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 set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubst1/getl".
19 include "csubst1/props.ma".
21 include "csubst0/getl.ma".
23 include "csubst0/props.ma".
25 include "subst1/props.ma".
27 include "drop/props.ma".
29 theorem csubst1_getl_ge:
30 \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall
31 (c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c1
32 e) \to (getl n c2 e)))))))))
34 \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1:
35 C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1
36 c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c1 e) \to
37 (getl n c e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda
38 (c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2:
39 (getl n c1 e)).(csubst0_getl_ge i n H c1 c3 v H1 e H2))))) c2 H0))))))).
41 theorem csubst1_getl_lt:
42 \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall
43 (c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e1: C).((getl n c1
44 e1) \to (ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2:
45 C).(getl n c2 e2)))))))))))
47 \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1:
48 C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1
49 c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e1: C).((getl n c1 e1) \to
50 (ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2: C).(getl
51 n c e2)))))) (\lambda (e1: C).(\lambda (H1: (getl n c1 e1)).(eq_ind_r nat (S
52 (minus i (S n))) (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1
53 e2)) (\lambda (e2: C).(getl n c1 e2)))) (ex_intro2 C (\lambda (e2:
54 C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c1 e2)) e1
55 (csubst1_refl (S (minus i (S n))) v e1) H1) (minus i n) (minus_x_Sy i n H))))
56 (\lambda (c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e1: C).(\lambda
57 (H2: (getl n c1 e1)).(eq_ind_r nat (S (minus i (S n))) (\lambda (n0:
58 nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1 e2)) (\lambda (e2: C).(getl n
59 c3 e2)))) (let H3 \def (csubst0_getl_lt i n H c1 c3 v H1 e1 H2) in (or4_ind
60 (getl n c3 e1) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u:
61 T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b:
62 B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0
63 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
64 T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
65 B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind
66 b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u:
67 T).(getl n c3 (CHead e3 (Bind b) u)))))) (\lambda (_: B).(\lambda (e2:
68 C).(\lambda (e3: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3))))))
69 (ex4_5 B C C T T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda
70 (u: T).(\lambda (_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b:
71 B).(\lambda (_: C).(\lambda (e3: C).(\lambda (_: T).(\lambda (w: T).(getl n
72 c3 (CHead e3 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
73 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
74 (\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(\lambda
75 (_: T).(csubst0 (minus i (S n)) v e2 e3))))))) (ex2 C (\lambda (e2:
76 C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2)))
77 (\lambda (H4: (getl n c3 e1)).(ex_intro2 C (\lambda (e2: C).(csubst1 (S
78 (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2)) e1 (csubst1_refl
79 (S (minus i (S n))) v e1) H4)) (\lambda (H4: (ex3_4 B C T T (\lambda (b:
80 B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind
81 b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
82 T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
83 C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u
84 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u:
85 T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b:
86 B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0
87 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
88 T).(subst0 (minus i (S n)) v u w))))) (ex2 C (\lambda (e2: C).(csubst1 (S
89 (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0:
90 B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H5: (eq C e1
91 (CHead x1 (Bind x0) x2))).(\lambda (H6: (getl n c3 (CHead x1 (Bind x0)
92 x3))).(\lambda (H7: (subst0 (minus i (S n)) v x2 x3)).(eq_ind_r C (CHead x1
93 (Bind x0) x2) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S
94 n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2:
95 C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x2) e2)) (\lambda (e2:
96 C).(getl n c3 e2)) (CHead x1 (Bind x0) x3) (csubst1_sing (S (minus i (S n)))
97 v (CHead x1 (Bind x0) x2) (CHead x1 (Bind x0) x3) (csubst0_snd_bind x0 (minus
98 i (S n)) v x2 x3 H7 x1)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex3_4 B C C T
99 (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1
100 (CHead e2 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3:
101 C).(\lambda (u: T).(getl n c3 (CHead e3 (Bind b) u)))))) (\lambda (_:
102 B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(csubst0 (minus i (S n))
103 v e2 e3))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e2: C).(\lambda
104 (_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind b) u)))))) (\lambda (b:
105 B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u: T).(getl n c3 (CHead e3
106 (Bind b) u)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda
107 (_: T).(csubst0 (minus i (S n)) v e2 e3))))) (ex2 C (\lambda (e2: C).(csubst1
108 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0:
109 B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H5: (eq C e1
110 (CHead x1 (Bind x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0)
111 x3))).(\lambda (H7: (csubst0 (minus i (S n)) v x1 x2)).(eq_ind_r C (CHead x1
112 (Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S
113 n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2:
114 C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x3) e2)) (\lambda (e2:
115 C).(getl n c3 e2)) (CHead x2 (Bind x0) x3) (csubst1_sing (S (minus i (S n)))
116 v (CHead x1 (Bind x0) x3) (CHead x2 (Bind x0) x3) (csubst0_fst_bind x0 (minus
117 i (S n)) x1 x2 v H7 x3)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex4_5 B C C T
118 T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda
119 (_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b: B).(\lambda (_:
120 C).(\lambda (e3: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e3
121 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
122 (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
123 B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(\lambda (_: T).(csubst0
124 (minus i (S n)) v e2 e3)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda
125 (e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e2
126 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda
127 (_: T).(\lambda (w: T).(getl n c3 (CHead e3 (Bind b) w))))))) (\lambda (_:
128 B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
129 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3:
130 C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3))))))
131 (ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2:
132 C).(getl n c3 e2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
133 C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H5: (eq C e1 (CHead x1 (Bind
134 x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H7:
135 (subst0 (minus i (S n)) v x3 x4)).(\lambda (H8: (csubst0 (minus i (S n)) v x1
136 x2)).(eq_ind_r C (CHead x1 (Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2:
137 C).(csubst1 (S (minus i (S n))) v c e2)) (\lambda (e2: C).(getl n c3 e2))))
138 (ex_intro2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind
139 x0) x3) e2)) (\lambda (e2: C).(getl n c3 e2)) (CHead x2 (Bind x0) x4)
140 (csubst1_sing (S (minus i (S n))) v (CHead x1 (Bind x0) x3) (CHead x2 (Bind
141 x0) x4) (csubst0_both_bind x0 (minus i (S n)) v x3 x4 H7 x1 x2 H8)) H6) e1
142 H5)))))))))) H4)) H3)) (minus i n) (minus_x_Sy i n H)))))) c2 H0))))))).
144 theorem csubst1_getl_ge_back:
145 \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall
146 (c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c2
147 e) \to (getl n c1 e)))))))))
149 \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1:
150 C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1
151 c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c e) \to
152 (getl n c1 e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda
153 (c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2:
154 (getl n c3 e)).(csubst0_getl_ge_back i n H c1 c3 v H1 e H2))))) c2 H0))))))).
156 theorem getl_csubst1:
157 \forall (d: nat).(\forall (c: C).(\forall (e: C).(\forall (u: T).((getl d c
158 (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_:
159 C).(csubst1 d u c a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) d a0
162 \lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (c: C).(\forall (e:
163 C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C
164 (\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0:
165 C).(\lambda (a: C).(drop (S O) n a0 a))))))))) (\lambda (c: C).(C_ind
166 (\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind
167 Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0
168 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))))))) (\lambda
169 (n: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H: (getl O (CSort n)
170 (CHead e (Bind Abbr) u))).(getl_gen_sort n O (CHead e (Bind Abbr) u) H (ex2_2
171 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CSort n) a0))) (\lambda
172 (a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (c0: C).(\lambda
173 (H: ((\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind Abbr) u)) \to
174 (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) (\lambda
175 (a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))).(\lambda (k: K).(K_ind
176 (\lambda (k0: K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl O
177 (CHead c0 k0 t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0:
178 C).(\lambda (_: C).(csubst1 O u (CHead c0 k0 t) a0))) (\lambda (a0:
179 C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (b: B).(\lambda (t:
180 T).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Bind b)
181 t) (CHead e (Bind Abbr) u))).(let H1 \def (f_equal C C (\lambda (e0:
182 C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e |
183 (CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind Abbr) u) (CHead c0 (Bind b)
184 t) (clear_gen_bind b c0 (CHead e (Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind
185 b) t) (CHead e (Bind Abbr) u) H0))) in ((let H2 \def (f_equal C B (\lambda
186 (e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
187 Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B)
188 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead e
189 (Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e (Bind
190 Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) H0))) in
191 ((let H3 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda
192 (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0]))
193 (CHead e (Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e
194 (Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u)
195 H0))) in (\lambda (H4: (eq B Abbr b)).(\lambda (_: (eq C e c0)).(eq_ind_r T t
196 (\lambda (t0: T).(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O t0
197 (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0
198 a))))) (eq_ind B Abbr (\lambda (b0: B).(ex2_2 C C (\lambda (a0: C).(\lambda
199 (_: C).(csubst1 O t (CHead c0 (Bind b0) t) a0))) (\lambda (a0: C).(\lambda
200 (a: C).(drop (S O) O a0 a))))) (ex2_2_intro C C (\lambda (a0: C).(\lambda (_:
201 C).(csubst1 O t (CHead c0 (Bind Abbr) t) a0))) (\lambda (a0: C).(\lambda (a:
202 C).(drop (S O) O a0 a))) (CHead c0 (Bind Abbr) t) c0 (csubst1_refl O t (CHead
203 c0 (Bind Abbr) t)) (drop_drop (Bind Abbr) O c0 c0 (drop_refl c0) t)) b H4) u
204 H3)))) H2)) H1))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (e:
205 C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Flat f) t) (CHead e (Bind
206 Abbr) u))).(let H_x \def (subst1_ex u t O) in (let H1 \def H_x in (ex_ind T
207 (\lambda (t2: T).(subst1 O u t (lift (S O) O t2))) (ex2_2 C C (\lambda (a0:
208 C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda (a0:
209 C).(\lambda (a: C).(drop (S O) O a0 a)))) (\lambda (x: T).(\lambda (H2:
210 (subst1 O u t (lift (S O) O x))).(let H3 \def (H e u (getl_intro O c0 (CHead
211 e (Bind Abbr) u) c0 (drop_refl c0) (clear_gen_flat f c0 (CHead e (Bind Abbr)
212 u) t (getl_gen_O (CHead c0 (Flat f) t) (CHead e (Bind Abbr) u) H0)))) in
213 (ex2_2_ind C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0)))
214 (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) (ex2_2 C C (\lambda
215 (a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda
216 (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))) (\lambda (x0: C).(\lambda (x1:
217 C).(\lambda (H4: (csubst1 O u c0 x0)).(\lambda (H5: (drop (S O) O x0
218 x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CHead c0
219 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))
220 (CHead x0 (Flat f) (lift (S O) O x)) x1 (csubst1_flat f O u t (lift (S O) O
221 x) H2 c0 x0 H4) (drop_drop (Flat f) O x0 x1 H5 (lift (S O) O x))))))) H3))))
222 H1)))))))) k)))) c)) (\lambda (n: nat).(\lambda (H: ((\forall (c: C).(\forall
223 (e: C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C
224 (\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0:
225 C).(\lambda (a: C).(drop (S O) n a0 a)))))))))).(\lambda (c: C).(C_ind
226 (\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl (S n) c0 (CHead e
227 (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S
228 n) u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))))))))
229 (\lambda (n0: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl (S n)
230 (CSort n0) (CHead e (Bind Abbr) u))).(getl_gen_sort n0 (S n) (CHead e (Bind
231 Abbr) u) H0 (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u
232 (CSort n0) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0
233 a))))))))) (\lambda (c0: C).(\lambda (H0: ((\forall (e: C).(\forall (u:
234 T).((getl (S n) c0 (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0:
235 C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0: C).(\lambda (a:
236 C).(drop (S O) (S n) a0 a))))))))).(\lambda (k: K).(K_ind (\lambda (k0:
237 K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl (S n) (CHead c0 k0
238 t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_:
239 C).(csubst1 (S n) u (CHead c0 k0 t) a0))) (\lambda (a0: C).(\lambda (a:
240 C).(drop (S O) (S n) a0 a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda
241 (e: C).(\lambda (u: T).(\lambda (H1: (getl (S n) (CHead c0 (Bind b) t) (CHead
242 e (Bind Abbr) u))).(let H_x \def (subst1_ex u t n) in (let H2 \def H_x in
243 (ex_ind T (\lambda (t2: T).(subst1 n u t (lift (S O) n t2))) (ex2_2 C C
244 (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0)))
245 (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x:
246 T).(\lambda (H3: (subst1 n u t (lift (S O) n x))).(let H4 \def (H c0 e u
247 (getl_gen_S (Bind b) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C
248 (\lambda (a0: C).(\lambda (_: C).(csubst1 n u c0 a0))) (\lambda (a0:
249 C).(\lambda (a: C).(drop (S O) n a0 a))) (ex2_2 C C (\lambda (a0: C).(\lambda
250 (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda
251 (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1: C).(\lambda
252 (H5: (csubst1 n u c0 x0)).(\lambda (H6: (drop (S O) n x0 x1)).(ex2_2_intro C
253 C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t)
254 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (CHead x0
255 (Bind b) (lift (S O) n x)) (CHead x1 (Bind b) x) (csubst1_bind b n u t (lift
256 (S O) n x) H3 c0 x0 H5) (drop_skip_bind (S O) n x0 x1 H6 b x)))))) H4))))
257 H2)))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (e: C).(\lambda (u:
258 T).(\lambda (H1: (getl (S n) (CHead c0 (Flat f) t) (CHead e (Bind Abbr)
259 u))).(let H_x \def (subst1_ex u t (S n)) in (let H2 \def H_x in (ex_ind T
260 (\lambda (t2: T).(subst1 (S n) u t (lift (S O) (S n) t2))) (ex2_2 C C
261 (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0)))
262 (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x:
263 T).(\lambda (H3: (subst1 (S n) u t (lift (S O) (S n) x))).(let H4 \def (H0 e
264 u (getl_gen_S (Flat f) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C
265 (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0:
266 C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (ex2_2 C C (\lambda (a0:
267 C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) (\lambda (a0:
268 C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1:
269 C).(\lambda (H5: (csubst1 (S n) u c0 x0)).(\lambda (H6: (drop (S O) (S n) x0
270 x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u
271 (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S
272 n) a0 a))) (CHead x0 (Flat f) (lift (S O) (S n) x)) (CHead x1 (Flat f) x)
273 (csubst1_flat f (S n) u t (lift (S O) (S n) x) H3 c0 x0 H5) (drop_skip_flat
274 (S O) n x0 x1 H6 f x)))))) H4)))) H2)))))))) k)))) c)))) d).