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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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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/csubst0/defs.ma".
19 theorem csubst0_gen_sort:
20 \forall (x: C).(\forall (v: T).(\forall (i: nat).(\forall (n: nat).((csubst0
21 i v (CSort n) x) \to (\forall (P: Prop).P)))))
23 \lambda (x: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
24 (H: (csubst0 i v (CSort n) x)).(\lambda (P: Prop).(insert_eq C (CSort n)
25 (\lambda (c: C).(csubst0 i v c x)) (\lambda (_: C).P) (\lambda (y:
26 C).(\lambda (H0: (csubst0 i v y x)).(csubst0_ind (\lambda (_: nat).(\lambda
27 (_: T).(\lambda (c: C).(\lambda (_: C).((eq C c (CSort n)) \to P)))))
28 (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda
29 (u2: T).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(\lambda (H2: (eq
30 C (CHead c k u1) (CSort n))).(let H3 \def (eq_ind C (CHead c k u1) (\lambda
31 (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
32 \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H2) in
33 (False_ind P H3)))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (c1:
34 C).(\lambda (c2: C).(\lambda (v0: T).(\lambda (_: (csubst0 i0 v0 c1
35 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to P))).(\lambda (u: T).(\lambda
36 (H3: (eq C (CHead c1 k u) (CSort n))).(let H4 \def (eq_ind C (CHead c1 k u)
37 (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
38 \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in
39 (False_ind P H4))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0:
40 T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i0 v0 u1
41 u2)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csubst0 i0 v0 c1
42 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to P))).(\lambda (H4: (eq C (CHead
43 c1 k u1) (CSort n))).(let H5 \def (eq_ind C (CHead c1 k u1) (\lambda (ee:
44 C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
45 False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind P
46 H5))))))))))))) i v y x H0))) H)))))).
51 theorem csubst0_gen_head:
52 \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall
53 (v: T).(\forall (i: nat).((csubst0 i v (CHead c1 k u1) x) \to (or3 (ex3_2 T
54 nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2:
55 T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j:
56 nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq
57 nat i (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k
58 u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C
59 nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))))
60 (\lambda (u2: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k
61 u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
62 u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1
65 \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda
66 (v: T).(\lambda (i: nat).(\lambda (H: (csubst0 i v (CHead c1 k u1)
67 x)).(insert_eq C (CHead c1 k u1) (\lambda (c: C).(csubst0 i v c x)) (\lambda
68 (_: C).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k
69 j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda
70 (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_:
71 C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c2: C).(\lambda (_:
72 nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j
73 v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
74 nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c2: C).(\lambda (_:
75 nat).(eq C x (CHead c2 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
76 (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j:
77 nat).(csubst0 j v c1 c2))))))) (\lambda (y: C).(\lambda (H0: (csubst0 i v y
78 x)).(csubst0_ind (\lambda (n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda
79 (c0: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda (_:
80 T).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (u2: T).(\lambda (_:
81 nat).(eq C c0 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
82 t u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat n (s k
83 j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u1)))) (\lambda
84 (c2: C).(\lambda (j: nat).(csubst0 j t c1 c2)))) (ex4_3 T C nat (\lambda (_:
85 T).(\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j))))) (\lambda (u2:
86 T).(\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u2))))) (\lambda
87 (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j t u1 u2)))) (\lambda (_:
88 T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j t c1 c2))))))))))) (\lambda
89 (k0: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u0: T).(\lambda (u2:
90 T).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (c: C).(\lambda (H2: (eq C
91 (CHead c k0 u0) (CHead c1 k u1))).(let H3 \def (f_equal C C (\lambda (e:
92 C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c |
93 (CHead c0 _ _) \Rightarrow c0])) (CHead c k0 u0) (CHead c1 k u1) H2) in ((let
94 H4 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
95 with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c k0
96 u0) (CHead c1 k u1) H2) in ((let H5 \def (f_equal C T (\lambda (e: C).(match
97 e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _
98 t) \Rightarrow t])) (CHead c k0 u0) (CHead c1 k u1) H2) in (\lambda (H6: (eq
99 K k0 k)).(\lambda (H7: (eq C c c1)).(eq_ind_r C c1 (\lambda (c0: C).(or3
100 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))
101 (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c1 k u3))))
102 (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat
103 (\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c2:
104 C).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k u1)))) (\lambda (c2:
105 C).(\lambda (j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_:
106 T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda
107 (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k
108 u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1
109 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1
110 c2))))))) (let H8 \def (eq_ind T u0 (\lambda (t: T).(subst0 i0 v0 t u2)) H1
111 u1 H5) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat (\lambda (_:
112 T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u3: T).(\lambda
113 (_: nat).(eq C (CHead c1 k1 u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda
114 (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j:
115 nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C
116 (CHead c1 k1 u2) (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j:
117 nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
118 C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u3: T).(\lambda
119 (c2: C).(\lambda (_: nat).(eq C (CHead c1 k1 u2) (CHead c2 k u3))))) (\lambda
120 (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_:
121 T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 c2))))))) (or3_intro0
122 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j))))
123 (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3))))
124 (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat
125 (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c2:
126 C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k u1)))) (\lambda (c2:
127 C).(\lambda (j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_:
128 T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda
129 (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k
130 u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1
131 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1
132 c2))))) (ex3_2_intro T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0)
133 (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1
134 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3))) u2 i0
135 (refl_equal nat (s k i0)) (refl_equal C (CHead c1 k u2)) H8)) k0 H6)) c
136 H7)))) H4)) H3)))))))))) (\lambda (k0: K).(\lambda (i0: nat).(\lambda (c0:
137 C).(\lambda (c2: C).(\lambda (v0: T).(\lambda (H1: (csubst0 i0 v0 c0
138 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda
139 (_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u2: T).(\lambda (_:
140 nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
141 v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k
142 j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda
143 (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_:
144 T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u2:
145 T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
146 (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_:
147 T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda
148 (u: T).(\lambda (H3: (eq C (CHead c0 k0 u) (CHead c1 k u1))).(let H4 \def
149 (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
150 [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u)
151 (CHead c1 k u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in
152 C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _)
153 \Rightarrow k1])) (CHead c0 k0 u) (CHead c1 k u1) H3) in ((let H6 \def
154 (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
155 [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u)
156 (CHead c1 k u1) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0
157 c1)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex3_2 T nat (\lambda (_:
158 T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u2: T).(\lambda
159 (_: nat).(eq C (CHead c2 k0 t) (CHead c1 k u2)))) (\lambda (u2: T).(\lambda
160 (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j:
161 nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C
162 (CHead c2 k0 t) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j:
163 nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
164 C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u2: T).(\lambda
165 (c3: C).(\lambda (_: nat).(eq C (CHead c2 k0 t) (CHead c3 k u2))))) (\lambda
166 (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_:
167 T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (let H9 \def
168 (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat
169 (\lambda (_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u2:
170 T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda
171 (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j:
172 nat).(eq nat i0 (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
173 c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3
174 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k
175 j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3
176 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1
177 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1
178 c3)))))))) H2 c1 H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubst0
179 i0 v0 c c2)) H1 c1 H8) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat
180 (\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u2:
181 T).(\lambda (_: nat).(eq C (CHead c2 k1 u1) (CHead c1 k u2)))) (\lambda (u2:
182 T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_:
183 C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda
184 (_: nat).(eq C (CHead c2 k1 u1) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda
185 (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
186 C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u2: T).(\lambda
187 (c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u1) (CHead c3 k u2))))) (\lambda
188 (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_:
189 T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (or3_intro1
190 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j))))
191 (\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c1 k u2))))
192 (\lambda (u2: T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat
193 (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3:
194 C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k u1)))) (\lambda (c3:
195 C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_:
196 T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda
197 (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k
198 u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1
199 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1
200 c3))))) (ex3_2_intro C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0)
201 (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3
202 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))) c2 i0
203 (refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u1)) H10)) k0 H7))) u
204 H6)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (i0: nat).(\lambda (v0:
205 T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (subst0 i0 v0 u0
206 u2)).(\lambda (c0: C).(\lambda (c2: C).(\lambda (H2: (csubst0 i0 v0 c0
207 c2)).(\lambda (H3: (((eq C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda
208 (_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u3: T).(\lambda (_:
209 nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j
210 v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k
211 j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda
212 (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_:
213 T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u3:
214 T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u3))))) (\lambda
215 (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_:
216 T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda
217 (H4: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C
218 (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
219 \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k
220 u1) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return
221 (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _)
222 \Rightarrow k1])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H7 \def
223 (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
224 [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0)
225 (CHead c1 k u1) H4) in (\lambda (H8: (eq K k0 k)).(\lambda (H9: (eq C c0
226 c1)).(let H10 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to
227 (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i0 (s k j))))
228 (\lambda (u3: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3:
229 T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_:
230 C).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (c3: C).(\lambda (_:
231 nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0
232 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
233 nat).(eq nat i0 (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_:
234 nat).(eq C c2 (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda
235 (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
236 nat).(csubst0 j v0 c1 c3)))))))) H3 c1 H9) in (let H11 \def (eq_ind C c0
237 (\lambda (c: C).(csubst0 i0 v0 c c2)) H2 c1 H9) in (let H12 \def (eq_ind T u0
238 (\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H7) in (eq_ind_r K k (\lambda (k1:
239 K).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k
240 j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c1 k
241 u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat
242 (\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3:
243 C).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c3 k u1)))) (\lambda (c3:
244 C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_:
245 T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda
246 (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c3 k
247 u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1
248 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1
249 c3))))))) (or3_intro2 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat
250 (s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k u2)
251 (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3))))
252 (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))
253 (\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k u1))))
254 (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat
255 (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k
256 j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k
257 u2) (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j:
258 nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
259 nat).(csubst0 j v0 c1 c3))))) (ex4_3_intro T C nat (\lambda (_: T).(\lambda
260 (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3:
261 T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k
262 u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1
263 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1
264 c3)))) u2 c2 i0 (refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u2)) H12
265 H11)) k0 H8))))))) H6)) H5))))))))))))) i v y x H0))) H))))))).
270 theorem csubst0_gen_S_bind_2:
271 \forall (b: B).(\forall (x: C).(\forall (c2: C).(\forall (v: T).(\forall
272 (v2: T).(\forall (i: nat).((csubst0 (S i) v x (CHead c2 (Bind b) v2)) \to
273 (or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x
274 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2))
275 (\lambda (c1: C).(eq C x (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_:
276 C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_:
277 T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1
278 (Bind b) v1))))))))))))
280 \lambda (b: B).(\lambda (x: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
281 (v2: T).(\lambda (i: nat).(\lambda (H: (csubst0 (S i) v x (CHead c2 (Bind b)
282 v2))).(insert_eq C (CHead c2 (Bind b) v2) (\lambda (c: C).(csubst0 (S i) v x
283 c)) (\lambda (_: C).(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda
284 (v1: T).(eq C x (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i
285 v c1 c2)) (\lambda (c1: C).(eq C x (CHead c1 (Bind b) v2)))) (ex3_2 C T
286 (\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1:
287 C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1:
288 T).(eq C x (CHead c1 (Bind b) v1))))))) (\lambda (y: C).(\lambda (H0:
289 (csubst0 (S i) v x y)).(insert_eq nat (S i) (\lambda (n: nat).(csubst0 n v x
290 y)) (\lambda (_: nat).((eq C y (CHead c2 (Bind b) v2)) \to (or3 (ex2 T
291 (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x (CHead c2 (Bind
292 b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C
293 x (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1:
294 T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1
295 c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 (Bind b) v1))))))))
296 (\lambda (y0: nat).(\lambda (H1: (csubst0 y0 v x y)).(csubst0_ind (\lambda
297 (n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).((eq nat n (S i))
298 \to ((eq C c0 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1:
299 T).(subst0 i t v1 v2)) (\lambda (v1: T).(eq C c (CHead c2 (Bind b) v1))))
300 (ex2 C (\lambda (c1: C).(csubst0 i t c1 c2)) (\lambda (c1: C).(eq C c (CHead
301 c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i t v1
302 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i t c1 c2))) (\lambda (c1:
303 C).(\lambda (v1: T).(eq C c (CHead c1 (Bind b) v1)))))))))))) (\lambda (k:
304 K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2:
305 T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat
306 (s k i0) (S i))).(\lambda (H4: (eq C (CHead c k u2) (CHead c2 (Bind b)
307 v2))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
308 (_: C).C) with [(CSort _) \Rightarrow c | (CHead c0 _ _) \Rightarrow c0]))
309 (CHead c k u2) (CHead c2 (Bind b) v2) H4) in ((let H6 \def (f_equal C K
310 (\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _)
311 \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c k u2) (CHead c2
312 (Bind b) v2) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C
313 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t)
314 \Rightarrow t])) (CHead c k u2) (CHead c2 (Bind b) v2) H4) in (\lambda (H8:
315 (eq K k (Bind b))).(\lambda (H9: (eq C c c2)).(eq_ind_r C c2 (\lambda (c0:
316 C).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C
317 (CHead c0 k u1) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i
318 v0 c1 c2)) (\lambda (c1: C).(eq C (CHead c0 k u1) (CHead c1 (Bind b) v2))))
319 (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda
320 (c1: C).(\lambda (_: T).(csubst0 i v0 c1 c2))) (\lambda (c1: C).(\lambda (v1:
321 T).(eq C (CHead c0 k u1) (CHead c1 (Bind b) v1))))))) (let H10 \def (eq_ind T
322 u2 (\lambda (t: T).(subst0 i0 v0 u1 t)) H2 v2 H7) in (let H11 \def (eq_ind K
323 k (\lambda (k0: K).(eq nat (s k0 i0) (S i))) H3 (Bind b) H8) in (eq_ind_r K
324 (Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2))
325 (\lambda (v1: T).(eq C (CHead c2 k0 u1) (CHead c2 (Bind b) v1)))) (ex2 C
326 (\lambda (c1: C).(csubst0 i v0 c1 c2)) (\lambda (c1: C).(eq C (CHead c2 k0
327 u1) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1:
328 T).(subst0 i v0 v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v0 c1
329 c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c2 k0 u1) (CHead c1
330 (Bind b) v1))))))) (let H12 \def (f_equal nat nat (\lambda (e: nat).(match e
331 in nat return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n)
332 \Rightarrow n])) (S i0) (S i) H11) in (let H13 \def (eq_ind nat i0 (\lambda
333 (n: nat).(subst0 n v0 u1 v2)) H10 i H12) in (or3_intro0 (ex2 T (\lambda (v1:
334 T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c2 (Bind b) u1) (CHead
335 c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v0 c1 c2)) (\lambda
336 (c1: C).(eq C (CHead c2 (Bind b) u1) (CHead c1 (Bind b) v2)))) (ex3_2 C T
337 (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c1:
338 C).(\lambda (_: T).(csubst0 i v0 c1 c2))) (\lambda (c1: C).(\lambda (v1:
339 T).(eq C (CHead c2 (Bind b) u1) (CHead c1 (Bind b) v1))))) (ex_intro2 T
340 (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c2 (Bind
341 b) u1) (CHead c2 (Bind b) v1))) u1 H13 (refl_equal C (CHead c2 (Bind b)
342 u1)))))) k H8))) c H9)))) H6)) H5))))))))))) (\lambda (k: K).(\lambda (i0:
343 nat).(\lambda (c1: C).(\lambda (c0: C).(\lambda (v0: T).(\lambda (H2:
344 (csubst0 i0 v0 c1 c0)).(\lambda (H3: (((eq nat i0 (S i)) \to ((eq C c0 (CHead
345 c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2))
346 (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3:
347 C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2))))
348 (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda
349 (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1:
350 T).(eq C c1 (CHead c3 (Bind b) v1)))))))))).(\lambda (u: T).(\lambda (H4: (eq
351 nat (s k i0) (S i))).(\lambda (H5: (eq C (CHead c0 k u) (CHead c2 (Bind b)
352 v2))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
353 (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c]))
354 (CHead c0 k u) (CHead c2 (Bind b) v2) H5) in ((let H7 \def (f_equal C K
355 (\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _)
356 \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u) (CHead c2
357 (Bind b) v2) H5) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C
358 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
359 \Rightarrow t])) (CHead c0 k u) (CHead c2 (Bind b) v2) H5) in (\lambda (H9:
360 (eq K k (Bind b))).(\lambda (H10: (eq C c0 c2)).(eq_ind_r T v2 (\lambda (t:
361 T).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C
362 (CHead c1 k t) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i
363 v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 k t) (CHead c3 (Bind b) v2))))
364 (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda
365 (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1:
366 T).(eq C (CHead c1 k t) (CHead c3 (Bind b) v1))))))) (let H11 \def (eq_ind C
367 c0 (\lambda (c: C).((eq nat i0 (S i)) \to ((eq C c (CHead c2 (Bind b) v2))
368 \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C
369 c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2))
370 (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_:
371 C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_:
372 T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead
373 c3 (Bind b) v1))))))))) H3 c2 H10) in (let H12 \def (eq_ind C c0 (\lambda (c:
374 C).(csubst0 i0 v0 c1 c)) H2 c2 H10) in (let H13 \def (eq_ind K k (\lambda
375 (k0: K).(eq nat (s k0 i0) (S i))) H4 (Bind b) H9) in (eq_ind_r K (Bind b)
376 (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda
377 (v1: T).(eq C (CHead c1 k0 v2) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3:
378 C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 k0 v2) (CHead c3
379 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1
380 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3:
381 C).(\lambda (v1: T).(eq C (CHead c1 k0 v2) (CHead c3 (Bind b) v1))))))) (let
382 H14 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return (\lambda
383 (_: nat).nat) with [O \Rightarrow i0 | (S n) \Rightarrow n])) (S i0) (S i)
384 H13) in (let H15 \def (eq_ind nat i0 (\lambda (n: nat).((eq nat n (S i)) \to
385 ((eq C c2 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i
386 v0 v1 v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C
387 (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3
388 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1
389 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3:
390 C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind b) v1))))))))) H11 i H14) in
391 (let H16 \def (eq_ind nat i0 (\lambda (n: nat).(csubst0 n v0 c1 c2)) H12 i
392 H14) in (or3_intro1 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda
393 (v1: T).(eq C (CHead c1 (Bind b) v2) (CHead c2 (Bind b) v1)))) (ex2 C
394 (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 (Bind
395 b) v2) (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1:
396 T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3
397 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind b) v2) (CHead
398 c3 (Bind b) v1))))) (ex_intro2 C (\lambda (c3: C).(csubst0 i v0 c3 c2))
399 (\lambda (c3: C).(eq C (CHead c1 (Bind b) v2) (CHead c3 (Bind b) v2))) c1 H16
400 (refl_equal C (CHead c1 (Bind b) v2))))))) k H9)))) u H8)))) H7))
401 H6)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda
402 (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c1:
403 C).(\lambda (c0: C).(\lambda (H3: (csubst0 i0 v0 c1 c0)).(\lambda (H4: (((eq
404 nat i0 (S i)) \to ((eq C c0 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda
405 (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b)
406 v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C
407 c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1:
408 T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3
409 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind b)
410 v1)))))))))).(\lambda (H5: (eq nat (s k i0) (S i))).(\lambda (H6: (eq C
411 (CHead c0 k u2) (CHead c2 (Bind b) v2))).(let H7 \def (f_equal C C (\lambda
412 (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0
413 | (CHead c _ _) \Rightarrow c])) (CHead c0 k u2) (CHead c2 (Bind b) v2) H6)
414 in ((let H8 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
415 (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0]))
416 (CHead c0 k u2) (CHead c2 (Bind b) v2) H6) in ((let H9 \def (f_equal C T
417 (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
418 \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u2) (CHead c2
419 (Bind b) v2) H6) in (\lambda (H10: (eq K k (Bind b))).(\lambda (H11: (eq C c0
420 c2)).(let H12 \def (eq_ind C c0 (\lambda (c: C).((eq nat i0 (S i)) \to ((eq C
421 c (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1
422 v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3:
423 C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2))))
424 (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda
425 (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1:
426 T).(eq C c1 (CHead c3 (Bind b) v1))))))))) H4 c2 H11) in (let H13 \def
427 (eq_ind C c0 (\lambda (c: C).(csubst0 i0 v0 c1 c)) H3 c2 H11) in (let H14
428 \def (eq_ind T u2 (\lambda (t: T).(subst0 i0 v0 u1 t)) H2 v2 H9) in (let H15
429 \def (eq_ind K k (\lambda (k0: K).(eq nat (s k0 i0) (S i))) H5 (Bind b) H10)
430 in (eq_ind_r K (Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0
431 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c1 k0 u1) (CHead c2 (Bind b)
432 v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C
433 (CHead c1 k0 u1) (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_:
434 C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_:
435 T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1
436 k0 u1) (CHead c3 (Bind b) v1))))))) (let H16 \def (f_equal nat nat (\lambda
437 (e: nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i0
438 | (S n) \Rightarrow n])) (S i0) (S i) H15) in (let H17 \def (eq_ind nat i0
439 (\lambda (n: nat).((eq nat n (S i)) \to ((eq C c2 (CHead c2 (Bind b) v2)) \to
440 (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C c1
441 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2))
442 (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_:
443 C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_:
444 T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead
445 c3 (Bind b) v1))))))))) H12 i H16) in (let H18 \def (eq_ind nat i0 (\lambda
446 (n: nat).(csubst0 n v0 c1 c2)) H13 i H16) in (let H19 \def (eq_ind nat i0
447 (\lambda (n: nat).(subst0 n v0 u1 v2)) H14 i H16) in (or3_intro2 (ex2 T
448 (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c1 (Bind
449 b) u1) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3
450 c2)) (\lambda (c3: C).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v2))))
451 (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda
452 (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1:
453 T).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v1))))) (ex3_2_intro C T
454 (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3:
455 C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1:
456 T).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v1)))) c1 u1 H19 H18
457 (refl_equal C (CHead c1 (Bind b) u1)))))))) k H10)))))))) H8))
458 H7)))))))))))))) y0 v x y H1))) H0))) H))))))).
projects / helm.git / blob