1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props".
19 include "lift/fwd.ma".
23 theorem thead_x_lift_y_y:
24 \forall (k: K).(\forall (t: T).(\forall (v: T).(\forall (h: nat).(\forall
25 (d: nat).((eq T (THead k v (lift h d t)) t) \to (\forall (P: Prop).P))))))
27 \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (v:
28 T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift h d t0)) t0)
29 \to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda (v: T).(\lambda (h:
30 nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v (lift h d (TSort n)))
31 (TSort n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (lift h d
32 (TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
33 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
34 \Rightarrow True])) I (TSort n) H) in (False_ind P H0)))))))) (\lambda (n:
35 nat).(\lambda (v: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T
36 (THead k v (lift h d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def
37 (eq_ind T (THead k v (lift h d (TLRef n))) (\lambda (ee: T).(match ee in T
38 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
39 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in
40 (False_ind P H0)))))))) (\lambda (k0: K).(\lambda (t0: T).(\lambda (_:
41 ((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift
42 h d t0)) t0) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (H0:
43 ((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift
44 h d t1)) t1) \to (\forall (P: Prop).P))))))).(\lambda (v: T).(\lambda (h:
45 nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k v (lift h d (THead k0 t0
46 t1))) (THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K
47 (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
48 \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k]))
49 (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) in ((let H3 \def
50 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
51 [(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _)
52 \Rightarrow t])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1)
53 in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
54 (_: T).T) with [(TSort _) \Rightarrow (THead k0 ((let rec lref_map (f: ((nat
55 \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow
56 (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true
57 \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow
58 (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda
59 (x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat \to nat))) (d: nat)
60 (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef
61 i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false
62 \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u)
63 (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x h)) (s k0
64 d) t1)) | (TLRef _) \Rightarrow (THead k0 ((let rec lref_map (f: ((nat \to
65 nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow
66 (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true
67 \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow
68 (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda
69 (x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat \to nat))) (d: nat)
70 (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef
71 i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false
72 \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u)
73 (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x h)) (s k0
74 d) t1)) | (THead _ _ t) \Rightarrow t])) (THead k v (lift h d (THead k0 t0
75 t1))) (THead k0 t0 t1) H1) in (\lambda (_: (eq T v t0)).(\lambda (H6: (eq K k
76 k0)).(let H7 \def (eq_ind K k (\lambda (k: K).(\forall (v: T).(\forall (h:
77 nat).(\forall (d: nat).((eq T (THead k v (lift h d t1)) t1) \to (\forall (P:
78 Prop).P)))))) H0 k0 H6) in (let H8 \def (eq_ind T (lift h d (THead k0 t0 t1))
79 (\lambda (t: T).(eq T t t1)) H4 (THead k0 (lift h d t0) (lift h (s k0 d) t1))
80 (lift_head k0 t0 t1 h d)) in (H7 (lift h d t0) h (s k0 d) H8 P)))))) H3))
84 \forall (t: T).(\forall (d: nat).(eq T (lift O d t) t))
86 \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq T (lift O d t0)
87 t0))) (\lambda (n: nat).(\lambda (_: nat).(refl_equal T (TSort n)))) (\lambda
88 (n: nat).(\lambda (d: nat).(lt_le_e n d (eq T (lift O d (TLRef n)) (TLRef n))
89 (\lambda (H: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef
90 n))) (refl_equal T (TLRef n)) (lift O d (TLRef n)) (lift_lref_lt n O d H)))
91 (\lambda (H: (le d n)).(eq_ind_r T (TLRef (plus n O)) (\lambda (t0: T).(eq T
92 t0 (TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O)
93 (plus_n_O n))) (lift O d (TLRef n)) (lift_lref_ge n O d H)))))) (\lambda (k:
94 K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(eq T (lift O d t0)
95 t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (lift O d t1)
96 t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift O d t0) (lift O (s k d)
97 t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (sym_equal T (THead k t0 t1)
98 (THead k (lift O d t0) (lift O (s k d) t1)) (sym_equal T (THead k (lift O d
99 t0) (lift O (s k d) t1)) (THead k t0 t1) (sym_equal T (THead k t0 t1) (THead
100 k (lift O d t0) (lift O (s k d) t1)) (f_equal3 K T T T THead k k t0 (lift O d
101 t0) t1 (lift O (s k d) t1) (refl_equal K k) (sym_eq T (lift O d t0) t0 (H d))
102 (sym_eq T (lift O (s k d) t1) t1 (H0 (s k d))))))) (lift O d (THead k t0 t1))
103 (lift_head k t0 t1 O d)))))))) t).
105 theorem lift_lref_gt:
106 \forall (d: nat).(\forall (n: nat).((lt d n) \to (eq T (lift (S O) d (TLRef
107 (pred n))) (TLRef n))))
109 \lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt d n)).(eq_ind_r T (TLRef
110 (plus (pred n) (S O))) (\lambda (t: T).(eq T t (TLRef n))) (eq_ind nat (plus
111 (S O) (pred n)) (\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (eq_ind nat n
112 (\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (refl_equal T (TLRef n)) (S
113 (pred n)) (S_pred n d H)) (plus (pred n) (S O)) (plus_comm (S O) (pred n)))
114 (lift (S O) d (TLRef (pred n))) (lift_lref_ge (pred n) (S O) d (le_S_n d
115 (pred n) (eq_ind nat n (\lambda (n0: nat).(le (S d) n0)) H (S (pred n))
116 (S_pred n d H))))))).
119 \forall (x: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((eq T
120 (lift h d x) (lift h d t)) \to (eq T x t)))))
122 \lambda (x: T).(T_ind (\lambda (t: T).(\forall (t0: T).(\forall (h:
123 nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to (eq T t
124 t0)))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d:
125 nat).(\lambda (H: (eq T (lift h d (TSort n)) (lift h d t))).(let H0 \def
126 (eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H
127 (TSort n) (lift_sort n h d)) in (sym_eq T t (TSort n) (lift_gen_sort h d n t
128 H0)))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d:
129 nat).(\lambda (H: (eq T (lift h d (TLRef n)) (lift h d t))).(lt_le_e n d (eq
130 T (TLRef n) t) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d
131 (TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef n) (lift_lref_lt
132 n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_lt h d n (lt_le_trans n d
133 d H0 (le_n d)) t H1)))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift
134 h d (TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef (plus n h))
135 (lift_lref_ge n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_ge h d n H0
136 t H1)))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t:
137 T).(((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t)
138 (lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t:
139 T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to
140 (eq T t0 t)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d:
141 nat).((eq T (lift h d (THead k0 t t0)) (lift h d t1)) \to (eq T (THead k0 t
142 t0) t1)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H: ((\forall (t0:
143 T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to
144 (eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t: T).(\forall
145 (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to (eq T t0
146 t))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1:
147 (eq T (lift h d (THead (Bind b) t t0)) (lift h d t1))).(let H2 \def (eq_ind T
148 (lift h d (THead (Bind b) t t0)) (\lambda (t: T).(eq T t (lift h d t1))) H1
149 (THead (Bind b) (lift h d t) (lift h (S d) t0)) (lift_bind b t t0 h d)) in
150 (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Bind b) y
151 z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y))))
152 (\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) t0) (lift h (S d) z))))
153 (eq T (THead (Bind b) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda
154 (H3: (eq T t1 (THead (Bind b) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift
155 h d x0))).(\lambda (H5: (eq T (lift h (S d) t0) (lift h (S d) x1))).(eq_ind_r
156 T (THead (Bind b) x0 x1) (\lambda (t2: T).(eq T (THead (Bind b) t t0) t2))
157 (sym_equal T (THead (Bind b) x0 x1) (THead (Bind b) t t0) (sym_equal T (THead
158 (Bind b) t t0) (THead (Bind b) x0 x1) (sym_equal T (THead (Bind b) x0 x1)
159 (THead (Bind b) t t0) (f_equal3 K T T T THead (Bind b) (Bind b) x0 t x1 t0
160 (refl_equal K (Bind b)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T t0 x1 (H0 x1
161 h (S d) H5)))))) t1 H3)))))) (lift_gen_bind b (lift h d t) (lift h (S d) t0)
162 t1 h d H2)))))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H: ((\forall
163 (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d
164 t0)) \to (eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t:
165 T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to
166 (eq T t0 t))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d:
167 nat).(\lambda (H1: (eq T (lift h d (THead (Flat f) t t0)) (lift h d
168 t1))).(let H2 \def (eq_ind T (lift h d (THead (Flat f) t t0)) (\lambda (t:
169 T).(eq T t (lift h d t1))) H1 (THead (Flat f) (lift h d t) (lift h d t0))
170 (lift_flat f t t0 h d)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq
171 T t1 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d
172 t) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d t0) (lift
173 h d z)))) (eq T (THead (Flat f) t t0) t1) (\lambda (x0: T).(\lambda (x1:
174 T).(\lambda (H3: (eq T t1 (THead (Flat f) x0 x1))).(\lambda (H4: (eq T (lift
175 h d t) (lift h d x0))).(\lambda (H5: (eq T (lift h d t0) (lift h d
176 x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(eq T (THead (Flat
177 f) t t0) t2)) (sym_equal T (THead (Flat f) x0 x1) (THead (Flat f) t t0)
178 (sym_equal T (THead (Flat f) t t0) (THead (Flat f) x0 x1) (sym_equal T (THead
179 (Flat f) x0 x1) (THead (Flat f) t t0) (f_equal3 K T T T THead (Flat f) (Flat
180 f) x0 t x1 t0 (refl_equal K (Flat f)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T
181 t0 x1 (H0 x1 h d H5)))))) t1 H3)))))) (lift_gen_flat f (lift h d t) (lift h d
182 t0) t1 h d H2)))))))))))) k)) x).
184 theorem lift_gen_lift:
185 \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2:
186 nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1
187 t1) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1
188 t2))) (\lambda (t2: T).(eq T t1 (lift h2 d2 t2)))))))))))
190 \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: T).(\forall (h1:
191 nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to
192 ((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2:
193 T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2
194 t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (h1: nat).(\lambda
195 (h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda (_: (le d1
196 d2)).(\lambda (H0: (eq T (lift h1 d1 (TSort n)) (lift h2 (plus d2 h1)
197 x))).(let H1 \def (eq_ind T (lift h1 d1 (TSort n)) (\lambda (t: T).(eq T t
198 (lift h2 (plus d2 h1) x))) H0 (TSort n) (lift_sort n h1 d1)) in (eq_ind_r T
199 (TSort n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2)))
200 (\lambda (t2: T).(eq T (TSort n) (lift h2 d2 t2))))) (ex_intro2 T (\lambda
201 (t2: T).(eq T (TSort n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TSort n)
202 (lift h2 d2 t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T
203 (TSort n) t)) (refl_equal T (TSort n)) (lift h1 d1 (TSort n)) (lift_sort n h1
204 d1)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T
205 (TSort n)) (lift h2 d2 (TSort n)) (lift_sort n h2 d2))) x (lift_gen_sort h2
206 (plus d2 h1) n x H1))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda
207 (h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda
208 (H: (le d1 d2)).(\lambda (H0: (eq T (lift h1 d1 (TLRef n)) (lift h2 (plus d2
209 h1) x))).(lt_le_e n d1 (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2)))
210 (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) (\lambda (H1: (lt n
211 d1)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) (\lambda (t: T).(eq T t
212 (lift h2 (plus d2 h1) x))) H0 (TLRef n) (lift_lref_lt n h1 d1 H1)) in
213 (eq_ind_r T (TLRef n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift
214 h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) (ex_intro2 T
215 (\lambda (t2: T).(eq T (TLRef n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T
216 (TLRef n) (lift h2 d2 t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t:
217 T).(eq T (TLRef n) t)) (refl_equal T (TLRef n)) (lift h1 d1 (TLRef n))
218 (lift_lref_lt n h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef
219 n) t)) (refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2
220 (lt_le_trans n d1 d2 H1 H)))) x (lift_gen_lref_lt h2 (plus d2 h1) n
221 (lt_le_trans n d1 (plus d2 h1) H1 (le_plus_trans d1 d2 h1 H)) x H2))))
222 (\lambda (H1: (le d1 n)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n))
223 (\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H0 (TLRef (plus n h1))
224 (lift_lref_ge n h1 d1 H1)) in (lt_le_e n d2 (ex2 T (\lambda (t2: T).(eq T x
225 (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))
226 (\lambda (H3: (lt n d2)).(eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(ex2
227 T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n)
228 (lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq T (TLRef (plus n h1))
229 (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef
230 n) (eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(eq T (TLRef (plus n h1))
231 t)) (refl_equal T (TLRef (plus n h1))) (lift h1 d1 (TLRef n)) (lift_lref_ge n
232 h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t))
233 (refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 H3))) x
234 (lift_gen_lref_lt h2 (plus d2 h1) (plus n h1) (plus_lt_compat_r n d2 h1 H3) x
235 H2))) (\lambda (H3: (le d2 n)).(lt_le_e n (plus d2 h2) (ex2 T (\lambda (t2:
236 T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2
237 t2)))) (\lambda (H4: (lt n (plus d2 h2))).(lift_gen_lref_false h2 (plus d2
238 h1) (plus n h1) (le_S_n (plus d2 h1) (plus n h1) (lt_le_S (plus d2 h1) (S
239 (plus n h1)) (le_lt_n_Sm (plus d2 h1) (plus n h1) (plus_le_compat d2 n h1 h1
240 H3 (le_n h1))))) (eq_ind_r nat (plus (plus d2 h2) h1) (\lambda (n0: nat).(lt
241 (plus n h1) n0)) (lt_le_S (plus n h1) (plus (plus d2 h2) h1)
242 (plus_lt_compat_r n (plus d2 h2) h1 H4)) (plus (plus d2 h1) h2)
243 (plus_permute_2_in_3 d2 h1 h2)) x H2 (ex2 T (\lambda (t2: T).(eq T x (lift h1
244 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))))) (\lambda (H4:
245 (le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus n h1) (\lambda (n:
246 nat).(eq T (TLRef n) (lift h2 (plus d2 h1) x))) H2 (plus (minus (plus n h1)
247 h2) h2) (le_plus_minus_sym h2 (plus n h1) (le_plus_trans h2 n h1
248 (le_trans_plus_r d2 h2 n H4)))) in (eq_ind_r T (TLRef (minus (plus n h1) h2))
249 (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda
250 (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq
251 T (TLRef (minus (plus n h1) h2)) (lift h1 d1 t2))) (\lambda (t2: T).(eq T
252 (TLRef n) (lift h2 d2 t2))) (TLRef (minus n h2)) (eq_ind_r nat (plus (minus n
253 h2) h1) (\lambda (n0: nat).(eq T (TLRef n0) (lift h1 d1 (TLRef (minus n
254 h2))))) (eq_ind_r T (TLRef (plus (minus n h2) h1)) (\lambda (t: T).(eq T
255 (TLRef (plus (minus n h2) h1)) t)) (refl_equal T (TLRef (plus (minus n h2)
256 h1))) (lift h1 d1 (TLRef (minus n h2))) (lift_lref_ge (minus n h2) h1 d1
257 (le_trans d1 d2 (minus n h2) H (le_minus d2 n h2 H4)))) (minus (plus n h1)
258 h2) (le_minus_plus h2 n (le_trans_plus_r d2 h2 n H4) h1)) (eq_ind_r nat (plus
259 (minus n h2) h2) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2 d2 (TLRef
260 (minus n0 h2))))) (eq_ind_r T (TLRef (plus (minus (plus (minus n h2) h2) h2)
261 h2)) (\lambda (t: T).(eq T (TLRef (plus (minus n h2) h2)) t)) (f_equal nat T
262 TLRef (plus (minus n h2) h2) (plus (minus (plus (minus n h2) h2) h2) h2)
263 (f_equal2 nat nat nat plus (minus n h2) (minus (plus (minus n h2) h2) h2) h2
264 h2 (sym_eq nat (minus (plus (minus n h2) h2) h2) (minus n h2) (minus_plus_r
265 (minus n h2) h2)) (refl_equal nat h2))) (lift h2 d2 (TLRef (minus (plus
266 (minus n h2) h2) h2))) (lift_lref_ge (minus (plus (minus n h2) h2) h2) h2 d2
267 (le_minus d2 (plus (minus n h2) h2) h2 (plus_le_compat d2 (minus n h2) h2 h2
268 (le_minus d2 n h2 H4) (le_n h2))))) n (le_plus_minus_sym h2 n
269 (le_trans_plus_r d2 h2 n H4)))) x (lift_gen_lref_ge h2 (plus d2 h1) (minus
270 (plus n h1) h2) (arith0 h2 d2 n H4 h1) x H5)))))))))))))))))) (\lambda (k:
271 K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (h1: nat).(\forall
272 (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift
273 h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift
274 h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))))))))))))).(\lambda
275 (t0: T).(\lambda (H0: ((\forall (x: T).(\forall (h1: nat).(\forall (h2:
276 nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1
277 t0) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1
278 t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2 t2))))))))))))).(\lambda (x:
279 T).(\lambda (h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2:
280 nat).(\lambda (H1: (le d1 d2)).(\lambda (H2: (eq T (lift h1 d1 (THead k t
281 t0)) (lift h2 (plus d2 h1) x))).(K_ind (\lambda (k0: K).((eq T (lift h1 d1
282 (THead k0 t t0)) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T
283 x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead k0 t t0) (lift h2 d2
284 t2)))))) (\lambda (b: B).(\lambda (H3: (eq T (lift h1 d1 (THead (Bind b) t
285 t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind T (lift h1 d1 (THead
286 (Bind b) t t0)) (\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H3 (THead
287 (Bind b) (lift h1 d1 t) (lift h1 (S d1) t0)) (lift_bind b t t0 h1 d1)) in
288 (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y
289 z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift h2 (plus d2
290 h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 (S d1) t0) (lift h2
291 (S (plus d2 h1)) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2)))
292 (\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2)))) (\lambda (x0:
293 T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Bind b) x0 x1))).(\lambda
294 (H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T
295 (lift h1 (S d1) t0) (lift h2 (S (plus d2 h1)) x1))).(eq_ind_r T (THead (Bind
296 b) x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h1 d1 t3)))
297 (\lambda (t3: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (ex2_ind T
298 (\lambda (t2: T).(eq T x0 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2
299 d2 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) x0 x1) (lift h1 d1
300 t2))) (\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2))))
301 (\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9: (eq T
302 t (lift h2 d2 x2))).(eq_ind_r T (lift h1 d1 x2) (\lambda (t2: T).(ex2 T
303 (\lambda (t3: T).(eq T (THead (Bind b) t2 x1) (lift h1 d1 t3))) (\lambda (t3:
304 T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2
305 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1
306 d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) t2 t0)
307 (lift h2 d2 t3))))) (let H10 \def (refl_equal nat (plus (S d2) h1)) in (let
308 H11 \def (eq_ind nat (S (plus d2 h1)) (\lambda (n: nat).(eq T (lift h1 (S d1)
309 t0) (lift h2 n x1))) H7 (plus (S d2) h1) H10) in (ex2_ind T (\lambda (t2:
310 T).(eq T x1 (lift h1 (S d1) t2))) (\lambda (t2: T).(eq T t0 (lift h2 (S d2)
311 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) x1) (lift
312 h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) t0) (lift
313 h2 d2 t2)))) (\lambda (x3: T).(\lambda (H12: (eq T x1 (lift h1 (S d1)
314 x3))).(\lambda (H13: (eq T t0 (lift h2 (S d2) x3))).(eq_ind_r T (lift h1 (S
315 d1) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift
316 h1 d1 x2) t2) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift
317 h2 d2 x2) t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 (S d2) x3) (\lambda
318 (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1 d1 x2) (lift
319 h1 (S d1) x3)) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift
320 h2 d2 x2) t2) (lift h2 d2 t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead
321 (Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) (lift h1 d1 t2))) (\lambda (t2:
322 T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) (lift h2 d2
323 t2))) (THead (Bind b) x2 x3) (eq_ind_r T (THead (Bind b) (lift h1 d1 x2)
324 (lift h1 (S d1) x3)) (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2)
325 (lift h1 (S d1) x3)) t2)) (refl_equal T (THead (Bind b) (lift h1 d1 x2) (lift
326 h1 (S d1) x3))) (lift h1 d1 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h1
327 d1)) (eq_ind_r T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3))
328 (\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3))
329 t2)) (refl_equal T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)))
330 (lift h2 d2 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h2 d2))) t0 H13) x1
331 H12)))) (H0 x1 h1 h2 (S d1) (S d2) (le_S_n (S d1) (S d2) (lt_le_S (S d1) (S
332 (S d2)) (lt_n_S d1 (S d2) (le_lt_n_Sm d1 d2 H1)))) H11)))) t H9) x0 H8)))) (H
333 x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_bind b (lift h1 d1 t) (lift h1 (S
334 d1) t0) x h2 (plus d2 h1) H4))))) (\lambda (f: F).(\lambda (H3: (eq T (lift
335 h1 d1 (THead (Flat f) t t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind
336 T (lift h1 d1 (THead (Flat f) t t0)) (\lambda (t: T).(eq T t (lift h2 (plus
337 d2 h1) x))) H3 (THead (Flat f) (lift h1 d1 t) (lift h1 d1 t0)) (lift_flat f t
338 t0 h1 d1)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead
339 (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift
340 h2 (plus d2 h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 d1 t0)
341 (lift h2 (plus d2 h1) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2)))
342 (\lambda (t2: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t2)))) (\lambda (x0:
343 T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Flat f) x0 x1))).(\lambda
344 (H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T
345 (lift h1 d1 t0) (lift h2 (plus d2 h1) x1))).(eq_ind_r T (THead (Flat f) x0
346 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h1 d1 t3)))
347 (\lambda (t3: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t3))))) (ex2_ind T
348 (\lambda (t2: T).(eq T x0 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2
349 d2 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Flat f) x0 x1) (lift h1 d1
350 t2))) (\lambda (t2: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t2))))
351 (\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9: (eq T
352 t (lift h2 d2 x2))).(eq_ind_r T (lift h1 d1 x2) (\lambda (t2: T).(ex2 T
353 (\lambda (t3: T).(eq T (THead (Flat f) t2 x1) (lift h1 d1 t3))) (\lambda (t3:
354 T).(eq T (THead (Flat f) t t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2
355 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) (lift h1
356 d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t2 t0)
357 (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h1 d1 t2)))
358 (\lambda (t2: T).(eq T t0 (lift h2 d2 t2))) (ex2 T (\lambda (t2: T).(eq T
359 (THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T
360 (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 t2)))) (\lambda (x3:
361 T).(\lambda (H10: (eq T x1 (lift h1 d1 x3))).(\lambda (H11: (eq T t0 (lift h2
362 d2 x3))).(eq_ind_r T (lift h1 d1 x3) (\lambda (t2: T).(ex2 T (\lambda (t3:
363 T).(eq T (THead (Flat f) (lift h1 d1 x2) t2) (lift h1 d1 t3))) (\lambda (t3:
364 T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 t3))))) (eq_ind_r T
365 (lift h2 d2 x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat
366 f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1 t3))) (\lambda (t3: T).(eq T
367 (THead (Flat f) (lift h2 d2 x2) t2) (lift h2 d2 t3))))) (ex_intro2 T (\lambda
368 (t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1
369 t2))) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3))
370 (lift h2 d2 t2))) (THead (Flat f) x2 x3) (eq_ind_r T (THead (Flat f) (lift h1
371 d1 x2) (lift h1 d1 x3)) (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1
372 x2) (lift h1 d1 x3)) t2)) (refl_equal T (THead (Flat f) (lift h1 d1 x2) (lift
373 h1 d1 x3))) (lift h1 d1 (THead (Flat f) x2 x3)) (lift_flat f x2 x3 h1 d1))
374 (eq_ind_r T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3)) (\lambda (t2:
375 T).(eq T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3)) t2)) (refl_equal T
376 (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3))) (lift h2 d2 (THead (Flat f)
377 x2 x3)) (lift_flat f x2 x3 h2 d2))) t0 H11) x1 H10)))) (H0 x1 h1 h2 d1 d2 H1
378 H7)) t H9) x0 H8)))) (H x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_flat f
379 (lift h1 d1 t) (lift h1 d1 t0) x h2 (plus d2 h1) H4))))) k H2)))))))))))))
383 \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d:
384 nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e
385 (lift h d t)) (lift (plus k h) d t))))))))
387 \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k:
388 nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to
389 (eq T (lift k e (lift h d t0)) (lift (plus k h) d t0))))))))) (\lambda (n:
390 nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e:
391 nat).(\lambda (_: (le e (plus d h))).(\lambda (_: (le d e)).(eq_ind_r T
392 (TSort n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TSort
393 n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d
394 (TSort n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0))
395 (refl_equal T (TSort n)) (lift (plus k h) d (TSort n)) (lift_sort n (plus k
396 h) d)) (lift k e (TSort n)) (lift_sort n k e)) (lift h d (TSort n))
397 (lift_sort n h d))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (k:
398 nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H: (le e (plus d
399 h))).(\lambda (H0: (le d e)).(lt_le_e n d (eq T (lift k e (lift h d (TLRef
400 n))) (lift (plus k h) d (TLRef n))) (\lambda (H1: (lt n d)).(eq_ind_r T
401 (TLRef n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TLRef
402 n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d
403 (TLRef n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0))
404 (refl_equal T (TLRef n)) (lift (plus k h) d (TLRef n)) (lift_lref_lt n (plus
405 k h) d H1)) (lift k e (TLRef n)) (lift_lref_lt n k e (lt_le_trans n d e H1
406 H0))) (lift h d (TLRef n)) (lift_lref_lt n h d H1))) (\lambda (H1: (le d
407 n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (lift k e t0) (lift
408 (plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus (plus n h) k)) (\lambda
409 (t0: T).(eq T t0 (lift (plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus n
410 (plus k h))) (\lambda (t0: T).(eq T (TLRef (plus (plus n h) k)) t0)) (f_equal
411 nat T TLRef (plus (plus n h) k) (plus n (plus k h))
412 (plus_permute_2_in_3_assoc n h k)) (lift (plus k h) d (TLRef n))
413 (lift_lref_ge n (plus k h) d H1)) (lift k e (TLRef (plus n h))) (lift_lref_ge
414 (plus n h) k e (le_trans e (plus d h) (plus n h) H (plus_le_compat d n h h H1
415 (le_n h))))) (lift h d (TLRef n)) (lift_lref_ge n h d H1))))))))))) (\lambda
416 (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k:
417 nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to
418 (eq T (lift k e (lift h d t0)) (lift (plus k h) d t0)))))))))).(\lambda (t1:
419 T).(\lambda (H0: ((\forall (h: nat).(\forall (k: nat).(\forall (d:
420 nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e
421 (lift h d t1)) (lift (plus k h) d t1)))))))))).(\lambda (h: nat).(\lambda
422 (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e (plus d
423 h))).(\lambda (H2: (le d e)).(eq_ind_r T (THead k (lift h d t0) (lift h (s k
424 d) t1)) (\lambda (t2: T).(eq T (lift k0 e t2) (lift (plus k0 h) d (THead k t0
425 t1)))) (eq_ind_r T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift
426 h (s k d) t1))) (\lambda (t2: T).(eq T t2 (lift (plus k0 h) d (THead k t0
427 t1)))) (eq_ind_r T (THead k (lift (plus k0 h) d t0) (lift (plus k0 h) (s k d)
428 t1)) (\lambda (t2: T).(eq T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k
429 e) (lift h (s k d) t1))) t2)) (f_equal3 K T T T THead k k (lift k0 e (lift h
430 d t0)) (lift (plus k0 h) d t0) (lift k0 (s k e) (lift h (s k d) t1)) (lift
431 (plus k0 h) (s k d) t1) (refl_equal K k) (H h k0 d e H1 H2) (H0 h k0 (s k d)
432 (s k e) (eq_ind nat (s k (plus d h)) (\lambda (n: nat).(le (s k e) n)) (s_le
433 k e (plus d h) H1) (plus (s k d) h) (s_plus k d h)) (s_le k d e H2))) (lift
434 (plus k0 h) d (THead k t0 t1)) (lift_head k t0 t1 (plus k0 h) d)) (lift k0 e
435 (THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k (lift h d t0) (lift
436 h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) (lift_head k t0 t1 h
440 \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d:
441 nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t))
442 (lift k e (lift h d t))))))))
444 \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k:
445 nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k
446 d) (lift k e t0)) (lift k e (lift h d t0))))))))) (\lambda (n: nat).(\lambda
447 (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (_:
448 (le e d)).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (lift h (plus k d) t0)
449 (lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq
450 T t0 (lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0:
451 T).(eq T (TSort n) (lift k e t0))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq
452 T (TSort n) t0)) (refl_equal T (TSort n)) (lift k e (TSort n)) (lift_sort n k
453 e)) (lift h d (TSort n)) (lift_sort n h d)) (lift h (plus k d) (TSort n))
454 (lift_sort n h (plus k d))) (lift k e (TSort n)) (lift_sort n k e))))))))
455 (\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d:
456 nat).(\lambda (e: nat).(\lambda (H: (le e d)).(lt_le_e n e (eq T (lift h
457 (plus k d) (lift k e (TLRef n))) (lift k e (lift h d (TLRef n)))) (\lambda
458 (H0: (lt n e)).(let H1 \def (lt_le_trans n e d H0 H) in (eq_ind_r T (TLRef n)
459 (\lambda (t0: T).(eq T (lift h (plus k d) t0) (lift k e (lift h d (TLRef
460 n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift k e (lift h d
461 (TLRef n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) (lift k
462 e t0))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0))
463 (refl_equal T (TLRef n)) (lift k e (TLRef n)) (lift_lref_lt n k e H0)) (lift
464 h d (TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus k d) (TLRef n))
465 (lift_lref_lt n h (plus k d) (lt_le_trans n d (plus k d) H1 (le_plus_r k
466 d)))) (lift k e (TLRef n)) (lift_lref_lt n k e H0)))) (\lambda (H0: (le e
467 n)).(eq_ind_r T (TLRef (plus n k)) (\lambda (t0: T).(eq T (lift h (plus k d)
468 t0) (lift k e (lift h d (TLRef n))))) (eq_ind_r nat (plus d k) (\lambda (n0:
469 nat).(eq T (lift h n0 (TLRef (plus n k))) (lift k e (lift h d (TLRef n)))))
470 (lt_le_e n d (eq T (lift h (plus d k) (TLRef (plus n k))) (lift k e (lift h d
471 (TLRef n)))) (\lambda (H1: (lt n d)).(eq_ind_r T (TLRef (plus n k)) (\lambda
472 (t0: T).(eq T t0 (lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef n)
473 (\lambda (t0: T).(eq T (TLRef (plus n k)) (lift k e t0))) (eq_ind_r T (TLRef
474 (plus n k)) (\lambda (t0: T).(eq T (TLRef (plus n k)) t0)) (refl_equal T
475 (TLRef (plus n k))) (lift k e (TLRef n)) (lift_lref_ge n k e H0)) (lift h d
476 (TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus d k) (TLRef (plus n k)))
477 (lift_lref_lt (plus n k) h (plus d k) (lt_le_S (plus n k) (plus d k)
478 (plus_lt_compat_r n d k H1))))) (\lambda (H1: (le d n)).(eq_ind_r T (TLRef
479 (plus (plus n k) h)) (\lambda (t0: T).(eq T t0 (lift k e (lift h d (TLRef
480 n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (TLRef (plus
481 (plus n k) h)) (lift k e t0))) (eq_ind_r T (TLRef (plus (plus n h) k))
482 (\lambda (t0: T).(eq T (TLRef (plus (plus n k) h)) t0)) (f_equal nat T TLRef
483 (plus (plus n k) h) (plus (plus n h) k) (sym_eq nat (plus (plus n h) k) (plus
484 (plus n k) h) (plus_permute_2_in_3 n h k))) (lift k e (TLRef (plus n h)))
485 (lift_lref_ge (plus n h) k e (le_S_n e (plus n h) (lt_le_S e (S (plus n h))
486 (le_lt_n_Sm e (plus n h) (le_plus_trans e n h H0)))))) (lift h d (TLRef n))
487 (lift_lref_ge n h d H1)) (lift h (plus d k) (TLRef (plus n k))) (lift_lref_ge
488 (plus n k) h (plus d k) (le_S_n (plus d k) (plus n k) (lt_le_S (plus d k) (S
489 (plus n k)) (le_lt_n_Sm (plus d k) (plus n k) (plus_le_compat d n k k H1
490 (le_n k))))))))) (plus k d) (plus_comm k d)) (lift k e (TLRef n))
491 (lift_lref_ge n k e H0)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda
492 (H: ((\forall (h: nat).(\forall (k: nat).(\forall (d: nat).(\forall (e:
493 nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t0)) (lift k e (lift h
494 d t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k:
495 nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k
496 d) (lift k e t1)) (lift k e (lift h d t1)))))))))).(\lambda (h: nat).(\lambda
497 (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e
498 d)).(eq_ind_r T (THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2:
499 T).(eq T (lift h (plus k0 d) t2) (lift k0 e (lift h d (THead k t0 t1)))))
500 (eq_ind_r T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus
501 k0 d)) (lift k0 (s k e) t1))) (\lambda (t2: T).(eq T t2 (lift k0 e (lift h d
502 (THead k t0 t1))))) (eq_ind_r T (THead k (lift h d t0) (lift h (s k d) t1))
503 (\lambda (t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h
504 (s k (plus k0 d)) (lift k0 (s k e) t1))) (lift k0 e t2))) (eq_ind_r T (THead
505 k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h (s k d) t1))) (\lambda
506 (t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus
507 k0 d)) (lift k0 (s k e) t1))) t2)) (eq_ind_r nat (plus k0 (s k d)) (\lambda
508 (n: nat).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h n (lift
509 k0 (s k e) t1))) (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h
510 (s k d) t1))))) (f_equal3 K T T T THead k k (lift h (plus k0 d) (lift k0 e
511 t0)) (lift k0 e (lift h d t0)) (lift h (plus k0 (s k d)) (lift k0 (s k e)
512 t1)) (lift k0 (s k e) (lift h (s k d) t1)) (refl_equal K k) (H h k0 d e H1)
513 (H0 h k0 (s k d) (s k e) (s_le k e d H1))) (s k (plus k0 d)) (s_plus_sym k k0
514 d)) (lift k0 e (THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k
515 (lift h d t0) (lift h (s k d) t1) k0 e)) (lift h d (THead k t0 t1))
516 (lift_head k t0 t1 h d)) (lift h (plus k0 d) (THead k (lift k0 e t0) (lift k0
517 (s k e) t1))) (lift_head k (lift k0 e t0) (lift k0 (s k e) t1) h (plus k0
518 d))) (lift k0 e (THead k t0 t1)) (lift_head k t0 t1 k0 e)))))))))))) t).