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 "tlist/defs.ma".
21 include "lift/fwd.ma".
25 theorem thead_x_lift_y_y:
26 \forall (k: K).(\forall (t: T).(\forall (v: T).(\forall (h: nat).(\forall
27 (d: nat).((eq T (THead k v (lift h d t)) t) \to (\forall (P: Prop).P))))))
29 \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (v:
30 T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift h d t0)) t0)
31 \to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda (v: T).(\lambda (h:
32 nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v (lift h d (TSort n)))
33 (TSort n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (lift h d
34 (TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
35 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
36 \Rightarrow True])) I (TSort n) H) in (False_ind P H0)))))))) (\lambda (n:
37 nat).(\lambda (v: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T
38 (THead k v (lift h d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def
39 (eq_ind T (THead k v (lift h d (TLRef n))) (\lambda (ee: T).(match ee in T
40 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
41 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in
42 (False_ind P H0)))))))) (\lambda (k0: K).(\lambda (t0: T).(\lambda (_:
43 ((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift
44 h d t0)) t0) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (H0:
45 ((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift
46 h d t1)) t1) \to (\forall (P: Prop).P))))))).(\lambda (v: T).(\lambda (h:
47 nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k v (lift h d (THead k0 t0
48 t1))) (THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K
49 (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
50 \Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1]))
51 (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) in ((let H3 \def
52 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
53 [(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t2 _)
54 \Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1)
55 H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return
56 (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead k0 ((let rec lref_map
57 (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort
58 n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0)
59 with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3)
60 \Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in
61 lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat
62 \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort n)
63 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with
64 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3)
65 \Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in
66 lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (TLRef _) \Rightarrow
67 (THead k0 ((let rec lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T
68 \def (match t2 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
69 (TLRef (match (blt i d0) with [true \Rightarrow i | false \Rightarrow (f
70 i)])) | (THead k1 u t3) \Rightarrow (THead k1 (lref_map f d0 u) (lref_map f
71 (s k1 d0) t3))]) in lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec
72 lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with
73 [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i
74 d0) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3)
75 \Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in
76 lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (THead _ _ t2)
77 \Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1)
78 H1) in (\lambda (_: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def
79 (eq_ind K k (\lambda (k1: K).(\forall (v0: T).(\forall (h0: nat).(\forall
80 (d0: nat).((eq T (THead k1 v0 (lift h0 d0 t1)) t1) \to (\forall (P0:
81 Prop).P0)))))) H0 k0 H6) in (let H8 \def (eq_ind T (lift h d (THead k0 t0
82 t1)) (\lambda (t2: T).(eq T t2 t1)) H4 (THead k0 (lift h d t0) (lift h (s k0
83 d) t1)) (lift_head k0 t0 t1 h d)) in (H7 (lift h d t0) h (s k0 d) H8 P))))))
84 H3)) H2)))))))))))) t)).
87 \forall (t: T).(\forall (d: nat).(eq T (lift O d t) t))
89 \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq T (lift O d t0)
90 t0))) (\lambda (n: nat).(\lambda (_: nat).(refl_equal T (TSort n)))) (\lambda
91 (n: nat).(\lambda (d: nat).(lt_le_e n d (eq T (lift O d (TLRef n)) (TLRef n))
92 (\lambda (H: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef
93 n))) (refl_equal T (TLRef n)) (lift O d (TLRef n)) (lift_lref_lt n O d H)))
94 (\lambda (H: (le d n)).(eq_ind_r T (TLRef (plus n O)) (\lambda (t0: T).(eq T
95 t0 (TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O)
96 (plus_n_O n))) (lift O d (TLRef n)) (lift_lref_ge n O d H)))))) (\lambda (k:
97 K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(eq T (lift O d t0)
98 t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (lift O d t1)
99 t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift O d t0) (lift O (s k d)
100 t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (sym_eq T (THead k t0 t1)
101 (THead k (lift O d t0) (lift O (s k d) t1)) (sym_eq T (THead k (lift O d t0)
102 (lift O (s k d) t1)) (THead k t0 t1) (sym_eq T (THead k t0 t1) (THead k (lift
103 O d t0) (lift O (s k d) t1)) (f_equal3 K T T T THead k k t0 (lift O d t0) t1
104 (lift O (s k d) t1) (refl_equal K k) (sym_eq T (lift O d t0) t0 (H d))
105 (sym_eq T (lift O (s k d) t1) t1 (H0 (s k d))))))) (lift O d (THead k t0 t1))
106 (lift_head k t0 t1 O d)))))))) t).
108 theorem lift_lref_gt:
109 \forall (d: nat).(\forall (n: nat).((lt d n) \to (eq T (lift (S O) d (TLRef
110 (pred n))) (TLRef n))))
112 \lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt d n)).(eq_ind_r T (TLRef
113 (plus (pred n) (S O))) (\lambda (t: T).(eq T t (TLRef n))) (eq_ind nat (plus
114 (S O) (pred n)) (\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (eq_ind nat n
115 (\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (refl_equal T (TLRef n)) (S
116 (pred n)) (S_pred n d H)) (plus (pred n) (S O)) (plus_comm (S O) (pred n)))
117 (lift (S O) d (TLRef (pred n))) (lift_lref_ge (pred n) (S O) d (le_S_n d
118 (pred n) (eq_ind nat n (\lambda (n0: nat).(le (S d) n0)) H (S (pred n))
119 (S_pred n d H))))))).
122 \forall (h: nat).(\forall (d: nat).(\forall (v: T).(\forall (vs: TList).(eq
123 TList (lifts h d (TApp vs v)) (TApp (lifts h d vs) (lift h d v))))))
125 \lambda (h: nat).(\lambda (d: nat).(\lambda (v: T).(\lambda (vs:
126 TList).(TList_ind (\lambda (t: TList).(eq TList (lifts h d (TApp t v)) (TApp
127 (lifts h d t) (lift h d v)))) (refl_equal TList (TCons (lift h d v) TNil))
128 (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq TList (lifts h d (TApp
129 t0 v)) (TApp (lifts h d t0) (lift h d v)))).(eq_ind_r TList (TApp (lifts h d
130 t0) (lift h d v)) (\lambda (t1: TList).(eq TList (TCons (lift h d t) t1)
131 (TCons (lift h d t) (TApp (lifts h d t0) (lift h d v))))) (refl_equal TList
132 (TCons (lift h d t) (TApp (lifts h d t0) (lift h d v)))) (lifts h d (TApp t0
136 \forall (x: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((eq T
137 (lift h d x) (lift h d t)) \to (eq T x t)))))
139 \lambda (x: T).(T_ind (\lambda (t: T).(\forall (t0: T).(\forall (h:
140 nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to (eq T t
141 t0)))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d:
142 nat).(\lambda (H: (eq T (lift h d (TSort n)) (lift h d t))).(let H0 \def
143 (eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H
144 (TSort n) (lift_sort n h d)) in (sym_eq T t (TSort n) (lift_gen_sort h d n t
145 H0)))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d:
146 nat).(\lambda (H: (eq T (lift h d (TLRef n)) (lift h d t))).(lt_le_e n d (eq
147 T (TLRef n) t) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d
148 (TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef n) (lift_lref_lt
149 n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_lt h d n (lt_le_trans n d
150 d H0 (le_n d)) t H1)))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift
151 h d (TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef (plus n h))
152 (lift_lref_ge n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_ge h d n H0
153 t H1)))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t:
154 T).(((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t)
155 (lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t1:
156 T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1))
157 \to (eq T t0 t1)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d:
158 nat).((eq T (lift h d (THead k0 t t0)) (lift h d t1)) \to (eq T (THead k0 t
159 t0) t1)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H: ((\forall (t0:
160 T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to
161 (eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall
162 (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0
163 t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1:
164 (eq T (lift h d (THead (Bind b) t t0)) (lift h d t1))).(let H2 \def (eq_ind T
165 (lift h d (THead (Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1
166 (THead (Bind b) (lift h d t) (lift h (S d) t0)) (lift_bind b t t0 h d)) in
167 (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Bind b) y
168 z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y))))
169 (\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) t0) (lift h (S d) z))))
170 (eq T (THead (Bind b) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda
171 (H3: (eq T t1 (THead (Bind b) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift
172 h d x0))).(\lambda (H5: (eq T (lift h (S d) t0) (lift h (S d) x1))).(eq_ind_r
173 T (THead (Bind b) x0 x1) (\lambda (t2: T).(eq T (THead (Bind b) t t0) t2))
174 (sym_eq T (THead (Bind b) x0 x1) (THead (Bind b) t t0) (sym_eq T (THead (Bind
175 b) t t0) (THead (Bind b) x0 x1) (sym_eq T (THead (Bind b) x0 x1) (THead (Bind
176 b) t t0) (f_equal3 K T T T THead (Bind b) (Bind b) x0 t x1 t0 (refl_equal K
177 (Bind b)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T t0 x1 (H0 x1 h (S d)
178 H5)))))) t1 H3)))))) (lift_gen_bind b (lift h d t) (lift h (S d) t0) t1 h d
179 H2)))))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H: ((\forall (t0:
180 T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to
181 (eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall
182 (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0
183 t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1:
184 (eq T (lift h d (THead (Flat f) t t0)) (lift h d t1))).(let H2 \def (eq_ind T
185 (lift h d (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1
186 (THead (Flat f) (lift h d t) (lift h d t0)) (lift_flat f t t0 h d)) in
187 (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Flat f) y
188 z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y))))
189 (\lambda (_: T).(\lambda (z: T).(eq T (lift h d t0) (lift h d z)))) (eq T
190 (THead (Flat f) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq
191 T t1 (THead (Flat f) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift h d
192 x0))).(\lambda (H5: (eq T (lift h d t0) (lift h d x1))).(eq_ind_r T (THead
193 (Flat f) x0 x1) (\lambda (t2: T).(eq T (THead (Flat f) t t0) t2)) (sym_eq T
194 (THead (Flat f) x0 x1) (THead (Flat f) t t0) (sym_eq T (THead (Flat f) t t0)
195 (THead (Flat f) x0 x1) (sym_eq T (THead (Flat f) x0 x1) (THead (Flat f) t t0)
196 (f_equal3 K T T T THead (Flat f) (Flat f) x0 t x1 t0 (refl_equal K (Flat f))
197 (sym_eq T t x0 (H x0 h d H4)) (sym_eq T t0 x1 (H0 x1 h d H5)))))) t1 H3))))))
198 (lift_gen_flat f (lift h d t) (lift h d t0) t1 h d H2)))))))))))) k)) x).
200 theorem lift_gen_lift:
201 \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2:
202 nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1
203 t1) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1
204 t2))) (\lambda (t2: T).(eq T t1 (lift h2 d2 t2)))))))))))
206 \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: T).(\forall (h1:
207 nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to
208 ((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2:
209 T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2
210 t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (h1: nat).(\lambda
211 (h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda (_: (le d1
212 d2)).(\lambda (H0: (eq T (lift h1 d1 (TSort n)) (lift h2 (plus d2 h1)
213 x))).(let H1 \def (eq_ind T (lift h1 d1 (TSort n)) (\lambda (t: T).(eq T t
214 (lift h2 (plus d2 h1) x))) H0 (TSort n) (lift_sort n h1 d1)) in (eq_ind_r T
215 (TSort n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2)))
216 (\lambda (t2: T).(eq T (TSort n) (lift h2 d2 t2))))) (ex_intro2 T (\lambda
217 (t2: T).(eq T (TSort n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TSort n)
218 (lift h2 d2 t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T
219 (TSort n) t)) (refl_equal T (TSort n)) (lift h1 d1 (TSort n)) (lift_sort n h1
220 d1)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T
221 (TSort n)) (lift h2 d2 (TSort n)) (lift_sort n h2 d2))) x (lift_gen_sort h2
222 (plus d2 h1) n x H1))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda
223 (h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda
224 (H: (le d1 d2)).(\lambda (H0: (eq T (lift h1 d1 (TLRef n)) (lift h2 (plus d2
225 h1) x))).(lt_le_e n d1 (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2)))
226 (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) (\lambda (H1: (lt n
227 d1)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) (\lambda (t: T).(eq T t
228 (lift h2 (plus d2 h1) x))) H0 (TLRef n) (lift_lref_lt n h1 d1 H1)) in
229 (eq_ind_r T (TLRef n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift
230 h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) (ex_intro2 T
231 (\lambda (t2: T).(eq T (TLRef n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T
232 (TLRef n) (lift h2 d2 t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t:
233 T).(eq T (TLRef n) t)) (refl_equal T (TLRef n)) (lift h1 d1 (TLRef n))
234 (lift_lref_lt n h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef
235 n) t)) (refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2
236 (lt_le_trans n d1 d2 H1 H)))) x (lift_gen_lref_lt h2 (plus d2 h1) n
237 (lt_le_trans n d1 (plus d2 h1) H1 (le_plus_trans d1 d2 h1 H)) x H2))))
238 (\lambda (H1: (le d1 n)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n))
239 (\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H0 (TLRef (plus n h1))
240 (lift_lref_ge n h1 d1 H1)) in (lt_le_e n d2 (ex2 T (\lambda (t2: T).(eq T x
241 (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))
242 (\lambda (H3: (lt n d2)).(eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(ex2
243 T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n)
244 (lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq T (TLRef (plus n h1))
245 (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef
246 n) (eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(eq T (TLRef (plus n h1))
247 t)) (refl_equal T (TLRef (plus n h1))) (lift h1 d1 (TLRef n)) (lift_lref_ge n
248 h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t))
249 (refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 H3))) x
250 (lift_gen_lref_lt h2 (plus d2 h1) (plus n h1) (plus_lt_compat_r n d2 h1 H3) x
251 H2))) (\lambda (H3: (le d2 n)).(lt_le_e n (plus d2 h2) (ex2 T (\lambda (t2:
252 T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2
253 t2)))) (\lambda (H4: (lt n (plus d2 h2))).(lift_gen_lref_false h2 (plus d2
254 h1) (plus n h1) (le_S_n (plus d2 h1) (plus n h1) (lt_le_S (plus d2 h1) (S
255 (plus n h1)) (le_lt_n_Sm (plus d2 h1) (plus n h1) (plus_le_compat d2 n h1 h1
256 H3 (le_n h1))))) (eq_ind_r nat (plus (plus d2 h2) h1) (\lambda (n0: nat).(lt
257 (plus n h1) n0)) (lt_le_S (plus n h1) (plus (plus d2 h2) h1)
258 (plus_lt_compat_r n (plus d2 h2) h1 H4)) (plus (plus d2 h1) h2)
259 (plus_permute_2_in_3 d2 h1 h2)) x H2 (ex2 T (\lambda (t2: T).(eq T x (lift h1
260 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))))) (\lambda (H4:
261 (le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus n h1) (\lambda (n0:
262 nat).(eq T (TLRef n0) (lift h2 (plus d2 h1) x))) H2 (plus (minus (plus n h1)
263 h2) h2) (le_plus_minus_sym h2 (plus n h1) (le_plus_trans h2 n h1
264 (le_trans_plus_r d2 h2 n H4)))) in (eq_ind_r T (TLRef (minus (plus n h1) h2))
265 (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda
266 (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq
267 T (TLRef (minus (plus n h1) h2)) (lift h1 d1 t2))) (\lambda (t2: T).(eq T
268 (TLRef n) (lift h2 d2 t2))) (TLRef (minus n h2)) (eq_ind_r nat (plus (minus n
269 h2) h1) (\lambda (n0: nat).(eq T (TLRef n0) (lift h1 d1 (TLRef (minus n
270 h2))))) (eq_ind_r T (TLRef (plus (minus n h2) h1)) (\lambda (t: T).(eq T
271 (TLRef (plus (minus n h2) h1)) t)) (refl_equal T (TLRef (plus (minus n h2)
272 h1))) (lift h1 d1 (TLRef (minus n h2))) (lift_lref_ge (minus n h2) h1 d1
273 (le_trans d1 d2 (minus n h2) H (le_minus d2 n h2 H4)))) (minus (plus n h1)
274 h2) (le_minus_plus h2 n (le_trans_plus_r d2 h2 n H4) h1)) (eq_ind_r nat (plus
275 (minus n h2) h2) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2 d2 (TLRef
276 (minus n0 h2))))) (eq_ind_r T (TLRef (plus (minus (plus (minus n h2) h2) h2)
277 h2)) (\lambda (t: T).(eq T (TLRef (plus (minus n h2) h2)) t)) (f_equal nat T
278 TLRef (plus (minus n h2) h2) (plus (minus (plus (minus n h2) h2) h2) h2)
279 (f_equal2 nat nat nat plus (minus n h2) (minus (plus (minus n h2) h2) h2) h2
280 h2 (sym_eq nat (minus (plus (minus n h2) h2) h2) (minus n h2) (minus_plus_r
281 (minus n h2) h2)) (refl_equal nat h2))) (lift h2 d2 (TLRef (minus (plus
282 (minus n h2) h2) h2))) (lift_lref_ge (minus (plus (minus n h2) h2) h2) h2 d2
283 (le_minus d2 (plus (minus n h2) h2) h2 (plus_le_compat d2 (minus n h2) h2 h2
284 (le_minus d2 n h2 H4) (le_n h2))))) n (le_plus_minus_sym h2 n
285 (le_trans_plus_r d2 h2 n H4)))) x (lift_gen_lref_ge h2 (plus d2 h1) (minus
286 (plus n h1) h2) (arith0 h2 d2 n H4 h1) x H5)))))))))))))))))) (\lambda (k:
287 K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (h1: nat).(\forall
288 (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift
289 h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift
290 h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))))))))))))).(\lambda
291 (t0: T).(\lambda (H0: ((\forall (x: T).(\forall (h1: nat).(\forall (h2:
292 nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1
293 t0) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1
294 t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2 t2))))))))))))).(\lambda (x:
295 T).(\lambda (h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2:
296 nat).(\lambda (H1: (le d1 d2)).(\lambda (H2: (eq T (lift h1 d1 (THead k t
297 t0)) (lift h2 (plus d2 h1) x))).(K_ind (\lambda (k0: K).((eq T (lift h1 d1
298 (THead k0 t t0)) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T
299 x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead k0 t t0) (lift h2 d2
300 t2)))))) (\lambda (b: B).(\lambda (H3: (eq T (lift h1 d1 (THead (Bind b) t
301 t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind T (lift h1 d1 (THead
302 (Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h2 (plus d2 h1) x))) H3
303 (THead (Bind b) (lift h1 d1 t) (lift h1 (S d1) t0)) (lift_bind b t t0 h1 d1))
304 in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y
305 z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift h2 (plus d2
306 h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 (S d1) t0) (lift h2
307 (S (plus d2 h1)) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2)))
308 (\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2)))) (\lambda (x0:
309 T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Bind b) x0 x1))).(\lambda
310 (H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T
311 (lift h1 (S d1) t0) (lift h2 (S (plus d2 h1)) x1))).(eq_ind_r T (THead (Bind
312 b) x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h1 d1 t3)))
313 (\lambda (t3: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (ex2_ind T
314 (\lambda (t2: T).(eq T x0 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2
315 d2 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) x0 x1) (lift h1 d1
316 t2))) (\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2))))
317 (\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9: (eq T
318 t (lift h2 d2 x2))).(eq_ind_r T (lift h1 d1 x2) (\lambda (t2: T).(ex2 T
319 (\lambda (t3: T).(eq T (THead (Bind b) t2 x1) (lift h1 d1 t3))) (\lambda (t3:
320 T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2
321 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1
322 d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) t2 t0)
323 (lift h2 d2 t3))))) (let H10 \def (refl_equal nat (plus (S d2) h1)) in (let
324 H11 \def (eq_ind nat (S (plus d2 h1)) (\lambda (n: nat).(eq T (lift h1 (S d1)
325 t0) (lift h2 n x1))) H7 (plus (S d2) h1) H10) in (ex2_ind T (\lambda (t2:
326 T).(eq T x1 (lift h1 (S d1) t2))) (\lambda (t2: T).(eq T t0 (lift h2 (S d2)
327 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) x1) (lift
328 h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) t0) (lift
329 h2 d2 t2)))) (\lambda (x3: T).(\lambda (H12: (eq T x1 (lift h1 (S d1)
330 x3))).(\lambda (H13: (eq T t0 (lift h2 (S d2) x3))).(eq_ind_r T (lift h1 (S
331 d1) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift
332 h1 d1 x2) t2) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift
333 h2 d2 x2) t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 (S d2) x3) (\lambda
334 (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1 d1 x2) (lift
335 h1 (S d1) x3)) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift
336 h2 d2 x2) t2) (lift h2 d2 t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead
337 (Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) (lift h1 d1 t2))) (\lambda (t2:
338 T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) (lift h2 d2
339 t2))) (THead (Bind b) x2 x3) (eq_ind_r T (THead (Bind b) (lift h1 d1 x2)
340 (lift h1 (S d1) x3)) (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2)
341 (lift h1 (S d1) x3)) t2)) (refl_equal T (THead (Bind b) (lift h1 d1 x2) (lift
342 h1 (S d1) x3))) (lift h1 d1 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h1
343 d1)) (eq_ind_r T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3))
344 (\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3))
345 t2)) (refl_equal T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)))
346 (lift h2 d2 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h2 d2))) t0 H13) x1
347 H12)))) (H0 x1 h1 h2 (S d1) (S d2) (le_S_n (S d1) (S d2) (lt_le_S (S d1) (S
348 (S d2)) (lt_n_S d1 (S d2) (le_lt_n_Sm d1 d2 H1)))) H11)))) t H9) x0 H8)))) (H
349 x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_bind b (lift h1 d1 t) (lift h1 (S
350 d1) t0) x h2 (plus d2 h1) H4))))) (\lambda (f: F).(\lambda (H3: (eq T (lift
351 h1 d1 (THead (Flat f) t t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind
352 T (lift h1 d1 (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift h2 (plus
353 d2 h1) x))) H3 (THead (Flat f) (lift h1 d1 t) (lift h1 d1 t0)) (lift_flat f t
354 t0 h1 d1)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead
355 (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift
356 h2 (plus d2 h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 d1 t0)
357 (lift h2 (plus d2 h1) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2)))
358 (\lambda (t2: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t2)))) (\lambda (x0:
359 T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Flat f) x0 x1))).(\lambda
360 (H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T
361 (lift h1 d1 t0) (lift h2 (plus d2 h1) x1))).(eq_ind_r T (THead (Flat f) x0
362 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h1 d1 t3)))
363 (\lambda (t3: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t3))))) (ex2_ind T
364 (\lambda (t2: T).(eq T x0 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2
365 d2 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Flat f) x0 x1) (lift h1 d1
366 t2))) (\lambda (t2: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t2))))
367 (\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9: (eq T
368 t (lift h2 d2 x2))).(eq_ind_r T (lift h1 d1 x2) (\lambda (t2: T).(ex2 T
369 (\lambda (t3: T).(eq T (THead (Flat f) t2 x1) (lift h1 d1 t3))) (\lambda (t3:
370 T).(eq T (THead (Flat f) t t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2
371 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) (lift h1
372 d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t2 t0)
373 (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h1 d1 t2)))
374 (\lambda (t2: T).(eq T t0 (lift h2 d2 t2))) (ex2 T (\lambda (t2: T).(eq T
375 (THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T
376 (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 t2)))) (\lambda (x3:
377 T).(\lambda (H10: (eq T x1 (lift h1 d1 x3))).(\lambda (H11: (eq T t0 (lift h2
378 d2 x3))).(eq_ind_r T (lift h1 d1 x3) (\lambda (t2: T).(ex2 T (\lambda (t3:
379 T).(eq T (THead (Flat f) (lift h1 d1 x2) t2) (lift h1 d1 t3))) (\lambda (t3:
380 T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 t3))))) (eq_ind_r T
381 (lift h2 d2 x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat
382 f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1 t3))) (\lambda (t3: T).(eq T
383 (THead (Flat f) (lift h2 d2 x2) t2) (lift h2 d2 t3))))) (ex_intro2 T (\lambda
384 (t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1
385 t2))) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3))
386 (lift h2 d2 t2))) (THead (Flat f) x2 x3) (eq_ind_r T (THead (Flat f) (lift h1
387 d1 x2) (lift h1 d1 x3)) (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1
388 x2) (lift h1 d1 x3)) t2)) (refl_equal T (THead (Flat f) (lift h1 d1 x2) (lift
389 h1 d1 x3))) (lift h1 d1 (THead (Flat f) x2 x3)) (lift_flat f x2 x3 h1 d1))
390 (eq_ind_r T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3)) (\lambda (t2:
391 T).(eq T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3)) t2)) (refl_equal T
392 (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3))) (lift h2 d2 (THead (Flat f)
393 x2 x3)) (lift_flat f x2 x3 h2 d2))) t0 H11) x1 H10)))) (H0 x1 h1 h2 d1 d2 H1
394 H7)) t H9) x0 H8)))) (H x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_flat f
395 (lift h1 d1 t) (lift h1 d1 t0) x h2 (plus d2 h1) H4))))) k H2)))))))))))))
399 \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d:
400 nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e
401 (lift h d t)) (lift (plus k h) d t))))))))
403 \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k:
404 nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to
405 (eq T (lift k e (lift h d t0)) (lift (plus k h) d t0))))))))) (\lambda (n:
406 nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e:
407 nat).(\lambda (_: (le e (plus d h))).(\lambda (_: (le d e)).(eq_ind_r T
408 (TSort n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TSort
409 n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d
410 (TSort n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0))
411 (refl_equal T (TSort n)) (lift (plus k h) d (TSort n)) (lift_sort n (plus k
412 h) d)) (lift k e (TSort n)) (lift_sort n k e)) (lift h d (TSort n))
413 (lift_sort n h d))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (k:
414 nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H: (le e (plus d
415 h))).(\lambda (H0: (le d e)).(lt_le_e n d (eq T (lift k e (lift h d (TLRef
416 n))) (lift (plus k h) d (TLRef n))) (\lambda (H1: (lt n d)).(eq_ind_r T
417 (TLRef n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TLRef
418 n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d
419 (TLRef n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0))
420 (refl_equal T (TLRef n)) (lift (plus k h) d (TLRef n)) (lift_lref_lt n (plus
421 k h) d H1)) (lift k e (TLRef n)) (lift_lref_lt n k e (lt_le_trans n d e H1
422 H0))) (lift h d (TLRef n)) (lift_lref_lt n h d H1))) (\lambda (H1: (le d
423 n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (lift k e t0) (lift
424 (plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus (plus n h) k)) (\lambda
425 (t0: T).(eq T t0 (lift (plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus n
426 (plus k h))) (\lambda (t0: T).(eq T (TLRef (plus (plus n h) k)) t0)) (f_equal
427 nat T TLRef (plus (plus n h) k) (plus n (plus k h))
428 (plus_permute_2_in_3_assoc n h k)) (lift (plus k h) d (TLRef n))
429 (lift_lref_ge n (plus k h) d H1)) (lift k e (TLRef (plus n h))) (lift_lref_ge
430 (plus n h) k e (le_trans e (plus d h) (plus n h) H (plus_le_compat d n h h H1
431 (le_n h))))) (lift h d (TLRef n)) (lift_lref_ge n h d H1))))))))))) (\lambda
432 (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k0:
433 nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to
434 (eq T (lift k0 e (lift h d t0)) (lift (plus k0 h) d t0)))))))))).(\lambda
435 (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: nat).(\forall (d:
436 nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k0 e
437 (lift h d t1)) (lift (plus k0 h) d t1)))))))))).(\lambda (h: nat).(\lambda
438 (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e (plus d
439 h))).(\lambda (H2: (le d e)).(eq_ind_r T (THead k (lift h d t0) (lift h (s k
440 d) t1)) (\lambda (t2: T).(eq T (lift k0 e t2) (lift (plus k0 h) d (THead k t0
441 t1)))) (eq_ind_r T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift
442 h (s k d) t1))) (\lambda (t2: T).(eq T t2 (lift (plus k0 h) d (THead k t0
443 t1)))) (eq_ind_r T (THead k (lift (plus k0 h) d t0) (lift (plus k0 h) (s k d)
444 t1)) (\lambda (t2: T).(eq T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k
445 e) (lift h (s k d) t1))) t2)) (f_equal3 K T T T THead k k (lift k0 e (lift h
446 d t0)) (lift (plus k0 h) d t0) (lift k0 (s k e) (lift h (s k d) t1)) (lift
447 (plus k0 h) (s k d) t1) (refl_equal K k) (H h k0 d e H1 H2) (H0 h k0 (s k d)
448 (s k e) (eq_ind nat (s k (plus d h)) (\lambda (n: nat).(le (s k e) n)) (s_le
449 k e (plus d h) H1) (plus (s k d) h) (s_plus k d h)) (s_le k d e H2))) (lift
450 (plus k0 h) d (THead k t0 t1)) (lift_head k t0 t1 (plus k0 h) d)) (lift k0 e
451 (THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k (lift h d t0) (lift
452 h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) (lift_head k t0 t1 h
456 \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d:
457 nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t))
458 (lift k e (lift h d t))))))))
460 \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k:
461 nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k
462 d) (lift k e t0)) (lift k e (lift h d t0))))))))) (\lambda (n: nat).(\lambda
463 (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (_:
464 (le e d)).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (lift h (plus k d) t0)
465 (lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq
466 T t0 (lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0:
467 T).(eq T (TSort n) (lift k e t0))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq
468 T (TSort n) t0)) (refl_equal T (TSort n)) (lift k e (TSort n)) (lift_sort n k
469 e)) (lift h d (TSort n)) (lift_sort n h d)) (lift h (plus k d) (TSort n))
470 (lift_sort n h (plus k d))) (lift k e (TSort n)) (lift_sort n k e))))))))
471 (\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d:
472 nat).(\lambda (e: nat).(\lambda (H: (le e d)).(lt_le_e n e (eq T (lift h
473 (plus k d) (lift k e (TLRef n))) (lift k e (lift h d (TLRef n)))) (\lambda
474 (H0: (lt n e)).(let H1 \def (lt_le_trans n e d H0 H) in (eq_ind_r T (TLRef n)
475 (\lambda (t0: T).(eq T (lift h (plus k d) t0) (lift k e (lift h d (TLRef
476 n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift k e (lift h d
477 (TLRef n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) (lift k
478 e t0))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0))
479 (refl_equal T (TLRef n)) (lift k e (TLRef n)) (lift_lref_lt n k e H0)) (lift
480 h d (TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus k d) (TLRef n))
481 (lift_lref_lt n h (plus k d) (lt_le_trans n d (plus k d) H1 (le_plus_r k
482 d)))) (lift k e (TLRef n)) (lift_lref_lt n k e H0)))) (\lambda (H0: (le e
483 n)).(eq_ind_r T (TLRef (plus n k)) (\lambda (t0: T).(eq T (lift h (plus k d)
484 t0) (lift k e (lift h d (TLRef n))))) (eq_ind_r nat (plus d k) (\lambda (n0:
485 nat).(eq T (lift h n0 (TLRef (plus n k))) (lift k e (lift h d (TLRef n)))))
486 (lt_le_e n d (eq T (lift h (plus d k) (TLRef (plus n k))) (lift k e (lift h d
487 (TLRef n)))) (\lambda (H1: (lt n d)).(eq_ind_r T (TLRef (plus n k)) (\lambda
488 (t0: T).(eq T t0 (lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef n)
489 (\lambda (t0: T).(eq T (TLRef (plus n k)) (lift k e t0))) (eq_ind_r T (TLRef
490 (plus n k)) (\lambda (t0: T).(eq T (TLRef (plus n k)) t0)) (refl_equal T
491 (TLRef (plus n k))) (lift k e (TLRef n)) (lift_lref_ge n k e H0)) (lift h d
492 (TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus d k) (TLRef (plus n k)))
493 (lift_lref_lt (plus n k) h (plus d k) (lt_le_S (plus n k) (plus d k)
494 (plus_lt_compat_r n d k H1))))) (\lambda (H1: (le d n)).(eq_ind_r T (TLRef
495 (plus (plus n k) h)) (\lambda (t0: T).(eq T t0 (lift k e (lift h d (TLRef
496 n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (TLRef (plus
497 (plus n k) h)) (lift k e t0))) (eq_ind_r T (TLRef (plus (plus n h) k))
498 (\lambda (t0: T).(eq T (TLRef (plus (plus n k) h)) t0)) (f_equal nat T TLRef
499 (plus (plus n k) h) (plus (plus n h) k) (sym_eq nat (plus (plus n h) k) (plus
500 (plus n k) h) (plus_permute_2_in_3 n h k))) (lift k e (TLRef (plus n h)))
501 (lift_lref_ge (plus n h) k e (le_S_n e (plus n h) (lt_le_S e (S (plus n h))
502 (le_lt_n_Sm e (plus n h) (le_plus_trans e n h H0)))))) (lift h d (TLRef n))
503 (lift_lref_ge n h d H1)) (lift h (plus d k) (TLRef (plus n k))) (lift_lref_ge
504 (plus n k) h (plus d k) (le_S_n (plus d k) (plus n k) (lt_le_S (plus d k) (S
505 (plus n k)) (le_lt_n_Sm (plus d k) (plus n k) (plus_le_compat d n k k H1
506 (le_n k))))))))) (plus k d) (plus_comm k d)) (lift k e (TLRef n))
507 (lift_lref_ge n k e H0)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda
508 (H: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: nat).(\forall (e:
509 nat).((le e d) \to (eq T (lift h (plus k0 d) (lift k0 e t0)) (lift k0 e (lift
510 h d t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall
511 (k0: nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h
512 (plus k0 d) (lift k0 e t1)) (lift k0 e (lift h d t1)))))))))).(\lambda (h:
513 nat).(\lambda (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le
514 e d)).(eq_ind_r T (THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2:
515 T).(eq T (lift h (plus k0 d) t2) (lift k0 e (lift h d (THead k t0 t1)))))
516 (eq_ind_r T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus
517 k0 d)) (lift k0 (s k e) t1))) (\lambda (t2: T).(eq T t2 (lift k0 e (lift h d
518 (THead k t0 t1))))) (eq_ind_r T (THead k (lift h d t0) (lift h (s k d) t1))
519 (\lambda (t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h
520 (s k (plus k0 d)) (lift k0 (s k e) t1))) (lift k0 e t2))) (eq_ind_r T (THead
521 k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h (s k d) t1))) (\lambda
522 (t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus
523 k0 d)) (lift k0 (s k e) t1))) t2)) (eq_ind_r nat (plus k0 (s k d)) (\lambda
524 (n: nat).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h n (lift
525 k0 (s k e) t1))) (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h
526 (s k d) t1))))) (f_equal3 K T T T THead k k (lift h (plus k0 d) (lift k0 e
527 t0)) (lift k0 e (lift h d t0)) (lift h (plus k0 (s k d)) (lift k0 (s k e)
528 t1)) (lift k0 (s k e) (lift h (s k d) t1)) (refl_equal K k) (H h k0 d e H1)
529 (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
530 d)) (lift k0 e (THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k
531 (lift h d t0) (lift h (s k d) t1) k0 e)) (lift h d (THead k t0 t1))
532 (lift_head k t0 t1 h d)) (lift h (plus k0 d) (THead k (lift k0 e t0) (lift k0
533 (s k e) t1))) (lift_head k (lift k0 e t0) (lift k0 (s k e) t1) h (plus k0
534 d))) (lift k0 e (THead k t0 t1)) (lift_head k t0 t1 k0 e)))))))))))) t).