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