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 *********************)
19 include "getl/props.ma".
21 include "clear/drop.ma".
26 \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h:
27 nat).((getl h c (CHead e (Bind b) u)) \to (drop (S h) O c e))))))
29 \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e:
30 C).(\forall (u: T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to
31 (drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u:
32 T).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) (CHead e (Bind b)
33 u))).(getl_gen_sort n h (CHead e (Bind b) u) H (drop (S h) O (CSort n)
34 e))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u:
35 T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to (drop (S h) O c0
36 e))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u:
37 T).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t)
38 (CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e))) (\lambda (H0:
39 (getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear
40 (CHead c0 k0 t) (CHead e (Bind b) u)) \to (drop (S O) O (CHead c0 k0 t) e)))
41 (\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e (Bind
42 b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
43 (\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c1 _ _) \Rightarrow
44 c1])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0
45 (CHead e (Bind b) u) t H1)) in ((let H3 \def (f_equal C B (\lambda (e0:
46 C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b |
47 (CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with
48 [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u)
49 (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in
50 ((let H4 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda
51 (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0]))
52 (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e
53 (Bind b) u) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C e
54 c0)).(eq_ind_r C c0 (\lambda (c1: C).(drop (S O) O (CHead c0 (Bind b0) t)
55 c1)) (eq_ind B b (\lambda (b1: B).(drop (S O) O (CHead c0 (Bind b1) t) c0))
56 (drop_drop (Bind b) O c0 c0 (drop_refl c0) t) b0 H5) e H6)))) H3)) H2))))
57 (\lambda (f: F).(\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e (Bind b)
58 u))).(drop_clear_O b (CHead c0 (Flat f) t) e u (clear_flat c0 (CHead e (Bind
59 b) u) (clear_gen_flat f c0 (CHead e (Bind b) u) t H1) f t) e O (drop_refl
60 e)))) k (getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) (\lambda (n:
61 nat).(\lambda (_: (((getl n (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S
62 n) O (CHead c0 k t) e)))).(\lambda (H1: (getl (S n) (CHead c0 k t) (CHead e
63 (Bind b) u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0:
64 nat).(drop n0 O c0 e)) (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind b) u) t
65 n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)).
67 theorem getl_drop_conf_lt:
68 \forall (b: B).(\forall (c: C).(\forall (c0: C).(\forall (u: T).(\forall (i:
69 nat).((getl i c (CHead c0 (Bind b) u)) \to (\forall (e: C).(\forall (h:
70 nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda
71 (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
72 C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
73 h d c0 e0)))))))))))))
75 \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1:
76 C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead c1 (Bind b) u)) \to
77 (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i d))
78 c0 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
79 (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda
80 (_: T).(\lambda (e0: C).(drop h d c1 e0))))))))))))) (\lambda (n:
81 nat).(\lambda (c0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (getl i
82 (CSort n) (CHead c0 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda
83 (d: nat).(\lambda (_: (drop h (S (plus i d)) (CSort n) e)).(getl_gen_sort n i
84 (CHead c0 (Bind b) u) H (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
85 (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b)
86 v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c0 e0)))))))))))))) (\lambda
87 (c0: C).(\lambda (H: ((\forall (c1: C).(\forall (u: T).(\forall (i:
88 nat).((getl i c0 (CHead c1 (Bind b) u)) \to (\forall (e: C).(\forall (h:
89 nat).(\forall (d: nat).((drop h (S (plus i d)) c0 e) \to (ex3_2 T C (\lambda
90 (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
91 C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
92 h d c1 e0)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c1:
93 C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i (CHead c0 k t)
94 (CHead c1 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d:
95 nat).(\lambda (H1: (drop h (S (plus i d)) (CHead c0 k t) e)).(let H2 \def
96 (getl_gen_all (CHead c0 k t) (CHead c1 (Bind b) u) i H0) in (ex2_ind C
97 (\lambda (e0: C).(drop i O (CHead c0 k t) e0)) (\lambda (e0: C).(clear e0
98 (CHead c1 (Bind b) u))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
99 (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b)
100 v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x:
101 C).(\lambda (H3: (drop i O (CHead c0 k t) x)).(\lambda (H4: (clear x (CHead
102 c1 (Bind b) u))).(C_ind (\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to
103 ((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_:
104 C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead
105 e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))
106 (\lambda (n: nat).(\lambda (_: (drop i O (CHead c0 k t) (CSort n))).(\lambda
107 (H6: (clear (CSort n) (CHead c1 (Bind b) u))).(clear_gen_sort (CHead c1 (Bind
108 b) u) n H6 (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
109 (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda
110 (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) (\lambda (x0: C).(\lambda
111 (IHx: (((drop i O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u))
112 \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
113 (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda
114 (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(\lambda (k0: K).(\lambda
115 (t0: T).(\lambda (H5: (drop i O (CHead c0 k t) (CHead x0 k0 t0))).(\lambda
116 (H6: (clear (CHead x0 k0 t0) (CHead c1 (Bind b) u))).(K_ind (\lambda (k1:
117 K).((drop i O (CHead c0 k t) (CHead x0 k1 t0)) \to ((clear (CHead x0 k1 t0)
118 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
119 (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b)
120 v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) (\lambda (b0:
121 B).(\lambda (H7: (drop i O (CHead c0 k t) (CHead x0 (Bind b0) t0))).(\lambda
122 (H8: (clear (CHead x0 (Bind b0) t0) (CHead c1 (Bind b) u))).(let H9 \def
123 (f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
124 [(CSort _) \Rightarrow c1 | (CHead c2 _ _) \Rightarrow c2])) (CHead c1 (Bind
125 b) u) (CHead x0 (Bind b0) t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0
126 H8)) in ((let H10 \def (f_equal C B (\lambda (e0: C).(match e0 in C return
127 (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k1 _) \Rightarrow
128 (match k1 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 |
129 (Flat _) \Rightarrow b])])) (CHead c1 (Bind b) u) (CHead x0 (Bind b0) t0)
130 (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 H8)) in ((let H11 \def
131 (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with
132 [(CSort _) \Rightarrow u | (CHead _ _ t1) \Rightarrow t1])) (CHead c1 (Bind
133 b) u) (CHead x0 (Bind b0) t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0
134 H8)) in (\lambda (H12: (eq B b b0)).(\lambda (H13: (eq C c1 x0)).(let H14
135 \def (eq_ind_r T t0 (\lambda (t1: T).(drop i O (CHead c0 k t) (CHead x0 (Bind
136 b0) t1))) H7 u H11) in (let H15 \def (eq_ind_r B b0 (\lambda (b1: B).(drop i
137 O (CHead c0 k t) (CHead x0 (Bind b1) u))) H14 b H12) in (let H16 \def
138 (eq_ind_r C x0 (\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to ((clear c2
139 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
140 (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b)
141 v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) IHx c1 H13) in
142 (let H17 \def (eq_ind_r C x0 (\lambda (c2: C).(drop i O (CHead c0 k t) (CHead
143 c2 (Bind b) u))) H15 c1 H13) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
144 C).(eq T u (lift h (r (Bind b) d) v)))) (\lambda (v: T).(\lambda (e0:
145 C).(drop i O e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0:
146 C).(drop h (r (Bind b) d) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_:
147 C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead
148 e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))
149 (\lambda (x1: T).(\lambda (x2: C).(\lambda (H18: (eq T u (lift h (r (Bind b)
150 d) x1))).(\lambda (H19: (drop i O e (CHead x2 (Bind b) x1))).(\lambda (H20:
151 (drop h (r (Bind b) d) c1 x2)).(let H21 \def (eq_ind T u (\lambda (t1:
152 T).((drop i O (CHead c0 k t) c1) \to ((clear c1 (CHead c1 (Bind b) t1)) \to
153 (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda
154 (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_:
155 T).(\lambda (e0: C).(drop h d c1 e0))))))) H16 (lift h (r (Bind b) d) x1)
156 H18) in (eq_ind_r T (lift h (r (Bind b) d) x1) (\lambda (t1: T).(ex3_2 T C
157 (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v:
158 T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_:
159 T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v:
160 T).(\lambda (_: C).(eq T (lift h (r (Bind b) d) x1) (lift h d v)))) (\lambda
161 (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_:
162 T).(\lambda (e0: C).(drop h d c1 e0))) x1 x2 (refl_equal T (lift h d x1))
163 (getl_intro i e (CHead x2 (Bind b) x1) (CHead x2 (Bind b) x1) H19 (clear_bind
164 b x2 x1)) H20) u H18))))))) (drop_conf_lt (Bind b) i u c1 (CHead c0 k t) H17
165 e h d H1))))))))) H10)) H9))))) (\lambda (f: F).(\lambda (H7: (drop i O
166 (CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda (H8: (clear (CHead x0 (Flat
167 f) t0) (CHead c1 (Bind b) u))).(nat_ind (\lambda (n: nat).((drop h (S (plus n
168 d)) (CHead c0 k t) e) \to ((drop n O (CHead c0 k t) (CHead x0 (Flat f) t0))
169 \to ((((drop n O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to
170 (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda
171 (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_:
172 T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C (\lambda (v:
173 T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
174 C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
175 h d c1 e0)))))))) (\lambda (H9: (drop h (S (plus O d)) (CHead c0 k t)
176 e)).(\lambda (H10: (drop O O (CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda
177 (IHx0: (((drop O O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u))
178 \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
179 (\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda
180 (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(let H11 \def (f_equal C C
181 (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _)
182 \Rightarrow c0 | (CHead c2 _ _) \Rightarrow c2])) (CHead c0 k t) (CHead x0
183 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead x0 (Flat f) t0) H10)) in
184 ((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda
185 (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k1 _) \Rightarrow k1]))
186 (CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead x0
187 (Flat f) t0) H10)) in ((let H13 \def (f_equal C T (\lambda (e0: C).(match e0
188 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t1)
189 \Rightarrow t1])) (CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead
190 c0 k t) (CHead x0 (Flat f) t0) H10)) in (\lambda (H14: (eq K k (Flat
191 f))).(\lambda (H15: (eq C c0 x0)).(let H16 \def (eq_ind_r C x0 (\lambda (c2:
192 C).(clear c2 (CHead c1 (Bind b) u))) (clear_gen_flat f x0 (CHead c1 (Bind b)
193 u) t0 H8) c0 H15) in (let H17 \def (eq_ind_r C x0 (\lambda (c2: C).((drop O O
194 (CHead c0 k t) c2) \to ((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C
195 (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
196 T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_:
197 T).(\lambda (e0: C).(drop h d c1 e0))))))) IHx0 c0 H15) in (let H18 \def
198 (eq_ind K k (\lambda (k1: K).((drop O O (CHead c0 k1 t) c0) \to ((clear c0
199 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
200 (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b)
201 v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (Flat f)
202 H14) in (let H19 \def (eq_ind K k (\lambda (k1: K).(drop h (S (plus O d))
203 (CHead c0 k1 t) e)) H9 (Flat f) H14) in (ex3_2_ind C T (\lambda (e0:
204 C).(\lambda (v: T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda
205 (v: T).(eq T t (lift h (r (Flat f) (plus O d)) v)))) (\lambda (e0:
206 C).(\lambda (_: T).(drop h (r (Flat f) (plus O d)) c0 e0))) (ex3_2 T C
207 (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
208 T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_:
209 T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: C).(\lambda (x2:
210 T).(\lambda (H20: (eq C e (CHead x1 (Flat f) x2))).(\lambda (H21: (eq T t
211 (lift h (r (Flat f) (plus O d)) x2))).(\lambda (H22: (drop h (r (Flat f)
212 (plus O d)) c0 x1)).(let H23 \def (f_equal T T (\lambda (e0: T).e0) t (lift h
213 (r (Flat f) (plus O d)) x2) H21) in (let H24 \def (eq_ind C e (\lambda (c2:
214 C).((drop O O (CHead c0 (Flat f) t) c0) \to ((clear c0 (CHead c1 (Bind b) u))
215 \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
216 (\lambda (v: T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) v)))) (\lambda
217 (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H18 (CHead x1 (Flat f) x2)
218 H20) in (eq_ind_r C (CHead x1 (Flat f) x2) (\lambda (c2: C).(ex3_2 T C
219 (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
220 T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) v)))) (\lambda (_:
221 T).(\lambda (e0: C).(drop h d c1 e0))))) (let H25 \def (eq_ind T t (\lambda
222 (t1: T).((drop O O (CHead c0 (Flat f) t1) c0) \to ((clear c0 (CHead c1 (Bind
223 b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
224 (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0
225 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H24
226 (lift h (S d) x2) H23) in (let H26 \def (H c1 u O (getl_intro O c0 (CHead c1
227 (Bind b) u) c0 (drop_refl c0) H16) x1 h d H22) in (ex3_2_ind T C (\lambda (v:
228 T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
229 C).(getl O x1 (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
230 h d c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d
231 v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead
232 e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))
233 (\lambda (x3: T).(\lambda (x4: C).(\lambda (H27: (eq T u (lift h d
234 x3))).(\lambda (H28: (getl O x1 (CHead x4 (Bind b) x3))).(\lambda (H29: (drop
235 h d c1 x4)).(let H30 \def (eq_ind T u (\lambda (t1: T).((drop O O (CHead c0
236 (Flat f) (lift h (S d) x2)) c0) \to ((clear c0 (CHead c1 (Bind b) t1)) \to
237 (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda
238 (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 (Bind b)
239 v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H25 (lift h d
240 x3) H27) in (let H31 \def (eq_ind T u (\lambda (t1: T).(clear c0 (CHead c1
241 (Bind b) t1))) H16 (lift h d x3) H27) in (eq_ind_r T (lift h d x3) (\lambda
242 (t1: T).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v))))
243 (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0
244 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))
245 (ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T (lift h d x3) (lift h
246 d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2)
247 (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))
248 x3 x4 (refl_equal T (lift h d x3)) (getl_flat x1 (CHead x4 (Bind b) x3) O H28
249 f x2) H29) u H27)))))))) H26))) e H20)))))))) (drop_gen_skip_l c0 e t h (plus
250 O d) (Flat f) H19))))))))) H12)) H11))))) (\lambda (i0: nat).(\lambda (IHi:
251 (((drop h (S (plus i0 d)) (CHead c0 k t) e) \to ((drop i0 O (CHead c0 k t)
252 (CHead x0 (Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear x0
253 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
254 (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 e (CHead e0 (Bind
255 b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T
256 C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
257 T).(\lambda (e0: C).(getl i0 e (CHead e0 (Bind b) v)))) (\lambda (_:
258 T).(\lambda (e0: C).(drop h d c1 e0))))))))).(\lambda (H9: (drop h (S (plus
259 (S i0) d)) (CHead c0 k t) e)).(\lambda (H10: (drop (S i0) O (CHead c0 k t)
260 (CHead x0 (Flat f) t0))).(\lambda (IHx0: (((drop (S i0) O (CHead c0 k t) x0)
261 \to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda
262 (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0)
263 e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1
264 e0)))))))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0
265 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k (plus (S i0) d))
266 v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus (S i0) d)) c0 e0)))
267 (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda
268 (v: T).(\lambda (e0: C).(getl (S i0) e (CHead e0 (Bind b) v)))) (\lambda (_:
269 T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: C).(\lambda (x2:
270 T).(\lambda (H11: (eq C e (CHead x1 k x2))).(\lambda (H12: (eq T t (lift h (r
271 k (plus (S i0) d)) x2))).(\lambda (H13: (drop h (r k (plus (S i0) d)) c0
272 x1)).(let H14 \def (f_equal T T (\lambda (e0: T).e0) t (lift h (r k (plus (S
273 i0) d)) x2) H12) in (let H15 \def (eq_ind C e (\lambda (c2: C).((drop h (S
274 (plus i0 d)) (CHead c0 k t) c2) \to ((drop i0 O (CHead c0 k t) (CHead x0
275 (Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear x0 (CHead c1
276 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d
277 v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v))))
278 (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C
279 (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
280 T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v)))) (\lambda (_:
281 T).(\lambda (e0: C).(drop h d c1 e0)))))))) IHi (CHead x1 k x2) H11) in (let
282 H16 \def (eq_ind C e (\lambda (c2: C).((drop (S i0) O (CHead c0 k t) x0) \to
283 ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_:
284 C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2
285 (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1
286 e0))))))) IHx0 (CHead x1 k x2) H11) in (eq_ind_r C (CHead x1 k x2) (\lambda
287 (c2: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
288 (\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2 (CHead e0 (Bind b) v))))
289 (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H17 \def (eq_ind T
290 t (\lambda (t1: T).((drop (S i0) O (CHead c0 k t1) x0) \to ((clear x0 (CHead
291 c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift
292 h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2)
293 (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1
294 e0))))))) H16 (lift h (r k (S (plus i0 d))) x2) H14) in (let H18 \def (eq_ind
295 T t (\lambda (t1: T).((drop h (S (plus i0 d)) (CHead c0 k t1) (CHead x1 k
296 x2)) \to ((drop i0 O (CHead c0 k t1) (CHead x0 (Flat f) t0)) \to ((((drop i0
297 O (CHead c0 k t1) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C
298 (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
299 T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v))))
300 (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C
301 (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
302 T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v))))
303 (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) H15 (lift h (r k (S
304 (plus i0 d))) x2) H14) in (let H19 \def (eq_ind nat (r k (plus (S i0) d))
305 (\lambda (n: nat).(drop h n c0 x1)) H13 (plus (r k (S i0)) d) (r_plus k (S
306 i0) d)) in (let H20 \def (eq_ind nat (r k (S i0)) (\lambda (n: nat).(drop h
307 (plus n d) c0 x1)) H19 (S (r k i0)) (r_S k i0)) in (let H21 \def (H c1 u (r k
308 i0) (getl_intro (r k i0) c0 (CHead c1 (Bind b) u) (CHead x0 (Flat f) t0)
309 (drop_gen_drop k c0 (CHead x0 (Flat f) t0) t i0 H10) (clear_flat x0 (CHead c1
310 (Bind b) u) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) f t0)) x1 h d
311 H20) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d
312 v)))) (\lambda (v: T).(\lambda (e0: C).(getl (r k i0) x1 (CHead e0 (Bind b)
313 v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) (ex3_2 T C (\lambda
314 (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
315 C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_:
316 T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x3: T).(\lambda (x4:
317 C).(\lambda (H22: (eq T u (lift h d x3))).(\lambda (H23: (getl (r k i0) x1
318 (CHead x4 (Bind b) x3))).(\lambda (H24: (drop h d c1 x4)).(let H25 \def
319 (eq_ind T u (\lambda (t1: T).((drop (S i0) O (CHead c0 k (lift h (r k (S
320 (plus i0 d))) x2)) x0) \to ((clear x0 (CHead c1 (Bind b) t1)) \to (ex3_2 T C
321 (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v:
322 T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v))))
323 (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (lift h d x3)
324 H22) in (let H26 \def (eq_ind T u (\lambda (t1: T).(clear x0 (CHead c1 (Bind
325 b) t1))) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) (lift h d x3) H22)
326 in (eq_ind_r T (lift h d x3) (\lambda (t1: T).(ex3_2 T C (\lambda (v:
327 T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0:
328 C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_:
329 T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v:
330 T).(\lambda (_: C).(eq T (lift h d x3) (lift h d v)))) (\lambda (v:
331 T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v))))
332 (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x3 x4 (refl_equal T (lift
333 h d x3)) (getl_head k i0 x1 (CHead x4 (Bind b) x3) H23 x2) H24) u H22))))))))
334 H21)))))) e H11))))))))) (drop_gen_skip_l c0 e t h (plus (S i0) d) k
335 H9))))))) i H1 H7 IHx)))) k0 H5 H6))))))) x H3 H4)))) H2)))))))))))))) c)).
337 theorem getl_drop_conf_ge:
338 \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall
339 (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d
340 h) i) \to (getl (minus i h) e a)))))))))
342 \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c
343 a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h
344 d c e)).(\lambda (H1: (le (plus d h) i)).(let H2 \def (getl_gen_all c a i H)
345 in (ex2_ind C (\lambda (e0: C).(drop i O c e0)) (\lambda (e0: C).(clear e0
346 a)) (getl (minus i h) e a) (\lambda (x: C).(\lambda (H3: (drop i O c
347 x)).(\lambda (H4: (clear x a)).(getl_intro (minus i h) e a x (drop_conf_ge i
348 x c H3 e h d H0 H1) H4)))) H2)))))))))).
350 theorem getl_conf_ge_drop:
351 \forall (b: B).(\forall (c1: C).(\forall (e: C).(\forall (u: T).(\forall (i:
352 nat).((getl i c1 (CHead e (Bind b) u)) \to (\forall (c2: C).((drop (S O) i c1
353 c2) \to (drop i O c2 e))))))))
355 \lambda (b: B).(\lambda (c1: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i:
356 nat).(\lambda (H: (getl i c1 (CHead e (Bind b) u))).(\lambda (c2: C).(\lambda
357 (H0: (drop (S O) i c1 c2)).(let H3 \def (eq_ind nat (minus (S i) (S O))
358 (\lambda (n: nat).(drop n O c2 e)) (drop_conf_ge (S i) e c1 (getl_drop b c1 e
359 u i H) c2 (S O) i H0 (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(le n (S
360 i))) (le_n (S i)) (plus i (S O)) (plus_comm i (S O)))) i (minus_Sx_SO i)) in
363 theorem getl_drop_conf_rev:
364 \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to
365 (\forall (b: B).(\forall (c2: C).(\forall (v2: T).(\forall (i: nat).((getl i
366 c2 (CHead e2 (Bind b) v2)) \to (ex2 C (\lambda (c1: C).(drop j O c1 c2))
367 (\lambda (c1: C).(drop (S i) j c1 e1)))))))))))
369 \lambda (j: nat).(\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop j O e1
370 e2)).(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(\lambda (i:
371 nat).(\lambda (H0: (getl i c2 (CHead e2 (Bind b) v2))).(drop_conf_rev j e1 e2
372 H c2 (S i) (getl_drop b c2 e2 v2 i H0)))))))))).
374 theorem drop_getl_trans_lt:
375 \forall (i: nat).(\forall (d: nat).((lt i d) \to (\forall (c1: C).(\forall
376 (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (b: B).(\forall (e2:
377 C).(\forall (v: T).((getl i c2 (CHead e2 (Bind b) v)) \to (ex2 C (\lambda
378 (e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda
379 (e1: C).(drop h (minus d (S i)) e1 e2)))))))))))))
381 \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (lt i d)).(\lambda (c1:
382 C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1
383 c2)).(\lambda (b: B).(\lambda (e2: C).(\lambda (v: T).(\lambda (H1: (getl i
384 c2 (CHead e2 (Bind b) v))).(let H2 \def (getl_gen_all c2 (CHead e2 (Bind b)
385 v) i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) (\lambda (e:
386 C).(clear e (CHead e2 (Bind b) v))) (ex2 C (\lambda (e1: C).(getl i c1 (CHead
387 e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h (minus d
388 (S i)) e1 e2))) (\lambda (x: C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4:
389 (clear x (CHead e2 (Bind b) v))).(ex2_ind C (\lambda (e1: C).(drop i O c1
390 e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex2 C (\lambda (e1:
391 C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1:
392 C).(drop h (minus d (S i)) e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O
393 c1 x0)).(\lambda (H6: (drop h (minus d i) x0 x)).(let H7 \def (eq_ind nat
394 (minus d i) (\lambda (n: nat).(drop h n x0 x)) H6 (S (minus d (S i)))
395 (minus_x_Sy d i H)) in (let H8 \def (drop_clear_S x x0 h (minus d (S i)) H7 b
396 e2 v H4) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind b) (lift h
397 (minus d (S i)) v)))) (\lambda (c3: C).(drop h (minus d (S i)) c3 e2)) (ex2 C
398 (\lambda (e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v))))
399 (\lambda (e1: C).(drop h (minus d (S i)) e1 e2))) (\lambda (x1: C).(\lambda
400 (H9: (clear x0 (CHead x1 (Bind b) (lift h (minus d (S i)) v)))).(\lambda
401 (H10: (drop h (minus d (S i)) x1 e2)).(ex_intro2 C (\lambda (e1: C).(getl i
402 c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h
403 (minus d (S i)) e1 e2)) x1 (getl_intro i c1 (CHead x1 (Bind b) (lift h (minus
404 d (S i)) v)) x0 H5 H9) H10)))) H8)))))) (drop_trans_le i d (le_S_n i d (le_S
405 (S i) d H)) c1 c2 h H0 x H3))))) H2)))))))))))).
407 theorem drop_getl_trans_le:
408 \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall
409 (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2
410 e2) \to (ex3_2 C C (\lambda (e0: C).(\lambda (_: C).(drop i O c1 e0)))
411 (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) (\lambda (_:
412 C).(\lambda (e1: C).(clear e1 e2))))))))))))
414 \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (le i d)).(\lambda (c1:
415 C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1
416 c2)).(\lambda (e2: C).(\lambda (H1: (getl i c2 e2)).(let H2 \def
417 (getl_gen_all c2 e2 i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e))
418 (\lambda (e: C).(clear e e2)) (ex3_2 C C (\lambda (e0: C).(\lambda (_:
419 C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i)
420 e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 e2)))) (\lambda (x:
421 C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(let H5 \def
422 (drop_trans_le i d H c1 c2 h H0 x H3) in (ex2_ind C (\lambda (e1: C).(drop i
423 O c1 e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex3_2 C C (\lambda
424 (e0: C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1:
425 C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1
426 e2)))) (\lambda (x0: C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h
427 (minus d i) x0 x)).(ex3_2_intro C C (\lambda (e0: C).(\lambda (_: C).(drop i
428 O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1)))
429 (\lambda (_: C).(\lambda (e1: C).(clear e1 e2))) x0 x H6 H7 H4)))) H5)))))
432 theorem drop_getl_trans_ge:
433 \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d:
434 nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 e2)
435 \to ((le d i) \to (getl (plus i h) c1 e2)))))))))
437 \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (d:
438 nat).(\lambda (h: nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2:
439 C).(\lambda (H0: (getl i c2 e2)).(\lambda (H1: (le d i)).(let H2 \def
440 (getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: C).(drop i O c2 e))
441 (\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) (\lambda (x:
442 C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(getl_intro
443 (plus i h) c1 e2 x (drop_trans_ge i c1 c2 d h H x H3 H1) H4)))) H2)))))))))).
445 theorem getl_drop_trans:
446 \forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl h c1 c2) \to
447 (\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 e2) \to (drop (S (plus i
450 \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (h:
451 nat).((getl h c c2) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c2
452 e2) \to (drop (S (plus i h)) O c e2)))))))) (\lambda (n: nat).(\lambda (c2:
453 C).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) c2)).(\lambda (e2:
454 C).(\lambda (i: nat).(\lambda (_: (drop (S i) O c2 e2)).(getl_gen_sort n h c2
455 H (drop (S (plus i h)) O (CSort n) e2))))))))) (\lambda (c2: C).(\lambda
456 (IHc: ((\forall (c3: C).(\forall (h: nat).((getl h c2 c3) \to (\forall (e2:
457 C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O c2
458 e2))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(\forall
459 (c3: C).(\forall (h: nat).((getl h (CHead c2 k0 t) c3) \to (\forall (e2:
460 C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O (CHead
461 c2 k0 t) e2))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3:
462 C).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c2 (Bind b)
463 t) c3) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop
464 (S (plus i n)) O (CHead c2 (Bind b) t) e2)))))) (\lambda (H: (getl O (CHead
465 c2 (Bind b) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H0: (drop (S
466 i) O c3 e2)).(let H1 \def (eq_ind C c3 (\lambda (c: C).(drop (S i) O c e2))
467 H0 (CHead c2 (Bind b) t) (clear_gen_bind b c2 c3 t (getl_gen_O (CHead c2
468 (Bind b) t) c3 H))) in (eq_ind nat i (\lambda (n: nat).(drop (S n) O (CHead
469 c2 (Bind b) t) e2)) (drop_drop (Bind b) i c2 e2 (drop_gen_drop (Bind b) c2 e2
470 t i H1) t) (plus i O) (plus_n_O i))))))) (\lambda (n: nat).(\lambda (_:
471 (((getl n (CHead c2 (Bind b) t) c3) \to (\forall (e2: C).(\forall (i:
472 nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Bind b) t)
473 e2))))))).(\lambda (H0: (getl (S n) (CHead c2 (Bind b) t) c3)).(\lambda (e2:
474 C).(\lambda (i: nat).(\lambda (H1: (drop (S i) O c3 e2)).(eq_ind nat (plus (S
475 i) n) (\lambda (n0: nat).(drop (S n0) O (CHead c2 (Bind b) t) e2)) (drop_drop
476 (Bind b) (plus (S i) n) c2 e2 (IHc c3 n (getl_gen_S (Bind b) c2 c3 t n H0) e2
477 i H1) t) (plus i (S n)) (plus_Snm_nSm i n)))))))) h))))) (\lambda (f:
478 F).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: nat).(nat_ind (\lambda (n:
479 nat).((getl n (CHead c2 (Flat f) t) c3) \to (\forall (e2: C).(\forall (i:
480 nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Flat f) t)
481 e2)))))) (\lambda (H: (getl O (CHead c2 (Flat f) t) c3)).(\lambda (e2:
482 C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O c3 e2)).(drop_drop (Flat f)
483 (plus i O) c2 e2 (IHc c3 O (getl_intro O c2 c3 c2 (drop_refl c2)
484 (clear_gen_flat f c2 c3 t (getl_gen_O (CHead c2 (Flat f) t) c3 H))) e2 i H0)
485 t))))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead c2 (Flat f) t) c3) \to
486 (\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i
487 n)) O (CHead c2 (Flat f) t) e2))))))).(\lambda (H0: (getl (S n) (CHead c2
488 (Flat f) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H1: (drop (S i)
489 O c3 e2)).(drop_drop (Flat f) (plus i (S n)) c2 e2 (IHc c3 (S n) (getl_gen_S
490 (Flat f) c2 c3 t n H0) e2 i H1) t))))))) h))))) k)))) c1).
projects / helm.git / blob