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 "sc3/defs.ma".
21 include "sn3/lift1.ma".
23 include "nf2/lift1.ma".
25 include "csuba/arity.ma".
27 include "arity/lift1.ma".
29 include "arity/aprem.ma".
31 include "llt/props.ma".
33 include "drop1/getl.ma".
35 include "drop1/props.ma".
37 include "lift1/props.ma".
39 theorem sc3_arity_gen:
40 \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((sc3 g a c
41 t) \to (arity g c t a)))))
43 \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(A_ind
44 (\lambda (a0: A).((sc3 g a0 c t) \to (arity g c t a0))) (\lambda (n:
45 nat).(\lambda (n0: nat).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c
46 t))).(let H0 \def H in (and_ind (arity g c t (ASort n n0)) (sn3 c t) (arity g
47 c t (ASort n n0)) (\lambda (H1: (arity g c t (ASort n n0))).(\lambda (_: (sn3
48 c t)).H1)) H0))))) (\lambda (a0: A).(\lambda (_: (((sc3 g a0 c t) \to (arity
49 g c t a0)))).(\lambda (a1: A).(\lambda (_: (((sc3 g a1 c t) \to (arity g c t
50 a1)))).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d:
51 C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
52 \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H1 in
53 (and_ind (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g
54 a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat
55 Appl) w (lift1 is t)))))))) (arity g c t (AHead a0 a1)) (\lambda (H3: (arity
56 g c t (AHead a0 a1))).(\lambda (_: ((\forall (d: C).(\forall (w: T).((sc3 g
57 a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat
58 Appl) w (lift1 is t)))))))))).H3)) H2))))))) a)))).
61 \forall (g: G).(\forall (a1: A).(\forall (c: C).(\forall (t: T).((sc3 g a1 c
62 t) \to (\forall (a2: A).((leq g a1 a2) \to (sc3 g a2 c t)))))))
64 \lambda (g: G).(\lambda (a1: A).(llt_wf_ind (\lambda (a: A).(\forall (c:
65 C).(\forall (t: T).((sc3 g a c t) \to (\forall (a2: A).((leq g a a2) \to (sc3
66 g a2 c t))))))) (\lambda (a2: A).(A_ind (\lambda (a: A).(((\forall (a3:
67 A).((llt a3 a) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 c t) \to
68 (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t))))))))) \to (\forall (c:
69 C).(\forall (t: T).((sc3 g a c t) \to (\forall (a3: A).((leq g a a3) \to (sc3
70 g a3 c t)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (_: ((\forall
71 (a3: A).((llt a3 (ASort n n0)) \to (\forall (c: C).(\forall (t: T).((sc3 g a3
72 c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t)))))))))).(\lambda
73 (c: C).(\lambda (t: T).(\lambda (H0: (land (arity g c t (ASort n n0)) (sn3 c
74 t))).(\lambda (a3: A).(\lambda (H1: (leq g (ASort n n0) a3)).(let H2 \def H0
75 in (and_ind (arity g c t (ASort n n0)) (sn3 c t) (sc3 g a3 c t) (\lambda (H3:
76 (arity g c t (ASort n n0))).(\lambda (H4: (sn3 c t)).(let H_y \def
77 (arity_repl g c t (ASort n n0) H3 a3 H1) in (let H_x \def (leq_gen_sort g n
78 n0 a3 H1) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2:
79 nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a3 (ASort h2 n2))))) (\lambda
80 (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort n n0) k)
81 (aplus g (ASort h2 n2) k))))) (sc3 g a3 c t) (\lambda (x0: nat).(\lambda (x1:
82 nat).(\lambda (x2: nat).(\lambda (H6: (eq A a3 (ASort x1 x0))).(\lambda (_:
83 (eq A (aplus g (ASort n n0) x2) (aplus g (ASort x1 x0) x2))).(let H8 \def
84 (eq_ind A a3 (\lambda (a: A).(arity g c t a)) H_y (ASort x1 x0) H6) in
85 (eq_ind_r A (ASort x1 x0) (\lambda (a: A).(sc3 g a c t)) (conj (arity g c t
86 (ASort x1 x0)) (sn3 c t) H8 H4) a3 H6))))))) H5)))))) H2)))))))))) (\lambda
87 (a: A).(\lambda (_: ((((\forall (a3: A).((llt a3 a) \to (\forall (c:
88 C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to
89 (sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a c t) \to
90 (\forall (a3: A).((leq g a a3) \to (sc3 g a3 c t))))))))).(\lambda (a0:
91 A).(\lambda (H0: ((((\forall (a3: A).((llt a3 a0) \to (\forall (c:
92 C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to
93 (sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a0 c t)
94 \to (\forall (a3: A).((leq g a0 a3) \to (sc3 g a3 c t))))))))).(\lambda (H1:
95 ((\forall (a3: A).((llt a3 (AHead a a0)) \to (\forall (c: C).(\forall (t:
96 T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c
97 t)))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H2: (land (arity g c t
98 (AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall
99 (is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is
100 t)))))))))).(\lambda (a3: A).(\lambda (H3: (leq g (AHead a a0) a3)).(let H4
101 \def H2 in (and_ind (arity g c t (AHead a a0)) (\forall (d: C).(\forall (w:
102 T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
103 (THead (Flat Appl) w (lift1 is t)))))))) (sc3 g a3 c t) (\lambda (H5: (arity
104 g c t (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a
105 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat
106 Appl) w (lift1 is t)))))))))).(let H_x \def (leq_gen_head g a a0 a3 H3) in
107 (let H7 \def H_x in (ex3_2_ind A A (\lambda (a4: A).(\lambda (a5: A).(eq A a3
108 (AHead a4 a5)))) (\lambda (a4: A).(\lambda (_: A).(leq g a a4))) (\lambda (_:
109 A).(\lambda (a5: A).(leq g a0 a5))) (sc3 g a3 c t) (\lambda (x0: A).(\lambda
110 (x1: A).(\lambda (H8: (eq A a3 (AHead x0 x1))).(\lambda (H9: (leq g a
111 x0)).(\lambda (H10: (leq g a0 x1)).(eq_ind_r A (AHead x0 x1) (\lambda (a4:
112 A).(sc3 g a4 c t)) (conj (arity g c t (AHead x0 x1)) (\forall (d: C).(\forall
113 (w: T).((sc3 g x0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g x1
114 d (THead (Flat Appl) w (lift1 is t)))))))) (arity_repl g c t (AHead a a0) H5
115 (AHead x0 x1) (leq_head g a x0 H9 a0 x1 H10)) (\lambda (d: C).(\lambda (w:
116 T).(\lambda (H11: (sc3 g x0 d w)).(\lambda (is: PList).(\lambda (H12: (drop1
117 is d c)).(H0 (\lambda (a4: A).(\lambda (H13: (llt a4 a0)).(\lambda (c0:
118 C).(\lambda (t0: T).(\lambda (H14: (sc3 g a4 c0 t0)).(\lambda (a5:
119 A).(\lambda (H15: (leq g a4 a5)).(H1 a4 (llt_trans a4 a0 (AHead a a0) H13
120 (llt_head_dx a a0)) c0 t0 H14 a5 H15)))))))) d (THead (Flat Appl) w (lift1 is
121 t)) (H6 d w (H1 x0 (llt_repl g a x0 H9 (AHead a a0) (llt_head_sx a a0)) d w
122 H11 a (leq_sym g a x0 H9)) is H12) x1 H10))))))) a3 H8)))))) H7)))))
123 H4)))))))))))) a2)) a1)).
126 \forall (g: G).(\forall (a: A).(\forall (e: C).(\forall (t: T).((sc3 g a e
127 t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e)
128 \to (sc3 g a c (lift h d t))))))))))
130 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (e:
131 C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h:
132 nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d t))))))))))
133 (\lambda (n: nat).(\lambda (n0: nat).(\lambda (e: C).(\lambda (t: T).(\lambda
134 (H: (land (arity g e t (ASort n n0)) (sn3 e t))).(\lambda (c: C).(\lambda (h:
135 nat).(\lambda (d: nat).(\lambda (H0: (drop h d c e)).(let H1 \def H in
136 (and_ind (arity g e t (ASort n n0)) (sn3 e t) (land (arity g c (lift h d t)
137 (ASort n n0)) (sn3 c (lift h d t))) (\lambda (H2: (arity g e t (ASort n
138 n0))).(\lambda (H3: (sn3 e t)).(conj (arity g c (lift h d t) (ASort n n0))
139 (sn3 c (lift h d t)) (arity_lift g e t (ASort n n0) H2 c h d H0) (sn3_lift e
140 t H3 c h d H0)))) H1))))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (e:
141 C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h:
142 nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d
143 t))))))))))).(\lambda (a1: A).(\lambda (_: ((\forall (e: C).(\forall (t:
144 T).((sc3 g a1 e t) \to (\forall (c: C).(\forall (h: nat).(\forall (d:
145 nat).((drop h d c e) \to (sc3 g a1 c (lift h d t))))))))))).(\lambda (e:
146 C).(\lambda (t: T).(\lambda (H1: (land (arity g e t (AHead a0 a1)) (\forall
147 (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d
148 e) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(\lambda (c:
149 C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h d c e)).(let H3
150 \def H1 in (and_ind (arity g e t (AHead a0 a1)) (\forall (d0: C).(\forall (w:
151 T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 e) \to (sc3 g a1
152 d0 (THead (Flat Appl) w (lift1 is t)))))))) (land (arity g c (lift h d t)
153 (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall
154 (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
155 (lift h d t)))))))))) (\lambda (H4: (arity g e t (AHead a0 a1))).(\lambda
156 (H5: ((\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is:
157 PList).((drop1 is d0 e) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
158 t)))))))))).(conj (arity g c (lift h d t) (AHead a0 a1)) (\forall (d0:
159 C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c)
160 \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (lift h d t)))))))))
161 (arity_lift g e t (AHead a0 a1) H4 c h d H2) (\lambda (d0: C).(\lambda (w:
162 T).(\lambda (H6: (sc3 g a0 d0 w)).(\lambda (is: PList).(\lambda (H7: (drop1
163 is d0 c)).(let H_y \def (H5 d0 w H6 (PConsTail is h d)) in (eq_ind T (lift1
164 (PConsTail is h d) t) (\lambda (t0: T).(sc3 g a1 d0 (THead (Flat Appl) w
165 t0))) (H_y (drop1_cons_tail c e h d H2 is d0 H7)) (lift1 is (lift h d t))
166 (lift1_cons_tail t h d is))))))))))) H3))))))))))))) a)).
169 \forall (g: G).(\forall (e: C).(\forall (a: A).(\forall (hds:
170 PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 hds c e)
171 \to (sc3 g a c (lift1 hds t)))))))))
173 \lambda (g: G).(\lambda (e: C).(\lambda (a: A).(\lambda (hds:
174 PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (t: T).((sc3 g
175 a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t))))))) (\lambda (c:
176 C).(\lambda (t: T).(\lambda (H: (sc3 g a e t)).(\lambda (H0: (drop1 PNil c
177 e)).(let H1 \def (match H0 in drop1 return (\lambda (p: PList).(\lambda (c0:
178 C).(\lambda (c1: C).(\lambda (_: (drop1 p c0 c1)).((eq PList p PNil) \to ((eq
179 C c0 c) \to ((eq C c1 e) \to (sc3 g a c t)))))))) with [(drop1_nil c0)
180 \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c0
181 c)).(\lambda (H3: (eq C c0 e)).(eq_ind C c (\lambda (c1: C).((eq C c1 e) \to
182 (sc3 g a c t))) (\lambda (H4: (eq C c e)).(eq_ind C e (\lambda (c1: C).(sc3 g
183 a c1 t)) H c (sym_eq C c e H4))) c0 (sym_eq C c0 c H2) H3)))) | (drop1_cons
184 c1 c2 h d H1 c3 hds0 H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds0)
185 PNil)).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def
186 (eq_ind PList (PCons h d hds0) (\lambda (e0: PList).(match e0 in PList return
187 (\lambda (_: PList).Prop) with [PNil \Rightarrow False | (PCons _ _ _)
188 \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c) \to ((eq C c3 e)
189 \to ((drop h d c1 c2) \to ((drop1 hds0 c2 c3) \to (sc3 g a c t))))) H6)) H4
190 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c) (refl_equal C
191 e))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda
192 (H: ((\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 p c e) \to
193 (sc3 g a c (lift1 p t)))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H0:
194 (sc3 g a e t)).(\lambda (H1: (drop1 (PCons n n0 p) c e)).(let H2 \def (match
195 H1 in drop1 return (\lambda (p0: PList).(\lambda (c0: C).(\lambda (c1:
196 C).(\lambda (_: (drop1 p0 c0 c1)).((eq PList p0 (PCons n n0 p)) \to ((eq C c0
197 c) \to ((eq C c1 e) \to (sc3 g a c (lift n n0 (lift1 p t)))))))))) with
198 [(drop1_nil c0) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0
199 p))).(\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq C c0 e)).((let H5 \def
200 (eq_ind PList PNil (\lambda (e0: PList).(match e0 in PList return (\lambda
201 (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow
202 False])) I (PCons n n0 p) H2) in (False_ind ((eq C c0 c) \to ((eq C c0 e) \to
203 (sc3 g a c (lift n n0 (lift1 p t))))) H5)) H3 H4)))) | (drop1_cons c1 c2 h d
204 H2 c3 hds0 H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds0) (PCons n
205 n0 p))).(\lambda (H5: (eq C c1 c)).(\lambda (H6: (eq C c3 e)).((let H7 \def
206 (f_equal PList PList (\lambda (e0: PList).(match e0 in PList return (\lambda
207 (_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p0) \Rightarrow
208 p0])) (PCons h d hds0) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat
209 (\lambda (e0: PList).(match e0 in PList return (\lambda (_: PList).nat) with
210 [PNil \Rightarrow d | (PCons _ n1 _) \Rightarrow n1])) (PCons h d hds0)
211 (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e0:
212 PList).(match e0 in PList return (\lambda (_: PList).nat) with [PNil
213 \Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) (PCons h d hds0) (PCons n n0
214 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds0
215 p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1 hds0
216 c2 c3) \to (sc3 g a c (lift n n0 (lift1 p t)))))))))) (\lambda (H10: (eq nat
217 d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds0 p) \to ((eq C c1 c)
218 \to ((eq C c3 e) \to ((drop n n1 c1 c2) \to ((drop1 hds0 c2 c3) \to (sc3 g a
219 c (lift n n0 (lift1 p t))))))))) (\lambda (H11: (eq PList hds0 p)).(eq_ind
220 PList p (\lambda (p0: PList).((eq C c1 c) \to ((eq C c3 e) \to ((drop n n0 c1
221 c2) \to ((drop1 p0 c2 c3) \to (sc3 g a c (lift n n0 (lift1 p t))))))))
222 (\lambda (H12: (eq C c1 c)).(eq_ind C c (\lambda (c0: C).((eq C c3 e) \to
223 ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to (sc3 g a c (lift n n0 (lift1 p
224 t))))))) (\lambda (H13: (eq C c3 e)).(eq_ind C e (\lambda (c0: C).((drop n n0
225 c c2) \to ((drop1 p c2 c0) \to (sc3 g a c (lift n n0 (lift1 p t))))))
226 (\lambda (H14: (drop n n0 c c2)).(\lambda (H15: (drop1 p c2 e)).(sc3_lift g a
227 c2 (lift1 p t) (H c2 t H0 H15) c n n0 H14))) c3 (sym_eq C c3 e H13))) c1
228 (sym_eq C c1 c H12))) hds0 (sym_eq PList hds0 p H11))) d (sym_eq nat d n0
229 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal
230 PList (PCons n n0 p)) (refl_equal C c) (refl_equal C e))))))))))) hds)))).
233 \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (i:
234 nat).(\forall (d: C).(\forall (v: T).(\forall (c: C).((sc3 g a c (THeads
235 (Flat Appl) vs (lift (S i) O v))) \to ((getl i c (CHead d (Bind Abbr) v)) \to
236 (sc3 g a c (THeads (Flat Appl) vs (TLRef i)))))))))))
238 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs:
239 TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c:
240 C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c
241 (CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs (TLRef
242 i))))))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (vs:
243 TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(\lambda (c:
244 C).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs (lift (S i) O v))
245 (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (lift (S i) O v))))).(\lambda
246 (H0: (getl i c (CHead d (Bind Abbr) v))).(let H1 \def H in (and_ind (arity g
247 c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0)) (sn3 c (THeads (Flat
248 Appl) vs (lift (S i) O v))) (land (arity g c (THeads (Flat Appl) vs (TLRef
249 i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))) (\lambda (H2:
250 (arity g c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0))).(\lambda
251 (H3: (sn3 c (THeads (Flat Appl) vs (lift (S i) O v)))).(conj (arity g c
252 (THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs
253 (TLRef i))) (arity_appls_abbr g c d v i H0 vs (ASort n n0) H2)
254 (sn3_appls_abbr c d v i H0 vs H3)))) H1))))))))))) (\lambda (a0: A).(\lambda
255 (_: ((\forall (vs: TList).(\forall (i: nat).(\forall (d: C).(\forall (v:
256 T).(\forall (c: C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to
257 ((getl i c (CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs
258 (TLRef i)))))))))))).(\lambda (a1: A).(\lambda (H0: ((\forall (vs:
259 TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c:
260 C).((sc3 g a1 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c
261 (CHead d (Bind Abbr) v)) \to (sc3 g a1 c (THeads (Flat Appl) vs (TLRef
262 i)))))))))))).(\lambda (vs: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda
263 (v: T).(\lambda (c: C).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs
264 (lift (S i) O v)) (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0
265 d0 w) \to (\forall (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat
266 Appl) w (lift1 is (THeads (Flat Appl) vs (lift (S i) O v)))))))))))).(\lambda
267 (H2: (getl i c (CHead d (Bind Abbr) v))).(let H3 \def H1 in (and_ind (arity g
268 c (THeads (Flat Appl) vs (lift (S i) O v)) (AHead a0 a1)) (\forall (d0:
269 C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c)
270 \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift
271 (S i) O v)))))))))) (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead
272 a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is:
273 PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
274 (THeads (Flat Appl) vs (TLRef i))))))))))) (\lambda (H4: (arity g c (THeads
275 (Flat Appl) vs (lift (S i) O v)) (AHead a0 a1))).(\lambda (H5: ((\forall (d0:
276 C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c)
277 \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift
278 (S i) O v)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (TLRef i))
279 (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall
280 (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
281 (THeads (Flat Appl) vs (TLRef i)))))))))) (arity_appls_abbr g c d v i H2 vs
282 (AHead a0 a1) H4) (\lambda (d0: C).(\lambda (w: T).(\lambda (H6: (sc3 g a0 d0
283 w)).(\lambda (is: PList).(\lambda (H7: (drop1 is d0 c)).(let H_x \def
284 (drop1_getl_trans is c d0 H7 Abbr d v i H2) in (let H8 \def H_x in (ex2_ind C
285 (\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: C).(getl (trans is
286 i) d0 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) v)))) (sc3 g a1 d0 (THead
287 (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (x:
288 C).(\lambda (_: (drop1 (ptrans is i) x d)).(\lambda (H10: (getl (trans is i)
289 d0 (CHead x (Bind Abbr) (lift1 (ptrans is i) v)))).(let H_y \def (H0 (TCons w
290 (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is
291 (TLRef i))) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w t))) (eq_ind_r
292 T (TLRef (trans is i)) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w
293 (THeads (Flat Appl) (lifts1 is vs) t)))) (H_y (trans is i) x (lift1 (ptrans
294 is i) v) d0 (eq_ind T (lift1 is (lift (S i) O v)) (\lambda (t: T).(sc3 g a1
295 d0 (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 is vs) t)))) (eq_ind T
296 (lift1 is (THeads (Flat Appl) vs (lift (S i) O v))) (\lambda (t: T).(sc3 g a1
297 d0 (THead (Flat Appl) w t))) (H5 d0 w H6 is H7) (THeads (Flat Appl) (lifts1
298 is vs) (lift1 is (lift (S i) O v))) (lifts1_flat Appl is (lift (S i) O v)
299 vs)) (lift (S (trans is i)) O (lift1 (ptrans is i) v)) (lift1_free is i v))
300 H10) (lift1 is (TLRef i)) (lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs
301 (TLRef i))) (lifts1_flat Appl is (TLRef i) vs)))))) H8)))))))))))
305 \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall
306 (u: T).((sc3 g (asucc g a) c (THeads (Flat Appl) vs u)) \to (\forall (t:
307 T).((sc3 g a c (THeads (Flat Appl) vs t)) \to (sc3 g a c (THeads (Flat Appl)
308 vs (THead (Flat Cast) u t))))))))))
310 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs:
311 TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat
312 Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to
313 (sc3 g a0 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (\lambda
314 (n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u:
315 T).(\lambda (H: (sc3 g (match n with [O \Rightarrow (ASort O (next g n0)) |
316 (S h) \Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u))).(\lambda (t:
317 T).(\lambda (H0: (land (arity g c (THeads (Flat Appl) vs t) (ASort n n0))
318 (sn3 c (THeads (Flat Appl) vs t)))).(nat_ind (\lambda (n1: nat).((sc3 g
319 (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow
320 (ASort h n0)]) c (THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads
321 (Flat Appl) vs t) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land
322 (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0))
323 (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))) (\lambda (H1:
324 (sc3 g (ASort O (next g n0)) c (THeads (Flat Appl) vs u))).(\lambda (H2:
325 (land (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads (Flat
326 Appl) vs t)))).(let H3 \def H1 in (and_ind (arity g c (THeads (Flat Appl) vs
327 u) (ASort O (next g n0))) (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c
328 (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads
329 (Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads
330 (Flat Appl) vs u) (ASort O (next g n0)))).(\lambda (H5: (sn3 c (THeads (Flat
331 Appl) vs u))).(let H6 \def H2 in (and_ind (arity g c (THeads (Flat Appl) vs
332 t) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity g c (THeads
333 (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat
334 Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat
335 Appl) vs t) (ASort O n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs
336 t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort
337 O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))
338 (arity_appls_cast g c u t vs (ASort O n0) H4 H7) (sn3_appls_cast c vs u H5 t
339 H8)))) H6)))) H3)))) (\lambda (n1: nat).(\lambda (_: (((sc3 g (match n1 with
340 [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) c
341 (THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads (Flat Appl) vs t)
342 (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land (arity g c
343 (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) (sn3 c (THeads
344 (Flat Appl) vs (THead (Flat Cast) u t)))))))).(\lambda (H1: (sc3 g (ASort n1
345 n0) c (THeads (Flat Appl) vs u))).(\lambda (H2: (land (arity g c (THeads
346 (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let
347 H3 \def H1 in (and_ind (arity g c (THeads (Flat Appl) vs u) (ASort n1 n0))
348 (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c (THeads (Flat Appl) vs
349 (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs
350 (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u)
351 (ASort n1 n0))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def
352 H2 in (and_ind (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c
353 (THeads (Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead
354 (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead
355 (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort
356 (S n1) n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g
357 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c
358 (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (arity_appls_cast g c u t vs
359 (ASort (S n1) n0) H4 H7) (sn3_appls_cast c vs u H5 t H8)))) H6)))) H3)))))) n
360 H H0))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (vs: TList).(\forall
361 (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to
362 (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c
363 (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (a1:
364 A).(\lambda (H0: ((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3
365 g (asucc g a1) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a1 c
366 (THeads (Flat Appl) vs t)) \to (sc3 g a1 c (THeads (Flat Appl) vs (THead
367 (Flat Cast) u t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u:
368 T).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc
369 g a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
370 PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1
371 is (THeads (Flat Appl) vs u))))))))))).(\lambda (t: T).(\lambda (H2: (land
372 (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall
373 (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1
374 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))))).(let H3
375 \def H1 in (and_ind (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g
376 a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
377 PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1
378 is (THeads (Flat Appl) vs u))))))))) (land (arity g c (THeads (Flat Appl) vs
379 (THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3
380 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead
381 (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u
382 t))))))))))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (AHead a0
383 (asucc g a1)))).(\lambda (H5: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d
384 w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead
385 (Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))))).(let H6 \def H2
386 in (and_ind (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d:
387 C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
388 \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs
389 t))))))))) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t))
390 (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall
391 (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is
392 (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))) (\lambda (H7: (arity
393 g c (THeads (Flat Appl) vs t) (AHead a0 a1))).(\lambda (H8: ((\forall (d:
394 C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
395 \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs
396 t))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t))
397 (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall
398 (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is
399 (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (arity_appls_cast g c
400 u t vs (AHead a0 a1) H4 H7) (\lambda (d: C).(\lambda (w: T).(\lambda (H9:
401 (sc3 g a0 d w)).(\lambda (is: PList).(\lambda (H10: (drop1 is d c)).(let H_y
402 \def (H0 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1
403 is vs) (lift1 is (THead (Flat Cast) u t))) (\lambda (t0: T).(sc3 g a1 d
404 (THead (Flat Appl) w t0))) (eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1
405 is t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat Appl)
406 (lifts1 is vs) t0)))) (H_y d (lift1 is u) (eq_ind T (lift1 is (THeads (Flat
407 Appl) vs u)) (\lambda (t0: T).(sc3 g (asucc g a1) d (THead (Flat Appl) w
408 t0))) (H5 d w H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is u))
409 (lifts1_flat Appl is u vs)) (lift1 is t) (eq_ind T (lift1 is (THeads (Flat
410 Appl) vs t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w t0))) (H8 d w
411 H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is t)) (lifts1_flat Appl
412 is t vs))) (lift1 is (THead (Flat Cast) u t)) (lift1_flat Cast is u t))
413 (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (lifts1_flat Appl
414 is (THead (Flat Cast) u t) vs))))))))))) H6)))) H3)))))))))))) a)).
416 theorem sc3_props__sc3_sn3_abst:
417 \forall (g: G).(\forall (a: A).(land (\forall (c: C).(\forall (t: T).((sc3 g
418 a c t) \to (sn3 c t)))) (\forall (vs: TList).(\forall (i: nat).(let t \def
419 (THeads (Flat Appl) vs (TLRef i)) in (\forall (c: C).((arity g c t a) \to
420 ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a c t))))))))))
422 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(land (\forall (c:
423 C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall (vs:
424 TList).(\forall (i: nat).(let t \def (THeads (Flat Appl) vs (TLRef i)) in
425 (\forall (c: C).((arity g c t a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to
426 (sc3 g a0 c t)))))))))) (\lambda (n: nat).(\lambda (n0: nat).(conj (\forall
427 (c: C).(\forall (t: T).((land (arity g c t (ASort n n0)) (sn3 c t)) \to (sn3
428 c t)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c
429 (THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) \to ((nf2 c (TLRef i)) \to
430 ((sns3 c vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n
431 n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))) (\lambda (c:
432 C).(\lambda (t: T).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c
433 t))).(let H0 \def H in (and_ind (arity g c t (ASort n n0)) (sn3 c t) (sn3 c
434 t) (\lambda (_: (arity g c t (ASort n n0))).(\lambda (H2: (sn3 c t)).H2))
435 H0))))) (\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H:
436 (arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n n0))).(\lambda (H0:
437 (nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(conj (arity g c (THeads (Flat
438 Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i))) H
439 (sn3_appls_lref c i H0 vs H1))))))))))) (\lambda (a0: A).(\lambda (H: (land
440 (\forall (c: C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall
441 (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads (Flat Appl)
442 vs (TLRef i)) a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a0 c
443 (THeads (Flat Appl) vs (TLRef i))))))))))).(\lambda (a1: A).(\lambda (H0:
444 (land (\forall (c: C).(\forall (t: T).((sc3 g a1 c t) \to (sn3 c t))))
445 (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads
446 (Flat Appl) vs (TLRef i)) a1) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to
447 (sc3 g a1 c (THeads (Flat Appl) vs (TLRef i))))))))))).(conj (\forall (c:
448 C).(\forall (t: T).((land (arity g c t (AHead a0 a1)) (\forall (d:
449 C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
450 \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t))))))))) \to (sn3 c t))))
451 (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads
452 (Flat Appl) vs (TLRef i)) (AHead a0 a1)) \to ((nf2 c (TLRef i)) \to ((sns3 c
453 vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))
454 (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
455 PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads
456 (Flat Appl) vs (TLRef i))))))))))))))))) (\lambda (c: C).(\lambda (t:
457 T).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall
458 (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1
459 d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H in (and_ind
460 (\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 t0))))
461 (\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads
462 (Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to
463 (sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) (\lambda (_:
464 ((\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0
465 t0)))))).(\lambda (H4: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0:
466 C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i))
467 \to ((sns3 c0 vs) \to (sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef
468 i))))))))))).(let H5 \def H0 in (and_ind (\forall (c0: C).(\forall (t0:
469 T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))) (\forall (vs: TList).(\forall (i:
470 nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a1) \to
471 ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 (THeads (Flat Appl) vs
472 (TLRef i))))))))) (sn3 c t) (\lambda (H6: ((\forall (c0: C).(\forall (t0:
473 T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))))).(\lambda (_: ((\forall (vs:
474 TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs
475 (TLRef i)) a1) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0
476 (THeads (Flat Appl) vs (TLRef i))))))))))).(let H8 \def H1 in (and_ind (arity
477 g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to
478 (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w
479 (lift1 is t)))))))) (sn3 c t) (\lambda (H9: (arity g c t (AHead a0
480 a1))).(\lambda (H10: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to
481 (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w
482 (lift1 is t)))))))))).(let H_y \def (arity_aprem g c t (AHead a0 a1) H9 O a0)
483 in (let H11 \def (H_y (aprem_zero a0 a1)) in (ex2_3_ind C T nat (\lambda (d:
484 C).(\lambda (_: T).(\lambda (j: nat).(drop j O d c)))) (\lambda (d:
485 C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g a0))))) (sn3 c t)
486 (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H12: (drop x2
487 O x0 c)).(\lambda (H13: (arity g x0 x1 (asucc g a0))).(let H_y0 \def (H10
488 (CHead x0 (Bind Abst) x1) (TLRef O) (H4 TNil O (CHead x0 (Bind Abst) x1)
489 (arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1) a0
490 H13) (nf2_lref_abst (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1))
491 I) (PCons (S x2) O PNil)) in (let H_y1 \def (H6 (CHead x0 (Bind Abst) x1)
492 (THead (Flat Appl) (TLRef O) (lift (S x2) O t)) (H_y0 (drop1_cons (CHead x0
493 (Bind Abst) x1) c (S x2) O (drop_drop (Bind Abst) x2 x0 c H12 x1) c PNil
494 (drop1_nil c)))) in (let H_x \def (sn3_gen_flat Appl (CHead x0 (Bind Abst)
495 x1) (TLRef O) (lift (S x2) O t) H_y1) in (let H14 \def H_x in (and_ind (sn3
496 (CHead x0 (Bind Abst) x1) (TLRef O)) (sn3 (CHead x0 (Bind Abst) x1) (lift (S
497 x2) O t)) (sn3 c t) (\lambda (_: (sn3 (CHead x0 (Bind Abst) x1) (TLRef
498 O))).(\lambda (H16: (sn3 (CHead x0 (Bind Abst) x1) (lift (S x2) O
499 t))).(sn3_gen_lift (CHead x0 (Bind Abst) x1) t (S x2) O H16 c (drop_drop
500 (Bind Abst) x2 x0 c H12 x1)))) H14)))))))))) H11))))) H8)))) H5)))) H2)))))
501 (\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H1: (arity g
502 c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))).(\lambda (H2: (nf2 c
503 (TLRef i))).(\lambda (H3: (sns3 c vs)).(conj (arity g c (THeads (Flat Appl)
504 vs (TLRef i)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w)
505 \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w
506 (lift1 is (THeads (Flat Appl) vs (TLRef i)))))))))) H1 (\lambda (d:
507 C).(\lambda (w: T).(\lambda (H4: (sc3 g a0 d w)).(\lambda (is:
508 PList).(\lambda (H5: (drop1 is d c)).(let H6 \def H in (and_ind (\forall (c0:
509 C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))) (\forall (vs0:
510 TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl)
511 vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a0
512 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a1 d (THead (Flat Appl)
513 w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (H7: ((\forall (c0:
514 C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))))).(\lambda (_:
515 ((\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0
516 (THeads (Flat Appl) vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3
517 c0 vs0) \to (sc3 g a0 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))))).(let H9
518 \def H0 in (and_ind (\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) \to
519 (sn3 c0 t)))) (\forall (vs0: TList).(\forall (i0: nat).(\forall (c0:
520 C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) \to ((nf2 c0 (TLRef
521 i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat Appl) vs0 (TLRef
522 i0))))))))) (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs
523 (TLRef i))))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t)
524 \to (sn3 c0 t)))))).(\lambda (H11: ((\forall (vs0: TList).(\forall (i0:
525 nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1)
526 \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat
527 Appl) vs0 (TLRef i0))))))))))).(let H_y \def (H11 (TCons w (lifts1 is vs)))
528 in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i)))
529 (\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w t))) (eq_ind_r T (TLRef
530 (trans is i)) (\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat
531 Appl) (lifts1 is vs) t)))) (H_y (trans is i) d (eq_ind T (lift1 is (TLRef i))
532 (\lambda (t: T).(arity g d (THead (Flat Appl) w (THeads (Flat Appl) (lifts1
533 is vs) t)) a1)) (eq_ind T (lift1 is (THeads (Flat Appl) vs (TLRef i)))
534 (\lambda (t: T).(arity g d (THead (Flat Appl) w t) a1)) (arity_appl g d w a0
535 (sc3_arity_gen g d w a0 H4) (lift1 is (THeads (Flat Appl) vs (TLRef i))) a1
536 (arity_lift1 g (AHead a0 a1) c is d (THeads (Flat Appl) vs (TLRef i)) H5 H1))
537 (THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) (lifts1_flat Appl is
538 (TLRef i) vs)) (TLRef (trans is i)) (lift1_lref is i)) (eq_ind T (lift1 is
539 (TLRef i)) (\lambda (t: T).(nf2 d t)) (nf2_lift1 c is d (TLRef i) H5 H2)
540 (TLRef (trans is i)) (lift1_lref is i)) (conj (sn3 d w) (sns3 d (lifts1 is
541 vs)) (H7 d w H4) (sns3_lifts1 c is d H5 vs H3))) (lift1 is (TLRef i))
542 (lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs (TLRef i))) (lifts1_flat
543 Appl is (TLRef i) vs))))) H9)))) H6))))))))))))))))))) a)).
546 \forall (g: G).(\forall (a: A).(\forall (c: C).(\forall (t: T).((sc3 g a c
549 \lambda (g: G).(\lambda (a: A).(\lambda (c: C).(\lambda (t: T).(\lambda (H:
550 (sc3 g a c t)).(let H_x \def (sc3_props__sc3_sn3_abst g a) in (let H0 \def
551 H_x in (and_ind (\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3
552 c0 t0)))) (\forall (vs: TList).(\forall (i: nat).(let t0 \def (THeads (Flat
553 Appl) vs (TLRef i)) in (\forall (c0: C).((arity g c0 t0 a) \to ((nf2 c0
554 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a c0 t0)))))))) (sn3 c t) (\lambda
555 (H1: ((\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 c0
556 t0)))))).(\lambda (_: ((\forall (vs: TList).(\forall (i: nat).(let t0 \def
557 (THeads (Flat Appl) vs (TLRef i)) in (\forall (c0: C).((arity g c0 t0 a) \to
558 ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a c0 t0)))))))))).(H1 c t
562 \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall
563 (i: nat).((arity g c (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c (TLRef
564 i)) \to ((sns3 c vs) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i))))))))))
566 \lambda (g: G).(\lambda (a: A).(\lambda (vs: TList).(\lambda (c: C).(\lambda
567 (i: nat).(\lambda (H: (arity g c (THeads (Flat Appl) vs (TLRef i))
568 a)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(let H_x \def
569 (sc3_props__sc3_sn3_abst g a) in (let H2 \def H_x in (and_ind (\forall (c0:
570 C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0 t)))) (\forall (vs0:
571 TList).(\forall (i0: nat).(let t \def (THeads (Flat Appl) vs0 (TLRef i0)) in
572 (\forall (c0: C).((arity g c0 t a) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0
573 vs0) \to (sc3 g a c0 t)))))))) (sc3 g a c (THeads (Flat Appl) vs (TLRef i)))
574 (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0
575 t)))))).(\lambda (H4: ((\forall (vs0: TList).(\forall (i0: nat).(let t \def
576 (THeads (Flat Appl) vs0 (TLRef i0)) in (\forall (c0: C).((arity g c0 t a) \to
577 ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a c0 t)))))))))).(H4 vs i
578 c H H0 H1))) H2)))))))))).
581 \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1:
582 A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v:
583 T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts
584 (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a2 c (THeads (Flat Appl) vs
585 (THead (Bind b) v t)))))))))))))
587 \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda
588 (a1: A).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (vs: TList).(\forall
589 (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads
590 (Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads
591 (Flat Appl) vs (THead (Bind b) v t)))))))))) (\lambda (n: nat).(\lambda (n0:
592 nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t:
593 T).(\lambda (H0: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl)
594 (lifts (S O) O vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat
595 Appl) (lifts (S O) O vs) t)))).(\lambda (H1: (sc3 g a1 c v)).(let H2 \def H0
596 in (and_ind (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O
597 vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S
598 O) O vs) t)) (land (arity g c (THeads (Flat Appl) vs (THead (Bind b) v t))
599 (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t)))) (\lambda
600 (H3: (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)
601 (ASort n n0))).(\lambda (H4: (sn3 (CHead c (Bind b) v) (THeads (Flat Appl)
602 (lifts (S O) O vs) t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Bind
603 b) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t)))
604 (arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 H1) t vs (ASort n n0)
605 H3) (sn3_appls_bind b H c v (sc3_sn3 g a1 c v H1) vs t H4)))) H2))))))))))
606 (\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall (c: C).(\forall
607 (v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads (Flat Appl)
608 (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads (Flat Appl)
609 vs (THead (Bind b) v t))))))))))).(\lambda (a0: A).(\lambda (H1: ((\forall
610 (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 (CHead
611 c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v)
612 \to (sc3 g a0 c (THeads (Flat Appl) vs (THead (Bind b) v
613 t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda
614 (t: T).(\lambda (H2: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl)
615 (lifts (S O) O vs) t) (AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a
616 d w) \to (\forall (is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g
617 a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs)
618 t))))))))))).(\lambda (H3: (sc3 g a1 c v)).(let H4 \def H2 in (and_ind (arity
619 g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) (AHead a
620 a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is:
621 PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat Appl)
622 w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))) (land (arity g
623 c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) (\forall (d:
624 C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c)
625 \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead
626 (Bind b) v t))))))))))) (\lambda (H5: (arity g (CHead c (Bind b) v) (THeads
627 (Flat Appl) (lifts (S O) O vs) t) (AHead a a0))).(\lambda (H6: ((\forall (d:
628 C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d
629 (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads
630 (Flat Appl) (lifts (S O) O vs) t))))))))))).(conj (arity g c (THeads (Flat
631 Appl) vs (THead (Bind b) v t)) (AHead a a0)) (\forall (d: C).(\forall (w:
632 T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
633 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Bind b) v
634 t)))))))))) (arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 H3) t vs
635 (AHead a a0) H5) (\lambda (d: C).(\lambda (w: T).(\lambda (H7: (sc3 g a d
636 w)).(\lambda (is: PList).(\lambda (H8: (drop1 is d c)).(let H_y \def (H1
637 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs)
638 (lift1 is (THead (Bind b) v t))) (\lambda (t0: T).(sc3 g a0 d (THead (Flat
639 Appl) w t0))) (eq_ind_r T (THead (Bind b) (lift1 is v) (lift1 (Ss is) t))
640 (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w (THeads (Flat Appl) (lifts1
641 is vs) t0)))) (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind TList (lifts1 (Ss
642 is) (lifts (S O) O vs)) (\lambda (t0: TList).(sc3 g a0 (CHead d (Bind b)
643 (lift1 is v)) (THead (Flat Appl) (lift (S O) O w) (THeads (Flat Appl) t0
644 (lift1 (Ss is) t))))) (eq_ind T (lift1 (Ss is) (THeads (Flat Appl) (lifts (S
645 O) O vs) t)) (\lambda (t0: T).(sc3 g a0 (CHead d (Bind b) (lift1 is v))
646 (THead (Flat Appl) (lift (S O) O w) t0))) (H6 (CHead d (Bind b) (lift1 is v))
647 (lift (S O) O w) (sc3_lift g a d w H7 (CHead d (Bind b) (lift1 is v)) (S O) O
648 (drop_drop (Bind b) O d d (drop_refl d) (lift1 is v))) (Ss is)
649 (drop1_skip_bind b c is d v H8)) (THeads (Flat Appl) (lifts1 (Ss is) (lifts
650 (S O) O vs)) (lift1 (Ss is) t)) (lifts1_flat Appl (Ss is) t (lifts (S O) O
651 vs))) (lifts (S O) O (lifts1 is vs)) (lifts1_xhg is vs)) (sc3_lift1 g c a1 is
652 d v H3 H8)) (lift1 is (THead (Bind b) v t)) (lift1_bind b is v t)) (lift1 is
653 (THeads (Flat Appl) vs (THead (Bind b) v t))) (lifts1_flat Appl is (THead
654 (Bind b) v t) vs))))))))))) H4)))))))))))) a2))))).
657 \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (vs:
658 TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 c (THeads
659 (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w:
660 T).((sc3 g (asucc g a1) c w) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead
661 (Flat Appl) v (THead (Bind Abst) w t))))))))))))))
663 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(A_ind (\lambda (a:
664 A).(\forall (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3
665 g a c (THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v)
666 \to (\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a c (THeads (Flat
667 Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))))))))))) (\lambda
668 (n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v:
669 T).(\lambda (t: T).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs
670 (THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead
671 (Bind Abbr) v t))))).(\lambda (H0: (sc3 g a1 c v)).(\lambda (w: T).(\lambda
672 (H1: (sc3 g (asucc g a1) c w)).(let H2 \def H in (and_ind (arity g c (THeads
673 (Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat
674 Appl) vs (THead (Bind Abbr) v t))) (land (arity g c (THeads (Flat Appl) vs
675 (THead (Flat Appl) v (THead (Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads
676 (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H3:
677 (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n
678 n0))).(\lambda (H4: (sn3 c (THeads (Flat Appl) vs (THead (Bind Abbr) v
679 t)))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v (THead
680 (Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat
681 Appl) v (THead (Bind Abst) w t)))) (arity_appls_appl g c v a1 (sc3_arity_gen
682 g c v a1 H0) w (sc3_arity_gen g c w (asucc g a1) H1) t vs (ASort n n0) H3)
683 (sn3_appls_beta c v t vs H4 w (sc3_sn3 g (asucc g a1) c w H1)))))
684 H2)))))))))))) (\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall
685 (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a c (THeads (Flat Appl) vs
686 (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: T).((sc3 g
687 (asucc g a1) c w) \to (sc3 g a c (THeads (Flat Appl) vs (THead (Flat Appl) v
688 (THead (Bind Abst) w t)))))))))))))).(\lambda (a0: A).(\lambda (H0: ((\forall
689 (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 c
690 (THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to
691 (\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a0 c (THeads (Flat Appl)
692 vs (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))))))).(\lambda (vs:
693 TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H1: (land
694 (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (AHead a a0))
695 (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is:
696 PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads
697 (Flat Appl) vs (THead (Bind Abbr) v t)))))))))))).(\lambda (H2: (sc3 g a1 c
698 v)).(\lambda (w: T).(\lambda (H3: (sc3 g (asucc g a1) c w)).(let H4 \def H1
699 in (and_ind (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (AHead
700 a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall (is:
701 PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is
702 (THeads (Flat Appl) vs (THead (Bind Abbr) v t)))))))))) (land (arity g c
703 (THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))) (AHead
704 a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall (is:
705 PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is
706 (THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w
707 t)))))))))))) (\lambda (H5: (arity g c (THeads (Flat Appl) vs (THead (Bind
708 Abbr) v t)) (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w0:
709 T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
710 (THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Bind Abbr) v
711 t)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v
712 (THead (Bind Abst) w t))) (AHead a a0)) (\forall (d: C).(\forall (w0:
713 T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
714 (THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Flat Appl) v
715 (THead (Bind Abst) w t))))))))))) (arity_appls_appl g c v a1 (sc3_arity_gen g
716 c v a1 H2) w (sc3_arity_gen g c w (asucc g a1) H3) t vs (AHead a a0) H5)
717 (\lambda (d: C).(\lambda (w0: T).(\lambda (H7: (sc3 g a d w0)).(\lambda (is:
718 PList).(\lambda (H8: (drop1 is d c)).(eq_ind_r T (THeads (Flat Appl) (lifts1
719 is vs) (lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t)))) (\lambda
720 (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 t0))) (eq_ind_r T (THead (Flat
721 Appl) (lift1 is v) (lift1 is (THead (Bind Abst) w t))) (\lambda (t0: T).(sc3
722 g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) t0))))
723 (eq_ind_r T (THead (Bind Abst) (lift1 is w) (lift1 (Ss is) t)) (\lambda (t0:
724 T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs)
725 (THead (Flat Appl) (lift1 is v) t0))))) (let H_y \def (H0 (TCons w0 (lifts1
726 is vs))) in (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind T (lift1 is (THead
727 (Bind Abbr) v t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads
728 (Flat Appl) (lifts1 is vs) t0)))) (eq_ind T (lift1 is (THeads (Flat Appl) vs
729 (THead (Bind Abbr) v t))) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0
730 t0))) (H6 d w0 H7 is H8) (THeads (Flat Appl) (lifts1 is vs) (lift1 is (THead
731 (Bind Abbr) v t))) (lifts1_flat Appl is (THead (Bind Abbr) v t) vs)) (THead
732 (Bind Abbr) (lift1 is v) (lift1 (Ss is) t)) (lift1_bind Abbr is v t))
733 (sc3_lift1 g c a1 is d v H2 H8) (lift1 is w) (sc3_lift1 g c (asucc g a1) is d
734 w H3 H8))) (lift1 is (THead (Bind Abst) w t)) (lift1_bind Abst is w t))
735 (lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t))) (lift1_flat Appl is
736 v (THead (Bind Abst) w t))) (lift1 is (THeads (Flat Appl) vs (THead (Flat
737 Appl) v (THead (Bind Abst) w t)))) (lifts1_flat Appl is (THead (Flat Appl) v
738 (THead (Bind Abst) w t)) vs)))))))))) H4)))))))))))))) a2))).
projects / helm.git / blob