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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 *)
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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/subst0/fwd".
19 include "subst0/defs.ma".
21 include "lift/props.ma".
23 theorem subst0_inv_coq:
24 \forall (i: nat).(\forall (v: T).(\forall (t1: T).(\forall (t2: T).(\forall
25 (P: ((nat \to (T \to (T \to (T \to Prop)))))).((((subst0 i v t1 t2) \to
26 (\forall (v0: T).(\forall (i0: nat).((eq nat i0 i) \to ((eq T v0 v) \to ((eq
27 T (TLRef i0) t1) \to ((eq T (lift (S i0) O v0) t2) \to (P i v t1 t2)))))))))
28 \to ((((subst0 i v t1 t2) \to (\forall (v0: T).(\forall (u2: T).(\forall (u1:
29 T).(\forall (i0: nat).(\forall (t: T).(\forall (k: K).((eq nat i0 i) \to ((eq
30 T v0 v) \to ((eq T (THead k u1 t) t1) \to ((eq T (THead k u2 t) t2) \to
31 ((subst0 i0 v0 u1 u2) \to (P i v t1 t2)))))))))))))) \to ((((subst0 i v t1
32 t2) \to (\forall (k: K).(\forall (v0: T).(\forall (t0: T).(\forall (t3:
33 T).(\forall (i0: nat).(\forall (u: T).((eq nat i0 i) \to ((eq T v0 v) \to
34 ((eq T (THead k u t3) t1) \to ((eq T (THead k u t0) t2) \to ((subst0 (s k i0)
35 v0 t3 t0) \to (P i v t1 t2)))))))))))))) \to ((((subst0 i v t1 t2) \to
36 (\forall (v0: T).(\forall (u1: T).(\forall (u2: T).(\forall (i0:
37 nat).(\forall (k: K).(\forall (t0: T).(\forall (t3: T).((eq nat i0 i) \to
38 ((eq T v0 v) \to ((eq T (THead k u1 t0) t1) \to ((eq T (THead k u2 t3) t2)
39 \to ((subst0 i0 v0 u1 u2) \to ((subst0 (s k i0) v0 t0 t3) \to (P i v t1
40 t2)))))))))))))))) \to ((subst0 i v t1 t2) \to (P i v t1 t2))))))))))
42 \lambda (i: nat).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
43 (P: ((nat \to (T \to (T \to (T \to Prop)))))).(\lambda (H: (((subst0 i v t1
44 t2) \to (\forall (v0: T).(\forall (i0: nat).((eq nat i0 i) \to ((eq T v0 v)
45 \to ((eq T (TLRef i0) t1) \to ((eq T (lift (S i0) O v0) t2) \to (P i v t1
46 t2)))))))))).(\lambda (H0: (((subst0 i v t1 t2) \to (\forall (v0: T).(\forall
47 (u2: T).(\forall (u1: T).(\forall (i0: nat).(\forall (t: T).(\forall (k:
48 K).((eq nat i0 i) \to ((eq T v0 v) \to ((eq T (THead k u1 t) t1) \to ((eq T
49 (THead k u2 t) t2) \to ((subst0 i0 v0 u1 u2) \to (P i v t1
50 t2))))))))))))))).(\lambda (H1: (((subst0 i v t1 t2) \to (\forall (k:
51 K).(\forall (v0: T).(\forall (t0: T).(\forall (t3: T).(\forall (i0:
52 nat).(\forall (u: T).((eq nat i0 i) \to ((eq T v0 v) \to ((eq T (THead k u
53 t3) t1) \to ((eq T (THead k u t0) t2) \to ((subst0 (s k i0) v0 t3 t0) \to (P
54 i v t1 t2))))))))))))))).(\lambda (H2: (((subst0 i v t1 t2) \to (\forall (v0:
55 T).(\forall (u1: T).(\forall (u2: T).(\forall (i0: nat).(\forall (k:
56 K).(\forall (t0: T).(\forall (t3: T).((eq nat i0 i) \to ((eq T v0 v) \to ((eq
57 T (THead k u1 t0) t1) \to ((eq T (THead k u2 t3) t2) \to ((subst0 i0 v0 u1
58 u2) \to ((subst0 (s k i0) v0 t0 t3) \to (P i v t1
59 t2))))))))))))))))).(\lambda (H3: (subst0 i v t1 t2)).(let H4 \def (match H3
60 in subst0 return (\lambda (n: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda
61 (t3: T).(\lambda (_: (subst0 n t t0 t3)).((eq nat n i) \to ((eq T t v) \to
62 ((eq T t0 t1) \to ((eq T t3 t2) \to (P i v t1 t2)))))))))) with [(subst0_lref
63 v0 i0) \Rightarrow (\lambda (H4: (eq nat i0 i)).(\lambda (H5: (eq T v0
64 v)).(\lambda (H6: (eq T (TLRef i0) t1)).(\lambda (H7: (eq T (lift (S i0) O
65 v0) t2)).(H H3 v0 i0 H4 H5 H6 H7))))) | (subst0_fst v0 u2 u1 i0 H4 t k)
66 \Rightarrow (\lambda (H5: (eq nat i0 i)).(\lambda (H6: (eq T v0 v)).(\lambda
67 (H7: (eq T (THead k u1 t) t1)).(\lambda (H8: (eq T (THead k u2 t) t2)).(H0 H3
68 v0 u2 u1 i0 t k H5 H6 H7 H8 H4))))) | (subst0_snd k v0 t0 t3 i0 H4 u)
69 \Rightarrow (\lambda (H5: (eq nat i0 i)).(\lambda (H6: (eq T v0 v)).(\lambda
70 (H7: (eq T (THead k u t3) t1)).(\lambda (H8: (eq T (THead k u t0) t2)).(H1 H3
71 k v0 t0 t3 i0 u H5 H6 H7 H8 H4))))) | (subst0_both v0 u1 u2 i0 H4 k t0 t3 H5)
72 \Rightarrow (\lambda (H6: (eq nat i0 i)).(\lambda (H7: (eq T v0 v)).(\lambda
73 (H8: (eq T (THead k u1 t0) t1)).(\lambda (H9: (eq T (THead k u2 t3) t2)).(H2
74 H3 v0 u1 u2 i0 k t0 t3 H6 H7 H8 H9 H4 H5)))))]) in (H4 (refl_equal nat i)
75 (refl_equal T v) (refl_equal T t1) (refl_equal T t2)))))))))))).
77 theorem subst0_gen_sort:
78 \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0
79 i v (TSort n) x) \to (\forall (P: Prop).P)))))
81 \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
82 (H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(subst0_inv_coq i v (TSort
83 n) x (\lambda (_: nat).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).P))))
84 (\lambda (H0: (subst0 i v (TSort n) x)).(\lambda (v0: T).(\lambda (i0:
85 nat).(\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3:
86 (eq T (TLRef i0) (TSort n))).(\lambda (H4: (eq T (lift (S i0) O v0) x)).(let
87 H5 \def (eq_ind nat i0 (\lambda (n0: nat).(eq T (lift (S n0) O v0) x)) H4 i
88 H1) in (let H6 \def (eq_ind nat i0 (\lambda (n0: nat).(eq T (TLRef n0) (TSort
89 n))) H3 i H1) in (let H7 \def (eq_ind T v0 (\lambda (t: T).(eq T (lift (S i)
90 O t) x)) H5 v H2) in (let H8 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v
91 (TSort n) t)) H0 (lift (S i) O v) H7) in (let H9 \def (eq_ind_r T x (\lambda
92 (t: T).(subst0 i v (TSort n) t)) H (lift (S i) O v) H7) in (let H10 \def
93 (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
94 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
95 (THead _ _ _) \Rightarrow False])) I (TSort n) H6) in (False_ind P
96 H10)))))))))))))) (\lambda (H0: (subst0 i v (TSort n) x)).(\lambda (v0:
97 T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (t:
98 T).(\lambda (k: K).(\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0
99 v)).(\lambda (H3: (eq T (THead k u1 t) (TSort n))).(\lambda (H4: (eq T (THead
100 k u2 t) x)).(\lambda (H5: (subst0 i0 v0 u1 u2)).(let H6 \def (eq_ind nat i0
101 (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H5 i H1) in (let H7 \def (eq_ind T
102 v0 (\lambda (t0: T).(subst0 i t0 u1 u2)) H6 v H2) in (let H8 \def (eq_ind_r T
103 x (\lambda (t0: T).(subst0 i v (TSort n) t0)) H0 (THead k u2 t) H4) in (let
104 H9 \def (eq_ind_r T x (\lambda (t0: T).(subst0 i v (TSort n) t0)) H (THead k
105 u2 t) H4) in (let H10 \def (eq_ind T (THead k u1 t) (\lambda (ee: T).(match
106 ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
107 (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n)
108 H3) in (False_ind P H10)))))))))))))))))) (\lambda (H0: (subst0 i v (TSort n)
109 x)).(\lambda (k: K).(\lambda (v0: T).(\lambda (t0: T).(\lambda (t3:
110 T).(\lambda (i0: nat).(\lambda (u: T).(\lambda (H1: (eq nat i0 i)).(\lambda
111 (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u t3) (TSort n))).(\lambda
112 (H4: (eq T (THead k u t0) x)).(\lambda (H5: (subst0 (s k i0) v0 t3 t0)).(let
113 H6 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 (s k n0) v0 t3 t0)) H5 i
114 H1) in (let H7 \def (eq_ind T v0 (\lambda (t: T).(subst0 (s k i) t t3 t0)) H6
115 v H2) in (let H8 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (TSort n) t))
116 H0 (THead k u t0) H4) in (let H9 \def (eq_ind_r T x (\lambda (t: T).(subst0 i
117 v (TSort n) t)) H (THead k u t0) H4) in (let H10 \def (eq_ind T (THead k u
118 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
119 _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
120 \Rightarrow True])) I (TSort n) H3) in (False_ind P H10))))))))))))))))))
121 (\lambda (H0: (subst0 i v (TSort n) x)).(\lambda (v0: T).(\lambda (u1:
122 T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (k: K).(\lambda (t0:
123 T).(\lambda (t3: T).(\lambda (H2: (eq nat i0 i)).(\lambda (H3: (eq T v0
124 v)).(\lambda (H4: (eq T (THead k u1 t0) (TSort n))).(\lambda (H5: (eq T
125 (THead k u2 t3) x)).(\lambda (H1: (subst0 i0 v0 u1 u2)).(\lambda (H6: (subst0
126 (s k i0) v0 t0 t3)).(let H7 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 (s
127 k n0) v0 t0 t3)) H6 i H2) in (let H8 \def (eq_ind nat i0 (\lambda (n0:
128 nat).(subst0 n0 v0 u1 u2)) H1 i H2) in (let H9 \def (eq_ind T v0 (\lambda (t:
129 T).(subst0 (s k i) t t0 t3)) H7 v H3) in (let H10 \def (eq_ind T v0 (\lambda
130 (t: T).(subst0 i t u1 u2)) H8 v H3) in (let H11 \def (eq_ind_r T x (\lambda
131 (t: T).(subst0 i v (TSort n) t)) H0 (THead k u2 t3) H5) in (let H12 \def
132 (eq_ind_r T x (\lambda (t: T).(subst0 i v (TSort n) t)) H (THead k u2 t3) H5)
133 in (let H13 \def (eq_ind T (THead k u1 t0) (\lambda (ee: T).(match ee in T
134 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
135 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H4) in
136 (False_ind P H13)))))))))))))))))))))) H)))))).
138 theorem subst0_gen_lref:
139 \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0
140 i v (TLRef n) x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))
142 \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
143 (H: (subst0 i v (TLRef n) x)).(subst0_inv_coq i v (TLRef n) x (\lambda (n0:
144 nat).(\lambda (t: T).(\lambda (_: T).(\lambda (t1: T).(land (eq nat n n0) (eq
145 T t1 (lift (S n) O t))))))) (\lambda (H0: (subst0 i v (TLRef n) x)).(\lambda
146 (v0: T).(\lambda (i0: nat).(\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T
147 v0 v)).(\lambda (H3: (eq T (TLRef i0) (TLRef n))).(\lambda (H4: (eq T (lift
148 (S i0) O v0) x)).(let H5 \def (eq_ind nat i0 (\lambda (n0: nat).(eq T (lift
149 (S n0) O v0) x)) H4 i H1) in (let H6 \def (eq_ind nat i0 (\lambda (n0:
150 nat).(eq T (TLRef n0) (TLRef n))) H3 i H1) in (let H7 \def (eq_ind T v0
151 (\lambda (t: T).(eq T (lift (S i) O t) x)) H5 v H2) in (let H8 \def (eq_ind_r
152 T x (\lambda (t: T).(subst0 i v (TLRef n) t)) H0 (lift (S i) O v) H7) in (let
153 H9 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (TLRef n) t)) H (lift (S i)
154 O v) H7) in (eq_ind T (lift (S i) O v) (\lambda (t: T).(land (eq nat n i) (eq
155 T t (lift (S n) O v)))) (let H10 \def (f_equal T nat (\lambda (e: T).(match e
156 in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | (TLRef n0)
157 \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H6) in
158 (let H11 \def (eq_ind_r nat n (\lambda (n0: nat).(subst0 i v (TLRef n0) (lift
159 (S i) O v))) H8 i H10) in (let H12 \def (eq_ind_r nat n (\lambda (n0:
160 nat).(subst0 i v (TLRef n0) (lift (S i) O v))) H9 i H10) in (eq_ind nat i
161 (\lambda (n0: nat).(land (eq nat n0 i) (eq T (lift (S i) O v) (lift (S n0) O
162 v)))) (conj (eq nat i i) (eq T (lift (S i) O v) (lift (S i) O v)) (refl_equal
163 nat i) (refl_equal T (lift (S i) O v))) n H10)))) x H7))))))))))))) (\lambda
164 (H0: (subst0 i v (TLRef n) x)).(\lambda (v0: T).(\lambda (u2: T).(\lambda
165 (u1: T).(\lambda (i0: nat).(\lambda (t: T).(\lambda (k: K).(\lambda (H1: (eq
166 nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u1 t)
167 (TLRef n))).(\lambda (H4: (eq T (THead k u2 t) x)).(\lambda (H5: (subst0 i0
168 v0 u1 u2)).(let H6 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1
169 u2)) H5 i H1) in (let H7 \def (eq_ind T v0 (\lambda (t0: T).(subst0 i t0 u1
170 u2)) H6 v H2) in (let H8 \def (eq_ind_r T x (\lambda (t0: T).(subst0 i v
171 (TLRef n) t0)) H0 (THead k u2 t) H4) in (let H9 \def (eq_ind_r T x (\lambda
172 (t0: T).(subst0 i v (TLRef n) t0)) H (THead k u2 t) H4) in (eq_ind T (THead k
173 u2 t) (\lambda (t0: T).(land (eq nat n i) (eq T t0 (lift (S n) O v)))) (let
174 H10 \def (eq_ind T (THead k u1 t) (\lambda (ee: T).(match ee in T return
175 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
176 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H3) in
177 (False_ind (land (eq nat n i) (eq T (THead k u2 t) (lift (S n) O v))) H10)) x
178 H4))))))))))))))))) (\lambda (H0: (subst0 i v (TLRef n) x)).(\lambda (k:
179 K).(\lambda (v0: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0:
180 nat).(\lambda (u: T).(\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0
181 v)).(\lambda (H3: (eq T (THead k u t3) (TLRef n))).(\lambda (H4: (eq T (THead
182 k u t0) x)).(\lambda (H5: (subst0 (s k i0) v0 t3 t0)).(let H6 \def (eq_ind
183 nat i0 (\lambda (n0: nat).(subst0 (s k n0) v0 t3 t0)) H5 i H1) in (let H7
184 \def (eq_ind T v0 (\lambda (t: T).(subst0 (s k i) t t3 t0)) H6 v H2) in (let
185 H8 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (TLRef n) t)) H0 (THead k u
186 t0) H4) in (let H9 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (TLRef n)
187 t)) H (THead k u t0) H4) in (eq_ind T (THead k u t0) (\lambda (t: T).(land
188 (eq nat n i) (eq T t (lift (S n) O v)))) (let H10 \def (eq_ind T (THead k u
189 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
190 _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
191 \Rightarrow True])) I (TLRef n) H3) in (False_ind (land (eq nat n i) (eq T
192 (THead k u t0) (lift (S n) O v))) H10)) x H4))))))))))))))))) (\lambda (H0:
193 (subst0 i v (TLRef n) x)).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2:
194 T).(\lambda (i0: nat).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3:
195 T).(\lambda (H2: (eq nat i0 i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq
196 T (THead k u1 t0) (TLRef n))).(\lambda (H5: (eq T (THead k u2 t3)
197 x)).(\lambda (H1: (subst0 i0 v0 u1 u2)).(\lambda (H6: (subst0 (s k i0) v0 t0
198 t3)).(let H7 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 (s k n0) v0 t0
199 t3)) H6 i H2) in (let H8 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 n0 v0
200 u1 u2)) H1 i H2) in (let H9 \def (eq_ind T v0 (\lambda (t: T).(subst0 (s k i)
201 t t0 t3)) H7 v H3) in (let H10 \def (eq_ind T v0 (\lambda (t: T).(subst0 i t
202 u1 u2)) H8 v H3) in (let H11 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v
203 (TLRef n) t)) H0 (THead k u2 t3) H5) in (let H12 \def (eq_ind_r T x (\lambda
204 (t: T).(subst0 i v (TLRef n) t)) H (THead k u2 t3) H5) in (eq_ind T (THead k
205 u2 t3) (\lambda (t: T).(land (eq nat n i) (eq T t (lift (S n) O v)))) (let
206 H13 \def (eq_ind T (THead k u1 t0) (\lambda (ee: T).(match ee in T return
207 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
208 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H4) in
209 (False_ind (land (eq nat n i) (eq T (THead k u2 t3) (lift (S n) O v))) H13))
210 x H5))))))))))))))))))))) H))))).
212 theorem subst0_gen_head:
213 \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall
214 (x: T).(\forall (i: nat).((subst0 i v (THead k u1 t1) x) \to (or3 (ex2 T
215 (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
216 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2:
217 T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
218 T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1
219 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))))
221 \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda
222 (x: T).(\lambda (i: nat).(\lambda (H: (subst0 i v (THead k u1 t1)
223 x)).(subst0_inv_coq i v (THead k u1 t1) x (\lambda (n: nat).(\lambda (t:
224 T).(\lambda (_: T).(\lambda (t2: T).(or3 (ex2 T (\lambda (u2: T).(eq T t2
225 (THead k u2 t1))) (\lambda (u2: T).(subst0 n t u1 u2))) (ex2 T (\lambda (t3:
226 T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k n) t t1 t3)))
227 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3))))
228 (\lambda (u2: T).(\lambda (_: T).(subst0 n t u1 u2))) (\lambda (_:
229 T).(\lambda (t3: T).(subst0 (s k n) t t1 t3))))))))) (\lambda (H0: (subst0 i
230 v (THead k u1 t1) x)).(\lambda (v0: T).(\lambda (i0: nat).(\lambda (H1: (eq
231 nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (TLRef i0) (THead k
232 u1 t1))).(\lambda (H4: (eq T (lift (S i0) O v0) x)).(let H5 \def (eq_ind nat
233 i0 (\lambda (n: nat).(eq T (lift (S n) O v0) x)) H4 i H1) in (let H6 \def
234 (eq_ind nat i0 (\lambda (n: nat).(eq T (TLRef n) (THead k u1 t1))) H3 i H1)
235 in (let H7 \def (eq_ind T v0 (\lambda (t: T).(eq T (lift (S i) O t) x)) H5 v
236 H2) in (let H8 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (THead k u1 t1)
237 t)) H0 (lift (S i) O v) H7) in (let H9 \def (eq_ind_r T x (\lambda (t:
238 T).(subst0 i v (THead k u1 t1) t)) H (lift (S i) O v) H7) in (eq_ind T (lift
239 (S i) O v) (\lambda (t: T).(or3 (ex2 T (\lambda (u2: T).(eq T t (THead k u2
240 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T t
241 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T
242 (\lambda (u2: T).(\lambda (t2: T).(eq T t (THead k u2 t2)))) (\lambda (u2:
243 T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2:
244 T).(subst0 (s k i) v t1 t2)))))) (let H10 \def (eq_ind T (TLRef i) (\lambda
245 (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
246 \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
247 False])) I (THead k u1 t1) H6) in (False_ind (or3 (ex2 T (\lambda (u2: T).(eq
248 T (lift (S i) O v) (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)))
249 (ex2 T (\lambda (t2: T).(eq T (lift (S i) O v) (THead k u1 t2))) (\lambda
250 (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
251 T).(eq T (lift (S i) O v) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
252 T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1
253 t2))))) H10)) x H7))))))))))))) (\lambda (H0: (subst0 i v (THead k u1 t1)
254 x)).(\lambda (v0: T).(\lambda (u2: T).(\lambda (u0: T).(\lambda (i0:
255 nat).(\lambda (t: T).(\lambda (k0: K).(\lambda (H1: (eq nat i0 i)).(\lambda
256 (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k0 u0 t) (THead k u1
257 t1))).(\lambda (H4: (eq T (THead k0 u2 t) x)).(\lambda (H5: (subst0 i0 v0 u0
258 u2)).(let H6 \def (eq_ind nat i0 (\lambda (n: nat).(subst0 n v0 u0 u2)) H5 i
259 H1) in (let H7 \def (eq_ind T v0 (\lambda (t0: T).(subst0 i t0 u0 u2)) H6 v
260 H2) in (let H8 \def (eq_ind_r T x (\lambda (t0: T).(subst0 i v (THead k u1
261 t1) t0)) H0 (THead k0 u2 t) H4) in (let H9 \def (eq_ind_r T x (\lambda (t0:
262 T).(subst0 i v (THead k u1 t1) t0)) H (THead k0 u2 t) H4) in (eq_ind T (THead
263 k0 u2 t) (\lambda (t0: T).(or3 (ex2 T (\lambda (u3: T).(eq T t0 (THead k u3
264 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T t0
265 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T
266 (\lambda (u3: T).(\lambda (t2: T).(eq T t0 (THead k u3 t2)))) (\lambda (u3:
267 T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2:
268 T).(subst0 (s k i) v t1 t2)))))) (let H10 \def (f_equal T K (\lambda (e:
269 T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 |
270 (TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u0 t)
271 (THead k u1 t1) H3) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e
272 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
273 \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead k0 u0 t) (THead k u1
274 t1) H3) in ((let H12 \def (f_equal T T (\lambda (e: T).(match e in T return
275 (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t |
276 (THead _ _ t0) \Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in
277 (\lambda (H13: (eq T u0 u1)).(\lambda (H14: (eq K k0 k)).(let H15 \def
278 (eq_ind K k0 (\lambda (k1: K).(subst0 i v (THead k u1 t1) (THead k1 u2 t)))
279 H9 k H14) in (let H16 \def (eq_ind K k0 (\lambda (k1: K).(subst0 i v (THead k
280 u1 t1) (THead k1 u2 t))) H8 k H14) in (eq_ind_r K k (\lambda (k1: K).(or3
281 (ex2 T (\lambda (u3: T).(eq T (THead k1 u2 t) (THead k u3 t1))) (\lambda (u3:
282 T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1 u2 t) (THead
283 k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda
284 (u3: T).(\lambda (t2: T).(eq T (THead k1 u2 t) (THead k u3 t2)))) (\lambda
285 (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2:
286 T).(subst0 (s k i) v t1 t2)))))) (let H17 \def (eq_ind T t (\lambda (t0:
287 T).(subst0 i v (THead k u1 t1) (THead k u2 t0))) H15 t1 H12) in (let H18 \def
288 (eq_ind T t (\lambda (t0: T).(subst0 i v (THead k u1 t1) (THead k u2 t0)))
289 H16 t1 H12) in (eq_ind_r T t1 (\lambda (t0: T).(or3 (ex2 T (\lambda (u3:
290 T).(eq T (THead k u2 t0) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1
291 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k u2 t0) (THead k u1 t2)))
292 (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3:
293 T).(\lambda (t2: T).(eq T (THead k u2 t0) (THead k u3 t2)))) (\lambda (u3:
294 T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2:
295 T).(subst0 (s k i) v t1 t2)))))) (let H19 \def (eq_ind T u0 (\lambda (t0:
296 T).(subst0 i v t0 u2)) H7 u1 H13) in (or3_intro0 (ex2 T (\lambda (u3: T).(eq
297 T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3)))
298 (ex2 T (\lambda (t2: T).(eq T (THead k u2 t1) (THead k u1 t2))) (\lambda (t2:
299 T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2:
300 T).(eq T (THead k u2 t1) (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_:
301 T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1
302 t2)))) (ex_intro2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 t1)))
303 (\lambda (u3: T).(subst0 i v u1 u3)) u2 (refl_equal T (THead k u2 t1)) H19)))
304 t H12))) k0 H14)))))) H11)) H10)) x H4))))))))))))))))) (\lambda (H0: (subst0
305 i v (THead k u1 t1) x)).(\lambda (k0: K).(\lambda (v0: T).(\lambda (t0:
306 T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (u: T).(\lambda (H1: (eq nat
307 i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k0 u t3) (THead
308 k u1 t1))).(\lambda (H4: (eq T (THead k0 u t0) x)).(\lambda (H5: (subst0 (s
309 k0 i0) v0 t3 t0)).(let H6 \def (eq_ind nat i0 (\lambda (n: nat).(subst0 (s k0
310 n) v0 t3 t0)) H5 i H1) in (let H7 \def (eq_ind T v0 (\lambda (t: T).(subst0
311 (s k0 i) t t3 t0)) H6 v H2) in (let H8 \def (eq_ind_r T x (\lambda (t:
312 T).(subst0 i v (THead k u1 t1) t)) H0 (THead k0 u t0) H4) in (let H9 \def
313 (eq_ind_r T x (\lambda (t: T).(subst0 i v (THead k u1 t1) t)) H (THead k0 u
314 t0) H4) in (eq_ind T (THead k0 u t0) (\lambda (t: T).(or3 (ex2 T (\lambda
315 (u2: T).(eq T t (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2
316 T (\lambda (t2: T).(eq T t (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i)
317 v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t (THead k u2
318 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_:
319 T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))) (let H10 \def (f_equal T K
320 (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
321 \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1]))
322 (THead k0 u t3) (THead k u1 t1) H3) in ((let H11 \def (f_equal T T (\lambda
323 (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u
324 | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead k0 u t3)
325 (THead k u1 t1) H3) in ((let H12 \def (f_equal T T (\lambda (e: T).(match e
326 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
327 \Rightarrow t3 | (THead _ _ t) \Rightarrow t])) (THead k0 u t3) (THead k u1
328 t1) H3) in (\lambda (H13: (eq T u u1)).(\lambda (H14: (eq K k0 k)).(let H15
329 \def (eq_ind T u (\lambda (t: T).(subst0 i v (THead k u1 t1) (THead k0 t
330 t0))) H9 u1 H13) in (let H16 \def (eq_ind T u (\lambda (t: T).(subst0 i v
331 (THead k u1 t1) (THead k0 t t0))) H8 u1 H13) in (eq_ind_r T u1 (\lambda (t:
332 T).(or3 (ex2 T (\lambda (u2: T).(eq T (THead k0 t t0) (THead k u2 t1)))
333 (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k0
334 t t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T
335 T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k0 t t0) (THead k u2 t2))))
336 (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_:
337 T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))) (let H17 \def (eq_ind T t3
338 (\lambda (t: T).(subst0 (s k0 i) v t t0)) H7 t1 H12) in (let H18 \def (eq_ind
339 K k0 (\lambda (k1: K).(subst0 i v (THead k u1 t1) (THead k1 u1 t0))) H15 k
340 H14) in (let H19 \def (eq_ind K k0 (\lambda (k1: K).(subst0 i v (THead k u1
341 t1) (THead k1 u1 t0))) H16 k H14) in (let H20 \def (eq_ind K k0 (\lambda (k1:
342 K).(subst0 (s k1 i) v t1 t0)) H17 k H14) in (eq_ind_r K k (\lambda (k1:
343 K).(or3 (ex2 T (\lambda (u2: T).(eq T (THead k1 u1 t0) (THead k u2 t1)))
344 (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k1
345 u1 t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T
346 T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k1 u1 t0) (THead k u2 t2))))
347 (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_:
348 T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))) (or3_intro1 (ex2 T (\lambda
349 (u2: T).(eq T (THead k u1 t0) (THead k u2 t1))) (\lambda (u2: T).(subst0 i v
350 u1 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k u1 t0) (THead k u1 t2)))
351 (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2:
352 T).(\lambda (t2: T).(eq T (THead k u1 t0) (THead k u2 t2)))) (\lambda (u2:
353 T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2:
354 T).(subst0 (s k i) v t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (THead k
355 u1 t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2)) t0
356 (refl_equal T (THead k u1 t0)) H20)) k0 H14))))) u H13)))))) H11)) H10)) x
357 H4))))))))))))))))) (\lambda (H0: (subst0 i v (THead k u1 t1) x)).(\lambda
358 (v0: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (k0:
359 K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H2: (eq nat i0 i)).(\lambda
360 (H3: (eq T v0 v)).(\lambda (H4: (eq T (THead k0 u0 t0) (THead k u1
361 t1))).(\lambda (H5: (eq T (THead k0 u2 t3) x)).(\lambda (H1: (subst0 i0 v0 u0
362 u2)).(\lambda (H6: (subst0 (s k0 i0) v0 t0 t3)).(let H7 \def (eq_ind nat i0
363 (\lambda (n: nat).(subst0 (s k0 n) v0 t0 t3)) H6 i H2) in (let H8 \def
364 (eq_ind nat i0 (\lambda (n: nat).(subst0 n v0 u0 u2)) H1 i H2) in (let H9
365 \def (eq_ind T v0 (\lambda (t: T).(subst0 (s k0 i) t t0 t3)) H7 v H3) in (let
366 H10 \def (eq_ind T v0 (\lambda (t: T).(subst0 i t u0 u2)) H8 v H3) in (let
367 H11 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (THead k u1 t1) t)) H0
368 (THead k0 u2 t3) H5) in (let H12 \def (eq_ind_r T x (\lambda (t: T).(subst0 i
369 v (THead k u1 t1) t)) H (THead k0 u2 t3) H5) in (eq_ind T (THead k0 u2 t3)
370 (\lambda (t: T).(or3 (ex2 T (\lambda (u3: T).(eq T t (THead k u3 t1)))
371 (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T t (THead
372 k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda
373 (u3: T).(\lambda (t2: T).(eq T t (THead k u3 t2)))) (\lambda (u3: T).(\lambda
374 (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i)
375 v t1 t2)))))) (let H13 \def (f_equal T K (\lambda (e: T).(match e in T return
376 (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0
377 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u0 t0) (THead k u1 t1) H4) in
378 ((let H14 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
379 T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t
380 _) \Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H4) in ((let H15 \def
381 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
382 [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
383 \Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H4) in (\lambda (H16: (eq T
384 u0 u1)).(\lambda (H17: (eq K k0 k)).(let H18 \def (eq_ind T t0 (\lambda (t:
385 T).(subst0 (s k0 i) v t t3)) H9 t1 H15) in (let H19 \def (eq_ind K k0
386 (\lambda (k1: K).(subst0 i v (THead k u1 t1) (THead k1 u2 t3))) H12 k H17) in
387 (let H20 \def (eq_ind K k0 (\lambda (k1: K).(subst0 i v (THead k u1 t1)
388 (THead k1 u2 t3))) H11 k H17) in (let H21 \def (eq_ind K k0 (\lambda (k1:
389 K).(subst0 (s k1 i) v t1 t3)) H18 k H17) in (eq_ind_r K k (\lambda (k1:
390 K).(or3 (ex2 T (\lambda (u3: T).(eq T (THead k1 u2 t3) (THead k u3 t1)))
391 (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1
392 u2 t3) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T
393 T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k1 u2 t3) (THead k u3 t2))))
394 (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_:
395 T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))) (let H22 \def (eq_ind T u0
396 (\lambda (t: T).(subst0 i v t u2)) H10 u1 H16) in (or3_intro2 (ex2 T (\lambda
397 (u3: T).(eq T (THead k u2 t3) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v
398 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k u2 t3) (THead k u1 t2)))
399 (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3:
400 T).(\lambda (t2: T).(eq T (THead k u2 t3) (THead k u3 t2)))) (\lambda (u3:
401 T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2:
402 T).(subst0 (s k i) v t1 t2)))) (ex3_2_intro T T (\lambda (u3: T).(\lambda
403 (t2: T).(eq T (THead k u2 t3) (THead k u3 t2)))) (\lambda (u3: T).(\lambda
404 (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i)
405 v t1 t2))) u2 t3 (refl_equal T (THead k u2 t3)) H22 H21))) k0 H17))))))))
406 H14)) H13)) x H5))))))))))))))))))))) H))))))).
408 theorem subst0_gen_lift_lt:
409 \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
410 (h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t1)
411 x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
412 (t2: T).(subst0 i u t1 t2)))))))))
414 \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x:
415 T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift h d
416 u) (lift h (S (plus i d)) t) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h
417 (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2))))))))) (\lambda (n:
418 nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
419 nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S (plus i d)) (TSort n))
420 x)).(let H0 \def (eq_ind T (lift h (S (plus i d)) (TSort n)) (\lambda (t:
421 T).(subst0 i (lift h d u) t x)) H (TSort n) (lift_sort n h (S (plus i d))))
422 in (subst0_gen_sort (lift h d u) x i n H0 (ex2 T (\lambda (t2: T).(eq T x
423 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TSort n)
424 t2))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: nat).(\lambda
425 (h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S
426 (plus i d)) (TLRef n)) x)).(lt_le_e n (S (plus i d)) (ex2 T (\lambda (t2:
427 T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef
428 n) t2))) (\lambda (H0: (lt n (S (plus i d)))).(let H1 \def (eq_ind T (lift h
429 (S (plus i d)) (TLRef n)) (\lambda (t: T).(subst0 i (lift h d u) t x)) H
430 (TLRef n) (lift_lref_lt n h (S (plus i d)) H0)) in (and_ind (eq nat n i) (eq
431 T x (lift (S n) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S
432 (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2:
433 (eq nat n i)).(\lambda (H3: (eq T x (lift (S n) O (lift h d u)))).(eq_ind_r T
434 (lift (S n) O (lift h d u)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t
435 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))))
436 (eq_ind_r nat i (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S n0)
437 O (lift h d u)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
438 (TLRef n0) t2)))) (eq_ind T (lift h (plus (S i) d) (lift (S i) O u)) (\lambda
439 (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) (\lambda
440 (t2: T).(subst0 i u (TLRef i) t2)))) (ex_intro2 T (\lambda (t2: T).(eq T
441 (lift h (S (plus i d)) (lift (S i) O u)) (lift h (S (plus i d)) t2)))
442 (\lambda (t2: T).(subst0 i u (TLRef i) t2)) (lift (S i) O u) (refl_equal T
443 (lift h (S (plus i d)) (lift (S i) O u))) (subst0_lref u i)) (lift (S i) O
444 (lift h d u)) (lift_d u h (S i) d O (le_O_n d))) n H2) x H3)))
445 (subst0_gen_lref (lift h d u) x i n H1)))) (\lambda (H0: (le (S (plus i d))
446 n)).(let H1 \def (eq_ind T (lift h (S (plus i d)) (TLRef n)) (\lambda (t:
447 T).(subst0 i (lift h d u) t x)) H (TLRef (plus n h)) (lift_lref_ge n h (S
448 (plus i d)) H0)) in (and_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n
449 h)) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
450 t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: (eq nat
451 (plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n h)) O (lift h d
452 u)))).(let H4 \def (eq_ind_r nat i (\lambda (n0: nat).(le (S (plus n0 d)) n))
453 H0 (plus n h) H2) in (le_false n (plus (plus n h) d) (ex2 T (\lambda (t2:
454 T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef
455 n) t2))) (le_plus_trans n (plus n h) d (le_plus_l n h)) H4))))
456 (subst0_gen_lref (lift h d u) x i (plus n h) H1))))))))))) (\lambda (k:
457 K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall
458 (h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t)
459 x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
460 (t2: T).(subst0 i u t t2)))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall
461 (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift
462 h d u) (lift h (S (plus i d)) t0) x) \to (ex2 T (\lambda (t2: T).(eq T x
463 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t0
464 t2)))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
465 nat).(\lambda (H1: (subst0 i (lift h d u) (lift h (S (plus i d)) (THead k t
466 t0)) x)).(let H2 \def (eq_ind T (lift h (S (plus i d)) (THead k t t0))
467 (\lambda (t2: T).(subst0 i (lift h d u) t2 x)) H1 (THead k (lift h (S (plus i
468 d)) t) (lift h (s k (S (plus i d))) t0)) (lift_head k t t0 h (S (plus i d))))
469 in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k (S (plus
470 i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S (plus i d))
471 t) u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t)
472 t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i
473 d))) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k
474 u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S
475 (plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h
476 d u) (lift h (s k (S (plus i d))) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x
477 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0)
478 t2))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k
479 (S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S
480 (plus i d)) t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h
481 (s k (S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h
482 (S (plus i d)) t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
483 t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0:
484 T).(\lambda (H4: (eq T x (THead k x0 (lift h (s k (S (plus i d)))
485 t0)))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) t)
486 x0)).(eq_ind_r T (THead k x0 (lift h (s k (S (plus i d))) t0)) (\lambda (t2:
487 T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda
488 (t3: T).(subst0 i u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T
489 x0 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T
490 (\lambda (t2: T).(eq T (THead k x0 (lift h (s k (S (plus i d))) t0)) (lift h
491 (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))
492 (\lambda (x1: T).(\lambda (H6: (eq T x0 (lift h (S (plus i d)) x1))).(\lambda
493 (H7: (subst0 i u t x1)).(eq_ind_r T (lift h (S (plus i d)) x1) (\lambda (t2:
494 T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k (S (plus i d)))
495 t0)) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0)
496 t3)))) (eq_ind T (lift h (S (plus i d)) (THead k x1 t0)) (\lambda (t2:
497 T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda
498 (t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T
499 (lift h (S (plus i d)) (THead k x1 t0)) (lift h (S (plus i d)) t2))) (\lambda
500 (t2: T).(subst0 i u (THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h
501 (S (plus i d)) (THead k x1 t0))) (subst0_fst u x1 t i H7 t0 k)) (THead k
502 (lift h (S (plus i d)) x1) (lift h (s k (S (plus i d))) t0)) (lift_head k x1
503 t0 h (S (plus i d)))) x0 H6)))) (H x0 i h d H5)) x H4)))) H3)) (\lambda (H3:
504 (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) t2)))
505 (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d)))
506 t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i
507 d)) t) t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S
508 (plus i d))) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
509 t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0:
510 T).(\lambda (H4: (eq T x (THead k (lift h (S (plus i d)) t) x0))).(\lambda
511 (H5: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) t0)
512 x0)).(eq_ind_r T (THead k (lift h (S (plus i d)) t) x0) (\lambda (t2: T).(ex2
513 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3:
514 T).(subst0 i u (THead k t t0) t3)))) (let H6 \def (eq_ind nat (s k (S (plus i
515 d))) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x0)) H5 (S
516 (s k (plus i d))) (s_S k (plus i d))) in (let H7 \def (eq_ind nat (s k (plus
517 i d)) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x0))
518 H6 (plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x0
519 (lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2))
520 (ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) x0) (lift h
521 (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))
522 (\lambda (x1: T).(\lambda (H8: (eq T x0 (lift h (S (plus (s k i) d))
523 x1))).(\lambda (H9: (subst0 (s k i) u t0 x1)).(eq_ind_r T (lift h (S (plus (s
524 k i) d)) x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h
525 (S (plus i d)) t) t2) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i
526 u (THead k t t0) t3)))) (eq_ind nat (s k (plus i d)) (\lambda (n: nat).(ex2 T
527 (\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h (S n) x1))
528 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0)
529 t2)))) (eq_ind nat (s k (S (plus i d))) (\lambda (n: nat).(ex2 T (\lambda
530 (t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h n x1)) (lift h (S
531 (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind
532 T (lift h (S (plus i d)) (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda
533 (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u
534 (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift h (S (plus i
535 d)) (THead k t x1)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
536 (THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h (S (plus i d))
537 (THead k t x1))) (subst0_snd k u x1 t0 i H9 t)) (THead k (lift h (S (plus i
538 d)) t) (lift h (s k (S (plus i d))) x1)) (lift_head k t x1 h (S (plus i d))))
539 (S (s k (plus i d))) (s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x0
540 H8)))) (H0 x0 (s k i) h d H7)))) x H4)))) H3)) (\lambda (H3: (ex3_2 T T
541 (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2:
542 T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S (plus i d)) t) u2)))
543 (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S
544 (plus i d))) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq
545 T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d
546 u) (lift h (S (plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0
547 (s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) t2))) (ex2 T (\lambda
548 (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
549 (THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T x
550 (THead k x0 x1))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d))
551 t) x0)).(\lambda (H6: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i
552 d))) t0) x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda
553 (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u
554 (THead k t t0) t3)))) (let H7 \def (eq_ind nat (s k (S (plus i d))) (\lambda
555 (n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x1)) H6 (S (s k (plus i
556 d))) (s_S k (plus i d))) in (let H8 \def (eq_ind nat (s k (plus i d))
557 (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x1)) H7
558 (plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x1
559 (lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2))
560 (ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h (S (plus i d)) t2)))
561 (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x2: T).(\lambda
562 (H9: (eq T x1 (lift h (S (plus (s k i) d)) x2))).(\lambda (H10: (subst0 (s k
563 i) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (S (plus i d))
564 t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T (\lambda (t2: T).(eq T
565 (THead k x0 x1) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
566 (THead k t t0) t2))) (\lambda (x3: T).(\lambda (H11: (eq T x0 (lift h (S
567 (plus i d)) x3))).(\lambda (H12: (subst0 i u t x3)).(eq_ind_r T (lift h (S
568 (plus i d)) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2
569 x1) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0)
570 t3)))) (eq_ind_r T (lift h (S (plus (s k i) d)) x2) (\lambda (t2: T).(ex2 T
571 (\lambda (t3: T).(eq T (THead k (lift h (S (plus i d)) x3) t2) (lift h (S
572 (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (eq_ind
573 nat (s k (plus i d)) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k
574 (lift h (S (plus i d)) x3) (lift h (S n) x2)) (lift h (S (plus i d)) t2)))
575 (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind nat (s k (S (plus
576 i d))) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S
577 (plus i d)) x3) (lift h n x2)) (lift h (S (plus i d)) t2))) (\lambda (t2:
578 T).(subst0 i u (THead k t t0) t2)))) (eq_ind T (lift h (S (plus i d)) (THead
579 k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus
580 i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T
581 (\lambda (t2: T).(eq T (lift h (S (plus i d)) (THead k x3 x2)) (lift h (S
582 (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)) (THead k
583 x3 x2) (refl_equal T (lift h (S (plus i d)) (THead k x3 x2))) (subst0_both u
584 t x3 i H12 k t0 x2 H10)) (THead k (lift h (S (plus i d)) x3) (lift h (s k (S
585 (plus i d))) x2)) (lift_head k x3 x2 h (S (plus i d)))) (S (s k (plus i d)))
586 (s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x1 H9) x0 H11)))) (H x0
587 i h d H5))))) (H0 x1 (s k i) h d H8)))) x H4)))))) H3)) (subst0_gen_head k
588 (lift h d u) (lift h (S (plus i d)) t) (lift h (s k (S (plus i d))) t0) x i
589 H2))))))))))))) t1)).
591 theorem subst0_gen_lift_false:
592 \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall
593 (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u
594 (lift h d t) x) \to (\forall (P: Prop).P)))))))))
596 \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).(\forall (x:
597 T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i
598 (plus d h)) \to ((subst0 i u (lift h d t0) x) \to (\forall (P:
599 Prop).P)))))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (x: T).(\lambda
600 (h: nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (_: (le d i)).(\lambda
601 (_: (lt i (plus d h))).(\lambda (H1: (subst0 i u (lift h d (TSort n))
602 x)).(\lambda (P: Prop).(let H2 \def (eq_ind T (lift h d (TSort n)) (\lambda
603 (t0: T).(subst0 i u t0 x)) H1 (TSort n) (lift_sort n h d)) in
604 (subst0_gen_sort u x i n H2 P)))))))))))) (\lambda (n: nat).(\lambda (u:
605 T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (i:
606 nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d h))).(\lambda (H1:
607 (subst0 i u (lift h d (TLRef n)) x)).(\lambda (P: Prop).(lt_le_e n d P
608 (\lambda (H2: (lt n d)).(let H3 \def (eq_ind T (lift h d (TLRef n)) (\lambda
609 (t0: T).(subst0 i u t0 x)) H1 (TLRef n) (lift_lref_lt n h d H2)) in (and_ind
610 (eq nat n i) (eq T x (lift (S n) O u)) P (\lambda (H4: (eq nat n i)).(\lambda
611 (_: (eq T x (lift (S n) O u))).(let H6 \def (eq_ind nat n (\lambda (n0:
612 nat).(lt n0 d)) H2 i H4) in (le_false d i P H H6)))) (subst0_gen_lref u x i n
613 H3)))) (\lambda (H2: (le d n)).(let H3 \def (eq_ind T (lift h d (TLRef n))
614 (\lambda (t0: T).(subst0 i u t0 x)) H1 (TLRef (plus n h)) (lift_lref_ge n h d
615 H2)) in (and_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n h)) O u)) P
616 (\lambda (H4: (eq nat (plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n
617 h)) O u))).(let H6 \def (eq_ind_r nat i (\lambda (n0: nat).(lt n0 (plus d
618 h))) H0 (plus n h) H4) in (le_false d n P H2 (lt_le_S n d (simpl_lt_plus_r h
619 n d H6)))))) (subst0_gen_lref u x i (plus n h) H3))))))))))))))) (\lambda (k:
620 K).(\lambda (t0: T).(\lambda (H: ((\forall (u: T).(\forall (x: T).(\forall
621 (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h))
622 \to ((subst0 i u (lift h d t0) x) \to (\forall (P:
623 Prop).P))))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (u: T).(\forall
624 (x: T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to
625 ((lt i (plus d h)) \to ((subst0 i u (lift h d t1) x) \to (\forall (P:
626 Prop).P))))))))))).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
627 (d: nat).(\lambda (i: nat).(\lambda (H1: (le d i)).(\lambda (H2: (lt i (plus
628 d h))).(\lambda (H3: (subst0 i u (lift h d (THead k t0 t1)) x)).(\lambda (P:
629 Prop).(let H4 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2:
630 T).(subst0 i u t2 x)) H3 (THead k (lift h d t0) (lift h (s k d) t1))
631 (lift_head k t0 t1 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k
632 u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2)))
633 (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2:
634 T).(subst0 (s k i) u (lift h (s k d) t1) t2))) (ex3_2 T T (\lambda (u2:
635 T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
636 T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0
637 (s k i) u (lift h (s k d) t1) t2)))) P (\lambda (H5: (ex2 T (\lambda (u2:
638 T).(eq T x (THead k u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u
639 (lift h d t0) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h
640 (s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2)) P (\lambda
641 (x0: T).(\lambda (_: (eq T x (THead k x0 (lift h (s k d) t1)))).(\lambda (H7:
642 (subst0 i u (lift h d t0) x0)).(H u x0 h d i H1 H2 H7 P)))) H5)) (\lambda
643 (H5: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda
644 (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2)))).(ex2_ind T (\lambda (t2:
645 T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: T).(subst0 (s k i) u
646 (lift h (s k d) t1) t2)) P (\lambda (x0: T).(\lambda (_: (eq T x (THead k
647 (lift h d t0) x0))).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t1)
648 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))
649 (\lambda (n: nat).(lt (s k i) n)) (lt_le_S (s k i) (s k (plus d h)) (s_lt k i
650 (plus d h) H2)) (plus (s k d) h) (s_plus k d h)) H7 P)))) H5)) (\lambda (H5:
651 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2))))
652 (\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_:
653 T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))))).(ex3_2_ind
654 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda
655 (u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_:
656 T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))) P (\lambda
657 (x0: T).(\lambda (x1: T).(\lambda (_: (eq T x (THead k x0 x1))).(\lambda (H7:
658 (subst0 i u (lift h d t0) x0)).(\lambda (_: (subst0 (s k i) u (lift h (s k d)
659 t1) x1)).(H u x0 h d i H1 H2 H7 P)))))) H5)) (subst0_gen_head k u (lift h d
660 t0) (lift h (s k d) t1) x i H4))))))))))))))))) t).
662 theorem subst0_gen_lift_ge:
663 \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
664 (h: nat).(\forall (d: nat).((subst0 i u (lift h d t1) x) \to ((le (plus d h)
665 i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
666 T).(subst0 (minus i h) u t1 t2))))))))))
668 \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x:
669 T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h
670 d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d
671 t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)))))))))) (\lambda (n:
672 nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
673 nat).(\lambda (H: (subst0 i u (lift h d (TSort n)) x)).(\lambda (_: (le (plus
674 d h) i)).(let H1 \def (eq_ind T (lift h d (TSort n)) (\lambda (t: T).(subst0
675 i u t x)) H (TSort n) (lift_sort n h d)) in (subst0_gen_sort u x i n H1 (ex2
676 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i
677 h) u (TSort n) t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i:
678 nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i u (lift h d
679 (TLRef n)) x)).(\lambda (H0: (le (plus d h) i)).(lt_le_e n d (ex2 T (\lambda
680 (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef
681 n) t2))) (\lambda (H1: (lt n d)).(let H2 \def (eq_ind T (lift h d (TLRef n))
682 (\lambda (t: T).(subst0 i u t x)) H (TLRef n) (lift_lref_lt n h d H1)) in
683 (and_ind (eq nat n i) (eq T x (lift (S n) O u)) (ex2 T (\lambda (t2: T).(eq T
684 x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef n) t2)))
685 (\lambda (H3: (eq nat n i)).(\lambda (_: (eq T x (lift (S n) O u))).(let H5
686 \def (eq_ind nat n (\lambda (n0: nat).(lt n0 d)) H1 i H3) in (le_false (plus
687 d h) i (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
688 T).(subst0 (minus i h) u (TLRef n) t2))) H0 (le_plus_trans (S i) d h H5)))))
689 (subst0_gen_lref u x i n H2)))) (\lambda (H1: (le d n)).(let H2 \def (eq_ind
690 T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t x)) H (TLRef (plus n h))
691 (lift_lref_ge n h d H1)) in (and_ind (eq nat (plus n h) i) (eq T x (lift (S
692 (plus n h)) O u)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
693 (t2: T).(subst0 (minus i h) u (TLRef n) t2))) (\lambda (H3: (eq nat (plus n
694 h) i)).(\lambda (H4: (eq T x (lift (S (plus n h)) O u))).(eq_ind nat (plus n
695 h) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2)))
696 (\lambda (t2: T).(subst0 (minus n0 h) u (TLRef n) t2)))) (eq_ind_r T (lift (S
697 (plus n h)) O u) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d
698 t2))) (\lambda (t2: T).(subst0 (minus (plus n h) h) u (TLRef n) t2))))
699 (eq_ind_r nat n (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S
700 (plus n h)) O u) (lift h d t2))) (\lambda (t2: T).(subst0 n0 u (TLRef n)
701 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift (S (plus n h)) O u) (lift h
702 d t2))) (\lambda (t2: T).(subst0 n u (TLRef n) t2)) (lift (S n) O u)
703 (eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t: T).(eq T (lift (S (plus n
704 h)) O u) t)) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(eq T (lift (S n0) O
705 u) (lift (plus h (S n)) O u))) (eq_ind_r nat (plus h (S n)) (\lambda (n0:
706 nat).(eq T (lift n0 O u) (lift (plus h (S n)) O u))) (refl_equal T (lift
707 (plus h (S n)) O u)) (S (plus h n)) (plus_n_Sm h n)) (plus n h) (plus_comm n
708 h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans_plus_r O d
709 (plus O (S n)) (plus_le_compat O O d (S n) (le_n O) (le_S d n H1))) (le_O_n
710 d))) (subst0_lref u n)) (minus (plus n h) h) (minus_plus_r n h)) x H4) i
711 H3))) (subst0_gen_lref u x i (plus n h) H2)))))))))))) (\lambda (k:
712 K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall
713 (h: nat).(\forall (d: nat).((subst0 i u (lift h d t) x) \to ((le (plus d h)
714 i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
715 T).(subst0 (minus i h) u t t2))))))))))).(\lambda (t0: T).(\lambda (H0:
716 ((\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d:
717 nat).((subst0 i u (lift h d t0) x) \to ((le (plus d h) i) \to (ex2 T (\lambda
718 (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t0
719 t2))))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda
720 (d: nat).(\lambda (H1: (subst0 i u (lift h d (THead k t t0)) x)).(\lambda
721 (H2: (le (plus d h) i)).(let H3 \def (eq_ind T (lift h d (THead k t t0))
722 (\lambda (t2: T).(subst0 i u t2 x)) H1 (THead k (lift h d t) (lift h (s k d)
723 t0)) (lift_head k t t0 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x
724 (THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u (lift h d t)
725 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda
726 (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))) (ex3_2 T T (\lambda (u2:
727 T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
728 T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s
729 k i) u (lift h (s k d) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x (lift h d
730 t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda
731 (H4: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k d) t0))))
732 (\lambda (u2: T).(subst0 i u (lift h d t) u2)))).(ex2_ind T (\lambda (u2:
733 T).(eq T x (THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u
734 (lift h d t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
735 (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda
736 (H5: (eq T x (THead k x0 (lift h (s k d) t0)))).(\lambda (H6: (subst0 i u
737 (lift h d t) x0)).(eq_ind_r T (THead k x0 (lift h (s k d) t0)) (\lambda (t2:
738 T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0
739 (minus i h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T x0
740 (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)) (ex2 T (\lambda
741 (t2: T).(eq T (THead k x0 (lift h (s k d) t0)) (lift h d t2))) (\lambda (t2:
742 T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x1: T).(\lambda (H7:
743 (eq T x0 (lift h d x1))).(\lambda (H8: (subst0 (minus i h) u t x1)).(eq_ind_r
744 T (lift h d x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2
745 (lift h (s k d) t0)) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u
746 (THead k t t0) t3)))) (eq_ind T (lift h d (THead k x1 t0)) (\lambda (t2:
747 T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0
748 (minus i h) u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift
749 h d (THead k x1 t0)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u
750 (THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h d (THead k x1 t0)))
751 (subst0_fst u x1 t (minus i h) H8 t0 k)) (THead k (lift h d x1) (lift h (s k
752 d) t0)) (lift_head k x1 t0 h d)) x0 H7)))) (H x0 i h d H6 H2)) x H5)))) H4))
753 (\lambda (H4: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2)))
754 (\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2)))).(ex2_ind T
755 (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0
756 (s k i) u (lift h (s k d) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d
757 t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda
758 (x0: T).(\lambda (H5: (eq T x (THead k (lift h d t) x0))).(\lambda (H6:
759 (subst0 (s k i) u (lift h (s k d) t0) x0)).(eq_ind_r T (THead k (lift h d t)
760 x0) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3)))
761 (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (ex2_ind T
762 (\lambda (t2: T).(eq T x0 (lift h (s k d) t2))) (\lambda (t2: T).(subst0
763 (minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T (THead k (lift h d
764 t) x0) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0)
765 t2))) (\lambda (x1: T).(\lambda (H7: (eq T x0 (lift h (s k d) x1))).(\lambda
766 (H8: (subst0 (minus (s k i) h) u t0 x1)).(eq_ind_r T (lift h (s k d) x1)
767 (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h d t) t2)
768 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3))))
769 (eq_ind T (lift h d (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda (t3:
770 T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t
771 t0) t3)))) (let H9 \def (eq_ind_r nat (minus (s k i) h) (\lambda (n:
772 nat).(subst0 n u t0 x1)) H8 (s k (minus i h)) (s_minus k i h (le_trans_plus_r
773 d h i H2))) in (ex_intro2 T (\lambda (t2: T).(eq T (lift h d (THead k t x1))
774 (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))
775 (THead k t x1) (refl_equal T (lift h d (THead k t x1))) (subst0_snd k u x1 t0
776 (minus i h) H9 t))) (THead k (lift h d t) (lift h (s k d) x1)) (lift_head k t
777 x1 h d)) x0 H7)))) (H0 x0 (s k i) h (s k d) H6 (eq_ind nat (s k (plus d h))
778 (\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i H2) (plus (s k d) h)
779 (s_plus k d h)))) x H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2:
780 T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
781 T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s
782 k i) u (lift h (s k d) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
783 (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i
784 u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) u (lift
785 h (s k d) t0) t2))) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
786 (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda
787 (x1: T).(\lambda (H5: (eq T x (THead k x0 x1))).(\lambda (H6: (subst0 i u
788 (lift h d t) x0)).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t0)
789 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq
790 T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0)
791 t3)))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h (s k d) t2))) (\lambda
792 (t2: T).(subst0 (minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T
793 (THead k x0 x1) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead
794 k t t0) t2))) (\lambda (x2: T).(\lambda (H8: (eq T x1 (lift h (s k d)
795 x2))).(\lambda (H9: (subst0 (minus (s k i) h) u t0 x2)).(ex2_ind T (\lambda
796 (t2: T).(eq T x0 (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t
797 t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h d t2))) (\lambda
798 (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x3: T).(\lambda
799 (H10: (eq T x0 (lift h d x3))).(\lambda (H11: (subst0 (minus i h) u t
800 x3)).(eq_ind_r T (lift h d x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T
801 (THead k t2 x1) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead
802 k t t0) t3)))) (eq_ind_r T (lift h (s k d) x2) (\lambda (t2: T).(ex2 T
803 (\lambda (t3: T).(eq T (THead k (lift h d x3) t2) (lift h d t3))) (\lambda
804 (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind T (lift h d
805 (THead k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d
806 t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (let H12
807 \def (eq_ind_r nat (minus (s k i) h) (\lambda (n: nat).(subst0 n u t0 x2)) H9
808 (s k (minus i h)) (s_minus k i h (le_trans_plus_r d h i H2))) in (ex_intro2 T
809 (\lambda (t2: T).(eq T (lift h d (THead k x3 x2)) (lift h d t2))) (\lambda
810 (t2: T).(subst0 (minus i h) u (THead k t t0) t2)) (THead k x3 x2) (refl_equal
811 T (lift h d (THead k x3 x2))) (subst0_both u t x3 (minus i h) H11 k t0 x2
812 H12))) (THead k (lift h d x3) (lift h (s k d) x2)) (lift_head k x3 x2 h d))
813 x1 H8) x0 H10)))) (H x0 i h d H6 H2))))) (H0 x1 (s k i) h (s k d) H7 (eq_ind
814 nat (s k (plus d h)) (\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i
815 H2) (plus (s k d) h) (s_plus k d h)))) x H5)))))) H4)) (subst0_gen_head k u
816 (lift h d t) (lift h (s k d) t0) x i H3)))))))))))))) t1)).
projects / helm.git / blob