: T).(\lambda (e0:
770 C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda (y1: T).(\lambda
771 (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
772 T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda
773 (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H8:
774 (eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9: (getl n a (CHead x1
775 (Bind Abst) x0))).(\lambda (H10: (drop (S O) (minus d0 (S n)) d x1)).(let H11
776 \def (eq_ind T u (\lambda (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall
777 (d1: nat).((getl d1 d (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop
778 (S O) d1 d a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift
779 (S O) d1 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d1
780 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift
781 (S O) (minus d0 (S n)) x0) H8) in (let H12 \def (eq_ind T u (\lambda (t0:
782 T).(ty3 g d t0 t)) H1 (lift (S O) (minus d0 (S n)) x0) H8) in (eq_ind_r T
783 (lift (S O) (minus d0 (S n)) x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
784 T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
785 T).(\lambda (y2: T).(eq T (lift (S n) O t0) (lift (S O) d0 y2)))) (\lambda
786 (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H13 \def (H11 e u0 (minus
787 d0 (S n)) (getl_gen_S (Bind Abst) d (CHead e (Bind Void) u0) u (minus d0 (S
788 n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T
789 (lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) y1)))) (\lambda
790 (_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n)) y2)))) (\lambda
791 (y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T (\lambda (y1:
792 T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
793 T).(\lambda (y2: T).(eq T (lift (S n) O (lift (S O) (minus d0 (S n)) x0))
794 (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
795 (\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T (lift (S O) (minus d0
796 (S n)) x0) (lift (S O) (minus d0 (S n)) x2))).(\lambda (H15: (eq T t (lift (S
797 O) (minus d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let H17 \def
798 (eq_ind T t (\lambda (t0: T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t0))
799 H12 (lift (S O) (minus d0 (S n)) x3) H15) in (let H18 \def (eq_ind_r T x2
800 (\lambda (t0: T).(ty3 g x1 t0 x3)) H16 x0 (lift_inj x0 x2 (S O) (minus d0 (S
801 n)) H14)) in (eq_ind T (lift (S O) (plus (S n) (minus d0 (S n))) (lift (S n)
802 O x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
803 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0
804 (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))
805 (eq_ind nat d0 (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
806 T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq
807 T (lift (S O) n0 (lift (S n) O x0)) (lift (S O) d0 y2)))) (\lambda (y1:
808 T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1:
809 T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
810 T).(\lambda (y2: T).(eq T (lift (S O) d0 (lift (S n) O x0)) (lift (S O) d0
811 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S
812 n) O x0) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0))
813 (refl_equal T (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O) d0
814 H5)) (refl_equal T (lift (S O) d0 (lift (S n) O x0))) (ty3_abst g n a x1 x0
815 H9 x3 H18)) (plus (S n) (minus d0 (S n))) (le_plus_minus (S n) d0 H5)) (lift
816 (S n) O (lift (S O) (minus d0 (S n)) x0)) (lift_d x0 (S O) (S n) (minus d0 (S
817 n)) O (le_O_n (minus d0 (S n)))))))))))) H13)) u H8))))))))
818 (getl_drop_conf_lt Abst c0 d u n H0 a (S O) (minus d0 (S n)) H7))))) (\lambda
819 (H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0 (\lambda (n0: nat).(drop (S
820 O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r nat d0 (\lambda (n0:
821 nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in (eq_ind nat n
822 (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef
823 n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O
824 u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
825 y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl
826 n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n
827 H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind
828 Abst) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with
829 [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return
830 (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
831 (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True |
832 Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e (Bind
833 Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Void) u0)
834 H7)) in (False_ind (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef
835 n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O
836 u) (lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
837 H9))) d0 H5)))) (\lambda (H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S
838 O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
839 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift
840 (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
841 y1 y2))))) (eq_ind nat (plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2
842 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1))))
843 (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2))))
844 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus
845 (minus n (S O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
846 T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_:
847 T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda
848 (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1:
849 T).(\lambda (_: T).(eq T (TLRef (plus (minus n (S O)) (S O))) (lift (S O) d0
850 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0
851 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S
852 O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda
853 (t0: T).(eq T (TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef
854 (plus (minus n (S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O))))
855 (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T
856 (lift (plus (S O) n) O u) (\lambda (t0: T).(eq T (lift (S n) O u) t0))
857 (refl_equal T (lift (S n) O u)) (lift (S O) d0 (lift n O u)) (lift_free u n
858 (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) (plus O n) H5)) (le_O_n d0)))
859 (eq_ind_r nat (S (minus n (S O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n
860 (S O))) (lift n0 O u))) (ty3_abst g (minus n (S O)) a d u (getl_drop_conf_ge
861 n (CHead d (Bind Abst) u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0)
862 (\lambda (n0: nat).(le n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1)
863 n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n
864 (S O))) (plus_sym (S O) (minus n (S O)))) (S (plus O (minus n (S O))))
865 (refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n
866 (le_lt_trans O d0 n (le_O_n d0) H5))))))))))))))))))) (\lambda (c0:
867 C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda
868 (H1: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e
869 (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T
870 (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_:
871 T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
872 (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (b: B).(\lambda (t3: T).(\lambda
873 (t4: T).(\lambda (H2: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H3:
874 ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d (CHead c0 (Bind
875 b) u) (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d (CHead c0
876 (Bind b) u) a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift
877 (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2))))
878 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e:
879 C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind
880 Void) u0))).(\lambda (a: C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def
881 (H1 e u0 d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T
882 u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d
883 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T
884 (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind b) u t3) (lift (S O) d
885 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) u t4) (lift (S
886 O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda
887 (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T u (lift (S O) d x0))).(\lambda
888 (H8: (eq T t (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def
889 (eq_ind T t (\lambda (t0: T).(ty3 g c0 u t0)) H0 (lift (S O) d x1) H8) in
890 (let H11 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x1)))
891 H10 (lift (S O) d x0) H7) in (let H12 \def (eq_ind T u (\lambda (t0:
892 T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 (CHead c0
893 (Bind b) t0) (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0
894 (CHead c0 (Bind b) t0) a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
895 T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4
896 (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1
897 y2))))))))))) H3 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T u (\lambda
898 (t0: T).(ty3 g (CHead c0 (Bind b) t0) t3 t4)) H2 (lift (S O) d x0) H7) in
899 (eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
900 T).(\lambda (_: T).(eq T (THead (Bind b) t0 t3) (lift (S O) d y1)))) (\lambda
901 (_: T).(\lambda (y2: T).(eq T (THead (Bind b) t0 t4) (lift (S O) d y2))))
902 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H12 e u0
903 (S d) (getl_head (Bind b) d c0 (CHead e (Bind Void) u0) H4 (lift (S O) d x0))
904 (CHead a (Bind b) x0) (drop_skip_bind (S O) d c0 a H5 b x0)) in (ex3_2_ind T
905 T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift (S O) (S d) y1)))) (\lambda
906 (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) (S d) y2)))) (\lambda (y1:
907 T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T (\lambda
908 (y1: T).(\lambda (_: T).(eq T (THead (Bind b) (lift (S O) d x0) t3) (lift (S
909 O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S
910 O) d x0) t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
911 y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S
912 O) (S d) x2))).(\lambda (H16: (eq T t4 (lift (S O) (S d) x3))).(\lambda (H17:
913 (ty3 g (CHead a (Bind b) x0) x2 x3)).(eq_ind_r T (lift (S O) (S d) x3)
914 (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead
915 (Bind b) (lift (S O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda
916 (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y2))))
917 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r T (lift (S O)
918 (S d) x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
919 (THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y1)))) (\lambda (_:
920 T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d)
921 x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
922 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind
923 b) (lift (S O) d x0) (lift (S O) (S d) x2)) (lift (S O) d y1)))) (\lambda (_:
924 T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d)
925 x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
926 (THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (sym_eq T (lift (S O) d (THead
927 (Bind b) x0 x2)) (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x2))
928 (lift_bind b x0 x2 (S O) d)) (sym_eq T (lift (S O) d (THead (Bind b) x0 x3))
929 (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x3)) (lift_bind b x0 x3
930 (S O) d)) (ty3_bind g a x0 x1 H9 b x2 x3 H17)) t3 H15) t4 H16)))))) H14)) u
931 H7)))))))))) H6)))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda
932 (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: ((\forall (e: C).(\forall
933 (u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall
934 (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
935 T).(eq T w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T u
936 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
937 y2)))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v
938 (THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0:
939 T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall (a:
940 C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
941 v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind
942 Abst) u t) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
943 y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda
944 (H4: (getl d c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H5:
945 (drop (S O) d c0 a)).(let H6 \def (H3 e u0 d H4 a H5) in (ex3_2_ind T T
946 (\lambda (y1: T).(\lambda (_: T).(eq T v (lift (S O) d y1)))) (\lambda (_:
947 T).(\lambda (y2: T).(eq T (THead (Bind Abst) u t) (lift (S O) d y2))))
948 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1:
949 T).(\lambda (_: T).(eq T (THead (Flat Appl) w v) (lift (S O) d y1))))
950 (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind
951 Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
952 y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T v (lift (S O)
953 d x0))).(\lambda (H8: (eq T (THead (Bind Abst) u t) (lift (S O) d
954 x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def (eq_ind T v (\lambda (t0:
955 T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H2 (lift (S O) d x0) H7) in
956 (eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
957 T).(\lambda (_: T).(eq T (THead (Flat Appl) w t0) (lift (S O) d y1))))
958 (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind
959 Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
960 y1 y2))))) (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x1 (THead
961 (Bind Abst) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift (S O) d
962 y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift (S O) (S d) z)))) (ex3_2
963 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d
964 x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat
965 Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1:
966 T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3:
967 T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x2 x3))).(\lambda (H12: (eq T u
968 (lift (S O) d x2))).(\lambda (H13: (eq T t (lift (S O) (S d) x3))).(let H14
969 \def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H9 (THead (Bind Abst) x2
970 x3) H11) in (eq_ind_r T (lift (S O) (S d) x3) (\lambda (t0: T).(ex3_2 T T
971 (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d
972 x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat
973 Appl) w (THead (Bind Abst) u t0)) (lift (S O) d y2)))) (\lambda (y1:
974 T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H15 \def (eq_ind T u (\lambda
975 (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 c0
976 (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to
977 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d0 y1))))
978 (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1:
979 T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H1 (lift (S O) d x2) H12) in
980 (eq_ind_r T (lift (S O) d x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
981 T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O)
982 d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead
983 (Bind Abst) t0 (lift (S O) (S d) x3))) (lift (S O) d y2)))) (\lambda (y1:
984 T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (H15 e u0 d H4 a H5) in
985 (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d y1))))
986 (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2))))
987 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1:
988 T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O)
989 d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead
990 (Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2))))
991 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4:
992 T).(\lambda (x5: T).(\lambda (H17: (eq T w (lift (S O) d x4))).(\lambda (H18:
993 (eq T (lift (S O) d x2) (lift (S O) d x5))).(\lambda (H19: (ty3 g a x4
994 x5)).(eq_ind_r T (lift (S O) d x4) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
995 T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (lift (S O) d x0)) (lift (S O)
996 d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) t0 (THead
997 (Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2))))
998 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H20 \def (eq_ind_r
999 T x5 (\lambda (t0: T).(ty3 g a x4 t0)) H19 x2 (lift_inj x2 x5 (S O) d H18))
1000 in (eq_ind T (lift (S O) d (THead (Bind Abst) x2 x3)) (\lambda (t0: T).(ex3_2
1001 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) (lift (S O) d
1002 x4) (lift (S O) d x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
1003 T).(eq T (THead (Flat Appl) (lift (S O) d x4) t0) (lift (S O) d y2))))
1004 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d
1005 (THead (Flat Appl) x4 x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
1006 T).(\lambda (_: T).(eq T t0 (lift (S O) d y1)))) (\lambda (_: T).(\lambda
1007 (y2: T).(eq T (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d (THead (Bind
1008 Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g
1009 a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst)
1010 x2 x3))) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
1011 (lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1)))) (\lambda (_:
1012 T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
1013 (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_:
1014 T).(eq T (lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1))))
1015 (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d (THead (Flat Appl) x4
1016 (THead (Bind Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
1017 (y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x4 x0) (THead (Flat Appl) x4
1018 (THead (Bind Abst) x2 x3)) (refl_equal T (lift (S O) d (THead (Flat Appl) x4
1019 x0))) (refl_equal T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) x2
1020 x3)))) (ty3_appl g a x4 x2 H20 x0 x3 H14)) (THead (Flat Appl) (lift (S O) d
1021 x4) (lift (S O) d (THead (Bind Abst) x2 x3))) (lift_flat Appl x4 (THead (Bind
1022 Abst) x2 x3) (S O) d)) (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d
1023 x0)) (lift_flat Appl x4 x0 (S O) d)) (THead (Bind Abst) (lift (S O) d x2)
1024 (lift (S O) (S d) x3)) (lift_bind Abst x2 x3 (S O) d))) w H17)))))) H16)) u
1025 H12)) t H13))))))) (lift_gen_bind Abst u t x1 (S O) d H8)) v H7)))))))
1026 H6))))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4:
1027 T).(\lambda (H0: (ty3 g c0 t3 t4)).(\lambda (H1: ((\forall (e: C).(\forall
1028 (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to (\forall
1029 (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
1030 T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4
1031 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
1032 y2)))))))))))).(\lambda (t0: T).(\lambda (H2: (ty3 g c0 t4 t0)).(\lambda (H3:
1033 ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind
1034 Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda
1035 (y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) (\lambda (_:
1036 T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
1037 (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda
1038 (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind Void) u))).(\lambda (a:
1039 C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def (H3 e u d H4 a H5) in
1040 (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1))))
1041 (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1:
1042 T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda
1043 (_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_:
1044 T).(\lambda (y2: T).(eq T (THead (Flat Cast) t0 t4) (lift (S O) d y2))))
1045 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0:
1046 T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (lift (S O) d x0))).(\lambda (H8:
1047 (eq T t0 (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def
1048 (eq_ind T t0 (\lambda (t: T).(ty3 g c0 t4 t)) H2 (lift (S O) d x1) H8) in
1049 (eq_ind_r T (lift (S O) d x1) (\lambda (t: T).(ex3_2 T T (\lambda (y1:
1050 T).(\lambda (_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1))))
1051 (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) t t4) (lift (S O) d
1052 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H11 \def
1053 (eq_ind T t4 (\lambda (t: T).(ty3 g c0 t (lift (S O) d x1))) H10 (lift (S O)
1054 d x0) H7) in (let H12 \def (eq_ind T t4 (\lambda (t: T).(\forall (e0:
1055 C).(\forall (u0: T).(\forall (d0: nat).((getl d0 c0 (CHead e0 (Bind Void)
1056 u0)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to (ex3_2 T T (\lambda (y1:
1057 T).(\lambda (_: T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
1058 (y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
1059 g a0 y1 y2))))))))))) H1 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T t4
1060 (\lambda (t: T).(ty3 g c0 t3 t)) H0 (lift (S O) d x0) H7) in (eq_ind_r T
1061 (lift (S O) d x0) (\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
1062 T).(eq T (THead (Flat Cast) t t3) (lift (S O) d y1)))) (\lambda (_:
1063 T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1) t) (lift (S O)
1064 d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def
1065 (H12 e u d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T
1066 t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d
1067 x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
1068 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) (lift (S
1069 O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T
1070 (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O) d y2))))
1071 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2:
1072 T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S O) d x2))).(\lambda
1073 (H16: (eq T (lift (S O) d x0) (lift (S O) d x3))).(\lambda (H17: (ty3 g a x2
1074 x3)).(let H18 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t (lift (S O) d
1075 x0))) H13 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda
1076 (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast)
1077 (lift (S O) d x0) t) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
1078 T).(eq T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O)
1079 d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H19 \def
1080 (eq_ind_r T x3 (\lambda (t: T).(ty3 g a x2 t)) H17 x0 (lift_inj x0 x3 (S O) d
1081 H16)) in (eq_ind T (lift (S O) d (THead (Flat Cast) x0 x2)) (\lambda (t:
1082 T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1))))
1083 (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1)
1084 (lift (S O) d x0)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2:
1085 T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Cast) x1 x0))
1086 (\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O)
1087 d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda
1088 (y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g
1089 a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S
1090 O) d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda
1091 (y2: T).(eq T (lift (S O) d (THead (Flat Cast) x1 x0)) (lift (S O) d y2))))
1092 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) x0 x2)
1093 (THead (Flat Cast) x1 x0) (refl_equal T (lift (S O) d (THead (Flat Cast) x0
1094 x2))) (refl_equal T (lift (S O) d (THead (Flat Cast) x1 x0))) (ty3_cast g a
1095 x2 x0 H19 x1 H9)) (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0))
1096 (lift_flat Cast x1 x0 (S O) d)) (THead (Flat Cast) (lift (S O) d x0) (lift (S
1097 O) d x2)) (lift_flat Cast x0 x2 (S O) d))) t3 H15))))))) H14)) t4 H7)))) t0
1098 H8))))))) H6)))))))))))))))) c t1 t2 H))))).
1100 Initial nodes: 13105