1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "Basic-1/sc3/defs.ma".
19 include "Basic-1/sn3/lift1.ma".
21 include "Basic-1/nf2/lift1.ma".
23 include "Basic-1/csuba/arity.ma".
25 include "Basic-1/arity/lift1.ma".
27 include "Basic-1/arity/aprem.ma".
29 include "Basic-1/llt/props.ma".
31 include "Basic-1/drop1/getl.ma".
33 include "Basic-1/drop1/props.ma".
35 include "Basic-1/lift1/props.ma".
37 theorem sc3_arity_gen:
38 \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((sc3 g a c
39 t) \to (arity g c t a)))))
41 \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(A_ind
42 (\lambda (a0: A).((sc3 g a0 c t) \to (arity g c t a0))) (\lambda (n:
43 nat).(\lambda (n0: nat).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c
44 t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (arity
45 g c t (ASort n n0)) (\lambda (H1: (arity g c t (ASort n n0))).(\lambda (_:
46 (sn3 c t)).H1)) H0))))) (\lambda (a0: A).(\lambda (_: (((sc3 g a0 c t) \to
47 (arity g c t a0)))).(\lambda (a1: A).(\lambda (_: (((sc3 g a1 c t) \to (arity
48 g c t a1)))).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d:
49 C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
50 \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H1 in
51 (land_ind (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g
52 a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat
53 Appl) w (lift1 is t)))))))) (arity g c t (AHead a0 a1)) (\lambda (H3: (arity
54 g c t (AHead a0 a1))).(\lambda (_: ((\forall (d: C).(\forall (w: T).((sc3 g
55 a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat
56 Appl) w (lift1 is t)))))))))).H3)) H2))))))) a)))).
62 \forall (g: G).(\forall (a1: A).(\forall (c: C).(\forall (t: T).((sc3 g a1 c
63 t) \to (\forall (a2: A).((leq g a1 a2) \to (sc3 g a2 c t)))))))
65 \lambda (g: G).(\lambda (a1: A).(llt_wf_ind (\lambda (a: A).(\forall (c:
66 C).(\forall (t: T).((sc3 g a c t) \to (\forall (a2: A).((leq g a a2) \to (sc3
67 g a2 c t))))))) (\lambda (a2: A).(A_ind (\lambda (a: A).(((\forall (a3:
68 A).((llt a3 a) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 c t) \to
69 (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t))))))))) \to (\forall (c:
70 C).(\forall (t: T).((sc3 g a c t) \to (\forall (a3: A).((leq g a a3) \to (sc3
71 g a3 c t)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (_: ((\forall
72 (a3: A).((llt a3 (ASort n n0)) \to (\forall (c: C).(\forall (t: T).((sc3 g a3
73 c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t)))))))))).(\lambda
74 (c: C).(\lambda (t: T).(\lambda (H0: (land (arity g c t (ASort n n0)) (sn3 c
75 t))).(\lambda (a3: A).(\lambda (H1: (leq g (ASort n n0) a3)).(let H2 \def H0
76 in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sc3 g a3 c t) (\lambda
77 (H3: (arity g c t (ASort n n0))).(\lambda (H4: (sn3 c t)).(let H_y \def
78 (arity_repl g c t (ASort n n0) H3 a3 H1) in (let H_x \def (leq_gen_sort1 g n
79 n0 a3 H1) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2:
80 nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort n n0) k)
81 (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
82 (_: nat).(eq A a3 (ASort h2 n2))))) (sc3 g a3 c t) (\lambda (x0:
83 nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A (aplus g (ASort
84 n n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A a3 (ASort x1
85 x0))).(let H8 \def (f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H7) in
86 (let H9 \def (eq_ind A a3 (\lambda (a: A).(arity g c t a)) H_y (ASort x1 x0)
87 H8) in (eq_ind_r A (ASort x1 x0) (\lambda (a: A).(sc3 g a c t)) (conj (arity
88 g c t (ASort x1 x0)) (sn3 c t) H9 H4) a3 H8)))))))) H5)))))) H2))))))))))
89 (\lambda (a: A).(\lambda (_: ((((\forall (a3: A).((llt a3 a) \to (\forall (c:
90 C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to
91 (sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a c t) \to
92 (\forall (a3: A).((leq g a a3) \to (sc3 g a3 c t))))))))).(\lambda (a0:
93 A).(\lambda (H0: ((((\forall (a3: A).((llt a3 a0) \to (\forall (c:
94 C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to
95 (sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a0 c t)
96 \to (\forall (a3: A).((leq g a0 a3) \to (sc3 g a3 c t))))))))).(\lambda (H1:
97 ((\forall (a3: A).((llt a3 (AHead a a0)) \to (\forall (c: C).(\forall (t:
98 T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c
99 t)))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H2: (land (arity g c t
100 (AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall
101 (is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is
102 t)))))))))).(\lambda (a3: A).(\lambda (H3: (leq g (AHead a a0) a3)).(let H4
103 \def H2 in (land_ind (arity g c t (AHead a a0)) (\forall (d: C).(\forall (w:
104 T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
105 (THead (Flat Appl) w (lift1 is t)))))))) (sc3 g a3 c t) (\lambda (H5: (arity
106 g c t (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a
107 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat
108 Appl) w (lift1 is t)))))))))).(let H_x \def (leq_gen_head1 g a a0 a3 H3) in
109 (let H7 \def H_x in (ex3_2_ind A A (\lambda (a4: A).(\lambda (_: A).(leq g a
110 a4))) (\lambda (_: A).(\lambda (a5: A).(leq g a0 a5))) (\lambda (a4:
111 A).(\lambda (a5: A).(eq A a3 (AHead a4 a5)))) (sc3 g a3 c t) (\lambda (x0:
112 A).(\lambda (x1: A).(\lambda (H8: (leq g a x0)).(\lambda (H9: (leq g a0
113 x1)).(\lambda (H10: (eq A a3 (AHead x0 x1))).(let H11 \def (f_equal A A
114 (\lambda (e: A).e) a3 (AHead x0 x1) H10) in (eq_ind_r A (AHead x0 x1)
115 (\lambda (a4: A).(sc3 g a4 c t)) (conj (arity g c t (AHead x0 x1)) (\forall
116 (d: C).(\forall (w: T).((sc3 g x0 d w) \to (\forall (is: PList).((drop1 is d
117 c) \to (sc3 g x1 d (THead (Flat Appl) w (lift1 is t)))))))) (arity_repl g c t
118 (AHead a a0) H5 (AHead x0 x1) (leq_head g a x0 H8 a0 x1 H9)) (\lambda (d:
119 C).(\lambda (w: T).(\lambda (H12: (sc3 g x0 d w)).(\lambda (is:
120 PList).(\lambda (H13: (drop1 is d c)).(H0 (\lambda (a4: A).(\lambda (H14:
121 (llt a4 a0)).(\lambda (c0: C).(\lambda (t0: T).(\lambda (H15: (sc3 g a4 c0
122 t0)).(\lambda (a5: A).(\lambda (H16: (leq g a4 a5)).(H1 a4 (llt_trans a4 a0
123 (AHead a a0) H14 (llt_head_dx a a0)) c0 t0 H15 a5 H16)))))))) d (THead (Flat
124 Appl) w (lift1 is t)) (H6 d w (H1 x0 (llt_repl g a x0 H8 (AHead a a0)
125 (llt_head_sx a a0)) d w H12 a (leq_sym g a x0 H8)) is H13) x1 H9))))))) a3
126 H11))))))) H7))))) H4)))))))))))) a2)) a1)).
132 \forall (g: G).(\forall (a: A).(\forall (e: C).(\forall (t: T).((sc3 g a e
133 t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e)
134 \to (sc3 g a c (lift h d t))))))))))
136 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (e:
137 C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h:
138 nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d t))))))))))
139 (\lambda (n: nat).(\lambda (n0: nat).(\lambda (e: C).(\lambda (t: T).(\lambda
140 (H: (land (arity g e t (ASort n n0)) (sn3 e t))).(\lambda (c: C).(\lambda (h:
141 nat).(\lambda (d: nat).(\lambda (H0: (drop h d c e)).(let H1 \def H in
142 (land_ind (arity g e t (ASort n n0)) (sn3 e t) (land (arity g c (lift h d t)
143 (ASort n n0)) (sn3 c (lift h d t))) (\lambda (H2: (arity g e t (ASort n
144 n0))).(\lambda (H3: (sn3 e t)).(conj (arity g c (lift h d t) (ASort n n0))
145 (sn3 c (lift h d t)) (arity_lift g e t (ASort n n0) H2 c h d H0) (sn3_lift e
146 t H3 c h d H0)))) H1))))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (e:
147 C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h:
148 nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d
149 t))))))))))).(\lambda (a1: A).(\lambda (_: ((\forall (e: C).(\forall (t:
150 T).((sc3 g a1 e t) \to (\forall (c: C).(\forall (h: nat).(\forall (d:
151 nat).((drop h d c e) \to (sc3 g a1 c (lift h d t))))))))))).(\lambda (e:
152 C).(\lambda (t: T).(\lambda (H1: (land (arity g e t (AHead a0 a1)) (\forall
153 (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d
154 e) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(\lambda (c:
155 C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h d c e)).(let H3
156 \def H1 in (land_ind (arity g e t (AHead a0 a1)) (\forall (d0: C).(\forall
157 (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 e) \to (sc3 g
158 a1 d0 (THead (Flat Appl) w (lift1 is t)))))))) (land (arity g c (lift h d t)
159 (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall
160 (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
161 (lift h d t)))))))))) (\lambda (H4: (arity g e t (AHead a0 a1))).(\lambda
162 (H5: ((\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is:
163 PList).((drop1 is d0 e) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
164 t)))))))))).(conj (arity g c (lift h d t) (AHead a0 a1)) (\forall (d0:
165 C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c)
166 \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (lift h d t)))))))))
167 (arity_lift g e t (AHead a0 a1) H4 c h d H2) (\lambda (d0: C).(\lambda (w:
168 T).(\lambda (H6: (sc3 g a0 d0 w)).(\lambda (is: PList).(\lambda (H7: (drop1
169 is d0 c)).(let H_y \def (H5 d0 w H6 (PConsTail is h d)) in (eq_ind T (lift1
170 (PConsTail is h d) t) (\lambda (t0: T).(sc3 g a1 d0 (THead (Flat Appl) w
171 t0))) (H_y (drop1_cons_tail c e h d H2 is d0 H7)) (lift1 is (lift h d t))
172 (lift1_cons_tail t h d is))))))))))) H3))))))))))))) a)).
178 \forall (g: G).(\forall (e: C).(\forall (a: A).(\forall (hds:
179 PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 hds c e)
180 \to (sc3 g a c (lift1 hds t)))))))))
182 \lambda (g: G).(\lambda (e: C).(\lambda (a: A).(\lambda (hds:
183 PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (t: T).((sc3 g
184 a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t))))))) (\lambda (c:
185 C).(\lambda (t: T).(\lambda (H: (sc3 g a e t)).(\lambda (H0: (drop1 PNil c
186 e)).(let H_y \def (drop1_gen_pnil c e H0) in (eq_ind_r C e (\lambda (c0:
187 C).(sc3 g a c0 t)) H c H_y)))))) (\lambda (n: nat).(\lambda (n0:
188 nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (t: T).((sc3
189 g a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t)))))))).(\lambda (c:
190 C).(\lambda (t: T).(\lambda (H0: (sc3 g a e t)).(\lambda (H1: (drop1 (PCons n
191 n0 p) c e)).(let H_x \def (drop1_gen_pcons c e p n n0 H1) in (let H2 \def H_x
192 in (ex2_ind C (\lambda (c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2
193 e)) (sc3 g a c (lift n n0 (lift1 p t))) (\lambda (x: C).(\lambda (H3: (drop n
194 n0 c x)).(\lambda (H4: (drop1 p x e)).(sc3_lift g a x (lift1 p t) (H x t H0
195 H4) c n n0 H3)))) H2))))))))))) hds)))).
201 \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (i:
202 nat).(\forall (d: C).(\forall (v: T).(\forall (c: C).((sc3 g a c (THeads
203 (Flat Appl) vs (lift (S i) O v))) \to ((getl i c (CHead d (Bind Abbr) v)) \to
204 (sc3 g a c (THeads (Flat Appl) vs (TLRef i)))))))))))
206 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs:
207 TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c:
208 C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c
209 (CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs (TLRef
210 i))))))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (vs:
211 TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(\lambda (c:
212 C).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs (lift (S i) O v))
213 (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (lift (S i) O v))))).(\lambda
214 (H0: (getl i c (CHead d (Bind Abbr) v))).(let H1 \def H in (land_ind (arity g
215 c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0)) (sn3 c (THeads (Flat
216 Appl) vs (lift (S i) O v))) (land (arity g c (THeads (Flat Appl) vs (TLRef
217 i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))) (\lambda (H2:
218 (arity g c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0))).(\lambda
219 (H3: (sn3 c (THeads (Flat Appl) vs (lift (S i) O v)))).(conj (arity g c
220 (THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs
221 (TLRef i))) (arity_appls_abbr g c d v i H0 vs (ASort n n0) H2)
222 (sn3_appls_abbr c d v i H0 vs H3)))) H1))))))))))) (\lambda (a0: A).(\lambda
223 (_: ((\forall (vs: TList).(\forall (i: nat).(\forall (d: C).(\forall (v:
224 T).(\forall (c: C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to
225 ((getl i c (CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs
226 (TLRef i)))))))))))).(\lambda (a1: A).(\lambda (H0: ((\forall (vs:
227 TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c:
228 C).((sc3 g a1 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c
229 (CHead d (Bind Abbr) v)) \to (sc3 g a1 c (THeads (Flat Appl) vs (TLRef
230 i)))))))))))).(\lambda (vs: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda
231 (v: T).(\lambda (c: C).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs
232 (lift (S i) O v)) (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0
233 d0 w) \to (\forall (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat
234 Appl) w (lift1 is (THeads (Flat Appl) vs (lift (S i) O v)))))))))))).(\lambda
235 (H2: (getl i c (CHead d (Bind Abbr) v))).(let H3 \def H1 in (land_ind (arity
236 g c (THeads (Flat Appl) vs (lift (S i) O v)) (AHead a0 a1)) (\forall (d0:
237 C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c)
238 \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift
239 (S i) O v)))))))))) (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead
240 a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is:
241 PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
242 (THeads (Flat Appl) vs (TLRef i))))))))))) (\lambda (H4: (arity g c (THeads
243 (Flat Appl) vs (lift (S i) O v)) (AHead a0 a1))).(\lambda (H5: ((\forall (d0:
244 C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c)
245 \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift
246 (S i) O v)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (TLRef i))
247 (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall
248 (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
249 (THeads (Flat Appl) vs (TLRef i)))))))))) (arity_appls_abbr g c d v i H2 vs
250 (AHead a0 a1) H4) (\lambda (d0: C).(\lambda (w: T).(\lambda (H6: (sc3 g a0 d0
251 w)).(\lambda (is: PList).(\lambda (H7: (drop1 is d0 c)).(let H_x \def
252 (drop1_getl_trans is c d0 H7 Abbr d v i H2) in (let H8 \def H_x in (ex2_ind C
253 (\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: C).(getl (trans is
254 i) d0 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) v)))) (sc3 g a1 d0 (THead
255 (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (x:
256 C).(\lambda (_: (drop1 (ptrans is i) x d)).(\lambda (H10: (getl (trans is i)
257 d0 (CHead x (Bind Abbr) (lift1 (ptrans is i) v)))).(let H_y \def (H0 (TCons w
258 (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is
259 (TLRef i))) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w t))) (eq_ind_r
260 T (TLRef (trans is i)) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w
261 (THeads (Flat Appl) (lifts1 is vs) t)))) (H_y (trans is i) x (lift1 (ptrans
262 is i) v) d0 (eq_ind T (lift1 is (lift (S i) O v)) (\lambda (t: T).(sc3 g a1
263 d0 (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 is vs) t)))) (eq_ind T
264 (lift1 is (THeads (Flat Appl) vs (lift (S i) O v))) (\lambda (t: T).(sc3 g a1
265 d0 (THead (Flat Appl) w t))) (H5 d0 w H6 is H7) (THeads (Flat Appl) (lifts1
266 is vs) (lift1 is (lift (S i) O v))) (lifts1_flat Appl is (lift (S i) O v)
267 vs)) (lift (S (trans is i)) O (lift1 (ptrans is i) v)) (lift1_free is i v))
268 H10) (lift1 is (TLRef i)) (lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs
269 (TLRef i))) (lifts1_flat Appl is (TLRef i) vs)))))) H8)))))))))))
276 \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall
277 (u: T).((sc3 g (asucc g a) c (THeads (Flat Appl) vs u)) \to (\forall (t:
278 T).((sc3 g a c (THeads (Flat Appl) vs t)) \to (sc3 g a c (THeads (Flat Appl)
279 vs (THead (Flat Cast) u t))))))))))
281 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs:
282 TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat
283 Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to
284 (sc3 g a0 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (\lambda
285 (n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u:
286 T).(\lambda (H: (sc3 g (match n with [O \Rightarrow (ASort O (next g n0)) |
287 (S h) \Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u))).(\lambda (t:
288 T).(\lambda (H0: (land (arity g c (THeads (Flat Appl) vs t) (ASort n n0))
289 (sn3 c (THeads (Flat Appl) vs t)))).(nat_ind (\lambda (n1: nat).((sc3 g
290 (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow
291 (ASort h n0)]) c (THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads
292 (Flat Appl) vs t) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land
293 (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0))
294 (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))) (\lambda (H1:
295 (sc3 g (ASort O (next g n0)) c (THeads (Flat Appl) vs u))).(\lambda (H2:
296 (land (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads (Flat
297 Appl) vs t)))).(let H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs
298 u) (ASort O (next g n0))) (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c
299 (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads
300 (Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads
301 (Flat Appl) vs u) (ASort O (next g n0)))).(\lambda (H5: (sn3 c (THeads (Flat
302 Appl) vs u))).(let H6 \def H2 in (land_ind (arity g c (THeads (Flat Appl) vs
303 t) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity g c (THeads
304 (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat
305 Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat
306 Appl) vs t) (ASort O n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs
307 t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort
308 O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))
309 (arity_appls_cast g c u t vs (ASort O n0) H4 H7) (sn3_appls_cast c vs u H5 t
310 H8)))) H6)))) H3)))) (\lambda (n1: nat).(\lambda (_: (((sc3 g (match n1 with
311 [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) c
312 (THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads (Flat Appl) vs t)
313 (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land (arity g c
314 (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) (sn3 c (THeads
315 (Flat Appl) vs (THead (Flat Cast) u t)))))))).(\lambda (H1: (sc3 g (ASort n1
316 n0) c (THeads (Flat Appl) vs u))).(\lambda (H2: (land (arity g c (THeads
317 (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let
318 H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (ASort n1 n0))
319 (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c (THeads (Flat Appl) vs
320 (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs
321 (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u)
322 (ASort n1 n0))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def
323 H2 in (land_ind (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3
324 c (THeads (Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead
325 (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead
326 (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort
327 (S n1) n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g
328 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c
329 (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (arity_appls_cast g c u t vs
330 (ASort (S n1) n0) H4 H7) (sn3_appls_cast c vs u H5 t H8)))) H6)))) H3)))))) n
331 H H0))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (vs: TList).(\forall
332 (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to
333 (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c
334 (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (a1:
335 A).(\lambda (H0: ((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3
336 g (asucc g a1) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a1 c
337 (THeads (Flat Appl) vs t)) \to (sc3 g a1 c (THeads (Flat Appl) vs (THead
338 (Flat Cast) u t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u:
339 T).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc
340 g a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
341 PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1
342 is (THeads (Flat Appl) vs u))))))))))).(\lambda (t: T).(\lambda (H2: (land
343 (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall
344 (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1
345 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))))).(let H3
346 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g
347 a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
348 PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1
349 is (THeads (Flat Appl) vs u))))))))) (land (arity g c (THeads (Flat Appl) vs
350 (THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3
351 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead
352 (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u
353 t))))))))))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (AHead a0
354 (asucc g a1)))).(\lambda (H5: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d
355 w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead
356 (Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))))).(let H6 \def H2
357 in (land_ind (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d:
358 C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
359 \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs
360 t))))))))) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t))
361 (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall
362 (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is
363 (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))) (\lambda (H7: (arity
364 g c (THeads (Flat Appl) vs t) (AHead a0 a1))).(\lambda (H8: ((\forall (d:
365 C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
366 \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs
367 t))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t))
368 (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall
369 (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is
370 (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (arity_appls_cast g c
371 u t vs (AHead a0 a1) H4 H7) (\lambda (d: C).(\lambda (w: T).(\lambda (H9:
372 (sc3 g a0 d w)).(\lambda (is: PList).(\lambda (H10: (drop1 is d c)).(let H_y
373 \def (H0 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1
374 is vs) (lift1 is (THead (Flat Cast) u t))) (\lambda (t0: T).(sc3 g a1 d
375 (THead (Flat Appl) w t0))) (eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1
376 is t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat Appl)
377 (lifts1 is vs) t0)))) (H_y d (lift1 is u) (eq_ind T (lift1 is (THeads (Flat
378 Appl) vs u)) (\lambda (t0: T).(sc3 g (asucc g a1) d (THead (Flat Appl) w
379 t0))) (H5 d w H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is u))
380 (lifts1_flat Appl is u vs)) (lift1 is t) (eq_ind T (lift1 is (THeads (Flat
381 Appl) vs t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w t0))) (H8 d w
382 H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is t)) (lifts1_flat Appl
383 is t vs))) (lift1 is (THead (Flat Cast) u t)) (lift1_flat Cast is u t))
384 (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (lifts1_flat Appl
385 is (THead (Flat Cast) u t) vs))))))))))) H6)))) H3)))))))))))) a)).
390 theorem sc3_props__sc3_sn3_abst:
391 \forall (g: G).(\forall (a: A).(land (\forall (c: C).(\forall (t: T).((sc3 g
392 a c t) \to (sn3 c t)))) (\forall (vs: TList).(\forall (i: nat).(let t \def
393 (THeads (Flat Appl) vs (TLRef i)) in (\forall (c: C).((arity g c t a) \to
394 ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a c t))))))))))
396 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(land (\forall (c:
397 C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall (vs:
398 TList).(\forall (i: nat).(let t \def (THeads (Flat Appl) vs (TLRef i)) in
399 (\forall (c: C).((arity g c t a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to
400 (sc3 g a0 c t)))))))))) (\lambda (n: nat).(\lambda (n0: nat).(conj (\forall
401 (c: C).(\forall (t: T).((land (arity g c t (ASort n n0)) (sn3 c t)) \to (sn3
402 c t)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c
403 (THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) \to ((nf2 c (TLRef i)) \to
404 ((sns3 c vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n
405 n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))) (\lambda (c:
406 C).(\lambda (t: T).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c
407 t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sn3 c
408 t) (\lambda (_: (arity g c t (ASort n n0))).(\lambda (H2: (sn3 c t)).H2))
409 H0))))) (\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H:
410 (arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n n0))).(\lambda (H0:
411 (nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(conj (arity g c (THeads (Flat
412 Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i))) H
413 (sn3_appls_lref c i H0 vs H1))))))))))) (\lambda (a0: A).(\lambda (H: (land
414 (\forall (c: C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall
415 (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads (Flat Appl)
416 vs (TLRef i)) a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a0 c
417 (THeads (Flat Appl) vs (TLRef i))))))))))).(\lambda (a1: A).(\lambda (H0:
418 (land (\forall (c: C).(\forall (t: T).((sc3 g a1 c t) \to (sn3 c t))))
419 (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads
420 (Flat Appl) vs (TLRef i)) a1) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to
421 (sc3 g a1 c (THeads (Flat Appl) vs (TLRef i))))))))))).(conj (\forall (c:
422 C).(\forall (t: T).((land (arity g c t (AHead a0 a1)) (\forall (d:
423 C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
424 \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t))))))))) \to (sn3 c t))))
425 (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads
426 (Flat Appl) vs (TLRef i)) (AHead a0 a1)) \to ((nf2 c (TLRef i)) \to ((sns3 c
427 vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))
428 (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
429 PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads
430 (Flat Appl) vs (TLRef i))))))))))))))))) (\lambda (c: C).(\lambda (t:
431 T).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall
432 (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1
433 d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H in (land_ind
434 (\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 t0))))
435 (\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads
436 (Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to
437 (sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) (\lambda (_:
438 ((\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0
439 t0)))))).(\lambda (H4: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0:
440 C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i))
441 \to ((sns3 c0 vs) \to (sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef
442 i))))))))))).(let H5 \def H0 in (land_ind (\forall (c0: C).(\forall (t0:
443 T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))) (\forall (vs: TList).(\forall (i:
444 nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a1) \to
445 ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 (THeads (Flat Appl) vs
446 (TLRef i))))))))) (sn3 c t) (\lambda (H6: ((\forall (c0: C).(\forall (t0:
447 T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))))).(\lambda (_: ((\forall (vs:
448 TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs
449 (TLRef i)) a1) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0
450 (THeads (Flat Appl) vs (TLRef i))))))))))).(let H8 \def H1 in (land_ind
451 (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w)
452 \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w
453 (lift1 is t)))))))) (sn3 c t) (\lambda (H9: (arity g c t (AHead a0
454 a1))).(\lambda (H10: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to
455 (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w
456 (lift1 is t)))))))))).(let H_y \def (arity_aprem g c t (AHead a0 a1) H9 O a0)
457 in (let H11 \def (H_y (aprem_zero a0 a1)) in (ex2_3_ind C T nat (\lambda (d:
458 C).(\lambda (_: T).(\lambda (j: nat).(drop j O d c)))) (\lambda (d:
459 C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g a0))))) (sn3 c t)
460 (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H12: (drop x2
461 O x0 c)).(\lambda (H13: (arity g x0 x1 (asucc g a0))).(let H_y0 \def (H10
462 (CHead x0 (Bind Abst) x1) (TLRef O) (H4 TNil O (CHead x0 (Bind Abst) x1)
463 (arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1) a0
464 H13) (nf2_lref_abst (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1))
465 I) (PCons (S x2) O PNil)) in (let H_y1 \def (H6 (CHead x0 (Bind Abst) x1)
466 (THead (Flat Appl) (TLRef O) (lift (S x2) O t)) (H_y0 (drop1_cons (CHead x0
467 (Bind Abst) x1) c (S x2) O (drop_drop (Bind Abst) x2 x0 c H12 x1) c PNil
468 (drop1_nil c)))) in (let H_x \def (sn3_gen_flat Appl (CHead x0 (Bind Abst)
469 x1) (TLRef O) (lift (S x2) O t) H_y1) in (let H14 \def H_x in (land_ind (sn3
470 (CHead x0 (Bind Abst) x1) (TLRef O)) (sn3 (CHead x0 (Bind Abst) x1) (lift (S
471 x2) O t)) (sn3 c t) (\lambda (_: (sn3 (CHead x0 (Bind Abst) x1) (TLRef
472 O))).(\lambda (H16: (sn3 (CHead x0 (Bind Abst) x1) (lift (S x2) O
473 t))).(sn3_gen_lift (CHead x0 (Bind Abst) x1) t (S x2) O H16 c (drop_drop
474 (Bind Abst) x2 x0 c H12 x1)))) H14)))))))))) H11))))) H8)))) H5)))) H2)))))
475 (\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H1: (arity g
476 c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))).(\lambda (H2: (nf2 c
477 (TLRef i))).(\lambda (H3: (sns3 c vs)).(conj (arity g c (THeads (Flat Appl)
478 vs (TLRef i)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w)
479 \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w
480 (lift1 is (THeads (Flat Appl) vs (TLRef i)))))))))) H1 (\lambda (d:
481 C).(\lambda (w: T).(\lambda (H4: (sc3 g a0 d w)).(\lambda (is:
482 PList).(\lambda (H5: (drop1 is d c)).(let H6 \def H in (land_ind (\forall
483 (c0: C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))) (\forall (vs0:
484 TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl)
485 vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a0
486 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a1 d (THead (Flat Appl)
487 w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (H7: ((\forall (c0:
488 C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))))).(\lambda (_:
489 ((\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0
490 (THeads (Flat Appl) vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3
491 c0 vs0) \to (sc3 g a0 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))))).(let H9
492 \def H0 in (land_ind (\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) \to
493 (sn3 c0 t)))) (\forall (vs0: TList).(\forall (i0: nat).(\forall (c0:
494 C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) \to ((nf2 c0 (TLRef
495 i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat Appl) vs0 (TLRef
496 i0))))))))) (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs
497 (TLRef i))))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t)
498 \to (sn3 c0 t)))))).(\lambda (H11: ((\forall (vs0: TList).(\forall (i0:
499 nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1)
500 \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat
501 Appl) vs0 (TLRef i0))))))))))).(let H_y \def (H11 (TCons w (lifts1 is vs)))
502 in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i)))
503 (\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w t))) (eq_ind_r T (TLRef
504 (trans is i)) (\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat
505 Appl) (lifts1 is vs) t)))) (H_y (trans is i) d (eq_ind T (lift1 is (TLRef i))
506 (\lambda (t: T).(arity g d (THead (Flat Appl) w (THeads (Flat Appl) (lifts1
507 is vs) t)) a1)) (eq_ind T (lift1 is (THeads (Flat Appl) vs (TLRef i)))
508 (\lambda (t: T).(arity g d (THead (Flat Appl) w t) a1)) (arity_appl g d w a0
509 (sc3_arity_gen g d w a0 H4) (lift1 is (THeads (Flat Appl) vs (TLRef i))) a1
510 (arity_lift1 g (AHead a0 a1) c is d (THeads (Flat Appl) vs (TLRef i)) H5 H1))
511 (THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) (lifts1_flat Appl is
512 (TLRef i) vs)) (TLRef (trans is i)) (lift1_lref is i)) (eq_ind T (lift1 is
513 (TLRef i)) (\lambda (t: T).(nf2 d t)) (nf2_lift1 c is d (TLRef i) H5 H2)
514 (TLRef (trans is i)) (lift1_lref is i)) (conj (sn3 d w) (sns3 d (lifts1 is
515 vs)) (H7 d w H4) (sns3_lifts1 c is d H5 vs H3))) (lift1 is (TLRef i))
516 (lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs (TLRef i))) (lifts1_flat
517 Appl is (TLRef i) vs))))) H9)))) H6))))))))))))))))))) a)).
523 \forall (g: G).(\forall (a: A).(\forall (c: C).(\forall (t: T).((sc3 g a c
526 \lambda (g: G).(\lambda (a: A).(\lambda (c: C).(\lambda (t: T).(\lambda (H:
527 (sc3 g a c t)).(let H_x \def (sc3_props__sc3_sn3_abst g a) in (let H0 \def
528 H_x in (land_ind (\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3
529 c0 t0)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g
530 c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0
531 vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t)
532 (\lambda (H1: ((\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 c0
533 t0)))))).(\lambda (_: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0:
534 C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i))
535 \to ((sns3 c0 vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef
536 i))))))))))).(H1 c t H))) H0))))))).
542 \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall
543 (i: nat).((arity g c (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c (TLRef
544 i)) \to ((sns3 c vs) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i))))))))))
546 \lambda (g: G).(\lambda (a: A).(\lambda (vs: TList).(\lambda (c: C).(\lambda
547 (i: nat).(\lambda (H: (arity g c (THeads (Flat Appl) vs (TLRef i))
548 a)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(let H_x \def
549 (sc3_props__sc3_sn3_abst g a) in (let H2 \def H_x in (land_ind (\forall (c0:
550 C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0 t)))) (\forall (vs0:
551 TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl)
552 vs0 (TLRef i0)) a) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a
553 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a c (THeads (Flat Appl)
554 vs (TLRef i))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a c0 t)
555 \to (sn3 c0 t)))))).(\lambda (H4: ((\forall (vs0: TList).(\forall (i0:
556 nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a) \to
557 ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a c0 (THeads (Flat Appl)
558 vs0 (TLRef i0))))))))))).(H4 vs i c H H0 H1))) H2)))))))))).
564 \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1:
565 A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v:
566 T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts
567 (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a2 c (THeads (Flat Appl) vs
568 (THead (Bind b) v t)))))))))))))
570 \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda
571 (a1: A).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (vs: TList).(\forall
572 (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads
573 (Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads
574 (Flat Appl) vs (THead (Bind b) v t)))))))))) (\lambda (n: nat).(\lambda (n0:
575 nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t:
576 T).(\lambda (H0: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl)
577 (lifts (S O) O vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat
578 Appl) (lifts (S O) O vs) t)))).(\lambda (H1: (sc3 g a1 c v)).(let H2 \def H0
579 in (land_ind (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O
580 vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S
581 O) O vs) t)) (land (arity g c (THeads (Flat Appl) vs (THead (Bind b) v t))
582 (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t)))) (\lambda
583 (H3: (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)
584 (ASort n n0))).(\lambda (H4: (sn3 (CHead c (Bind b) v) (THeads (Flat Appl)
585 (lifts (S O) O vs) t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Bind
586 b) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t)))
587 (arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 H1) t vs (ASort n n0)
588 H3) (sn3_appls_bind b H c v (sc3_sn3 g a1 c v H1) vs t H4)))) H2))))))))))
589 (\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall (c: C).(\forall
590 (v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads (Flat Appl)
591 (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads (Flat Appl)
592 vs (THead (Bind b) v t))))))))))).(\lambda (a0: A).(\lambda (H1: ((\forall
593 (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 (CHead
594 c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v)
595 \to (sc3 g a0 c (THeads (Flat Appl) vs (THead (Bind b) v
596 t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda
597 (t: T).(\lambda (H2: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl)
598 (lifts (S O) O vs) t) (AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a
599 d w) \to (\forall (is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g
600 a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs)
601 t))))))))))).(\lambda (H3: (sc3 g a1 c v)).(let H4 \def H2 in (land_ind
602 (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)
603 (AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall
604 (is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat
605 Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))) (land
606 (arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0))
607 (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is:
608 PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads
609 (Flat Appl) vs (THead (Bind b) v t))))))))))) (\lambda (H5: (arity g (CHead c
610 (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) (AHead a a0))).(\lambda
611 (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is:
612 PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat Appl)
613 w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))))).(conj (arity
614 g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) (\forall (d:
615 C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c)
616 \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead
617 (Bind b) v t)))))))))) (arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1
618 H3) t vs (AHead a a0) H5) (\lambda (d: C).(\lambda (w: T).(\lambda (H7: (sc3
619 g a d w)).(\lambda (is: PList).(\lambda (H8: (drop1 is d c)).(let H_y \def
620 (H1 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is
621 vs) (lift1 is (THead (Bind b) v t))) (\lambda (t0: T).(sc3 g a0 d (THead
622 (Flat Appl) w t0))) (eq_ind_r T (THead (Bind b) (lift1 is v) (lift1 (Ss is)
623 t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w (THeads (Flat Appl)
624 (lifts1 is vs) t0)))) (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind TList
625 (lifts1 (Ss is) (lifts (S O) O vs)) (\lambda (t0: TList).(sc3 g a0 (CHead d
626 (Bind b) (lift1 is v)) (THead (Flat Appl) (lift (S O) O w) (THeads (Flat
627 Appl) t0 (lift1 (Ss is) t))))) (eq_ind T (lift1 (Ss is) (THeads (Flat Appl)
628 (lifts (S O) O vs) t)) (\lambda (t0: T).(sc3 g a0 (CHead d (Bind b) (lift1 is
629 v)) (THead (Flat Appl) (lift (S O) O w) t0))) (H6 (CHead d (Bind b) (lift1 is
630 v)) (lift (S O) O w) (sc3_lift g a d w H7 (CHead d (Bind b) (lift1 is v)) (S
631 O) O (drop_drop (Bind b) O d d (drop_refl d) (lift1 is v))) (Ss is)
632 (drop1_skip_bind b c is d v H8)) (THeads (Flat Appl) (lifts1 (Ss is) (lifts
633 (S O) O vs)) (lift1 (Ss is) t)) (lifts1_flat Appl (Ss is) t (lifts (S O) O
634 vs))) (lifts (S O) O (lifts1 is vs)) (lifts1_xhg is vs)) (sc3_lift1 g c a1 is
635 d v H3 H8)) (lift1 is (THead (Bind b) v t)) (lift1_bind b is v t)) (lift1 is
636 (THeads (Flat Appl) vs (THead (Bind b) v t))) (lifts1_flat Appl is (THead
637 (Bind b) v t) vs))))))))))) H4)))))))))))) a2))))).
643 \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (vs:
644 TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 c (THeads
645 (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w:
646 T).((sc3 g (asucc g a1) c w) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead
647 (Flat Appl) v (THead (Bind Abst) w t))))))))))))))
649 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(A_ind (\lambda (a:
650 A).(\forall (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3
651 g a c (THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v)
652 \to (\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a c (THeads (Flat
653 Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))))))))))) (\lambda
654 (n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v:
655 T).(\lambda (t: T).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs
656 (THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead
657 (Bind Abbr) v t))))).(\lambda (H0: (sc3 g a1 c v)).(\lambda (w: T).(\lambda
658 (H1: (sc3 g (asucc g a1) c w)).(let H2 \def H in (land_ind (arity g c (THeads
659 (Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat
660 Appl) vs (THead (Bind Abbr) v t))) (land (arity g c (THeads (Flat Appl) vs
661 (THead (Flat Appl) v (THead (Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads
662 (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H3:
663 (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n
664 n0))).(\lambda (H4: (sn3 c (THeads (Flat Appl) vs (THead (Bind Abbr) v
665 t)))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v (THead
666 (Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat
667 Appl) v (THead (Bind Abst) w t)))) (arity_appls_appl g c v a1 (sc3_arity_gen
668 g c v a1 H0) w (sc3_arity_gen g c w (asucc g a1) H1) t vs (ASort n n0) H3)
669 (sn3_appls_beta c v t vs H4 w (sc3_sn3 g (asucc g a1) c w H1)))))
670 H2)))))))))))) (\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall
671 (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a c (THeads (Flat Appl) vs
672 (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: T).((sc3 g
673 (asucc g a1) c w) \to (sc3 g a c (THeads (Flat Appl) vs (THead (Flat Appl) v
674 (THead (Bind Abst) w t)))))))))))))).(\lambda (a0: A).(\lambda (H0: ((\forall
675 (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 c
676 (THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to
677 (\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a0 c (THeads (Flat Appl)
678 vs (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))))))).(\lambda (vs:
679 TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H1: (land
680 (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (AHead a a0))
681 (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is:
682 PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads
683 (Flat Appl) vs (THead (Bind Abbr) v t)))))))))))).(\lambda (H2: (sc3 g a1 c
684 v)).(\lambda (w: T).(\lambda (H3: (sc3 g (asucc g a1) c w)).(let H4 \def H1
685 in (land_ind (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t))
686 (AHead a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall
687 (is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is
688 (THeads (Flat Appl) vs (THead (Bind Abbr) v t)))))))))) (land (arity g c
689 (THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))) (AHead
690 a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall (is:
691 PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is
692 (THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w
693 t)))))))))))) (\lambda (H5: (arity g c (THeads (Flat Appl) vs (THead (Bind
694 Abbr) v t)) (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w0:
695 T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
696 (THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Bind Abbr) v
697 t)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v
698 (THead (Bind Abst) w t))) (AHead a a0)) (\forall (d: C).(\forall (w0:
699 T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
700 (THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Flat Appl) v
701 (THead (Bind Abst) w t))))))))))) (arity_appls_appl g c v a1 (sc3_arity_gen g
702 c v a1 H2) w (sc3_arity_gen g c w (asucc g a1) H3) t vs (AHead a a0) H5)
703 (\lambda (d: C).(\lambda (w0: T).(\lambda (H7: (sc3 g a d w0)).(\lambda (is:
704 PList).(\lambda (H8: (drop1 is d c)).(eq_ind_r T (THeads (Flat Appl) (lifts1
705 is vs) (lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t)))) (\lambda
706 (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 t0))) (eq_ind_r T (THead (Flat
707 Appl) (lift1 is v) (lift1 is (THead (Bind Abst) w t))) (\lambda (t0: T).(sc3
708 g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) t0))))
709 (eq_ind_r T (THead (Bind Abst) (lift1 is w) (lift1 (Ss is) t)) (\lambda (t0:
710 T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs)
711 (THead (Flat Appl) (lift1 is v) t0))))) (let H_y \def (H0 (TCons w0 (lifts1
712 is vs))) in (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind T (lift1 is (THead
713 (Bind Abbr) v t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads
714 (Flat Appl) (lifts1 is vs) t0)))) (eq_ind T (lift1 is (THeads (Flat Appl) vs
715 (THead (Bind Abbr) v t))) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0
716 t0))) (H6 d w0 H7 is H8) (THeads (Flat Appl) (lifts1 is vs) (lift1 is (THead
717 (Bind Abbr) v t))) (lifts1_flat Appl is (THead (Bind Abbr) v t) vs)) (THead
718 (Bind Abbr) (lift1 is v) (lift1 (Ss is) t)) (lift1_bind Abbr is v t))
719 (sc3_lift1 g c a1 is d v H2 H8) (lift1 is w) (sc3_lift1 g c (asucc g a1) is d
720 w H3 H8))) (lift1 is (THead (Bind Abst) w t)) (lift1_bind Abst is w t))
721 (lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t))) (lift1_flat Appl is
722 v (THead (Bind Abst) w t))) (lift1 is (THeads (Flat Appl) vs (THead (Flat
723 Appl) v (THead (Bind Abst) w t)))) (lifts1_flat Appl is (THead (Flat Appl) v
724 (THead (Bind Abst) w t)) vs)))))))))) H4)))))))))))))) a2))).