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 "csubc/defs.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 H1 \def (match H0 in
29 csubc return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csubc ? c
30 c0)).((eq C c (CHead e (Bind b) u)) \to ((eq C c0 c2) \to (ex2 C (\lambda
31 (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u)
32 e2)))))))) with [(csubc_sort n) \Rightarrow (\lambda (H1: (eq C (CSort n)
33 (CHead e (Bind b) u))).(\lambda (H2: (eq C (CSort n) c2)).((let H3 \def
34 (eq_ind C (CSort n) (\lambda (e0: C).(match e0 in C return (\lambda (_:
35 C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
36 False])) I (CHead e (Bind b) u) H1) in (False_ind ((eq C (CSort n) c2) \to
37 (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead e
38 (Bind b) u) e2)))) H3)) H2))) | (csubc_head c0 c3 H1 k v) \Rightarrow
39 (\lambda (H2: (eq C (CHead c0 k v) (CHead e (Bind b) u))).(\lambda (H3: (eq C
40 (CHead c3 k v) c2)).((let H4 \def (f_equal C T (\lambda (e0: C).(match e0 in
41 C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t)
42 \Rightarrow t])) (CHead c0 k v) (CHead e (Bind b) u) H2) in ((let H5 \def
43 (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with
44 [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k v)
45 (CHead e (Bind b) u) H2) in ((let H6 \def (f_equal C C (\lambda (e0:
46 C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
47 (CHead c _ _) \Rightarrow c])) (CHead c0 k v) (CHead e (Bind b) u) H2) in
48 (eq_ind C e (\lambda (c: C).((eq K k (Bind b)) \to ((eq T v u) \to ((eq C
49 (CHead c3 k v) c2) \to ((csubc g c c3) \to (ex2 C (\lambda (e2: C).(clear c2
50 e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2)))))))) (\lambda (H7:
51 (eq K k (Bind b))).(eq_ind K (Bind b) (\lambda (k0: K).((eq T v u) \to ((eq C
52 (CHead c3 k0 v) c2) \to ((csubc g e c3) \to (ex2 C (\lambda (e2: C).(clear c2
53 e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))))))) (\lambda (H8:
54 (eq T v u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c3 (Bind b) t) c2) \to
55 ((csubc g e c3) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2:
56 C).(csubc g (CHead e (Bind b) u) e2)))))) (\lambda (H9: (eq C (CHead c3 (Bind
57 b) u) c2)).(eq_ind C (CHead c3 (Bind b) u) (\lambda (c: C).((csubc g e c3)
58 \to (ex2 C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead e
59 (Bind b) u) e2))))) (\lambda (H10: (csubc g e c3)).(ex_intro2 C (\lambda (e2:
60 C).(clear (CHead c3 (Bind b) u) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind
61 b) u) e2)) (CHead c3 (Bind b) u) (clear_bind b c3 u) (csubc_head g e c3 H10
62 (Bind b) u))) c2 H9)) v (sym_eq T v u H8))) k (sym_eq K k (Bind b) H7))) c0
63 (sym_eq C c0 e H6))) H5)) H4)) H3 H1))) | (csubc_abst c0 c3 H1 v a H2 w H3)
64 \Rightarrow (\lambda (H4: (eq C (CHead c0 (Bind Abst) v) (CHead e (Bind b)
65 u))).(\lambda (H5: (eq C (CHead c3 (Bind Abbr) w) c2)).((let H6 \def (f_equal
66 C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _)
67 \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind Abst) v)
68 (CHead e (Bind b) u) H4) in ((let H7 \def (f_equal C B (\lambda (e0:
69 C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst
70 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
71 [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (CHead c0 (Bind
72 Abst) v) (CHead e (Bind b) u) H4) in ((let H8 \def (f_equal C C (\lambda (e0:
73 C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
74 (CHead c _ _) \Rightarrow c])) (CHead c0 (Bind Abst) v) (CHead e (Bind b) u)
75 H4) in (eq_ind C e (\lambda (c: C).((eq B Abst b) \to ((eq T v u) \to ((eq C
76 (CHead c3 (Bind Abbr) w) c2) \to ((csubc g c c3) \to ((sc3 g (asucc g a) c v)
77 \to ((sc3 g a c3 w) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2:
78 C).(csubc g (CHead e (Bind b) u) e2)))))))))) (\lambda (H9: (eq B Abst
79 b)).(eq_ind B Abst (\lambda (b0: B).((eq T v u) \to ((eq C (CHead c3 (Bind
80 Abbr) w) c2) \to ((csubc g e c3) \to ((sc3 g (asucc g a) e v) \to ((sc3 g a
81 c3 w) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g
82 (CHead e (Bind b0) u) e2))))))))) (\lambda (H10: (eq T v u)).(eq_ind T u
83 (\lambda (t: T).((eq C (CHead c3 (Bind Abbr) w) c2) \to ((csubc g e c3) \to
84 ((sc3 g (asucc g a) e t) \to ((sc3 g a c3 w) \to (ex2 C (\lambda (e2:
85 C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead e (Bind Abst) u)
86 e2)))))))) (\lambda (H11: (eq C (CHead c3 (Bind Abbr) w) c2)).(eq_ind C
87 (CHead c3 (Bind Abbr) w) (\lambda (c: C).((csubc g e c3) \to ((sc3 g (asucc g
88 a) e u) \to ((sc3 g a c3 w) \to (ex2 C (\lambda (e2: C).(clear c e2))
89 (\lambda (e2: C).(csubc g (CHead e (Bind Abst) u) e2))))))) (\lambda (H12:
90 (csubc g e c3)).(\lambda (H13: (sc3 g (asucc g a) e u)).(\lambda (H14: (sc3 g
91 a c3 w)).(ex_intro2 C (\lambda (e2: C).(clear (CHead c3 (Bind Abbr) w) e2))
92 (\lambda (e2: C).(csubc g (CHead e (Bind Abst) u) e2)) (CHead c3 (Bind Abbr)
93 w) (clear_bind Abbr c3 w) (csubc_abst g e c3 H12 u a H13 w H14))))) c2 H11))
94 v (sym_eq T v u H10))) b H9)) c0 (sym_eq C c0 e H8))) H7)) H6)) H5 H1 H2
95 H3)))]) in (H1 (refl_equal C (CHead e (Bind b) u)) (refl_equal C c2))))))))
96 (\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1:
97 ((\forall (c2: C).((csubc g e c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2))
98 (\lambda (e2: C).(csubc g c e2))))))).(\lambda (f: F).(\lambda (u:
99 T).(\lambda (c2: C).(\lambda (H2: (csubc g (CHead e (Flat f) u) c2)).(let H3
100 \def (match H2 in csubc return (\lambda (c0: C).(\lambda (c3: C).(\lambda (_:
101 (csubc ? c0 c3)).((eq C c0 (CHead e (Flat f) u)) \to ((eq C c3 c2) \to (ex2 C
102 (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2)))))))) with
103 [(csubc_sort n) \Rightarrow (\lambda (H3: (eq C (CSort n) (CHead e (Flat f)
104 u))).(\lambda (H4: (eq C (CSort n) c2)).((let H5 \def (eq_ind C (CSort n)
105 (\lambda (e0: C).(match e0 in C return (\lambda (_: C).Prop) with [(CSort _)
106 \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e (Flat f) u)
107 H3) in (False_ind ((eq C (CSort n) c2) \to (ex2 C (\lambda (e2: C).(clear c2
108 e2)) (\lambda (e2: C).(csubc g c e2)))) H5)) H4))) | (csubc_head c0 c3 H3 k
109 v) \Rightarrow (\lambda (H4: (eq C (CHead c0 k v) (CHead e (Flat f)
110 u))).(\lambda (H5: (eq C (CHead c3 k v) c2)).((let H6 \def (f_equal C T
111 (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _)
112 \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c0 k v) (CHead e (Flat
113 f) u) H4) in ((let H7 \def (f_equal C K (\lambda (e0: C).(match e0 in C
114 return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _)
115 \Rightarrow k0])) (CHead c0 k v) (CHead e (Flat f) u) H4) in ((let H8 \def
116 (f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
117 [(CSort _) \Rightarrow c0 | (CHead c4 _ _) \Rightarrow c4])) (CHead c0 k v)
118 (CHead e (Flat f) u) H4) in (eq_ind C e (\lambda (c4: C).((eq K k (Flat f))
119 \to ((eq T v u) \to ((eq C (CHead c3 k v) c2) \to ((csubc g c4 c3) \to (ex2 C
120 (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))))))))
121 (\lambda (H9: (eq K k (Flat f))).(eq_ind K (Flat f) (\lambda (k0: K).((eq T v
122 u) \to ((eq C (CHead c3 k0 v) c2) \to ((csubc g e c3) \to (ex2 C (\lambda
123 (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))))))) (\lambda (H10:
124 (eq T v u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c3 (Flat f) t) c2) \to
125 ((csubc g e c3) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2:
126 C).(csubc g c e2)))))) (\lambda (H11: (eq C (CHead c3 (Flat f) u)
127 c2)).(eq_ind C (CHead c3 (Flat f) u) (\lambda (c4: C).((csubc g e c3) \to
128 (ex2 C (\lambda (e2: C).(clear c4 e2)) (\lambda (e2: C).(csubc g c e2)))))
129 (\lambda (H12: (csubc g e c3)).(let H_x \def (H1 c3 H12) in (let H13 \def H_x
130 in (ex2_ind C (\lambda (e2: C).(clear c3 e2)) (\lambda (e2: C).(csubc g c
131 e2)) (ex2 C (\lambda (e2: C).(clear (CHead c3 (Flat f) u) e2)) (\lambda (e2:
132 C).(csubc g c e2))) (\lambda (x: C).(\lambda (H14: (clear c3 x)).(\lambda
133 (H15: (csubc g c x)).(ex_intro2 C (\lambda (e2: C).(clear (CHead c3 (Flat f)
134 u) e2)) (\lambda (e2: C).(csubc g c e2)) x (clear_flat c3 x H14 f u) H15))))
135 H13)))) c2 H11)) v (sym_eq T v u H10))) k (sym_eq K k (Flat f) H9))) c0
136 (sym_eq C c0 e H8))) H7)) H6)) H5 H3))) | (csubc_abst c0 c3 H3 v a H4 w H5)
137 \Rightarrow (\lambda (H6: (eq C (CHead c0 (Bind Abst) v) (CHead e (Flat f)
138 u))).(\lambda (H7: (eq C (CHead c3 (Bind Abbr) w) c2)).((let H8 \def (eq_ind
139 C (CHead c0 (Bind Abst) v) (\lambda (e0: C).(match e0 in C return (\lambda
140 (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow
141 (match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True |
142 (Flat _) \Rightarrow False])])) I (CHead e (Flat f) u) H6) in (False_ind ((eq
143 C (CHead c3 (Bind Abbr) w) c2) \to ((csubc g c0 c3) \to ((sc3 g (asucc g a)
144 c0 v) \to ((sc3 g a c3 w) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda
145 (e2: C).(csubc g c e2))))))) H8)) H7 H3 H4 H5)))]) in (H3 (refl_equal C
146 (CHead e (Flat f) u)) (refl_equal C c2))))))))))) c1 e1 H)))).