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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/arity".
19 include "csubc/props.ma".
21 theorem sc3_arity_csubc:
22 \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
23 t a) \to (\forall (d1: C).(\forall (is: PList).((drop1 is d1 c1) \to (\forall
24 (c2: C).((csubc g d1 c2) \to (sc3 g a c2 (lift1 is t)))))))))))
26 \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
27 (arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
28 A).(\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2:
29 C).((csubc g d1 c2) \to (sc3 g a0 c2 (lift1 is t0)))))))))) (\lambda (c:
30 C).(\lambda (n: nat).(\lambda (d1: C).(\lambda (is: PList).(\lambda (_:
31 (drop1 is d1 c)).(\lambda (c2: C).(\lambda (_: (csubc g d1 c2)).(eq_ind_r T
32 (TSort n) (\lambda (t0: T).(land (arity g c2 t0 (ASort O n)) (sn3 c2 t0)))
33 (conj (arity g c2 (TSort n) (ASort O n)) (sn3 c2 (TSort n)) (arity_sort g c2
34 n) (sn3_nf2 c2 (TSort n) (nf2_sort c2 n))) (lift1 is (TSort n)) (lift1_sort n
35 is))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
36 nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0:
37 A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (d1: C).(\forall
38 (is: PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g
39 a0 c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda
40 (H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(let
41 H_x \def (drop1_getl_trans is c d1 H3 Abbr d u i H0) in (let H5 \def H_x in
42 (ex2_ind C (\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2:
43 C).(getl (trans is i) d1 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) u))))
44 (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x: C).(\lambda (_: (drop1
45 (ptrans is i) x d)).(\lambda (H7: (getl (trans is i) d1 (CHead x (Bind Abbr)
46 (lift1 (ptrans is i) u)))).(let H_x0 \def (csubc_getl_conf g d1 (CHead x
47 (Bind Abbr) (lift1 (ptrans is i) u)) (trans is i) H7 c2 H4) in (let H8 \def
48 H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans is i) c2 e2)) (\lambda (e2:
49 C).(csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) e2)) (sc3 g a0 c2
50 (lift1 is (TLRef i))) (\lambda (x0: C).(\lambda (H9: (getl (trans is i) c2
51 x0)).(\lambda (H10: (csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u))
52 x0)).(let H11 \def (match H10 in csubc return (\lambda (c0: C).(\lambda (c3:
53 C).(\lambda (_: (csubc ? c0 c3)).((eq C c0 (CHead x (Bind Abbr) (lift1
54 (ptrans is i) u))) \to ((eq C c3 x0) \to (sc3 g a0 c2 (lift1 is (TLRef
55 i)))))))) with [(csubc_sort n) \Rightarrow (\lambda (H11: (eq C (CSort n)
56 (CHead x (Bind Abbr) (lift1 (ptrans is i) u)))).(\lambda (H12: (eq C (CSort
57 n) x0)).((let H13 \def (eq_ind C (CSort n) (\lambda (e: C).(match e in C
58 return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
59 \Rightarrow False])) I (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) H11) in
60 (False_ind ((eq C (CSort n) x0) \to (sc3 g a0 c2 (lift1 is (TLRef i)))) H13))
61 H12))) | (csubc_head c0 c3 H11 k v) \Rightarrow (\lambda (H12: (eq C (CHead
62 c0 k v) (CHead x (Bind Abbr) (lift1 (ptrans is i) u)))).(\lambda (H13: (eq C
63 (CHead c3 k v) x0)).((let H14 \def (f_equal C T (\lambda (e: C).(match e in C
64 return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t0)
65 \Rightarrow t0])) (CHead c0 k v) (CHead x (Bind Abbr) (lift1 (ptrans is i)
66 u)) H12) in ((let H15 \def (f_equal C K (\lambda (e: C).(match e in C return
67 (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow
68 k0])) (CHead c0 k v) (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) H12) in
69 ((let H16 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_:
70 C).C) with [(CSort _) \Rightarrow c0 | (CHead c4 _ _) \Rightarrow c4]))
71 (CHead c0 k v) (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) H12) in (eq_ind
72 C x (\lambda (c4: C).((eq K k (Bind Abbr)) \to ((eq T v (lift1 (ptrans is i)
73 u)) \to ((eq C (CHead c3 k v) x0) \to ((csubc g c4 c3) \to (sc3 g a0 c2
74 (lift1 is (TLRef i)))))))) (\lambda (H17: (eq K k (Bind Abbr))).(eq_ind K
75 (Bind Abbr) (\lambda (k0: K).((eq T v (lift1 (ptrans is i) u)) \to ((eq C
76 (CHead c3 k0 v) x0) \to ((csubc g x c3) \to (sc3 g a0 c2 (lift1 is (TLRef
77 i))))))) (\lambda (H18: (eq T v (lift1 (ptrans is i) u))).(eq_ind T (lift1
78 (ptrans is i) u) (\lambda (t0: T).((eq C (CHead c3 (Bind Abbr) t0) x0) \to
79 ((csubc g x c3) \to (sc3 g a0 c2 (lift1 is (TLRef i)))))) (\lambda (H19: (eq
80 C (CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)) x0)).(eq_ind C (CHead c3
81 (Bind Abbr) (lift1 (ptrans is i) u)) (\lambda (_: C).((csubc g x c3) \to (sc3
82 g a0 c2 (lift1 is (TLRef i))))) (\lambda (_: (csubc g x c3)).(let H21 \def
83 (eq_ind_r C x0 (\lambda (c4: C).(getl (trans is i) c2 c4)) H9 (CHead c3 (Bind
84 Abbr) (lift1 (ptrans is i) u)) H19) in (let H_y \def (sc3_abbr g a0 TNil) in
85 (eq_ind_r T (TLRef (trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y
86 (trans is i) c3 (lift1 (ptrans is i) u) c2 (eq_ind T (lift1 is (lift (S i) O
87 u)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (eq_ind T (lift1 (PConsTail is (S i)
88 O) u) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H2 d1 (PConsTail is (S i) O)
89 (drop1_cons_tail c d (S i) O (getl_drop Abbr c d u i H0) is d1 H3) c2 H4)
90 (lift1 is (lift (S i) O u)) (lift1_cons_tail u (S i) O is)) (lift (S (trans
91 is i)) O (lift1 (ptrans is i) u)) (lift1_free is i u)) H21) (lift1 is (TLRef
92 i)) (lift1_lref is i))))) x0 H19)) v (sym_eq T v (lift1 (ptrans is i) u)
93 H18))) k (sym_eq K k (Bind Abbr) H17))) c0 (sym_eq C c0 x H16))) H15)) H14))
94 H13 H11))) | (csubc_abst c0 c3 H11 v a1 H12 w H13) \Rightarrow (\lambda (H14:
95 (eq C (CHead c0 (Bind Abst) v) (CHead x (Bind Abbr) (lift1 (ptrans is i)
96 u)))).(\lambda (H15: (eq C (CHead c3 (Bind Abbr) w) x0)).((let H16 \def
97 (eq_ind C (CHead c0 (Bind Abst) v) (\lambda (e: C).(match e in C return
98 (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
99 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
100 \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
101 False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
102 \Rightarrow False])])) I (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) H14)
103 in (False_ind ((eq C (CHead c3 (Bind Abbr) w) x0) \to ((csubc g c0 c3) \to
104 ((sc3 g (asucc g a1) c0 v) \to ((sc3 g a1 c3 w) \to (sc3 g a0 c2 (lift1 is
105 (TLRef i))))))) H16)) H15 H11 H12 H13)))]) in (H11 (refl_equal C (CHead x
106 (Bind Abbr) (lift1 (ptrans is i) u))) (refl_equal C x0)))))) H8))))))
107 H5)))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda
108 (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0:
109 A).(\lambda (H1: (arity g d u (asucc g a0))).(\lambda (_: ((\forall (d1:
110 C).(\forall (is: PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1
111 c2) \to (sc3 g (asucc g a0) c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda
112 (is: PList).(\lambda (H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4:
113 (csubc g d1 c2)).(let H5 \def H0 in (let H_x \def (drop1_getl_trans is c d1
114 H3 Abst d u i H5) in (let H6 \def H_x in (ex2_ind C (\lambda (e2: C).(drop1
115 (ptrans is i) e2 d)) (\lambda (e2: C).(getl (trans is i) d1 (CHead e2 (Bind
116 Abst) (lift1 (ptrans is i) u)))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda
117 (x: C).(\lambda (H7: (drop1 (ptrans is i) x d)).(\lambda (H8: (getl (trans is
118 i) d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)))).(let H_x0 \def
119 (csubc_getl_conf g d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)) (trans is
120 i) H8 c2 H4) in (let H9 \def H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans
121 is i) c2 e2)) (\lambda (e2: C).(csubc g (CHead x (Bind Abst) (lift1 (ptrans
122 is i) u)) e2)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x0: C).(\lambda
123 (H10: (getl (trans is i) c2 x0)).(\lambda (H11: (csubc g (CHead x (Bind Abst)
124 (lift1 (ptrans is i) u)) x0)).(let H12 \def (match H11 in csubc return
125 (\lambda (c0: C).(\lambda (c3: C).(\lambda (_: (csubc ? c0 c3)).((eq C c0
126 (CHead x (Bind Abst) (lift1 (ptrans is i) u))) \to ((eq C c3 x0) \to (sc3 g
127 a0 c2 (lift1 is (TLRef i)))))))) with [(csubc_sort n) \Rightarrow (\lambda
128 (H12: (eq C (CSort n) (CHead x (Bind Abst) (lift1 (ptrans is i)
129 u)))).(\lambda (H13: (eq C (CSort n) x0)).((let H14 \def (eq_ind C (CSort n)
130 (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _)
131 \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead x (Bind Abst)
132 (lift1 (ptrans is i) u)) H12) in (False_ind ((eq C (CSort n) x0) \to (sc3 g
133 a0 c2 (lift1 is (TLRef i)))) H14)) H13))) | (csubc_head c0 c3 H12 k v)
134 \Rightarrow (\lambda (H13: (eq C (CHead c0 k v) (CHead x (Bind Abst) (lift1
135 (ptrans is i) u)))).(\lambda (H14: (eq C (CHead c3 k v) x0)).((let H15 \def
136 (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
137 [(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k v)
138 (CHead x (Bind Abst) (lift1 (ptrans is i) u)) H13) in ((let H16 \def (f_equal
139 C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _)
140 \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k v) (CHead x
141 (Bind Abst) (lift1 (ptrans is i) u)) H13) in ((let H17 \def (f_equal C C
142 (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
143 \Rightarrow c0 | (CHead c4 _ _) \Rightarrow c4])) (CHead c0 k v) (CHead x
144 (Bind Abst) (lift1 (ptrans is i) u)) H13) in (eq_ind C x (\lambda (c4:
145 C).((eq K k (Bind Abst)) \to ((eq T v (lift1 (ptrans is i) u)) \to ((eq C
146 (CHead c3 k v) x0) \to ((csubc g c4 c3) \to (sc3 g a0 c2 (lift1 is (TLRef
147 i)))))))) (\lambda (H18: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda
148 (k0: K).((eq T v (lift1 (ptrans is i) u)) \to ((eq C (CHead c3 k0 v) x0) \to
149 ((csubc g x c3) \to (sc3 g a0 c2 (lift1 is (TLRef i))))))) (\lambda (H19: (eq
150 T v (lift1 (ptrans is i) u))).(eq_ind T (lift1 (ptrans is i) u) (\lambda (t0:
151 T).((eq C (CHead c3 (Bind Abst) t0) x0) \to ((csubc g x c3) \to (sc3 g a0 c2
152 (lift1 is (TLRef i)))))) (\lambda (H20: (eq C (CHead c3 (Bind Abst) (lift1
153 (ptrans is i) u)) x0)).(eq_ind C (CHead c3 (Bind Abst) (lift1 (ptrans is i)
154 u)) (\lambda (_: C).((csubc g x c3) \to (sc3 g a0 c2 (lift1 is (TLRef i)))))
155 (\lambda (_: (csubc g x c3)).(let H22 \def (eq_ind_r C x0 (\lambda (c4:
156 C).(getl (trans is i) c2 c4)) H10 (CHead c3 (Bind Abst) (lift1 (ptrans is i)
157 u)) H20) in (let H_y \def (sc3_abst g a0 TNil) in (eq_ind_r T (TLRef (trans
158 is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y c2 (trans is i)
159 (csubc_arity_conf g d1 c2 H4 (TLRef (trans is i)) a0 (eq_ind T (lift1 is
160 (TLRef i)) (\lambda (t0: T).(arity g d1 t0 a0)) (arity_lift1 g a0 c is d1
161 (TLRef i) H3 (arity_abst g c d u i H0 a0 H1)) (TLRef (trans is i))
162 (lift1_lref is i))) (nf2_lref_abst c2 c3 (lift1 (ptrans is i) u) (trans is i)
163 H22) I) (lift1 is (TLRef i)) (lift1_lref is i))))) x0 H20)) v (sym_eq T v
164 (lift1 (ptrans is i) u) H19))) k (sym_eq K k (Bind Abst) H18))) c0 (sym_eq C
165 c0 x H17))) H16)) H15)) H14 H12))) | (csubc_abst c0 c3 H12 v a1 H13 w H14)
166 \Rightarrow (\lambda (H15: (eq C (CHead c0 (Bind Abst) v) (CHead x (Bind
167 Abst) (lift1 (ptrans is i) u)))).(\lambda (H16: (eq C (CHead c3 (Bind Abbr)
168 w) x0)).((let H17 \def (f_equal C T (\lambda (e: C).(match e in C return
169 (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow
170 t0])) (CHead c0 (Bind Abst) v) (CHead x (Bind Abst) (lift1 (ptrans is i) u))
171 H15) in ((let H18 \def (f_equal C C (\lambda (e: C).(match e in C return
172 (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c4 _ _)
173 \Rightarrow c4])) (CHead c0 (Bind Abst) v) (CHead x (Bind Abst) (lift1
174 (ptrans is i) u)) H15) in (eq_ind C x (\lambda (c4: C).((eq T v (lift1
175 (ptrans is i) u)) \to ((eq C (CHead c3 (Bind Abbr) w) x0) \to ((csubc g c4
176 c3) \to ((sc3 g (asucc g a1) c4 v) \to ((sc3 g a1 c3 w) \to (sc3 g a0 c2
177 (lift1 is (TLRef i))))))))) (\lambda (H19: (eq T v (lift1 (ptrans is i)
178 u))).(eq_ind T (lift1 (ptrans is i) u) (\lambda (t0: T).((eq C (CHead c3
179 (Bind Abbr) w) x0) \to ((csubc g x c3) \to ((sc3 g (asucc g a1) x t0) \to
180 ((sc3 g a1 c3 w) \to (sc3 g a0 c2 (lift1 is (TLRef i)))))))) (\lambda (H20:
181 (eq C (CHead c3 (Bind Abbr) w) x0)).(eq_ind C (CHead c3 (Bind Abbr) w)
182 (\lambda (_: C).((csubc g x c3) \to ((sc3 g (asucc g a1) x (lift1 (ptrans is
183 i) u)) \to ((sc3 g a1 c3 w) \to (sc3 g a0 c2 (lift1 is (TLRef i)))))))
184 (\lambda (_: (csubc g x c3)).(\lambda (H22: (sc3 g (asucc g a1) x (lift1
185 (ptrans is i) u))).(\lambda (H23: (sc3 g a1 c3 w)).(let H24 \def (eq_ind_r C
186 x0 (\lambda (c4: C).(getl (trans is i) c2 c4)) H10 (CHead c3 (Bind Abbr) w)
187 H20) in (let H_y \def (sc3_abbr g a0 TNil) in (eq_ind_r T (TLRef (trans is
188 i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y (trans is i) c3 w c2 (let H_y0
189 \def (arity_lift1 g (asucc g a0) d (ptrans is i) x u H7 H1) in (let H_y1 \def
190 (sc3_arity_gen g x (lift1 (ptrans is i) u) (asucc g a1) H22) in (sc3_repl g
191 a1 c2 (lift (S (trans is i)) O w) (sc3_lift g a1 c3 w H23 c2 (S (trans is i))
192 O (getl_drop Abbr c2 c3 w (trans is i) H24)) a0 (asucc_inj g a1 a0
193 (arity_mono g x (lift1 (ptrans is i) u) (asucc g a1) H_y1 (asucc g a0)
194 H_y0))))) H24) (lift1 is (TLRef i)) (lift1_lref is i))))))) x0 H20)) v
195 (sym_eq T v (lift1 (ptrans is i) u) H19))) c0 (sym_eq C c0 x H18))) H17)) H16
196 H12 H13 H14)))]) in (H12 (refl_equal C (CHead x (Bind Abst) (lift1 (ptrans is
197 i) u))) (refl_equal C x0)))))) H9)))))) H6))))))))))))))))) (\lambda (b:
198 B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u:
199 T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall
200 (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g
201 d1 c2) \to (sc3 g a1 c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda (a2:
202 A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: ((\forall
203 (d1: C).(\forall (is: PList).((drop1 is d1 (CHead c (Bind b) u)) \to (\forall
204 (c2: C).((csubc g d1 c2) \to (sc3 g a2 c2 (lift1 is t0))))))))).(\lambda (d1:
205 C).(\lambda (is: PList).(\lambda (H5: (drop1 is d1 c)).(\lambda (c2:
206 C).(\lambda (H6: (csubc g d1 c2)).(let H_y \def (sc3_bind g b H0 a1 a2 TNil)
207 in (eq_ind_r T (THead (Bind b) (lift1 is u) (lift1 (Ss is) t0)) (\lambda (t1:
208 T).(sc3 g a2 c2 t1)) (H_y c2 (lift1 is u) (lift1 (Ss is) t0) (H4 (CHead d1
209 (Bind b) (lift1 is u)) (Ss is) (drop1_skip_bind b c is d1 u H5) (CHead c2
210 (Bind b) (lift1 is u)) (csubc_head g d1 c2 H6 (Bind b) (lift1 is u))) (H2 d1
211 is H5 c2 H6)) (lift1 is (THead (Bind b) u t0)) (lift1_bind b is u
212 t0))))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1:
213 A).(\lambda (H0: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (d1:
214 C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1
215 c2) \to (sc3 g (asucc g a1) c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda
216 (a2: A).(\lambda (H2: (arity g (CHead c (Bind Abst) u) t0 a2)).(\lambda (H3:
217 ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 (CHead c (Bind Abst) u))
218 \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a2 c2 (lift1 is
219 t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H4: (drop1 is d1
220 c)).(\lambda (c2: C).(\lambda (H5: (csubc g d1 c2)).(eq_ind_r T (THead (Bind
221 Abst) (lift1 is u) (lift1 (Ss is) t0)) (\lambda (t1: T).(land (arity g c2 t1
222 (AHead a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall
223 (is0: PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1
224 is0 t1)))))))))) (conj (arity g c2 (THead (Bind Abst) (lift1 is u) (lift1 (Ss
225 is) t0)) (AHead a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to
226 (\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat Appl) w
227 (lift1 is0 (THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0))))))))))
228 (csubc_arity_conf g d1 c2 H5 (THead (Bind Abst) (lift1 is u) (lift1 (Ss is)
229 t0)) (AHead a1 a2) (arity_head g d1 (lift1 is u) a1 (arity_lift1 g (asucc g
230 a1) c is d1 u H4 H0) (lift1 (Ss is) t0) a2 (arity_lift1 g a2 (CHead c (Bind
231 Abst) u) (Ss is) (CHead d1 (Bind Abst) (lift1 is u)) t0 (drop1_skip_bind Abst
232 c is d1 u H4) H2))) (\lambda (d: C).(\lambda (w: T).(\lambda (H6: (sc3 g a1 d
233 w)).(\lambda (is0: PList).(\lambda (H7: (drop1 is0 d c2)).(eq_ind_r T (THead
234 (Bind Abst) (lift1 is0 (lift1 is u)) (lift1 (Ss is0) (lift1 (Ss is) t0)))
235 (\lambda (t1: T).(sc3 g a2 d (THead (Flat Appl) w t1))) (let H8 \def
236 (sc3_appl g a1 a2 TNil) in (H8 d w (lift1 (Ss is0) (lift1 (Ss is) t0)) (let
237 H_y \def (sc3_bind g Abbr (\lambda (H9: (eq B Abbr Abst)).(not_abbr_abst H9))
238 a1 a2 TNil) in (H_y d w (lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_x \def
239 (csubc_drop1_conf_rev g is0 d c2 H7 d1 H5) in (let H9 \def H_x in (ex2_ind C
240 (\lambda (c3: C).(drop1 is0 c3 d1)) (\lambda (c3: C).(csubc g c3 d)) (sc3 g
241 a2 (CHead d (Bind Abbr) w) (lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (x:
242 C).(\lambda (H10: (drop1 is0 x d1)).(\lambda (H11: (csubc g x d)).(eq_ind_r T
243 (lift1 (papp (Ss is0) (Ss is)) t0) (\lambda (t1: T).(sc3 g a2 (CHead d (Bind
244 Abbr) w) t1)) (eq_ind_r PList (Ss (papp is0 is)) (\lambda (p: PList).(sc3 g
245 a2 (CHead d (Bind Abbr) w) (lift1 p t0))) (H3 (CHead x (Bind Abst) (lift1
246 (papp is0 is) u)) (Ss (papp is0 is)) (drop1_skip_bind Abst c (papp is0 is) x
247 u (drop1_trans is0 x d1 H10 is c H4)) (CHead d (Bind Abbr) w) (csubc_abst g x
248 d H11 (lift1 (papp is0 is) u) a1 (H1 x (papp is0 is) (drop1_trans is0 x d1
249 H10 is c H4) x (csubc_refl g x)) w H6)) (papp (Ss is0) (Ss is)) (papp_ss is0
250 is)) (lift1 (Ss is0) (lift1 (Ss is) t0)) (lift1_lift1 (Ss is0) (Ss is)
251 t0))))) H9))) H6)) H6 (lift1 is0 (lift1 is u)) (sc3_lift1 g c2 (asucc g a1)
252 is0 d (lift1 is u) (H1 d1 is H4 c2 H5) H7))) (lift1 is0 (THead (Bind Abst)
253 (lift1 is u) (lift1 (Ss is) t0))) (lift1_bind Abst is0 (lift1 is u) (lift1
254 (Ss is) t0))))))))) (lift1 is (THead (Bind Abst) u t0)) (lift1_bind Abst is u
255 t0)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda
256 (_: (arity g c u a1)).(\lambda (H1: ((\forall (d1: C).(\forall (is:
257 PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a1
258 c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity
259 g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (d1: C).(\forall (is:
260 PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g
261 (AHead a1 a2) c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is:
262 PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g
263 d1 c2)).(let H_y \def (H1 d1 is H4 c2 H5) in (let H_y0 \def (H3 d1 is H4 c2
264 H5) in (let H6 \def H_y0 in (and_ind (arity g c2 (lift1 is t0) (AHead a1 a2))
265 (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0:
266 PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0
267 (lift1 is t0))))))))) (sc3 g a2 c2 (lift1 is (THead (Flat Appl) u t0)))
268 (\lambda (_: (arity g c2 (lift1 is t0) (AHead a1 a2))).(\lambda (H8:
269 ((\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0:
270 PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0
271 (lift1 is t0))))))))))).(let H_y1 \def (H8 c2 (lift1 is u) H_y PNil) in
272 (eq_ind_r T (THead (Flat Appl) (lift1 is u) (lift1 is t0)) (\lambda (t1:
273 T).(sc3 g a2 c2 t1)) (H_y1 (drop1_nil c2)) (lift1 is (THead (Flat Appl) u
274 t0)) (lift1_flat Appl is u t0))))) H6)))))))))))))))))) (\lambda (c:
275 C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g
276 a0))).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c)
277 \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (asucc g a0) c2 (lift1 is
278 u))))))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3:
279 ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2:
280 C).((csubc g d1 c2) \to (sc3 g a0 c2 (lift1 is t0))))))))).(\lambda (d1:
281 C).(\lambda (is: PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2:
282 C).(\lambda (H5: (csubc g d1 c2)).(let H_y \def (sc3_cast g a0 TNil) in
283 (eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1 is t0)) (\lambda (t1:
284 T).(sc3 g a0 c2 t1)) (H_y c2 (lift1 is u) (H1 d1 is H4 c2 H5) (lift1 is t0)
285 (H3 d1 is H4 c2 H5)) (lift1 is (THead (Flat Cast) u t0)) (lift1_flat Cast is
286 u t0)))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1:
287 A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (d1: C).(\forall
288 (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g
289 a1 c2 (lift1 is t0))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1
290 a2)).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H3: (drop1 is d1
291 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(sc3_repl g a1 c2 (lift1
292 is t0) (H1 d1 is H3 c2 H4) a2 H2))))))))))))) c1 t a H))))).
295 \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
296 a) \to (sc3 g a c t)))))
298 \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
299 (arity g c t a)).(let H_y \def (sc3_arity_csubc g c t a H c PNil) in (H_y
300 (drop1_nil c) c (csubc_refl g c))))))).
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