X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Fleq%2Fasucc.ma;h=42005d6ba6fb90f8d2f57672b7ed4a32f484f936;hb=57ae1762497a5f3ea75740e2908e04adb8642cc2;hp=d79f266be7efda2d9b435277444808163c8f0491;hpb=e8656c819b0b5e7bea7b4da244015b480af5f0f5;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma b/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma index d79f266be..42005d6ba 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma @@ -16,737 +16,432 @@ include "basic_1/leq/props.ma". -theorem asucc_repl: +lemma asucc_repl: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g (asucc g a1) (asucc g a2))))) \def \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(let TMP_3 \def (\lambda (a: A).(\lambda (a0: A).(let TMP_1 \def (asucc -g a) in (let TMP_2 \def (asucc g a0) in (leq g TMP_1 TMP_2))))) in (let -TMP_186 \def (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: -nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort -h1 n1) k) (aplus g (ASort h2 n2) k))).(let TMP_8 \def (\lambda (n: nat).((eq -A (aplus g (ASort n n1) k) (aplus g (ASort h2 n2) k)) \to (let TMP_5 \def -(match n with [O \Rightarrow (let TMP_4 \def (next g n1) in (ASort O TMP_4)) -| (S h) \Rightarrow (ASort h n1)]) in (let TMP_7 \def (match h2 with [O -\Rightarrow (let TMP_6 \def (next g n2) in (ASort O TMP_6)) | (S h) -\Rightarrow (ASort h n2)]) in (leq g TMP_5 TMP_7))))) in (let TMP_97 \def -(\lambda (H1: (eq A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(let -TMP_13 \def (\lambda (n: nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort -n n2) k)) \to (let TMP_9 \def (next g n1) in (let TMP_10 \def (ASort O TMP_9) -in (let TMP_12 \def (match n with [O \Rightarrow (let TMP_11 \def (next g n2) -in (ASort O TMP_11)) | (S h) \Rightarrow (ASort h n2)]) in (leq g TMP_10 -TMP_12)))))) in (let TMP_54 \def (\lambda (H2: (eq A (aplus g (ASort O n1) k) -(aplus g (ASort O n2) k))).(let TMP_14 \def (next g n1) in (let TMP_15 \def -(next g n2) in (let TMP_16 \def (ASort O n1) in (let TMP_17 \def (S k) in -(let TMP_18 \def (aplus g TMP_16 TMP_17) in (let TMP_22 \def (\lambda (a: -A).(let TMP_19 \def (next g n2) in (let TMP_20 \def (ASort O TMP_19) in (let -TMP_21 \def (aplus g TMP_20 k) in (eq A a TMP_21))))) in (let TMP_23 \def -(ASort O n2) in (let TMP_24 \def (S k) in (let TMP_25 \def (aplus g TMP_23 -TMP_24) in (let TMP_29 \def (\lambda (a: A).(let TMP_26 \def (ASort O n1) in -(let TMP_27 \def (S k) in (let TMP_28 \def (aplus g TMP_26 TMP_27) in (eq A -TMP_28 a))))) in (let TMP_30 \def (ASort O n2) in (let TMP_31 \def (aplus g -TMP_30 k) in (let TMP_36 \def (\lambda (a: A).(let TMP_32 \def (asucc g a) in -(let TMP_33 \def (ASort O n2) in (let TMP_34 \def (aplus g TMP_33 k) in (let -TMP_35 \def (asucc g TMP_34) in (eq A TMP_32 TMP_35)))))) in (let TMP_37 \def -(ASort O n2) in (let TMP_38 \def (aplus g TMP_37 k) in (let TMP_39 \def -(asucc g TMP_38) in (let TMP_40 \def (refl_equal A TMP_39) in (let TMP_41 -\def (ASort O n1) in (let TMP_42 \def (aplus g TMP_41 k) in (let TMP_43 \def -(eq_ind_r A TMP_31 TMP_36 TMP_40 TMP_42 H2) in (let TMP_44 \def (next g n2) -in (let TMP_45 \def (ASort O TMP_44) in (let TMP_46 \def (aplus g TMP_45 k) -in (let TMP_47 \def (aplus_sort_O_S_simpl g n2 k) in (let TMP_48 \def (eq_ind -A TMP_25 TMP_29 TMP_43 TMP_46 TMP_47) in (let TMP_49 \def (next g n1) in (let -TMP_50 \def (ASort O TMP_49) in (let TMP_51 \def (aplus g TMP_50 k) in (let -TMP_52 \def (aplus_sort_O_S_simpl g n1 k) in (let TMP_53 \def (eq_ind A -TMP_18 TMP_22 TMP_48 TMP_51 TMP_52) in (leq_sort g O O TMP_14 TMP_15 k -TMP_53)))))))))))))))))))))))))))))))) in (let TMP_96 \def (\lambda (h3: +a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g (asucc g a) (asucc g +a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: +nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g +(ASort h2 n2) k))).(nat_ind (\lambda (n: nat).((eq A (aplus g (ASort n n1) k) +(aplus g (ASort h2 n2) k)) \to (leq g (match n with [O \Rightarrow (ASort O +(next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow +(ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) (\lambda (H1: (eq +A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda (n: +nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort n n2) k)) \to (leq g +(ASort O (next g n1)) (match n with [O \Rightarrow (ASort O (next g n2)) | (S +h) \Rightarrow (ASort h n2)])))) (\lambda (H2: (eq A (aplus g (ASort O n1) k) +(aplus g (ASort O n2) k))).(leq_sort g O O (next g n1) (next g n2) k (eq_ind +A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O +(next g n2)) k))) (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq +A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort O n2) k) +(\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k)))) +(refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort O n1) k) +H2) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)) (aplus g +(ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))) (\lambda (h3: nat).(\lambda (_: (((eq A (aplus g (ASort O n1) k) (aplus g (ASort h3 n2) k)) \to (leq g (ASort O (next g n1)) (match h3 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H2: (eq A (aplus g -(ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(let TMP_55 \def (next g n1) -in (let TMP_56 \def (ASort O n1) in (let TMP_57 \def (S k) in (let TMP_58 -\def (aplus g TMP_56 TMP_57) in (let TMP_61 \def (\lambda (a: A).(let TMP_59 -\def (ASort h3 n2) in (let TMP_60 \def (aplus g TMP_59 k) in (eq A a -TMP_60)))) in (let TMP_62 \def (S h3) in (let TMP_63 \def (ASort TMP_62 n2) -in (let TMP_64 \def (S k) in (let TMP_65 \def (aplus g TMP_63 TMP_64) in (let -TMP_69 \def (\lambda (a: A).(let TMP_66 \def (ASort O n1) in (let TMP_67 \def -(S k) in (let TMP_68 \def (aplus g TMP_66 TMP_67) in (eq A TMP_68 a))))) in -(let TMP_70 \def (S h3) in (let TMP_71 \def (ASort TMP_70 n2) in (let TMP_72 -\def (aplus g TMP_71 k) in (let TMP_78 \def (\lambda (a: A).(let TMP_73 \def -(asucc g a) in (let TMP_74 \def (S h3) in (let TMP_75 \def (ASort TMP_74 n2) -in (let TMP_76 \def (aplus g TMP_75 k) in (let TMP_77 \def (asucc g TMP_76) -in (eq A TMP_73 TMP_77))))))) in (let TMP_79 \def (S h3) in (let TMP_80 \def -(ASort TMP_79 n2) in (let TMP_81 \def (aplus g TMP_80 k) in (let TMP_82 \def -(asucc g TMP_81) in (let TMP_83 \def (refl_equal A TMP_82) in (let TMP_84 -\def (ASort O n1) in (let TMP_85 \def (aplus g TMP_84 k) in (let TMP_86 \def -(eq_ind_r A TMP_72 TMP_78 TMP_83 TMP_85 H2) in (let TMP_87 \def (ASort h3 n2) -in (let TMP_88 \def (aplus g TMP_87 k) in (let TMP_89 \def -(aplus_sort_S_S_simpl g n2 h3 k) in (let TMP_90 \def (eq_ind A TMP_65 TMP_69 -TMP_86 TMP_88 TMP_89) in (let TMP_91 \def (next g n1) in (let TMP_92 \def -(ASort O TMP_91) in (let TMP_93 \def (aplus g TMP_92 k) in (let TMP_94 \def -(aplus_sort_O_S_simpl g n1 k) in (let TMP_95 \def (eq_ind A TMP_58 TMP_61 -TMP_90 TMP_93 TMP_94) in (leq_sort g O h3 TMP_55 n2 k -TMP_95))))))))))))))))))))))))))))))))))) in (nat_ind TMP_13 TMP_54 TMP_96 h2 -H1))))) in (let TMP_185 \def (\lambda (h3: nat).(\lambda (IHh1: (((eq A -(aplus g (ASort h3 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (match h3 -with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow (ASort h n1)]) -(match h2 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow -(ASort h n2)]))))).(\lambda (H1: (eq A (aplus g (ASort (S h3) n1) k) (aplus g -(ASort h2 n2) k))).(let TMP_101 \def (\lambda (n: nat).((eq A (aplus g (ASort -(S h3) n1) k) (aplus g (ASort n n2) k)) \to ((((eq A (aplus g (ASort h3 n1) -k) (aplus g (ASort n n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort -O (next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match n with [O -\Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) \to -(let TMP_98 \def (ASort h3 n1) in (let TMP_100 \def (match n with [O -\Rightarrow (let TMP_99 \def (next g n2) in (ASort O TMP_99)) | (S h) -\Rightarrow (ASort h n2)]) in (leq g TMP_98 TMP_100)))))) in (let TMP_141 -\def (\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2) +(ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(leq_sort g O h3 (next g n1) +n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g +(ASort h3 n2) k))) (eq_ind A (aplus g (ASort (S h3) n2) (S k)) (\lambda (a: +A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S h3) +n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S h3) n2) +k)))) (refl_equal A (asucc g (aplus g (ASort (S h3) n2) k))) (aplus g (ASort +O n1) k) H2) (aplus g (ASort h3 n2) k) (aplus_sort_S_S_simpl g n2 h3 k)) +(aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))))) h2 H1)) +(\lambda (h3: nat).(\lambda (IHh1: (((eq A (aplus g (ASort h3 n1) k) (aplus g +(ASort h2 n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next g +n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow (ASort +O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H1: (eq A +(aplus g (ASort (S h3) n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda +(n: nat).((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort n n2) k)) \to +((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort n n2) k)) \to (leq g +(match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow +(ASort h n1)]) (match n with [O \Rightarrow (ASort O (next g n2)) | (S h) +\Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match n with [O +\Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))) +(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2) k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort O n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) -\Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(let TMP_102 \def (next -g n2) in (let TMP_103 \def (ASort O n2) in (let TMP_104 \def (S k) in (let -TMP_105 \def (aplus g TMP_103 TMP_104) in (let TMP_108 \def (\lambda (a: -A).(let TMP_106 \def (ASort h3 n1) in (let TMP_107 \def (aplus g TMP_106 k) -in (eq A TMP_107 a)))) in (let TMP_109 \def (S h3) in (let TMP_110 \def -(ASort TMP_109 n1) in (let TMP_111 \def (S k) in (let TMP_112 \def (aplus g -TMP_110 TMP_111) in (let TMP_116 \def (\lambda (a: A).(let TMP_113 \def -(ASort O n2) in (let TMP_114 \def (S k) in (let TMP_115 \def (aplus g TMP_113 -TMP_114) in (eq A a TMP_115))))) in (let TMP_117 \def (ASort O n2) in (let -TMP_118 \def (aplus g TMP_117 k) in (let TMP_123 \def (\lambda (a: A).(let -TMP_119 \def (asucc g a) in (let TMP_120 \def (ASort O n2) in (let TMP_121 -\def (aplus g TMP_120 k) in (let TMP_122 \def (asucc g TMP_121) in (eq A -TMP_119 TMP_122)))))) in (let TMP_124 \def (ASort O n2) in (let TMP_125 \def -(aplus g TMP_124 k) in (let TMP_126 \def (asucc g TMP_125) in (let TMP_127 -\def (refl_equal A TMP_126) in (let TMP_128 \def (S h3) in (let TMP_129 \def -(ASort TMP_128 n1) in (let TMP_130 \def (aplus g TMP_129 k) in (let TMP_131 -\def (eq_ind_r A TMP_118 TMP_123 TMP_127 TMP_130 H2) in (let TMP_132 \def -(ASort h3 n1) in (let TMP_133 \def (aplus g TMP_132 k) in (let TMP_134 \def -(aplus_sort_S_S_simpl g n1 h3 k) in (let TMP_135 \def (eq_ind A TMP_112 -TMP_116 TMP_131 TMP_133 TMP_134) in (let TMP_136 \def (next g n2) in (let -TMP_137 \def (ASort O TMP_136) in (let TMP_138 \def (aplus g TMP_137 k) in -(let TMP_139 \def (aplus_sort_O_S_simpl g n2 k) in (let TMP_140 \def (eq_ind -A TMP_105 TMP_108 TMP_135 TMP_138 TMP_139) in (leq_sort g h3 O n1 TMP_102 k -TMP_140))))))))))))))))))))))))))))))))) in (let TMP_184 \def (\lambda (h4: -nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort h4 -n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k)) \to -(leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) +\Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(leq_sort g h3 O n1 +(next g n2) k (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A +(aplus g (ASort h3 n1) k) a)) (eq_ind A (aplus g (ASort (S h3) n1) (S k)) +(\lambda (a: A).(eq A a (aplus g (ASort O n2) (S k)))) (eq_ind_r A (aplus g +(ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O +n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort +(S h3) n1) k) H2) (aplus g (ASort h3 n1) k) (aplus_sort_S_S_simpl g n1 h3 k)) +(aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k))))) (\lambda +(h4: nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort +h4 n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k)) +\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h4 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match h4 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))))).(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort (S h4) n2) k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort (S h4) n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next -g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(let TMP_142 \def -(S h3) in (let TMP_143 \def (ASort TMP_142 n1) in (let TMP_144 \def (S k) in -(let TMP_145 \def (aplus g TMP_143 TMP_144) in (let TMP_148 \def (\lambda (a: -A).(let TMP_146 \def (ASort h4 n2) in (let TMP_147 \def (aplus g TMP_146 k) -in (eq A a TMP_147)))) in (let TMP_149 \def (S h4) in (let TMP_150 \def -(ASort TMP_149 n2) in (let TMP_151 \def (S k) in (let TMP_152 \def (aplus g -TMP_150 TMP_151) in (let TMP_157 \def (\lambda (a: A).(let TMP_153 \def (S -h3) in (let TMP_154 \def (ASort TMP_153 n1) in (let TMP_155 \def (S k) in -(let TMP_156 \def (aplus g TMP_154 TMP_155) in (eq A TMP_156 a)))))) in (let -TMP_158 \def (S h4) in (let TMP_159 \def (ASort TMP_158 n2) in (let TMP_160 -\def (aplus g TMP_159 k) in (let TMP_166 \def (\lambda (a: A).(let TMP_161 -\def (asucc g a) in (let TMP_162 \def (S h4) in (let TMP_163 \def (ASort -TMP_162 n2) in (let TMP_164 \def (aplus g TMP_163 k) in (let TMP_165 \def -(asucc g TMP_164) in (eq A TMP_161 TMP_165))))))) in (let TMP_167 \def (S h4) -in (let TMP_168 \def (ASort TMP_167 n2) in (let TMP_169 \def (aplus g TMP_168 -k) in (let TMP_170 \def (asucc g TMP_169) in (let TMP_171 \def (refl_equal A -TMP_170) in (let TMP_172 \def (S h3) in (let TMP_173 \def (ASort TMP_172 n1) -in (let TMP_174 \def (aplus g TMP_173 k) in (let TMP_175 \def (eq_ind_r A -TMP_160 TMP_166 TMP_171 TMP_174 H2) in (let TMP_176 \def (ASort h4 n2) in -(let TMP_177 \def (aplus g TMP_176 k) in (let TMP_178 \def -(aplus_sort_S_S_simpl g n2 h4 k) in (let TMP_179 \def (eq_ind A TMP_152 -TMP_157 TMP_175 TMP_177 TMP_178) in (let TMP_180 \def (ASort h3 n1) in (let -TMP_181 \def (aplus g TMP_180 k) in (let TMP_182 \def (aplus_sort_S_S_simpl g -n1 h3 k) in (let TMP_183 \def (eq_ind A TMP_145 TMP_148 TMP_179 TMP_181 -TMP_182) in (leq_sort g h3 h4 n1 n2 k -TMP_183)))))))))))))))))))))))))))))))))))) in (nat_ind TMP_101 TMP_141 -TMP_184 h2 H1 IHh1))))))) in (nat_ind TMP_8 TMP_97 TMP_185 h1 H0)))))))))) in -(let TMP_189 \def (\lambda (a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 -a4)).(\lambda (_: (leq g (asucc g a3) (asucc g a4))).(\lambda (a5: -A).(\lambda (a6: A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g (asucc g -a5) (asucc g a6))).(let TMP_187 \def (asucc g a5) in (let TMP_188 \def (asucc -g a6) in (leq_head g a3 a4 H0 TMP_187 TMP_188 H3))))))))))) in (leq_ind g -TMP_3 TMP_186 TMP_189 a1 a2 H))))))). +g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(leq_sort g h3 h4 +n1 n2 k (eq_ind A (aplus g (ASort (S h3) n1) (S k)) (\lambda (a: A).(eq A a +(aplus g (ASort h4 n2) k))) (eq_ind A (aplus g (ASort (S h4) n2) (S k)) +(\lambda (a: A).(eq A (aplus g (ASort (S h3) n1) (S k)) a)) (eq_ind_r A +(aplus g (ASort (S h4) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g +(aplus g (ASort (S h4) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S +h4) n2) k))) (aplus g (ASort (S h3) n1) k) H2) (aplus g (ASort h4 n2) k) +(aplus_sort_S_S_simpl g n2 h4 k)) (aplus g (ASort h3 n1) k) +(aplus_sort_S_S_simpl g n1 h3 k))))))) h2 H1 IHh1)))) h1 H0))))))) (\lambda +(a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 a4)).(\lambda (_: (leq g +(asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: +(leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g +a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))). -theorem asucc_inj: +lemma asucc_inj: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc g a2)) \to (leq g a1 a2)))) \def - \lambda (g: G).(\lambda (a1: A).(let TMP_1 \def (\lambda (a: A).(\forall -(a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) in (let TMP_315 -\def (\lambda (n: nat).(\lambda (n0: nat).(\lambda (a2: A).(let TMP_3 \def -(\lambda (a: A).((leq g (asucc g (ASort n n0)) (asucc g a)) \to (let TMP_2 -\def (ASort n n0) in (leq g TMP_2 a)))) in (let TMP_260 \def (\lambda (n1: -nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0)) (asucc g -(ASort n1 n2)))).(let TMP_6 \def (\lambda (n3: nat).((leq g (asucc g (ASort -n3 n0)) (asucc g (ASort n1 n2))) \to (let TMP_4 \def (ASort n3 n0) in (let -TMP_5 \def (ASort n1 n2) in (leq g TMP_4 TMP_5))))) in (let TMP_133 \def -(\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1 n2)))).(let -TMP_9 \def (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g (ASort -n3 n2))) \to (let TMP_7 \def (ASort O n0) in (let TMP_8 \def (ASort n3 n2) in -(leq g TMP_7 TMP_8))))) in (let TMP_73 \def (\lambda (H1: (leq g (asucc g -(ASort O n0)) (asucc g (ASort O n2)))).(let TMP_10 \def (next g n0) in (let -TMP_11 \def (next g n2) in (let TMP_12 \def (ASort O TMP_11) in (let H_x \def -(leq_gen_sort1 g O TMP_10 TMP_12 H1) in (let H2 \def H_x in (let TMP_18 \def -(\lambda (n3: nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_13 \def -(next g n0) in (let TMP_14 \def (ASort O TMP_13) in (let TMP_15 \def (aplus g -TMP_14 k) in (let TMP_16 \def (ASort h2 n3) in (let TMP_17 \def (aplus g -TMP_16 k) in (eq A TMP_15 TMP_17))))))))) in (let TMP_22 \def (\lambda (n3: -nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_19 \def (next g n2) in -(let TMP_20 \def (ASort O TMP_19) in (let TMP_21 \def (ASort h2 n3) in (eq A -TMP_20 TMP_21))))))) in (let TMP_23 \def (ASort O n0) in (let TMP_24 \def -(ASort O n2) in (let TMP_25 \def (leq g TMP_23 TMP_24) in (let TMP_72 \def -(\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A -(aplus g (ASort O (next g n0)) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: -(eq A (ASort O (next g n2)) (ASort x1 x0))).(let TMP_26 \def (\lambda (e: -A).(match e with [(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow O])) -in (let TMP_27 \def (next g n2) in (let TMP_28 \def (ASort O TMP_27) in (let -TMP_29 \def (ASort x1 x0) in (let H5 \def (f_equal A nat TMP_26 TMP_28 TMP_29 -H4) in (let TMP_31 \def (\lambda (e: A).(match e with [(ASort _ n3) -\Rightarrow n3 | (AHead _ _) \Rightarrow (let TMP_30 \def (match g with -[(mk_G next _) \Rightarrow next]) in (TMP_30 n2))])) in (let TMP_32 \def -(next g n2) in (let TMP_33 \def (ASort O TMP_32) in (let TMP_34 \def (ASort -x1 x0) in (let H6 \def (f_equal A nat TMP_31 TMP_33 TMP_34 H4) in (let TMP_71 -\def (\lambda (H7: (eq nat O x1)).(let TMP_40 \def (\lambda (n3: nat).(let -TMP_35 \def (next g n0) in (let TMP_36 \def (ASort O TMP_35) in (let TMP_37 -\def (aplus g TMP_36 x2) in (let TMP_38 \def (ASort n3 x0) in (let TMP_39 -\def (aplus g TMP_38 x2) in (eq A TMP_37 TMP_39))))))) in (let H8 \def -(eq_ind_r nat x1 TMP_40 H3 O H7) in (let TMP_46 \def (\lambda (n3: nat).(let -TMP_41 \def (next g n0) in (let TMP_42 \def (ASort O TMP_41) in (let TMP_43 -\def (aplus g TMP_42 x2) in (let TMP_44 \def (ASort O n3) in (let TMP_45 \def -(aplus g TMP_44 x2) in (eq A TMP_43 TMP_45))))))) in (let TMP_47 \def (next g -n2) in (let H9 \def (eq_ind_r nat x0 TMP_46 H8 TMP_47 H6) in (let TMP_48 \def -(next g n0) in (let TMP_49 \def (ASort O TMP_48) in (let TMP_50 \def (aplus g -TMP_49 x2) in (let TMP_54 \def (\lambda (a: A).(let TMP_51 \def (next g n2) -in (let TMP_52 \def (ASort O TMP_51) in (let TMP_53 \def (aplus g TMP_52 x2) -in (eq A a TMP_53))))) in (let TMP_55 \def (ASort O n0) in (let TMP_56 \def -(S x2) in (let TMP_57 \def (aplus g TMP_55 TMP_56) in (let TMP_58 \def -(aplus_sort_O_S_simpl g n0 x2) in (let H10 \def (eq_ind_r A TMP_50 TMP_54 H9 -TMP_57 TMP_58) in (let TMP_59 \def (next g n2) in (let TMP_60 \def (ASort O -TMP_59) in (let TMP_61 \def (aplus g TMP_60 x2) in (let TMP_65 \def (\lambda -(a: A).(let TMP_62 \def (ASort O n0) in (let TMP_63 \def (S x2) in (let -TMP_64 \def (aplus g TMP_62 TMP_63) in (eq A TMP_64 a))))) in (let TMP_66 -\def (ASort O n2) in (let TMP_67 \def (S x2) in (let TMP_68 \def (aplus g -TMP_66 TMP_67) in (let TMP_69 \def (aplus_sort_O_S_simpl g n2 x2) in (let H11 -\def (eq_ind_r A TMP_61 TMP_65 H10 TMP_68 TMP_69) in (let TMP_70 \def (S x2) -in (leq_sort g O O n0 n2 TMP_70 H11)))))))))))))))))))))))))) in (TMP_71 -H5))))))))))))))))) in (ex2_3_ind nat nat nat TMP_18 TMP_22 TMP_25 TMP_72 -H2))))))))))))) in (let TMP_132 \def (\lambda (n3: nat).(\lambda (_: (((leq g -(asucc g (ASort O n0)) (asucc g (ASort n3 n2))) \to (leq g (ASort O n0) -(ASort n3 n2))))).(\lambda (H1: (leq g (asucc g (ASort O n0)) (asucc g (ASort -(S n3) n2)))).(let TMP_74 \def (next g n0) in (let TMP_75 \def (ASort n3 n2) -in (let H_x \def (leq_gen_sort1 g O TMP_74 TMP_75 H1) in (let H2 \def H_x in -(let TMP_81 \def (\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(let -TMP_76 \def (next g n0) in (let TMP_77 \def (ASort O TMP_76) in (let TMP_78 -\def (aplus g TMP_77 k) in (let TMP_79 \def (ASort h2 n4) in (let TMP_80 \def -(aplus g TMP_79 k) in (eq A TMP_78 TMP_80))))))))) in (let TMP_84 \def -(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_82 \def -(ASort n3 n2) in (let TMP_83 \def (ASort h2 n4) in (eq A TMP_82 TMP_83)))))) -in (let TMP_85 \def (ASort O n0) in (let TMP_86 \def (S n3) in (let TMP_87 -\def (ASort TMP_86 n2) in (let TMP_88 \def (leq g TMP_85 TMP_87) in (let -TMP_131 \def (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: -nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort -x1 x0) x2))).(\lambda (H4: (eq A (ASort n3 n2) (ASort x1 x0))).(let TMP_89 -\def (\lambda (e: A).(match e with [(ASort n4 _) \Rightarrow n4 | (AHead _ _) -\Rightarrow n3])) in (let TMP_90 \def (ASort n3 n2) in (let TMP_91 \def -(ASort x1 x0) in (let H5 \def (f_equal A nat TMP_89 TMP_90 TMP_91 H4) in (let -TMP_92 \def (\lambda (e: A).(match e with [(ASort _ n4) \Rightarrow n4 | -(AHead _ _) \Rightarrow n2])) in (let TMP_93 \def (ASort n3 n2) in (let -TMP_94 \def (ASort x1 x0) in (let H6 \def (f_equal A nat TMP_92 TMP_93 TMP_94 -H4) in (let TMP_130 \def (\lambda (H7: (eq nat n3 x1)).(let TMP_100 \def -(\lambda (n4: nat).(let TMP_95 \def (next g n0) in (let TMP_96 \def (ASort O -TMP_95) in (let TMP_97 \def (aplus g TMP_96 x2) in (let TMP_98 \def (ASort n4 -x0) in (let TMP_99 \def (aplus g TMP_98 x2) in (eq A TMP_97 TMP_99))))))) in -(let H8 \def (eq_ind_r nat x1 TMP_100 H3 n3 H7) in (let TMP_106 \def (\lambda -(n4: nat).(let TMP_101 \def (next g n0) in (let TMP_102 \def (ASort O -TMP_101) in (let TMP_103 \def (aplus g TMP_102 x2) in (let TMP_104 \def -(ASort n3 n4) in (let TMP_105 \def (aplus g TMP_104 x2) in (eq A TMP_103 -TMP_105))))))) in (let H9 \def (eq_ind_r nat x0 TMP_106 H8 n2 H6) in (let -TMP_107 \def (next g n0) in (let TMP_108 \def (ASort O TMP_107) in (let -TMP_109 \def (aplus g TMP_108 x2) in (let TMP_112 \def (\lambda (a: A).(let -TMP_110 \def (ASort n3 n2) in (let TMP_111 \def (aplus g TMP_110 x2) in (eq A -a TMP_111)))) in (let TMP_113 \def (ASort O n0) in (let TMP_114 \def (S x2) -in (let TMP_115 \def (aplus g TMP_113 TMP_114) in (let TMP_116 \def -(aplus_sort_O_S_simpl g n0 x2) in (let H10 \def (eq_ind_r A TMP_109 TMP_112 -H9 TMP_115 TMP_116) in (let TMP_117 \def (ASort n3 n2) in (let TMP_118 \def -(aplus g TMP_117 x2) in (let TMP_122 \def (\lambda (a: A).(let TMP_119 \def -(ASort O n0) in (let TMP_120 \def (S x2) in (let TMP_121 \def (aplus g -TMP_119 TMP_120) in (eq A TMP_121 a))))) in (let TMP_123 \def (S n3) in (let -TMP_124 \def (ASort TMP_123 n2) in (let TMP_125 \def (S x2) in (let TMP_126 -\def (aplus g TMP_124 TMP_125) in (let TMP_127 \def (aplus_sort_S_S_simpl g -n2 n3 x2) in (let H11 \def (eq_ind_r A TMP_118 TMP_122 H10 TMP_126 TMP_127) -in (let TMP_128 \def (S n3) in (let TMP_129 \def (S x2) in (leq_sort g O -TMP_128 n0 n2 TMP_129 H11)))))))))))))))))))))))))) in (TMP_130 -H5))))))))))))))) in (ex2_3_ind nat nat nat TMP_81 TMP_84 TMP_88 TMP_131 -H2))))))))))))))) in (nat_ind TMP_9 TMP_73 TMP_132 n1 H0))))) in (let TMP_259 -\def (\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0)) -(asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda -(H0: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(let -TMP_137 \def (\lambda (n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g -(ASort n4 n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 -n2))) \to (leq g (ASort n3 n0) (ASort n4 n2)))) \to (let TMP_134 \def (S n3) -in (let TMP_135 \def (ASort TMP_134 n0) in (let TMP_136 \def (ASort n4 n2) in -(leq g TMP_135 TMP_136))))))) in (let TMP_200 \def (\lambda (H1: (leq g -(asucc g (ASort (S n3) n0)) (asucc g (ASort O n2)))).(\lambda (_: (((leq g -(asucc g (ASort n3 n0)) (asucc g (ASort O n2))) \to (leq g (ASort n3 n0) -(ASort O n2))))).(let TMP_138 \def (next g n2) in (let TMP_139 \def (ASort O -TMP_138) in (let H_x \def (leq_gen_sort1 g n3 n0 TMP_139 H1) in (let H2 \def -H_x in (let TMP_144 \def (\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: -nat).(let TMP_140 \def (ASort n3 n0) in (let TMP_141 \def (aplus g TMP_140 k) -in (let TMP_142 \def (ASort h2 n4) in (let TMP_143 \def (aplus g TMP_142 k) -in (eq A TMP_141 TMP_143)))))))) in (let TMP_148 \def (\lambda (n4: -nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_145 \def (next g n2) in -(let TMP_146 \def (ASort O TMP_145) in (let TMP_147 \def (ASort h2 n4) in (eq -A TMP_146 TMP_147))))))) in (let TMP_149 \def (S n3) in (let TMP_150 \def -(ASort TMP_149 n0) in (let TMP_151 \def (ASort O n2) in (let TMP_152 \def -(leq g TMP_150 TMP_151) in (let TMP_199 \def (\lambda (x0: nat).(\lambda (x1: + \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: +A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n: +nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g +(asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda +(n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0)) +(asucc g (ASort n1 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort +n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2)))) +(\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1 +n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g +(ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2)))) (\lambda (H1: (leq g +(asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H_x \def (leq_gen_sort1 +g O (next g n0) (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind +nat nat nat (\lambda (n3: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A +(aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n3) k))))) (\lambda (n3: +nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort +h2 n3))))) (leq g (ASort O n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1: +nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) +x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) +(ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with +[(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow O])) (ASort O (next g +n2)) (ASort x1 x0) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match +e with [(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow ((match g with +[(mk_G next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) +H4) in (\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda +(n3: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 x0) +x2))) H3 O H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n3: nat).(eq A +(aplus g (ASort O (next g n0)) x2) (aplus g (ASort O n3) x2))) H8 (next g n2) +H6) in (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) (\lambda +(a: A).(eq A a (aplus g (ASort O (next g n2)) x2))) H9 (aplus g (ASort O n0) +(S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g +(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) +a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in +(leq_sort g O O n0 n2 (S x2) H11))))))) H5))))))) H2)))) (\lambda (n3: +nat).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2))) +\to (leq g (ASort O n0) (ASort n3 n2))))).(\lambda (H1: (leq g (asucc g +(ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H_x \def (leq_gen_sort1 g O +(next g n0) (ASort n3 n2) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat +(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort +O (next g n0)) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda +(h2: nat).(\lambda (_: nat).(eq A (ASort n3 n2) (ASort h2 n4))))) (leq g +(ASort O n0) (ASort (S n3) n2)) (\lambda (x0: nat).(\lambda (x1: +nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) +x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n3 n2) (ASort x1 +x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort n4 _) +\Rightarrow n4 | (AHead _ _) \Rightarrow n3])) (ASort n3 n2) (ASort x1 x0) +H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort _ +n4) \Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort n3 n2) (ASort x1 +x0) H4) in (\lambda (H7: (eq nat n3 x1)).(let H8 \def (eq_ind_r nat x1 +(\lambda (n4: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort +n4 x0) x2))) H3 n3 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: +nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 n4) x2))) H8 +n2 H6) in (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) +(\lambda (a: A).(eq A a (aplus g (ASort n3 n2) x2))) H9 (aplus g (ASort O n0) +(S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g +(ASort n3 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) a)) H10 +(aplus g (ASort (S n3) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n3 x2)) in +(leq_sort g O (S n3) n0 n2 (S x2) H11))))))) H5))))))) H2)))))) n1 H0)) +(\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0)) (asucc g +(ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda (H0: (leq +g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(nat_ind (\lambda +(n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to +((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort +n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 n2))))) +(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O +n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort O n2))) +\to (leq g (ASort n3 n0) (ASort O n2))))).(let H_x \def (leq_gen_sort1 g n3 +n0 (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat +(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort +n3 n0) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda (h2: +nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort h2 n4))))) (leq g +(ASort (S n3) n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) (ASort x1 -x0))).(let TMP_153 \def (\lambda (e: A).(match e with [(ASort n4 _) -\Rightarrow n4 | (AHead _ _) \Rightarrow O])) in (let TMP_154 \def (next g -n2) in (let TMP_155 \def (ASort O TMP_154) in (let TMP_156 \def (ASort x1 x0) -in (let H5 \def (f_equal A nat TMP_153 TMP_155 TMP_156 H4) in (let TMP_158 -\def (\lambda (e: A).(match e with [(ASort _ n4) \Rightarrow n4 | (AHead _ _) -\Rightarrow (let TMP_157 \def (match g with [(mk_G next _) \Rightarrow next]) -in (TMP_157 n2))])) in (let TMP_159 \def (next g n2) in (let TMP_160 \def -(ASort O TMP_159) in (let TMP_161 \def (ASort x1 x0) in (let H6 \def (f_equal -A nat TMP_158 TMP_160 TMP_161 H4) in (let TMP_198 \def (\lambda (H7: (eq nat -O x1)).(let TMP_166 \def (\lambda (n4: nat).(let TMP_162 \def (ASort n3 n0) -in (let TMP_163 \def (aplus g TMP_162 x2) in (let TMP_164 \def (ASort n4 x0) -in (let TMP_165 \def (aplus g TMP_164 x2) in (eq A TMP_163 TMP_165)))))) in -(let H8 \def (eq_ind_r nat x1 TMP_166 H3 O H7) in (let TMP_171 \def (\lambda -(n4: nat).(let TMP_167 \def (ASort n3 n0) in (let TMP_168 \def (aplus g -TMP_167 x2) in (let TMP_169 \def (ASort O n4) in (let TMP_170 \def (aplus g -TMP_169 x2) in (eq A TMP_168 TMP_170)))))) in (let TMP_172 \def (next g n2) -in (let H9 \def (eq_ind_r nat x0 TMP_171 H8 TMP_172 H6) in (let TMP_173 \def -(ASort n3 n0) in (let TMP_174 \def (aplus g TMP_173 x2) in (let TMP_178 \def -(\lambda (a: A).(let TMP_175 \def (next g n2) in (let TMP_176 \def (ASort O -TMP_175) in (let TMP_177 \def (aplus g TMP_176 x2) in (eq A a TMP_177))))) in -(let TMP_179 \def (S n3) in (let TMP_180 \def (ASort TMP_179 n0) in (let -TMP_181 \def (S x2) in (let TMP_182 \def (aplus g TMP_180 TMP_181) in (let -TMP_183 \def (aplus_sort_S_S_simpl g n0 n3 x2) in (let H10 \def (eq_ind_r A -TMP_174 TMP_178 H9 TMP_182 TMP_183) in (let TMP_184 \def (next g n2) in (let -TMP_185 \def (ASort O TMP_184) in (let TMP_186 \def (aplus g TMP_185 x2) in -(let TMP_191 \def (\lambda (a: A).(let TMP_187 \def (S n3) in (let TMP_188 -\def (ASort TMP_187 n0) in (let TMP_189 \def (S x2) in (let TMP_190 \def -(aplus g TMP_188 TMP_189) in (eq A TMP_190 a)))))) in (let TMP_192 \def -(ASort O n2) in (let TMP_193 \def (S x2) in (let TMP_194 \def (aplus g -TMP_192 TMP_193) in (let TMP_195 \def (aplus_sort_O_S_simpl g n2 x2) in (let -H11 \def (eq_ind_r A TMP_186 TMP_191 H10 TMP_194 TMP_195) in (let TMP_196 -\def (S n3) in (let TMP_197 \def (S x2) in (leq_sort g TMP_196 O n0 n2 -TMP_197 H11))))))))))))))))))))))))))) in (TMP_198 H5))))))))))))))))) in -(ex2_3_ind nat nat nat TMP_144 TMP_148 TMP_152 TMP_199 H2)))))))))))))) in -(let TMP_258 \def (\lambda (n4: nat).(\lambda (_: (((leq g (asucc g (ASort (S -n3) n0)) (asucc g (ASort n4 n2))) \to ((((leq g (asucc g (ASort n3 n0)) -(asucc g (ASort n4 n2))) \to (leq g (ASort n3 n0) (ASort n4 n2)))) \to (leq g -(ASort (S n3) n0) (ASort n4 n2)))))).(\lambda (H1: (leq g (asucc g (ASort (S -n3) n0)) (asucc g (ASort (S n4) n2)))).(\lambda (_: (((leq g (asucc g (ASort -n3 n0)) (asucc g (ASort (S n4) n2))) \to (leq g (ASort n3 n0) (ASort (S n4) -n2))))).(let TMP_201 \def (ASort n4 n2) in (let H_x \def (leq_gen_sort1 g n3 -n0 TMP_201 H1) in (let H2 \def H_x in (let TMP_206 \def (\lambda (n5: -nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_202 \def (ASort n3 n0) in -(let TMP_203 \def (aplus g TMP_202 k) in (let TMP_204 \def (ASort h2 n5) in -(let TMP_205 \def (aplus g TMP_204 k) in (eq A TMP_203 TMP_205)))))))) in -(let TMP_209 \def (\lambda (n5: nat).(\lambda (h2: nat).(\lambda (_: -nat).(let TMP_207 \def (ASort n4 n2) in (let TMP_208 \def (ASort h2 n5) in -(eq A TMP_207 TMP_208)))))) in (let TMP_210 \def (S n3) in (let TMP_211 \def -(ASort TMP_210 n0) in (let TMP_212 \def (S n4) in (let TMP_213 \def (ASort -TMP_212 n2) in (let TMP_214 \def (leq g TMP_211 TMP_213) in (let TMP_257 \def -(\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A -(aplus g (ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A -(ASort n4 n2) (ASort x1 x0))).(let TMP_215 \def (\lambda (e: A).(match e with -[(ASort n5 _) \Rightarrow n5 | (AHead _ _) \Rightarrow n4])) in (let TMP_216 -\def (ASort n4 n2) in (let TMP_217 \def (ASort x1 x0) in (let H5 \def -(f_equal A nat TMP_215 TMP_216 TMP_217 H4) in (let TMP_218 \def (\lambda (e: -A).(match e with [(ASort _ n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2])) -in (let TMP_219 \def (ASort n4 n2) in (let TMP_220 \def (ASort x1 x0) in (let -H6 \def (f_equal A nat TMP_218 TMP_219 TMP_220 H4) in (let TMP_256 \def -(\lambda (H7: (eq nat n4 x1)).(let TMP_225 \def (\lambda (n5: nat).(let -TMP_221 \def (ASort n3 n0) in (let TMP_222 \def (aplus g TMP_221 x2) in (let -TMP_223 \def (ASort n5 x0) in (let TMP_224 \def (aplus g TMP_223 x2) in (eq A -TMP_222 TMP_224)))))) in (let H8 \def (eq_ind_r nat x1 TMP_225 H3 n4 H7) in -(let TMP_230 \def (\lambda (n5: nat).(let TMP_226 \def (ASort n3 n0) in (let -TMP_227 \def (aplus g TMP_226 x2) in (let TMP_228 \def (ASort n4 n5) in (let -TMP_229 \def (aplus g TMP_228 x2) in (eq A TMP_227 TMP_229)))))) in (let H9 -\def (eq_ind_r nat x0 TMP_230 H8 n2 H6) in (let TMP_231 \def (ASort n3 n0) in -(let TMP_232 \def (aplus g TMP_231 x2) in (let TMP_235 \def (\lambda (a: -A).(let TMP_233 \def (ASort n4 n2) in (let TMP_234 \def (aplus g TMP_233 x2) -in (eq A a TMP_234)))) in (let TMP_236 \def (S n3) in (let TMP_237 \def -(ASort TMP_236 n0) in (let TMP_238 \def (S x2) in (let TMP_239 \def (aplus g -TMP_237 TMP_238) in (let TMP_240 \def (aplus_sort_S_S_simpl g n0 n3 x2) in -(let H10 \def (eq_ind_r A TMP_232 TMP_235 H9 TMP_239 TMP_240) in (let TMP_241 -\def (ASort n4 n2) in (let TMP_242 \def (aplus g TMP_241 x2) in (let TMP_247 -\def (\lambda (a: A).(let TMP_243 \def (S n3) in (let TMP_244 \def (ASort -TMP_243 n0) in (let TMP_245 \def (S x2) in (let TMP_246 \def (aplus g TMP_244 -TMP_245) in (eq A TMP_246 a)))))) in (let TMP_248 \def (S n4) in (let TMP_249 -\def (ASort TMP_248 n2) in (let TMP_250 \def (S x2) in (let TMP_251 \def -(aplus g TMP_249 TMP_250) in (let TMP_252 \def (aplus_sort_S_S_simpl g n2 n4 -x2) in (let H11 \def (eq_ind_r A TMP_242 TMP_247 H10 TMP_251 TMP_252) in (let -TMP_253 \def (S n3) in (let TMP_254 \def (S n4) in (let TMP_255 \def (S x2) -in (leq_sort g TMP_253 TMP_254 n0 n2 TMP_255 H11))))))))))))))))))))))))))) -in (TMP_256 H5))))))))))))))) in (ex2_3_ind nat nat nat TMP_206 TMP_209 -TMP_214 TMP_257 H2)))))))))))))))) in (nat_ind TMP_137 TMP_200 TMP_258 n1 H0 -IHn))))))) in (nat_ind TMP_6 TMP_133 TMP_259 n H))))))) in (let TMP_314 \def -(\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n n0)) (asucc g a)) \to -(leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda (H0: (((leq g (asucc g -(ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0) a0)))).(\lambda (H1: (leq -g (asucc g (ASort n n0)) (asucc g (AHead a a0)))).(let TMP_263 \def (\lambda -(n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 -n0) a))) \to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g -(ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a -a0))) \to (let TMP_261 \def (ASort n1 n0) in (let TMP_262 \def (AHead a a0) -in (leq g TMP_261 TMP_262))))))) in (let TMP_288 \def (\lambda (_: (((leq g -(asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O n0) a)))).(\lambda -(_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq g (ASort O n0) -a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g (AHead a -a0)))).(let TMP_264 \def (next g n0) in (let TMP_265 \def (asucc g a0) in -(let TMP_266 \def (AHead a TMP_265) in (let H_x \def (leq_gen_sort1 g O -TMP_264 TMP_266 H4) in (let H5 \def H_x in (let TMP_272 \def (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_267 \def (next g n0) in -(let TMP_268 \def (ASort O TMP_267) in (let TMP_269 \def (aplus g TMP_268 k) -in (let TMP_270 \def (ASort h2 n2) in (let TMP_271 \def (aplus g TMP_270 k) -in (eq A TMP_269 TMP_271))))))))) in (let TMP_276 \def (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_273 \def (asucc g a0) in -(let TMP_274 \def (AHead a TMP_273) in (let TMP_275 \def (ASort h2 n2) in (eq -A TMP_274 TMP_275))))))) in (let TMP_277 \def (ASort O n0) in (let TMP_278 -\def (AHead a a0) in (let TMP_279 \def (leq g TMP_277 TMP_278) in (let -TMP_287 \def (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: -nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort x1 -x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1 x0))).(let -TMP_280 \def (asucc g a0) in (let TMP_281 \def (AHead a TMP_280) in (let -TMP_282 \def (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False | -(AHead _ _) \Rightarrow True])) in (let TMP_283 \def (ASort x1 x0) in (let H8 -\def (eq_ind A TMP_281 TMP_282 I TMP_283 H7) in (let TMP_284 \def (ASort O -n0) in (let TMP_285 \def (AHead a a0) in (let TMP_286 \def (leq g TMP_284 -TMP_285) in (False_ind TMP_286 H8)))))))))))))) in (ex2_3_ind nat nat nat -TMP_272 TMP_276 TMP_279 TMP_287 H5))))))))))))))) in (let TMP_313 \def -(\lambda (n1: nat).(\lambda (_: (((((leq g (asucc g (ASort n1 n0)) (asucc g +x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort n4 _) +\Rightarrow n4 | (AHead _ _) \Rightarrow O])) (ASort O (next g n2)) (ASort x1 +x0) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort +_ n4) \Rightarrow n4 | (AHead _ _) \Rightarrow ((match g with [(mk_G next _) +\Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in (\lambda +(H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n4: nat).(eq A +(aplus g (ASort n3 n0) x2) (aplus g (ASort n4 x0) x2))) H3 O H7) in (let H9 +\def (eq_ind_r nat x0 (\lambda (n4: nat).(eq A (aplus g (ASort n3 n0) x2) +(aplus g (ASort O n4) x2))) H8 (next g n2) H6) in (let H10 \def (eq_ind_r A +(aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g (ASort O (next g +n2)) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) (aplus_sort_S_S_simpl g n0 +n3 x2)) in (let H11 \def (eq_ind_r A (aplus g (ASort O (next g n2)) x2) +(\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2)) a)) H10 (aplus g +(ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in (leq_sort g (S n3) O +n0 n2 (S x2) H11))))))) H5))))))) H2))))) (\lambda (n4: nat).(\lambda (_: +(((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to ((((leq g +(asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort n3 n0) +(ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 n2)))))).(\lambda +(H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S n4) +n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S n4) +n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H_x \def +(leq_gen_sort1 g n3 n0 (ASort n4 n2) H1) in (let H2 \def H_x in (ex2_3_ind +nat nat nat (\lambda (n5: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A +(aplus g (ASort n3 n0) k) (aplus g (ASort h2 n5) k))))) (\lambda (n5: +nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort n4 n2) (ASort h2 +n5))))) (leq g (ASort (S n3) n0) (ASort (S n4) n2)) (\lambda (x0: +nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g +(ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n4 +n2) (ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with +[(ASort n5 _) \Rightarrow n5 | (AHead _ _) \Rightarrow n4])) (ASort n4 n2) +(ASort x1 x0) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e +with [(ASort _ n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2])) (ASort n4 +n2) (ASort x1 x0) H4) in (\lambda (H7: (eq nat n4 x1)).(let H8 \def (eq_ind_r +nat x1 (\lambda (n5: nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n5 +x0) x2))) H3 n4 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n5: nat).(eq A +(aplus g (ASort n3 n0) x2) (aplus g (ASort n4 n5) x2))) H8 n2 H6) in (let H10 +\def (eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g +(ASort n4 n2) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) +(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g +(ASort n4 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2)) +a)) H10 (aplus g (ASort (S n4) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n4 x2)) +in (leq_sort g (S n3) (S n4) n0 n2 (S x2) H11))))))) H5))))))) H2))))))) n1 +H0 IHn)))) n H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n +n0)) (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda +(H0: (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0) +a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g (AHead a +a0)))).(nat_ind (\lambda (n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1 -n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a +n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0)))))) +(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O +n0) a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq +g (ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g +(AHead a a0)))).(let H_x \def (leq_gen_sort1 g O (next g n0) (AHead a (asucc +g a0)) H4) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: +nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort O (next g +n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: +nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0)) (ASort h2 n2))))) (leq g +(ASort O n0) (AHead a a0)) (\lambda (x0: nat).(\lambda (x1: nat).(\lambda +(x2: nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g +(ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1 +x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda (ee: A).(match +ee with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I +(ASort x1 x0) H7) in (False_ind (leq g (ASort O n0) (AHead a a0)) H8))))))) +H5)))))) (\lambda (n1: nat).(\lambda (_: (((((leq g (asucc g (ASort n1 n0)) +(asucc g a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 +n0)) (asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort +n1 n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1) n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g (asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0) a0)))).(\lambda (H4: (leq g -(asucc g (ASort (S n1) n0)) (asucc g (AHead a a0)))).(let TMP_289 \def (asucc -g a0) in (let TMP_290 \def (AHead a TMP_289) in (let H_x \def (leq_gen_sort1 -g n1 n0 TMP_290 H4) in (let H5 \def H_x in (let TMP_295 \def (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_291 \def (ASort n1 n0) in -(let TMP_292 \def (aplus g TMP_291 k) in (let TMP_293 \def (ASort h2 n2) in -(let TMP_294 \def (aplus g TMP_293 k) in (eq A TMP_292 TMP_294)))))))) in -(let TMP_299 \def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: -nat).(let TMP_296 \def (asucc g a0) in (let TMP_297 \def (AHead a TMP_296) in -(let TMP_298 \def (ASort h2 n2) in (eq A TMP_297 TMP_298))))))) in (let -TMP_300 \def (S n1) in (let TMP_301 \def (ASort TMP_300 n0) in (let TMP_302 -\def (AHead a a0) in (let TMP_303 \def (leq g TMP_301 TMP_302) in (let -TMP_312 \def (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: -nat).(\lambda (_: (eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0) -x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1 x0))).(let TMP_304 -\def (asucc g a0) in (let TMP_305 \def (AHead a TMP_304) in (let TMP_306 \def -(\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _) -\Rightarrow True])) in (let TMP_307 \def (ASort x1 x0) in (let H8 \def -(eq_ind A TMP_305 TMP_306 I TMP_307 H7) in (let TMP_308 \def (S n1) in (let -TMP_309 \def (ASort TMP_308 n0) in (let TMP_310 \def (AHead a a0) in (let -TMP_311 \def (leq g TMP_309 TMP_310) in (False_ind TMP_311 H8))))))))))))))) -in (ex2_3_ind nat nat nat TMP_295 TMP_299 TMP_303 TMP_312 H5))))))))))))))))) -in (nat_ind TMP_263 TMP_288 TMP_313 n H H0 H1))))))))) in (A_ind TMP_3 -TMP_260 TMP_314 a2))))))) in (let TMP_396 \def (\lambda (a: A).(\lambda (_: -((\forall (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq g a -a2))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0) -(asucc g a2)) \to (leq g a0 a2))))).(\lambda (a2: A).(let TMP_317 \def -(\lambda (a3: A).((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (let -TMP_316 \def (AHead a a0) in (leq g TMP_316 a3)))) in (let TMP_364 \def +(asucc g (ASort (S n1) n0)) (asucc g (AHead a a0)))).(let H_x \def +(leq_gen_sort1 g n1 n0 (AHead a (asucc g a0)) H4) in (let H5 \def H_x in +(ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: +nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda +(n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0)) +(ASort h2 n2))))) (leq g (ASort (S n1) n0) (AHead a a0)) (\lambda (x0: +nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A (aplus g (ASort +n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g +a0)) (ASort x1 x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda +(ee: A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _) +\Rightarrow True])) I (ASort x1 x0) H7) in (False_ind (leq g (ASort (S n1) +n0) (AHead a a0)) H8))))))) H5)))))))) n H H0 H1)))))) a2)))) (\lambda (a: +A).(\lambda (_: ((\forall (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq +g a a2))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g +a0) (asucc g a2)) \to (leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3: +A).((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a -a0)) (asucc g (ASort n n0)))).(let TMP_320 \def (\lambda (n1: nat).((leq g -(asucc g (AHead a a0)) (asucc g (ASort n1 n0))) \to (let TMP_318 \def (AHead -a a0) in (let TMP_319 \def (ASort n1 n0) in (leq g TMP_318 TMP_319))))) in -(let TMP_342 \def (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort -O n0)))).(let TMP_321 \def (asucc g a0) in (let TMP_322 \def (next g n0) in -(let TMP_323 \def (ASort O TMP_322) in (let H_x \def (leq_gen_head1 g a -TMP_321 TMP_323 H2) in (let H3 \def H_x in (let TMP_324 \def (\lambda (a3: -A).(\lambda (_: A).(leq g a a3))) in (let TMP_326 \def (\lambda (_: -A).(\lambda (a4: A).(let TMP_325 \def (asucc g a0) in (leq g TMP_325 a4)))) -in (let TMP_330 \def (\lambda (a3: A).(\lambda (a4: A).(let TMP_327 \def -(next g n0) in (let TMP_328 \def (ASort O TMP_327) in (let TMP_329 \def -(AHead a3 a4) in (eq A TMP_328 TMP_329)))))) in (let TMP_331 \def (AHead a -a0) in (let TMP_332 \def (ASort O n0) in (let TMP_333 \def (leq g TMP_331 -TMP_332) in (let TMP_341 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: -(leq g a x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda (H6: (eq A -(ASort O (next g n0)) (AHead x0 x1))).(let TMP_334 \def (next g n0) in (let -TMP_335 \def (ASort O TMP_334) in (let TMP_336 \def (\lambda (ee: A).(match -ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) in -(let TMP_337 \def (AHead x0 x1) in (let H7 \def (eq_ind A TMP_335 TMP_336 I -TMP_337 H6) in (let TMP_338 \def (AHead a a0) in (let TMP_339 \def (ASort O -n0) in (let TMP_340 \def (leq g TMP_338 TMP_339) in (False_ind TMP_340 -H7)))))))))))))) in (ex3_2_ind A A TMP_324 TMP_326 TMP_330 TMP_333 TMP_341 -H3)))))))))))))) in (let TMP_363 \def (\lambda (n1: nat).(\lambda (_: (((leq -g (asucc g (AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) -(ASort n1 n0))))).(\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort -(S n1) n0)))).(let TMP_343 \def (asucc g a0) in (let TMP_344 \def (ASort n1 -n0) in (let H_x \def (leq_gen_head1 g a TMP_343 TMP_344 H2) in (let H3 \def -H_x in (let TMP_345 \def (\lambda (a3: A).(\lambda (_: A).(leq g a a3))) in -(let TMP_347 \def (\lambda (_: A).(\lambda (a4: A).(let TMP_346 \def (asucc g -a0) in (leq g TMP_346 a4)))) in (let TMP_350 \def (\lambda (a3: A).(\lambda -(a4: A).(let TMP_348 \def (ASort n1 n0) in (let TMP_349 \def (AHead a3 a4) in -(eq A TMP_348 TMP_349))))) in (let TMP_351 \def (AHead a a0) in (let TMP_352 -\def (S n1) in (let TMP_353 \def (ASort TMP_352 n0) in (let TMP_354 \def (leq -g TMP_351 TMP_353) in (let TMP_362 \def (\lambda (x0: A).(\lambda (x1: +a0)) (asucc g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g +(AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 +n0)))) (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O +n0)))).(let H_x \def (leq_gen_head1 g a (asucc g a0) (ASort O (next g n0)) +H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: +A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq g (asucc g a0) a4))) +(\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O (next g n0)) (AHead a3 +a4)))) (leq g (AHead a a0) (ASort O n0)) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda -(H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let TMP_355 \def (ASort n1 n0) in -(let TMP_356 \def (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow -True | (AHead _ _) \Rightarrow False])) in (let TMP_357 \def (AHead x0 x1) in -(let H7 \def (eq_ind A TMP_355 TMP_356 I TMP_357 H6) in (let TMP_358 \def -(AHead a a0) in (let TMP_359 \def (S n1) in (let TMP_360 \def (ASort TMP_359 -n0) in (let TMP_361 \def (leq g TMP_358 TMP_360) in (False_ind TMP_361 -H7)))))))))))))) in (ex3_2_ind A A TMP_345 TMP_347 TMP_350 TMP_354 TMP_362 -H3)))))))))))))))) in (nat_ind TMP_320 TMP_342 TMP_363 n H1))))))) in (let -TMP_395 \def (\lambda (a3: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) -(asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda (a4: A).(\lambda (_: -(((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g (AHead a a0) -a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g (AHead a3 -a4)))).(let TMP_365 \def (asucc g a0) in (let TMP_366 \def (asucc g a4) in -(let TMP_367 \def (AHead a3 TMP_366) in (let H_x \def (leq_gen_head1 g a -TMP_365 TMP_367 H3) in (let H4 \def H_x in (let TMP_368 \def (\lambda (a5: -A).(\lambda (_: A).(leq g a a5))) in (let TMP_370 \def (\lambda (_: -A).(\lambda (a6: A).(let TMP_369 \def (asucc g a0) in (leq g TMP_369 a6)))) -in (let TMP_374 \def (\lambda (a5: A).(\lambda (a6: A).(let TMP_371 \def -(asucc g a4) in (let TMP_372 \def (AHead a3 TMP_371) in (let TMP_373 \def -(AHead a5 a6) in (eq A TMP_372 TMP_373)))))) in (let TMP_375 \def (AHead a -a0) in (let TMP_376 \def (AHead a3 a4) in (let TMP_377 \def (leq g TMP_375 -TMP_376) in (let TMP_394 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (H5: +(H6: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H7 \def (eq_ind A +(ASort O (next g n0)) (\lambda (ee: A).(match ee with [(ASort _ _) +\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in +(False_ind (leq g (AHead a a0) (ASort O n0)) H7))))))) H3)))) (\lambda (n1: +nat).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0))) +\to (leq g (AHead a a0) (ASort n1 n0))))).(\lambda (H2: (leq g (asucc g +(AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H_x \def (leq_gen_head1 g a +(asucc g a0) (ASort n1 n0) H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda +(a3: A).(\lambda (_: A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq +g (asucc g a0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0) +(AHead a3 a4)))) (leq g (AHead a a0) (ASort (S n1) n0)) (\lambda (x0: +A).(\lambda (x1: A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g +a0) x1)).(\lambda (H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let H7 \def +(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee with [(ASort _ _) +\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in +(False_ind (leq g (AHead a a0) (ASort (S n1) n0)) H7))))))) H3)))))) n H1)))) +(\lambda (a3: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a3)) +\to (leq g (AHead a a0) a3)))).(\lambda (a4: A).(\lambda (_: (((leq g (asucc +g (AHead a a0)) (asucc g a4)) \to (leq g (AHead a a0) a4)))).(\lambda (H3: +(leq g (asucc g (AHead a a0)) (asucc g (AHead a3 a4)))).(let H_x \def +(leq_gen_head1 g a (asucc g a0) (AHead a3 (asucc g a4)) H3) in (let H4 \def +H_x in (ex3_2_ind A A (\lambda (a5: A).(\lambda (_: A).(leq g a a5))) +(\lambda (_: A).(\lambda (a6: A).(leq g (asucc g a0) a6))) (\lambda (a5: +A).(\lambda (a6: A).(eq A (AHead a3 (asucc g a4)) (AHead a5 a6)))) (leq g +(AHead a a0) (AHead a3 a4)) (\lambda (x0: A).(\lambda (x1: A).(\lambda (H5: (leq g a x0)).(\lambda (H6: (leq g (asucc g a0) x1)).(\lambda (H7: (eq A -(AHead a3 (asucc g a4)) (AHead x0 x1))).(let TMP_378 \def (\lambda (e: -A).(match e with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow a5])) -in (let TMP_379 \def (asucc g a4) in (let TMP_380 \def (AHead a3 TMP_379) in -(let TMP_381 \def (AHead x0 x1) in (let H8 \def (f_equal A A TMP_378 TMP_380 -TMP_381 H7) in (let TMP_384 \def (\lambda (e: A).(match e with [(ASort _ _) -\Rightarrow (asucc g a4) | (AHead _ a5) \Rightarrow a5])) in (let TMP_385 -\def (asucc g a4) in (let TMP_386 \def (AHead a3 TMP_385) in (let TMP_387 -\def (AHead x0 x1) in (let H9 \def (f_equal A A TMP_384 TMP_386 TMP_387 H7) -in (let TMP_393 \def (\lambda (H10: (eq A a3 x0)).(let TMP_389 \def (\lambda -(a5: A).(let TMP_388 \def (asucc g a0) in (leq g TMP_388 a5))) in (let -TMP_390 \def (asucc g a4) in (let H11 \def (eq_ind_r A x1 TMP_389 H6 TMP_390 -H9) in (let TMP_391 \def (\lambda (a5: A).(leq g a a5)) in (let H12 \def -(eq_ind_r A x0 TMP_391 H5 a3 H10) in (let TMP_392 \def (H0 a4 H11) in -(leq_head g a a3 H12 a0 a4 TMP_392)))))))) in (TMP_393 H8))))))))))))))))) in -(ex3_2_ind A A TMP_368 TMP_370 TMP_374 TMP_377 TMP_394 H4)))))))))))))))))) -in (A_ind TMP_317 TMP_364 TMP_395 a2))))))))) in (A_ind TMP_1 TMP_315 TMP_396 -a1))))). +(AHead a3 (asucc g a4)) (AHead x0 x1))).(let H8 \def (f_equal A A (\lambda +(e: A).(match e with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow +a5])) (AHead a3 (asucc g a4)) (AHead x0 x1) H7) in ((let H9 \def (f_equal A A +(\lambda (e: A).(match e with [(ASort _ _) \Rightarrow (asucc g a4) | (AHead +_ a5) \Rightarrow a5])) (AHead a3 (asucc g a4)) (AHead x0 x1) H7) in (\lambda +(H10: (eq A a3 x0)).(let H11 \def (eq_ind_r A x1 (\lambda (a5: A).(leq g +(asucc g a0) a5)) H6 (asucc g a4) H9) in (let H12 \def (eq_ind_r A x0 +(\lambda (a5: A).(leq g a a5)) H5 a3 H10) in (leq_head g a a3 H12 a0 a4 (H0 +a4 H11)))))) H8))))))) H4)))))))) a2)))))) a1)). -theorem leq_asucc: +lemma leq_asucc: \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g a0))))) \def - \lambda (g: G).(\lambda (a: A).(let TMP_3 \def (\lambda (a0: A).(let TMP_2 -\def (\lambda (a1: A).(let TMP_1 \def (asucc g a1) in (leq g a0 TMP_1))) in -(ex A TMP_2))) in (let TMP_11 \def (\lambda (n: nat).(\lambda (n0: nat).(let -TMP_6 \def (\lambda (a0: A).(let TMP_4 \def (ASort n n0) in (let TMP_5 \def -(asucc g a0) in (leq g TMP_4 TMP_5)))) in (let TMP_7 \def (S n) in (let TMP_8 -\def (ASort TMP_7 n0) in (let TMP_9 \def (ASort n n0) in (let TMP_10 \def -(leq_refl g TMP_9) in (ex_intro A TMP_6 TMP_8 TMP_10)))))))) in (let TMP_26 -\def (\lambda (a0: A).(\lambda (_: (ex A (\lambda (a1: A).(leq g a0 (asucc g -a1))))).(\lambda (a1: A).(\lambda (H0: (ex A (\lambda (a2: A).(leq g a1 -(asucc g a2))))).(let H1 \def H0 in (let TMP_13 \def (\lambda (a2: A).(let -TMP_12 \def (asucc g a2) in (leq g a1 TMP_12))) in (let TMP_16 \def (\lambda -(a2: A).(let TMP_14 \def (AHead a0 a1) in (let TMP_15 \def (asucc g a2) in -(leq g TMP_14 TMP_15)))) in (let TMP_17 \def (ex A TMP_16) in (let TMP_25 -\def (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc g x))).(let TMP_20 \def -(\lambda (a2: A).(let TMP_18 \def (AHead a0 a1) in (let TMP_19 \def (asucc g -a2) in (leq g TMP_18 TMP_19)))) in (let TMP_21 \def (AHead a0 x) in (let -TMP_22 \def (leq_refl g a0) in (let TMP_23 \def (asucc g x) in (let TMP_24 -\def (leq_head g a0 a0 TMP_22 a1 TMP_23 H2) in (ex_intro A TMP_20 TMP_21 -TMP_24)))))))) in (ex_ind A TMP_13 TMP_17 TMP_25 H1)))))))))) in (A_ind TMP_3 -TMP_11 TMP_26 a))))). + \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(ex A (\lambda (a1: +A).(leq g a0 (asucc g a1))))) (\lambda (n: nat).(\lambda (n0: nat).(ex_intro +A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0) +(leq_refl g (ASort n n0))))) (\lambda (a0: A).(\lambda (_: (ex A (\lambda +(a1: A).(leq g a0 (asucc g a1))))).(\lambda (a1: A).(\lambda (H0: (ex A +(\lambda (a2: A).(leq g a1 (asucc g a2))))).(let H1 \def H0 in (ex_ind A +(\lambda (a2: A).(leq g a1 (asucc g a2))) (ex A (\lambda (a2: A).(leq g +(AHead a0 a1) (asucc g a2)))) (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc +g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2))) +(AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1)))))) +a)). -theorem leq_ahead_asucc_false: +lemma leq_ahead_asucc_false: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) (asucc g a1)) \to (\forall (P: Prop).P)))) \def - \lambda (g: G).(\lambda (a1: A).(let TMP_1 \def (\lambda (a: A).(\forall -(a2: A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) in -(let TMP_34 \def (\lambda (n: nat).(\lambda (n0: nat).(\lambda (a2: -A).(\lambda (H: (leq g (AHead (ASort n n0) a2) (match n with [O \Rightarrow -(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]))).(\lambda (P: -Prop).(let TMP_2 \def (\lambda (n1: nat).((leq g (AHead (ASort n1 n0) a2) -(match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow -(ASort h n0)])) \to P)) in (let TMP_18 \def (\lambda (H0: (leq g (AHead -(ASort O n0) a2) (ASort O (next g n0)))).(let TMP_3 \def (ASort O n0) in (let -TMP_4 \def (next g n0) in (let TMP_5 \def (ASort O TMP_4) in (let H_x \def -(leq_gen_head1 g TMP_3 a2 TMP_5 H0) in (let H1 \def H_x in (let TMP_7 \def -(\lambda (a3: A).(\lambda (_: A).(let TMP_6 \def (ASort O n0) in (leq g TMP_6 -a3)))) in (let TMP_8 \def (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) in -(let TMP_12 \def (\lambda (a3: A).(\lambda (a4: A).(let TMP_9 \def (next g -n0) in (let TMP_10 \def (ASort O TMP_9) in (let TMP_11 \def (AHead a3 a4) in -(eq A TMP_10 TMP_11)))))) in (let TMP_17 \def (\lambda (x0: A).(\lambda (x1: + \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: +A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda +(n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead +(ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) +\Rightarrow (ASort h n0)]))).(\lambda (P: Prop).(nat_ind (\lambda (n1: +nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O +(next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P)) (\lambda (H0: (leq g +(AHead (ASort O n0) a2) (ASort O (next g n0)))).(let H_x \def (leq_gen_head1 +g (ASort O n0) a2 (ASort O (next g n0)) H0) in (let H1 \def H_x in (ex3_2_ind +A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: +A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A +(ASort O (next g n0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g (ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda -(H4: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let TMP_13 \def (next g n0) -in (let TMP_14 \def (ASort O TMP_13) in (let TMP_15 \def (\lambda (ee: -A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) in (let TMP_16 \def (AHead x0 x1) in (let H5 \def (eq_ind A TMP_14 -TMP_15 I TMP_16 H4) in (False_ind P H5))))))))))) in (ex3_2_ind A A TMP_7 -TMP_8 TMP_12 P TMP_17 H1))))))))))) in (let TMP_33 \def (\lambda (n1: -nat).(\lambda (_: (((leq g (AHead (ASort n1 n0) a2) (match n1 with [O -\Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to -P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort n1 n0))).(let -TMP_19 \def (S n1) in (let TMP_20 \def (ASort TMP_19 n0) in (let TMP_21 \def -(ASort n1 n0) in (let H_x \def (leq_gen_head1 g TMP_20 a2 TMP_21 H0) in (let -H1 \def H_x in (let TMP_24 \def (\lambda (a3: A).(\lambda (_: A).(let TMP_22 -\def (S n1) in (let TMP_23 \def (ASort TMP_22 n0) in (leq g TMP_23 a3))))) in -(let TMP_25 \def (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) in (let -TMP_28 \def (\lambda (a3: A).(\lambda (a4: A).(let TMP_26 \def (ASort n1 n0) -in (let TMP_27 \def (AHead a3 a4) in (eq A TMP_26 TMP_27))))) in (let TMP_32 -\def (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S n1) n0) -x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort n1 n0) (AHead x0 -x1))).(let TMP_29 \def (ASort n1 n0) in (let TMP_30 \def (\lambda (ee: +(H4: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H5 \def (eq_ind A +(ASort O (next g n0)) (\lambda (ee: A).(match ee with [(ASort _ _) +\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in +(False_ind P H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead +(ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) +\Rightarrow (ASort h n0)])) \to P))).(\lambda (H0: (leq g (AHead (ASort (S +n1) n0) a2) (ASort n1 n0))).(let H_x \def (leq_gen_head1 g (ASort (S n1) n0) +a2 (ASort n1 n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: +A).(\lambda (_: A).(leq g (ASort (S n1) n0) a3))) (\lambda (_: A).(\lambda +(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0) +(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g +(ASort (S n1) n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort +n1 n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) in (let TMP_31 \def (AHead x0 x1) in (let H5 \def (eq_ind A TMP_29 -TMP_30 I TMP_31 H4) in (False_ind P H5)))))))))) in (ex3_2_ind A A TMP_24 -TMP_25 TMP_28 P TMP_32 H1))))))))))))) in (nat_ind TMP_2 TMP_18 TMP_33 n -H))))))))) in (let TMP_61 \def (\lambda (a: A).(\lambda (_: ((\forall (a2: -A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P))))).(\lambda -(a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a0 a2) (asucc g a0)) -\to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead -(AHead a a0) a2) (AHead a (asucc g a0)))).(\lambda (P: Prop).(let TMP_35 \def -(AHead a a0) in (let TMP_36 \def (asucc g a0) in (let TMP_37 \def (AHead a -TMP_36) in (let H_x \def (leq_gen_head1 g TMP_35 a2 TMP_37 H1) in (let H2 -\def H_x in (let TMP_39 \def (\lambda (a3: A).(\lambda (_: A).(let TMP_38 -\def (AHead a a0) in (leq g TMP_38 a3)))) in (let TMP_40 \def (\lambda (_: -A).(\lambda (a4: A).(leq g a2 a4))) in (let TMP_44 \def (\lambda (a3: -A).(\lambda (a4: A).(let TMP_41 \def (asucc g a0) in (let TMP_42 \def (AHead -a TMP_41) in (let TMP_43 \def (AHead a3 a4) in (eq A TMP_42 TMP_43)))))) in -(let TMP_60 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (H3: (leq g -(AHead a a0) x0)).(\lambda (H4: (leq g a2 x1)).(\lambda (H5: (eq A (AHead a -(asucc g a0)) (AHead x0 x1))).(let TMP_45 \def (\lambda (e: A).(match e with -[(ASort _ _) \Rightarrow a | (AHead a3 _) \Rightarrow a3])) in (let TMP_46 -\def (asucc g a0) in (let TMP_47 \def (AHead a TMP_46) in (let TMP_48 \def -(AHead x0 x1) in (let H6 \def (f_equal A A TMP_45 TMP_47 TMP_48 H5) in (let -TMP_51 \def (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow (asucc g -a0) | (AHead _ a3) \Rightarrow a3])) in (let TMP_52 \def (asucc g a0) in (let -TMP_53 \def (AHead a TMP_52) in (let TMP_54 \def (AHead x0 x1) in (let H7 -\def (f_equal A A TMP_51 TMP_53 TMP_54 H5) in (let TMP_59 \def (\lambda (H8: -(eq A a x0)).(let TMP_55 \def (\lambda (a3: A).(leq g a2 a3)) in (let TMP_56 -\def (asucc g a0) in (let H9 \def (eq_ind_r A x1 TMP_55 H4 TMP_56 H7) in (let -TMP_58 \def (\lambda (a3: A).(let TMP_57 \def (AHead a a0) in (leq g TMP_57 -a3))) in (let H10 \def (eq_ind_r A x0 TMP_58 H3 a H8) in (leq_ahead_false_1 g -a a0 H10 P))))))) in (TMP_59 H6))))))))))))))))) in (ex3_2_ind A A TMP_39 -TMP_40 TMP_44 P TMP_60 H2))))))))))))))))) in (A_ind TMP_1 TMP_34 TMP_61 -a1))))). +False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n H)))))) +(\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a a2) (asucc g +a)) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall +(a2: A).((leq g (AHead a0 a2) (asucc g a0)) \to (\forall (P: +Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2) +(AHead a (asucc g a0)))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g +(AHead a a0) a2 (AHead a (asucc g a0)) H1) in (let H2 \def H_x in (ex3_2_ind +A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) (\lambda (_: +A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A +(AHead a (asucc g a0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: +A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2 +x1)).(\lambda (H5: (eq A (AHead a (asucc g a0)) (AHead x0 x1))).(let H6 \def +(f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a | +(AHead a3 _) \Rightarrow a3])) (AHead a (asucc g a0)) (AHead x0 x1) H5) in +((let H7 \def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) +\Rightarrow (asucc g a0) | (AHead _ a3) \Rightarrow a3])) (AHead a (asucc g +a0)) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def (eq_ind_r A +x1 (\lambda (a3: A).(leq g a2 a3)) H4 (asucc g a0) H7) in (let H10 \def +(eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a a0) a3)) H3 a H8) in +(leq_ahead_false_1 g a a0 H10 P))))) H6))))))) H2)))))))))) a1)). -theorem leq_asucc_false: +lemma leq_asucc_false: \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P: Prop).P))) \def - \lambda (g: G).(\lambda (a: A).(let TMP_1 \def (\lambda (a0: A).((leq g -(asucc g a0) a0) \to (\forall (P: Prop).P))) in (let TMP_103 \def (\lambda -(n: nat).(\lambda (n0: nat).(\lambda (H: (leq g (match n with [O \Rightarrow -(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n -n0))).(\lambda (P: Prop).(let TMP_2 \def (\lambda (n1: nat).((leq g (match n1 -with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) -(ASort n1 n0)) \to P)) in (let TMP_50 \def (\lambda (H0: (leq g (ASort O -(next g n0)) (ASort O n0))).(let TMP_3 \def (next g n0) in (let TMP_4 \def -(ASort O n0) in (let H_x \def (leq_gen_sort1 g O TMP_3 TMP_4 H0) in (let H1 -\def H_x in (let TMP_10 \def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(k: nat).(let TMP_5 \def (next g n0) in (let TMP_6 \def (ASort O TMP_5) in -(let TMP_7 \def (aplus g TMP_6 k) in (let TMP_8 \def (ASort h2 n2) in (let -TMP_9 \def (aplus g TMP_8 k) in (eq A TMP_7 TMP_9))))))))) in (let TMP_13 -\def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_11 \def -(ASort O n0) in (let TMP_12 \def (ASort h2 n2) in (eq A TMP_11 TMP_12)))))) -in (let TMP_49 \def (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: -nat).(\lambda (H2: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort -x1 x0) x2))).(\lambda (H3: (eq A (ASort O n0) (ASort x1 x0))).(let TMP_14 -\def (\lambda (e: A).(match e with [(ASort n1 _) \Rightarrow n1 | (AHead _ _) -\Rightarrow O])) in (let TMP_15 \def (ASort O n0) in (let TMP_16 \def (ASort -x1 x0) in (let H4 \def (f_equal A nat TMP_14 TMP_15 TMP_16 H3) in (let TMP_17 -\def (\lambda (e: A).(match e with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) -\Rightarrow n0])) in (let TMP_18 \def (ASort O n0) in (let TMP_19 \def (ASort -x1 x0) in (let H5 \def (f_equal A nat TMP_17 TMP_18 TMP_19 H3) in (let TMP_48 -\def (\lambda (H6: (eq nat O x1)).(let TMP_25 \def (\lambda (n1: nat).(let -TMP_20 \def (next g n0) in (let TMP_21 \def (ASort O TMP_20) in (let TMP_22 -\def (aplus g TMP_21 x2) in (let TMP_23 \def (ASort n1 x0) in (let TMP_24 -\def (aplus g TMP_23 x2) in (eq A TMP_22 TMP_24))))))) in (let H7 \def -(eq_ind_r nat x1 TMP_25 H2 O H6) in (let TMP_31 \def (\lambda (n1: nat).(let -TMP_26 \def (next g n0) in (let TMP_27 \def (ASort O TMP_26) in (let TMP_28 -\def (aplus g TMP_27 x2) in (let TMP_29 \def (ASort O n1) in (let TMP_30 \def -(aplus g TMP_29 x2) in (eq A TMP_28 TMP_30))))))) in (let H8 \def (eq_ind_r -nat x0 TMP_31 H7 n0 H5) in (let TMP_32 \def (next g n0) in (let TMP_33 \def -(ASort O TMP_32) in (let TMP_34 \def (aplus g TMP_33 x2) in (let TMP_37 \def -(\lambda (a0: A).(let TMP_35 \def (ASort O n0) in (let TMP_36 \def (aplus g -TMP_35 x2) in (eq A a0 TMP_36)))) in (let TMP_38 \def (ASort O n0) in (let -TMP_39 \def (S x2) in (let TMP_40 \def (aplus g TMP_38 TMP_39) in (let TMP_41 -\def (aplus_sort_O_S_simpl g n0 x2) in (let H9 \def (eq_ind_r A TMP_34 TMP_37 -H8 TMP_40 TMP_41) in (let TMP_42 \def (S x2) in (let TMP_43 \def (ASort O n0) -in (let H_y \def (aplus_inj g TMP_42 x2 TMP_43 H9) in (let TMP_44 \def -(\lambda (n1: nat).(le n1 x2)) in (let TMP_45 \def (le_n x2) in (let TMP_46 -\def (S x2) in (let TMP_47 \def (eq_ind_r nat x2 TMP_44 TMP_45 TMP_46 H_y) in -(le_Sx_x x2 TMP_47 P)))))))))))))))))))))) in (TMP_48 H4))))))))))))))) in -(ex2_3_ind nat nat nat TMP_10 TMP_13 P TMP_49 H1))))))))) in (let TMP_102 -\def (\lambda (n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow + \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0) +a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda +(H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) +\Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).(nat_ind +(\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O (next g +n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P)) (\lambda (H0: +(leq g (ASort O (next g n0)) (ASort O n0))).(let H_x \def (leq_gen_sort1 g O +(next g n0) (ASort O n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat +(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort +O (next g n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda +(h2: nat).(\lambda (_: nat).(eq A (ASort O n0) (ASort h2 n2))))) P (\lambda +(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g +(ASort O (next g n0)) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A +(ASort O n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: +A).(match e with [(ASort n1 _) \Rightarrow n1 | (AHead _ _) \Rightarrow O])) +(ASort O n0) (ASort x1 x0) H3) in ((let H5 \def (f_equal A nat (\lambda (e: +A).(match e with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) \Rightarrow n0])) +(ASort O n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat O x1)).(let H7 \def +(eq_ind_r nat x1 (\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) +(aplus g (ASort n1 x0) x2))) H2 O H6) in (let H8 \def (eq_ind_r nat x0 +(\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort O +n1) x2))) H7 n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort O (next g +n0)) x2) (\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) x2))) H8 (aplus g +(ASort O n0) (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H_y \def +(aplus_inj g (S x2) x2 (ASort O n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 +(\lambda (n1: nat).(le n1 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) +H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to -P))).(\lambda (H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let TMP_51 \def -(S n1) in (let TMP_52 \def (ASort TMP_51 n0) in (let H_x \def (leq_gen_sort1 -g n1 n0 TMP_52 H0) in (let H1 \def H_x in (let TMP_57 \def (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_53 \def (ASort n1 n0) in -(let TMP_54 \def (aplus g TMP_53 k) in (let TMP_55 \def (ASort h2 n2) in (let -TMP_56 \def (aplus g TMP_55 k) in (eq A TMP_54 TMP_56)))))))) in (let TMP_61 -\def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_58 \def -(S n1) in (let TMP_59 \def (ASort TMP_58 n0) in (let TMP_60 \def (ASort h2 -n2) in (eq A TMP_59 TMP_60))))))) in (let TMP_101 \def (\lambda (x0: -nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g -(ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A (ASort (S -n1) n0) (ASort x1 x0))).(let TMP_62 \def (\lambda (e: A).(match e with -[(ASort n2 _) \Rightarrow n2 | (AHead _ _) \Rightarrow (S n1)])) in (let -TMP_63 \def (S n1) in (let TMP_64 \def (ASort TMP_63 n0) in (let TMP_65 \def -(ASort x1 x0) in (let H4 \def (f_equal A nat TMP_62 TMP_64 TMP_65 H3) in (let -TMP_66 \def (\lambda (e: A).(match e with [(ASort _ n2) \Rightarrow n2 | -(AHead _ _) \Rightarrow n0])) in (let TMP_67 \def (S n1) in (let TMP_68 \def -(ASort TMP_67 n0) in (let TMP_69 \def (ASort x1 x0) in (let H5 \def (f_equal -A nat TMP_66 TMP_68 TMP_69 H3) in (let TMP_100 \def (\lambda (H6: (eq nat (S -n1) x1)).(let TMP_74 \def (\lambda (n2: nat).(let TMP_70 \def (ASort n1 n0) -in (let TMP_71 \def (aplus g TMP_70 x2) in (let TMP_72 \def (ASort n2 x0) in -(let TMP_73 \def (aplus g TMP_72 x2) in (eq A TMP_71 TMP_73)))))) in (let -TMP_75 \def (S n1) in (let H7 \def (eq_ind_r nat x1 TMP_74 H2 TMP_75 H6) in -(let TMP_81 \def (\lambda (n2: nat).(let TMP_76 \def (ASort n1 n0) in (let -TMP_77 \def (aplus g TMP_76 x2) in (let TMP_78 \def (S n1) in (let TMP_79 -\def (ASort TMP_78 n2) in (let TMP_80 \def (aplus g TMP_79 x2) in (eq A -TMP_77 TMP_80))))))) in (let H8 \def (eq_ind_r nat x0 TMP_81 H7 n0 H5) in -(let TMP_82 \def (ASort n1 n0) in (let TMP_83 \def (aplus g TMP_82 x2) in -(let TMP_87 \def (\lambda (a0: A).(let TMP_84 \def (S n1) in (let TMP_85 \def -(ASort TMP_84 n0) in (let TMP_86 \def (aplus g TMP_85 x2) in (eq A a0 -TMP_86))))) in (let TMP_88 \def (S n1) in (let TMP_89 \def (ASort TMP_88 n0) -in (let TMP_90 \def (S x2) in (let TMP_91 \def (aplus g TMP_89 TMP_90) in -(let TMP_92 \def (aplus_sort_S_S_simpl g n0 n1 x2) in (let H9 \def (eq_ind_r -A TMP_83 TMP_87 H8 TMP_91 TMP_92) in (let TMP_93 \def (S x2) in (let TMP_94 -\def (S n1) in (let TMP_95 \def (ASort TMP_94 n0) in (let H_y \def (aplus_inj -g TMP_93 x2 TMP_95 H9) in (let TMP_96 \def (\lambda (n2: nat).(le n2 x2)) in -(let TMP_97 \def (le_n x2) in (let TMP_98 \def (S x2) in (let TMP_99 \def -(eq_ind_r nat x2 TMP_96 TMP_97 TMP_98 H_y) in (le_Sx_x x2 TMP_99 -P)))))))))))))))))))))))) in (TMP_100 H4))))))))))))))))) in (ex2_3_ind nat -nat nat TMP_57 TMP_61 P TMP_101 H1))))))))))) in (nat_ind TMP_2 TMP_50 -TMP_102 n H)))))))) in (let TMP_123 \def (\lambda (a0: A).(\lambda (_: (((leq -g (asucc g a0) a0) \to (\forall (P: Prop).P)))).(\lambda (a1: A).(\lambda -(H0: (((leq g (asucc g a1) a1) \to (\forall (P: Prop).P)))).(\lambda (H1: -(leq g (AHead a0 (asucc g a1)) (AHead a0 a1))).(\lambda (P: Prop).(let -TMP_104 \def (asucc g a1) in (let TMP_105 \def (AHead a0 a1) in (let H_x \def -(leq_gen_head1 g a0 TMP_104 TMP_105 H1) in (let H2 \def H_x in (let TMP_106 -\def (\lambda (a3: A).(\lambda (_: A).(leq g a0 a3))) in (let TMP_108 \def -(\lambda (_: A).(\lambda (a4: A).(let TMP_107 \def (asucc g a1) in (leq g -TMP_107 a4)))) in (let TMP_111 \def (\lambda (a3: A).(\lambda (a4: A).(let -TMP_109 \def (AHead a0 a1) in (let TMP_110 \def (AHead a3 a4) in (eq A -TMP_109 TMP_110))))) in (let TMP_122 \def (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H3: (leq g a0 x0)).(\lambda (H4: (leq g (asucc g a1) -x1)).(\lambda (H5: (eq A (AHead a0 a1) (AHead x0 x1))).(let TMP_112 \def -(\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a0 | (AHead a2 _) -\Rightarrow a2])) in (let TMP_113 \def (AHead a0 a1) in (let TMP_114 \def -(AHead x0 x1) in (let H6 \def (f_equal A A TMP_112 TMP_113 TMP_114 H5) in -(let TMP_115 \def (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a1 | -(AHead _ a2) \Rightarrow a2])) in (let TMP_116 \def (AHead a0 a1) in (let -TMP_117 \def (AHead x0 x1) in (let H7 \def (f_equal A A TMP_115 TMP_116 -TMP_117 H5) in (let TMP_121 \def (\lambda (H8: (eq A a0 x0)).(let TMP_119 -\def (\lambda (a2: A).(let TMP_118 \def (asucc g a1) in (leq g TMP_118 a2))) -in (let H9 \def (eq_ind_r A x1 TMP_119 H4 a1 H7) in (let TMP_120 \def -(\lambda (a2: A).(leq g a0 a2)) in (let H10 \def (eq_ind_r A x0 TMP_120 H3 a0 -H8) in (H0 H9 P)))))) in (TMP_121 H6))))))))))))))) in (ex3_2_ind A A TMP_106 -TMP_108 TMP_111 P TMP_122 H2))))))))))))))) in (A_ind TMP_1 TMP_103 TMP_123 -a))))). +P))).(\lambda (H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let H_x \def +(leq_gen_sort1 g n1 n0 (ASort (S n1) n0) H0) in (let H1 \def H_x in +(ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: +nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda +(n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort (S n1) n0) (ASort +h2 n2))))) P (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: +nat).(\lambda (H2: (eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0) +x2))).(\lambda (H3: (eq A (ASort (S n1) n0) (ASort x1 x0))).(let H4 \def +(f_equal A nat (\lambda (e: A).(match e with [(ASort n2 _) \Rightarrow n2 | +(AHead _ _) \Rightarrow (S n1)])) (ASort (S n1) n0) (ASort x1 x0) H3) in +((let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort _ n2) +\Rightarrow n2 | (AHead _ _) \Rightarrow n0])) (ASort (S n1) n0) (ASort x1 +x0) H3) in (\lambda (H6: (eq nat (S n1) x1)).(let H7 \def (eq_ind_r nat x1 +(\lambda (n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort n2 x0) +x2))) H2 (S n1) H6) in (let H8 \def (eq_ind_r nat x0 (\lambda (n2: nat).(eq A +(aplus g (ASort n1 n0) x2) (aplus g (ASort (S n1) n2) x2))) H7 n0 H5) in (let +H9 \def (eq_ind_r A (aplus g (ASort n1 n0) x2) (\lambda (a0: A).(eq A a0 +(aplus g (ASort (S n1) n0) x2))) H8 (aplus g (ASort (S n1) n0) (S x2)) +(aplus_sort_S_S_simpl g n0 n1 x2)) in (let H_y \def (aplus_inj g (S x2) x2 +(ASort (S n1) n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n2: nat).(le +n2 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))))) n H))))) (\lambda +(a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to (\forall (P: +Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g a1) a1) \to +(\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g a1)) (AHead +a0 a1))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g a0 (asucc g a1) +(AHead a0 a1) H1) in (let H2 \def H_x in (ex3_2_ind A A (\lambda (a3: +A).(\lambda (_: A).(leq g a0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g +(asucc g a1) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a0 a1) +(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (H3: (leq g a0 +x0)).(\lambda (H4: (leq g (asucc g a1) x1)).(\lambda (H5: (eq A (AHead a0 a1) +(AHead x0 x1))).(let H6 \def (f_equal A A (\lambda (e: A).(match e with +[(ASort _ _) \Rightarrow a0 | (AHead a2 _) \Rightarrow a2])) (AHead a0 a1) +(AHead x0 x1) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e with +[(ASort _ _) \Rightarrow a1 | (AHead _ a2) \Rightarrow a2])) (AHead a0 a1) +(AHead x0 x1) H5) in (\lambda (H8: (eq A a0 x0)).(let H9 \def (eq_ind_r A x1 +(\lambda (a2: A).(leq g (asucc g a1) a2)) H4 a1 H7) in (let H10 \def +(eq_ind_r A x0 (\lambda (a2: A).(leq g a0 a2)) H3 a0 H8) in (H0 H9 P))))) +H6))))))) H2))))))))) a)).