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/subst0/defs.ma".
19 include "LambdaDelta-1/lift/props.ma".
21 theorem subst0_gen_sort:
22 \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0
23 i v (TSort n) x) \to (\forall (P: Prop).P)))))
25 \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
26 (H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(insert_eq T (TSort n)
27 (\lambda (t: T).(subst0 i v t x)) (\lambda (_: T).P) (\lambda (y: T).(\lambda
28 (H0: (subst0 i v y x)).(subst0_ind (\lambda (_: nat).(\lambda (_: T).(\lambda
29 (t0: T).(\lambda (_: T).((eq T t0 (TSort n)) \to P))))) (\lambda (_:
30 T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TSort n))).(let H2 \def
31 (eq_ind T (TLRef i0) (\lambda (ee: T).(match ee in T return (\lambda (_:
32 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
33 (THead _ _ _) \Rightarrow False])) I (TSort n) H1) in (False_ind P H2)))))
34 (\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0:
35 nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n))
36 \to P))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T (THead k u1 t)
37 (TSort n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda (ee: T).(match ee
38 in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
39 _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in
40 (False_ind P H4))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2:
41 T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v0 t1
42 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (u: T).(\lambda
43 (H3: (eq T (THead k u t1) (TSort n))).(let H4 \def (eq_ind T (THead k u t1)
44 (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
45 \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
46 True])) I (TSort n) H3) in (False_ind P H4))))))))))) (\lambda (v0:
47 T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0
48 i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to P))).(\lambda (k:
49 K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (s k i0) v0 t1
50 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (H5: (eq T (THead k
51 u1 t1) (TSort n))).(let H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee:
52 T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
53 False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
54 (TSort n) H5) in (False_ind P H6)))))))))))))) i v y x H0))) H)))))).
56 theorem subst0_gen_lref:
57 \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0
58 i v (TLRef n) x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))
60 \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
61 (H: (subst0 i v (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(subst0
62 i v t x)) (\lambda (_: T).(land (eq nat n i) (eq T x (lift (S n) O v))))
63 (\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda (n0:
64 nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).((eq T t0 (TLRef n))
65 \to (land (eq nat n n0) (eq T t1 (lift (S n) O t)))))))) (\lambda (v0:
66 T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TLRef n))).(let H2 \def
67 (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
68 [(TSort _) \Rightarrow i0 | (TLRef n0) \Rightarrow n0 | (THead _ _ _)
69 \Rightarrow i0])) (TLRef i0) (TLRef n) H1) in (eq_ind_r nat n (\lambda (n0:
70 nat).(land (eq nat n n0) (eq T (lift (S n0) O v0) (lift (S n) O v0)))) (conj
71 (eq nat n n) (eq T (lift (S n) O v0) (lift (S n) O v0)) (refl_equal nat n)
72 (refl_equal T (lift (S n) O v0))) i0 H2))))) (\lambda (v0: T).(\lambda (u2:
73 T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1
74 u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2
75 (lift (S n) O v0)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T
76 (THead k u1 t) (TLRef n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda
77 (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
78 \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
79 True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u2
80 t) (lift (S n) O v0))) H4))))))))))) (\lambda (k: K).(\lambda (v0:
81 T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0
82 (s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (land (eq nat n (s
83 k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda (u: T).(\lambda (H3: (eq T
84 (THead k u t1) (TLRef n))).(let H4 \def (eq_ind T (THead k u t1) (\lambda
85 (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
86 \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
87 True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u
88 t2) (lift (S n) O v0))) H4))))))))))) (\lambda (v0: T).(\lambda (u1:
89 T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1
90 u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2
91 (lift (S n) O v0)))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2:
92 T).(\lambda (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef
93 n)) \to (land (eq nat n (s k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda
94 (H5: (eq T (THead k u1 t1) (TLRef n))).(let H6 \def (eq_ind T (THead k u1 t1)
95 (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
96 \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
97 True])) I (TLRef n) H5) in (False_ind (land (eq nat n i0) (eq T (THead k u2
98 t2) (lift (S n) O v0))) H6)))))))))))))) i v y x H0))) H))))).
100 theorem subst0_gen_head:
101 \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall
102 (x: T).(\forall (i: nat).((subst0 i v (THead k u1 t1) x) \to (or3 (ex2 T
103 (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
104 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2:
105 T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
106 T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1
107 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))))
109 \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda
110 (x: T).(\lambda (i: nat).(\lambda (H: (subst0 i v (THead k u1 t1)
111 x)).(insert_eq T (THead k u1 t1) (\lambda (t: T).(subst0 i v t x)) (\lambda
112 (_: T).(or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2:
113 T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2)))
114 (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2:
115 T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
116 T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1
117 t2)))))) (\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda
118 (n: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t2: T).((eq T t0 (THead k
119 u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda
120 (u2: T).(subst0 n t u1 u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1
121 t3))) (\lambda (t3: T).(subst0 (s k n) t t1 t3))) (ex3_2 T T (\lambda (u2:
122 T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_:
123 T).(subst0 n t u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k n) t t1
124 t3)))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef
125 i0) (THead k u1 t1))).(let H2 \def (eq_ind T (TLRef i0) (\lambda (ee:
126 T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
127 False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
128 (THead k u1 t1) H1) in (False_ind (or3 (ex2 T (\lambda (u2: T).(eq T (lift (S
129 i0) O v0) (THead k u2 t1))) (\lambda (u2: T).(subst0 i0 v0 u1 u2))) (ex2 T
130 (\lambda (t2: T).(eq T (lift (S i0) O v0) (THead k u1 t2))) (\lambda (t2:
131 T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
132 T).(eq T (lift (S i0) O v0) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
133 T).(subst0 i0 v0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0)
134 v0 t1 t2))))) H2))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u0:
135 T).(\lambda (i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq
136 T u0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3
137 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T
138 u2 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T
139 (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3:
140 T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2:
141 T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (t: T).(\lambda (k0:
142 K).(\lambda (H3: (eq T (THead k0 u0 t) (THead k u1 t1))).(let H4 \def
143 (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
144 [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _)
145 \Rightarrow k1])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H5 \def
146 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
147 [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t0 _)
148 \Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H6 \def
149 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
150 [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0)
151 \Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in (\lambda (H7: (eq T
152 u0 u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T
153 (\lambda (u3: T).(eq T (THead k1 u2 t) (THead k u3 t1))) (\lambda (u3:
154 T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1 u2 t)
155 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T
156 (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k1 u2 t) (THead k u3 t2))))
157 (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_:
158 T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (eq_ind_r T t1 (\lambda
159 (t0: T).(or3 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t0) (THead k u3 t1)))
160 (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead
161 k u2 t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))
162 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t0) (THead k
163 u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda
164 (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (let H9 \def (eq_ind
165 T u0 (\lambda (t0: T).((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda
166 (u3: T).(eq T u2 (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3)))
167 (ex2 T (\lambda (t2: T).(eq T u2 (THead k u1 t2))) (\lambda (t2: T).(subst0
168 (s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2
169 (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3)))
170 (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))))))) H2 u1 H7)
171 in (let H10 \def (eq_ind T u0 (\lambda (t0: T).(subst0 i0 v0 t0 u2)) H1 u1
172 H7) in (or3_intro0 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3
173 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T
174 (THead k u2 t1) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1
175 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t1)
176 (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3)))
177 (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))) (ex_intro2 T
178 (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3:
179 T).(subst0 i0 v0 u1 u3)) u2 (refl_equal T (THead k u2 t1)) H10)))) t H6) k0
180 H8)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (v0: T).(\lambda (t2:
181 T).(\lambda (t0: T).(\lambda (i0: nat).(\lambda (H1: (subst0 (s k0 i0) v0 t0
182 t2)).(\lambda (H2: (((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2:
183 T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2)))
184 (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0
185 (s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
186 T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0
187 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1
188 t3)))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1
189 t1))).(let H4 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda
190 (_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead
191 k1 _ _) \Rightarrow k1])) (THead k0 u t0) (THead k u1 t1) H3) in ((let H5
192 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
193 with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _)
194 \Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in ((let H6 \def
195 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
196 [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
197 \Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in (\lambda (H7: (eq T u
198 u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex2 T
199 (\lambda (u2: T).(eq T (THead k0 t t2) (THead k u2 t1))) (\lambda (u2:
200 T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k0 t t2)
201 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T
202 (\lambda (u2: T).(\lambda (t3: T).(eq T (THead k0 t t2) (THead k u2 t3))))
203 (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_:
204 T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (let H9 \def (eq_ind T t0
205 (\lambda (t: T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2:
206 T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2)))
207 (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0
208 (s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
209 T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0
210 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1
211 t3))))))) H2 t1 H6) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s
212 k0 i0) v0 t t2)) H1 t1 H6) in (let H11 \def (eq_ind K k0 (\lambda (k1:
213 K).((eq T t1 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2
214 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k1 i0) v0 u1 u2))) (ex2 T
215 (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s
216 k1 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
217 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1
218 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1
219 t3))))))) H9 k H8) in (let H12 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s
220 k1 i0) v0 t1 t2)) H10 k H8) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T
221 (\lambda (u2: T).(eq T (THead k1 u1 t2) (THead k u2 t1))) (\lambda (u2:
222 T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k1 u1 t2)
223 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T
224 (\lambda (u2: T).(\lambda (t3: T).(eq T (THead k1 u1 t2) (THead k u2 t3))))
225 (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_:
226 T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (or3_intro1 (ex2 T
227 (\lambda (u2: T).(eq T (THead k u1 t2) (THead k u2 t1))) (\lambda (u2:
228 T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k u1 t2)
229 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T
230 (\lambda (u2: T).(\lambda (t3: T).(eq T (THead k u1 t2) (THead k u2 t3))))
231 (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_:
232 T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))) (ex_intro2 T (\lambda (t3:
233 T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0)
234 v0 t1 t3)) t2 (refl_equal T (THead k u1 t2)) H12)) k0 H8))))) u H7)))) H5))
235 H4))))))))))) (\lambda (v0: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda
236 (i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq T u0 (THead
237 k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1)))
238 (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T u2
239 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T
240 (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3:
241 T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2:
242 T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (k0: K).(\lambda (t0:
243 T).(\lambda (t2: T).(\lambda (H3: (subst0 (s k0 i0) v0 t0 t2)).(\lambda (H4:
244 (((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead
245 k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda
246 (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0))
247 v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3
248 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3)))
249 (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1
250 t3)))))))).(\lambda (H5: (eq T (THead k0 u0 t0) (THead k u1 t1))).(let H6
251 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
252 with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _)
253 \Rightarrow k1])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H7 \def
254 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
255 [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _)
256 \Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H8 \def
257 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
258 [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
259 \Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in (\lambda (H9: (eq T
260 u0 u1)).(\lambda (H10: (eq K k0 k)).(let H11 \def (eq_ind T t0 (\lambda (t:
261 T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead
262 k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda
263 (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0))
264 v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3
265 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3)))
266 (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))))))) H4
267 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s k0 i0) v0 t
268 t2)) H3 t1 H8) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq T t1
269 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3 t1)))
270 (\lambda (u3: T).(subst0 (s k1 i0) v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T
271 t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3)))
272 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 t3))))
273 (\lambda (u3: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1 u3))) (\lambda (_:
274 T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))))))) H11 k H10) in
275 (let H14 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s k1 i0) v0 t1 t2)) H12
276 k H10) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T (\lambda (u3: T).(eq T
277 (THead k1 u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3)))
278 (ex2 T (\lambda (t3: T).(eq T (THead k1 u2 t2) (THead k u1 t3))) (\lambda
279 (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda
280 (t3: T).(eq T (THead k1 u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda
281 (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k
282 i0) v0 t1 t3)))))) (let H15 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead
283 k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1)))
284 (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T u2
285 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T
286 (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead k u3 t3)))) (\lambda (u3:
287 T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3:
288 T).(subst0 (s k i0) v0 t1 t3))))))) H2 u1 H9) in (let H16 \def (eq_ind T u0
289 (\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H9) in (or3_intro2 (ex2 T (\lambda
290 (u3: T).(eq T (THead k u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0
291 v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead k u2 t2) (THead k u1 t3)))
292 (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3:
293 T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3:
294 T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3:
295 T).(subst0 (s k i0) v0 t1 t3)))) (ex3_2_intro T T (\lambda (u3: T).(\lambda
296 (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda
297 (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k
298 i0) v0 t1 t3))) u2 t2 (refl_equal T (THead k u2 t2)) H16 H14)))) k0
299 H10)))))))) H7)) H6)))))))))))))) i v y x H0))) H))))))).
301 theorem subst0_gen_lift_lt:
302 \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
303 (h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t1)
304 x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
305 (t2: T).(subst0 i u t1 t2)))))))))
307 \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x:
308 T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift h d
309 u) (lift h (S (plus i d)) t) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h
310 (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2))))))))) (\lambda (n:
311 nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
312 nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S (plus i d)) (TSort n))
313 x)).(let H0 \def (eq_ind T (lift h (S (plus i d)) (TSort n)) (\lambda (t:
314 T).(subst0 i (lift h d u) t x)) H (TSort n) (lift_sort n h (S (plus i d))))
315 in (subst0_gen_sort (lift h d u) x i n H0 (ex2 T (\lambda (t2: T).(eq T x
316 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TSort n)
317 t2))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: nat).(\lambda
318 (h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S
319 (plus i d)) (TLRef n)) x)).(lt_le_e n (S (plus i d)) (ex2 T (\lambda (t2:
320 T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef
321 n) t2))) (\lambda (H0: (lt n (S (plus i d)))).(let H1 \def (eq_ind T (lift h
322 (S (plus i d)) (TLRef n)) (\lambda (t: T).(subst0 i (lift h d u) t x)) H
323 (TLRef n) (lift_lref_lt n h (S (plus i d)) H0)) in (land_ind (eq nat n i) (eq
324 T x (lift (S n) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S
325 (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2:
326 (eq nat n i)).(\lambda (H3: (eq T x (lift (S n) O (lift h d u)))).(eq_ind_r T
327 (lift (S n) O (lift h d u)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t
328 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))))
329 (eq_ind_r nat i (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S n0)
330 O (lift h d u)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
331 (TLRef n0) t2)))) (eq_ind T (lift h (plus (S i) d) (lift (S i) O u)) (\lambda
332 (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) (\lambda
333 (t2: T).(subst0 i u (TLRef i) t2)))) (ex_intro2 T (\lambda (t2: T).(eq T
334 (lift h (S (plus i d)) (lift (S i) O u)) (lift h (S (plus i d)) t2)))
335 (\lambda (t2: T).(subst0 i u (TLRef i) t2)) (lift (S i) O u) (refl_equal T
336 (lift h (S (plus i d)) (lift (S i) O u))) (subst0_lref u i)) (lift (S i) O
337 (lift h d u)) (lift_d u h (S i) d O (le_O_n d))) n H2) x H3)))
338 (subst0_gen_lref (lift h d u) x i n H1)))) (\lambda (H0: (le (S (plus i d))
339 n)).(let H1 \def (eq_ind T (lift h (S (plus i d)) (TLRef n)) (\lambda (t:
340 T).(subst0 i (lift h d u) t x)) H (TLRef (plus n h)) (lift_lref_ge n h (S
341 (plus i d)) H0)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n
342 h)) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
343 t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: (eq nat
344 (plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n h)) O (lift h d
345 u)))).(let H4 \def (eq_ind_r nat i (\lambda (n0: nat).(le (S (plus n0 d)) n))
346 H0 (plus n h) H2) in (le_false n (plus (plus n h) d) (ex2 T (\lambda (t2:
347 T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef
348 n) t2))) (le_plus_trans n (plus n h) d (le_plus_l n h)) H4))))
349 (subst0_gen_lref (lift h d u) x i (plus n h) H1))))))))))) (\lambda (k:
350 K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall
351 (h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t)
352 x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
353 (t2: T).(subst0 i u t t2)))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall
354 (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift
355 h d u) (lift h (S (plus i d)) t0) x) \to (ex2 T (\lambda (t2: T).(eq T x
356 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t0
357 t2)))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
358 nat).(\lambda (H1: (subst0 i (lift h d u) (lift h (S (plus i d)) (THead k t
359 t0)) x)).(let H2 \def (eq_ind T (lift h (S (plus i d)) (THead k t t0))
360 (\lambda (t2: T).(subst0 i (lift h d u) t2 x)) H1 (THead k (lift h (S (plus i
361 d)) t) (lift h (s k (S (plus i d))) t0)) (lift_head k t t0 h (S (plus i d))))
362 in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k (S (plus
363 i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S (plus i d))
364 t) u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t)
365 t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i
366 d))) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k
367 u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S
368 (plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h
369 d u) (lift h (s k (S (plus i d))) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x
370 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0)
371 t2))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k
372 (S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S
373 (plus i d)) t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h
374 (s k (S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h
375 (S (plus i d)) t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
376 t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0:
377 T).(\lambda (H4: (eq T x (THead k x0 (lift h (s k (S (plus i d)))
378 t0)))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) t)
379 x0)).(eq_ind_r T (THead k x0 (lift h (s k (S (plus i d))) t0)) (\lambda (t2:
380 T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda
381 (t3: T).(subst0 i u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T
382 x0 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T
383 (\lambda (t2: T).(eq T (THead k x0 (lift h (s k (S (plus i d))) t0)) (lift h
384 (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))
385 (\lambda (x1: T).(\lambda (H6: (eq T x0 (lift h (S (plus i d)) x1))).(\lambda
386 (H7: (subst0 i u t x1)).(eq_ind_r T (lift h (S (plus i d)) x1) (\lambda (t2:
387 T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k (S (plus i d)))
388 t0)) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0)
389 t3)))) (eq_ind T (lift h (S (plus i d)) (THead k x1 t0)) (\lambda (t2:
390 T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda
391 (t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T
392 (lift h (S (plus i d)) (THead k x1 t0)) (lift h (S (plus i d)) t2))) (\lambda
393 (t2: T).(subst0 i u (THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h
394 (S (plus i d)) (THead k x1 t0))) (subst0_fst u x1 t i H7 t0 k)) (THead k
395 (lift h (S (plus i d)) x1) (lift h (s k (S (plus i d))) t0)) (lift_head k x1
396 t0 h (S (plus i d)))) x0 H6)))) (H x0 i h d H5)) x H4)))) H3)) (\lambda (H3:
397 (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) t2)))
398 (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d)))
399 t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i
400 d)) t) t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S
401 (plus i d))) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
402 t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0:
403 T).(\lambda (H4: (eq T x (THead k (lift h (S (plus i d)) t) x0))).(\lambda
404 (H5: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) t0)
405 x0)).(eq_ind_r T (THead k (lift h (S (plus i d)) t) x0) (\lambda (t2: T).(ex2
406 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3:
407 T).(subst0 i u (THead k t t0) t3)))) (let H6 \def (eq_ind nat (s k (S (plus i
408 d))) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x0)) H5 (S
409 (s k (plus i d))) (s_S k (plus i d))) in (let H7 \def (eq_ind nat (s k (plus
410 i d)) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x0))
411 H6 (plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x0
412 (lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2))
413 (ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) x0) (lift h
414 (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))
415 (\lambda (x1: T).(\lambda (H8: (eq T x0 (lift h (S (plus (s k i) d))
416 x1))).(\lambda (H9: (subst0 (s k i) u t0 x1)).(eq_ind_r T (lift h (S (plus (s
417 k i) d)) x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h
418 (S (plus i d)) t) t2) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i
419 u (THead k t t0) t3)))) (eq_ind nat (s k (plus i d)) (\lambda (n: nat).(ex2 T
420 (\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h (S n) x1))
421 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0)
422 t2)))) (eq_ind nat (s k (S (plus i d))) (\lambda (n: nat).(ex2 T (\lambda
423 (t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h n x1)) (lift h (S
424 (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind
425 T (lift h (S (plus i d)) (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda
426 (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u
427 (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift h (S (plus i
428 d)) (THead k t x1)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
429 (THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h (S (plus i d))
430 (THead k t x1))) (subst0_snd k u x1 t0 i H9 t)) (THead k (lift h (S (plus i
431 d)) t) (lift h (s k (S (plus i d))) x1)) (lift_head k t x1 h (S (plus i d))))
432 (S (s k (plus i d))) (s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x0
433 H8)))) (H0 x0 (s k i) h d H7)))) x H4)))) H3)) (\lambda (H3: (ex3_2 T T
434 (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2:
435 T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S (plus i d)) t) u2)))
436 (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S
437 (plus i d))) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq
438 T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d
439 u) (lift h (S (plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0
440 (s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) t2))) (ex2 T (\lambda
441 (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
442 (THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T x
443 (THead k x0 x1))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d))
444 t) x0)).(\lambda (H6: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i
445 d))) t0) x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda
446 (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u
447 (THead k t t0) t3)))) (let H7 \def (eq_ind nat (s k (S (plus i d))) (\lambda
448 (n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x1)) H6 (S (s k (plus i
449 d))) (s_S k (plus i d))) in (let H8 \def (eq_ind nat (s k (plus i d))
450 (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x1)) H7
451 (plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x1
452 (lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2))
453 (ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h (S (plus i d)) t2)))
454 (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x2: T).(\lambda
455 (H9: (eq T x1 (lift h (S (plus (s k i) d)) x2))).(\lambda (H10: (subst0 (s k
456 i) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (S (plus i d))
457 t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T (\lambda (t2: T).(eq T
458 (THead k x0 x1) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
459 (THead k t t0) t2))) (\lambda (x3: T).(\lambda (H11: (eq T x0 (lift h (S
460 (plus i d)) x3))).(\lambda (H12: (subst0 i u t x3)).(eq_ind_r T (lift h (S
461 (plus i d)) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2
462 x1) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0)
463 t3)))) (eq_ind_r T (lift h (S (plus (s k i) d)) x2) (\lambda (t2: T).(ex2 T
464 (\lambda (t3: T).(eq T (THead k (lift h (S (plus i d)) x3) t2) (lift h (S
465 (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (eq_ind
466 nat (s k (plus i d)) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k
467 (lift h (S (plus i d)) x3) (lift h (S n) x2)) (lift h (S (plus i d)) t2)))
468 (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind nat (s k (S (plus
469 i d))) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S
470 (plus i d)) x3) (lift h n x2)) (lift h (S (plus i d)) t2))) (\lambda (t2:
471 T).(subst0 i u (THead k t t0) t2)))) (eq_ind T (lift h (S (plus i d)) (THead
472 k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus
473 i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T
474 (\lambda (t2: T).(eq T (lift h (S (plus i d)) (THead k x3 x2)) (lift h (S
475 (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)) (THead k
476 x3 x2) (refl_equal T (lift h (S (plus i d)) (THead k x3 x2))) (subst0_both u
477 t x3 i H12 k t0 x2 H10)) (THead k (lift h (S (plus i d)) x3) (lift h (s k (S
478 (plus i d))) x2)) (lift_head k x3 x2 h (S (plus i d)))) (S (s k (plus i d)))
479 (s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x1 H9) x0 H11)))) (H x0
480 i h d H5))))) (H0 x1 (s k i) h d H8)))) x H4)))))) H3)) (subst0_gen_head k
481 (lift h d u) (lift h (S (plus i d)) t) (lift h (s k (S (plus i d))) t0) x i
482 H2))))))))))))) t1)).
484 theorem subst0_gen_lift_false:
485 \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall
486 (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u
487 (lift h d t) x) \to (\forall (P: Prop).P)))))))))
489 \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).(\forall (x:
490 T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i
491 (plus d h)) \to ((subst0 i u (lift h d t0) x) \to (\forall (P:
492 Prop).P)))))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (x: T).(\lambda
493 (h: nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (_: (le d i)).(\lambda
494 (_: (lt i (plus d h))).(\lambda (H1: (subst0 i u (lift h d (TSort n))
495 x)).(\lambda (P: Prop).(let H2 \def (eq_ind T (lift h d (TSort n)) (\lambda
496 (t0: T).(subst0 i u t0 x)) H1 (TSort n) (lift_sort n h d)) in
497 (subst0_gen_sort u x i n H2 P)))))))))))) (\lambda (n: nat).(\lambda (u:
498 T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (i:
499 nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d h))).(\lambda (H1:
500 (subst0 i u (lift h d (TLRef n)) x)).(\lambda (P: Prop).(lt_le_e n d P
501 (\lambda (H2: (lt n d)).(let H3 \def (eq_ind T (lift h d (TLRef n)) (\lambda
502 (t0: T).(subst0 i u t0 x)) H1 (TLRef n) (lift_lref_lt n h d H2)) in (land_ind
503 (eq nat n i) (eq T x (lift (S n) O u)) P (\lambda (H4: (eq nat n i)).(\lambda
504 (_: (eq T x (lift (S n) O u))).(let H6 \def (eq_ind nat n (\lambda (n0:
505 nat).(lt n0 d)) H2 i H4) in (le_false d i P H H6)))) (subst0_gen_lref u x i n
506 H3)))) (\lambda (H2: (le d n)).(let H3 \def (eq_ind T (lift h d (TLRef n))
507 (\lambda (t0: T).(subst0 i u t0 x)) H1 (TLRef (plus n h)) (lift_lref_ge n h d
508 H2)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n h)) O u)) P
509 (\lambda (H4: (eq nat (plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n
510 h)) O u))).(let H6 \def (eq_ind_r nat i (\lambda (n0: nat).(lt n0 (plus d
511 h))) H0 (plus n h) H4) in (le_false d n P H2 (lt_le_S n d (simpl_lt_plus_r h
512 n d H6)))))) (subst0_gen_lref u x i (plus n h) H3))))))))))))))) (\lambda (k:
513 K).(\lambda (t0: T).(\lambda (H: ((\forall (u: T).(\forall (x: T).(\forall
514 (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h))
515 \to ((subst0 i u (lift h d t0) x) \to (\forall (P:
516 Prop).P))))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (u: T).(\forall
517 (x: T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to
518 ((lt i (plus d h)) \to ((subst0 i u (lift h d t1) x) \to (\forall (P:
519 Prop).P))))))))))).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
520 (d: nat).(\lambda (i: nat).(\lambda (H1: (le d i)).(\lambda (H2: (lt i (plus
521 d h))).(\lambda (H3: (subst0 i u (lift h d (THead k t0 t1)) x)).(\lambda (P:
522 Prop).(let H4 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2:
523 T).(subst0 i u t2 x)) H3 (THead k (lift h d t0) (lift h (s k d) t1))
524 (lift_head k t0 t1 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k
525 u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2)))
526 (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2:
527 T).(subst0 (s k i) u (lift h (s k d) t1) t2))) (ex3_2 T T (\lambda (u2:
528 T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
529 T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0
530 (s k i) u (lift h (s k d) t1) t2)))) P (\lambda (H5: (ex2 T (\lambda (u2:
531 T).(eq T x (THead k u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u
532 (lift h d t0) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h
533 (s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2)) P (\lambda
534 (x0: T).(\lambda (_: (eq T x (THead k x0 (lift h (s k d) t1)))).(\lambda (H7:
535 (subst0 i u (lift h d t0) x0)).(H u x0 h d i H1 H2 H7 P)))) H5)) (\lambda
536 (H5: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda
537 (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2)))).(ex2_ind T (\lambda (t2:
538 T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: T).(subst0 (s k i) u
539 (lift h (s k d) t1) t2)) P (\lambda (x0: T).(\lambda (_: (eq T x (THead k
540 (lift h d t0) x0))).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t1)
541 x0)).(H0 u x0 h (s k d) (s k i) (s_le k d i H1) (eq_ind nat (s k (plus d h))
542 (\lambda (n: nat).(lt (s k i) n)) (lt_le_S (s k i) (s k (plus d h)) (s_lt k i
543 (plus d h) H2)) (plus (s k d) h) (s_plus k d h)) H7 P)))) H5)) (\lambda (H5:
544 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2))))
545 (\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_:
546 T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))))).(ex3_2_ind
547 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda
548 (u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_:
549 T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))) P (\lambda
550 (x0: T).(\lambda (x1: T).(\lambda (_: (eq T x (THead k x0 x1))).(\lambda (H7:
551 (subst0 i u (lift h d t0) x0)).(\lambda (_: (subst0 (s k i) u (lift h (s k d)
552 t1) x1)).(H u x0 h d i H1 H2 H7 P)))))) H5)) (subst0_gen_head k u (lift h d
553 t0) (lift h (s k d) t1) x i H4))))))))))))))))) t).
555 theorem subst0_gen_lift_ge:
556 \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
557 (h: nat).(\forall (d: nat).((subst0 i u (lift h d t1) x) \to ((le (plus d h)
558 i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
559 T).(subst0 (minus i h) u t1 t2))))))))))
561 \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x:
562 T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h
563 d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d
564 t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)))))))))) (\lambda (n:
565 nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
566 nat).(\lambda (H: (subst0 i u (lift h d (TSort n)) x)).(\lambda (_: (le (plus
567 d h) i)).(let H1 \def (eq_ind T (lift h d (TSort n)) (\lambda (t: T).(subst0
568 i u t x)) H (TSort n) (lift_sort n h d)) in (subst0_gen_sort u x i n H1 (ex2
569 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i
570 h) u (TSort n) t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i:
571 nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i u (lift h d
572 (TLRef n)) x)).(\lambda (H0: (le (plus d h) i)).(lt_le_e n d (ex2 T (\lambda
573 (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef
574 n) t2))) (\lambda (H1: (lt n d)).(let H2 \def (eq_ind T (lift h d (TLRef n))
575 (\lambda (t: T).(subst0 i u t x)) H (TLRef n) (lift_lref_lt n h d H1)) in
576 (land_ind (eq nat n i) (eq T x (lift (S n) O u)) (ex2 T (\lambda (t2: T).(eq
577 T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef n) t2)))
578 (\lambda (H3: (eq nat n i)).(\lambda (_: (eq T x (lift (S n) O u))).(let H5
579 \def (eq_ind nat n (\lambda (n0: nat).(lt n0 d)) H1 i H3) in (le_false (plus
580 d h) i (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
581 T).(subst0 (minus i h) u (TLRef n) t2))) H0 (le_plus_trans (S i) d h H5)))))
582 (subst0_gen_lref u x i n H2)))) (\lambda (H1: (le d n)).(let H2 \def (eq_ind
583 T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t x)) H (TLRef (plus n h))
584 (lift_lref_ge n h d H1)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S
585 (plus n h)) O u)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
586 (t2: T).(subst0 (minus i h) u (TLRef n) t2))) (\lambda (H3: (eq nat (plus n
587 h) i)).(\lambda (H4: (eq T x (lift (S (plus n h)) O u))).(eq_ind nat (plus n
588 h) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2)))
589 (\lambda (t2: T).(subst0 (minus n0 h) u (TLRef n) t2)))) (eq_ind_r T (lift (S
590 (plus n h)) O u) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d
591 t2))) (\lambda (t2: T).(subst0 (minus (plus n h) h) u (TLRef n) t2))))
592 (eq_ind_r nat n (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S
593 (plus n h)) O u) (lift h d t2))) (\lambda (t2: T).(subst0 n0 u (TLRef n)
594 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift (S (plus n h)) O u) (lift h
595 d t2))) (\lambda (t2: T).(subst0 n u (TLRef n) t2)) (lift (S n) O u)
596 (eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t: T).(eq T (lift (S (plus n
597 h)) O u) t)) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(eq T (lift (S n0) O
598 u) (lift (plus h (S n)) O u))) (eq_ind_r nat (plus h (S n)) (\lambda (n0:
599 nat).(eq T (lift n0 O u) (lift (plus h (S n)) O u))) (refl_equal T (lift
600 (plus h (S n)) O u)) (S (plus h n)) (plus_n_Sm h n)) (plus n h) (plus_sym n
601 h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans_plus_r O d
602 (plus O (S n)) (le_plus_plus O O d (S n) (le_n O) (le_S d n H1))) (le_O_n
603 d))) (subst0_lref u n)) (minus (plus n h) h) (minus_plus_r n h)) x H4) i
604 H3))) (subst0_gen_lref u x i (plus n h) H2)))))))))))) (\lambda (k:
605 K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall
606 (h: nat).(\forall (d: nat).((subst0 i u (lift h d t) x) \to ((le (plus d h)
607 i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
608 T).(subst0 (minus i h) u t t2))))))))))).(\lambda (t0: T).(\lambda (H0:
609 ((\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d:
610 nat).((subst0 i u (lift h d t0) x) \to ((le (plus d h) i) \to (ex2 T (\lambda
611 (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t0
612 t2))))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda
613 (d: nat).(\lambda (H1: (subst0 i u (lift h d (THead k t t0)) x)).(\lambda
614 (H2: (le (plus d h) i)).(let H3 \def (eq_ind T (lift h d (THead k t t0))
615 (\lambda (t2: T).(subst0 i u t2 x)) H1 (THead k (lift h d t) (lift h (s k d)
616 t0)) (lift_head k t t0 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x
617 (THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u (lift h d t)
618 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda
619 (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))) (ex3_2 T T (\lambda (u2:
620 T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
621 T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s
622 k i) u (lift h (s k d) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x (lift h d
623 t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda
624 (H4: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k d) t0))))
625 (\lambda (u2: T).(subst0 i u (lift h d t) u2)))).(ex2_ind T (\lambda (u2:
626 T).(eq T x (THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u
627 (lift h d t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
628 (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda
629 (H5: (eq T x (THead k x0 (lift h (s k d) t0)))).(\lambda (H6: (subst0 i u
630 (lift h d t) x0)).(eq_ind_r T (THead k x0 (lift h (s k d) t0)) (\lambda (t2:
631 T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0
632 (minus i h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T x0
633 (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)) (ex2 T (\lambda
634 (t2: T).(eq T (THead k x0 (lift h (s k d) t0)) (lift h d t2))) (\lambda (t2:
635 T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x1: T).(\lambda (H7:
636 (eq T x0 (lift h d x1))).(\lambda (H8: (subst0 (minus i h) u t x1)).(eq_ind_r
637 T (lift h d x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2
638 (lift h (s k d) t0)) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u
639 (THead k t t0) t3)))) (eq_ind T (lift h d (THead k x1 t0)) (\lambda (t2:
640 T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0
641 (minus i h) u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift
642 h d (THead k x1 t0)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u
643 (THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h d (THead k x1 t0)))
644 (subst0_fst u x1 t (minus i h) H8 t0 k)) (THead k (lift h d x1) (lift h (s k
645 d) t0)) (lift_head k x1 t0 h d)) x0 H7)))) (H x0 i h d H6 H2)) x H5)))) H4))
646 (\lambda (H4: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2)))
647 (\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2)))).(ex2_ind T
648 (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0
649 (s k i) u (lift h (s k d) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d
650 t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda
651 (x0: T).(\lambda (H5: (eq T x (THead k (lift h d t) x0))).(\lambda (H6:
652 (subst0 (s k i) u (lift h (s k d) t0) x0)).(eq_ind_r T (THead k (lift h d t)
653 x0) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3)))
654 (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (ex2_ind T
655 (\lambda (t2: T).(eq T x0 (lift h (s k d) t2))) (\lambda (t2: T).(subst0
656 (minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T (THead k (lift h d
657 t) x0) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0)
658 t2))) (\lambda (x1: T).(\lambda (H7: (eq T x0 (lift h (s k d) x1))).(\lambda
659 (H8: (subst0 (minus (s k i) h) u t0 x1)).(eq_ind_r T (lift h (s k d) x1)
660 (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h d t) t2)
661 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3))))
662 (eq_ind T (lift h d (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda (t3:
663 T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t
664 t0) t3)))) (let H9 \def (eq_ind_r nat (minus (s k i) h) (\lambda (n:
665 nat).(subst0 n u t0 x1)) H8 (s k (minus i h)) (s_minus k i h (le_trans_plus_r
666 d h i H2))) in (ex_intro2 T (\lambda (t2: T).(eq T (lift h d (THead k t x1))
667 (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))
668 (THead k t x1) (refl_equal T (lift h d (THead k t x1))) (subst0_snd k u x1 t0
669 (minus i h) H9 t))) (THead k (lift h d t) (lift h (s k d) x1)) (lift_head k t
670 x1 h d)) x0 H7)))) (H0 x0 (s k i) h (s k d) H6 (eq_ind nat (s k (plus d h))
671 (\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i H2) (plus (s k d) h)
672 (s_plus k d h)))) x H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2:
673 T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
674 T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s
675 k i) u (lift h (s k d) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
676 (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i
677 u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) u (lift
678 h (s k d) t0) t2))) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
679 (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda
680 (x1: T).(\lambda (H5: (eq T x (THead k x0 x1))).(\lambda (H6: (subst0 i u
681 (lift h d t) x0)).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t0)
682 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq
683 T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0)
684 t3)))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h (s k d) t2))) (\lambda
685 (t2: T).(subst0 (minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T
686 (THead k x0 x1) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead
687 k t t0) t2))) (\lambda (x2: T).(\lambda (H8: (eq T x1 (lift h (s k d)
688 x2))).(\lambda (H9: (subst0 (minus (s k i) h) u t0 x2)).(ex2_ind T (\lambda
689 (t2: T).(eq T x0 (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t
690 t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h d t2))) (\lambda
691 (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x3: T).(\lambda
692 (H10: (eq T x0 (lift h d x3))).(\lambda (H11: (subst0 (minus i h) u t
693 x3)).(eq_ind_r T (lift h d x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T
694 (THead k t2 x1) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead
695 k t t0) t3)))) (eq_ind_r T (lift h (s k d) x2) (\lambda (t2: T).(ex2 T
696 (\lambda (t3: T).(eq T (THead k (lift h d x3) t2) (lift h d t3))) (\lambda
697 (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind T (lift h d
698 (THead k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d
699 t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (let H12
700 \def (eq_ind_r nat (minus (s k i) h) (\lambda (n: nat).(subst0 n u t0 x2)) H9
701 (s k (minus i h)) (s_minus k i h (le_trans_plus_r d h i H2))) in (ex_intro2 T
702 (\lambda (t2: T).(eq T (lift h d (THead k x3 x2)) (lift h d t2))) (\lambda
703 (t2: T).(subst0 (minus i h) u (THead k t t0) t2)) (THead k x3 x2) (refl_equal
704 T (lift h d (THead k x3 x2))) (subst0_both u t x3 (minus i h) H11 k t0 x2
705 H12))) (THead k (lift h d x3) (lift h (s k d) x2)) (lift_head k x3 x2 h d))
706 x1 H8) x0 H10)))) (H x0 i h d H6 H2))))) (H0 x1 (s k i) h (s k d) H7 (eq_ind
707 nat (s k (plus d h)) (\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i
708 H2) (plus (s k d) h) (s_plus k d h)))) x H5)))))) H4)) (subst0_gen_head k u
709 (lift h d t) (lift h (s k d) t0) x i H3)))))))))))))) t1)).
projects / helm.git / blob