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/LambdaDelta-1/csuba/arity".
19 include "csuba/getl.ma".
21 include "csuba/props.ma".
23 include "arity/props.ma".
28 \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
29 t a) \to (\forall (c2: C).((csuba g c1 c2) \to (arity g c2 t a)))))))
31 \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
32 (arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
33 A).(\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0)))))) (\lambda (c:
34 C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (csuba g c
35 c2)).(arity_sort g c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u:
36 T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr)
37 u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall
38 (c2: C).((csuba g d c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda
39 (H3: (csuba g c c2)).(let H4 \def (csuba_getl_abbr g c d u i H0 c2 H3) in
40 (ex2_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda
41 (d2: C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda
42 (H5: (getl i c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (csuba g d
43 x)).(arity_abbr g c2 x u i H5 a0 (H2 x H6))))) H4)))))))))))) (\lambda (c:
44 C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
45 (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc
46 g a0))).(\lambda (H2: ((\forall (c2: C).((csuba g d c2) \to (arity g c2 u
47 (asucc g a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c c2)).(let H4 \def
48 (csuba_getl_abst g c d u i H0 c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(getl
49 i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2))) (ex4_3 C T
50 A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2
51 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
52 d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc
53 g a1))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2
54 a1))))) (arity g c2 (TLRef i) a0) (\lambda (H5: (ex2 C (\lambda (d2: C).(getl
55 i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2)))).(ex2_ind C
56 (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
57 C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H6:
58 (getl i c2 (CHead x (Bind Abst) u))).(\lambda (H7: (csuba g d x)).(arity_abst
59 g c2 x u i H6 a0 (H2 x H7))))) H5)) (\lambda (H5: (ex4_3 C T A (\lambda (d2:
60 C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2)))))
61 (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d d2)))) (\lambda
62 (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc g a1)))))
63 (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2
64 a1)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
65 A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
66 T).(\lambda (_: A).(csuba g d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda
67 (a1: A).(arity g d u (asucc g a1))))) (\lambda (d2: C).(\lambda (u2:
68 T).(\lambda (a1: A).(arity g d2 u2 a1)))) (arity g c2 (TLRef i) a0) (\lambda
69 (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H6: (getl i c2 (CHead x0
70 (Bind Abbr) x1))).(\lambda (_: (csuba g d x0)).(\lambda (H8: (arity g d u
71 (asucc g x2))).(\lambda (H9: (arity g x0 x1 x2)).(arity_repl g c2 (TLRef i)
72 x2 (arity_abbr g c2 x0 x1 i H6 x2 H9) a0 (asucc_inj g x2 a0 (arity_mono g d u
73 (asucc g x2) H8 (asucc g a0) H1)))))))))) H5)) H4)))))))))))) (\lambda (b:
74 B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u:
75 T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall
76 (c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda
77 (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H4:
78 ((\forall (c2: C).((csuba g (CHead c (Bind b) u) c2) \to (arity g c2 t0
79 a2))))).(\lambda (c2: C).(\lambda (H5: (csuba g c c2)).(arity_bind g b H0 c2
80 u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u) (csuba_head g c c2 H5 (Bind
81 b) u)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1:
82 A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c2:
83 C).((csuba g c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda (t0:
84 T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0
85 a2)).(\lambda (H3: ((\forall (c2: C).((csuba g (CHead c (Bind Abst) u) c2)
86 \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (csuba g c
87 c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u)
88 (csuba_head g c c2 H4 (Bind Abst) u)))))))))))))) (\lambda (c: C).(\lambda
89 (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1:
90 ((\forall (c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0:
91 T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3:
92 ((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 (AHead a1
93 a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c c2)).(arity_appl g c2 u a1
94 (H1 c2 H4) t0 a2 (H3 c2 H4))))))))))))) (\lambda (c: C).(\lambda (u:
95 T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1:
96 ((\forall (c2: C).((csuba g c c2) \to (arity g c2 u (asucc g
97 a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3:
98 ((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0))))).(\lambda (c2:
99 C).(\lambda (H4: (csuba g c c2)).(arity_cast g c2 u a0 (H1 c2 H4) t0 (H3 c2
100 H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_:
101 (arity g c t0 a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c c2) \to (arity
102 g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2:
103 C).(\lambda (H3: (csuba g c c2)).(arity_repl g c2 t0 a1 (H1 c2 H3) a2
104 H2)))))))))) c1 t a H))))).
106 theorem csuba_arity_rev:
107 \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
108 t a) \to (\forall (c2: C).((csuba g c2 c1) \to (arity g c2 t a)))))))
110 \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
111 (arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
112 A).(\forall (c2: C).((csuba g c2 c) \to (arity g c2 t0 a0)))))) (\lambda (c:
113 C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (csuba g c2
114 c)).(arity_sort g c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u:
115 T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr)
116 u))).(\lambda (a0: A).(\lambda (H1: (arity g d u a0)).(\lambda (H2: ((\forall
117 (c2: C).((csuba g c2 d) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda
118 (H3: (csuba g c2 c)).(let H4 \def (csuba_getl_abbr_rev g c d u i H0 c2 H3) in
119 (or_ind (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u)))
120 (\lambda (d2: C).(csuba g d2 d))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
121 T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
122 C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d)))) (\lambda (d2:
123 C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) (\lambda
124 (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u a1))))) (arity g c2
125 (TLRef i) a0) (\lambda (H5: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2
126 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2:
127 C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d))
128 (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H6: (getl i c2 (CHead x
129 (Bind Abbr) u))).(\lambda (H7: (csuba g x d)).(arity_abbr g c2 x u i H6 a0
130 (H2 x H7))))) H5)) (\lambda (H5: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
131 T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
132 C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d)))) (\lambda (d2:
133 C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) (\lambda
134 (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u a1)))))).(ex4_3_ind C T
135 A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2
136 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
137 d2 d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2
138 (asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d
139 u a1)))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda
140 (x2: A).(\lambda (H6: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (_:
141 (csuba g x0 d)).(\lambda (H8: (arity g x0 x1 (asucc g x2))).(\lambda (H9:
142 (arity g d u x2)).(arity_repl g c2 (TLRef i) x2 (arity_abst g c2 x0 x1 i H6
143 x2 H8) a0 (arity_mono g d u x2 H9 a0 H1))))))))) H5)) H4)))))))))))) (\lambda
144 (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl
145 i c (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u
146 (asucc g a0))).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to (arity g
147 c2 u (asucc g a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2 c)).(let H4
148 \def (csuba_getl_abst_rev g c d u i H0 c2 H3) in (ex2_ind C (\lambda (d2:
149 C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d))
150 (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x
151 (Bind Abst) u))).(\lambda (H6: (csuba g x d)).(arity_abst g c2 x u i H5 a0
152 (H2 x H6))))) H4)))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b
153 Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity
154 g c u a1)).(\lambda (H2: ((\forall (c2: C).((csuba g c2 c) \to (arity g c2 u
155 a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c
156 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c2: C).((csuba g c2 (CHead c
157 (Bind b) u)) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H5: (csuba
158 g c2 c)).(arity_bind g b H0 c2 u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b)
159 u) (csuba_head g c2 c H5 (Bind b) u)))))))))))))))) (\lambda (c: C).(\lambda
160 (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda
161 (H1: ((\forall (c2: C).((csuba g c2 c) \to (arity g c2 u (asucc g
162 a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c
163 (Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c2: C).((csuba g c2 (CHead c
164 (Bind Abst) u)) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4:
165 (csuba g c2 c)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind
166 Abst) u) (csuba_head g c2 c H4 (Bind Abst) u)))))))))))))) (\lambda (c:
167 C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda
168 (H1: ((\forall (c2: C).((csuba g c2 c) \to (arity g c2 u a1))))).(\lambda
169 (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda
170 (H3: ((\forall (c2: C).((csuba g c2 c) \to (arity g c2 t0 (AHead a1
171 a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2 c)).(arity_appl g c2 u a1
172 (H1 c2 H4) t0 a2 (H3 c2 H4))))))))))))) (\lambda (c: C).(\lambda (u:
173 T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1:
174 ((\forall (c2: C).((csuba g c2 c) \to (arity g c2 u (asucc g
175 a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3:
176 ((\forall (c2: C).((csuba g c2 c) \to (arity g c2 t0 a0))))).(\lambda (c2:
177 C).(\lambda (H4: (csuba g c2 c)).(arity_cast g c2 u a0 (H1 c2 H4) t0 (H3 c2
178 H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_:
179 (arity g c t0 a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c2 c) \to (arity
180 g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2:
181 C).(\lambda (H3: (csuba g c2 c)).(arity_repl g c2 t0 a1 (H1 c2 H3) a2
182 H2)))))))))) c1 t a H))))).
184 theorem arity_appls_appl:
185 \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (a1: A).((arity g c
186 v a1) \to (\forall (u: T).((arity g c u (asucc g a1)) \to (\forall (t:
187 T).(\forall (vs: TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) vs
188 (THead (Bind Abbr) v t)) a2) \to (arity g c (THeads (Flat Appl) vs (THead
189 (Flat Appl) v (THead (Bind Abst) u t))) a2)))))))))))
191 \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H:
192 (arity g c v a1)).(\lambda (u: T).(\lambda (H0: (arity g c u (asucc g
193 a1))).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0:
194 TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) t0 (THead (Bind Abbr)
195 v t)) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead
196 (Bind Abst) u t))) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g c (THead
197 (Bind Abbr) v t) a2)).(let H_x \def (arity_gen_bind Abbr (\lambda (H2: (eq B
198 Abbr Abst)).(not_abbr_abst H2)) g c v t a2 H1) in (let H2 \def H_x in
199 (ex2_ind A (\lambda (a3: A).(arity g c v a3)) (\lambda (_: A).(arity g (CHead
200 c (Bind Abbr) v) t a2)) (arity g c (THead (Flat Appl) v (THead (Bind Abst) u
201 t)) a2) (\lambda (x: A).(\lambda (_: (arity g c v x)).(\lambda (H4: (arity g
202 (CHead c (Bind Abbr) v) t a2)).(arity_appl g c v a1 H (THead (Bind Abst) u t)
203 a2 (arity_head g c u a1 H0 t a2 (csuba_arity_rev g (CHead c (Bind Abbr) v) t
204 a2 H4 (CHead c (Bind Abst) u) (csuba_abst g c c (csuba_refl g c) u a1 H0 v
205 H))))))) H2))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H1:
206 ((\forall (a2: A).((arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))
207 a2) \to (arity g c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind
208 Abst) u t))) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g c (THead (Flat
209 Appl) t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))) a2)).(let H3 \def
210 (arity_gen_appl g c t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) a2 H2)
211 in (ex2_ind A (\lambda (a3: A).(arity g c t0 a3)) (\lambda (a3: A).(arity g c
212 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) (AHead a3 a2))) (arity g c
213 (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead
214 (Bind Abst) u t)))) a2) (\lambda (x: A).(\lambda (H4: (arity g c t0
215 x)).(\lambda (H5: (arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))
216 (AHead x a2))).(arity_appl g c t0 x H4 (THeads (Flat Appl) t1 (THead (Flat
217 Appl) v (THead (Bind Abst) u t))) a2 (H1 (AHead x a2) H5))))) H3)))))))