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 "LambdaDelta-1/ty3/props.ma".
19 include "LambdaDelta-1/pc3/subst1.ma".
21 include "LambdaDelta-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)).(ex3_2_ind T T (\lambda (u2:
463 T).(\lambda (t3: T).(eq T (lift (S O) d x1) (THead (Bind Abst) u2 t3))))
464 (\lambda (u2: T).(\lambda (_: T).(subst1 d u0 u u2))) (\lambda (_:
465 T).(\lambda (t3: T).(subst1 (s (Bind Abst) d) u0 t t3))) (ex3_2 T T (\lambda
466 (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d
467 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w
468 (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2:
469 T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H15: (eq T
470 (lift (S O) d x1) (THead (Bind Abst) x4 x5))).(\lambda (H16: (subst1 d u0 u
471 x4)).(\lambda (H17: (subst1 (s (Bind Abst) d) u0 t x5)).(let H18 \def (sym_eq
472 T (lift (S O) d x1) (THead (Bind Abst) x4 x5) H15) in (ex3_2_ind T T (\lambda
473 (y: T).(\lambda (z: T).(eq T x1 (THead (Bind Abst) y z)))) (\lambda (y:
474 T).(\lambda (_: T).(eq T x4 (lift (S O) d y)))) (\lambda (_: T).(\lambda (z:
475 T).(eq T x5 (lift (S O) (S d) z)))) (ex3_2 T T (\lambda (y1: T).(\lambda (_:
476 T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_:
477 T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u
478 t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
479 (\lambda (x6: T).(\lambda (x7: T).(\lambda (H19: (eq T x1 (THead (Bind Abst)
480 x6 x7))).(\lambda (H20: (eq T x4 (lift (S O) d x6))).(\lambda (H21: (eq T x5
481 (lift (S O) (S d) x7))).(let H22 \def (eq_ind T x5 (\lambda (t0: T).(subst1
482 (s (Bind Abst) d) u0 t t0)) H17 (lift (S O) (S d) x7) H21) in (let H23 \def
483 (eq_ind T x4 (\lambda (t0: T).(subst1 d u0 u t0)) H16 (lift (S O) d x6) H20)
484 in (let H24 \def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H10 (THead
485 (Bind Abst) x6 x7) H19) in (let H25 \def (eq_ind T x6 (\lambda (t0: T).(ty3 g
486 a x0 (THead (Bind Abst) t0 x7))) H24 x3 (subst1_confluence_lift u x6 u0 d H23
487 x3 H13)) in (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0
488 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
489 T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d
490 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Appl)
491 x2 x0) (THead (Flat Appl) x2 (THead (Bind Abst) x3 x7)) (eq_ind_r T (THead
492 (Flat Appl) (lift (S O) d x2) (lift (S O) d x0)) (\lambda (t0: T).(subst1 d
493 u0 (THead (Flat Appl) w v) t0)) (subst1_head u0 w (lift (S O) d x2) d H12
494 (Flat Appl) v (lift (S O) d x0) H8) (lift (S O) d (THead (Flat Appl) x2 x0))
495 (lift_flat Appl x2 x0 (S O) d)) (eq_ind_r T (THead (Flat Appl) (lift (S O) d
496 x2) (lift (S O) d (THead (Bind Abst) x3 x7))) (\lambda (t0: T).(subst1 d u0
497 (THead (Flat Appl) w (THead (Bind Abst) u t)) t0)) (subst1_head u0 w (lift (S
498 O) d x2) d H12 (Flat Appl) (THead (Bind Abst) u t) (lift (S O) d (THead (Bind
499 Abst) x3 x7)) (eq_ind_r T (THead (Bind Abst) (lift (S O) d x3) (lift (S O) (S
500 d) x7)) (\lambda (t0: T).(subst1 (s (Flat Appl) d) u0 (THead (Bind Abst) u t)
501 t0)) (subst1_head u0 u (lift (S O) d x3) (s (Flat Appl) d) H13 (Bind Abst) t
502 (lift (S O) (S d) x7) H22) (lift (S O) d (THead (Bind Abst) x3 x7))
503 (lift_bind Abst x3 x7 (S O) d))) (lift (S O) d (THead (Flat Appl) x2 (THead
504 (Bind Abst) x3 x7))) (lift_flat Appl x2 (THead (Bind Abst) x3 x7) (S O) d))
505 (ty3_appl g a x2 x3 H14 x0 x7 H25))))))))))) (lift_gen_bind Abst x4 x5 x1 (S
506 O) d H18)))))))) (subst1_gen_head (Bind Abst) u0 u t (lift (S O) d x1) d
507 H9))))))) H11))))))) H7))))))))))))))))))) (\lambda (c0: C).(\lambda (t3:
508 T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: ((\forall
509 (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u))
510 \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d
511 a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t3 (lift (S
512 O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u t4 (lift (S O) d
513 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
514 y2)))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda
515 (H3: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e
516 (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a:
517 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))))).
556 theorem ty3_gen_cvoid:
557 \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
558 t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c
559 (CHead e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c a) \to (ex3_2 T
560 T (\lambda (y1: T).(\lambda (_: T).(eq T t1 (lift (S O) d y1)))) (\lambda (_:
561 T).(\lambda (y2: T).(eq T t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
562 (y2: T).(ty3 g a y1 y2))))))))))))))
564 \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
565 (H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda
566 (t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead
567 e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T
568 (\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) (\lambda (_:
569 T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
570 (y2: T).(ty3 g a y1 y2))))))))))))) (\lambda (c0: C).(\lambda (t3:
571 T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e:
572 C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to
573 (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda
574 (_: T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t
575 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
576 y2)))))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 u
577 t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl
578 d c0 (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to
579 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1))))
580 (\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1:
581 T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (H4: (pc3 c0 t4
582 t3)).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H5: (getl d
583 c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H6: (drop (S O) d c0
584 a)).(let H7 \def (H3 e u0 d H5 a H6) in (ex3_2_ind T T (\lambda (y1:
585 T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
586 T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
587 y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d
588 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2))))
589 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0:
590 T).(\lambda (x1: T).(\lambda (H8: (eq T u (lift (S O) d x0))).(\lambda (H9:
591 (eq T t4 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def
592 (eq_ind T t4 (\lambda (t0: T).(pc3 c0 t0 t3)) H4 (lift (S O) d x1) H9) in
593 (let H12 \def (eq_ind T t4 (\lambda (t0: T).(ty3 g c0 u t0)) H2 (lift (S O) d
594 x1) H9) in (let H13 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S
595 O) d x1))) H12 (lift (S O) d x0) H8) in (eq_ind_r T (lift (S O) d x0)
596 (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift
597 (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2))))
598 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H1 e u0
599 d H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift
600 (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d y2))))
601 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1:
602 T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1)))) (\lambda (_:
603 T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
604 (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15:
605 (eq T t3 (lift (S O) d x2))).(\lambda (H16: (eq T t (lift (S O) d
606 x3))).(\lambda (H17: (ty3 g a x2 x3)).(let H18 \def (eq_ind T t (\lambda (t0:
607 T).(ty3 g c0 t3 t0)) H0 (lift (S O) d x3) H16) in (let H19 \def (eq_ind T t3
608 (\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x3))) H18 (lift (S O) d x2) H15)
609 in (let H20 \def (eq_ind T t3 (\lambda (t0: T).(pc3 c0 (lift (S O) d x1) t0))
610 H11 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda (t0:
611 T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift
612 (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2))))
613 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T
614 (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1))))
615 (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2))))
616 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 (refl_equal T (lift
617 (S O) d x0)) (refl_equal T (lift (S O) d x2)) (ty3_conv g a x2 x3 H17 x0 x1
618 H10 (pc3_gen_lift c0 x1 x2 (S O) d H20 a H6))) t3 H15))))))))) H14)) u
619 H8))))))))) H7)))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda
620 (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (_: (getl d c0 (CHead e
621 (Bind Void) u))).(\lambda (a: C).(\lambda (_: (drop (S O) d c0
622 a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TSort m) (lift
623 (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (TSort (next g m))
624 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
625 (TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t: T).(eq T
626 (TSort m) t)) (refl_equal T (TSort m)) (lift (S O) d (TSort m)) (lift_sort m
627 (S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(eq T (TSort (next g
628 m)) t)) (refl_equal T (TSort (next g m))) (lift (S O) d (TSort (next g m)))
629 (lift_sort (next g m) (S O) d)) (ty3_sort g a m)))))))))) (\lambda (n:
630 nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n
631 c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u
632 t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: T).(\forall (d0: nat).((getl
633 d0 d (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d0 d a) \to
634 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d0 y1))))
635 (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1:
636 T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0:
637 T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void)
638 u0))).(\lambda (a: C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0
639 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0
640 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0
641 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt
642 n d0)).(let H6 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0
643 (CHead d (Bind Abbr) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e
644 (Bind Void) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 (le_S_n n d0 (le_S (S n)
645 d0 H5))) (S (minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind
646 nat d0 (\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 (S (plus n (minus d0 (S
647 n)))) (lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
648 C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0:
649 C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
650 C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda (y1: T).(\lambda
651 (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
652 T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda
653 (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H8:
654 (eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9: (getl n a (CHead x1
655 (Bind Abbr) x0))).(\lambda (H10: (drop (S O) (minus d0 (S n)) d x1)).(let H11
656 \def (eq_ind T u (\lambda (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall
657 (d1: nat).((getl d1 d (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop
658 (S O) d1 d a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift
659 (S O) d1 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d1
660 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift
661 (S O) (minus d0 (S n)) x0) H8) in (let H12 \def (eq_ind T u (\lambda (t0:
662 T).(ty3 g d t0 t)) H1 (lift (S O) (minus d0 (S n)) x0) H8) in (let H13 \def
663 (H11 e u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Void) u0)
664 u (minus d0 (S n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda
665 (_: T).(eq T (lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n))
666 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n))
667 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T
668 (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1))))
669 (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2))))
670 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2:
671 T).(\lambda (x3: T).(\lambda (H14: (eq T (lift (S O) (minus d0 (S n)) x0)
672 (lift (S O) (minus d0 (S n)) x2))).(\lambda (H15: (eq T t (lift (S O) (minus
673 d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let H17 \def (eq_ind T t
674 (\lambda (t0: T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t0)) H12 (lift (S
675 O) (minus d0 (S n)) x3) H15) in (eq_ind_r T (lift (S O) (minus d0 (S n)) x3)
676 (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n)
677 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O
678 t0) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
679 y2))))) (let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(ty3 g x1 t0 x3)) H16
680 x0 (lift_inj x0 x2 (S O) (minus d0 (S n)) H14)) in (eq_ind T (lift (S O)
681 (plus (S n) (minus d0 (S n))) (lift (S n) O x3)) (\lambda (t0: T).(ex3_2 T T
682 (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1))))
683 (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1:
684 T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat d0 (\lambda (n0:
685 nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O)
686 d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) n0 (lift (S n) O
687 x3)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
688 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n)
689 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d0
690 (lift (S n) O x3)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2:
691 T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x3) (eq_ind_r T (TLRef n)
692 (\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d0
693 (TLRef n)) (lift_lref_lt n (S O) d0 H5)) (refl_equal T (lift (S O) d0 (lift
694 (S n) O x3))) (ty3_abbr g n a x1 x0 H9 x3 H18)) (plus (S n) (minus d0 (S n)))
695 (le_plus_minus (S n) d0 H5)) (lift (S n) O (lift (S O) (minus d0 (S n)) x3))
696 (lift_d x3 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n)))))) t
697 H15))))))) H13))))))))) (getl_drop_conf_lt Abbr c0 d u n H0 a (S O) (minus d0
698 (S n)) H7))))) (\lambda (H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0
699 (\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r
700 nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in
701 (eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
702 T).(eq T (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq
703 T (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2:
704 T).(ty3 g a y1 y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abbr) u)
705 (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0
706 (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def
707 (eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return
708 (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
709 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
710 \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
711 True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
712 \Rightarrow False])])) I (CHead e (Bind Void) u0) (getl_mono c0 (CHead d
713 (Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (False_ind (ex3_2 T T
714 (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) n y1))))
715 (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) n y2))))
716 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) H9))) d0 H5)))) (\lambda
717 (H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S O)))) (\lambda (n0:
718 nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O)
719 d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O)
720 d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat
721 (plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
722 T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_:
723 T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda
724 (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S
725 O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq
726 T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T
727 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
728 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef
729 (plus (minus n (S O)) (S O))) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
730 (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
731 T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S O))) (lift n O t)
732 (eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda (t0: T).(eq T
733 (TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef (plus (minus n
734 (S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) (lift_lref_ge (minus
735 n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T (lift (plus (S O) n) O
736 t) (\lambda (t0: T).(eq T (lift (S n) O t) t0)) (refl_equal T (lift (S n) O
737 t)) (lift (S O) d0 (lift n O t)) (lift_free t n (S O) O d0 (le_S_n d0 (plus O
738 n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S
739 O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t)))
740 (ty3_abbr g (minus n (S O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr)
741 u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le
742 n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n
743 (le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n (S O))) (plus_sym
744 (S O) (minus n (S O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus
745 O (minus n (S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0)
746 H5))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
747 C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst)
748 u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e:
749 C).(\forall (u0: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Void) u0))
750 \to (\forall (a: C).((drop (S O) d0 d a) \to (ex3_2 T T (\lambda (y1:
751 T).(\lambda (_: T).(eq T u (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
752 (y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
753 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d0:
754 nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void) u0))).(\lambda (a:
755 C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda
756 (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
757 T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda
758 (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt n d0)).(let H6
759 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 (CHead d (Bind
760 Abst) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e (Bind Void) u0)
761 c0 H3 (CHead d (Bind Abst) u) n H0 (le_S_n n d0 (le_S (S n) d0 H5))) (S
762 (minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind nat d0
763 (\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 (S (plus n (minus d0 (S n))))
764 (lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
765 C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0:
766 C).(getl n a (CHead e0 (Bind Abst) v)))) (\lambda (_: T).(\lambda (e0:
767 C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda (y1: T).(\lambda
768 (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
769 T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda
770 (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H8:
771 (eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9: (getl n a (CHead x1
772 (Bind Abst) x0))).(\lambda (H10: (drop (S O) (minus d0 (S n)) d x1)).(let H11
773 \def (eq_ind T u (\lambda (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall
774 (d1: nat).((getl d1 d (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop
775 (S O) d1 d a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift
776 (S O) d1 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d1
777 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift
778 (S O) (minus d0 (S n)) x0) H8) in (let H12 \def (eq_ind T u (\lambda (t0:
779 T).(ty3 g d t0 t)) H1 (lift (S O) (minus d0 (S n)) x0) H8) in (eq_ind_r T
780 (lift (S O) (minus d0 (S n)) x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
781 T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
782 T).(\lambda (y2: T).(eq T (lift (S n) O t0) (lift (S O) d0 y2)))) (\lambda
783 (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H13 \def (H11 e u0 (minus
784 d0 (S n)) (getl_gen_S (Bind Abst) d (CHead e (Bind Void) u0) u (minus d0 (S
785 n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T
786 (lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) y1)))) (\lambda
787 (_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n)) y2)))) (\lambda
788 (y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T (\lambda (y1:
789 T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
790 T).(\lambda (y2: T).(eq T (lift (S n) O (lift (S O) (minus d0 (S n)) x0))
791 (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
792 (\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T (lift (S O) (minus d0
793 (S n)) x0) (lift (S O) (minus d0 (S n)) x2))).(\lambda (H15: (eq T t (lift (S
794 O) (minus d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let H17 \def
795 (eq_ind T t (\lambda (t0: T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t0))
796 H12 (lift (S O) (minus d0 (S n)) x3) H15) in (let H18 \def (eq_ind_r T x2
797 (\lambda (t0: T).(ty3 g x1 t0 x3)) H16 x0 (lift_inj x0 x2 (S O) (minus d0 (S
798 n)) H14)) in (eq_ind T (lift (S O) (plus (S n) (minus d0 (S n))) (lift (S n)
799 O x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
800 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0
801 (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))
802 (eq_ind nat d0 (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
803 T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq
804 T (lift (S O) n0 (lift (S n) O x0)) (lift (S O) d0 y2)))) (\lambda (y1:
805 T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1:
806 T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
807 T).(\lambda (y2: T).(eq T (lift (S O) d0 (lift (S n) O x0)) (lift (S O) d0
808 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S
809 n) O x0) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0))
810 (refl_equal T (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O) d0
811 H5)) (refl_equal T (lift (S O) d0 (lift (S n) O x0))) (ty3_abst g n a x1 x0
812 H9 x3 H18)) (plus (S n) (minus d0 (S n))) (le_plus_minus (S n) d0 H5)) (lift
813 (S n) O (lift (S O) (minus d0 (S n)) x0)) (lift_d x0 (S O) (S n) (minus d0 (S
814 n)) O (le_O_n (minus d0 (S n)))))))))))) H13)) u H8))))))))
815 (getl_drop_conf_lt Abst c0 d u n H0 a (S O) (minus d0 (S n)) H7))))) (\lambda
816 (H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0 (\lambda (n0: nat).(drop (S
817 O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r nat d0 (\lambda (n0:
818 nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in (eq_ind nat n
819 (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef
820 n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O
821 u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
822 y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl
823 n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n
824 H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind
825 Abst) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with
826 [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return
827 (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
828 (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True |
829 Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e (Bind
830 Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Void) u0)
831 H7)) in (False_ind (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef
832 n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O
833 u) (lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
834 H9))) d0 H5)))) (\lambda (H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S
835 O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
836 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift
837 (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
838 y1 y2))))) (eq_ind nat (plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2
839 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1))))
840 (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2))))
841 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus
842 (minus n (S O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
843 T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_:
844 T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda
845 (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1:
846 T).(\lambda (_: T).(eq T (TLRef (plus (minus n (S O)) (S O))) (lift (S O) d0
847 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0
848 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S
849 O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda
850 (t0: T).(eq T (TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef
851 (plus (minus n (S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O))))
852 (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T
853 (lift (plus (S O) n) O u) (\lambda (t0: T).(eq T (lift (S n) O u) t0))
854 (refl_equal T (lift (S n) O u)) (lift (S O) d0 (lift n O u)) (lift_free u n
855 (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) (plus O n) H5)) (le_O_n d0)))
856 (eq_ind_r nat (S (minus n (S O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n
857 (S O))) (lift n0 O u))) (ty3_abst g (minus n (S O)) a d u (getl_drop_conf_ge
858 n (CHead d (Bind Abst) u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0)
859 (\lambda (n0: nat).(le n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1)
860 n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n
861 (S O))) (plus_sym (S O) (minus n (S O)))) (S (plus O (minus n (S O))))
862 (refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n
863 (le_lt_trans O d0 n (le_O_n d0) H5))))))))))))))))))) (\lambda (c0:
864 C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda
865 (H1: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e
866 (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T
867 (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_:
868 T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
869 (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (b: B).(\lambda (t3: T).(\lambda
870 (t4: T).(\lambda (H2: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H3:
871 ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d (CHead c0 (Bind
872 b) u) (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d (CHead c0
873 (Bind b) u) a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift
874 (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2))))
875 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e:
876 C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind
877 Void) u0))).(\lambda (a: C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def
878 (H1 e u0 d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T
879 u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d
880 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T
881 (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind b) u t3) (lift (S O) d
882 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) u t4) (lift (S
883 O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda
884 (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T u (lift (S O) d x0))).(\lambda
885 (H8: (eq T t (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def
886 (eq_ind T t (\lambda (t0: T).(ty3 g c0 u t0)) H0 (lift (S O) d x1) H8) in
887 (let H11 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x1)))
888 H10 (lift (S O) d x0) H7) in (let H12 \def (eq_ind T u (\lambda (t0:
889 T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 (CHead c0
890 (Bind b) t0) (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0
891 (CHead c0 (Bind b) t0) a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
892 T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4
893 (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1
894 y2))))))))))) H3 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T u (\lambda
895 (t0: T).(ty3 g (CHead c0 (Bind b) t0) t3 t4)) H2 (lift (S O) d x0) H7) in
896 (eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
897 T).(\lambda (_: T).(eq T (THead (Bind b) t0 t3) (lift (S O) d y1)))) (\lambda
898 (_: T).(\lambda (y2: T).(eq T (THead (Bind b) t0 t4) (lift (S O) d y2))))
899 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H12 e u0
900 (S d) (getl_head (Bind b) d c0 (CHead e (Bind Void) u0) H4 (lift (S O) d x0))
901 (CHead a (Bind b) x0) (drop_skip_bind (S O) d c0 a H5 b x0)) in (ex3_2_ind T
902 T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift (S O) (S d) y1)))) (\lambda
903 (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) (S d) y2)))) (\lambda (y1:
904 T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T (\lambda
905 (y1: T).(\lambda (_: T).(eq T (THead (Bind b) (lift (S O) d x0) t3) (lift (S
906 O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S
907 O) d x0) t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
908 y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S
909 O) (S d) x2))).(\lambda (H16: (eq T t4 (lift (S O) (S d) x3))).(\lambda (H17:
910 (ty3 g (CHead a (Bind b) x0) x2 x3)).(eq_ind_r T (lift (S O) (S d) x3)
911 (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead
912 (Bind b) (lift (S O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda
913 (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y2))))
914 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r T (lift (S O)
915 (S d) x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
916 (THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y1)))) (\lambda (_:
917 T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d)
918 x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
919 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind
920 b) (lift (S O) d x0) (lift (S O) (S d) x2)) (lift (S O) d y1)))) (\lambda (_:
921 T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d)
922 x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
923 (THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (sym_eq T (lift (S O) d (THead
924 (Bind b) x0 x2)) (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x2))
925 (lift_bind b x0 x2 (S O) d)) (sym_eq T (lift (S O) d (THead (Bind b) x0 x3))
926 (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x3)) (lift_bind b x0 x3
927 (S O) d)) (ty3_bind g a x0 x1 H9 b x2 x3 H17)) t3 H15) t4 H16)))))) H14)) u
928 H7)))))))))) H6)))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda
929 (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: ((\forall (e: C).(\forall
930 (u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall
931 (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
932 T).(eq T w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T u
933 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
934 y2)))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v
935 (THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0:
936 T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall (a:
937 C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
938 v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind
939 Abst) u t) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
940 y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda
941 (H4: (getl d c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H5:
942 (drop (S O) d c0 a)).(let H6 \def (H3 e u0 d H4 a H5) in (ex3_2_ind T T
943 (\lambda (y1: T).(\lambda (_: T).(eq T v (lift (S O) d y1)))) (\lambda (_:
944 T).(\lambda (y2: T).(eq T (THead (Bind Abst) u t) (lift (S O) d y2))))
945 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1:
946 T).(\lambda (_: T).(eq T (THead (Flat Appl) w v) (lift (S O) d y1))))
947 (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind
948 Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
949 y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T v (lift (S O)
950 d x0))).(\lambda (H8: (eq T (THead (Bind Abst) u t) (lift (S O) d
951 x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def (eq_ind T v (\lambda (t0:
952 T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H2 (lift (S O) d x0) H7) in
953 (eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
954 T).(\lambda (_: T).(eq T (THead (Flat Appl) w t0) (lift (S O) d y1))))
955 (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind
956 Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
957 y1 y2))))) (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x1 (THead
958 (Bind Abst) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift (S O) d
959 y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift (S O) (S d) z)))) (ex3_2
960 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d
961 x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat
962 Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1:
963 T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3:
964 T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x2 x3))).(\lambda (H12: (eq T u
965 (lift (S O) d x2))).(\lambda (H13: (eq T t (lift (S O) (S d) x3))).(let H14
966 \def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H9 (THead (Bind Abst) x2
967 x3) H11) in (eq_ind_r T (lift (S O) (S d) x3) (\lambda (t0: T).(ex3_2 T T
968 (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d
969 x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat
970 Appl) w (THead (Bind Abst) u t0)) (lift (S O) d y2)))) (\lambda (y1:
971 T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H15 \def (eq_ind T u (\lambda
972 (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 c0
973 (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to
974 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d0 y1))))
975 (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1:
976 T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H1 (lift (S O) d x2) H12) in
977 (eq_ind_r T (lift (S O) d x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
978 T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O)
979 d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead
980 (Bind Abst) t0 (lift (S O) (S d) x3))) (lift (S O) d y2)))) (\lambda (y1:
981 T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (H15 e u0 d H4 a H5) in
982 (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d y1))))
983 (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2))))
984 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1:
985 T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O)
986 d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead
987 (Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2))))
988 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4:
989 T).(\lambda (x5: T).(\lambda (H17: (eq T w (lift (S O) d x4))).(\lambda (H18:
990 (eq T (lift (S O) d x2) (lift (S O) d x5))).(\lambda (H19: (ty3 g a x4
991 x5)).(eq_ind_r T (lift (S O) d x4) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
992 T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (lift (S O) d x0)) (lift (S O)
993 d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) t0 (THead
994 (Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2))))
995 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H20 \def (eq_ind_r
996 T x5 (\lambda (t0: T).(ty3 g a x4 t0)) H19 x2 (lift_inj x2 x5 (S O) d H18))
997 in (eq_ind T (lift (S O) d (THead (Bind Abst) x2 x3)) (\lambda (t0: T).(ex3_2
998 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) (lift (S O) d
999 x4) (lift (S O) d x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
1000 T).(eq T (THead (Flat Appl) (lift (S O) d x4) t0) (lift (S O) d y2))))
1001 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d
1002 (THead (Flat Appl) x4 x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
1003 T).(\lambda (_: T).(eq T t0 (lift (S O) d y1)))) (\lambda (_: T).(\lambda
1004 (y2: T).(eq T (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d (THead (Bind
1005 Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g
1006 a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst)
1007 x2 x3))) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
1008 (lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1)))) (\lambda (_:
1009 T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
1010 (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_:
1011 T).(eq T (lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1))))
1012 (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d (THead (Flat Appl) x4
1013 (THead (Bind Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
1014 (y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x4 x0) (THead (Flat Appl) x4
1015 (THead (Bind Abst) x2 x3)) (refl_equal T (lift (S O) d (THead (Flat Appl) x4
1016 x0))) (refl_equal T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) x2
1017 x3)))) (ty3_appl g a x4 x2 H20 x0 x3 H14)) (THead (Flat Appl) (lift (S O) d
1018 x4) (lift (S O) d (THead (Bind Abst) x2 x3))) (lift_flat Appl x4 (THead (Bind
1019 Abst) x2 x3) (S O) d)) (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d
1020 x0)) (lift_flat Appl x4 x0 (S O) d)) (THead (Bind Abst) (lift (S O) d x2)
1021 (lift (S O) (S d) x3)) (lift_bind Abst x2 x3 (S O) d))) w H17)))))) H16)) u
1022 H12)) t H13))))))) (lift_gen_bind Abst u t x1 (S O) d H8)) v H7)))))))
1023 H6))))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4:
1024 T).(\lambda (H0: (ty3 g c0 t3 t4)).(\lambda (H1: ((\forall (e: C).(\forall
1025 (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to (\forall
1026 (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
1027 T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4
1028 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
1029 y2)))))))))))).(\lambda (t0: T).(\lambda (H2: (ty3 g c0 t4 t0)).(\lambda (H3:
1030 ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind
1031 Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda
1032 (y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) (\lambda (_:
1033 T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
1034 (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda
1035 (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind Void) u))).(\lambda (a:
1036 C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def (H3 e u d H4 a H5) in
1037 (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1))))
1038 (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1:
1039 T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda
1040 (_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_:
1041 T).(\lambda (y2: T).(eq T (THead (Flat Cast) t0 t4) (lift (S O) d y2))))
1042 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0:
1043 T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (lift (S O) d x0))).(\lambda (H8:
1044 (eq T t0 (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def
1045 (eq_ind T t0 (\lambda (t: T).(ty3 g c0 t4 t)) H2 (lift (S O) d x1) H8) in
1046 (eq_ind_r T (lift (S O) d x1) (\lambda (t: T).(ex3_2 T T (\lambda (y1:
1047 T).(\lambda (_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1))))
1048 (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) t t4) (lift (S O) d
1049 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H11 \def
1050 (eq_ind T t4 (\lambda (t: T).(ty3 g c0 t (lift (S O) d x1))) H10 (lift (S O)
1051 d x0) H7) in (let H12 \def (eq_ind T t4 (\lambda (t: T).(\forall (e0:
1052 C).(\forall (u0: T).(\forall (d0: nat).((getl d0 c0 (CHead e0 (Bind Void)
1053 u0)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to (ex3_2 T T (\lambda (y1:
1054 T).(\lambda (_: T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
1055 (y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
1056 g a0 y1 y2))))))))))) H1 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T t4
1057 (\lambda (t: T).(ty3 g c0 t3 t)) H0 (lift (S O) d x0) H7) in (eq_ind_r T
1058 (lift (S O) d x0) (\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
1059 T).(eq T (THead (Flat Cast) t t3) (lift (S O) d y1)))) (\lambda (_:
1060 T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1) t) (lift (S O)
1061 d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def
1062 (H12 e u d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T
1063 t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d
1064 x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
1065 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) (lift (S
1066 O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T
1067 (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O) d y2))))
1068 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2:
1069 T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S O) d x2))).(\lambda
1070 (H16: (eq T (lift (S O) d x0) (lift (S O) d x3))).(\lambda (H17: (ty3 g a x2
1071 x3)).(let H18 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t (lift (S O) d
1072 x0))) H13 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda
1073 (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast)
1074 (lift (S O) d x0) t) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
1075 T).(eq T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O)
1076 d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H19 \def
1077 (eq_ind_r T x3 (\lambda (t: T).(ty3 g a x2 t)) H17 x0 (lift_inj x0 x3 (S O) d
1078 H16)) in (eq_ind T (lift (S O) d (THead (Flat Cast) x0 x2)) (\lambda (t:
1079 T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1))))
1080 (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1)
1081 (lift (S O) d x0)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2:
1082 T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Cast) x1 x0))
1083 (\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O)
1084 d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda
1085 (y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g
1086 a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S
1087 O) d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda
1088 (y2: T).(eq T (lift (S O) d (THead (Flat Cast) x1 x0)) (lift (S O) d y2))))
1089 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) x0 x2)
1090 (THead (Flat Cast) x1 x0) (refl_equal T (lift (S O) d (THead (Flat Cast) x0
1091 x2))) (refl_equal T (lift (S O) d (THead (Flat Cast) x1 x0))) (ty3_cast g a
1092 x2 x0 H19 x1 H9)) (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0))
1093 (lift_flat Cast x1 x0 (S O) d)) (THead (Flat Cast) (lift (S O) d x0) (lift (S
1094 O) d x2)) (lift_flat Cast x0 x2 (S O) d))) t3 H15))))))) H14)) t4 H7)))) t0
1095 H8))))))) H6)))))))))))))))) c t1 t2 H))))).
projects / helm.git / blob