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