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/csubst1/defs.ma".
19 include "basic_1/csubst0/fwd.ma".
21 include "basic_1/subst1/defs.ma".
23 include "basic_1/s/fwd.ma".
26 \forall (i: nat).(\forall (v: T).(\forall (c1: C).(\forall (P: ((C \to
27 Prop))).((P c1) \to (((\forall (c2: C).((csubst0 i v c1 c2) \to (P c2)))) \to
28 (\forall (c: C).((csubst1 i v c1 c) \to (P c))))))))
30 \lambda (i: nat).(\lambda (v: T).(\lambda (c1: C).(\lambda (P: ((C \to
31 Prop))).(\lambda (f: (P c1)).(\lambda (f0: ((\forall (c2: C).((csubst0 i v c1
32 c2) \to (P c2))))).(\lambda (c: C).(\lambda (c0: (csubst1 i v c1 c)).(match
33 c0 with [csubst1_refl \Rightarrow f | (csubst1_sing x x0) \Rightarrow (f0 x
36 theorem csubst1_gen_head:
37 \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall
38 (v: T).(\forall (i: nat).((csubst1 (s k i) v (CHead c1 k u1) x) \to (ex3_2 T
39 C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 k u2)))) (\lambda (u2:
40 T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2:
41 C).(csubst1 i v c1 c2))))))))))
43 \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda
44 (v: T).(\lambda (i: nat).(\lambda (H: (csubst1 (s k i) v (CHead c1 k u1)
45 x)).(let TMP_1 \def (s k i) in (let TMP_2 \def (CHead c1 k u1) in (let TMP_7
46 \def (\lambda (c: C).(let TMP_4 \def (\lambda (u2: T).(\lambda (c2: C).(let
47 TMP_3 \def (CHead c2 k u2) in (eq C c TMP_3)))) in (let TMP_5 \def (\lambda
48 (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) in (let TMP_6 \def (\lambda (_:
49 T).(\lambda (c2: C).(csubst1 i v c1 c2))) in (ex3_2 T C TMP_4 TMP_5
50 TMP_6))))) in (let TMP_10 \def (\lambda (u2: T).(\lambda (c2: C).(let TMP_8
51 \def (CHead c1 k u1) in (let TMP_9 \def (CHead c2 k u2) in (eq C TMP_8
52 TMP_9))))) in (let TMP_11 \def (\lambda (u2: T).(\lambda (_: C).(subst1 i v
53 u1 u2))) in (let TMP_12 \def (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1
54 c2))) in (let TMP_13 \def (CHead c1 k u1) in (let TMP_14 \def (refl_equal C
55 TMP_13) in (let TMP_15 \def (subst1_refl i v u1) in (let TMP_16 \def
56 (csubst1_refl i v c1) in (let TMP_17 \def (ex3_2_intro T C TMP_10 TMP_11
57 TMP_12 u1 c1 TMP_14 TMP_15 TMP_16) in (let TMP_138 \def (\lambda (c2:
58 C).(\lambda (H0: (csubst0 (s k i) v (CHead c1 k u1) c2)).(let TMP_18 \def (s
59 k i) in (let H1 \def (csubst0_gen_head k c1 c2 u1 v TMP_18 H0) in (let TMP_21
60 \def (\lambda (_: T).(\lambda (j: nat).(let TMP_19 \def (s k i) in (let
61 TMP_20 \def (s k j) in (eq nat TMP_19 TMP_20))))) in (let TMP_23 \def
62 (\lambda (u2: T).(\lambda (_: nat).(let TMP_22 \def (CHead c1 k u2) in (eq C
63 c2 TMP_22)))) in (let TMP_24 \def (\lambda (u2: T).(\lambda (j: nat).(subst0
64 j v u1 u2))) in (let TMP_25 \def (ex3_2 T nat TMP_21 TMP_23 TMP_24) in (let
65 TMP_28 \def (\lambda (_: C).(\lambda (j: nat).(let TMP_26 \def (s k i) in
66 (let TMP_27 \def (s k j) in (eq nat TMP_26 TMP_27))))) in (let TMP_30 \def
67 (\lambda (c3: C).(\lambda (_: nat).(let TMP_29 \def (CHead c3 k u1) in (eq C
68 c2 TMP_29)))) in (let TMP_31 \def (\lambda (c3: C).(\lambda (j: nat).(csubst0
69 j v c1 c3))) in (let TMP_32 \def (ex3_2 C nat TMP_28 TMP_30 TMP_31) in (let
70 TMP_35 \def (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(let TMP_33
71 \def (s k i) in (let TMP_34 \def (s k j) in (eq nat TMP_33 TMP_34)))))) in
72 (let TMP_37 \def (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(let
73 TMP_36 \def (CHead c3 k u2) in (eq C c2 TMP_36))))) in (let TMP_38 \def
74 (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) in
75 (let TMP_39 \def (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0
76 j v c1 c3)))) in (let TMP_40 \def (ex4_3 T C nat TMP_35 TMP_37 TMP_38 TMP_39)
77 in (let TMP_42 \def (\lambda (u2: T).(\lambda (c3: C).(let TMP_41 \def (CHead
78 c3 k u2) in (eq C c2 TMP_41)))) in (let TMP_43 \def (\lambda (u2: T).(\lambda
79 (_: C).(subst1 i v u1 u2))) in (let TMP_44 \def (\lambda (_: T).(\lambda (c3:
80 C).(csubst1 i v c1 c3))) in (let TMP_45 \def (ex3_2 T C TMP_42 TMP_43 TMP_44)
81 in (let TMP_75 \def (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j:
82 nat).(eq nat (s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2
83 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1
84 u2))))).(let TMP_48 \def (\lambda (_: T).(\lambda (j: nat).(let TMP_46 \def
85 (s k i) in (let TMP_47 \def (s k j) in (eq nat TMP_46 TMP_47))))) in (let
86 TMP_50 \def (\lambda (u2: T).(\lambda (_: nat).(let TMP_49 \def (CHead c1 k
87 u2) in (eq C c2 TMP_49)))) in (let TMP_51 \def (\lambda (u2: T).(\lambda (j:
88 nat).(subst0 j v u1 u2))) in (let TMP_53 \def (\lambda (u2: T).(\lambda (c3:
89 C).(let TMP_52 \def (CHead c3 k u2) in (eq C c2 TMP_52)))) in (let TMP_54
90 \def (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) in (let TMP_55
91 \def (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) in (let TMP_56
92 \def (ex3_2 T C TMP_53 TMP_54 TMP_55) in (let TMP_74 \def (\lambda (x0:
93 T).(\lambda (x1: nat).(\lambda (H3: (eq nat (s k i) (s k x1))).(\lambda (H4:
94 (eq C c2 (CHead c1 k x0))).(\lambda (H5: (subst0 x1 v u1 x0)).(let TMP_57
95 \def (CHead c1 k x0) in (let TMP_62 \def (\lambda (c: C).(let TMP_59 \def
96 (\lambda (u2: T).(\lambda (c3: C).(let TMP_58 \def (CHead c3 k u2) in (eq C c
97 TMP_58)))) in (let TMP_60 \def (\lambda (u2: T).(\lambda (_: C).(subst1 i v
98 u1 u2))) in (let TMP_61 \def (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1
99 c3))) in (ex3_2 T C TMP_59 TMP_60 TMP_61))))) in (let H_y \def (s_inj k i x1
100 H3) in (let TMP_63 \def (\lambda (n: nat).(subst0 n v u1 x0)) in (let H6 \def
101 (eq_ind_r nat x1 TMP_63 H5 i H_y) in (let TMP_66 \def (\lambda (u2:
102 T).(\lambda (c3: C).(let TMP_64 \def (CHead c1 k x0) in (let TMP_65 \def
103 (CHead c3 k u2) in (eq C TMP_64 TMP_65))))) in (let TMP_67 \def (\lambda (u2:
104 T).(\lambda (_: C).(subst1 i v u1 u2))) in (let TMP_68 \def (\lambda (_:
105 T).(\lambda (c3: C).(csubst1 i v c1 c3))) in (let TMP_69 \def (CHead c1 k x0)
106 in (let TMP_70 \def (refl_equal C TMP_69) in (let TMP_71 \def (subst1_single
107 i v u1 x0 H6) in (let TMP_72 \def (csubst1_refl i v c1) in (let TMP_73 \def
108 (ex3_2_intro T C TMP_66 TMP_67 TMP_68 x0 c1 TMP_70 TMP_71 TMP_72) in
109 (eq_ind_r C TMP_57 TMP_62 TMP_73 c2 H4))))))))))))))))))) in (ex3_2_ind T nat
110 TMP_48 TMP_50 TMP_51 TMP_56 TMP_74 H2)))))))))) in (let TMP_105 \def (\lambda
111 (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k
112 j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda
113 (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))).(let TMP_78 \def (\lambda
114 (_: C).(\lambda (j: nat).(let TMP_76 \def (s k i) in (let TMP_77 \def (s k j)
115 in (eq nat TMP_76 TMP_77))))) in (let TMP_80 \def (\lambda (c3: C).(\lambda
116 (_: nat).(let TMP_79 \def (CHead c3 k u1) in (eq C c2 TMP_79)))) in (let
117 TMP_81 \def (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))) in (let
118 TMP_83 \def (\lambda (u2: T).(\lambda (c3: C).(let TMP_82 \def (CHead c3 k
119 u2) in (eq C c2 TMP_82)))) in (let TMP_84 \def (\lambda (u2: T).(\lambda (_:
120 C).(subst1 i v u1 u2))) in (let TMP_85 \def (\lambda (_: T).(\lambda (c3:
121 C).(csubst1 i v c1 c3))) in (let TMP_86 \def (ex3_2 T C TMP_83 TMP_84 TMP_85)
122 in (let TMP_104 \def (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq
123 nat (s k i) (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k u1))).(\lambda (H5:
124 (csubst0 x1 v c1 x0)).(let TMP_87 \def (CHead x0 k u1) in (let TMP_92 \def
125 (\lambda (c: C).(let TMP_89 \def (\lambda (u2: T).(\lambda (c3: C).(let
126 TMP_88 \def (CHead c3 k u2) in (eq C c TMP_88)))) in (let TMP_90 \def
127 (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) in (let TMP_91 \def
128 (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) in (ex3_2 T C TMP_89
129 TMP_90 TMP_91))))) in (let H_y \def (s_inj k i x1 H3) in (let TMP_93 \def
130 (\lambda (n: nat).(csubst0 n v c1 x0)) in (let H6 \def (eq_ind_r nat x1
131 TMP_93 H5 i H_y) in (let TMP_96 \def (\lambda (u2: T).(\lambda (c3: C).(let
132 TMP_94 \def (CHead x0 k u1) in (let TMP_95 \def (CHead c3 k u2) in (eq C
133 TMP_94 TMP_95))))) in (let TMP_97 \def (\lambda (u2: T).(\lambda (_:
134 C).(subst1 i v u1 u2))) in (let TMP_98 \def (\lambda (_: T).(\lambda (c3:
135 C).(csubst1 i v c1 c3))) in (let TMP_99 \def (CHead x0 k u1) in (let TMP_100
136 \def (refl_equal C TMP_99) in (let TMP_101 \def (subst1_refl i v u1) in (let
137 TMP_102 \def (csubst1_sing i v c1 x0 H6) in (let TMP_103 \def (ex3_2_intro T
138 C TMP_96 TMP_97 TMP_98 u1 x0 TMP_100 TMP_101 TMP_102) in (eq_ind_r C TMP_87
139 TMP_92 TMP_103 c2 H4))))))))))))))))))) in (ex3_2_ind C nat TMP_78 TMP_80
140 TMP_81 TMP_86 TMP_104 H2)))))))))) in (let TMP_137 \def (\lambda (H2: (ex4_3
141 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s
142 k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
143 c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v
144 u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1
145 c3)))))).(let TMP_108 \def (\lambda (_: T).(\lambda (_: C).(\lambda (j:
146 nat).(let TMP_106 \def (s k i) in (let TMP_107 \def (s k j) in (eq nat
147 TMP_106 TMP_107)))))) in (let TMP_110 \def (\lambda (u2: T).(\lambda (c3:
148 C).(\lambda (_: nat).(let TMP_109 \def (CHead c3 k u2) in (eq C c2
149 TMP_109))))) in (let TMP_111 \def (\lambda (u2: T).(\lambda (_: C).(\lambda
150 (j: nat).(subst0 j v u1 u2)))) in (let TMP_112 \def (\lambda (_: T).(\lambda
151 (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) in (let TMP_114 \def
152 (\lambda (u2: T).(\lambda (c3: C).(let TMP_113 \def (CHead c3 k u2) in (eq C
153 c2 TMP_113)))) in (let TMP_115 \def (\lambda (u2: T).(\lambda (_: C).(subst1
154 i v u1 u2))) in (let TMP_116 \def (\lambda (_: T).(\lambda (c3: C).(csubst1 i
155 v c1 c3))) in (let TMP_117 \def (ex3_2 T C TMP_114 TMP_115 TMP_116) in (let
156 TMP_136 \def (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda
157 (H3: (eq nat (s k i) (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 k
158 x0))).(\lambda (H5: (subst0 x2 v u1 x0)).(\lambda (H6: (csubst0 x2 v c1
159 x1)).(let TMP_118 \def (CHead x1 k x0) in (let TMP_123 \def (\lambda (c:
160 C).(let TMP_120 \def (\lambda (u2: T).(\lambda (c3: C).(let TMP_119 \def
161 (CHead c3 k u2) in (eq C c TMP_119)))) in (let TMP_121 \def (\lambda (u2:
162 T).(\lambda (_: C).(subst1 i v u1 u2))) in (let TMP_122 \def (\lambda (_:
163 T).(\lambda (c3: C).(csubst1 i v c1 c3))) in (ex3_2 T C TMP_120 TMP_121
164 TMP_122))))) in (let H_y \def (s_inj k i x2 H3) in (let TMP_124 \def (\lambda
165 (n: nat).(csubst0 n v c1 x1)) in (let H7 \def (eq_ind_r nat x2 TMP_124 H6 i
166 H_y) in (let TMP_125 \def (\lambda (n: nat).(subst0 n v u1 x0)) in (let H8
167 \def (eq_ind_r nat x2 TMP_125 H5 i H_y) in (let TMP_128 \def (\lambda (u2:
168 T).(\lambda (c3: C).(let TMP_126 \def (CHead x1 k x0) in (let TMP_127 \def
169 (CHead c3 k u2) in (eq C TMP_126 TMP_127))))) in (let TMP_129 \def (\lambda
170 (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) in (let TMP_130 \def (\lambda
171 (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) in (let TMP_131 \def (CHead x1
172 k x0) in (let TMP_132 \def (refl_equal C TMP_131) in (let TMP_133 \def
173 (subst1_single i v u1 x0 H8) in (let TMP_134 \def (csubst1_sing i v c1 x1 H7)
174 in (let TMP_135 \def (ex3_2_intro T C TMP_128 TMP_129 TMP_130 x0 x1 TMP_132
175 TMP_133 TMP_134) in (eq_ind_r C TMP_118 TMP_123 TMP_135 c2
176 H4))))))))))))))))))))))) in (ex4_3_ind T C nat TMP_108 TMP_110 TMP_111
177 TMP_112 TMP_117 TMP_136 H2))))))))))) in (or3_ind TMP_25 TMP_32 TMP_40 TMP_45
178 TMP_75 TMP_105 TMP_137 H1))))))))))))))))))))))))) in (csubst1_ind TMP_1 v
179 TMP_2 TMP_7 TMP_17 TMP_138 x H))))))))))))))))))).