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/ty3/props".
22 \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((ty3 g e
23 t1 t2) \to (\forall (c: C).(\forall (d: nat).(\forall (h: nat).((drop h d c
24 e) \to (ty3 g c (lift h d t1) (lift h d t2))))))))))
26 \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
27 (H: (ty3 g e t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda (t0:
28 T).(\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to
29 (ty3 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda (t0:
30 T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda (H1: ((\forall (c0:
31 C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h
32 d t0) (lift h d t)))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3
33 g c u t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d: nat).(\forall (h:
34 nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d
35 t3)))))))).(\lambda (H4: (pc3 c t3 t0)).(\lambda (c0: C).(\lambda (d:
36 nat).(\lambda (h: nat).(\lambda (H5: (drop h d c0 c)).(ty3_conv g c0 (lift h
37 d t0) (lift h d t) (H1 c0 d h H5) (lift h d u) (lift h d t3) (H3 c0 d h H5)
38 (pc3_lift c0 c h d H5 t3 t0 H4)))))))))))))))) (\lambda (c: C).(\lambda (m:
39 nat).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (_: (drop
40 h d c0 c)).(eq_ind_r T (TSort m) (\lambda (t: T).(ty3 g c0 t (lift h d (TSort
41 (next g m))))) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0
42 (TSort m) t)) (ty3_sort g c0 m) (lift h d (TSort (next g m))) (lift_sort
43 (next g m) h d)) (lift h d (TSort m)) (lift_sort m h d)))))))) (\lambda (n:
44 nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c
45 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u
46 t)).(\lambda (H2: ((\forall (c0: C).(\forall (d0: nat).(\forall (h:
47 nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u) (lift h d0
48 t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: nat).(\lambda (H3:
49 (drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0
50 (lift (S n) O t))) (\lambda (H4: (lt n d0)).(let H5 \def (drop_getl_trans_le
51 n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) u) H0)
52 in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop n O c0 e0)))
53 (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n) e0 e1))) (\lambda (_:
54 C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (ty3 g c0 (lift h d0
55 (TLRef n)) (lift h d0 (lift (S n) O t))) (\lambda (x0: C).(\lambda (x1:
56 C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop h (minus d0 n) x0
57 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) u))).(let H9 \def (eq_ind
58 nat (minus d0 n) (\lambda (n0: nat).(drop h n0 x0 x1)) H7 (S (minus d0 (S
59 n))) (minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0
60 (S n)) H9 Abbr d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1
61 (Bind Abbr) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h (minus d0
62 (S n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O t)))
63 (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus
64 d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x d)).(eq_ind_r T
65 (TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t))))
66 (eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3 g c0 (TLRef
67 n) (lift h n0 (lift (S n) O t)))) (eq_ind_r T (lift (S n) O (lift h (minus d0
68 (S n)) t)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat d0 (\lambda
69 (_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S n)) t))))
70 (ty3_abbr g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0 (CHead x
71 (Bind Abbr) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus d0 (S n))
72 t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n)))
73 (le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S
74 n) O t)) (lift_d t h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0
75 (le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0
76 H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n
77 h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t)))) (eq_ind nat
78 (S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O
79 t)))) (eq_ind_r T (lift (plus h (S n)) O t) (\lambda (t0: T).(ty3 g c0 (TLRef
80 (plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0
81 (TLRef (plus n h)) (lift n0 O t))) (ty3_abbr g (plus n h) c0 d u
82 (drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) t H1) (plus
83 h (S n)) (plus_comm h (S n))) (lift h d0 (lift (S n) O t)) (lift_free t (S n)
84 h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O)
85 n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n))
86 (plus n (S O)) (plus_comm n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h
87 d0 H4)))))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d:
88 C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda
89 (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c0: C).(\forall
90 (d0: nat).(\forall (h: nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u)
91 (lift h d0 t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h:
92 nat).(\lambda (H3: (drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0
93 (TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda (H4: (lt n d0)).(let H5
94 \def (drop_getl_trans_le n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3
95 (CHead d (Bind Abst) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_:
96 C).(drop n O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n)
97 e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst)
98 u)))) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda
99 (x0: C).(\lambda (x1: C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop
100 h (minus d0 n) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) u))).(let
101 H9 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(drop h n0 x0 x1)) H7 (S
102 (minus d0 (S n))) (minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0
103 h (minus d0 (S n)) H9 Abst d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0
104 (CHead c1 (Bind Abst) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h
105 (minus d0 (S n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S
106 n) O u))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift
107 h (minus d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x
108 d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S
109 n) O u)))) (eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3
110 g c0 (TLRef n) (lift h n0 (lift (S n) O u)))) (eq_ind_r T (lift (S n) O (lift
111 h (minus d0 (S n)) u)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat
112 d0 (\lambda (_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S
113 n)) u)))) (ty3_abst g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0
114 (CHead x (Bind Abst) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus
115 d0 (S n)) t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n)))
116 (le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S
117 n) O u)) (lift_d u h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0
118 (le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0
119 H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n
120 h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O u)))) (eq_ind nat
121 (S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O
122 u)))) (eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t0: T).(ty3 g c0 (TLRef
123 (plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0
124 (TLRef (plus n h)) (lift n0 O u))) (ty3_abst g (plus n h) c0 d u
125 (drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) t H1) (plus
126 h (S n)) (plus_comm h (S n))) (lift h d0 (lift (S n) O u)) (lift_free u (S n)
127 h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O)
128 n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n))
129 (plus n (S O)) (plus_comm n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h
130 d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t:
131 T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c0: C).(\forall (d:
132 nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d
133 t)))))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3
134 g (CHead c (Bind b) u) t0 t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d:
135 nat).(\forall (h: nat).((drop h d c0 (CHead c (Bind b) u)) \to (ty3 g c0
136 (lift h d t0) (lift h d t3)))))))).(\lambda (t4: T).(\lambda (_: (ty3 g
137 (CHead c (Bind b) u) t3 t4)).(\lambda (H5: ((\forall (c0: C).(\forall (d:
138 nat).(\forall (h: nat).((drop h d c0 (CHead c (Bind b) u)) \to (ty3 g c0
139 (lift h d t3) (lift h d t4)))))))).(\lambda (c0: C).(\lambda (d:
140 nat).(\lambda (h: nat).(\lambda (H6: (drop h d c0 c)).(eq_ind_r T (THead
141 (Bind b) (lift h d u) (lift h (s (Bind b) d) t0)) (\lambda (t5: T).(ty3 g c0
142 t5 (lift h d (THead (Bind b) u t3)))) (eq_ind_r T (THead (Bind b) (lift h d
143 u) (lift h (s (Bind b) d) t3)) (\lambda (t5: T).(ty3 g c0 (THead (Bind b)
144 (lift h d u) (lift h (s (Bind b) d) t0)) t5)) (ty3_bind g c0 (lift h d u)
145 (lift h d t) (H1 c0 d h H6) b (lift h (S d) t0) (lift h (S d) t3) (H3 (CHead
146 c0 (Bind b) (lift h d u)) (S d) h (drop_skip_bind h d c0 c H6 b u)) (lift h
147 (S d) t4) (H5 (CHead c0 (Bind b) (lift h d u)) (S d) h (drop_skip_bind h d c0
148 c H6 b u))) (lift h d (THead (Bind b) u t3)) (lift_head (Bind b) u t3 h d))
149 (lift h d (THead (Bind b) u t0)) (lift_head (Bind b) u t0 h
150 d))))))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda
151 (_: (ty3 g c w u)).(\lambda (H1: ((\forall (c0: C).(\forall (d: nat).(\forall
152 (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d w) (lift h d
153 u)))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead
154 (Bind Abst) u t))).(\lambda (H3: ((\forall (c0: C).(\forall (d: nat).(\forall
155 (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d v) (lift h d (THead (Bind
156 Abst) u t))))))))).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h:
157 nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead (Flat Appl) (lift h d
158 w) (lift h (s (Flat Appl) d) v)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d
159 (THead (Flat Appl) w (THead (Bind Abst) u t))))) (eq_ind_r T (THead (Flat
160 Appl) (lift h d w) (lift h (s (Flat Appl) d) (THead (Bind Abst) u t)))
161 (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) (lift h d w) (lift h (s (Flat
162 Appl) d) v)) t0)) (eq_ind_r T (THead (Bind Abst) (lift h (s (Flat Appl) d) u)
163 (lift h (s (Bind Abst) (s (Flat Appl) d)) t)) (\lambda (t0: T).(ty3 g c0
164 (THead (Flat Appl) (lift h d w) (lift h (s (Flat Appl) d) v)) (THead (Flat
165 Appl) (lift h d w) t0))) (ty3_appl g c0 (lift h d w) (lift h d u) (H1 c0 d h
166 H4) (lift h d v) (lift h (S d) t) (eq_ind T (lift h d (THead (Bind Abst) u
167 t)) (\lambda (t0: T).(ty3 g c0 (lift h d v) t0)) (H3 c0 d h H4) (THead (Bind
168 Abst) (lift h d u) (lift h (S d) t)) (lift_bind Abst u t h d))) (lift h (s
169 (Flat Appl) d) (THead (Bind Abst) u t)) (lift_head (Bind Abst) u t h (s (Flat
170 Appl) d))) (lift h d (THead (Flat Appl) w (THead (Bind Abst) u t)))
171 (lift_head (Flat Appl) w (THead (Bind Abst) u t) h d)) (lift h d (THead (Flat
172 Appl) w v)) (lift_head (Flat Appl) w v h d))))))))))))))) (\lambda (c:
173 C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t0 t3)).(\lambda
174 (H1: ((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c)
175 \to (ty3 g c0 (lift h d t0) (lift h d t3)))))))).(\lambda (t4: T).(\lambda
176 (_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c0: C).(\forall (d:
177 nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d t3) (lift h d
178 t4)))))))).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H4:
179 (drop h d c0 c)).(eq_ind_r T (THead (Flat Cast) (lift h d t3) (lift h (s
180 (Flat Cast) d) t0)) (\lambda (t: T).(ty3 g c0 t (lift h d t3))) (ty3_cast g
181 c0 (lift h (s (Flat Cast) d) t0) (lift h d t3) (H1 c0 d h H4) (lift h d t4)
182 (H3 c0 d h H4)) (lift h d (THead (Flat Cast) t3 t0)) (lift_head (Flat Cast)
183 t3 t0 h d)))))))))))))) e t1 t2 H))))).
186 \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
187 t1 t2) \to (ex T (\lambda (t: T).(ty3 g c t2 t)))))))
189 \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
190 (H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda
191 (t0: T).(ex T (\lambda (t3: T).(ty3 g c0 t0 t3)))))) (\lambda (c0:
192 C).(\lambda (t0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t0 t)).(\lambda
193 (_: (ex T (\lambda (t3: T).(ty3 g c0 t t3)))).(\lambda (u: T).(\lambda (t3:
194 T).(\lambda (_: (ty3 g c0 u t3)).(\lambda (_: (ex T (\lambda (t4: T).(ty3 g
195 c0 t3 t4)))).(\lambda (_: (pc3 c0 t3 t0)).(ex_intro T (\lambda (t4: T).(ty3 g
196 c0 t0 t4)) t H0))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro T
197 (\lambda (t: T).(ty3 g c0 (TSort (next g m)) t)) (TSort (next g (next g m)))
198 (ty3_sort g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
199 C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr)
200 u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex T (\lambda
201 (t0: T).(ty3 g d t t0)))).(let H3 \def H2 in (ex_ind T (\lambda (t0: T).(ty3
202 g d t t0)) (ex T (\lambda (t0: T).(ty3 g c0 (lift (S n) O t) t0))) (\lambda
203 (x: T).(\lambda (H4: (ty3 g d t x)).(ex_intro T (\lambda (t0: T).(ty3 g c0
204 (lift (S n) O t) t0)) (lift (S n) O x) (ty3_lift g d t x H4 c0 O (S n)
205 (getl_drop Abbr c0 d u n H0))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0:
206 C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind
207 Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (_: (ex T
208 (\lambda (t0: T).(ty3 g d t t0)))).(ex_intro T (\lambda (t0: T).(ty3 g c0
209 (lift (S n) O u) t0)) (lift (S n) O t) (ty3_lift g d u t H1 c0 O (S n)
210 (getl_drop Abst c0 d u n H0))))))))))) (\lambda (c0: C).(\lambda (u:
211 T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda (_: (ex T (\lambda
212 (t0: T).(ty3 g c0 t t0)))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3:
213 T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (_: (ex T
214 (\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)))).(\lambda (t4:
215 T).(\lambda (H4: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H5: (ex T
216 (\lambda (t5: T).(ty3 g (CHead c0 (Bind b) u) t4 t5)))).(let H6 \def H5 in
217 (ex_ind T (\lambda (t5: T).(ty3 g (CHead c0 (Bind b) u) t4 t5)) (ex T
218 (\lambda (t5: T).(ty3 g c0 (THead (Bind b) u t3) t5))) (\lambda (x:
219 T).(\lambda (H7: (ty3 g (CHead c0 (Bind b) u) t4 x)).(ex_intro T (\lambda
220 (t5: T).(ty3 g c0 (THead (Bind b) u t3) t5)) (THead (Bind b) u t4) (ty3_bind
221 g c0 u t H0 b t3 t4 H4 x H7)))) H6))))))))))))))) (\lambda (c0: C).(\lambda
222 (w: T).(\lambda (u: T).(\lambda (H0: (ty3 g c0 w u)).(\lambda (H1: (ex T
223 (\lambda (t: T).(ty3 g c0 u t)))).(\lambda (v: T).(\lambda (t: T).(\lambda
224 (_: (ty3 g c0 v (THead (Bind Abst) u t))).(\lambda (H3: (ex T (\lambda (t0:
225 T).(ty3 g c0 (THead (Bind Abst) u t) t0)))).(let H4 \def H1 in (ex_ind T
226 (\lambda (t0: T).(ty3 g c0 u t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead
227 (Flat Appl) w (THead (Bind Abst) u t)) t0))) (\lambda (x: T).(\lambda (_:
228 (ty3 g c0 u x)).(let H6 \def H3 in (ex_ind T (\lambda (t0: T).(ty3 g c0
229 (THead (Bind Abst) u t) t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat
230 Appl) w (THead (Bind Abst) u t)) t0))) (\lambda (x0: T).(\lambda (H7: (ty3 g
231 c0 (THead (Bind Abst) u t) x0)).(ex4_3_ind T T T (\lambda (t3: T).(\lambda
232 (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u t3) x0)))) (\lambda (_:
233 T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c0 u t0)))) (\lambda (t3:
234 T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind Abst) u) t t3))))
235 (\lambda (t3: T).(\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind
236 Abst) u) t3 t4)))) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w
237 (THead (Bind Abst) u t)) t0))) (\lambda (x1: T).(\lambda (x2: T).(\lambda
238 (x3: T).(\lambda (_: (pc3 c0 (THead (Bind Abst) u x1) x0)).(\lambda (H9: (ty3
239 g c0 u x2)).(\lambda (H10: (ty3 g (CHead c0 (Bind Abst) u) t x1)).(\lambda
240 (H11: (ty3 g (CHead c0 (Bind Abst) u) x1 x3)).(ex_intro T (\lambda (t0:
241 T).(ty3 g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) t0)) (THead (Flat
242 Appl) w (THead (Bind Abst) u x1)) (ty3_appl g c0 w u H0 (THead (Bind Abst) u
243 t) x1 (ty3_bind g c0 u x2 H9 Abst t x1 H10 x3 H11)))))))))) (ty3_gen_bind g
244 Abst c0 u t x0 H7)))) H6)))) H4))))))))))) (\lambda (c0: C).(\lambda (t0:
245 T).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t0 t3)).(\lambda (H1: (ex T
246 (\lambda (t: T).(ty3 g c0 t3 t)))).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3
247 t4)).(\lambda (_: (ex T (\lambda (t: T).(ty3 g c0 t4 t)))).H1)))))))) c t1 t2
251 \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u
252 t1) \to (\forall (t2: T).((ty3 g c u t2) \to (pc3 c t1 t2)))))))
254 \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H:
255 (ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
256 T).(\forall (t2: T).((ty3 g c0 t t2) \to (pc3 c0 t0 t2)))))) (\lambda (c0:
257 C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda
258 (_: ((\forall (t3: T).((ty3 g c0 t2 t3) \to (pc3 c0 t t3))))).(\lambda (u0:
259 T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H3: ((\forall
260 (t3: T).((ty3 g c0 u0 t3) \to (pc3 c0 t0 t3))))).(\lambda (H4: (pc3 c0 t0
261 t2)).(\lambda (t3: T).(\lambda (H5: (ty3 g c0 u0 t3)).(pc3_t t0 c0 t2 (pc3_s
262 c0 t2 t0 H4) t3 (H3 t3 H5)))))))))))))) (\lambda (c0: C).(\lambda (m:
263 nat).(\lambda (t2: T).(\lambda (H0: (ty3 g c0 (TSort m) t2)).(ty3_gen_sort g
264 c0 t2 m H0))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda
265 (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u0))).(\lambda (t:
266 T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall (t2: T).((ty3 g d u0
267 t2) \to (pc3 d t t2))))).(\lambda (t2: T).(\lambda (H3: (ty3 g c0 (TLRef n)
268 t2)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
269 T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda
270 (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1:
271 T).(\lambda (t0: T).(ty3 g e u1 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
272 (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e:
273 C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1)))))
274 (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))) (pc3 c0
275 (lift (S n) O t) t2) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (_:
276 T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda
277 (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e:
278 C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T
279 (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0)
280 t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e
281 (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g
282 e u1 t0)))) (pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1:
283 T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O x2) t2)).(\lambda
284 (H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g x0 x1
285 x2)).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n
286 c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n
287 H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e:
288 C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
289 (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind
290 Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr)
291 x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return
292 (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t0)
293 \Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1)
294 (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in
295 (\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0:
296 T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H8 u0 H10) in (let H13 \def
297 (eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def
298 (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0))) H12 d
299 H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3 g c1 u0 x2)) H13 d
300 H11) in (pc3_t (lift (S n) O x2) c0 (lift (S n) O t) (pc3_lift c0 d (S n) O
301 (getl_drop Abbr c0 d u0 n H14) t x2 (H2 x2 H15)) t2 H5))))))) H9)))))))))
302 H4)) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_:
303 T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda
304 (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1:
305 T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_:
306 C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda
307 (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst)
308 u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))
309 (pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2:
310 T).(\lambda (_: (pc3 c0 (lift (S n) O x1) t2)).(\lambda (H6: (getl n c0
311 (CHead x0 (Bind Abst) x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def
312 (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead
313 x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0
314 (Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abbr) u0)
315 (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
316 \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
317 (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
318 B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void
319 \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abst)
320 x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1)
321 H6)) in (False_ind (pc3 c0 (lift (S n) O t) t2) H9))))))))) H4))
322 (ty3_gen_lref g c0 t2 n H3)))))))))))) (\lambda (n: nat).(\lambda (c0:
323 C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind
324 Abst) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_:
325 ((\forall (t2: T).((ty3 g d u0 t2) \to (pc3 d t t2))))).(\lambda (t2:
326 T).(\lambda (H3: (ty3 g c0 (TLRef n) t2)).(or_ind (ex3_3 C T T (\lambda (_:
327 C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda
328 (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
329 u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))))
330 (ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift
331 (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n
332 c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda
333 (t0: T).(ty3 g e u1 t0))))) (pc3 c0 (lift (S n) O u0) t2) (\lambda (H4:
334 (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift
335 (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n
336 c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda
337 (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_:
338 T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda
339 (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e:
340 C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O
341 u0) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3
342 c0 (lift (S n) O x2) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abbr)
343 x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind
344 Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1)
345 (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in
346 (let H9 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee: C).(match ee in
347 C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k
348 _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
349 \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
350 False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
351 \Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d
352 (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind (pc3 c0
353 (lift (S n) O u0) t2) H9))))))))) H4)) (\lambda (H4: (ex3_3 C T T (\lambda
354 (_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2))))
355 (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
356 Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1
357 t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_:
358 T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda
359 (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1:
360 T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O u0) t2) (\lambda
361 (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O
362 x1) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7:
363 (ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda
364 (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d
365 (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (f_equal
366 C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
367 \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abst) u0)
368 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead
369 x0 (Bind Abst) x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match
370 e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _
371 t0) \Rightarrow t0])) (CHead d (Bind Abst) u0) (CHead x0 (Bind Abst) x1)
372 (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in
373 (\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0:
374 T).(getl n c0 (CHead x0 (Bind Abst) t0))) H8 u0 H10) in (let H13 \def
375 (eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def
376 (eq_ind_r T x1 (\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)) H5 u0 H10) in
377 (let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind
378 Abst) u0))) H12 d H11) in (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3
379 g c1 u0 x2)) H13 d H11) in H14))))))) H9))))))))) H4)) (ty3_gen_lref g c0 t2
380 n H3)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda
381 (_: (ty3 g c0 u0 t)).(\lambda (_: ((\forall (t2: T).((ty3 g c0 u0 t2) \to
382 (pc3 c0 t t2))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t2: T).(\lambda
383 (_: (ty3 g (CHead c0 (Bind b) u0) t0 t2)).(\lambda (H3: ((\forall (t3:
384 T).((ty3 g (CHead c0 (Bind b) u0) t0 t3) \to (pc3 (CHead c0 (Bind b) u0) t2
385 t3))))).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t2
386 t3)).(\lambda (_: ((\forall (t4: T).((ty3 g (CHead c0 (Bind b) u0) t2 t4) \to
387 (pc3 (CHead c0 (Bind b) u0) t3 t4))))).(\lambda (t4: T).(\lambda (H6: (ty3 g
388 c0 (THead (Bind b) u0 t0) t4)).(ex4_3_ind T T T (\lambda (t5: T).(\lambda (_:
389 T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u0 t5) t4)))) (\lambda (_:
390 T).(\lambda (t6: T).(\lambda (_: T).(ty3 g c0 u0 t6)))) (\lambda (t5:
391 T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u0) t0 t5))))
392 (\lambda (t5: T).(\lambda (_: T).(\lambda (t7: T).(ty3 g (CHead c0 (Bind b)
393 u0) t5 t7)))) (pc3 c0 (THead (Bind b) u0 t2) t4) (\lambda (x0: T).(\lambda
394 (x1: T).(\lambda (x2: T).(\lambda (H7: (pc3 c0 (THead (Bind b) u0 x0)
395 t4)).(\lambda (_: (ty3 g c0 u0 x1)).(\lambda (H9: (ty3 g (CHead c0 (Bind b)
396 u0) t0 x0)).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) x0 x2)).(pc3_t (THead
397 (Bind b) u0 x0) c0 (THead (Bind b) u0 t2) (pc3_head_2 c0 u0 t2 x0 (Bind b)
398 (H3 x0 H9)) t4 H7)))))))) (ty3_gen_bind g b c0 u0 t0 t4 H6)))))))))))))))))
399 (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda (_: (ty3 g c0 w
400 u0)).(\lambda (_: ((\forall (t2: T).((ty3 g c0 w t2) \to (pc3 c0 u0
401 t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind
402 Abst) u0 t))).(\lambda (H3: ((\forall (t2: T).((ty3 g c0 v t2) \to (pc3 c0
403 (THead (Bind Abst) u0 t) t2))))).(\lambda (t2: T).(\lambda (H4: (ty3 g c0
404 (THead (Flat Appl) w v) t2)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t0:
405 T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u1 t0)) t2))) (\lambda
406 (u1: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u1 t0)))) (\lambda
407 (u1: T).(\lambda (_: T).(ty3 g c0 w u1))) (pc3 c0 (THead (Flat Appl) w (THead
408 (Bind Abst) u0 t)) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (pc3
409 c0 (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) t2)).(\lambda (H6: (ty3 g
410 c0 v (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c0 w x0)).(pc3_t (THead
411 (Flat Appl) w (THead (Bind Abst) x0 x1)) c0 (THead (Flat Appl) w (THead (Bind
412 Abst) u0 t)) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) (THead (Bind Abst) x0
413 x1) (H3 (THead (Bind Abst) x0 x1) H6) w Appl) t2 H5)))))) (ty3_gen_appl g c0
414 w v t2 H4))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2:
415 T).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (_: ((\forall (t3: T).((ty3 g c0
416 t0 t3) \to (pc3 c0 t2 t3))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2
417 t3)).(\lambda (_: ((\forall (t4: T).((ty3 g c0 t2 t4) \to (pc3 c0 t3
418 t4))))).(\lambda (t4: T).(\lambda (H4: (ty3 g c0 (THead (Flat Cast) t2 t0)
419 t4)).(and_ind (pc3 c0 t2 t4) (ty3 g c0 t0 t2) (pc3 c0 t2 t4) (\lambda (H5:
420 (pc3 c0 t2 t4)).(\lambda (_: (ty3 g c0 t0 t2)).H5)) (ty3_gen_cast g c0 t0 t2
421 t4 H4)))))))))))) c u t1 H))))).