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 include "basic_1/T/fwd.ma".
19 include "basic_1/tlt/defs.ma".
22 \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n:
23 nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((le v w) \to
24 (\forall (n: nat).(le (wadd f v n) (wadd g w n))))))))
26 \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H:
27 ((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w:
28 nat).(\lambda (H0: (le v w)).(\lambda (n: nat).(let TMP_3 \def (\lambda (n0:
29 nat).(let TMP_1 \def (wadd f v n0) in (let TMP_2 \def (wadd g w n0) in (le
30 TMP_1 TMP_2)))) in (let TMP_4 \def (\lambda (n0: nat).(\lambda (_: (le (wadd
31 f v n0) (wadd g w n0))).(H n0))) in (nat_ind TMP_3 H0 TMP_4 n))))))))).
34 \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n:
35 nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((lt v w) \to
36 (\forall (n: nat).(le (wadd f v n) (wadd g w n))))))))
38 \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H:
39 ((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w:
40 nat).(\lambda (H0: (lt v w)).(\lambda (n: nat).(let TMP_3 \def (\lambda (n0:
41 nat).(let TMP_1 \def (wadd f v n0) in (let TMP_2 \def (wadd g w n0) in (le
42 TMP_1 TMP_2)))) in (let TMP_4 \def (S v) in (let TMP_5 \def (S w) in (let
43 TMP_6 \def (S v) in (let TMP_7 \def (S TMP_6) in (let TMP_8 \def (S w) in
44 (let TMP_9 \def (S v) in (let TMP_10 \def (le_n_S TMP_9 w H0) in (let TMP_11
45 \def (le_S TMP_7 TMP_8 TMP_10) in (let TMP_12 \def (le_S_n TMP_4 TMP_5
46 TMP_11) in (let TMP_13 \def (le_S_n v w TMP_12) in (let TMP_14 \def (\lambda
47 (n0: nat).(\lambda (_: (le (wadd f v n0) (wadd g w n0))).(H n0))) in (nat_ind
48 TMP_3 TMP_13 TMP_14 n))))))))))))))))))).
51 \forall (n: nat).(eq nat (wadd (\lambda (_: nat).O) O n) O)
53 \lambda (n: nat).(let TMP_3 \def (\lambda (n0: nat).(let TMP_1 \def (\lambda
54 (_: nat).O) in (let TMP_2 \def (wadd TMP_1 O n0) in (eq nat TMP_2 O)))) in
55 (let TMP_4 \def (refl_equal nat O) in (let TMP_5 \def (\lambda (n0:
56 nat).(\lambda (_: (eq nat (wadd (\lambda (_: nat).O) O n0) O)).(refl_equal
57 nat O))) in (nat_ind TMP_3 TMP_4 TMP_5 n)))).
60 \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
61 nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t)
64 \lambda (t: T).(let TMP_3 \def (\lambda (t0: T).(\forall (f: ((nat \to
65 nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n))))
66 \to (let TMP_1 \def (weight_map f t0) in (let TMP_2 \def (weight_map g t0) in
67 (le TMP_1 TMP_2))))))) in (let TMP_6 \def (\lambda (n: nat).(\lambda (f:
68 ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall (n0:
69 nat).(le (f n0) (g n0))))).(let TMP_4 \def (TSort n) in (let TMP_5 \def
70 (weight_map g TMP_4) in (le_O_n TMP_5))))))) in (let TMP_7 \def (\lambda (n:
71 nat).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H:
72 ((\forall (n0: nat).(le (f n0) (g n0))))).(H n))))) in (let TMP_144 \def
73 (\lambda (k: K).(let TMP_12 \def (\lambda (k0: K).(\forall (t0: T).(((\forall
74 (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f
75 n) (g n)))) \to (le (weight_map f t0) (weight_map g t0)))))) \to (\forall
76 (t1: T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
77 nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t1)
78 (weight_map g t1)))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat
79 \to nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (let TMP_8 \def (THead
80 k0 t0 t1) in (let TMP_9 \def (weight_map f TMP_8) in (let TMP_10 \def (THead
81 k0 t0 t1) in (let TMP_11 \def (weight_map g TMP_10) in (le TMP_9
82 TMP_11))))))))))))) in (let TMP_129 \def (\lambda (b: B).(let TMP_43 \def
83 (\lambda (b0: B).(\forall (t0: T).(((\forall (f: ((nat \to nat))).(\forall
84 (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le
85 (weight_map f t0) (weight_map g t0)))))) \to (\forall (t1: T).(((\forall (f:
86 ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n)
87 (g n)))) \to (le (weight_map f t1) (weight_map g t1)))))) \to (\forall (f:
88 ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n)
89 (g n)))) \to (let TMP_27 \def (match b0 with [Abbr \Rightarrow (let TMP_21
90 \def (weight_map f t0) in (let TMP_22 \def (weight_map f t0) in (let TMP_23
91 \def (S TMP_22) in (let TMP_24 \def (wadd f TMP_23) in (let TMP_25 \def
92 (weight_map TMP_24 t1) in (let TMP_26 \def (plus TMP_21 TMP_25) in (S
93 TMP_26))))))) | Abst \Rightarrow (let TMP_17 \def (weight_map f t0) in (let
94 TMP_18 \def (wadd f O) in (let TMP_19 \def (weight_map TMP_18 t1) in (let
95 TMP_20 \def (plus TMP_17 TMP_19) in (S TMP_20))))) | Void \Rightarrow (let
96 TMP_13 \def (weight_map f t0) in (let TMP_14 \def (wadd f O) in (let TMP_15
97 \def (weight_map TMP_14 t1) in (let TMP_16 \def (plus TMP_13 TMP_15) in (S
98 TMP_16)))))]) in (let TMP_42 \def (match b0 with [Abbr \Rightarrow (let
99 TMP_36 \def (weight_map g t0) in (let TMP_37 \def (weight_map g t0) in (let
100 TMP_38 \def (S TMP_37) in (let TMP_39 \def (wadd g TMP_38) in (let TMP_40
101 \def (weight_map TMP_39 t1) in (let TMP_41 \def (plus TMP_36 TMP_40) in (S
102 TMP_41))))))) | Abst \Rightarrow (let TMP_32 \def (weight_map g t0) in (let
103 TMP_33 \def (wadd g O) in (let TMP_34 \def (weight_map TMP_33 t1) in (let
104 TMP_35 \def (plus TMP_32 TMP_34) in (S TMP_35))))) | Void \Rightarrow (let
105 TMP_28 \def (weight_map g t0) in (let TMP_29 \def (wadd g O) in (let TMP_30
106 \def (weight_map TMP_29 t1) in (let TMP_31 \def (plus TMP_28 TMP_30) in (S
107 TMP_31)))))]) in (le TMP_27 TMP_42))))))))))) in (let TMP_84 \def (\lambda
108 (t0: T).(\lambda (H: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
109 nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t0)
110 (weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f: ((nat
111 \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g
112 n)))) \to (le (weight_map f t1) (weight_map g t1))))))).(\lambda (f: ((nat
113 \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le
114 (f n) (g n))))).(let TMP_44 \def (weight_map f t0) in (let TMP_45 \def
115 (weight_map f t0) in (let TMP_46 \def (S TMP_45) in (let TMP_47 \def (wadd f
116 TMP_46) in (let TMP_48 \def (weight_map TMP_47 t1) in (let TMP_49 \def (plus
117 TMP_44 TMP_48) in (let TMP_50 \def (weight_map g t0) in (let TMP_51 \def
118 (weight_map g t0) in (let TMP_52 \def (S TMP_51) in (let TMP_53 \def (wadd g
119 TMP_52) in (let TMP_54 \def (weight_map TMP_53 t1) in (let TMP_55 \def (plus
120 TMP_50 TMP_54) in (let TMP_56 \def (weight_map f t0) in (let TMP_57 \def
121 (weight_map g t0) in (let TMP_58 \def (weight_map f t0) in (let TMP_59 \def
122 (S TMP_58) in (let TMP_60 \def (wadd f TMP_59) in (let TMP_61 \def
123 (weight_map TMP_60 t1) in (let TMP_62 \def (weight_map g t0) in (let TMP_63
124 \def (S TMP_62) in (let TMP_64 \def (wadd g TMP_63) in (let TMP_65 \def
125 (weight_map TMP_64 t1) in (let TMP_66 \def (H f g H1) in (let TMP_67 \def
126 (weight_map f t0) in (let TMP_68 \def (S TMP_67) in (let TMP_69 \def (wadd f
127 TMP_68) in (let TMP_70 \def (weight_map g t0) in (let TMP_71 \def (S TMP_70)
128 in (let TMP_72 \def (wadd g TMP_71) in (let TMP_81 \def (\lambda (n:
129 nat).(let TMP_73 \def (weight_map f t0) in (let TMP_74 \def (S TMP_73) in
130 (let TMP_75 \def (weight_map g t0) in (let TMP_76 \def (S TMP_75) in (let
131 TMP_77 \def (weight_map f t0) in (let TMP_78 \def (weight_map g t0) in (let
132 TMP_79 \def (H f g H1) in (let TMP_80 \def (le_n_S TMP_77 TMP_78 TMP_79) in
133 (wadd_le f g H1 TMP_74 TMP_76 TMP_80 n)))))))))) in (let TMP_82 \def (H0
134 TMP_69 TMP_72 TMP_81) in (let TMP_83 \def (le_plus_plus TMP_56 TMP_57 TMP_61
135 TMP_65 TMP_66 TMP_82) in (le_n_S TMP_49 TMP_55
136 TMP_83)))))))))))))))))))))))))))))))))))))))) in (let TMP_106 \def (\lambda
137 (t0: T).(\lambda (H: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
138 nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t0)
139 (weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f: ((nat
140 \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g
141 n)))) \to (le (weight_map f t1) (weight_map g t1))))))).(\lambda (f: ((nat
142 \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le
143 (f n) (g n))))).(let TMP_85 \def (weight_map f t0) in (let TMP_86 \def (wadd
144 f O) in (let TMP_87 \def (weight_map TMP_86 t1) in (let TMP_88 \def (plus
145 TMP_85 TMP_87) in (let TMP_89 \def (weight_map g t0) in (let TMP_90 \def
146 (wadd g O) in (let TMP_91 \def (weight_map TMP_90 t1) in (let TMP_92 \def
147 (plus TMP_89 TMP_91) in (let TMP_93 \def (weight_map f t0) in (let TMP_94
148 \def (weight_map g t0) in (let TMP_95 \def (wadd f O) in (let TMP_96 \def
149 (weight_map TMP_95 t1) in (let TMP_97 \def (wadd g O) in (let TMP_98 \def
150 (weight_map TMP_97 t1) in (let TMP_99 \def (H f g H1) in (let TMP_100 \def
151 (wadd f O) in (let TMP_101 \def (wadd g O) in (let TMP_103 \def (\lambda (n:
152 nat).(let TMP_102 \def (le_O_n O) in (wadd_le f g H1 O O TMP_102 n))) in (let
153 TMP_104 \def (H0 TMP_100 TMP_101 TMP_103) in (let TMP_105 \def (le_plus_plus
154 TMP_93 TMP_94 TMP_96 TMP_98 TMP_99 TMP_104) in (le_n_S TMP_88 TMP_92
155 TMP_105)))))))))))))))))))))))))))) in (let TMP_128 \def (\lambda (t0:
156 T).(\lambda (H: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
157 nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t0)
158 (weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f: ((nat
159 \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g
160 n)))) \to (le (weight_map f t1) (weight_map g t1))))))).(\lambda (f: ((nat
161 \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le
162 (f n) (g n))))).(let TMP_107 \def (weight_map f t0) in (let TMP_108 \def
163 (wadd f O) in (let TMP_109 \def (weight_map TMP_108 t1) in (let TMP_110 \def
164 (plus TMP_107 TMP_109) in (let TMP_111 \def (weight_map g t0) in (let TMP_112
165 \def (wadd g O) in (let TMP_113 \def (weight_map TMP_112 t1) in (let TMP_114
166 \def (plus TMP_111 TMP_113) in (let TMP_115 \def (weight_map f t0) in (let
167 TMP_116 \def (weight_map g t0) in (let TMP_117 \def (wadd f O) in (let
168 TMP_118 \def (weight_map TMP_117 t1) in (let TMP_119 \def (wadd g O) in (let
169 TMP_120 \def (weight_map TMP_119 t1) in (let TMP_121 \def (H f g H1) in (let
170 TMP_122 \def (wadd f O) in (let TMP_123 \def (wadd g O) in (let TMP_125 \def
171 (\lambda (n: nat).(let TMP_124 \def (le_O_n O) in (wadd_le f g H1 O O TMP_124
172 n))) in (let TMP_126 \def (H0 TMP_122 TMP_123 TMP_125) in (let TMP_127 \def
173 (le_plus_plus TMP_115 TMP_116 TMP_118 TMP_120 TMP_121 TMP_126) in (le_n_S
174 TMP_110 TMP_114 TMP_127)))))))))))))))))))))))))))) in (B_ind TMP_43 TMP_84
175 TMP_106 TMP_128 b)))))) in (let TMP_143 \def (\lambda (_: F).(\lambda (t0:
176 T).(\lambda (H: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to
177 nat))).(((\forall (n: nat).(le (f0 n) (g n)))) \to (le (weight_map f0 t0)
178 (weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f0: ((nat
179 \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f0 n) (g
180 n)))) \to (le (weight_map f0 t1) (weight_map g t1))))))).(\lambda (f0: ((nat
181 \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le
182 (f0 n) (g n))))).(let TMP_130 \def (weight_map f0 t0) in (let TMP_131 \def
183 (weight_map f0 t1) in (let TMP_132 \def (plus TMP_130 TMP_131) in (let
184 TMP_133 \def (weight_map g t0) in (let TMP_134 \def (weight_map g t1) in (let
185 TMP_135 \def (plus TMP_133 TMP_134) in (let TMP_136 \def (weight_map f0 t0)
186 in (let TMP_137 \def (weight_map g t0) in (let TMP_138 \def (weight_map f0
187 t1) in (let TMP_139 \def (weight_map g t1) in (let TMP_140 \def (H f0 g H1)
188 in (let TMP_141 \def (H0 f0 g H1) in (let TMP_142 \def (le_plus_plus TMP_136
189 TMP_137 TMP_138 TMP_139 TMP_140 TMP_141) in (le_n_S TMP_132 TMP_135
190 TMP_142)))))))))))))))))))))) in (K_ind TMP_12 TMP_129 TMP_143 k))))) in
191 (T_ind TMP_3 TMP_6 TMP_7 TMP_144 t))))).
194 \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
195 nat))).(((\forall (n: nat).(eq nat (f n) (g n)))) \to (eq nat (weight_map f
196 t) (weight_map g t)))))
198 \lambda (t: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
199 nat))).(\lambda (H: ((\forall (n: nat).(eq nat (f n) (g n))))).(let TMP_1
200 \def (weight_map f t) in (let TMP_2 \def (weight_map g t) in (let TMP_10 \def
201 (\lambda (n: nat).(let TMP_3 \def (g n) in (let TMP_5 \def (\lambda (n0:
202 nat).(let TMP_4 \def (g n) in (le n0 TMP_4))) in (let TMP_6 \def (g n) in
203 (let TMP_7 \def (le_n TMP_6) in (let TMP_8 \def (f n) in (let TMP_9 \def (H
204 n) in (eq_ind_r nat TMP_3 TMP_5 TMP_7 TMP_8 TMP_9)))))))) in (let TMP_11 \def
205 (weight_le t f g TMP_10) in (let TMP_19 \def (\lambda (n: nat).(let TMP_12
206 \def (g n) in (let TMP_14 \def (\lambda (n0: nat).(let TMP_13 \def (g n) in
207 (le TMP_13 n0))) in (let TMP_15 \def (g n) in (let TMP_16 \def (le_n TMP_15)
208 in (let TMP_17 \def (f n) in (let TMP_18 \def (H n) in (eq_ind_r nat TMP_12
209 TMP_14 TMP_16 TMP_17 TMP_18)))))))) in (let TMP_20 \def (weight_le t g f
210 TMP_19) in (le_antisym TMP_1 TMP_2 TMP_11 TMP_20)))))))))).
212 theorem weight_add_O:
213 \forall (t: T).(eq nat (weight_map (wadd (\lambda (_: nat).O) O) t)
214 (weight_map (\lambda (_: nat).O) t))
216 \lambda (t: T).(let TMP_1 \def (\lambda (_: nat).O) in (let TMP_2 \def (wadd
217 TMP_1 O) in (let TMP_3 \def (\lambda (_: nat).O) in (let TMP_4 \def (\lambda
218 (n: nat).(wadd_O n)) in (weight_eq t TMP_2 TMP_3 TMP_4))))).
220 theorem weight_add_S:
221 \forall (t: T).(\forall (m: nat).(le (weight_map (wadd (\lambda (_: nat).O)
222 O) t) (weight_map (wadd (\lambda (_: nat).O) (S m)) t)))
224 \lambda (t: T).(\lambda (m: nat).(let TMP_1 \def (\lambda (_: nat).O) in
225 (let TMP_2 \def (wadd TMP_1 O) in (let TMP_3 \def (\lambda (_: nat).O) in
226 (let TMP_4 \def (S m) in (let TMP_5 \def (wadd TMP_3 TMP_4) in (let TMP_12
227 \def (\lambda (n: nat).(let TMP_6 \def (\lambda (_: nat).O) in (let TMP_7
228 \def (\lambda (_: nat).O) in (let TMP_8 \def (\lambda (_: nat).(le_O_n O)) in
229 (let TMP_9 \def (S m) in (let TMP_10 \def (S m) in (let TMP_11 \def (le_O_n
230 TMP_10) in (wadd_le TMP_6 TMP_7 TMP_8 O TMP_9 TMP_11 n)))))))) in (weight_le
231 t TMP_2 TMP_5 TMP_12)))))))).
234 \forall (v: T).(\forall (u: T).(\forall (t: T).((tlt u v) \to ((tlt v t) \to
237 \lambda (v: T).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (lt (weight u)
238 (weight v))).(\lambda (H0: (lt (weight v) (weight t))).(let TMP_1 \def
239 (weight u) in (let TMP_2 \def (weight v) in (let TMP_3 \def (weight t) in
240 (lt_trans TMP_1 TMP_2 TMP_3 H H0)))))))).
243 \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt u (THead k u t))))
245 \lambda (k: K).(let TMP_6 \def (\lambda (k0: K).(\forall (u: T).(\forall (t:
246 T).(let TMP_1 \def (\lambda (_: nat).O) in (let TMP_2 \def (weight_map TMP_1
247 u) in (let TMP_3 \def (\lambda (_: nat).O) in (let TMP_4 \def (THead k0 u t)
248 in (let TMP_5 \def (weight_map TMP_3 TMP_4) in (lt TMP_2 TMP_5))))))))) in
249 (let TMP_83 \def (\lambda (b: B).(let TMP_31 \def (\lambda (b0: B).(\forall
250 (u: T).(\forall (t: T).(let TMP_7 \def (\lambda (_: nat).O) in (let TMP_8
251 \def (weight_map TMP_7 u) in (let TMP_30 \def (match b0 with [Abbr
252 \Rightarrow (let TMP_21 \def (\lambda (_: nat).O) in (let TMP_22 \def
253 (weight_map TMP_21 u) in (let TMP_23 \def (\lambda (_: nat).O) in (let TMP_24
254 \def (\lambda (_: nat).O) in (let TMP_25 \def (weight_map TMP_24 u) in (let
255 TMP_26 \def (S TMP_25) in (let TMP_27 \def (wadd TMP_23 TMP_26) in (let
256 TMP_28 \def (weight_map TMP_27 t) in (let TMP_29 \def (plus TMP_22 TMP_28) in
257 (S TMP_29)))))))))) | Abst \Rightarrow (let TMP_15 \def (\lambda (_: nat).O)
258 in (let TMP_16 \def (weight_map TMP_15 u) in (let TMP_17 \def (\lambda (_:
259 nat).O) in (let TMP_18 \def (wadd TMP_17 O) in (let TMP_19 \def (weight_map
260 TMP_18 t) in (let TMP_20 \def (plus TMP_16 TMP_19) in (S TMP_20))))))) | Void
261 \Rightarrow (let TMP_9 \def (\lambda (_: nat).O) in (let TMP_10 \def
262 (weight_map TMP_9 u) in (let TMP_11 \def (\lambda (_: nat).O) in (let TMP_12
263 \def (wadd TMP_11 O) in (let TMP_13 \def (weight_map TMP_12 t) in (let TMP_14
264 \def (plus TMP_10 TMP_13) in (S TMP_14)))))))]) in (lt TMP_8 TMP_30))))))) in
265 (let TMP_52 \def (\lambda (u: T).(\lambda (t: T).(let TMP_32 \def (\lambda
266 (_: nat).O) in (let TMP_33 \def (weight_map TMP_32 u) in (let TMP_34 \def
267 (\lambda (_: nat).O) in (let TMP_35 \def (weight_map TMP_34 u) in (let TMP_36
268 \def (\lambda (_: nat).O) in (let TMP_37 \def (\lambda (_: nat).O) in (let
269 TMP_38 \def (weight_map TMP_37 u) in (let TMP_39 \def (S TMP_38) in (let
270 TMP_40 \def (wadd TMP_36 TMP_39) in (let TMP_41 \def (weight_map TMP_40 t) in
271 (let TMP_42 \def (plus TMP_35 TMP_41) in (let TMP_43 \def (\lambda (_:
272 nat).O) in (let TMP_44 \def (weight_map TMP_43 u) in (let TMP_45 \def
273 (\lambda (_: nat).O) in (let TMP_46 \def (\lambda (_: nat).O) in (let TMP_47
274 \def (weight_map TMP_46 u) in (let TMP_48 \def (S TMP_47) in (let TMP_49 \def
275 (wadd TMP_45 TMP_48) in (let TMP_50 \def (weight_map TMP_49 t) in (let TMP_51
276 \def (le_plus_l TMP_44 TMP_50) in (le_n_S TMP_33 TMP_42
277 TMP_51))))))))))))))))))))))) in (let TMP_67 \def (\lambda (u: T).(\lambda
278 (t: T).(let TMP_53 \def (\lambda (_: nat).O) in (let TMP_54 \def (weight_map
279 TMP_53 u) in (let TMP_55 \def (\lambda (_: nat).O) in (let TMP_56 \def
280 (weight_map TMP_55 u) in (let TMP_57 \def (\lambda (_: nat).O) in (let TMP_58
281 \def (wadd TMP_57 O) in (let TMP_59 \def (weight_map TMP_58 t) in (let TMP_60
282 \def (plus TMP_56 TMP_59) in (let TMP_61 \def (\lambda (_: nat).O) in (let
283 TMP_62 \def (weight_map TMP_61 u) in (let TMP_63 \def (\lambda (_: nat).O) in
284 (let TMP_64 \def (wadd TMP_63 O) in (let TMP_65 \def (weight_map TMP_64 t) in
285 (let TMP_66 \def (le_plus_l TMP_62 TMP_65) in (le_n_S TMP_54 TMP_60
286 TMP_66))))))))))))))))) in (let TMP_82 \def (\lambda (u: T).(\lambda (t:
287 T).(let TMP_68 \def (\lambda (_: nat).O) in (let TMP_69 \def (weight_map
288 TMP_68 u) in (let TMP_70 \def (\lambda (_: nat).O) in (let TMP_71 \def
289 (weight_map TMP_70 u) in (let TMP_72 \def (\lambda (_: nat).O) in (let TMP_73
290 \def (wadd TMP_72 O) in (let TMP_74 \def (weight_map TMP_73 t) in (let TMP_75
291 \def (plus TMP_71 TMP_74) in (let TMP_76 \def (\lambda (_: nat).O) in (let
292 TMP_77 \def (weight_map TMP_76 u) in (let TMP_78 \def (\lambda (_: nat).O) in
293 (let TMP_79 \def (wadd TMP_78 O) in (let TMP_80 \def (weight_map TMP_79 t) in
294 (let TMP_81 \def (le_plus_l TMP_77 TMP_80) in (le_n_S TMP_69 TMP_75
295 TMP_81))))))))))))))))) in (B_ind TMP_31 TMP_52 TMP_67 TMP_82 b)))))) in (let
296 TMP_96 \def (\lambda (_: F).(\lambda (u: T).(\lambda (t: T).(let TMP_84 \def
297 (\lambda (_: nat).O) in (let TMP_85 \def (weight_map TMP_84 u) in (let TMP_86
298 \def (\lambda (_: nat).O) in (let TMP_87 \def (weight_map TMP_86 u) in (let
299 TMP_88 \def (\lambda (_: nat).O) in (let TMP_89 \def (weight_map TMP_88 t) in
300 (let TMP_90 \def (plus TMP_87 TMP_89) in (let TMP_91 \def (\lambda (_:
301 nat).O) in (let TMP_92 \def (weight_map TMP_91 u) in (let TMP_93 \def
302 (\lambda (_: nat).O) in (let TMP_94 \def (weight_map TMP_93 t) in (let TMP_95
303 \def (le_plus_l TMP_92 TMP_94) in (le_n_S TMP_85 TMP_90
304 TMP_95)))))))))))))))) in (K_ind TMP_6 TMP_83 TMP_96 k)))).
307 \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt t (THead k u t))))
309 \lambda (k: K).(let TMP_6 \def (\lambda (k0: K).(\forall (u: T).(\forall (t:
310 T).(let TMP_1 \def (\lambda (_: nat).O) in (let TMP_2 \def (weight_map TMP_1
311 t) in (let TMP_3 \def (\lambda (_: nat).O) in (let TMP_4 \def (THead k0 u t)
312 in (let TMP_5 \def (weight_map TMP_3 TMP_4) in (lt TMP_2 TMP_5))))))))) in
313 (let TMP_161 \def (\lambda (b: B).(let TMP_31 \def (\lambda (b0: B).(\forall
314 (u: T).(\forall (t: T).(let TMP_7 \def (\lambda (_: nat).O) in (let TMP_8
315 \def (weight_map TMP_7 t) in (let TMP_30 \def (match b0 with [Abbr
316 \Rightarrow (let TMP_21 \def (\lambda (_: nat).O) in (let TMP_22 \def
317 (weight_map TMP_21 u) in (let TMP_23 \def (\lambda (_: nat).O) in (let TMP_24
318 \def (\lambda (_: nat).O) in (let TMP_25 \def (weight_map TMP_24 u) in (let
319 TMP_26 \def (S TMP_25) in (let TMP_27 \def (wadd TMP_23 TMP_26) in (let
320 TMP_28 \def (weight_map TMP_27 t) in (let TMP_29 \def (plus TMP_22 TMP_28) in
321 (S TMP_29)))))))))) | Abst \Rightarrow (let TMP_15 \def (\lambda (_: nat).O)
322 in (let TMP_16 \def (weight_map TMP_15 u) in (let TMP_17 \def (\lambda (_:
323 nat).O) in (let TMP_18 \def (wadd TMP_17 O) in (let TMP_19 \def (weight_map
324 TMP_18 t) in (let TMP_20 \def (plus TMP_16 TMP_19) in (S TMP_20))))))) | Void
325 \Rightarrow (let TMP_9 \def (\lambda (_: nat).O) in (let TMP_10 \def
326 (weight_map TMP_9 u) in (let TMP_11 \def (\lambda (_: nat).O) in (let TMP_12
327 \def (wadd TMP_11 O) in (let TMP_13 \def (weight_map TMP_12 t) in (let TMP_14
328 \def (plus TMP_10 TMP_13) in (S TMP_14)))))))]) in (lt TMP_8 TMP_30))))))) in
329 (let TMP_106 \def (\lambda (u: T).(\lambda (t: T).(let TMP_32 \def (\lambda
330 (_: nat).O) in (let TMP_33 \def (weight_map TMP_32 t) in (let TMP_34 \def
331 (\lambda (_: nat).O) in (let TMP_35 \def (weight_map TMP_34 t) in (let TMP_36
332 \def (S TMP_35) in (let TMP_37 \def (\lambda (_: nat).O) in (let TMP_38 \def
333 (weight_map TMP_37 u) in (let TMP_39 \def (\lambda (_: nat).O) in (let TMP_40
334 \def (\lambda (_: nat).O) in (let TMP_41 \def (weight_map TMP_40 u) in (let
335 TMP_42 \def (S TMP_41) in (let TMP_43 \def (wadd TMP_39 TMP_42) in (let
336 TMP_44 \def (weight_map TMP_43 t) in (let TMP_45 \def (plus TMP_38 TMP_44) in
337 (let TMP_46 \def (S TMP_45) in (let TMP_47 \def (\lambda (_: nat).O) in (let
338 TMP_48 \def (weight_map TMP_47 t) in (let TMP_49 \def (lt_n_Sn TMP_48) in
339 (let TMP_50 \def (\lambda (_: nat).O) in (let TMP_51 \def (weight_map TMP_50
340 t) in (let TMP_52 \def (\lambda (_: nat).O) in (let TMP_53 \def (weight_map
341 TMP_52 u) in (let TMP_54 \def (\lambda (_: nat).O) in (let TMP_55 \def
342 (\lambda (_: nat).O) in (let TMP_56 \def (weight_map TMP_55 u) in (let TMP_57
343 \def (S TMP_56) in (let TMP_58 \def (wadd TMP_54 TMP_57) in (let TMP_59 \def
344 (weight_map TMP_58 t) in (let TMP_60 \def (plus TMP_53 TMP_59) in (let TMP_61
345 \def (\lambda (_: nat).O) in (let TMP_62 \def (weight_map TMP_61 t) in (let
346 TMP_63 \def (\lambda (_: nat).O) in (let TMP_64 \def (\lambda (_: nat).O) in
347 (let TMP_65 \def (weight_map TMP_64 u) in (let TMP_66 \def (S TMP_65) in (let
348 TMP_67 \def (wadd TMP_63 TMP_66) in (let TMP_68 \def (weight_map TMP_67 t) in
349 (let TMP_69 \def (\lambda (_: nat).O) in (let TMP_70 \def (weight_map TMP_69
350 u) in (let TMP_71 \def (\lambda (_: nat).O) in (let TMP_72 \def (\lambda (_:
351 nat).O) in (let TMP_73 \def (weight_map TMP_72 u) in (let TMP_74 \def (S
352 TMP_73) in (let TMP_75 \def (wadd TMP_71 TMP_74) in (let TMP_76 \def
353 (weight_map TMP_75 t) in (let TMP_77 \def (plus TMP_70 TMP_76) in (let TMP_78
354 \def (\lambda (_: nat).O) in (let TMP_79 \def (wadd TMP_78 O) in (let TMP_80
355 \def (weight_map TMP_79 t) in (let TMP_87 \def (\lambda (n: nat).(let TMP_81
356 \def (\lambda (_: nat).O) in (let TMP_82 \def (\lambda (_: nat).O) in (let
357 TMP_83 \def (weight_map TMP_82 u) in (let TMP_84 \def (S TMP_83) in (let
358 TMP_85 \def (wadd TMP_81 TMP_84) in (let TMP_86 \def (weight_map TMP_85 t) in
359 (le n TMP_86)))))))) in (let TMP_88 \def (\lambda (_: nat).O) in (let TMP_89
360 \def (weight_map TMP_88 u) in (let TMP_90 \def (weight_add_S t TMP_89) in
361 (let TMP_91 \def (\lambda (_: nat).O) in (let TMP_92 \def (weight_map TMP_91
362 t) in (let TMP_93 \def (weight_add_O t) in (let TMP_94 \def (eq_ind nat
363 TMP_80 TMP_87 TMP_90 TMP_92 TMP_93) in (let TMP_95 \def (\lambda (_: nat).O)
364 in (let TMP_96 \def (weight_map TMP_95 u) in (let TMP_97 \def (\lambda (_:
365 nat).O) in (let TMP_98 \def (\lambda (_: nat).O) in (let TMP_99 \def
366 (weight_map TMP_98 u) in (let TMP_100 \def (S TMP_99) in (let TMP_101 \def
367 (wadd TMP_97 TMP_100) in (let TMP_102 \def (weight_map TMP_101 t) in (let
368 TMP_103 \def (le_plus_r TMP_96 TMP_102) in (let TMP_104 \def (le_trans TMP_62
369 TMP_68 TMP_77 TMP_94 TMP_103) in (let TMP_105 \def (le_n_S TMP_51 TMP_60
370 TMP_104) in (lt_le_trans TMP_33 TMP_36 TMP_46 TMP_49
371 TMP_105)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
372 in (let TMP_133 \def (\lambda (u: T).(\lambda (t: T).(let TMP_107 \def
373 (\lambda (_: nat).O) in (let TMP_108 \def (weight_map TMP_107 t) in (let
374 TMP_115 \def (\lambda (n: nat).(let TMP_109 \def (\lambda (_: nat).O) in (let
375 TMP_110 \def (weight_map TMP_109 t) in (let TMP_111 \def (\lambda (_: nat).O)
376 in (let TMP_112 \def (weight_map TMP_111 u) in (let TMP_113 \def (plus
377 TMP_112 n) in (let TMP_114 \def (S TMP_113) in (lt TMP_110 TMP_114)))))))) in
378 (let TMP_116 \def (\lambda (_: nat).O) in (let TMP_117 \def (weight_map
379 TMP_116 t) in (let TMP_118 \def (\lambda (_: nat).O) in (let TMP_119 \def
380 (weight_map TMP_118 u) in (let TMP_120 \def (\lambda (_: nat).O) in (let
381 TMP_121 \def (weight_map TMP_120 t) in (let TMP_122 \def (plus TMP_119
382 TMP_121) in (let TMP_123 \def (\lambda (_: nat).O) in (let TMP_124 \def
383 (weight_map TMP_123 u) in (let TMP_125 \def (\lambda (_: nat).O) in (let
384 TMP_126 \def (weight_map TMP_125 t) in (let TMP_127 \def (le_plus_r TMP_124
385 TMP_126) in (let TMP_128 \def (le_n_S TMP_117 TMP_122 TMP_127) in (let
386 TMP_129 \def (\lambda (_: nat).O) in (let TMP_130 \def (wadd TMP_129 O) in
387 (let TMP_131 \def (weight_map TMP_130 t) in (let TMP_132 \def (weight_add_O
388 t) in (eq_ind_r nat TMP_108 TMP_115 TMP_128 TMP_131
389 TMP_132))))))))))))))))))))))) in (let TMP_160 \def (\lambda (u: T).(\lambda
390 (t: T).(let TMP_134 \def (\lambda (_: nat).O) in (let TMP_135 \def
391 (weight_map TMP_134 t) in (let TMP_142 \def (\lambda (n: nat).(let TMP_136
392 \def (\lambda (_: nat).O) in (let TMP_137 \def (weight_map TMP_136 t) in (let
393 TMP_138 \def (\lambda (_: nat).O) in (let TMP_139 \def (weight_map TMP_138 u)
394 in (let TMP_140 \def (plus TMP_139 n) in (let TMP_141 \def (S TMP_140) in (lt
395 TMP_137 TMP_141)))))))) in (let TMP_143 \def (\lambda (_: nat).O) in (let
396 TMP_144 \def (weight_map TMP_143 t) in (let TMP_145 \def (\lambda (_: nat).O)
397 in (let TMP_146 \def (weight_map TMP_145 u) in (let TMP_147 \def (\lambda (_:
398 nat).O) in (let TMP_148 \def (weight_map TMP_147 t) in (let TMP_149 \def
399 (plus TMP_146 TMP_148) in (let TMP_150 \def (\lambda (_: nat).O) in (let
400 TMP_151 \def (weight_map TMP_150 u) in (let TMP_152 \def (\lambda (_: nat).O)
401 in (let TMP_153 \def (weight_map TMP_152 t) in (let TMP_154 \def (le_plus_r
402 TMP_151 TMP_153) in (let TMP_155 \def (le_n_S TMP_144 TMP_149 TMP_154) in
403 (let TMP_156 \def (\lambda (_: nat).O) in (let TMP_157 \def (wadd TMP_156 O)
404 in (let TMP_158 \def (weight_map TMP_157 t) in (let TMP_159 \def
405 (weight_add_O t) in (eq_ind_r nat TMP_135 TMP_142 TMP_155 TMP_158
406 TMP_159))))))))))))))))))))))) in (B_ind TMP_31 TMP_106 TMP_133 TMP_160
407 b)))))) in (let TMP_174 \def (\lambda (_: F).(\lambda (u: T).(\lambda (t:
408 T).(let TMP_162 \def (\lambda (_: nat).O) in (let TMP_163 \def (weight_map
409 TMP_162 t) in (let TMP_164 \def (\lambda (_: nat).O) in (let TMP_165 \def
410 (weight_map TMP_164 u) in (let TMP_166 \def (\lambda (_: nat).O) in (let
411 TMP_167 \def (weight_map TMP_166 t) in (let TMP_168 \def (plus TMP_165
412 TMP_167) in (let TMP_169 \def (\lambda (_: nat).O) in (let TMP_170 \def
413 (weight_map TMP_169 u) in (let TMP_171 \def (\lambda (_: nat).O) in (let
414 TMP_172 \def (weight_map TMP_171 t) in (let TMP_173 \def (le_plus_r TMP_170
415 TMP_172) in (le_n_S TMP_163 TMP_168 TMP_173)))))))))))))))) in (K_ind TMP_6
416 TMP_161 TMP_174 k)))).