1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "LambdaDelta-1/ty3/pr3_props.ma".
19 include "LambdaDelta-1/arity/pr3.ma".
21 include "LambdaDelta-1/asucc/fwd.ma".
24 \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
25 t1 t2) \to (ex2 A (\lambda (a1: A).(arity g c t1 a1)) (\lambda (a1: A).(arity
26 g c t2 (asucc g a1))))))))
28 \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
29 (H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda
30 (t0: T).(ex2 A (\lambda (a1: A).(arity g c0 t a1)) (\lambda (a1: A).(arity g
31 c0 t0 (asucc g a1))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t:
32 T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity
33 g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (u:
34 T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: (ex2 A
35 (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g
36 a1))))).(\lambda (H4: (pc3 c0 t4 t3)).(let H5 \def H1 in (ex2_ind A (\lambda
37 (a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1)))
38 (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3
39 (asucc g a1)))) (\lambda (x: A).(\lambda (H6: (arity g c0 t3 x)).(\lambda (_:
40 (arity g c0 t (asucc g x))).(let H8 \def H3 in (ex2_ind A (\lambda (a1:
41 A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A
42 (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g
43 a1)))) (\lambda (x0: A).(\lambda (H9: (arity g c0 u x0)).(\lambda (H10:
44 (arity g c0 t4 (asucc g x0))).(let H11 \def H4 in (ex2_ind T (\lambda (t0:
45 T).(pr3 c0 t4 t0)) (\lambda (t0: T).(pr3 c0 t3 t0)) (ex2 A (\lambda (a1:
46 A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g a1))))
47 (\lambda (x1: T).(\lambda (H12: (pr3 c0 t4 x1)).(\lambda (H13: (pr3 c0 t3
48 x1)).(ex_intro2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity
49 g c0 t3 (asucc g a1))) x0 H9 (arity_repl g c0 t3 x H6 (asucc g x0) (leq_sym g
50 (asucc g x0) x (arity_mono g c0 x1 (asucc g x0) (arity_sred_pr3 c0 t4 x1 H12
51 g (asucc g x0) H10) x (arity_sred_pr3 c0 t3 x1 H13 g x H6)))))))) H11)))))
52 H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro2 A
53 (\lambda (a1: A).(arity g c0 (TSort m) a1)) (\lambda (a1: A).(arity g c0
54 (TSort (next g m)) (asucc g a1))) (ASort O m) (arity_sort g c0 m) (arity_sort
55 g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
56 C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr)
57 u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A
58 (\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g
59 a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1))
60 (\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g
61 c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g
62 a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (H5: (arity g
63 d t (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1))
64 (\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g a1))) x (arity_abbr g
65 c0 d u n H0 x H4) (arity_lift g d t (asucc g x) H5 c0 (S n) O (getl_drop Abbr
66 c0 d u n H0)))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda
67 (d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst)
68 u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A
69 (\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g
70 a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1))
71 (\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g
72 c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g
73 a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (_: (arity g d
74 t (asucc g x))).(let H_x \def (leq_asucc g x) in (let H6 \def H_x in (ex_ind
75 A (\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g
76 c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g
77 a1)))) (\lambda (x0: A).(\lambda (H7: (leq g x (asucc g x0))).(ex_intro2 A
78 (\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0
79 (lift (S n) O u) (asucc g a1))) x0 (arity_abst g c0 d u n H0 x0 (arity_repl g
80 d u x H4 (asucc g x0) H7)) (arity_lift g d u (asucc g x0) (arity_repl g d u x
81 H4 (asucc g x0) H7) c0 (S n) O (getl_drop Abst c0 d u n H0))))) H6))))))
82 H3)))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (_:
83 (ty3 g c0 u t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity g c0 u a1))
84 (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (b: B).(\lambda (t3:
85 T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t3
86 t4)).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t3
87 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 (asucc g a1))))).(let
88 H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1:
89 A).(arity g c0 t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead
90 (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc
91 g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 u x)).(\lambda (_: (arity
92 g c0 t (asucc g x))).(let H7 \def H3 in (ex2_ind A (\lambda (a1: A).(arity g
93 (CHead c0 (Bind b) u) t3 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u)
94 t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b) u t3)
95 a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc g a1))))
96 (\lambda (x0: A).(\lambda (H8: (arity g (CHead c0 (Bind b) u) t3
97 x0)).(\lambda (H9: (arity g (CHead c0 (Bind b) u) t4 (asucc g x0))).(let H_x
98 \def (leq_asucc g x) in (let H10 \def H_x in (ex_ind A (\lambda (a0: A).(leq
99 g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b) u t3)
100 a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc g a1))))
101 (\lambda (x1: A).(\lambda (H11: (leq g x (asucc g x1))).(B_ind (\lambda (b0:
102 B).((arity g (CHead c0 (Bind b0) u) t3 x0) \to ((arity g (CHead c0 (Bind b0)
103 u) t4 (asucc g x0)) \to (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b0)
104 u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b0) u t4) (asucc g
105 a1))))))) (\lambda (H12: (arity g (CHead c0 (Bind Abbr) u) t3 x0)).(\lambda
106 (H13: (arity g (CHead c0 (Bind Abbr) u) t4 (asucc g x0))).(ex_intro2 A
107 (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u t3) a1)) (\lambda (a1:
108 A).(arity g c0 (THead (Bind Abbr) u t4) (asucc g a1))) x0 (arity_bind g Abbr
109 not_abbr_abst c0 u x H5 t3 x0 H12) (arity_bind g Abbr not_abbr_abst c0 u x H5
110 t4 (asucc g x0) H13)))) (\lambda (H12: (arity g (CHead c0 (Bind Abst) u) t3
111 x0)).(\lambda (H13: (arity g (CHead c0 (Bind Abst) u) t4 (asucc g
112 x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t3) a1))
113 (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t4) (asucc g a1))) (AHead
114 x1 x0) (arity_head g c0 u x1 (arity_repl g c0 u x H5 (asucc g x1) H11) t3 x0
115 H12) (arity_repl g c0 (THead (Bind Abst) u t4) (AHead x1 (asucc g x0))
116 (arity_head g c0 u x1 (arity_repl g c0 u x H5 (asucc g x1) H11) t4 (asucc g
117 x0) H13) (asucc g (AHead x1 x0)) (leq_refl g (asucc g (AHead x1 x0)))))))
118 (\lambda (H12: (arity g (CHead c0 (Bind Void) u) t3 x0)).(\lambda (H13:
119 (arity g (CHead c0 (Bind Void) u) t4 (asucc g x0))).(ex_intro2 A (\lambda
120 (a1: A).(arity g c0 (THead (Bind Void) u t3) a1)) (\lambda (a1: A).(arity g
121 c0 (THead (Bind Void) u t4) (asucc g a1))) x0 (arity_bind g Void
122 not_void_abst c0 u x H5 t3 x0 H12) (arity_bind g Void not_void_abst c0 u x H5
123 t4 (asucc g x0) H13)))) b H8 H9))) H10)))))) H7))))) H4)))))))))))) (\lambda
124 (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda
125 (H1: (ex2 A (\lambda (a1: A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0
126 u (asucc g a1))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v
127 (THead (Bind Abst) u t))).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g c0 v
128 a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g
129 a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 w a1))
130 (\lambda (a1: A).(arity g c0 u (asucc g a1))) (ex2 A (\lambda (a1: A).(arity
131 g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat
132 Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x: A).(\lambda
133 (H5: (arity g c0 w x)).(\lambda (H6: (arity g c0 u (asucc g x))).(let H7 \def
134 H3 in (ex2_ind A (\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity
135 g c0 (THead (Bind Abst) u t) (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g
136 c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat
137 Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x0: A).(\lambda
138 (H8: (arity g c0 v x0)).(\lambda (H9: (arity g c0 (THead (Bind Abst) u t)
139 (asucc g x0))).(let H10 \def (arity_gen_abst g c0 u t (asucc g x0) H9) in
140 (ex3_2_ind A A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g x0) (AHead a1
141 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1))))
142 (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))
143 (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda
144 (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g
145 a1)))) (\lambda (x1: A).(\lambda (x2: A).(\lambda (H11: (eq A (asucc g x0)
146 (AHead x1 x2))).(\lambda (H12: (arity g c0 u (asucc g x1))).(\lambda (H13:
147 (arity g (CHead c0 (Bind Abst) u) t x2)).(let H14 \def (sym_eq A (asucc g x0)
148 (AHead x1 x2) H11) in (let H15 \def (asucc_gen_head g x1 x2 x0 H14) in
149 (ex2_ind A (\lambda (a0: A).(eq A x0 (AHead x1 a0))) (\lambda (a0: A).(eq A
150 x2 (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v)
151 a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u
152 t)) (asucc g a1)))) (\lambda (x3: A).(\lambda (H16: (eq A x0 (AHead x1
153 x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def (eq_ind A x2
154 (\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13 (asucc g x3) H17)
155 in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v a)) H8 (AHead x1
156 x3) H16) in (ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v)
157 a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u
158 t)) (asucc g a1))) x3 (arity_appl g c0 w x1 (arity_repl g c0 w x H5 x1
159 (leq_sym g x1 x (asucc_inj g x1 x (arity_mono g c0 u (asucc g x1) H12 (asucc
160 g x) H6)))) v x3 H19) (arity_appl g c0 w x H5 (THead (Bind Abst) u t) (asucc
161 g x3) (arity_head g c0 u x H6 t (asucc g x3) H18)))))))) H15)))))))) H10)))))
162 H7))))) H4))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4:
163 T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: (ex2 A (\lambda (a1:
164 A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g
165 a1))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda (H3: (ex2 A
166 (\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g
167 a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 t3 a1))
168 (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity
169 g c0 (THead (Flat Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat
170 Cast) t0 t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 t3
171 x)).(\lambda (H6: (arity g c0 t4 (asucc g x))).(let H7 \def H3 in (ex2_ind A
172 (\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g
173 a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t4 t3) a1))
174 (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4) (asucc g a1))))
175 (\lambda (x0: A).(\lambda (H8: (arity g c0 t4 x0)).(\lambda (H9: (arity g c0
176 t0 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat
177 Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4)
178 (asucc g a1))) x (arity_cast g c0 t4 x H6 t3 H5) (arity_cast g c0 t0 (asucc g
179 x) (arity_repl g c0 t0 (asucc g x0) H9 (asucc g (asucc g x)) (asucc_repl g x0
180 (asucc g x) (arity_mono g c0 t4 x0 H8 (asucc g x) H6))) t4 H6))))) H7)))))
181 H4)))))))))) c t1 t2 H))))).