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/csubc/fwd.ma".
19 theorem csubc_clear_conf:
20 \forall (g: G).(\forall (c1: C).(\forall (e1: C).((clear c1 e1) \to (\forall
21 (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda
22 (e2: C).(csubc g e1 e2))))))))
24 \lambda (g: G).(\lambda (c1: C).(\lambda (e1: C).(\lambda (H: (clear c1
25 e1)).(clear_ind (\lambda (c: C).(\lambda (c0: C).(\forall (c2: C).((csubc g c
26 c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c0
27 e2))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (c2:
28 C).(\lambda (H0: (csubc g (CHead e (Bind b) u) c2)).(let H_x \def
29 (csubc_gen_head_l g e c2 u (Bind b) H0) in (let H1 \def H_x in (or3_ind (ex2
30 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g
31 e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K
32 (Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq
33 C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda
34 (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3
35 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g
36 a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2:
37 T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_:
38 C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda (b0: B).(\lambda
39 (_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3:
40 C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2: C).(clear c2 e2))
41 (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (H2: (ex2 C
42 (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g e
43 c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda
44 (c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2:
45 C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x: C).(\lambda (H3: (eq C c2
46 (CHead x (Bind b) u))).(\lambda (H4: (csubc g e x)).(eq_ind_r C (CHead x
47 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda
48 (e2: C).(csubc g (CHead e (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2:
49 C).(clear (CHead x (Bind b) u) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind
50 b) u) e2)) (CHead x (Bind b) u) (clear_bind b x u) (csubc_head g e x H4 (Bind
51 b) u)) c2 H3)))) H2)) (\lambda (H2: (ex5_3 C T A (\lambda (_: C).(\lambda (_:
52 T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda
53 (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3:
54 C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda
55 (_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda
56 (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_:
57 C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda
58 (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w)))))
59 (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda
60 (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda
61 (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda
62 (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2)))
63 (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H3: (eq K (Bind
64 b) (Bind Abst))).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda
65 (H5: (csubc g e x0)).(\lambda (H6: (sc3 g (asucc g x2) e u)).(\lambda (H7:
66 (sc3 g x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c: C).(ex2
67 C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b)
68 u) e2)))) (let H8 \def (f_equal K B (\lambda (e0: K).(match e0 in K return
69 (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b]))
70 (Bind b) (Bind Abst) H3) in (eq_ind_r B Abst (\lambda (b0: B).(ex2 C (\lambda
71 (e2: C).(clear (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g
72 (CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda (e2: C).(clear (CHead x0
73 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind Abst) u) e2))
74 (CHead x0 (Bind Abbr) x1) (clear_bind Abbr x0 x1) (csubc_abst g e x0 H5 u x2
75 H6 x1 H7)) b H8)) c2 H4))))))))) H2)) (\lambda (H2: (ex4_3 B C T (\lambda
76 (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0)
77 v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind b) (Bind
78 Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0
79 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g e
80 c3)))))).(ex4_3_ind B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2:
81 T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_:
82 C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda (b0: B).(\lambda
83 (_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3:
84 C).(\lambda (_: T).(csubc g e c3)))) (ex2 C (\lambda (e2: C).(clear c2 e2))
85 (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x0:
86 B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H3: (eq C c2 (CHead x1 (Bind
87 x0) x2))).(\lambda (H4: (eq K (Bind b) (Bind Void))).(\lambda (H5: (not (eq B
88 x0 Void))).(\lambda (H6: (csubc g e x1)).(eq_ind_r C (CHead x1 (Bind x0) x2)
89 (\lambda (c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc
90 g (CHead e (Bind b) u) e2)))) (let H7 \def (f_equal K B (\lambda (e0:
91 K).(match e0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 |
92 (Flat _) \Rightarrow b])) (Bind b) (Bind Void) H4) in (eq_ind_r B Void
93 (\lambda (b0: B).(ex2 C (\lambda (e2: C).(clear (CHead x1 (Bind x0) x2) e2))
94 (\lambda (e2: C).(csubc g (CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda
95 (e2: C).(clear (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g (CHead
96 e (Bind Void) u) e2)) (CHead x1 (Bind x0) x2) (clear_bind x0 x1 x2)
97 (csubc_void g e x1 H6 x0 H5 u x2)) b H7)) c2 H3)))))))) H2)) H1))))))))
98 (\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1:
99 ((\forall (c2: C).((csubc g e c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2))
100 (\lambda (e2: C).(csubc g c e2))))))).(\lambda (f: F).(\lambda (u:
101 T).(\lambda (c2: C).(\lambda (H2: (csubc g (CHead e (Flat f) u) c2)).(let H_x
102 \def (csubc_gen_head_l g e c2 u (Flat f) H2) in (let H3 \def H_x in (or3_ind
103 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3:
104 C).(csubc g e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_:
105 A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda
106 (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
107 T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda
108 (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda
109 (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3:
110 C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_:
111 B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda
112 (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
113 B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2:
114 C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (H4: (ex2 C
115 (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3: C).(csubc g e
116 c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda
117 (c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2:
118 C).(csubc g c e2))) (\lambda (x: C).(\lambda (H5: (eq C c2 (CHead x (Flat f)
119 u))).(\lambda (H6: (csubc g e x)).(eq_ind_r C (CHead x (Flat f) u) (\lambda
120 (c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) (\lambda (e2: C).(csubc g c
121 e2)))) (let H_x0 \def (H1 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda
122 (e2: C).(clear x e2)) (\lambda (e2: C).(csubc g c e2)) (ex2 C (\lambda (e2:
123 C).(clear (CHead x (Flat f) u) e2)) (\lambda (e2: C).(csubc g c e2)))
124 (\lambda (x0: C).(\lambda (H8: (clear x x0)).(\lambda (H9: (csubc g c
125 x0)).(ex_intro2 C (\lambda (e2: C).(clear (CHead x (Flat f) u) e2)) (\lambda
126 (e2: C).(csubc g c e2)) x0 (clear_flat x x0 H8 f u) H9)))) H7))) c2 H5))))
127 H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_:
128 A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda
129 (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
130 T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda
131 (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda
132 (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_:
133 T).(\lambda (_: A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda
134 (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3:
135 C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda
136 (_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda
137 (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda (e2: C).(clear c2
138 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: C).(\lambda (x1:
139 T).(\lambda (x2: A).(\lambda (H5: (eq K (Flat f) (Bind Abst))).(\lambda (H6:
140 (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (_: (csubc g e x0)).(\lambda
141 (_: (sc3 g (asucc g x2) e u)).(\lambda (_: (sc3 g x2 x0 x1)).(eq_ind_r C
142 (CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0
143 e2)) (\lambda (e2: C).(csubc g c e2)))) (let H10 \def (eq_ind K (Flat f)
144 (\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with [(Bind _)
145 \Rightarrow False | (Flat _) \Rightarrow True])) I (Bind Abst) H5) in
146 (False_ind (ex2 C (\lambda (e2: C).(clear (CHead x0 (Bind Abbr) x1) e2))
147 (\lambda (e2: C).(csubc g c e2))) H10)) c2 H6))))))))) H4)) (\lambda (H4:
148 (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2
149 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
150 T).(eq K (Flat f) (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda
151 (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_:
152 T).(csubc g e c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3:
153 C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_:
154 B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda
155 (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
156 B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3)))) (ex2 C (\lambda (e2:
157 C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: B).(\lambda
158 (x1: C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0)
159 x2))).(\lambda (H6: (eq K (Flat f) (Bind Void))).(\lambda (_: (not (eq B x0
160 Void))).(\lambda (_: (csubc g e x1)).(eq_ind_r C (CHead x1 (Bind x0) x2)
161 (\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) (\lambda (e2:
162 C).(csubc g c e2)))) (let H9 \def (eq_ind K (Flat f) (\lambda (ee: K).(match
163 ee in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat
164 _) \Rightarrow True])) I (Bind Void) H6) in (False_ind (ex2 C (\lambda (e2:
165 C).(clear (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g c e2))) H9))
166 c2 H5)))))))) H4)) H3))))))))))) c1 e1 H)))).