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 "basic_1/csubv/props.ma".
19 include "basic_1/csubv/fwd.ma".
21 include "basic_1/drop/fwd.ma".
23 theorem csubv_drop_conf:
24 \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (e1:
25 C).(\forall (h: nat).((drop h O c1 e1) \to (ex2 C (\lambda (e2: C).(csubv e1
26 e2)) (\lambda (e2: C).(drop h O c2 e2))))))))
28 \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(let TMP_3 \def
29 (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).(\forall (h: nat).((drop h
30 O c e1) \to (let TMP_1 \def (\lambda (e2: C).(csubv e1 e2)) in (let TMP_2
31 \def (\lambda (e2: C).(drop h O c0 e2)) in (ex2 C TMP_1 TMP_2)))))))) in (let
32 TMP_34 \def (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda
33 (H0: (drop h O (CSort n) e1)).(let TMP_4 \def (CSort n) in (let TMP_5 \def
34 (eq C e1 TMP_4) in (let TMP_6 \def (eq nat h O) in (let TMP_7 \def (eq nat O
35 O) in (let TMP_8 \def (\lambda (e2: C).(csubv e1 e2)) in (let TMP_10 \def
36 (\lambda (e2: C).(let TMP_9 \def (CSort n) in (drop h O TMP_9 e2))) in (let
37 TMP_11 \def (ex2 C TMP_8 TMP_10) in (let TMP_32 \def (\lambda (H1: (eq C e1
38 (CSort n))).(\lambda (H2: (eq nat h O)).(\lambda (_: (eq nat O O)).(let
39 TMP_15 \def (\lambda (n0: nat).(let TMP_12 \def (\lambda (e2: C).(csubv e1
40 e2)) in (let TMP_14 \def (\lambda (e2: C).(let TMP_13 \def (CSort n) in (drop
41 n0 O TMP_13 e2))) in (ex2 C TMP_12 TMP_14)))) in (let TMP_16 \def (CSort n)
42 in (let TMP_20 \def (\lambda (c: C).(let TMP_17 \def (\lambda (e2: C).(csubv
43 c e2)) in (let TMP_19 \def (\lambda (e2: C).(let TMP_18 \def (CSort n) in
44 (drop O O TMP_18 e2))) in (ex2 C TMP_17 TMP_19)))) in (let TMP_22 \def
45 (\lambda (e2: C).(let TMP_21 \def (CSort n) in (csubv TMP_21 e2))) in (let
46 TMP_24 \def (\lambda (e2: C).(let TMP_23 \def (CSort n) in (drop O O TMP_23
47 e2))) in (let TMP_25 \def (CSort n) in (let TMP_26 \def (CSort n) in (let
48 TMP_27 \def (csubv_refl TMP_26) in (let TMP_28 \def (CSort n) in (let TMP_29
49 \def (drop_refl TMP_28) in (let TMP_30 \def (ex_intro2 C TMP_22 TMP_24 TMP_25
50 TMP_27 TMP_29) in (let TMP_31 \def (eq_ind_r C TMP_16 TMP_20 TMP_30 e1 H1) in
51 (eq_ind_r nat O TMP_15 TMP_31 h H2)))))))))))))))) in (let TMP_33 \def
52 (drop_gen_sort n h O e1 H0) in (and3_ind TMP_5 TMP_6 TMP_7 TMP_11 TMP_32
53 TMP_33)))))))))))))) in (let TMP_85 \def (\lambda (c3: C).(\lambda (c4:
54 C).(\lambda (H0: (csubv c3 c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h:
55 nat).((drop h O c3 e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda
56 (e2: C).(drop h O c4 e2)))))))).(\lambda (v1: T).(\lambda (v2: T).(\lambda
57 (e1: C).(\lambda (h: nat).(\lambda (H2: (drop h O (CHead c3 (Bind Void) v1)
58 e1)).(let TMP_39 \def (\lambda (n: nat).((drop n O (CHead c3 (Bind Void) v1)
59 e1) \to (let TMP_35 \def (\lambda (e2: C).(csubv e1 e2)) in (let TMP_38 \def
60 (\lambda (e2: C).(let TMP_36 \def (Bind Void) in (let TMP_37 \def (CHead c4
61 TMP_36 v2) in (drop n O TMP_37 e2)))) in (ex2 C TMP_35 TMP_38))))) in (let
62 TMP_63 \def (\lambda (H3: (drop O O (CHead c3 (Bind Void) v1) e1)).(let
63 TMP_40 \def (Bind Void) in (let TMP_41 \def (CHead c3 TMP_40 v1) in (let
64 TMP_46 \def (\lambda (c: C).(let TMP_42 \def (\lambda (e2: C).(csubv c e2))
65 in (let TMP_45 \def (\lambda (e2: C).(let TMP_43 \def (Bind Void) in (let
66 TMP_44 \def (CHead c4 TMP_43 v2) in (drop O O TMP_44 e2)))) in (ex2 C TMP_42
67 TMP_45)))) in (let TMP_49 \def (\lambda (e2: C).(let TMP_47 \def (Bind Void)
68 in (let TMP_48 \def (CHead c3 TMP_47 v1) in (csubv TMP_48 e2)))) in (let
69 TMP_52 \def (\lambda (e2: C).(let TMP_50 \def (Bind Void) in (let TMP_51 \def
70 (CHead c4 TMP_50 v2) in (drop O O TMP_51 e2)))) in (let TMP_53 \def (Bind
71 Void) in (let TMP_54 \def (CHead c4 TMP_53 v2) in (let TMP_55 \def
72 (csubv_bind_same c3 c4 H0 Void v1 v2) in (let TMP_56 \def (Bind Void) in (let
73 TMP_57 \def (CHead c4 TMP_56 v2) in (let TMP_58 \def (drop_refl TMP_57) in
74 (let TMP_59 \def (ex_intro2 C TMP_49 TMP_52 TMP_54 TMP_55 TMP_58) in (let
75 TMP_60 \def (Bind Void) in (let TMP_61 \def (CHead c3 TMP_60 v1) in (let
76 TMP_62 \def (drop_gen_refl TMP_61 e1 H3) in (eq_ind C TMP_41 TMP_46 TMP_59 e1
77 TMP_62))))))))))))))))) in (let TMP_84 \def (\lambda (h0: nat).(\lambda (_:
78 (((drop h0 O (CHead c3 (Bind Void) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv
79 e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Bind Void) v2)
80 e2)))))).(\lambda (H3: (drop (S h0) O (CHead c3 (Bind Void) v1) e1)).(let
81 TMP_64 \def (Bind Void) in (let TMP_65 \def (r TMP_64 h0) in (let TMP_66 \def
82 (Bind Void) in (let TMP_67 \def (drop_gen_drop TMP_66 c3 e1 v1 h0 H3) in (let
83 H_x \def (H1 e1 TMP_65 TMP_67) in (let H4 \def H_x in (let TMP_68 \def
84 (\lambda (e2: C).(csubv e1 e2)) in (let TMP_69 \def (\lambda (e2: C).(drop h0
85 O c4 e2)) in (let TMP_70 \def (\lambda (e2: C).(csubv e1 e2)) in (let TMP_74
86 \def (\lambda (e2: C).(let TMP_71 \def (S h0) in (let TMP_72 \def (Bind Void)
87 in (let TMP_73 \def (CHead c4 TMP_72 v2) in (drop TMP_71 O TMP_73 e2))))) in
88 (let TMP_75 \def (ex2 C TMP_70 TMP_74) in (let TMP_83 \def (\lambda (x:
89 C).(\lambda (H5: (csubv e1 x)).(\lambda (H6: (drop h0 O c4 x)).(let TMP_76
90 \def (\lambda (e2: C).(csubv e1 e2)) in (let TMP_80 \def (\lambda (e2:
91 C).(let TMP_77 \def (S h0) in (let TMP_78 \def (Bind Void) in (let TMP_79
92 \def (CHead c4 TMP_78 v2) in (drop TMP_77 O TMP_79 e2))))) in (let TMP_81
93 \def (Bind Void) in (let TMP_82 \def (drop_drop TMP_81 h0 c4 x H6 v2) in
94 (ex_intro2 C TMP_76 TMP_80 x H5 TMP_82)))))))) in (ex2_ind C TMP_68 TMP_69
95 TMP_75 TMP_83 H4)))))))))))))))) in (nat_ind TMP_39 TMP_63 TMP_84 h
96 H2))))))))))))) in (let TMP_136 \def (\lambda (c3: C).(\lambda (c4:
97 C).(\lambda (H0: (csubv c3 c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h:
98 nat).((drop h O c3 e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda
99 (e2: C).(drop h O c4 e2)))))))).(\lambda (b1: B).(\lambda (H2: (not (eq B b1
100 Void))).(\lambda (b2: B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (e1:
101 C).(\lambda (h: nat).(\lambda (H3: (drop h O (CHead c3 (Bind b1) v1)
102 e1)).(let TMP_90 \def (\lambda (n: nat).((drop n O (CHead c3 (Bind b1) v1)
103 e1) \to (let TMP_86 \def (\lambda (e2: C).(csubv e1 e2)) in (let TMP_89 \def
104 (\lambda (e2: C).(let TMP_87 \def (Bind b2) in (let TMP_88 \def (CHead c4
105 TMP_87 v2) in (drop n O TMP_88 e2)))) in (ex2 C TMP_86 TMP_89))))) in (let
106 TMP_114 \def (\lambda (H4: (drop O O (CHead c3 (Bind b1) v1) e1)).(let TMP_91
107 \def (Bind b1) in (let TMP_92 \def (CHead c3 TMP_91 v1) in (let TMP_97 \def
108 (\lambda (c: C).(let TMP_93 \def (\lambda (e2: C).(csubv c e2)) in (let
109 TMP_96 \def (\lambda (e2: C).(let TMP_94 \def (Bind b2) in (let TMP_95 \def
110 (CHead c4 TMP_94 v2) in (drop O O TMP_95 e2)))) in (ex2 C TMP_93 TMP_96))))
111 in (let TMP_100 \def (\lambda (e2: C).(let TMP_98 \def (Bind b1) in (let
112 TMP_99 \def (CHead c3 TMP_98 v1) in (csubv TMP_99 e2)))) in (let TMP_103 \def
113 (\lambda (e2: C).(let TMP_101 \def (Bind b2) in (let TMP_102 \def (CHead c4
114 TMP_101 v2) in (drop O O TMP_102 e2)))) in (let TMP_104 \def (Bind b2) in
115 (let TMP_105 \def (CHead c4 TMP_104 v2) in (let TMP_106 \def (csubv_bind c3
116 c4 H0 b1 H2 b2 v1 v2) in (let TMP_107 \def (Bind b2) in (let TMP_108 \def
117 (CHead c4 TMP_107 v2) in (let TMP_109 \def (drop_refl TMP_108) in (let
118 TMP_110 \def (ex_intro2 C TMP_100 TMP_103 TMP_105 TMP_106 TMP_109) in (let
119 TMP_111 \def (Bind b1) in (let TMP_112 \def (CHead c3 TMP_111 v1) in (let
120 TMP_113 \def (drop_gen_refl TMP_112 e1 H4) in (eq_ind C TMP_92 TMP_97 TMP_110
121 e1 TMP_113))))))))))))))))) in (let TMP_135 \def (\lambda (h0: nat).(\lambda
122 (_: (((drop h0 O (CHead c3 (Bind b1) v1) e1) \to (ex2 C (\lambda (e2:
123 C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Bind b2) v2)
124 e2)))))).(\lambda (H4: (drop (S h0) O (CHead c3 (Bind b1) v1) e1)).(let
125 TMP_115 \def (Bind b1) in (let TMP_116 \def (r TMP_115 h0) in (let TMP_117
126 \def (Bind b1) in (let TMP_118 \def (drop_gen_drop TMP_117 c3 e1 v1 h0 H4) in
127 (let H_x \def (H1 e1 TMP_116 TMP_118) in (let H5 \def H_x in (let TMP_119
128 \def (\lambda (e2: C).(csubv e1 e2)) in (let TMP_120 \def (\lambda (e2:
129 C).(drop h0 O c4 e2)) in (let TMP_121 \def (\lambda (e2: C).(csubv e1 e2)) in
130 (let TMP_125 \def (\lambda (e2: C).(let TMP_122 \def (S h0) in (let TMP_123
131 \def (Bind b2) in (let TMP_124 \def (CHead c4 TMP_123 v2) in (drop TMP_122 O
132 TMP_124 e2))))) in (let TMP_126 \def (ex2 C TMP_121 TMP_125) in (let TMP_134
133 \def (\lambda (x: C).(\lambda (H6: (csubv e1 x)).(\lambda (H7: (drop h0 O c4
134 x)).(let TMP_127 \def (\lambda (e2: C).(csubv e1 e2)) in (let TMP_131 \def
135 (\lambda (e2: C).(let TMP_128 \def (S h0) in (let TMP_129 \def (Bind b2) in
136 (let TMP_130 \def (CHead c4 TMP_129 v2) in (drop TMP_128 O TMP_130 e2))))) in
137 (let TMP_132 \def (Bind b2) in (let TMP_133 \def (drop_drop TMP_132 h0 c4 x
138 H7 v2) in (ex_intro2 C TMP_127 TMP_131 x H6 TMP_133)))))))) in (ex2_ind C
139 TMP_119 TMP_120 TMP_126 TMP_134 H5)))))))))))))))) in (nat_ind TMP_90 TMP_114
140 TMP_135 h H3)))))))))))))))) in (let TMP_188 \def (\lambda (c3: C).(\lambda
141 (c4: C).(\lambda (H0: (csubv c3 c4)).(\lambda (H1: ((\forall (e1: C).(\forall
142 (h: nat).((drop h O c3 e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2))
143 (\lambda (e2: C).(drop h O c4 e2)))))))).(\lambda (f1: F).(\lambda (f2:
144 F).(\lambda (v1: T).(\lambda (v2: T).(\lambda (e1: C).(\lambda (h:
145 nat).(\lambda (H2: (drop h O (CHead c3 (Flat f1) v1) e1)).(let TMP_141 \def
146 (\lambda (n: nat).((drop n O (CHead c3 (Flat f1) v1) e1) \to (let TMP_137
147 \def (\lambda (e2: C).(csubv e1 e2)) in (let TMP_140 \def (\lambda (e2:
148 C).(let TMP_138 \def (Flat f2) in (let TMP_139 \def (CHead c4 TMP_138 v2) in
149 (drop n O TMP_139 e2)))) in (ex2 C TMP_137 TMP_140))))) in (let TMP_165 \def
150 (\lambda (H3: (drop O O (CHead c3 (Flat f1) v1) e1)).(let TMP_142 \def (Flat
151 f1) in (let TMP_143 \def (CHead c3 TMP_142 v1) in (let TMP_148 \def (\lambda
152 (c: C).(let TMP_144 \def (\lambda (e2: C).(csubv c e2)) in (let TMP_147 \def
153 (\lambda (e2: C).(let TMP_145 \def (Flat f2) in (let TMP_146 \def (CHead c4
154 TMP_145 v2) in (drop O O TMP_146 e2)))) in (ex2 C TMP_144 TMP_147)))) in (let
155 TMP_151 \def (\lambda (e2: C).(let TMP_149 \def (Flat f1) in (let TMP_150
156 \def (CHead c3 TMP_149 v1) in (csubv TMP_150 e2)))) in (let TMP_154 \def
157 (\lambda (e2: C).(let TMP_152 \def (Flat f2) in (let TMP_153 \def (CHead c4
158 TMP_152 v2) in (drop O O TMP_153 e2)))) in (let TMP_155 \def (Flat f2) in
159 (let TMP_156 \def (CHead c4 TMP_155 v2) in (let TMP_157 \def (csubv_flat c3
160 c4 H0 f1 f2 v1 v2) in (let TMP_158 \def (Flat f2) in (let TMP_159 \def (CHead
161 c4 TMP_158 v2) in (let TMP_160 \def (drop_refl TMP_159) in (let TMP_161 \def
162 (ex_intro2 C TMP_151 TMP_154 TMP_156 TMP_157 TMP_160) in (let TMP_162 \def
163 (Flat f1) in (let TMP_163 \def (CHead c3 TMP_162 v1) in (let TMP_164 \def
164 (drop_gen_refl TMP_163 e1 H3) in (eq_ind C TMP_143 TMP_148 TMP_161 e1
165 TMP_164))))))))))))))))) in (let TMP_187 \def (\lambda (h0: nat).(\lambda (_:
166 (((drop h0 O (CHead c3 (Flat f1) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv
167 e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Flat f2) v2) e2)))))).(\lambda
168 (H3: (drop (S h0) O (CHead c3 (Flat f1) v1) e1)).(let TMP_166 \def (Flat f1)
169 in (let TMP_167 \def (r TMP_166 h0) in (let TMP_168 \def (Flat f1) in (let
170 TMP_169 \def (drop_gen_drop TMP_168 c3 e1 v1 h0 H3) in (let H_x \def (H1 e1
171 TMP_167 TMP_169) in (let H4 \def H_x in (let TMP_170 \def (\lambda (e2:
172 C).(csubv e1 e2)) in (let TMP_172 \def (\lambda (e2: C).(let TMP_171 \def (S
173 h0) in (drop TMP_171 O c4 e2))) in (let TMP_173 \def (\lambda (e2: C).(csubv
174 e1 e2)) in (let TMP_177 \def (\lambda (e2: C).(let TMP_174 \def (S h0) in
175 (let TMP_175 \def (Flat f2) in (let TMP_176 \def (CHead c4 TMP_175 v2) in
176 (drop TMP_174 O TMP_176 e2))))) in (let TMP_178 \def (ex2 C TMP_173 TMP_177)
177 in (let TMP_186 \def (\lambda (x: C).(\lambda (H5: (csubv e1 x)).(\lambda
178 (H6: (drop (S h0) O c4 x)).(let TMP_179 \def (\lambda (e2: C).(csubv e1 e2))
179 in (let TMP_183 \def (\lambda (e2: C).(let TMP_180 \def (S h0) in (let
180 TMP_181 \def (Flat f2) in (let TMP_182 \def (CHead c4 TMP_181 v2) in (drop
181 TMP_180 O TMP_182 e2))))) in (let TMP_184 \def (Flat f2) in (let TMP_185 \def
182 (drop_drop TMP_184 h0 c4 x H6 v2) in (ex_intro2 C TMP_179 TMP_183 x H5
183 TMP_185)))))))) in (ex2_ind C TMP_170 TMP_172 TMP_178 TMP_186
184 H4)))))))))))))))) in (nat_ind TMP_141 TMP_165 TMP_187 h H2))))))))))))))) in
185 (csubv_ind TMP_3 TMP_34 TMP_85 TMP_136 TMP_188 c1 c2 H)))))))).