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