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/drop.ma".
19 theorem drop1_csubc_trans:
20 \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2:
21 C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C
22 (\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))
24 \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
25 (c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2
26 e1) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2
27 c1))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2
28 e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e2 e1)).(let H1 \def (match H in
29 drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda
30 (_: (drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 e2) \to
31 (ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c2
32 c1)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil
33 PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c e2)).(eq_ind C c2
34 (\lambda (c0: C).((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1
35 e1)) (\lambda (c1: C).(csubc g c2 c1))))) (\lambda (H4: (eq C c2 e2)).(eq_ind
36 C e2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda
37 (c1: C).(csubc g c0 c1)))) (let H5 \def (eq_ind_r C e2 (\lambda (c0:
38 C).(csubc g c0 e1)) H0 c2 H4) in (eq_ind C c2 (\lambda (c0: C).(ex2 C
39 (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c0 c1))))
40 (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g
41 c2 c1)) e1 (drop1_nil e1) H5) e2 H4)) c2 (sym_eq C c2 e2 H4))) c (sym_eq C c
42 c2 H2) H3)))) | (drop1_cons c1 c0 h d H1 c3 hds0 H2) \Rightarrow (\lambda
43 (H3: (eq PList (PCons h d hds0) PNil)).(\lambda (H4: (eq C c1 c2)).(\lambda
44 (H5: (eq C c3 e2)).((let H6 \def (eq_ind PList (PCons h d hds0) (\lambda (e:
45 PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil
46 \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in
47 (False_ind ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop h d c1 c0) \to ((drop1
48 hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 PNil c4 e1)) (\lambda (c4:
49 C).(csubc g c2 c4))))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList
50 PNil) (refl_equal C c2) (refl_equal C e2)))))))) (\lambda (n: nat).(\lambda
51 (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: C).(\forall (e2:
52 C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda
53 (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))).(\lambda
54 (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n n0 p) c2 e2)).(\lambda
55 (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2 \def (match H0 in drop1 return
56 (\lambda (p0: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p0
57 c c0)).((eq PList p0 (PCons n n0 p)) \to ((eq C c c2) \to ((eq C c0 e2) \to
58 (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc
59 g c2 c1)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList
60 PNil (PCons n n0 p))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c
61 e2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e in PList
62 return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _)
63 \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c2) \to ((eq
64 C c e2) \to (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda
65 (c1: C).(csubc g c2 c1))))) H5)) H3 H4)))) | (drop1_cons c1 c0 h d H2 c3 hds0
66 H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds0) (PCons n n0
67 p))).(\lambda (H5: (eq C c1 c2)).(\lambda (H6: (eq C c3 e2)).((let H7 \def
68 (f_equal PList PList (\lambda (e: PList).(match e in PList return (\lambda
69 (_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p0) \Rightarrow
70 p0])) (PCons h d hds0) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat
71 (\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with
72 [PNil \Rightarrow d | (PCons _ n1 _) \Rightarrow n1])) (PCons h d hds0)
73 (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e:
74 PList).(match e in PList return (\lambda (_: PList).nat) with [PNil
75 \Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) (PCons h d hds0) (PCons n n0
76 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds0
77 p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n1 d c1 c0) \to ((drop1
78 hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1))
79 (\lambda (c4: C).(csubc g c2 c4)))))))))) (\lambda (H10: (eq nat d
80 n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds0 p) \to ((eq C c1 c2)
81 \to ((eq C c3 e2) \to ((drop n n1 c1 c0) \to ((drop1 hds0 c0 c3) \to (ex2 C
82 (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c2
83 c4))))))))) (\lambda (H11: (eq PList hds0 p)).(eq_ind PList p (\lambda (p0:
84 PList).((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n n0 c1 c0) \to ((drop1 p0
85 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda
86 (c4: C).(csubc g c2 c4)))))))) (\lambda (H12: (eq C c1 c2)).(eq_ind C c2
87 (\lambda (c: C).((eq C c3 e2) \to ((drop n n0 c c0) \to ((drop1 p c0 c3) \to
88 (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc
89 g c2 c4))))))) (\lambda (H13: (eq C c3 e2)).(eq_ind C e2 (\lambda (c:
90 C).((drop n n0 c2 c0) \to ((drop1 p c0 c) \to (ex2 C (\lambda (c4: C).(drop1
91 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c2 c4)))))) (\lambda (H14:
92 (drop n n0 c2 c0)).(\lambda (H15: (drop1 p c0 e2)).(let H_x \def (H c0 e2 H15
93 e1 H1) in (let H16 \def H_x in (ex2_ind C (\lambda (c4: C).(drop1 p c4 e1))
94 (\lambda (c4: C).(csubc g c0 c4)) (ex2 C (\lambda (c4: C).(drop1 (PCons n n0
95 p) c4 e1)) (\lambda (c4: C).(csubc g c2 c4))) (\lambda (x: C).(\lambda (H17:
96 (drop1 p x e1)).(\lambda (H18: (csubc g c0 x)).(let H_x0 \def
97 (drop_csubc_trans g c2 c0 n0 n H14 x H18) in (let H19 \def H_x0 in (ex2_ind C
98 (\lambda (c4: C).(drop n n0 c4 x)) (\lambda (c4: C).(csubc g c2 c4)) (ex2 C
99 (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c2
100 c4))) (\lambda (x0: C).(\lambda (H20: (drop n n0 x0 x)).(\lambda (H21: (csubc
101 g c2 x0)).(ex_intro2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1))
102 (\lambda (c4: C).(csubc g c2 c4)) x0 (drop1_cons x0 x n n0 H20 e1 p H17)
103 H21)))) H19)))))) H16))))) c3 (sym_eq C c3 e2 H13))) c1 (sym_eq C c1 c2
104 H12))) hds0 (sym_eq PList hds0 p H11))) d (sym_eq nat d n0 H10))) h (sym_eq
105 nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n
106 n0 p)) (refl_equal C c2) (refl_equal C e2)))))))))))) hds)).
108 theorem csubc_drop1_conf_rev:
109 \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2:
110 C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C
111 (\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))
113 \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
114 (c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1
115 e2) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1
116 c2))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2
117 e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e1 e2)).(let H1 \def (match H in
118 drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda
119 (_: (drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 e2) \to
120 (ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1
121 c2)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil
122 PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c e2)).(eq_ind C c2
123 (\lambda (c0: C).((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1
124 e1)) (\lambda (c1: C).(csubc g c1 c2))))) (\lambda (H4: (eq C c2 e2)).(eq_ind
125 C e2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda
126 (c1: C).(csubc g c1 c0)))) (let H5 \def (eq_ind_r C e2 (\lambda (c0:
127 C).(csubc g e1 c0)) H0 c2 H4) in (eq_ind C c2 (\lambda (c0: C).(ex2 C
128 (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 c0))))
129 (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g
130 c1 c2)) e1 (drop1_nil e1) H5) e2 H4)) c2 (sym_eq C c2 e2 H4))) c (sym_eq C c
131 c2 H2) H3)))) | (drop1_cons c1 c0 h d H1 c3 hds0 H2) \Rightarrow (\lambda
132 (H3: (eq PList (PCons h d hds0) PNil)).(\lambda (H4: (eq C c1 c2)).(\lambda
133 (H5: (eq C c3 e2)).((let H6 \def (eq_ind PList (PCons h d hds0) (\lambda (e:
134 PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil
135 \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in
136 (False_ind ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop h d c1 c0) \to ((drop1
137 hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 PNil c4 e1)) (\lambda (c4:
138 C).(csubc g c4 c2))))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList
139 PNil) (refl_equal C c2) (refl_equal C e2)))))))) (\lambda (n: nat).(\lambda
140 (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: C).(\forall (e2:
141 C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda
142 (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))).(\lambda
143 (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n n0 p) c2 e2)).(\lambda
144 (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2 \def (match H0 in drop1 return
145 (\lambda (p0: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p0
146 c c0)).((eq PList p0 (PCons n n0 p)) \to ((eq C c c2) \to ((eq C c0 e2) \to
147 (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc
148 g c1 c2)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList
149 PNil (PCons n n0 p))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c
150 e2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e in PList
151 return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _)
152 \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c2) \to ((eq
153 C c e2) \to (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda
154 (c1: C).(csubc g c1 c2))))) H5)) H3 H4)))) | (drop1_cons c1 c0 h d H2 c3 hds0
155 H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds0) (PCons n n0
156 p))).(\lambda (H5: (eq C c1 c2)).(\lambda (H6: (eq C c3 e2)).((let H7 \def
157 (f_equal PList PList (\lambda (e: PList).(match e in PList return (\lambda
158 (_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p0) \Rightarrow
159 p0])) (PCons h d hds0) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat
160 (\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with
161 [PNil \Rightarrow d | (PCons _ n1 _) \Rightarrow n1])) (PCons h d hds0)
162 (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e:
163 PList).(match e in PList return (\lambda (_: PList).nat) with [PNil
164 \Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) (PCons h d hds0) (PCons n n0
165 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds0
166 p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n1 d c1 c0) \to ((drop1
167 hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1))
168 (\lambda (c4: C).(csubc g c4 c2)))))))))) (\lambda (H10: (eq nat d
169 n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds0 p) \to ((eq C c1 c2)
170 \to ((eq C c3 e2) \to ((drop n n1 c1 c0) \to ((drop1 hds0 c0 c3) \to (ex2 C
171 (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c4
172 c2))))))))) (\lambda (H11: (eq PList hds0 p)).(eq_ind PList p (\lambda (p0:
173 PList).((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n n0 c1 c0) \to ((drop1 p0
174 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda
175 (c4: C).(csubc g c4 c2)))))))) (\lambda (H12: (eq C c1 c2)).(eq_ind C c2
176 (\lambda (c: C).((eq C c3 e2) \to ((drop n n0 c c0) \to ((drop1 p c0 c3) \to
177 (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc
178 g c4 c2))))))) (\lambda (H13: (eq C c3 e2)).(eq_ind C e2 (\lambda (c:
179 C).((drop n n0 c2 c0) \to ((drop1 p c0 c) \to (ex2 C (\lambda (c4: C).(drop1
180 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c4 c2)))))) (\lambda (H14:
181 (drop n n0 c2 c0)).(\lambda (H15: (drop1 p c0 e2)).(let H_x \def (H c0 e2 H15
182 e1 H1) in (let H16 \def H_x in (ex2_ind C (\lambda (c4: C).(drop1 p c4 e1))
183 (\lambda (c4: C).(csubc g c4 c0)) (ex2 C (\lambda (c4: C).(drop1 (PCons n n0
184 p) c4 e1)) (\lambda (c4: C).(csubc g c4 c2))) (\lambda (x: C).(\lambda (H17:
185 (drop1 p x e1)).(\lambda (H18: (csubc g x c0)).(let H_x0 \def
186 (csubc_drop_conf_rev g c2 c0 n0 n H14 x H18) in (let H19 \def H_x0 in
187 (ex2_ind C (\lambda (c4: C).(drop n n0 c4 x)) (\lambda (c4: C).(csubc g c4
188 c2)) (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4:
189 C).(csubc g c4 c2))) (\lambda (x0: C).(\lambda (H20: (drop n n0 x0
190 x)).(\lambda (H21: (csubc g x0 c2)).(ex_intro2 C (\lambda (c4: C).(drop1
191 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c4 c2)) x0 (drop1_cons x0 x
192 n n0 H20 e1 p H17) H21)))) H19)))))) H16))))) c3 (sym_eq C c3 e2 H13))) c1
193 (sym_eq C c1 c2 H12))) hds0 (sym_eq PList hds0 p H11))) d (sym_eq nat d n0
194 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal
195 PList (PCons n n0 p)) (refl_equal C c2) (refl_equal C e2)))))))))))) hds)).