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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 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/lift/tlt".
19 include "lift/fwd.ma".
21 include "tlt/props.ma".
23 theorem lift_weight_map:
24 \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to
25 nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat
26 (weight_map f (lift h d t)) (weight_map f t))))))
28 \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d:
29 nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat
30 (f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f t0)))))))
31 (\lambda (n: nat).(\lambda (_: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
32 nat))).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (f m)
33 O))))).(refl_equal nat (weight_map f (TSort n)))))))) (\lambda (n:
34 nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
35 nat))).(\lambda (H: ((\forall (m: nat).((le d m) \to (eq nat (f m)
36 O))))).(lt_le_e n d (eq nat (weight_map f (lift h d (TLRef n))) (weight_map f
37 (TLRef n))) (\lambda (H0: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0:
38 T).(eq nat (weight_map f t0) (weight_map f (TLRef n)))) (refl_equal nat
39 (weight_map f (TLRef n))) (lift h d (TLRef n)) (lift_lref_lt n h d H0)))
40 (\lambda (H0: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq
41 nat (weight_map f t0) (weight_map f (TLRef n)))) (eq_ind_r nat O (\lambda
42 (n0: nat).(eq nat (f (plus n h)) n0)) (H (plus n h) (le_plus_trans d n h H0))
43 (f n) (H n H0)) (lift h d (TLRef n)) (lift_lref_ge n h d H0))))))))) (\lambda
44 (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d:
45 nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat
46 (f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f
47 t0)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (d:
48 nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat
49 (f m) O)))) \to (eq nat (weight_map f (lift h d t1)) (weight_map f
50 t1)))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
51 nat))).(\lambda (H1: ((\forall (m: nat).((le d m) \to (eq nat (f m)
52 O))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0
53 t1))) (weight_map f (THead k0 t0 t1)))) (\lambda (b: B).(eq_ind_r T (THead
54 (Bind b) (lift h d t0) (lift h (s (Bind b) d) t1)) (\lambda (t2: T).(eq nat
55 (weight_map f t2) (weight_map f (THead (Bind b) t0 t1)))) (B_ind (\lambda
56 (b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus (weight_map f (lift
57 h d t0)) (weight_map (wadd f (S (weight_map f (lift h d t0)))) (lift h (S d)
58 t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h d t0)) (weight_map
59 (wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S (plus (weight_map f
60 (lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))))]) (match b0 with
61 [Abbr \Rightarrow (S (plus (weight_map f t0) (weight_map (wadd f (S
62 (weight_map f t0))) t1))) | Abst \Rightarrow (S (plus (weight_map f t0)
63 (weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus (weight_map f t0)
64 (weight_map (wadd f O) t1)))]))) (eq_ind_r nat (weight_map f t0) (\lambda (n:
65 nat).(eq nat (S (plus n (weight_map (wadd f (S n)) (lift h (S d) t1)))) (S
66 (plus (weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)))))
67 (eq_ind_r nat (weight_map (wadd f (S (weight_map f t0))) t1) (\lambda (n:
68 nat).(eq nat (S (plus (weight_map f t0) n)) (S (plus (weight_map f t0)
69 (weight_map (wadd f (S (weight_map f t0))) t1))))) (refl_equal nat (S (plus
70 (weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1))))
71 (weight_map (wadd f (S (weight_map f t0))) (lift h (S d) t1)) (H0 h (S d)
72 (wadd f (S (weight_map f t0))) (\lambda (m: nat).(\lambda (H2: (le (S d)
73 m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d
74 n)) (eq nat (wadd f (S (weight_map f t0)) m) O) (\lambda (x: nat).(\lambda
75 (H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S x) (\lambda
76 (n: nat).(eq nat (wadd f (S (weight_map f t0)) n) O)) (H1 x H4) m H3))))
77 (le_gen_S d m H2)))))) (weight_map f (lift h d t0)) (H h d f H1)) (eq_ind_r
78 nat (weight_map (wadd f O) t1) (\lambda (n: nat).(eq nat (S (plus (weight_map
79 f (lift h d t0)) n)) (S (plus (weight_map f t0) (weight_map (wadd f O)
80 t1))))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map
81 (wadd f O) t1)) (plus (weight_map f t0) (weight_map (wadd f O) t1)) (f_equal2
82 nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) (weight_map
83 (wadd f O) t1) (weight_map (wadd f O) t1) (H h d f H1) (refl_equal nat
84 (weight_map (wadd f O) t1)))) (weight_map (wadd f O) (lift h (S d) t1)) (H0 h
85 (S d) (wadd f O) (\lambda (m: nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat
86 (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d n)) (eq nat (wadd
87 f O m) O) (\lambda (x: nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le
88 d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x
89 H4) m H3)))) (le_gen_S d m H2)))))) (eq_ind_r nat (weight_map (wadd f O) t1)
90 (\lambda (n: nat).(eq nat (S (plus (weight_map f (lift h d t0)) n)) (S (plus
91 (weight_map f t0) (weight_map (wadd f O) t1))))) (f_equal nat nat S (plus
92 (weight_map f (lift h d t0)) (weight_map (wadd f O) t1)) (plus (weight_map f
93 t0) (weight_map (wadd f O) t1)) (f_equal2 nat nat nat plus (weight_map f
94 (lift h d t0)) (weight_map f t0) (weight_map (wadd f O) t1) (weight_map (wadd
95 f O) t1) (H h d f H1) (refl_equal nat (weight_map (wadd f O) t1))))
96 (weight_map (wadd f O) (lift h (S d) t1)) (H0 h (S d) (wadd f O) (\lambda (m:
97 nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S
98 n))) (\lambda (n: nat).(le d n)) (eq nat (wadd f O m) O) (\lambda (x:
99 nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S
100 x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x H4) m H3)))) (le_gen_S d
101 m H2)))))) b) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind b) t0 t1 h
102 d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) (lift h (s
103 (Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) (weight_map f
104 (THead (Flat f0) t0 t1)))) (f_equal nat nat S (plus (weight_map f (lift h d
105 t0)) (weight_map f (lift h d t1))) (plus (weight_map f t0) (weight_map f t1))
106 (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0)
107 (weight_map f (lift h d t1)) (weight_map f t1) (H h d f H1) (H0 h d f H1)))
108 (lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d)))
112 \forall (t: T).(\forall (h: nat).(\forall (d: nat).(eq nat (weight (lift h d
115 \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(lift_weight_map t h d
116 (\lambda (_: nat).O) (\lambda (m: nat).(\lambda (_: (le d m)).(refl_equal nat
119 theorem lift_weight_add:
120 \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (d:
121 nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
122 (m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to
123 (((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) \to (eq nat
124 (weight_map f (lift h d t)) (weight_map g (lift (S h) d t)))))))))))
126 \lambda (w: nat).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h:
127 nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
128 nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat
129 (g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))
130 \to (eq nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d
131 t0))))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda
132 (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall (m:
133 nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d)
134 w)).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f
135 m)))))).(refl_equal nat (weight_map g (lift (S h) d (TSort n))))))))))))
136 (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
137 nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: ((\forall (m: nat).((lt m
138 d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) w)).(\lambda (H1:
139 ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))).(lt_le_e n d
140 (eq nat (weight_map f (lift h d (TLRef n))) (weight_map g (lift (S h) d
141 (TLRef n)))) (\lambda (H2: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0:
142 T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n)))))
143 (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq nat (weight_map f (TLRef n))
144 (weight_map g t0))) (sym_eq nat (g n) (f n) (H n H2)) (lift (S h) d (TLRef
145 n)) (lift_lref_lt n (S h) d H2)) (lift h d (TLRef n)) (lift_lref_lt n h d
146 H2))) (\lambda (H2: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0:
147 T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n)))))
148 (eq_ind_r T (TLRef (plus n (S h))) (\lambda (t0: T).(eq nat (weight_map f
149 (TLRef (plus n h))) (weight_map g t0))) (eq_ind nat (S (plus n h)) (\lambda
150 (n0: nat).(eq nat (f (plus n h)) (g n0))) (sym_eq nat (g (S (plus n h))) (f
151 (plus n h)) (H1 (plus n h) (le_plus_trans d n h H2))) (plus n (S h))
152 (plus_n_Sm n h)) (lift (S h) d (TLRef n)) (lift_lref_ge n (S h) d H2)) (lift
153 h d (TLRef n)) (lift_lref_ge n h d H2)))))))))))) (\lambda (k: K).(\lambda
154 (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat
155 \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to
156 (eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d
157 m) \to (eq nat (g (S m)) (f m))))) \to (eq nat (weight_map f (lift h d t0))
158 (weight_map g (lift (S h) d t0)))))))))))).(\lambda (t1: T).(\lambda (H0:
159 ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall
160 (g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f
161 m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g
162 (S m)) (f m))))) \to (eq nat (weight_map f (lift h d t1)) (weight_map g (lift
163 (S h) d t1)))))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat
164 \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (m:
165 nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (H2: (eq nat (g d)
166 w)).(\lambda (H3: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f
167 m)))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0
168 t1))) (weight_map g (lift (S h) d (THead k0 t0 t1))))) (\lambda (b:
169 B).(eq_ind_r T (THead (Bind b) (lift h d t0) (lift h (s (Bind b) d) t1))
170 (\lambda (t2: T).(eq nat (weight_map f t2) (weight_map g (lift (S h) d (THead
171 (Bind b) t0 t1))))) (eq_ind_r T (THead (Bind b) (lift (S h) d t0) (lift (S h)
172 (s (Bind b) d) t1)) (\lambda (t2: T).(eq nat (weight_map f (THead (Bind b)
173 (lift h d t0) (lift h (s (Bind b) d) t1))) (weight_map g t2))) (B_ind
174 (\lambda (b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus
175 (weight_map f (lift h d t0)) (weight_map (wadd f (S (weight_map f (lift h d
176 t0)))) (lift h (S d) t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h
177 d t0)) (weight_map (wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S
178 (plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d)
179 t1))))]) (match b0 with [Abbr \Rightarrow (S (plus (weight_map g (lift (S h)
180 d t0)) (weight_map (wadd g (S (weight_map g (lift (S h) d t0)))) (lift (S h)
181 (S d) t1)))) | Abst \Rightarrow (S (plus (weight_map g (lift (S h) d t0))
182 (weight_map (wadd g O) (lift (S h) (S d) t1)))) | Void \Rightarrow (S (plus
183 (weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d)
184 t1))))]))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map
185 (wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) t1))) (plus
186 (weight_map g (lift (S h) d t0)) (weight_map (wadd g (S (weight_map g (lift
187 (S h) d t0)))) (lift (S h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map
188 f (lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map (wadd f (S
189 (weight_map f (lift h d t0)))) (lift h (S d) t1)) (weight_map (wadd g (S
190 (weight_map g (lift (S h) d t0)))) (lift (S h) (S d) t1)) (H h d f g H1 H2
191 H3) (H0 h (S d) (wadd f (S (weight_map f (lift h d t0)))) (wadd g (S
192 (weight_map g (lift (S h) d t0)))) (\lambda (m: nat).(\lambda (H4: (lt m (S
193 d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0)))
194 (\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g (S (weight_map g (lift (S h) d
195 t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (H5: (eq nat m
196 O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift
197 (S h) d t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (f_equal nat
198 nat S (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t0)) (sym_eq
199 nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0)) (H h d f g
200 H1 H2 H3))) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S
201 m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat
202 m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g (S (weight_map g
203 (lift (S h) d t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda
204 (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r
205 nat (S x) (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift (S h) d
206 t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (H1 x H7) m H6))))
207 H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d)
208 m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d
209 n)) (eq nat (g m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (x:
210 nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S
211 x) (\lambda (n: nat).(eq nat (g n) (wadd f (S (weight_map f (lift h d t0)))
212 n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat S (plus
213 (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))) (plus
214 (weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d)
215 t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map g
216 (lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) (weight_map
217 (wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S d) (wadd f O)
218 (wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S d))).(or_ind (eq nat m O)
219 (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))
220 (eq nat (wadd g O m) (wadd f O m)) (\lambda (H5: (eq nat m O)).(eq_ind_r nat
221 O (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (refl_equal nat O) m
222 H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda
223 (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat m (S m0)))
224 (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O m) (wadd f O m)) (\lambda (x:
225 nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r nat (S
226 x) (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (H1 x H7) m H6))))
227 H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d)
228 m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d
229 n)) (eq nat (g m) (wadd f O m)) (\lambda (x: nat).(\lambda (H5: (eq nat m (S
230 x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g
231 n) (wadd f O n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat
232 S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d)
233 t1))) (plus (weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S
234 h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0))
235 (weight_map g (lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1))
236 (weight_map (wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S
237 d) (wadd f O) (wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S
238 d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0)))
239 (\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g O m) (wadd f O m)) (\lambda
240 (H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g O n)
241 (wadd f O n))) (refl_equal nat O) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0:
242 nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda
243 (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O
244 m) (wadd f O m)) (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda
245 (H7: (lt x d)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd g O n)
246 (wadd f O n))) (H1 x H7) m H6)))) H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m:
247 nat).(\lambda (H4: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S
248 n))) (\lambda (n: nat).(le d n)) (eq nat (g m) (wadd f O m)) (\lambda (x:
249 nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S
250 x) (\lambda (n: nat).(eq nat (g n) (wadd f O n))) (H3 x H6) m H5))))
251 (le_gen_S d m H4))))))) b) (lift (S h) d (THead (Bind b) t0 t1)) (lift_head
252 (Bind b) t0 t1 (S h) d)) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind
253 b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0)
254 (lift h (s (Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2)
255 (weight_map g (lift (S h) d (THead (Flat f0) t0 t1))))) (eq_ind_r T (THead
256 (Flat f0) (lift (S h) d t0) (lift (S h) (s (Flat f0) d) t1)) (\lambda (t2:
257 T).(eq nat (weight_map f (THead (Flat f0) (lift h d t0) (lift h (s (Flat f0)
258 d) t1))) (weight_map g t2))) (f_equal nat nat S (plus (weight_map f (lift h d
259 t0)) (weight_map f (lift h d t1))) (plus (weight_map g (lift (S h) d t0))
260 (weight_map g (lift (S h) d t1))) (f_equal2 nat nat nat plus (weight_map f
261 (lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t1))
262 (weight_map g (lift (S h) d t1)) (H h d f g H1 H2 H3) (H0 h d f g H1 H2 H3)))
263 (lift (S h) d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 (S h) d))
264 (lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d)))
267 theorem lift_weight_add_O:
268 \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (f: ((nat \to
269 nat))).(eq nat (weight_map f (lift h O t)) (weight_map (wadd f w) (lift (S h)
272 \lambda (w: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (f: ((nat \to
273 nat))).(lift_weight_add (plus (wadd f w O) O) t h O f (wadd f w) (\lambda (m:
274 nat).(\lambda (H: (lt m O)).(let H0 \def (match H in le return (\lambda (n:
275 nat).(\lambda (_: (le ? n)).((eq nat n O) \to (eq nat (wadd f w m) (f m)))))
276 with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) O)).(let H1 \def (eq_ind
277 nat (S m) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop)
278 with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind
279 (eq nat (wadd f w m) (f m)) H1))) | (le_S m0 H0) \Rightarrow (\lambda (H1:
280 (eq nat (S m0) O)).((let H2 \def (eq_ind nat (S m0) (\lambda (e: nat).(match
281 e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
282 \Rightarrow True])) I O H1) in (False_ind ((le (S m) m0) \to (eq nat (wadd f
283 w m) (f m))) H2)) H0))]) in (H0 (refl_equal nat O))))) (plus_n_O (wadd f w
284 O)) (\lambda (m: nat).(\lambda (_: (le O m)).(refl_equal nat (f m)))))))).
287 \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
288 (d: nat).(tlt t (THead k u (lift h d t)))))))
290 \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
291 (d: nat).(eq_ind nat (weight (lift h d t)) (\lambda (n: nat).(lt n (weight
292 (THead k u (lift h d t))))) (tlt_head_dx k u (lift h d t)) (weight t)
293 (lift_weight t h d)))))).