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/leq/props.ma".
20 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
21 (asucc g a1) (asucc g a2)))))
23 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
24 a2)).(let TMP_3 \def (\lambda (a: A).(\lambda (a0: A).(let TMP_1 \def (asucc
25 g a) in (let TMP_2 \def (asucc g a0) in (leq g TMP_1 TMP_2))))) in (let
26 TMP_186 \def (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1:
27 nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort
28 h1 n1) k) (aplus g (ASort h2 n2) k))).(let TMP_8 \def (\lambda (n: nat).((eq
29 A (aplus g (ASort n n1) k) (aplus g (ASort h2 n2) k)) \to (let TMP_5 \def
30 (match n with [O \Rightarrow (let TMP_4 \def (next g n1) in (ASort O TMP_4))
31 | (S h) \Rightarrow (ASort h n1)]) in (let TMP_7 \def (match h2 with [O
32 \Rightarrow (let TMP_6 \def (next g n2) in (ASort O TMP_6)) | (S h)
33 \Rightarrow (ASort h n2)]) in (leq g TMP_5 TMP_7))))) in (let TMP_97 \def
34 (\lambda (H1: (eq A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(let
35 TMP_13 \def (\lambda (n: nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort
36 n n2) k)) \to (let TMP_9 \def (next g n1) in (let TMP_10 \def (ASort O TMP_9)
37 in (let TMP_12 \def (match n with [O \Rightarrow (let TMP_11 \def (next g n2)
38 in (ASort O TMP_11)) | (S h) \Rightarrow (ASort h n2)]) in (leq g TMP_10
39 TMP_12)))))) in (let TMP_54 \def (\lambda (H2: (eq A (aplus g (ASort O n1) k)
40 (aplus g (ASort O n2) k))).(let TMP_14 \def (next g n1) in (let TMP_15 \def
41 (next g n2) in (let TMP_16 \def (ASort O n1) in (let TMP_17 \def (S k) in
42 (let TMP_18 \def (aplus g TMP_16 TMP_17) in (let TMP_22 \def (\lambda (a:
43 A).(let TMP_19 \def (next g n2) in (let TMP_20 \def (ASort O TMP_19) in (let
44 TMP_21 \def (aplus g TMP_20 k) in (eq A a TMP_21))))) in (let TMP_23 \def
45 (ASort O n2) in (let TMP_24 \def (S k) in (let TMP_25 \def (aplus g TMP_23
46 TMP_24) in (let TMP_29 \def (\lambda (a: A).(let TMP_26 \def (ASort O n1) in
47 (let TMP_27 \def (S k) in (let TMP_28 \def (aplus g TMP_26 TMP_27) in (eq A
48 TMP_28 a))))) in (let TMP_30 \def (ASort O n2) in (let TMP_31 \def (aplus g
49 TMP_30 k) in (let TMP_36 \def (\lambda (a: A).(let TMP_32 \def (asucc g a) in
50 (let TMP_33 \def (ASort O n2) in (let TMP_34 \def (aplus g TMP_33 k) in (let
51 TMP_35 \def (asucc g TMP_34) in (eq A TMP_32 TMP_35)))))) in (let TMP_37 \def
52 (ASort O n2) in (let TMP_38 \def (aplus g TMP_37 k) in (let TMP_39 \def
53 (asucc g TMP_38) in (let TMP_40 \def (refl_equal A TMP_39) in (let TMP_41
54 \def (ASort O n1) in (let TMP_42 \def (aplus g TMP_41 k) in (let TMP_43 \def
55 (eq_ind_r A TMP_31 TMP_36 TMP_40 TMP_42 H2) in (let TMP_44 \def (next g n2)
56 in (let TMP_45 \def (ASort O TMP_44) in (let TMP_46 \def (aplus g TMP_45 k)
57 in (let TMP_47 \def (aplus_sort_O_S_simpl g n2 k) in (let TMP_48 \def (eq_ind
58 A TMP_25 TMP_29 TMP_43 TMP_46 TMP_47) in (let TMP_49 \def (next g n1) in (let
59 TMP_50 \def (ASort O TMP_49) in (let TMP_51 \def (aplus g TMP_50 k) in (let
60 TMP_52 \def (aplus_sort_O_S_simpl g n1 k) in (let TMP_53 \def (eq_ind A
61 TMP_18 TMP_22 TMP_48 TMP_51 TMP_52) in (leq_sort g O O TMP_14 TMP_15 k
62 TMP_53)))))))))))))))))))))))))))))))) in (let TMP_96 \def (\lambda (h3:
63 nat).(\lambda (_: (((eq A (aplus g (ASort O n1) k) (aplus g (ASort h3 n2) k))
64 \to (leq g (ASort O (next g n1)) (match h3 with [O \Rightarrow (ASort O (next
65 g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H2: (eq A (aplus g
66 (ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(let TMP_55 \def (next g n1)
67 in (let TMP_56 \def (ASort O n1) in (let TMP_57 \def (S k) in (let TMP_58
68 \def (aplus g TMP_56 TMP_57) in (let TMP_61 \def (\lambda (a: A).(let TMP_59
69 \def (ASort h3 n2) in (let TMP_60 \def (aplus g TMP_59 k) in (eq A a
70 TMP_60)))) in (let TMP_62 \def (S h3) in (let TMP_63 \def (ASort TMP_62 n2)
71 in (let TMP_64 \def (S k) in (let TMP_65 \def (aplus g TMP_63 TMP_64) in (let
72 TMP_69 \def (\lambda (a: A).(let TMP_66 \def (ASort O n1) in (let TMP_67 \def
73 (S k) in (let TMP_68 \def (aplus g TMP_66 TMP_67) in (eq A TMP_68 a))))) in
74 (let TMP_70 \def (S h3) in (let TMP_71 \def (ASort TMP_70 n2) in (let TMP_72
75 \def (aplus g TMP_71 k) in (let TMP_78 \def (\lambda (a: A).(let TMP_73 \def
76 (asucc g a) in (let TMP_74 \def (S h3) in (let TMP_75 \def (ASort TMP_74 n2)
77 in (let TMP_76 \def (aplus g TMP_75 k) in (let TMP_77 \def (asucc g TMP_76)
78 in (eq A TMP_73 TMP_77))))))) in (let TMP_79 \def (S h3) in (let TMP_80 \def
79 (ASort TMP_79 n2) in (let TMP_81 \def (aplus g TMP_80 k) in (let TMP_82 \def
80 (asucc g TMP_81) in (let TMP_83 \def (refl_equal A TMP_82) in (let TMP_84
81 \def (ASort O n1) in (let TMP_85 \def (aplus g TMP_84 k) in (let TMP_86 \def
82 (eq_ind_r A TMP_72 TMP_78 TMP_83 TMP_85 H2) in (let TMP_87 \def (ASort h3 n2)
83 in (let TMP_88 \def (aplus g TMP_87 k) in (let TMP_89 \def
84 (aplus_sort_S_S_simpl g n2 h3 k) in (let TMP_90 \def (eq_ind A TMP_65 TMP_69
85 TMP_86 TMP_88 TMP_89) in (let TMP_91 \def (next g n1) in (let TMP_92 \def
86 (ASort O TMP_91) in (let TMP_93 \def (aplus g TMP_92 k) in (let TMP_94 \def
87 (aplus_sort_O_S_simpl g n1 k) in (let TMP_95 \def (eq_ind A TMP_58 TMP_61
88 TMP_90 TMP_93 TMP_94) in (leq_sort g O h3 TMP_55 n2 k
89 TMP_95))))))))))))))))))))))))))))))))))) in (nat_ind TMP_13 TMP_54 TMP_96 h2
90 H1))))) in (let TMP_185 \def (\lambda (h3: nat).(\lambda (IHh1: (((eq A
91 (aplus g (ASort h3 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (match h3
92 with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow (ASort h n1)])
93 (match h2 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow
94 (ASort h n2)]))))).(\lambda (H1: (eq A (aplus g (ASort (S h3) n1) k) (aplus g
95 (ASort h2 n2) k))).(let TMP_101 \def (\lambda (n: nat).((eq A (aplus g (ASort
96 (S h3) n1) k) (aplus g (ASort n n2) k)) \to ((((eq A (aplus g (ASort h3 n1)
97 k) (aplus g (ASort n n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort
98 O (next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match n with [O
99 \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) \to
100 (let TMP_98 \def (ASort h3 n1) in (let TMP_100 \def (match n with [O
101 \Rightarrow (let TMP_99 \def (next g n2) in (ASort O TMP_99)) | (S h)
102 \Rightarrow (ASort h n2)]) in (leq g TMP_98 TMP_100)))))) in (let TMP_141
103 \def (\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2)
104 k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort O n2) k))
105 \to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h)
106 \Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(let TMP_102 \def (next
107 g n2) in (let TMP_103 \def (ASort O n2) in (let TMP_104 \def (S k) in (let
108 TMP_105 \def (aplus g TMP_103 TMP_104) in (let TMP_108 \def (\lambda (a:
109 A).(let TMP_106 \def (ASort h3 n1) in (let TMP_107 \def (aplus g TMP_106 k)
110 in (eq A TMP_107 a)))) in (let TMP_109 \def (S h3) in (let TMP_110 \def
111 (ASort TMP_109 n1) in (let TMP_111 \def (S k) in (let TMP_112 \def (aplus g
112 TMP_110 TMP_111) in (let TMP_116 \def (\lambda (a: A).(let TMP_113 \def
113 (ASort O n2) in (let TMP_114 \def (S k) in (let TMP_115 \def (aplus g TMP_113
114 TMP_114) in (eq A a TMP_115))))) in (let TMP_117 \def (ASort O n2) in (let
115 TMP_118 \def (aplus g TMP_117 k) in (let TMP_123 \def (\lambda (a: A).(let
116 TMP_119 \def (asucc g a) in (let TMP_120 \def (ASort O n2) in (let TMP_121
117 \def (aplus g TMP_120 k) in (let TMP_122 \def (asucc g TMP_121) in (eq A
118 TMP_119 TMP_122)))))) in (let TMP_124 \def (ASort O n2) in (let TMP_125 \def
119 (aplus g TMP_124 k) in (let TMP_126 \def (asucc g TMP_125) in (let TMP_127
120 \def (refl_equal A TMP_126) in (let TMP_128 \def (S h3) in (let TMP_129 \def
121 (ASort TMP_128 n1) in (let TMP_130 \def (aplus g TMP_129 k) in (let TMP_131
122 \def (eq_ind_r A TMP_118 TMP_123 TMP_127 TMP_130 H2) in (let TMP_132 \def
123 (ASort h3 n1) in (let TMP_133 \def (aplus g TMP_132 k) in (let TMP_134 \def
124 (aplus_sort_S_S_simpl g n1 h3 k) in (let TMP_135 \def (eq_ind A TMP_112
125 TMP_116 TMP_131 TMP_133 TMP_134) in (let TMP_136 \def (next g n2) in (let
126 TMP_137 \def (ASort O TMP_136) in (let TMP_138 \def (aplus g TMP_137 k) in
127 (let TMP_139 \def (aplus_sort_O_S_simpl g n2 k) in (let TMP_140 \def (eq_ind
128 A TMP_105 TMP_108 TMP_135 TMP_138 TMP_139) in (leq_sort g h3 O n1 TMP_102 k
129 TMP_140))))))))))))))))))))))))))))))))) in (let TMP_184 \def (\lambda (h4:
130 nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort h4
131 n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k)) \to
132 (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h)
133 \Rightarrow (ASort h n1)]) (match h4 with [O \Rightarrow (ASort O (next g
134 n2)) | (S h) \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match h4
135 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h
136 n2)])))))).(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort
137 (S h4) n2) k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g
138 (ASort (S h4) n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next
139 g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(let TMP_142 \def
140 (S h3) in (let TMP_143 \def (ASort TMP_142 n1) in (let TMP_144 \def (S k) in
141 (let TMP_145 \def (aplus g TMP_143 TMP_144) in (let TMP_148 \def (\lambda (a:
142 A).(let TMP_146 \def (ASort h4 n2) in (let TMP_147 \def (aplus g TMP_146 k)
143 in (eq A a TMP_147)))) in (let TMP_149 \def (S h4) in (let TMP_150 \def
144 (ASort TMP_149 n2) in (let TMP_151 \def (S k) in (let TMP_152 \def (aplus g
145 TMP_150 TMP_151) in (let TMP_157 \def (\lambda (a: A).(let TMP_153 \def (S
146 h3) in (let TMP_154 \def (ASort TMP_153 n1) in (let TMP_155 \def (S k) in
147 (let TMP_156 \def (aplus g TMP_154 TMP_155) in (eq A TMP_156 a)))))) in (let
148 TMP_158 \def (S h4) in (let TMP_159 \def (ASort TMP_158 n2) in (let TMP_160
149 \def (aplus g TMP_159 k) in (let TMP_166 \def (\lambda (a: A).(let TMP_161
150 \def (asucc g a) in (let TMP_162 \def (S h4) in (let TMP_163 \def (ASort
151 TMP_162 n2) in (let TMP_164 \def (aplus g TMP_163 k) in (let TMP_165 \def
152 (asucc g TMP_164) in (eq A TMP_161 TMP_165))))))) in (let TMP_167 \def (S h4)
153 in (let TMP_168 \def (ASort TMP_167 n2) in (let TMP_169 \def (aplus g TMP_168
154 k) in (let TMP_170 \def (asucc g TMP_169) in (let TMP_171 \def (refl_equal A
155 TMP_170) in (let TMP_172 \def (S h3) in (let TMP_173 \def (ASort TMP_172 n1)
156 in (let TMP_174 \def (aplus g TMP_173 k) in (let TMP_175 \def (eq_ind_r A
157 TMP_160 TMP_166 TMP_171 TMP_174 H2) in (let TMP_176 \def (ASort h4 n2) in
158 (let TMP_177 \def (aplus g TMP_176 k) in (let TMP_178 \def
159 (aplus_sort_S_S_simpl g n2 h4 k) in (let TMP_179 \def (eq_ind A TMP_152
160 TMP_157 TMP_175 TMP_177 TMP_178) in (let TMP_180 \def (ASort h3 n1) in (let
161 TMP_181 \def (aplus g TMP_180 k) in (let TMP_182 \def (aplus_sort_S_S_simpl g
162 n1 h3 k) in (let TMP_183 \def (eq_ind A TMP_145 TMP_148 TMP_179 TMP_181
163 TMP_182) in (leq_sort g h3 h4 n1 n2 k
164 TMP_183)))))))))))))))))))))))))))))))))))) in (nat_ind TMP_101 TMP_141
165 TMP_184 h2 H1 IHh1))))))) in (nat_ind TMP_8 TMP_97 TMP_185 h1 H0)))))))))) in
166 (let TMP_189 \def (\lambda (a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3
167 a4)).(\lambda (_: (leq g (asucc g a3) (asucc g a4))).(\lambda (a5:
168 A).(\lambda (a6: A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g (asucc g
169 a5) (asucc g a6))).(let TMP_187 \def (asucc g a5) in (let TMP_188 \def (asucc
170 g a6) in (leq_head g a3 a4 H0 TMP_187 TMP_188 H3))))))))))) in (leq_ind g
171 TMP_3 TMP_186 TMP_189 a1 a2 H))))))).
174 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc
175 g a2)) \to (leq g a1 a2))))
177 \lambda (g: G).(\lambda (a1: A).(let TMP_1 \def (\lambda (a: A).(\forall
178 (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) in (let TMP_315
179 \def (\lambda (n: nat).(\lambda (n0: nat).(\lambda (a2: A).(let TMP_3 \def
180 (\lambda (a: A).((leq g (asucc g (ASort n n0)) (asucc g a)) \to (let TMP_2
181 \def (ASort n n0) in (leq g TMP_2 a)))) in (let TMP_260 \def (\lambda (n1:
182 nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0)) (asucc g
183 (ASort n1 n2)))).(let TMP_6 \def (\lambda (n3: nat).((leq g (asucc g (ASort
184 n3 n0)) (asucc g (ASort n1 n2))) \to (let TMP_4 \def (ASort n3 n0) in (let
185 TMP_5 \def (ASort n1 n2) in (leq g TMP_4 TMP_5))))) in (let TMP_133 \def
186 (\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1 n2)))).(let
187 TMP_9 \def (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g (ASort
188 n3 n2))) \to (let TMP_7 \def (ASort O n0) in (let TMP_8 \def (ASort n3 n2) in
189 (leq g TMP_7 TMP_8))))) in (let TMP_73 \def (\lambda (H1: (leq g (asucc g
190 (ASort O n0)) (asucc g (ASort O n2)))).(let TMP_10 \def (next g n0) in (let
191 TMP_11 \def (next g n2) in (let TMP_12 \def (ASort O TMP_11) in (let H_x \def
192 (leq_gen_sort1 g O TMP_10 TMP_12 H1) in (let H2 \def H_x in (let TMP_18 \def
193 (\lambda (n3: nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_13 \def
194 (next g n0) in (let TMP_14 \def (ASort O TMP_13) in (let TMP_15 \def (aplus g
195 TMP_14 k) in (let TMP_16 \def (ASort h2 n3) in (let TMP_17 \def (aplus g
196 TMP_16 k) in (eq A TMP_15 TMP_17))))))))) in (let TMP_22 \def (\lambda (n3:
197 nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_19 \def (next g n2) in
198 (let TMP_20 \def (ASort O TMP_19) in (let TMP_21 \def (ASort h2 n3) in (eq A
199 TMP_20 TMP_21))))))) in (let TMP_23 \def (ASort O n0) in (let TMP_24 \def
200 (ASort O n2) in (let TMP_25 \def (leq g TMP_23 TMP_24) in (let TMP_72 \def
201 (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A
202 (aplus g (ASort O (next g n0)) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4:
203 (eq A (ASort O (next g n2)) (ASort x1 x0))).(let TMP_26 \def (\lambda (e:
204 A).(match e with [(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow O]))
205 in (let TMP_27 \def (next g n2) in (let TMP_28 \def (ASort O TMP_27) in (let
206 TMP_29 \def (ASort x1 x0) in (let H5 \def (f_equal A nat TMP_26 TMP_28 TMP_29
207 H4) in (let TMP_31 \def (\lambda (e: A).(match e with [(ASort _ n3)
208 \Rightarrow n3 | (AHead _ _) \Rightarrow (let TMP_30 \def (match g with
209 [(mk_G next _) \Rightarrow next]) in (TMP_30 n2))])) in (let TMP_32 \def
210 (next g n2) in (let TMP_33 \def (ASort O TMP_32) in (let TMP_34 \def (ASort
211 x1 x0) in (let H6 \def (f_equal A nat TMP_31 TMP_33 TMP_34 H4) in (let TMP_71
212 \def (\lambda (H7: (eq nat O x1)).(let TMP_40 \def (\lambda (n3: nat).(let
213 TMP_35 \def (next g n0) in (let TMP_36 \def (ASort O TMP_35) in (let TMP_37
214 \def (aplus g TMP_36 x2) in (let TMP_38 \def (ASort n3 x0) in (let TMP_39
215 \def (aplus g TMP_38 x2) in (eq A TMP_37 TMP_39))))))) in (let H8 \def
216 (eq_ind_r nat x1 TMP_40 H3 O H7) in (let TMP_46 \def (\lambda (n3: nat).(let
217 TMP_41 \def (next g n0) in (let TMP_42 \def (ASort O TMP_41) in (let TMP_43
218 \def (aplus g TMP_42 x2) in (let TMP_44 \def (ASort O n3) in (let TMP_45 \def
219 (aplus g TMP_44 x2) in (eq A TMP_43 TMP_45))))))) in (let TMP_47 \def (next g
220 n2) in (let H9 \def (eq_ind_r nat x0 TMP_46 H8 TMP_47 H6) in (let TMP_48 \def
221 (next g n0) in (let TMP_49 \def (ASort O TMP_48) in (let TMP_50 \def (aplus g
222 TMP_49 x2) in (let TMP_54 \def (\lambda (a: A).(let TMP_51 \def (next g n2)
223 in (let TMP_52 \def (ASort O TMP_51) in (let TMP_53 \def (aplus g TMP_52 x2)
224 in (eq A a TMP_53))))) in (let TMP_55 \def (ASort O n0) in (let TMP_56 \def
225 (S x2) in (let TMP_57 \def (aplus g TMP_55 TMP_56) in (let TMP_58 \def
226 (aplus_sort_O_S_simpl g n0 x2) in (let H10 \def (eq_ind_r A TMP_50 TMP_54 H9
227 TMP_57 TMP_58) in (let TMP_59 \def (next g n2) in (let TMP_60 \def (ASort O
228 TMP_59) in (let TMP_61 \def (aplus g TMP_60 x2) in (let TMP_65 \def (\lambda
229 (a: A).(let TMP_62 \def (ASort O n0) in (let TMP_63 \def (S x2) in (let
230 TMP_64 \def (aplus g TMP_62 TMP_63) in (eq A TMP_64 a))))) in (let TMP_66
231 \def (ASort O n2) in (let TMP_67 \def (S x2) in (let TMP_68 \def (aplus g
232 TMP_66 TMP_67) in (let TMP_69 \def (aplus_sort_O_S_simpl g n2 x2) in (let H11
233 \def (eq_ind_r A TMP_61 TMP_65 H10 TMP_68 TMP_69) in (let TMP_70 \def (S x2)
234 in (leq_sort g O O n0 n2 TMP_70 H11)))))))))))))))))))))))))) in (TMP_71
235 H5))))))))))))))))) in (ex2_3_ind nat nat nat TMP_18 TMP_22 TMP_25 TMP_72
236 H2))))))))))))) in (let TMP_132 \def (\lambda (n3: nat).(\lambda (_: (((leq g
237 (asucc g (ASort O n0)) (asucc g (ASort n3 n2))) \to (leq g (ASort O n0)
238 (ASort n3 n2))))).(\lambda (H1: (leq g (asucc g (ASort O n0)) (asucc g (ASort
239 (S n3) n2)))).(let TMP_74 \def (next g n0) in (let TMP_75 \def (ASort n3 n2)
240 in (let H_x \def (leq_gen_sort1 g O TMP_74 TMP_75 H1) in (let H2 \def H_x in
241 (let TMP_81 \def (\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(let
242 TMP_76 \def (next g n0) in (let TMP_77 \def (ASort O TMP_76) in (let TMP_78
243 \def (aplus g TMP_77 k) in (let TMP_79 \def (ASort h2 n4) in (let TMP_80 \def
244 (aplus g TMP_79 k) in (eq A TMP_78 TMP_80))))))))) in (let TMP_84 \def
245 (\lambda (n4: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_82 \def
246 (ASort n3 n2) in (let TMP_83 \def (ASort h2 n4) in (eq A TMP_82 TMP_83))))))
247 in (let TMP_85 \def (ASort O n0) in (let TMP_86 \def (S n3) in (let TMP_87
248 \def (ASort TMP_86 n2) in (let TMP_88 \def (leq g TMP_85 TMP_87) in (let
249 TMP_131 \def (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2:
250 nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort
251 x1 x0) x2))).(\lambda (H4: (eq A (ASort n3 n2) (ASort x1 x0))).(let TMP_89
252 \def (\lambda (e: A).(match e with [(ASort n4 _) \Rightarrow n4 | (AHead _ _)
253 \Rightarrow n3])) in (let TMP_90 \def (ASort n3 n2) in (let TMP_91 \def
254 (ASort x1 x0) in (let H5 \def (f_equal A nat TMP_89 TMP_90 TMP_91 H4) in (let
255 TMP_92 \def (\lambda (e: A).(match e with [(ASort _ n4) \Rightarrow n4 |
256 (AHead _ _) \Rightarrow n2])) in (let TMP_93 \def (ASort n3 n2) in (let
257 TMP_94 \def (ASort x1 x0) in (let H6 \def (f_equal A nat TMP_92 TMP_93 TMP_94
258 H4) in (let TMP_130 \def (\lambda (H7: (eq nat n3 x1)).(let TMP_100 \def
259 (\lambda (n4: nat).(let TMP_95 \def (next g n0) in (let TMP_96 \def (ASort O
260 TMP_95) in (let TMP_97 \def (aplus g TMP_96 x2) in (let TMP_98 \def (ASort n4
261 x0) in (let TMP_99 \def (aplus g TMP_98 x2) in (eq A TMP_97 TMP_99))))))) in
262 (let H8 \def (eq_ind_r nat x1 TMP_100 H3 n3 H7) in (let TMP_106 \def (\lambda
263 (n4: nat).(let TMP_101 \def (next g n0) in (let TMP_102 \def (ASort O
264 TMP_101) in (let TMP_103 \def (aplus g TMP_102 x2) in (let TMP_104 \def
265 (ASort n3 n4) in (let TMP_105 \def (aplus g TMP_104 x2) in (eq A TMP_103
266 TMP_105))))))) in (let H9 \def (eq_ind_r nat x0 TMP_106 H8 n2 H6) in (let
267 TMP_107 \def (next g n0) in (let TMP_108 \def (ASort O TMP_107) in (let
268 TMP_109 \def (aplus g TMP_108 x2) in (let TMP_112 \def (\lambda (a: A).(let
269 TMP_110 \def (ASort n3 n2) in (let TMP_111 \def (aplus g TMP_110 x2) in (eq A
270 a TMP_111)))) in (let TMP_113 \def (ASort O n0) in (let TMP_114 \def (S x2)
271 in (let TMP_115 \def (aplus g TMP_113 TMP_114) in (let TMP_116 \def
272 (aplus_sort_O_S_simpl g n0 x2) in (let H10 \def (eq_ind_r A TMP_109 TMP_112
273 H9 TMP_115 TMP_116) in (let TMP_117 \def (ASort n3 n2) in (let TMP_118 \def
274 (aplus g TMP_117 x2) in (let TMP_122 \def (\lambda (a: A).(let TMP_119 \def
275 (ASort O n0) in (let TMP_120 \def (S x2) in (let TMP_121 \def (aplus g
276 TMP_119 TMP_120) in (eq A TMP_121 a))))) in (let TMP_123 \def (S n3) in (let
277 TMP_124 \def (ASort TMP_123 n2) in (let TMP_125 \def (S x2) in (let TMP_126
278 \def (aplus g TMP_124 TMP_125) in (let TMP_127 \def (aplus_sort_S_S_simpl g
279 n2 n3 x2) in (let H11 \def (eq_ind_r A TMP_118 TMP_122 H10 TMP_126 TMP_127)
280 in (let TMP_128 \def (S n3) in (let TMP_129 \def (S x2) in (leq_sort g O
281 TMP_128 n0 n2 TMP_129 H11)))))))))))))))))))))))))) in (TMP_130
282 H5))))))))))))))) in (ex2_3_ind nat nat nat TMP_81 TMP_84 TMP_88 TMP_131
283 H2))))))))))))))) in (nat_ind TMP_9 TMP_73 TMP_132 n1 H0))))) in (let TMP_259
284 \def (\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0))
285 (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda
286 (H0: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(let
287 TMP_137 \def (\lambda (n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g
288 (ASort n4 n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4
289 n2))) \to (leq g (ASort n3 n0) (ASort n4 n2)))) \to (let TMP_134 \def (S n3)
290 in (let TMP_135 \def (ASort TMP_134 n0) in (let TMP_136 \def (ASort n4 n2) in
291 (leq g TMP_135 TMP_136))))))) in (let TMP_200 \def (\lambda (H1: (leq g
292 (asucc g (ASort (S n3) n0)) (asucc g (ASort O n2)))).(\lambda (_: (((leq g
293 (asucc g (ASort n3 n0)) (asucc g (ASort O n2))) \to (leq g (ASort n3 n0)
294 (ASort O n2))))).(let TMP_138 \def (next g n2) in (let TMP_139 \def (ASort O
295 TMP_138) in (let H_x \def (leq_gen_sort1 g n3 n0 TMP_139 H1) in (let H2 \def
296 H_x in (let TMP_144 \def (\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k:
297 nat).(let TMP_140 \def (ASort n3 n0) in (let TMP_141 \def (aplus g TMP_140 k)
298 in (let TMP_142 \def (ASort h2 n4) in (let TMP_143 \def (aplus g TMP_142 k)
299 in (eq A TMP_141 TMP_143)))))))) in (let TMP_148 \def (\lambda (n4:
300 nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_145 \def (next g n2) in
301 (let TMP_146 \def (ASort O TMP_145) in (let TMP_147 \def (ASort h2 n4) in (eq
302 A TMP_146 TMP_147))))))) in (let TMP_149 \def (S n3) in (let TMP_150 \def
303 (ASort TMP_149 n0) in (let TMP_151 \def (ASort O n2) in (let TMP_152 \def
304 (leq g TMP_150 TMP_151) in (let TMP_199 \def (\lambda (x0: nat).(\lambda (x1:
305 nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort n3 n0) x2) (aplus
306 g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) (ASort x1
307 x0))).(let TMP_153 \def (\lambda (e: A).(match e with [(ASort n4 _)
308 \Rightarrow n4 | (AHead _ _) \Rightarrow O])) in (let TMP_154 \def (next g
309 n2) in (let TMP_155 \def (ASort O TMP_154) in (let TMP_156 \def (ASort x1 x0)
310 in (let H5 \def (f_equal A nat TMP_153 TMP_155 TMP_156 H4) in (let TMP_158
311 \def (\lambda (e: A).(match e with [(ASort _ n4) \Rightarrow n4 | (AHead _ _)
312 \Rightarrow (let TMP_157 \def (match g with [(mk_G next _) \Rightarrow next])
313 in (TMP_157 n2))])) in (let TMP_159 \def (next g n2) in (let TMP_160 \def
314 (ASort O TMP_159) in (let TMP_161 \def (ASort x1 x0) in (let H6 \def (f_equal
315 A nat TMP_158 TMP_160 TMP_161 H4) in (let TMP_198 \def (\lambda (H7: (eq nat
316 O x1)).(let TMP_166 \def (\lambda (n4: nat).(let TMP_162 \def (ASort n3 n0)
317 in (let TMP_163 \def (aplus g TMP_162 x2) in (let TMP_164 \def (ASort n4 x0)
318 in (let TMP_165 \def (aplus g TMP_164 x2) in (eq A TMP_163 TMP_165)))))) in
319 (let H8 \def (eq_ind_r nat x1 TMP_166 H3 O H7) in (let TMP_171 \def (\lambda
320 (n4: nat).(let TMP_167 \def (ASort n3 n0) in (let TMP_168 \def (aplus g
321 TMP_167 x2) in (let TMP_169 \def (ASort O n4) in (let TMP_170 \def (aplus g
322 TMP_169 x2) in (eq A TMP_168 TMP_170)))))) in (let TMP_172 \def (next g n2)
323 in (let H9 \def (eq_ind_r nat x0 TMP_171 H8 TMP_172 H6) in (let TMP_173 \def
324 (ASort n3 n0) in (let TMP_174 \def (aplus g TMP_173 x2) in (let TMP_178 \def
325 (\lambda (a: A).(let TMP_175 \def (next g n2) in (let TMP_176 \def (ASort O
326 TMP_175) in (let TMP_177 \def (aplus g TMP_176 x2) in (eq A a TMP_177))))) in
327 (let TMP_179 \def (S n3) in (let TMP_180 \def (ASort TMP_179 n0) in (let
328 TMP_181 \def (S x2) in (let TMP_182 \def (aplus g TMP_180 TMP_181) in (let
329 TMP_183 \def (aplus_sort_S_S_simpl g n0 n3 x2) in (let H10 \def (eq_ind_r A
330 TMP_174 TMP_178 H9 TMP_182 TMP_183) in (let TMP_184 \def (next g n2) in (let
331 TMP_185 \def (ASort O TMP_184) in (let TMP_186 \def (aplus g TMP_185 x2) in
332 (let TMP_191 \def (\lambda (a: A).(let TMP_187 \def (S n3) in (let TMP_188
333 \def (ASort TMP_187 n0) in (let TMP_189 \def (S x2) in (let TMP_190 \def
334 (aplus g TMP_188 TMP_189) in (eq A TMP_190 a)))))) in (let TMP_192 \def
335 (ASort O n2) in (let TMP_193 \def (S x2) in (let TMP_194 \def (aplus g
336 TMP_192 TMP_193) in (let TMP_195 \def (aplus_sort_O_S_simpl g n2 x2) in (let
337 H11 \def (eq_ind_r A TMP_186 TMP_191 H10 TMP_194 TMP_195) in (let TMP_196
338 \def (S n3) in (let TMP_197 \def (S x2) in (leq_sort g TMP_196 O n0 n2
339 TMP_197 H11))))))))))))))))))))))))))) in (TMP_198 H5))))))))))))))))) in
340 (ex2_3_ind nat nat nat TMP_144 TMP_148 TMP_152 TMP_199 H2)))))))))))))) in
341 (let TMP_258 \def (\lambda (n4: nat).(\lambda (_: (((leq g (asucc g (ASort (S
342 n3) n0)) (asucc g (ASort n4 n2))) \to ((((leq g (asucc g (ASort n3 n0))
343 (asucc g (ASort n4 n2))) \to (leq g (ASort n3 n0) (ASort n4 n2)))) \to (leq g
344 (ASort (S n3) n0) (ASort n4 n2)))))).(\lambda (H1: (leq g (asucc g (ASort (S
345 n3) n0)) (asucc g (ASort (S n4) n2)))).(\lambda (_: (((leq g (asucc g (ASort
346 n3 n0)) (asucc g (ASort (S n4) n2))) \to (leq g (ASort n3 n0) (ASort (S n4)
347 n2))))).(let TMP_201 \def (ASort n4 n2) in (let H_x \def (leq_gen_sort1 g n3
348 n0 TMP_201 H1) in (let H2 \def H_x in (let TMP_206 \def (\lambda (n5:
349 nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_202 \def (ASort n3 n0) in
350 (let TMP_203 \def (aplus g TMP_202 k) in (let TMP_204 \def (ASort h2 n5) in
351 (let TMP_205 \def (aplus g TMP_204 k) in (eq A TMP_203 TMP_205)))))))) in
352 (let TMP_209 \def (\lambda (n5: nat).(\lambda (h2: nat).(\lambda (_:
353 nat).(let TMP_207 \def (ASort n4 n2) in (let TMP_208 \def (ASort h2 n5) in
354 (eq A TMP_207 TMP_208)))))) in (let TMP_210 \def (S n3) in (let TMP_211 \def
355 (ASort TMP_210 n0) in (let TMP_212 \def (S n4) in (let TMP_213 \def (ASort
356 TMP_212 n2) in (let TMP_214 \def (leq g TMP_211 TMP_213) in (let TMP_257 \def
357 (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A
358 (aplus g (ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A
359 (ASort n4 n2) (ASort x1 x0))).(let TMP_215 \def (\lambda (e: A).(match e with
360 [(ASort n5 _) \Rightarrow n5 | (AHead _ _) \Rightarrow n4])) in (let TMP_216
361 \def (ASort n4 n2) in (let TMP_217 \def (ASort x1 x0) in (let H5 \def
362 (f_equal A nat TMP_215 TMP_216 TMP_217 H4) in (let TMP_218 \def (\lambda (e:
363 A).(match e with [(ASort _ n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2]))
364 in (let TMP_219 \def (ASort n4 n2) in (let TMP_220 \def (ASort x1 x0) in (let
365 H6 \def (f_equal A nat TMP_218 TMP_219 TMP_220 H4) in (let TMP_256 \def
366 (\lambda (H7: (eq nat n4 x1)).(let TMP_225 \def (\lambda (n5: nat).(let
367 TMP_221 \def (ASort n3 n0) in (let TMP_222 \def (aplus g TMP_221 x2) in (let
368 TMP_223 \def (ASort n5 x0) in (let TMP_224 \def (aplus g TMP_223 x2) in (eq A
369 TMP_222 TMP_224)))))) in (let H8 \def (eq_ind_r nat x1 TMP_225 H3 n4 H7) in
370 (let TMP_230 \def (\lambda (n5: nat).(let TMP_226 \def (ASort n3 n0) in (let
371 TMP_227 \def (aplus g TMP_226 x2) in (let TMP_228 \def (ASort n4 n5) in (let
372 TMP_229 \def (aplus g TMP_228 x2) in (eq A TMP_227 TMP_229)))))) in (let H9
373 \def (eq_ind_r nat x0 TMP_230 H8 n2 H6) in (let TMP_231 \def (ASort n3 n0) in
374 (let TMP_232 \def (aplus g TMP_231 x2) in (let TMP_235 \def (\lambda (a:
375 A).(let TMP_233 \def (ASort n4 n2) in (let TMP_234 \def (aplus g TMP_233 x2)
376 in (eq A a TMP_234)))) in (let TMP_236 \def (S n3) in (let TMP_237 \def
377 (ASort TMP_236 n0) in (let TMP_238 \def (S x2) in (let TMP_239 \def (aplus g
378 TMP_237 TMP_238) in (let TMP_240 \def (aplus_sort_S_S_simpl g n0 n3 x2) in
379 (let H10 \def (eq_ind_r A TMP_232 TMP_235 H9 TMP_239 TMP_240) in (let TMP_241
380 \def (ASort n4 n2) in (let TMP_242 \def (aplus g TMP_241 x2) in (let TMP_247
381 \def (\lambda (a: A).(let TMP_243 \def (S n3) in (let TMP_244 \def (ASort
382 TMP_243 n0) in (let TMP_245 \def (S x2) in (let TMP_246 \def (aplus g TMP_244
383 TMP_245) in (eq A TMP_246 a)))))) in (let TMP_248 \def (S n4) in (let TMP_249
384 \def (ASort TMP_248 n2) in (let TMP_250 \def (S x2) in (let TMP_251 \def
385 (aplus g TMP_249 TMP_250) in (let TMP_252 \def (aplus_sort_S_S_simpl g n2 n4
386 x2) in (let H11 \def (eq_ind_r A TMP_242 TMP_247 H10 TMP_251 TMP_252) in (let
387 TMP_253 \def (S n3) in (let TMP_254 \def (S n4) in (let TMP_255 \def (S x2)
388 in (leq_sort g TMP_253 TMP_254 n0 n2 TMP_255 H11)))))))))))))))))))))))))))
389 in (TMP_256 H5))))))))))))))) in (ex2_3_ind nat nat nat TMP_206 TMP_209
390 TMP_214 TMP_257 H2)))))))))))))))) in (nat_ind TMP_137 TMP_200 TMP_258 n1 H0
391 IHn))))))) in (nat_ind TMP_6 TMP_133 TMP_259 n H))))))) in (let TMP_314 \def
392 (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n n0)) (asucc g a)) \to
393 (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda (H0: (((leq g (asucc g
394 (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0) a0)))).(\lambda (H1: (leq
395 g (asucc g (ASort n n0)) (asucc g (AHead a a0)))).(let TMP_263 \def (\lambda
396 (n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1
397 n0) a))) \to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g
398 (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a
399 a0))) \to (let TMP_261 \def (ASort n1 n0) in (let TMP_262 \def (AHead a a0)
400 in (leq g TMP_261 TMP_262))))))) in (let TMP_288 \def (\lambda (_: (((leq g
401 (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O n0) a)))).(\lambda
402 (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq g (ASort O n0)
403 a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g (AHead a
404 a0)))).(let TMP_264 \def (next g n0) in (let TMP_265 \def (asucc g a0) in
405 (let TMP_266 \def (AHead a TMP_265) in (let H_x \def (leq_gen_sort1 g O
406 TMP_264 TMP_266 H4) in (let H5 \def H_x in (let TMP_272 \def (\lambda (n2:
407 nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_267 \def (next g n0) in
408 (let TMP_268 \def (ASort O TMP_267) in (let TMP_269 \def (aplus g TMP_268 k)
409 in (let TMP_270 \def (ASort h2 n2) in (let TMP_271 \def (aplus g TMP_270 k)
410 in (eq A TMP_269 TMP_271))))))))) in (let TMP_276 \def (\lambda (n2:
411 nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_273 \def (asucc g a0) in
412 (let TMP_274 \def (AHead a TMP_273) in (let TMP_275 \def (ASort h2 n2) in (eq
413 A TMP_274 TMP_275))))))) in (let TMP_277 \def (ASort O n0) in (let TMP_278
414 \def (AHead a a0) in (let TMP_279 \def (leq g TMP_277 TMP_278) in (let
415 TMP_287 \def (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2:
416 nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort x1
417 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1 x0))).(let
418 TMP_280 \def (asucc g a0) in (let TMP_281 \def (AHead a TMP_280) in (let
419 TMP_282 \def (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False |
420 (AHead _ _) \Rightarrow True])) in (let TMP_283 \def (ASort x1 x0) in (let H8
421 \def (eq_ind A TMP_281 TMP_282 I TMP_283 H7) in (let TMP_284 \def (ASort O
422 n0) in (let TMP_285 \def (AHead a a0) in (let TMP_286 \def (leq g TMP_284
423 TMP_285) in (False_ind TMP_286 H8)))))))))))))) in (ex2_3_ind nat nat nat
424 TMP_272 TMP_276 TMP_279 TMP_287 H5))))))))))))))) in (let TMP_313 \def
425 (\lambda (n1: nat).(\lambda (_: (((((leq g (asucc g (ASort n1 n0)) (asucc g
426 a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 n0))
427 (asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1
428 n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a
429 a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1) n0)) (asucc g a)) \to
430 (leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g (asucc g (ASort (S n1)
431 n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0) a0)))).(\lambda (H4: (leq g
432 (asucc g (ASort (S n1) n0)) (asucc g (AHead a a0)))).(let TMP_289 \def (asucc
433 g a0) in (let TMP_290 \def (AHead a TMP_289) in (let H_x \def (leq_gen_sort1
434 g n1 n0 TMP_290 H4) in (let H5 \def H_x in (let TMP_295 \def (\lambda (n2:
435 nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_291 \def (ASort n1 n0) in
436 (let TMP_292 \def (aplus g TMP_291 k) in (let TMP_293 \def (ASort h2 n2) in
437 (let TMP_294 \def (aplus g TMP_293 k) in (eq A TMP_292 TMP_294)))))))) in
438 (let TMP_299 \def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_:
439 nat).(let TMP_296 \def (asucc g a0) in (let TMP_297 \def (AHead a TMP_296) in
440 (let TMP_298 \def (ASort h2 n2) in (eq A TMP_297 TMP_298))))))) in (let
441 TMP_300 \def (S n1) in (let TMP_301 \def (ASort TMP_300 n0) in (let TMP_302
442 \def (AHead a a0) in (let TMP_303 \def (leq g TMP_301 TMP_302) in (let
443 TMP_312 \def (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2:
444 nat).(\lambda (_: (eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0)
445 x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1 x0))).(let TMP_304
446 \def (asucc g a0) in (let TMP_305 \def (AHead a TMP_304) in (let TMP_306 \def
447 (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _)
448 \Rightarrow True])) in (let TMP_307 \def (ASort x1 x0) in (let H8 \def
449 (eq_ind A TMP_305 TMP_306 I TMP_307 H7) in (let TMP_308 \def (S n1) in (let
450 TMP_309 \def (ASort TMP_308 n0) in (let TMP_310 \def (AHead a a0) in (let
451 TMP_311 \def (leq g TMP_309 TMP_310) in (False_ind TMP_311 H8)))))))))))))))
452 in (ex2_3_ind nat nat nat TMP_295 TMP_299 TMP_303 TMP_312 H5)))))))))))))))))
453 in (nat_ind TMP_263 TMP_288 TMP_313 n H H0 H1))))))))) in (A_ind TMP_3
454 TMP_260 TMP_314 a2))))))) in (let TMP_396 \def (\lambda (a: A).(\lambda (_:
455 ((\forall (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq g a
456 a2))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0)
457 (asucc g a2)) \to (leq g a0 a2))))).(\lambda (a2: A).(let TMP_317 \def
458 (\lambda (a3: A).((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (let
459 TMP_316 \def (AHead a a0) in (leq g TMP_316 a3)))) in (let TMP_364 \def
460 (\lambda (n: nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a
461 a0)) (asucc g (ASort n n0)))).(let TMP_320 \def (\lambda (n1: nat).((leq g
462 (asucc g (AHead a a0)) (asucc g (ASort n1 n0))) \to (let TMP_318 \def (AHead
463 a a0) in (let TMP_319 \def (ASort n1 n0) in (leq g TMP_318 TMP_319))))) in
464 (let TMP_342 \def (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort
465 O n0)))).(let TMP_321 \def (asucc g a0) in (let TMP_322 \def (next g n0) in
466 (let TMP_323 \def (ASort O TMP_322) in (let H_x \def (leq_gen_head1 g a
467 TMP_321 TMP_323 H2) in (let H3 \def H_x in (let TMP_324 \def (\lambda (a3:
468 A).(\lambda (_: A).(leq g a a3))) in (let TMP_326 \def (\lambda (_:
469 A).(\lambda (a4: A).(let TMP_325 \def (asucc g a0) in (leq g TMP_325 a4))))
470 in (let TMP_330 \def (\lambda (a3: A).(\lambda (a4: A).(let TMP_327 \def
471 (next g n0) in (let TMP_328 \def (ASort O TMP_327) in (let TMP_329 \def
472 (AHead a3 a4) in (eq A TMP_328 TMP_329)))))) in (let TMP_331 \def (AHead a
473 a0) in (let TMP_332 \def (ASort O n0) in (let TMP_333 \def (leq g TMP_331
474 TMP_332) in (let TMP_341 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (_:
475 (leq g a x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda (H6: (eq A
476 (ASort O (next g n0)) (AHead x0 x1))).(let TMP_334 \def (next g n0) in (let
477 TMP_335 \def (ASort O TMP_334) in (let TMP_336 \def (\lambda (ee: A).(match
478 ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) in
479 (let TMP_337 \def (AHead x0 x1) in (let H7 \def (eq_ind A TMP_335 TMP_336 I
480 TMP_337 H6) in (let TMP_338 \def (AHead a a0) in (let TMP_339 \def (ASort O
481 n0) in (let TMP_340 \def (leq g TMP_338 TMP_339) in (False_ind TMP_340
482 H7)))))))))))))) in (ex3_2_ind A A TMP_324 TMP_326 TMP_330 TMP_333 TMP_341
483 H3)))))))))))))) in (let TMP_363 \def (\lambda (n1: nat).(\lambda (_: (((leq
484 g (asucc g (AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0)
485 (ASort n1 n0))))).(\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort
486 (S n1) n0)))).(let TMP_343 \def (asucc g a0) in (let TMP_344 \def (ASort n1
487 n0) in (let H_x \def (leq_gen_head1 g a TMP_343 TMP_344 H2) in (let H3 \def
488 H_x in (let TMP_345 \def (\lambda (a3: A).(\lambda (_: A).(leq g a a3))) in
489 (let TMP_347 \def (\lambda (_: A).(\lambda (a4: A).(let TMP_346 \def (asucc g
490 a0) in (leq g TMP_346 a4)))) in (let TMP_350 \def (\lambda (a3: A).(\lambda
491 (a4: A).(let TMP_348 \def (ASort n1 n0) in (let TMP_349 \def (AHead a3 a4) in
492 (eq A TMP_348 TMP_349))))) in (let TMP_351 \def (AHead a a0) in (let TMP_352
493 \def (S n1) in (let TMP_353 \def (ASort TMP_352 n0) in (let TMP_354 \def (leq
494 g TMP_351 TMP_353) in (let TMP_362 \def (\lambda (x0: A).(\lambda (x1:
495 A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda
496 (H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let TMP_355 \def (ASort n1 n0) in
497 (let TMP_356 \def (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow
498 True | (AHead _ _) \Rightarrow False])) in (let TMP_357 \def (AHead x0 x1) in
499 (let H7 \def (eq_ind A TMP_355 TMP_356 I TMP_357 H6) in (let TMP_358 \def
500 (AHead a a0) in (let TMP_359 \def (S n1) in (let TMP_360 \def (ASort TMP_359
501 n0) in (let TMP_361 \def (leq g TMP_358 TMP_360) in (False_ind TMP_361
502 H7)))))))))))))) in (ex3_2_ind A A TMP_345 TMP_347 TMP_350 TMP_354 TMP_362
503 H3)))))))))))))))) in (nat_ind TMP_320 TMP_342 TMP_363 n H1))))))) in (let
504 TMP_395 \def (\lambda (a3: A).(\lambda (_: (((leq g (asucc g (AHead a a0))
505 (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda (a4: A).(\lambda (_:
506 (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g (AHead a a0)
507 a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g (AHead a3
508 a4)))).(let TMP_365 \def (asucc g a0) in (let TMP_366 \def (asucc g a4) in
509 (let TMP_367 \def (AHead a3 TMP_366) in (let H_x \def (leq_gen_head1 g a
510 TMP_365 TMP_367 H3) in (let H4 \def H_x in (let TMP_368 \def (\lambda (a5:
511 A).(\lambda (_: A).(leq g a a5))) in (let TMP_370 \def (\lambda (_:
512 A).(\lambda (a6: A).(let TMP_369 \def (asucc g a0) in (leq g TMP_369 a6))))
513 in (let TMP_374 \def (\lambda (a5: A).(\lambda (a6: A).(let TMP_371 \def
514 (asucc g a4) in (let TMP_372 \def (AHead a3 TMP_371) in (let TMP_373 \def
515 (AHead a5 a6) in (eq A TMP_372 TMP_373)))))) in (let TMP_375 \def (AHead a
516 a0) in (let TMP_376 \def (AHead a3 a4) in (let TMP_377 \def (leq g TMP_375
517 TMP_376) in (let TMP_394 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (H5:
518 (leq g a x0)).(\lambda (H6: (leq g (asucc g a0) x1)).(\lambda (H7: (eq A
519 (AHead a3 (asucc g a4)) (AHead x0 x1))).(let TMP_378 \def (\lambda (e:
520 A).(match e with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow a5]))
521 in (let TMP_379 \def (asucc g a4) in (let TMP_380 \def (AHead a3 TMP_379) in
522 (let TMP_381 \def (AHead x0 x1) in (let H8 \def (f_equal A A TMP_378 TMP_380
523 TMP_381 H7) in (let TMP_384 \def (\lambda (e: A).(match e with [(ASort _ _)
524 \Rightarrow (asucc g a4) | (AHead _ a5) \Rightarrow a5])) in (let TMP_385
525 \def (asucc g a4) in (let TMP_386 \def (AHead a3 TMP_385) in (let TMP_387
526 \def (AHead x0 x1) in (let H9 \def (f_equal A A TMP_384 TMP_386 TMP_387 H7)
527 in (let TMP_393 \def (\lambda (H10: (eq A a3 x0)).(let TMP_389 \def (\lambda
528 (a5: A).(let TMP_388 \def (asucc g a0) in (leq g TMP_388 a5))) in (let
529 TMP_390 \def (asucc g a4) in (let H11 \def (eq_ind_r A x1 TMP_389 H6 TMP_390
530 H9) in (let TMP_391 \def (\lambda (a5: A).(leq g a a5)) in (let H12 \def
531 (eq_ind_r A x0 TMP_391 H5 a3 H10) in (let TMP_392 \def (H0 a4 H11) in
532 (leq_head g a a3 H12 a0 a4 TMP_392)))))))) in (TMP_393 H8))))))))))))))))) in
533 (ex3_2_ind A A TMP_368 TMP_370 TMP_374 TMP_377 TMP_394 H4))))))))))))))))))
534 in (A_ind TMP_317 TMP_364 TMP_395 a2))))))))) in (A_ind TMP_1 TMP_315 TMP_396
538 \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g
541 \lambda (g: G).(\lambda (a: A).(let TMP_3 \def (\lambda (a0: A).(let TMP_2
542 \def (\lambda (a1: A).(let TMP_1 \def (asucc g a1) in (leq g a0 TMP_1))) in
543 (ex A TMP_2))) in (let TMP_11 \def (\lambda (n: nat).(\lambda (n0: nat).(let
544 TMP_6 \def (\lambda (a0: A).(let TMP_4 \def (ASort n n0) in (let TMP_5 \def
545 (asucc g a0) in (leq g TMP_4 TMP_5)))) in (let TMP_7 \def (S n) in (let TMP_8
546 \def (ASort TMP_7 n0) in (let TMP_9 \def (ASort n n0) in (let TMP_10 \def
547 (leq_refl g TMP_9) in (ex_intro A TMP_6 TMP_8 TMP_10)))))))) in (let TMP_26
548 \def (\lambda (a0: A).(\lambda (_: (ex A (\lambda (a1: A).(leq g a0 (asucc g
549 a1))))).(\lambda (a1: A).(\lambda (H0: (ex A (\lambda (a2: A).(leq g a1
550 (asucc g a2))))).(let H1 \def H0 in (let TMP_13 \def (\lambda (a2: A).(let
551 TMP_12 \def (asucc g a2) in (leq g a1 TMP_12))) in (let TMP_16 \def (\lambda
552 (a2: A).(let TMP_14 \def (AHead a0 a1) in (let TMP_15 \def (asucc g a2) in
553 (leq g TMP_14 TMP_15)))) in (let TMP_17 \def (ex A TMP_16) in (let TMP_25
554 \def (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc g x))).(let TMP_20 \def
555 (\lambda (a2: A).(let TMP_18 \def (AHead a0 a1) in (let TMP_19 \def (asucc g
556 a2) in (leq g TMP_18 TMP_19)))) in (let TMP_21 \def (AHead a0 x) in (let
557 TMP_22 \def (leq_refl g a0) in (let TMP_23 \def (asucc g x) in (let TMP_24
558 \def (leq_head g a0 a0 TMP_22 a1 TMP_23 H2) in (ex_intro A TMP_20 TMP_21
559 TMP_24)))))))) in (ex_ind A TMP_13 TMP_17 TMP_25 H1)))))))))) in (A_ind TMP_3
560 TMP_11 TMP_26 a))))).
562 theorem leq_ahead_asucc_false:
563 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2)
564 (asucc g a1)) \to (\forall (P: Prop).P))))
566 \lambda (g: G).(\lambda (a1: A).(let TMP_1 \def (\lambda (a: A).(\forall
567 (a2: A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) in
568 (let TMP_34 \def (\lambda (n: nat).(\lambda (n0: nat).(\lambda (a2:
569 A).(\lambda (H: (leq g (AHead (ASort n n0) a2) (match n with [O \Rightarrow
570 (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]))).(\lambda (P:
571 Prop).(let TMP_2 \def (\lambda (n1: nat).((leq g (AHead (ASort n1 n0) a2)
572 (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow
573 (ASort h n0)])) \to P)) in (let TMP_18 \def (\lambda (H0: (leq g (AHead
574 (ASort O n0) a2) (ASort O (next g n0)))).(let TMP_3 \def (ASort O n0) in (let
575 TMP_4 \def (next g n0) in (let TMP_5 \def (ASort O TMP_4) in (let H_x \def
576 (leq_gen_head1 g TMP_3 a2 TMP_5 H0) in (let H1 \def H_x in (let TMP_7 \def
577 (\lambda (a3: A).(\lambda (_: A).(let TMP_6 \def (ASort O n0) in (leq g TMP_6
578 a3)))) in (let TMP_8 \def (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) in
579 (let TMP_12 \def (\lambda (a3: A).(\lambda (a4: A).(let TMP_9 \def (next g
580 n0) in (let TMP_10 \def (ASort O TMP_9) in (let TMP_11 \def (AHead a3 a4) in
581 (eq A TMP_10 TMP_11)))))) in (let TMP_17 \def (\lambda (x0: A).(\lambda (x1:
582 A).(\lambda (_: (leq g (ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda
583 (H4: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let TMP_13 \def (next g n0)
584 in (let TMP_14 \def (ASort O TMP_13) in (let TMP_15 \def (\lambda (ee:
585 A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
586 False])) in (let TMP_16 \def (AHead x0 x1) in (let H5 \def (eq_ind A TMP_14
587 TMP_15 I TMP_16 H4) in (False_ind P H5))))))))))) in (ex3_2_ind A A TMP_7
588 TMP_8 TMP_12 P TMP_17 H1))))))))))) in (let TMP_33 \def (\lambda (n1:
589 nat).(\lambda (_: (((leq g (AHead (ASort n1 n0) a2) (match n1 with [O
590 \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to
591 P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort n1 n0))).(let
592 TMP_19 \def (S n1) in (let TMP_20 \def (ASort TMP_19 n0) in (let TMP_21 \def
593 (ASort n1 n0) in (let H_x \def (leq_gen_head1 g TMP_20 a2 TMP_21 H0) in (let
594 H1 \def H_x in (let TMP_24 \def (\lambda (a3: A).(\lambda (_: A).(let TMP_22
595 \def (S n1) in (let TMP_23 \def (ASort TMP_22 n0) in (leq g TMP_23 a3))))) in
596 (let TMP_25 \def (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) in (let
597 TMP_28 \def (\lambda (a3: A).(\lambda (a4: A).(let TMP_26 \def (ASort n1 n0)
598 in (let TMP_27 \def (AHead a3 a4) in (eq A TMP_26 TMP_27))))) in (let TMP_32
599 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S n1) n0)
600 x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort n1 n0) (AHead x0
601 x1))).(let TMP_29 \def (ASort n1 n0) in (let TMP_30 \def (\lambda (ee:
602 A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
603 False])) in (let TMP_31 \def (AHead x0 x1) in (let H5 \def (eq_ind A TMP_29
604 TMP_30 I TMP_31 H4) in (False_ind P H5)))))))))) in (ex3_2_ind A A TMP_24
605 TMP_25 TMP_28 P TMP_32 H1))))))))))))) in (nat_ind TMP_2 TMP_18 TMP_33 n
606 H))))))))) in (let TMP_61 \def (\lambda (a: A).(\lambda (_: ((\forall (a2:
607 A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P))))).(\lambda
608 (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a0 a2) (asucc g a0))
609 \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead
610 (AHead a a0) a2) (AHead a (asucc g a0)))).(\lambda (P: Prop).(let TMP_35 \def
611 (AHead a a0) in (let TMP_36 \def (asucc g a0) in (let TMP_37 \def (AHead a
612 TMP_36) in (let H_x \def (leq_gen_head1 g TMP_35 a2 TMP_37 H1) in (let H2
613 \def H_x in (let TMP_39 \def (\lambda (a3: A).(\lambda (_: A).(let TMP_38
614 \def (AHead a a0) in (leq g TMP_38 a3)))) in (let TMP_40 \def (\lambda (_:
615 A).(\lambda (a4: A).(leq g a2 a4))) in (let TMP_44 \def (\lambda (a3:
616 A).(\lambda (a4: A).(let TMP_41 \def (asucc g a0) in (let TMP_42 \def (AHead
617 a TMP_41) in (let TMP_43 \def (AHead a3 a4) in (eq A TMP_42 TMP_43)))))) in
618 (let TMP_60 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (H3: (leq g
619 (AHead a a0) x0)).(\lambda (H4: (leq g a2 x1)).(\lambda (H5: (eq A (AHead a
620 (asucc g a0)) (AHead x0 x1))).(let TMP_45 \def (\lambda (e: A).(match e with
621 [(ASort _ _) \Rightarrow a | (AHead a3 _) \Rightarrow a3])) in (let TMP_46
622 \def (asucc g a0) in (let TMP_47 \def (AHead a TMP_46) in (let TMP_48 \def
623 (AHead x0 x1) in (let H6 \def (f_equal A A TMP_45 TMP_47 TMP_48 H5) in (let
624 TMP_51 \def (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow (asucc g
625 a0) | (AHead _ a3) \Rightarrow a3])) in (let TMP_52 \def (asucc g a0) in (let
626 TMP_53 \def (AHead a TMP_52) in (let TMP_54 \def (AHead x0 x1) in (let H7
627 \def (f_equal A A TMP_51 TMP_53 TMP_54 H5) in (let TMP_59 \def (\lambda (H8:
628 (eq A a x0)).(let TMP_55 \def (\lambda (a3: A).(leq g a2 a3)) in (let TMP_56
629 \def (asucc g a0) in (let H9 \def (eq_ind_r A x1 TMP_55 H4 TMP_56 H7) in (let
630 TMP_58 \def (\lambda (a3: A).(let TMP_57 \def (AHead a a0) in (leq g TMP_57
631 a3))) in (let H10 \def (eq_ind_r A x0 TMP_58 H3 a H8) in (leq_ahead_false_1 g
632 a a0 H10 P))))))) in (TMP_59 H6))))))))))))))))) in (ex3_2_ind A A TMP_39
633 TMP_40 TMP_44 P TMP_60 H2))))))))))))))))) in (A_ind TMP_1 TMP_34 TMP_61
636 theorem leq_asucc_false:
637 \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P:
640 \lambda (g: G).(\lambda (a: A).(let TMP_1 \def (\lambda (a0: A).((leq g
641 (asucc g a0) a0) \to (\forall (P: Prop).P))) in (let TMP_103 \def (\lambda
642 (n: nat).(\lambda (n0: nat).(\lambda (H: (leq g (match n with [O \Rightarrow
643 (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n
644 n0))).(\lambda (P: Prop).(let TMP_2 \def (\lambda (n1: nat).((leq g (match n1
645 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)])
646 (ASort n1 n0)) \to P)) in (let TMP_50 \def (\lambda (H0: (leq g (ASort O
647 (next g n0)) (ASort O n0))).(let TMP_3 \def (next g n0) in (let TMP_4 \def
648 (ASort O n0) in (let H_x \def (leq_gen_sort1 g O TMP_3 TMP_4 H0) in (let H1
649 \def H_x in (let TMP_10 \def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
650 (k: nat).(let TMP_5 \def (next g n0) in (let TMP_6 \def (ASort O TMP_5) in
651 (let TMP_7 \def (aplus g TMP_6 k) in (let TMP_8 \def (ASort h2 n2) in (let
652 TMP_9 \def (aplus g TMP_8 k) in (eq A TMP_7 TMP_9))))))))) in (let TMP_13
653 \def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_11 \def
654 (ASort O n0) in (let TMP_12 \def (ASort h2 n2) in (eq A TMP_11 TMP_12))))))
655 in (let TMP_49 \def (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2:
656 nat).(\lambda (H2: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort
657 x1 x0) x2))).(\lambda (H3: (eq A (ASort O n0) (ASort x1 x0))).(let TMP_14
658 \def (\lambda (e: A).(match e with [(ASort n1 _) \Rightarrow n1 | (AHead _ _)
659 \Rightarrow O])) in (let TMP_15 \def (ASort O n0) in (let TMP_16 \def (ASort
660 x1 x0) in (let H4 \def (f_equal A nat TMP_14 TMP_15 TMP_16 H3) in (let TMP_17
661 \def (\lambda (e: A).(match e with [(ASort _ n1) \Rightarrow n1 | (AHead _ _)
662 \Rightarrow n0])) in (let TMP_18 \def (ASort O n0) in (let TMP_19 \def (ASort
663 x1 x0) in (let H5 \def (f_equal A nat TMP_17 TMP_18 TMP_19 H3) in (let TMP_48
664 \def (\lambda (H6: (eq nat O x1)).(let TMP_25 \def (\lambda (n1: nat).(let
665 TMP_20 \def (next g n0) in (let TMP_21 \def (ASort O TMP_20) in (let TMP_22
666 \def (aplus g TMP_21 x2) in (let TMP_23 \def (ASort n1 x0) in (let TMP_24
667 \def (aplus g TMP_23 x2) in (eq A TMP_22 TMP_24))))))) in (let H7 \def
668 (eq_ind_r nat x1 TMP_25 H2 O H6) in (let TMP_31 \def (\lambda (n1: nat).(let
669 TMP_26 \def (next g n0) in (let TMP_27 \def (ASort O TMP_26) in (let TMP_28
670 \def (aplus g TMP_27 x2) in (let TMP_29 \def (ASort O n1) in (let TMP_30 \def
671 (aplus g TMP_29 x2) in (eq A TMP_28 TMP_30))))))) in (let H8 \def (eq_ind_r
672 nat x0 TMP_31 H7 n0 H5) in (let TMP_32 \def (next g n0) in (let TMP_33 \def
673 (ASort O TMP_32) in (let TMP_34 \def (aplus g TMP_33 x2) in (let TMP_37 \def
674 (\lambda (a0: A).(let TMP_35 \def (ASort O n0) in (let TMP_36 \def (aplus g
675 TMP_35 x2) in (eq A a0 TMP_36)))) in (let TMP_38 \def (ASort O n0) in (let
676 TMP_39 \def (S x2) in (let TMP_40 \def (aplus g TMP_38 TMP_39) in (let TMP_41
677 \def (aplus_sort_O_S_simpl g n0 x2) in (let H9 \def (eq_ind_r A TMP_34 TMP_37
678 H8 TMP_40 TMP_41) in (let TMP_42 \def (S x2) in (let TMP_43 \def (ASort O n0)
679 in (let H_y \def (aplus_inj g TMP_42 x2 TMP_43 H9) in (let TMP_44 \def
680 (\lambda (n1: nat).(le n1 x2)) in (let TMP_45 \def (le_n x2) in (let TMP_46
681 \def (S x2) in (let TMP_47 \def (eq_ind_r nat x2 TMP_44 TMP_45 TMP_46 H_y) in
682 (le_Sx_x x2 TMP_47 P)))))))))))))))))))))) in (TMP_48 H4))))))))))))))) in
683 (ex2_3_ind nat nat nat TMP_10 TMP_13 P TMP_49 H1))))))))) in (let TMP_102
684 \def (\lambda (n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow
685 (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to
686 P))).(\lambda (H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let TMP_51 \def
687 (S n1) in (let TMP_52 \def (ASort TMP_51 n0) in (let H_x \def (leq_gen_sort1
688 g n1 n0 TMP_52 H0) in (let H1 \def H_x in (let TMP_57 \def (\lambda (n2:
689 nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_53 \def (ASort n1 n0) in
690 (let TMP_54 \def (aplus g TMP_53 k) in (let TMP_55 \def (ASort h2 n2) in (let
691 TMP_56 \def (aplus g TMP_55 k) in (eq A TMP_54 TMP_56)))))))) in (let TMP_61
692 \def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_58 \def
693 (S n1) in (let TMP_59 \def (ASort TMP_58 n0) in (let TMP_60 \def (ASort h2
694 n2) in (eq A TMP_59 TMP_60))))))) in (let TMP_101 \def (\lambda (x0:
695 nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g
696 (ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A (ASort (S
697 n1) n0) (ASort x1 x0))).(let TMP_62 \def (\lambda (e: A).(match e with
698 [(ASort n2 _) \Rightarrow n2 | (AHead _ _) \Rightarrow (S n1)])) in (let
699 TMP_63 \def (S n1) in (let TMP_64 \def (ASort TMP_63 n0) in (let TMP_65 \def
700 (ASort x1 x0) in (let H4 \def (f_equal A nat TMP_62 TMP_64 TMP_65 H3) in (let
701 TMP_66 \def (\lambda (e: A).(match e with [(ASort _ n2) \Rightarrow n2 |
702 (AHead _ _) \Rightarrow n0])) in (let TMP_67 \def (S n1) in (let TMP_68 \def
703 (ASort TMP_67 n0) in (let TMP_69 \def (ASort x1 x0) in (let H5 \def (f_equal
704 A nat TMP_66 TMP_68 TMP_69 H3) in (let TMP_100 \def (\lambda (H6: (eq nat (S
705 n1) x1)).(let TMP_74 \def (\lambda (n2: nat).(let TMP_70 \def (ASort n1 n0)
706 in (let TMP_71 \def (aplus g TMP_70 x2) in (let TMP_72 \def (ASort n2 x0) in
707 (let TMP_73 \def (aplus g TMP_72 x2) in (eq A TMP_71 TMP_73)))))) in (let
708 TMP_75 \def (S n1) in (let H7 \def (eq_ind_r nat x1 TMP_74 H2 TMP_75 H6) in
709 (let TMP_81 \def (\lambda (n2: nat).(let TMP_76 \def (ASort n1 n0) in (let
710 TMP_77 \def (aplus g TMP_76 x2) in (let TMP_78 \def (S n1) in (let TMP_79
711 \def (ASort TMP_78 n2) in (let TMP_80 \def (aplus g TMP_79 x2) in (eq A
712 TMP_77 TMP_80))))))) in (let H8 \def (eq_ind_r nat x0 TMP_81 H7 n0 H5) in
713 (let TMP_82 \def (ASort n1 n0) in (let TMP_83 \def (aplus g TMP_82 x2) in
714 (let TMP_87 \def (\lambda (a0: A).(let TMP_84 \def (S n1) in (let TMP_85 \def
715 (ASort TMP_84 n0) in (let TMP_86 \def (aplus g TMP_85 x2) in (eq A a0
716 TMP_86))))) in (let TMP_88 \def (S n1) in (let TMP_89 \def (ASort TMP_88 n0)
717 in (let TMP_90 \def (S x2) in (let TMP_91 \def (aplus g TMP_89 TMP_90) in
718 (let TMP_92 \def (aplus_sort_S_S_simpl g n0 n1 x2) in (let H9 \def (eq_ind_r
719 A TMP_83 TMP_87 H8 TMP_91 TMP_92) in (let TMP_93 \def (S x2) in (let TMP_94
720 \def (S n1) in (let TMP_95 \def (ASort TMP_94 n0) in (let H_y \def (aplus_inj
721 g TMP_93 x2 TMP_95 H9) in (let TMP_96 \def (\lambda (n2: nat).(le n2 x2)) in
722 (let TMP_97 \def (le_n x2) in (let TMP_98 \def (S x2) in (let TMP_99 \def
723 (eq_ind_r nat x2 TMP_96 TMP_97 TMP_98 H_y) in (le_Sx_x x2 TMP_99
724 P)))))))))))))))))))))))) in (TMP_100 H4))))))))))))))))) in (ex2_3_ind nat
725 nat nat TMP_57 TMP_61 P TMP_101 H1))))))))))) in (nat_ind TMP_2 TMP_50
726 TMP_102 n H)))))))) in (let TMP_123 \def (\lambda (a0: A).(\lambda (_: (((leq
727 g (asucc g a0) a0) \to (\forall (P: Prop).P)))).(\lambda (a1: A).(\lambda
728 (H0: (((leq g (asucc g a1) a1) \to (\forall (P: Prop).P)))).(\lambda (H1:
729 (leq g (AHead a0 (asucc g a1)) (AHead a0 a1))).(\lambda (P: Prop).(let
730 TMP_104 \def (asucc g a1) in (let TMP_105 \def (AHead a0 a1) in (let H_x \def
731 (leq_gen_head1 g a0 TMP_104 TMP_105 H1) in (let H2 \def H_x in (let TMP_106
732 \def (\lambda (a3: A).(\lambda (_: A).(leq g a0 a3))) in (let TMP_108 \def
733 (\lambda (_: A).(\lambda (a4: A).(let TMP_107 \def (asucc g a1) in (leq g
734 TMP_107 a4)))) in (let TMP_111 \def (\lambda (a3: A).(\lambda (a4: A).(let
735 TMP_109 \def (AHead a0 a1) in (let TMP_110 \def (AHead a3 a4) in (eq A
736 TMP_109 TMP_110))))) in (let TMP_122 \def (\lambda (x0: A).(\lambda (x1:
737 A).(\lambda (H3: (leq g a0 x0)).(\lambda (H4: (leq g (asucc g a1)
738 x1)).(\lambda (H5: (eq A (AHead a0 a1) (AHead x0 x1))).(let TMP_112 \def
739 (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a0 | (AHead a2 _)
740 \Rightarrow a2])) in (let TMP_113 \def (AHead a0 a1) in (let TMP_114 \def
741 (AHead x0 x1) in (let H6 \def (f_equal A A TMP_112 TMP_113 TMP_114 H5) in
742 (let TMP_115 \def (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a1 |
743 (AHead _ a2) \Rightarrow a2])) in (let TMP_116 \def (AHead a0 a1) in (let
744 TMP_117 \def (AHead x0 x1) in (let H7 \def (f_equal A A TMP_115 TMP_116
745 TMP_117 H5) in (let TMP_121 \def (\lambda (H8: (eq A a0 x0)).(let TMP_119
746 \def (\lambda (a2: A).(let TMP_118 \def (asucc g a1) in (leq g TMP_118 a2)))
747 in (let H9 \def (eq_ind_r A x1 TMP_119 H4 a1 H7) in (let TMP_120 \def
748 (\lambda (a2: A).(leq g a0 a2)) in (let H10 \def (eq_ind_r A x0 TMP_120 H3 a0
749 H8) in (H0 H9 P)))))) in (TMP_121 H6))))))))))))))) in (ex3_2_ind A A TMP_106
750 TMP_108 TMP_111 P TMP_122 H2))))))))))))))) in (A_ind TMP_1 TMP_103 TMP_123
projects / helm.git / blob