1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/props".
19 include "sc3/defs.ma".
21 include "sn3/lift1.ma".
23 include "nf2/lift1.ma".
25 include "arity/lift1.ma".
27 include "arity/aprem.ma".
29 include "llt/props.ma".
31 include "drop1/props.ma".
33 theorem sc3_arity_gen:
34 \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((sc3 g a c
35 t) \to (arity g c t a)))))
37 \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(A_ind
38 (\lambda (a0: A).((sc3 g a0 c t) \to (arity g c t a0))) (\lambda (n:
39 nat).(\lambda (n0: nat).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c
40 t))).(let H0 \def H in (and_ind (arity g c t (ASort n n0)) (sn3 c t) (arity g
41 c t (ASort n n0)) (\lambda (H1: (arity g c t (ASort n n0))).(\lambda (_: (sn3
42 c t)).H1)) H0))))) (\lambda (a0: A).(\lambda (_: (((sc3 g a0 c t) \to (arity
43 g c t a0)))).(\lambda (a1: A).(\lambda (_: (((sc3 g a1 c t) \to (arity g c t
44 a1)))).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d:
45 C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
46 \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H1 in
47 (and_ind (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g
48 a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat
49 Appl) w (lift1 is t)))))))) (arity g c t (AHead a0 a1)) (\lambda (H3: (arity
50 g c t (AHead a0 a1))).(\lambda (_: ((\forall (d: C).(\forall (w: T).((sc3 g
51 a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat
52 Appl) w (lift1 is t)))))))))).H3)) H2))))))) a)))).
55 \forall (g: G).(\forall (a1: A).(\forall (c: C).(\forall (t: T).((sc3 g a1 c
56 t) \to (\forall (a2: A).((leq g a1 a2) \to (sc3 g a2 c t)))))))
58 \lambda (g: G).(\lambda (a1: A).(llt_wf_ind (\lambda (a: A).(\forall (c:
59 C).(\forall (t: T).((sc3 g a c t) \to (\forall (a2: A).((leq g a a2) \to (sc3
60 g a2 c t))))))) (\lambda (a2: A).(A_ind (\lambda (a: A).(((\forall (a3:
61 A).((llt a3 a) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 c t) \to
62 (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t))))))))) \to (\forall (c:
63 C).(\forall (t: T).((sc3 g a c t) \to (\forall (a3: A).((leq g a a3) \to (sc3
64 g a3 c t)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (_: ((\forall
65 (a3: A).((llt a3 (ASort n n0)) \to (\forall (c: C).(\forall (t: T).((sc3 g a3
66 c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t)))))))))).(\lambda
67 (c: C).(\lambda (t: T).(\lambda (H0: (land (arity g c t (ASort n n0)) (sn3 c
68 t))).(\lambda (a3: A).(\lambda (H1: (leq g (ASort n n0) a3)).(let H2 \def H0
69 in (and_ind (arity g c t (ASort n n0)) (sn3 c t) (sc3 g a3 c t) (\lambda (H3:
70 (arity g c t (ASort n n0))).(\lambda (H4: (sn3 c t)).(let H_y \def
71 (arity_repl g c t (ASort n n0) H3 a3 H1) in (let H_x \def (leq_gen_sort g n
72 n0 a3 H1) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2:
73 nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a3 (ASort h2 n2))))) (\lambda
74 (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort n n0) k)
75 (aplus g (ASort h2 n2) k))))) (sc3 g a3 c t) (\lambda (x0: nat).(\lambda (x1:
76 nat).(\lambda (x2: nat).(\lambda (H6: (eq A a3 (ASort x1 x0))).(\lambda (_:
77 (eq A (aplus g (ASort n n0) x2) (aplus g (ASort x1 x0) x2))).(let H8 \def
78 (eq_ind A a3 (\lambda (a: A).(arity g c t a)) H_y (ASort x1 x0) H6) in
79 (eq_ind_r A (ASort x1 x0) (\lambda (a: A).(sc3 g a c t)) (conj (arity g c t
80 (ASort x1 x0)) (sn3 c t) H8 H4) a3 H6))))))) H5)))))) H2)))))))))) (\lambda
81 (a: A).(\lambda (_: ((((\forall (a3: A).((llt a3 a) \to (\forall (c:
82 C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to
83 (sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a c t) \to
84 (\forall (a3: A).((leq g a a3) \to (sc3 g a3 c t))))))))).(\lambda (a0:
85 A).(\lambda (H0: ((((\forall (a3: A).((llt a3 a0) \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 a0 c t)
88 \to (\forall (a3: A).((leq g a0 a3) \to (sc3 g a3 c t))))))))).(\lambda (H1:
89 ((\forall (a3: A).((llt a3 (AHead a a0)) \to (\forall (c: C).(\forall (t:
90 T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c
91 t)))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H2: (land (arity g c t
92 (AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall
93 (is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is
94 t)))))))))).(\lambda (a3: A).(\lambda (H3: (leq g (AHead a a0) a3)).(let H4
95 \def H2 in (and_ind (arity g c t (AHead a a0)) (\forall (d: C).(\forall (w:
96 T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
97 (THead (Flat Appl) w (lift1 is t)))))))) (sc3 g a3 c t) (\lambda (H5: (arity
98 g c t (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a
99 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat
100 Appl) w (lift1 is t)))))))))).(let H_x \def (leq_gen_head g a a0 a3 H3) in
101 (let H7 \def H_x in (ex3_2_ind A A (\lambda (a4: A).(\lambda (a5: A).(eq A a3
102 (AHead a4 a5)))) (\lambda (a4: A).(\lambda (_: A).(leq g a a4))) (\lambda (_:
103 A).(\lambda (a5: A).(leq g a0 a5))) (sc3 g a3 c t) (\lambda (x0: A).(\lambda
104 (x1: A).(\lambda (H8: (eq A a3 (AHead x0 x1))).(\lambda (H9: (leq g a
105 x0)).(\lambda (H10: (leq g a0 x1)).(eq_ind_r A (AHead x0 x1) (\lambda (a4:
106 A).(sc3 g a4 c t)) (conj (arity g c t (AHead x0 x1)) (\forall (d: C).(\forall
107 (w: T).((sc3 g x0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g x1
108 d (THead (Flat Appl) w (lift1 is t)))))))) (arity_repl g c t (AHead a a0) H5
109 (AHead x0 x1) (leq_head g a x0 H9 a0 x1 H10)) (\lambda (d: C).(\lambda (w:
110 T).(\lambda (H11: (sc3 g x0 d w)).(\lambda (is: PList).(\lambda (H12: (drop1
111 is d c)).(H0 (\lambda (a4: A).(\lambda (H13: (llt a4 a0)).(\lambda (c0:
112 C).(\lambda (t0: T).(\lambda (H14: (sc3 g a4 c0 t0)).(\lambda (a5:
113 A).(\lambda (H15: (leq g a4 a5)).(H1 a4 (llt_trans a4 a0 (AHead a a0) H13
114 (llt_head_dx a a0)) c0 t0 H14 a5 H15)))))))) d (THead (Flat Appl) w (lift1 is
115 t)) (H6 d w (H1 x0 (llt_repl g a x0 H9 (AHead a a0) (llt_head_sx a a0)) d w
116 H11 a (leq_sym g a x0 H9)) is H12) x1 H10))))))) a3 H8)))))) H7)))))
117 H4)))))))))))) a2)) a1)).
120 \forall (g: G).(\forall (a: A).(\forall (e: C).(\forall (t: T).((sc3 g a e
121 t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e)
122 \to (sc3 g a c (lift h d t))))))))))
124 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (e:
125 C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h:
126 nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d t))))))))))
127 (\lambda (n: nat).(\lambda (n0: nat).(\lambda (e: C).(\lambda (t: T).(\lambda
128 (H: (land (arity g e t (ASort n n0)) (sn3 e t))).(\lambda (c: C).(\lambda (h:
129 nat).(\lambda (d: nat).(\lambda (H0: (drop h d c e)).(let H1 \def H in
130 (and_ind (arity g e t (ASort n n0)) (sn3 e t) (land (arity g c (lift h d t)
131 (ASort n n0)) (sn3 c (lift h d t))) (\lambda (H2: (arity g e t (ASort n
132 n0))).(\lambda (H3: (sn3 e t)).(conj (arity g c (lift h d t) (ASort n n0))
133 (sn3 c (lift h d t)) (arity_lift g e t (ASort n n0) H2 c h d H0) (sn3_lift e
134 t H3 c h d H0)))) H1))))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (e:
135 C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h:
136 nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d
137 t))))))))))).(\lambda (a1: A).(\lambda (_: ((\forall (e: C).(\forall (t:
138 T).((sc3 g a1 e t) \to (\forall (c: C).(\forall (h: nat).(\forall (d:
139 nat).((drop h d c e) \to (sc3 g a1 c (lift h d t))))))))))).(\lambda (e:
140 C).(\lambda (t: T).(\lambda (H1: (land (arity g e t (AHead a0 a1)) (\forall
141 (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d
142 e) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(\lambda (c:
143 C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h d c e)).(let H3
144 \def H1 in (and_ind (arity g e t (AHead a0 a1)) (\forall (d0: C).(\forall (w:
145 T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 e) \to (sc3 g a1
146 d0 (THead (Flat Appl) w (lift1 is t)))))))) (land (arity g c (lift h d t)
147 (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall
148 (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
149 (lift h d t)))))))))) (\lambda (H4: (arity g e t (AHead a0 a1))).(\lambda
150 (H5: ((\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is:
151 PList).((drop1 is d0 e) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
152 t)))))))))).(conj (arity g c (lift h d t) (AHead a0 a1)) (\forall (d0:
153 C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c)
154 \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (lift h d t)))))))))
155 (arity_lift g e t (AHead a0 a1) H4 c h d H2) (\lambda (d0: C).(\lambda (w:
156 T).(\lambda (H6: (sc3 g a0 d0 w)).(\lambda (is: PList).(\lambda (H7: (drop1
157 is d0 c)).(let H_y \def (H5 d0 w H6 (PConsTail is h d)) in (eq_ind T (lift1
158 (PConsTail is h d) t) (\lambda (t0: T).(sc3 g a1 d0 (THead (Flat Appl) w
159 t0))) (H_y (drop1_cons_tail c e h d H2 is d0 H7)) (lift1 is (lift h d t))
160 (lift1_cons_tail t h d is))))))))))) H3))))))))))))) a)).
163 \forall (g: G).(\forall (e: C).(\forall (a: A).(\forall (hds:
164 PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 hds c e)
165 \to (sc3 g a c (lift1 hds t)))))))))
167 \lambda (g: G).(\lambda (e: C).(\lambda (a: A).(\lambda (hds:
168 PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (t: T).((sc3 g
169 a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t))))))) (\lambda (c:
170 C).(\lambda (t: T).(\lambda (H: (sc3 g a e t)).(\lambda (H0: (drop1 PNil c
171 e)).(let H1 \def (match H0 in drop1 return (\lambda (p: PList).(\lambda (c0:
172 C).(\lambda (c1: C).(\lambda (_: (drop1 p c0 c1)).((eq PList p PNil) \to ((eq
173 C c0 c) \to ((eq C c1 e) \to (sc3 g a c t)))))))) with [(drop1_nil c0)
174 \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c0
175 c)).(\lambda (H3: (eq C c0 e)).(eq_ind C c (\lambda (c1: C).((eq C c1 e) \to
176 (sc3 g a c t))) (\lambda (H4: (eq C c e)).(eq_ind C e (\lambda (c1: C).(sc3 g
177 a c1 t)) H c (sym_eq C c e H4))) c0 (sym_eq C c0 c H2) H3)))) | (drop1_cons
178 c1 c2 h d H1 c3 hds0 H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds0)
179 PNil)).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def
180 (eq_ind PList (PCons h d hds0) (\lambda (e0: PList).(match e0 in PList return
181 (\lambda (_: PList).Prop) with [PNil \Rightarrow False | (PCons _ _ _)
182 \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c) \to ((eq C c3 e)
183 \to ((drop h d c1 c2) \to ((drop1 hds0 c2 c3) \to (sc3 g a c t))))) H6)) H4
184 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c) (refl_equal C
185 e))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda
186 (H: ((\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 p c e) \to
187 (sc3 g a c (lift1 p t)))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H0:
188 (sc3 g a e t)).(\lambda (H1: (drop1 (PCons n n0 p) c e)).(let H2 \def (match
189 H1 in drop1 return (\lambda (p0: PList).(\lambda (c0: C).(\lambda (c1:
190 C).(\lambda (_: (drop1 p0 c0 c1)).((eq PList p0 (PCons n n0 p)) \to ((eq C c0
191 c) \to ((eq C c1 e) \to (sc3 g a c (lift n n0 (lift1 p t)))))))))) with
192 [(drop1_nil c0) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0
193 p))).(\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq C c0 e)).((let H5 \def
194 (eq_ind PList PNil (\lambda (e0: PList).(match e0 in PList return (\lambda
195 (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow
196 False])) I (PCons n n0 p) H2) in (False_ind ((eq C c0 c) \to ((eq C c0 e) \to
197 (sc3 g a c (lift n n0 (lift1 p t))))) H5)) H3 H4)))) | (drop1_cons c1 c2 h d
198 H2 c3 hds0 H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds0) (PCons n
199 n0 p))).(\lambda (H5: (eq C c1 c)).(\lambda (H6: (eq C c3 e)).((let H7 \def
200 (f_equal PList PList (\lambda (e0: PList).(match e0 in PList return (\lambda
201 (_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p0) \Rightarrow
202 p0])) (PCons h d hds0) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat
203 (\lambda (e0: PList).(match e0 in PList return (\lambda (_: PList).nat) with
204 [PNil \Rightarrow d | (PCons _ n1 _) \Rightarrow n1])) (PCons h d hds0)
205 (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e0:
206 PList).(match e0 in PList return (\lambda (_: PList).nat) with [PNil
207 \Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) (PCons h d hds0) (PCons n n0
208 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds0
209 p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1 hds0
210 c2 c3) \to (sc3 g a c (lift n n0 (lift1 p t)))))))))) (\lambda (H10: (eq nat
211 d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds0 p) \to ((eq C c1 c)
212 \to ((eq C c3 e) \to ((drop n n1 c1 c2) \to ((drop1 hds0 c2 c3) \to (sc3 g a
213 c (lift n n0 (lift1 p t))))))))) (\lambda (H11: (eq PList hds0 p)).(eq_ind
214 PList p (\lambda (p0: PList).((eq C c1 c) \to ((eq C c3 e) \to ((drop n n0 c1
215 c2) \to ((drop1 p0 c2 c3) \to (sc3 g a c (lift n n0 (lift1 p t))))))))
216 (\lambda (H12: (eq C c1 c)).(eq_ind C c (\lambda (c0: C).((eq C c3 e) \to
217 ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to (sc3 g a c (lift n n0 (lift1 p
218 t))))))) (\lambda (H13: (eq C c3 e)).(eq_ind C e (\lambda (c0: C).((drop n n0
219 c c2) \to ((drop1 p c2 c0) \to (sc3 g a c (lift n n0 (lift1 p t))))))
220 (\lambda (H14: (drop n n0 c c2)).(\lambda (H15: (drop1 p c2 e)).(sc3_lift g a
221 c2 (lift1 p t) (H c2 t H0 H15) c n n0 H14))) c3 (sym_eq C c3 e H13))) c1
222 (sym_eq C c1 c H12))) hds0 (sym_eq PList hds0 p H11))) d (sym_eq nat d n0
223 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal
224 PList (PCons n n0 p)) (refl_equal C c) (refl_equal C e))))))))))) hds)))).
227 \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (i:
228 nat).(\forall (d: C).(\forall (v: T).(\forall (c: C).((sc3 g a c (THeads
229 (Flat Appl) vs (lift (S i) O v))) \to ((getl i c (CHead d (Bind Abbr) v)) \to
230 (sc3 g a c (THeads (Flat Appl) vs (TLRef i)))))))))))
232 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs:
233 TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c:
234 C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c
235 (CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs (TLRef
236 i))))))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (vs:
237 TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(\lambda (c:
238 C).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs (lift (S i) O v))
239 (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (lift (S i) O v))))).(\lambda
240 (H0: (getl i c (CHead d (Bind Abbr) v))).(let H1 \def H in (and_ind (arity g
241 c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0)) (sn3 c (THeads (Flat
242 Appl) vs (lift (S i) O v))) (land (arity g c (THeads (Flat Appl) vs (TLRef
243 i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))) (\lambda (H2:
244 (arity g c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0))).(\lambda
245 (H3: (sn3 c (THeads (Flat Appl) vs (lift (S i) O v)))).(conj (arity g c
246 (THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs
247 (TLRef i))) (arity_appls_abbr g c d v i H0 vs (ASort n n0) H2)
248 (sn3_appls_abbr c d v i H0 vs H3)))) H1))))))))))) (\lambda (a0: A).(\lambda
249 (_: ((\forall (vs: TList).(\forall (i: nat).(\forall (d: C).(\forall (v:
250 T).(\forall (c: C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to
251 ((getl i c (CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs
252 (TLRef i)))))))))))).(\lambda (a1: A).(\lambda (H0: ((\forall (vs:
253 TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c:
254 C).((sc3 g a1 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c
255 (CHead d (Bind Abbr) v)) \to (sc3 g a1 c (THeads (Flat Appl) vs (TLRef
256 i)))))))))))).(\lambda (vs: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda
257 (v: T).(\lambda (c: C).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs
258 (lift (S i) O v)) (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0
259 d0 w) \to (\forall (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat
260 Appl) w (lift1 is (THeads (Flat Appl) vs (lift (S i) O v)))))))))))).(\lambda
261 (H2: (getl i c (CHead d (Bind Abbr) v))).(let H3 \def H1 in (and_ind (arity g
262 c (THeads (Flat Appl) vs (lift (S i) O v)) (AHead a0 a1)) (\forall (d0:
263 C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c)
264 \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift
265 (S i) O v)))))))))) (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead
266 a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is:
267 PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
268 (THeads (Flat Appl) vs (TLRef i))))))))))) (\lambda (H4: (arity g c (THeads
269 (Flat Appl) vs (lift (S i) O v)) (AHead a0 a1))).(\lambda (H5: ((\forall (d0:
270 C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c)
271 \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift
272 (S i) O v)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (TLRef i))
273 (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall
274 (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
275 (THeads (Flat Appl) vs (TLRef i)))))))))) (arity_appls_abbr g c d v i H2 vs
276 (AHead a0 a1) H4) (\lambda (d0: C).(\lambda (w: T).(\lambda (H6: (sc3 g a0 d0
277 w)).(\lambda (is: PList).(\lambda (H7: (drop1 is d0 c)).(let H_x \def
278 (drop1_getl_trans is c d0 H7 Abbr d v i H2) in (let H8 \def H_x in (ex_ind C
279 (\lambda (e2: C).(getl (trans is i) d0 (CHead e2 (Bind Abbr) (ctrans is i
280 v)))) (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs
281 (TLRef i))))) (\lambda (x: C).(\lambda (H9: (getl (trans is i) d0 (CHead x
282 (Bind Abbr) (ctrans is i v)))).(let H_y \def (H0 (TCons w (lifts1 is vs))) in
283 (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) (\lambda
284 (t: T).(sc3 g a1 d0 (THead (Flat Appl) w t))) (eq_ind_r T (TLRef (trans is
285 i)) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w (THeads (Flat Appl)
286 (lifts1 is vs) t)))) (H_y (trans is i) x (ctrans is i v) d0 (eq_ind T (lift1
287 is (lift (S i) O v)) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w
288 (THeads (Flat Appl) (lifts1 is vs) t)))) (eq_ind T (lift1 is (THeads (Flat
289 Appl) vs (lift (S i) O v))) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w
290 t))) (H5 d0 w H6 is H7) (THeads (Flat Appl) (lifts1 is vs) (lift1 is (lift (S
291 i) O v))) (lifts1_flat Appl is (lift (S i) O v) vs)) (lift (S (trans is i)) O
292 (ctrans is i v)) (lift1_free is i v)) H9) (lift1 is (TLRef i)) (lift1_lref is
293 i)) (lift1 is (THeads (Flat Appl) vs (TLRef i))) (lifts1_flat Appl is (TLRef
294 i) vs))))) H8))))))))))) H3))))))))))))) a)).
297 \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall
298 (u: T).((sc3 g (asucc g a) c (THeads (Flat Appl) vs u)) \to (\forall (t:
299 T).((sc3 g a c (THeads (Flat Appl) vs t)) \to (sc3 g a c (THeads (Flat Appl)
300 vs (THead (Flat Cast) u t))))))))))
302 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs:
303 TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat
304 Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to
305 (sc3 g a0 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (\lambda
306 (n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u:
307 T).(\lambda (H: (sc3 g (match n with [O \Rightarrow (ASort O (next g n0)) |
308 (S h) \Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u))).(\lambda (t:
309 T).(\lambda (H0: (land (arity g c (THeads (Flat Appl) vs t) (ASort n n0))
310 (sn3 c (THeads (Flat Appl) vs t)))).(nat_ind (\lambda (n1: nat).((sc3 g
311 (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow
312 (ASort h n0)]) c (THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads
313 (Flat Appl) vs t) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land
314 (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0))
315 (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))) (\lambda (H1:
316 (sc3 g (ASort O (next g n0)) c (THeads (Flat Appl) vs u))).(\lambda (H2:
317 (land (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads (Flat
318 Appl) vs t)))).(let H3 \def H1 in (and_ind (arity g c (THeads (Flat Appl) vs
319 u) (ASort O (next g n0))) (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c
320 (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads
321 (Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads
322 (Flat Appl) vs u) (ASort O (next g n0)))).(\lambda (H5: (sn3 c (THeads (Flat
323 Appl) vs u))).(let H6 \def H2 in (and_ind (arity g c (THeads (Flat Appl) vs
324 t) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity g c (THeads
325 (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat
326 Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat
327 Appl) vs t) (ASort O n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs
328 t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort
329 O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))
330 (arity_appls_cast g c u t vs (ASort O n0) H4 H7) (sn3_appls_cast c vs u H5 t
331 H8)))) H6)))) H3)))) (\lambda (n1: nat).(\lambda (_: (((sc3 g (match n1 with
332 [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) c
333 (THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads (Flat Appl) vs t)
334 (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land (arity g c
335 (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) (sn3 c (THeads
336 (Flat Appl) vs (THead (Flat Cast) u t)))))))).(\lambda (H1: (sc3 g (ASort n1
337 n0) c (THeads (Flat Appl) vs u))).(\lambda (H2: (land (arity g c (THeads
338 (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let
339 H3 \def H1 in (and_ind (arity g c (THeads (Flat Appl) vs u) (ASort n1 n0))
340 (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c (THeads (Flat Appl) vs
341 (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs
342 (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u)
343 (ASort n1 n0))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def
344 H2 in (and_ind (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c
345 (THeads (Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead
346 (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead
347 (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort
348 (S n1) n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g
349 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c
350 (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (arity_appls_cast g c u t vs
351 (ASort (S n1) n0) H4 H7) (sn3_appls_cast c vs u H5 t H8)))) H6)))) H3)))))) n
352 H H0))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (vs: TList).(\forall
353 (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to
354 (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c
355 (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (a1:
356 A).(\lambda (H0: ((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3
357 g (asucc g a1) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a1 c
358 (THeads (Flat Appl) vs t)) \to (sc3 g a1 c (THeads (Flat Appl) vs (THead
359 (Flat Cast) u t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u:
360 T).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc
361 g a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
362 PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1
363 is (THeads (Flat Appl) vs u))))))))))).(\lambda (t: T).(\lambda (H2: (land
364 (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall
365 (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1
366 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))))).(let H3
367 \def H1 in (and_ind (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g
368 a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
369 PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1
370 is (THeads (Flat Appl) vs u))))))))) (land (arity g c (THeads (Flat Appl) vs
371 (THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3
372 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead
373 (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u
374 t))))))))))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (AHead a0
375 (asucc g a1)))).(\lambda (H5: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d
376 w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead
377 (Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))))).(let H6 \def H2
378 in (and_ind (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d:
379 C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
380 \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs
381 t))))))))) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t))
382 (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall
383 (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is
384 (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))) (\lambda (H7: (arity
385 g c (THeads (Flat Appl) vs t) (AHead a0 a1))).(\lambda (H8: ((\forall (d:
386 C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
387 \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs
388 t))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t))
389 (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall
390 (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is
391 (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (arity_appls_cast g c
392 u t vs (AHead a0 a1) H4 H7) (\lambda (d: C).(\lambda (w: T).(\lambda (H9:
393 (sc3 g a0 d w)).(\lambda (is: PList).(\lambda (H10: (drop1 is d c)).(let H_y
394 \def (H0 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1
395 is vs) (lift1 is (THead (Flat Cast) u t))) (\lambda (t0: T).(sc3 g a1 d
396 (THead (Flat Appl) w t0))) (eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1
397 is t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat Appl)
398 (lifts1 is vs) t0)))) (H_y d (lift1 is u) (eq_ind T (lift1 is (THeads (Flat
399 Appl) vs u)) (\lambda (t0: T).(sc3 g (asucc g a1) d (THead (Flat Appl) w
400 t0))) (H5 d w H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is u))
401 (lifts1_flat Appl is u vs)) (lift1 is t) (eq_ind T (lift1 is (THeads (Flat
402 Appl) vs t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w t0))) (H8 d w
403 H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is t)) (lifts1_flat Appl
404 is t vs))) (lift1 is (THead (Flat Cast) u t)) (lift1_flat Cast is u t))
405 (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (lifts1_flat Appl
406 is (THead (Flat Cast) u t) vs))))))))))) H6)))) H3)))))))))))) a)).
408 theorem sc3_props__sc3_sn3_abst:
409 \forall (g: G).(\forall (a: A).(land (\forall (c: C).(\forall (t: T).((sc3 g
410 a c t) \to (sn3 c t)))) (\forall (vs: TList).(\forall (i: nat).(let t \def
411 (THeads (Flat Appl) vs (TLRef i)) in (\forall (c: C).((arity g c t a) \to
412 ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a c t))))))))))
414 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(land (\forall (c:
415 C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall (vs:
416 TList).(\forall (i: nat).(let t \def (THeads (Flat Appl) vs (TLRef i)) in
417 (\forall (c: C).((arity g c t a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to
418 (sc3 g a0 c t)))))))))) (\lambda (n: nat).(\lambda (n0: nat).(conj (\forall
419 (c: C).(\forall (t: T).((land (arity g c t (ASort n n0)) (sn3 c t)) \to (sn3
420 c t)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c
421 (THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) \to ((nf2 c (TLRef i)) \to
422 ((sns3 c vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n
423 n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))) (\lambda (c:
424 C).(\lambda (t: T).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c
425 t))).(let H0 \def H in (and_ind (arity g c t (ASort n n0)) (sn3 c t) (sn3 c
426 t) (\lambda (_: (arity g c t (ASort n n0))).(\lambda (H2: (sn3 c t)).H2))
427 H0))))) (\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H:
428 (arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n n0))).(\lambda (H0:
429 (nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(conj (arity g c (THeads (Flat
430 Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i))) H
431 (sn3_appls_lref c i H0 vs H1))))))))))) (\lambda (a0: A).(\lambda (H: (land
432 (\forall (c: C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall
433 (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads (Flat Appl)
434 vs (TLRef i)) a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a0 c
435 (THeads (Flat Appl) vs (TLRef i))))))))))).(\lambda (a1: A).(\lambda (H0:
436 (land (\forall (c: C).(\forall (t: T).((sc3 g a1 c t) \to (sn3 c t))))
437 (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads
438 (Flat Appl) vs (TLRef i)) a1) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to
439 (sc3 g a1 c (THeads (Flat Appl) vs (TLRef i))))))))))).(conj (\forall (c:
440 C).(\forall (t: T).((land (arity g c t (AHead a0 a1)) (\forall (d:
441 C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
442 \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is 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)) (AHead a0 a1)) \to ((nf2 c (TLRef i)) \to ((sns3 c
445 vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))
446 (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
447 PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads
448 (Flat Appl) vs (TLRef i))))))))))))))))) (\lambda (c: C).(\lambda (t:
449 T).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall
450 (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1
451 d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H in (and_ind
452 (\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 t0))))
453 (\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads
454 (Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to
455 (sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) (\lambda (_:
456 ((\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0
457 t0)))))).(\lambda (H4: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0:
458 C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i))
459 \to ((sns3 c0 vs) \to (sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef
460 i))))))))))).(let H5 \def H0 in (and_ind (\forall (c0: C).(\forall (t0:
461 T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))) (\forall (vs: TList).(\forall (i:
462 nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a1) \to
463 ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 (THeads (Flat Appl) vs
464 (TLRef i))))))))) (sn3 c t) (\lambda (H6: ((\forall (c0: C).(\forall (t0:
465 T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))))).(\lambda (_: ((\forall (vs:
466 TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs
467 (TLRef i)) a1) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0
468 (THeads (Flat Appl) vs (TLRef i))))))))))).(let H8 \def H1 in (and_ind (arity
469 g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to
470 (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w
471 (lift1 is t)))))))) (sn3 c t) (\lambda (H9: (arity g c t (AHead a0
472 a1))).(\lambda (H10: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to
473 (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w
474 (lift1 is t)))))))))).(let H_y \def (arity_aprem g c t (AHead a0 a1) H9 O a0)
475 in (let H11 \def (H_y (aprem_zero a0 a1)) in (ex2_3_ind C T nat (\lambda (d:
476 C).(\lambda (_: T).(\lambda (j: nat).(drop j O d c)))) (\lambda (d:
477 C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g a0))))) (sn3 c t)
478 (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H12: (drop x2
479 O x0 c)).(\lambda (H13: (arity g x0 x1 (asucc g a0))).(let H_y0 \def (H10
480 (CHead x0 (Bind Abst) x1) (TLRef O) (H4 TNil O (CHead x0 (Bind Abst) x1)
481 (arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1) a0
482 H13) (nf2_lref_abst (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1))
483 I) (PCons (S x2) O PNil)) in (let H_y1 \def (H6 (CHead x0 (Bind Abst) x1)
484 (THead (Flat Appl) (TLRef O) (lift (S x2) O t)) (H_y0 (drop1_cons (CHead x0
485 (Bind Abst) x1) c (S x2) O (drop_drop (Bind Abst) x2 x0 c H12 x1) c PNil
486 (drop1_nil c)))) in (let H_x \def (sn3_gen_flat Appl (CHead x0 (Bind Abst)
487 x1) (TLRef O) (lift (S x2) O t) H_y1) in (let H14 \def H_x in (and_ind (sn3
488 (CHead x0 (Bind Abst) x1) (TLRef O)) (sn3 (CHead x0 (Bind Abst) x1) (lift (S
489 x2) O t)) (sn3 c t) (\lambda (_: (sn3 (CHead x0 (Bind Abst) x1) (TLRef
490 O))).(\lambda (H16: (sn3 (CHead x0 (Bind Abst) x1) (lift (S x2) O
491 t))).(sn3_gen_lift (CHead x0 (Bind Abst) x1) t (S x2) O H16 c (drop_drop
492 (Bind Abst) x2 x0 c H12 x1)))) H14)))))))))) H11))))) H8)))) H5)))) H2)))))
493 (\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H1: (arity g
494 c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))).(\lambda (H2: (nf2 c
495 (TLRef i))).(\lambda (H3: (sns3 c vs)).(conj (arity g c (THeads (Flat Appl)
496 vs (TLRef i)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w)
497 \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w
498 (lift1 is (THeads (Flat Appl) vs (TLRef i)))))))))) H1 (\lambda (d:
499 C).(\lambda (w: T).(\lambda (H4: (sc3 g a0 d w)).(\lambda (is:
500 PList).(\lambda (H5: (drop1 is d c)).(let H6 \def H in (and_ind (\forall (c0:
501 C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))) (\forall (vs0:
502 TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl)
503 vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a0
504 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a1 d (THead (Flat Appl)
505 w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (H7: ((\forall (c0:
506 C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))))).(\lambda (_:
507 ((\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0
508 (THeads (Flat Appl) vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3
509 c0 vs0) \to (sc3 g a0 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))))).(let H9
510 \def H0 in (and_ind (\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) \to
511 (sn3 c0 t)))) (\forall (vs0: TList).(\forall (i0: nat).(\forall (c0:
512 C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) \to ((nf2 c0 (TLRef
513 i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat Appl) vs0 (TLRef
514 i0))))))))) (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs
515 (TLRef i))))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t)
516 \to (sn3 c0 t)))))).(\lambda (H11: ((\forall (vs0: TList).(\forall (i0:
517 nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1)
518 \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat
519 Appl) vs0 (TLRef i0))))))))))).(let H_y \def (H11 (TCons w (lifts1 is vs)))
520 in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i)))
521 (\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w t))) (eq_ind_r T (TLRef
522 (trans is i)) (\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat
523 Appl) (lifts1 is vs) t)))) (H_y (trans is i) d (eq_ind T (lift1 is (TLRef i))
524 (\lambda (t: T).(arity g d (THead (Flat Appl) w (THeads (Flat Appl) (lifts1
525 is vs) t)) a1)) (eq_ind T (lift1 is (THeads (Flat Appl) vs (TLRef i)))
526 (\lambda (t: T).(arity g d (THead (Flat Appl) w t) a1)) (arity_appl g d w a0
527 (sc3_arity_gen g d w a0 H4) (lift1 is (THeads (Flat Appl) vs (TLRef i))) a1
528 (arity_lift1 g (AHead a0 a1) c is d (THeads (Flat Appl) vs (TLRef i)) H5 H1))
529 (THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) (lifts1_flat Appl is
530 (TLRef i) vs)) (TLRef (trans is i)) (lift1_lref is i)) (eq_ind T (lift1 is
531 (TLRef i)) (\lambda (t: T).(nf2 d t)) (nf2_lift1 c is d (TLRef i) H5 H2)
532 (TLRef (trans is i)) (lift1_lref is i)) (conj (sn3 d w) (sns3 d (lifts1 is
533 vs)) (H7 d w H4) (sns3_lifts1 c is d H5 vs H3))) (lift1 is (TLRef i))
534 (lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs (TLRef i))) (lifts1_flat
535 Appl is (TLRef i) vs))))) H9)))) H6))))))))))))))))))) a)).
538 \forall (g: G).(\forall (a: A).(\forall (c: C).(\forall (t: T).((sc3 g a c
541 \lambda (g: G).(\lambda (a: A).(\lambda (c: C).(\lambda (t: T).(\lambda (H:
542 (sc3 g a c t)).(let H_x \def (sc3_props__sc3_sn3_abst g a) in (let H0 \def
543 H_x in (and_ind (\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3
544 c0 t0)))) (\forall (vs: TList).(\forall (i: nat).(let t0 \def (THeads (Flat
545 Appl) vs (TLRef i)) in (\forall (c0: C).((arity g c0 t0 a) \to ((nf2 c0
546 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a c0 t0)))))))) (sn3 c t) (\lambda
547 (H1: ((\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 c0
548 t0)))))).(\lambda (_: ((\forall (vs: TList).(\forall (i: nat).(let t0 \def
549 (THeads (Flat Appl) vs (TLRef i)) in (\forall (c0: C).((arity g c0 t0 a) \to
550 ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a c0 t0)))))))))).(H1 c t
554 \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall
555 (i: nat).((arity g c (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c (TLRef
556 i)) \to ((sns3 c vs) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i))))))))))
558 \lambda (g: G).(\lambda (a: A).(\lambda (vs: TList).(\lambda (c: C).(\lambda
559 (i: nat).(\lambda (H: (arity g c (THeads (Flat Appl) vs (TLRef i))
560 a)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(let H_x \def
561 (sc3_props__sc3_sn3_abst g a) in (let H2 \def H_x in (and_ind (\forall (c0:
562 C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0 t)))) (\forall (vs0:
563 TList).(\forall (i0: nat).(let t \def (THeads (Flat Appl) vs0 (TLRef i0)) in
564 (\forall (c0: C).((arity g c0 t a) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0
565 vs0) \to (sc3 g a c0 t)))))))) (sc3 g a c (THeads (Flat Appl) vs (TLRef i)))
566 (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0
567 t)))))).(\lambda (H4: ((\forall (vs0: TList).(\forall (i0: nat).(let t \def
568 (THeads (Flat Appl) vs0 (TLRef i0)) in (\forall (c0: C).((arity g c0 t a) \to
569 ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a c0 t)))))))))).(H4 vs i
570 c H H0 H1))) H2)))))))))).
573 \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1:
574 A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v:
575 T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts
576 (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a2 c (THeads (Flat Appl) vs
577 (THead (Bind b) v t)))))))))))))
579 \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda
580 (a1: A).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (vs: TList).(\forall
581 (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads
582 (Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads
583 (Flat Appl) vs (THead (Bind b) v t)))))))))) (\lambda (n: nat).(\lambda (n0:
584 nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t:
585 T).(\lambda (H0: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl)
586 (lifts (S O) O vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat
587 Appl) (lifts (S O) O vs) t)))).(\lambda (H1: (sc3 g a1 c v)).(let H2 \def H0
588 in (and_ind (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O
589 vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S
590 O) O vs) t)) (land (arity g c (THeads (Flat Appl) vs (THead (Bind b) v t))
591 (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t)))) (\lambda
592 (H3: (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)
593 (ASort n n0))).(\lambda (H4: (sn3 (CHead c (Bind b) v) (THeads (Flat Appl)
594 (lifts (S O) O vs) t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Bind
595 b) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t)))
596 (arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 H1) t vs (ASort n n0)
597 H3) (sn3_appls_bind b H c v (sc3_sn3 g a1 c v H1) vs t H4)))) H2))))))))))
598 (\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall (c: C).(\forall
599 (v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads (Flat Appl)
600 (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads (Flat Appl)
601 vs (THead (Bind b) v t))))))))))).(\lambda (a0: A).(\lambda (H1: ((\forall
602 (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 (CHead
603 c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v)
604 \to (sc3 g a0 c (THeads (Flat Appl) vs (THead (Bind b) v
605 t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda
606 (t: T).(\lambda (H2: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl)
607 (lifts (S O) O vs) t) (AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a
608 d w) \to (\forall (is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g
609 a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs)
610 t))))))))))).(\lambda (H3: (sc3 g a1 c v)).(let H4 \def H2 in (and_ind (arity
611 g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) (AHead a
612 a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is:
613 PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat Appl)
614 w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))) (land (arity g
615 c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) (\forall (d:
616 C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c)
617 \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead
618 (Bind b) v t))))))))))) (\lambda (H5: (arity g (CHead c (Bind b) v) (THeads
619 (Flat Appl) (lifts (S O) O vs) t) (AHead a a0))).(\lambda (H6: ((\forall (d:
620 C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d
621 (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads
622 (Flat Appl) (lifts (S O) O vs) t))))))))))).(conj (arity g c (THeads (Flat
623 Appl) vs (THead (Bind b) v t)) (AHead a a0)) (\forall (d: C).(\forall (w:
624 T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
625 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Bind b) v
626 t)))))))))) (arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 H3) t vs
627 (AHead a a0) H5) (\lambda (d: C).(\lambda (w: T).(\lambda (H7: (sc3 g a d
628 w)).(\lambda (is: PList).(\lambda (H8: (drop1 is d c)).(let H_y \def (H1
629 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs)
630 (lift1 is (THead (Bind b) v t))) (\lambda (t0: T).(sc3 g a0 d (THead (Flat
631 Appl) w t0))) (eq_ind_r T (THead (Bind b) (lift1 is v) (lift1 (Ss is) t))
632 (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w (THeads (Flat Appl) (lifts1
633 is vs) t0)))) (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind TList (lifts1 (Ss
634 is) (lifts (S O) O vs)) (\lambda (t0: TList).(sc3 g a0 (CHead d (Bind b)
635 (lift1 is v)) (THead (Flat Appl) (lift (S O) O w) (THeads (Flat Appl) t0
636 (lift1 (Ss is) t))))) (eq_ind T (lift1 (Ss is) (THeads (Flat Appl) (lifts (S
637 O) O vs) t)) (\lambda (t0: T).(sc3 g a0 (CHead d (Bind b) (lift1 is v))
638 (THead (Flat Appl) (lift (S O) O w) t0))) (H6 (CHead d (Bind b) (lift1 is v))
639 (lift (S O) O w) (sc3_lift g a d w H7 (CHead d (Bind b) (lift1 is v)) (S O) O
640 (drop_drop (Bind b) O d d (drop_refl d) (lift1 is v))) (Ss is)
641 (drop1_skip_bind b c is d v H8)) (THeads (Flat Appl) (lifts1 (Ss is) (lifts
642 (S O) O vs)) (lift1 (Ss is) t)) (lifts1_flat Appl (Ss is) t (lifts (S O) O
643 vs))) (lifts (S O) O (lifts1 is vs)) (lifts1_xhg is vs)) (sc3_lift1 g c a1 is
644 d v H3 H8)) (lift1 is (THead (Bind b) v t)) (lift1_bind b is v t)) (lift1 is
645 (THeads (Flat Appl) vs (THead (Bind b) v t))) (lifts1_flat Appl is (THead
646 (Bind b) v t) vs))))))))))) H4)))))))))))) a2))))).
649 \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (vs:
650 TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 c (THeads
651 (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w:
652 T).((sc3 g (asucc g a1) c w) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead
653 (Flat Appl) v (THead (Bind Abst) w t))))))))))))))
655 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(A_ind (\lambda (a:
656 A).(\forall (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3
657 g a c (THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v)
658 \to (\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a c (THeads (Flat
659 Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))))))))))) (\lambda
660 (n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v:
661 T).(\lambda (t: T).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs
662 (THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead
663 (Bind Abbr) v t))))).(\lambda (H0: (sc3 g a1 c v)).(\lambda (w: T).(\lambda
664 (H1: (sc3 g (asucc g a1) c w)).(let H2 \def H in (and_ind (arity g c (THeads
665 (Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat
666 Appl) vs (THead (Bind Abbr) v t))) (land (arity g c (THeads (Flat Appl) vs
667 (THead (Flat Appl) v (THead (Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads
668 (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H3:
669 (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n
670 n0))).(\lambda (H4: (sn3 c (THeads (Flat Appl) vs (THead (Bind Abbr) v
671 t)))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v (THead
672 (Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat
673 Appl) v (THead (Bind Abst) w t)))) (arity_appls_appl g c v a1 (sc3_arity_gen
674 g c v a1 H0) w (sc3_arity_gen g c w (asucc g a1) H1) t vs (ASort n n0) H3)
675 (sn3_appls_beta c v t vs H4 w (sc3_sn3 g (asucc g a1) c w H1)))))
676 H2)))))))))))) (\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall
677 (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a c (THeads (Flat Appl) vs
678 (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: T).((sc3 g
679 (asucc g a1) c w) \to (sc3 g a c (THeads (Flat Appl) vs (THead (Flat Appl) v
680 (THead (Bind Abst) w t)))))))))))))).(\lambda (a0: A).(\lambda (H0: ((\forall
681 (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 c
682 (THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to
683 (\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a0 c (THeads (Flat Appl)
684 vs (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))))))).(\lambda (vs:
685 TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H1: (land
686 (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (AHead a a0))
687 (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is:
688 PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads
689 (Flat Appl) vs (THead (Bind Abbr) v t)))))))))))).(\lambda (H2: (sc3 g a1 c
690 v)).(\lambda (w: T).(\lambda (H3: (sc3 g (asucc g a1) c w)).(let H4 \def H1
691 in (and_ind (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (AHead
692 a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall (is:
693 PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is
694 (THeads (Flat Appl) vs (THead (Bind Abbr) v t)))))))))) (land (arity g c
695 (THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))) (AHead
696 a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall (is:
697 PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is
698 (THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w
699 t)))))))))))) (\lambda (H5: (arity g c (THeads (Flat Appl) vs (THead (Bind
700 Abbr) v t)) (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w0:
701 T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
702 (THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Bind Abbr) v
703 t)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v
704 (THead (Bind Abst) w t))) (AHead a a0)) (\forall (d: C).(\forall (w0:
705 T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
706 (THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Flat Appl) v
707 (THead (Bind Abst) w t))))))))))) (arity_appls_appl g c v a1 (sc3_arity_gen g
708 c v a1 H2) w (sc3_arity_gen g c w (asucc g a1) H3) t vs (AHead a a0) H5)
709 (\lambda (d: C).(\lambda (w0: T).(\lambda (H7: (sc3 g a d w0)).(\lambda (is:
710 PList).(\lambda (H8: (drop1 is d c)).(eq_ind_r T (THeads (Flat Appl) (lifts1
711 is vs) (lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t)))) (\lambda
712 (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 t0))) (eq_ind_r T (THead (Flat
713 Appl) (lift1 is v) (lift1 is (THead (Bind Abst) w t))) (\lambda (t0: T).(sc3
714 g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) t0))))
715 (eq_ind_r T (THead (Bind Abst) (lift1 is w) (lift1 (Ss is) t)) (\lambda (t0:
716 T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs)
717 (THead (Flat Appl) (lift1 is v) t0))))) (let H_y \def (H0 (TCons w0 (lifts1
718 is vs))) in (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind T (lift1 is (THead
719 (Bind Abbr) v t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads
720 (Flat Appl) (lifts1 is vs) t0)))) (eq_ind T (lift1 is (THeads (Flat Appl) vs
721 (THead (Bind Abbr) v t))) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0
722 t0))) (H6 d w0 H7 is H8) (THeads (Flat Appl) (lifts1 is vs) (lift1 is (THead
723 (Bind Abbr) v t))) (lifts1_flat Appl is (THead (Bind Abbr) v t) vs)) (THead
724 (Bind Abbr) (lift1 is v) (lift1 (Ss is) t)) (lift1_bind Abbr is v t))
725 (sc3_lift1 g c a1 is d v H2 H8) (lift1 is w) (sc3_lift1 g c (asucc g a1) is d
726 w H3 H8))) (lift1 is (THead (Bind Abst) w t)) (lift1_bind Abst is w t))
727 (lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t))) (lift1_flat Appl is
728 v (THead (Bind Abst) w t))) (lift1 is (THeads (Flat Appl) vs (THead (Flat
729 Appl) v (THead (Bind Abst) w t)))) (lifts1_flat Appl is (THead (Flat Appl) v
730 (THead (Bind Abst) w t)) vs)))))))))) H4)))))))))))))) a2))).
projects / helm.git / blob