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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/clear/defs.ma".
19 theorem clear_gen_sort:
20 \forall (x: C).(\forall (n: nat).((clear (CSort n) x) \to (\forall (P:
23 \lambda (x: C).(\lambda (n: nat).(\lambda (H: (clear (CSort n) x)).(\lambda
24 (P: Prop).(let H0 \def (match H in clear return (\lambda (c: C).(\lambda (c0:
25 C).(\lambda (_: (clear c c0)).((eq C c (CSort n)) \to ((eq C c0 x) \to P)))))
26 with [(clear_bind b e u) \Rightarrow (\lambda (H0: (eq C (CHead e (Bind b) u)
27 (CSort n))).(\lambda (H1: (eq C (CHead e (Bind b) u) x)).((let H2 \def
28 (eq_ind C (CHead e (Bind b) u) (\lambda (e0: C).(match e0 in C return
29 (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _)
30 \Rightarrow True])) I (CSort n) H0) in (False_ind ((eq C (CHead e (Bind b) u)
31 x) \to P) H2)) H1))) | (clear_flat e c H0 f u) \Rightarrow (\lambda (H1: (eq
32 C (CHead e (Flat f) u) (CSort n))).(\lambda (H2: (eq C c x)).((let H3 \def
33 (eq_ind C (CHead e (Flat f) u) (\lambda (e0: C).(match e0 in C return
34 (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _)
35 \Rightarrow True])) I (CSort n) H1) in (False_ind ((eq C c x) \to ((clear e
36 c) \to P)) H3)) H2 H0)))]) in (H0 (refl_equal C (CSort n)) (refl_equal C
39 theorem clear_gen_bind:
40 \forall (b: B).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear
41 (CHead e (Bind b) u) x) \to (eq C x (CHead e (Bind b) u))))))
43 \lambda (b: B).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H:
44 (clear (CHead e (Bind b) u) x)).(let H0 \def (match H in clear return
45 (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (clear c c0)).((eq C c (CHead e
46 (Bind b) u)) \to ((eq C c0 x) \to (eq C x (CHead e (Bind b) u))))))) with
47 [(clear_bind b0 e0 u0) \Rightarrow (\lambda (H0: (eq C (CHead e0 (Bind b0)
48 u0) (CHead e (Bind b) u))).(\lambda (H1: (eq C (CHead e0 (Bind b0) u0)
49 x)).((let H2 \def (f_equal C T (\lambda (e1: C).(match e1 in C return
50 (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow
51 t])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H0) in ((let H3 \def
52 (f_equal C B (\lambda (e1: C).(match e1 in C return (\lambda (_: C).B) with
53 [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return
54 (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
55 b0])])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H0) in ((let H4 \def
56 (f_equal C C (\lambda (e1: C).(match e1 in C return (\lambda (_: C).C) with
57 [(CSort _) \Rightarrow e0 | (CHead c _ _) \Rightarrow c])) (CHead e0 (Bind
58 b0) u0) (CHead e (Bind b) u) H0) in (eq_ind C e (\lambda (c: C).((eq B b0 b)
59 \to ((eq T u0 u) \to ((eq C (CHead c (Bind b0) u0) x) \to (eq C x (CHead e
60 (Bind b) u)))))) (\lambda (H5: (eq B b0 b)).(eq_ind B b (\lambda (b1: B).((eq
61 T u0 u) \to ((eq C (CHead e (Bind b1) u0) x) \to (eq C x (CHead e (Bind b)
62 u))))) (\lambda (H6: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead e
63 (Bind b) t) x) \to (eq C x (CHead e (Bind b) u)))) (\lambda (H7: (eq C (CHead
64 e (Bind b) u) x)).(eq_ind C (CHead e (Bind b) u) (\lambda (c: C).(eq C c
65 (CHead e (Bind b) u))) (refl_equal C (CHead e (Bind b) u)) x H7)) u0 (sym_eq
66 T u0 u H6))) b0 (sym_eq B b0 b H5))) e0 (sym_eq C e0 e H4))) H3)) H2)) H1)))
67 | (clear_flat e0 c H0 f u0) \Rightarrow (\lambda (H1: (eq C (CHead e0 (Flat
68 f) u0) (CHead e (Bind b) u))).(\lambda (H2: (eq C c x)).((let H3 \def (eq_ind
69 C (CHead e0 (Flat f) u0) (\lambda (e1: C).(match e1 in C return (\lambda (_:
70 C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
71 k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat
72 _) \Rightarrow True])])) I (CHead e (Bind b) u) H1) in (False_ind ((eq C c x)
73 \to ((clear e0 c) \to (eq C x (CHead e (Bind b) u)))) H3)) H2 H0)))]) in (H0
74 (refl_equal C (CHead e (Bind b) u)) (refl_equal C x))))))).
76 theorem clear_gen_flat:
77 \forall (f: F).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear
78 (CHead e (Flat f) u) x) \to (clear e x)))))
80 \lambda (f: F).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H:
81 (clear (CHead e (Flat f) u) x)).(let H0 \def (match H in clear return
82 (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (clear c c0)).((eq C c (CHead e
83 (Flat f) u)) \to ((eq C c0 x) \to (clear e x)))))) with [(clear_bind b e0 u0)
84 \Rightarrow (\lambda (H0: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f)
85 u))).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) x)).((let H2 \def (eq_ind C
86 (CHead e0 (Bind b) u0) (\lambda (e1: C).(match e1 in C return (\lambda (_:
87 C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
88 k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat
89 _) \Rightarrow False])])) I (CHead e (Flat f) u) H0) in (False_ind ((eq C
90 (CHead e0 (Bind b) u0) x) \to (clear e x)) H2)) H1))) | (clear_flat e0 c H0
91 f0 u0) \Rightarrow (\lambda (H1: (eq C (CHead e0 (Flat f0) u0) (CHead e (Flat
92 f) u))).(\lambda (H2: (eq C c x)).((let H3 \def (f_equal C T (\lambda (e1:
93 C).(match e1 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
94 (CHead _ _ t) \Rightarrow t])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u)
95 H1) in ((let H4 \def (f_equal C F (\lambda (e1: C).(match e1 in C return
96 (\lambda (_: C).F) with [(CSort _) \Rightarrow f0 | (CHead _ k _) \Rightarrow
97 (match k in K return (\lambda (_: K).F) with [(Bind _) \Rightarrow f0 | (Flat
98 f1) \Rightarrow f1])])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u) H1) in
99 ((let H5 \def (f_equal C C (\lambda (e1: C).(match e1 in C return (\lambda
100 (_: C).C) with [(CSort _) \Rightarrow e0 | (CHead c0 _ _) \Rightarrow c0]))
101 (CHead e0 (Flat f0) u0) (CHead e (Flat f) u) H1) in (eq_ind C e (\lambda (c0:
102 C).((eq F f0 f) \to ((eq T u0 u) \to ((eq C c x) \to ((clear c0 c) \to (clear
103 e x)))))) (\lambda (H6: (eq F f0 f)).(eq_ind F f (\lambda (_: F).((eq T u0 u)
104 \to ((eq C c x) \to ((clear e c) \to (clear e x))))) (\lambda (H7: (eq T u0
105 u)).(eq_ind T u (\lambda (_: T).((eq C c x) \to ((clear e c) \to (clear e
106 x)))) (\lambda (H8: (eq C c x)).(eq_ind C x (\lambda (c0: C).((clear e c0)
107 \to (clear e x))) (\lambda (H9: (clear e x)).H9) c (sym_eq C c x H8))) u0
108 (sym_eq T u0 u H7))) f0 (sym_eq F f0 f H6))) e0 (sym_eq C e0 e H5))) H4))
109 H3)) H2 H0)))]) in (H0 (refl_equal C (CHead e (Flat f) u)) (refl_equal C
112 theorem clear_gen_flat_r:
113 \forall (f: F).(\forall (x: C).(\forall (e: C).(\forall (u: T).((clear x
114 (CHead e (Flat f) u)) \to (\forall (P: Prop).P)))))
116 \lambda (f: F).(\lambda (x: C).(\lambda (e: C).(\lambda (u: T).(\lambda (H:
117 (clear x (CHead e (Flat f) u))).(\lambda (P: Prop).(insert_eq C (CHead e
118 (Flat f) u) (\lambda (c: C).(clear x c)) (\lambda (_: C).P) (\lambda (y:
119 C).(\lambda (H0: (clear x y)).(clear_ind (\lambda (_: C).(\lambda (c0:
120 C).((eq C c0 (CHead e (Flat f) u)) \to P))) (\lambda (b: B).(\lambda (e0:
121 C).(\lambda (u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat
122 f) u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match
123 ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
124 (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
125 [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat
126 f) u) H1) in (False_ind P H2)))))) (\lambda (e0: C).(\lambda (c: C).(\lambda
127 (H1: (clear e0 c)).(\lambda (H2: (((eq C c (CHead e (Flat f) u)) \to
128 P))).(\lambda (_: F).(\lambda (_: T).(\lambda (H3: (eq C c (CHead e (Flat f)
129 u))).(let H4 \def (eq_ind C c (\lambda (c0: C).((eq C c0 (CHead e (Flat f)
130 u)) \to P)) H2 (CHead e (Flat f) u) H3) in (let H5 \def (eq_ind C c (\lambda
131 (c0: C).(clear e0 c0)) H1 (CHead e (Flat f) u) H3) in (H4 (refl_equal C
132 (CHead e (Flat f) u)))))))))))) x y H0))) H)))))).
134 theorem clear_gen_all:
135 \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (ex_3 B C T (\lambda (b:
136 B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u))))))))
138 \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (clear c1 c2)).(clear_ind
139 (\lambda (_: C).(\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda (e:
140 C).(\lambda (u: T).(eq C c0 (CHead e (Bind b) u)))))))) (\lambda (b:
141 B).(\lambda (e: C).(\lambda (u: T).(ex_3_intro B C T (\lambda (b0:
142 B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead e (Bind b) u) (CHead e0
143 (Bind b0) u0))))) b e u (refl_equal C (CHead e (Bind b) u)))))) (\lambda (e:
144 C).(\lambda (c: C).(\lambda (H0: (clear e c)).(\lambda (H1: (ex_3 B C T
145 (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(eq C c (CHead e0 (Bind b)
146 u))))))).(\lambda (_: F).(\lambda (_: T).(let H2 \def H1 in (ex_3_ind B C T
147 (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c (CHead e0 (Bind b)
148 u0))))) (ex_3 B C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c
149 (CHead e0 (Bind b) u0)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
150 T).(\lambda (H3: (eq C c (CHead x1 (Bind x0) x2))).(let H4 \def (eq_ind C c
151 (\lambda (c0: C).(clear e c0)) H0 (CHead x1 (Bind x0) x2) H3) in (eq_ind_r C
152 (CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda
153 (e0: C).(\lambda (u0: T).(eq C c0 (CHead e0 (Bind b) u0))))))) (ex_3_intro B
154 C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead x1 (Bind
155 x0) x2) (CHead e0 (Bind b) u0))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0)
156 x2))) c H3)))))) H2)))))))) c1 c2 H))).