X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Fpr0%2Fprops.ma;h=a99bdf288154c419401c6c4fb59921443c5c4c98;hb=5ff05b234ea985b91bfe6ef3e8dd54eeeb477e11;hp=d7f69d691abb71849d095ce10739f316e6dd9c78;hpb=e8656c819b0b5e7bea7b4da244015b480af5f0f5;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma index d7f69d691..a99bdf288 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma @@ -14,1745 +14,1002 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/pr0/defs.ma". +include "basic_1/pr0/fwd.ma". -include "Basic-1/subst0/subst0.ma". +include "basic_1/subst0/props.ma". theorem pr0_lift: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t1) (lift h d t2)))))) \def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t) -(lift h d t0)))))) (\lambda (t: T).(\lambda (h: nat).(\lambda (d: -nat).(pr0_refl (lift h d t))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda -(_: (pr0 u1 u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 -(lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda -(_: (pr0 t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 -(lift h d t3) (lift h d t4)))))).(\lambda (k: K).(\lambda (h: nat).(\lambda -(d: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) t3)) (\lambda (t: -T).(pr0 t (lift h d (THead k u2 t4)))) (eq_ind_r T (THead k (lift h d u2) -(lift h (s k d) t4)) (\lambda (t: T).(pr0 (THead k (lift h d u1) (lift h (s k -d) t3)) t)) (pr0_comp (lift h d u1) (lift h d u2) (H1 h d) (lift h (s k d) -t3) (lift h (s k d) t4) (H3 h (s k d)) k) (lift h d (THead k u2 t4)) -(lift_head k u2 t4 h d)) (lift h d (THead k u1 t3)) (lift_head k u1 t3 h -d))))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(let TMP_3 \def +(\lambda (t: T).(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).(let +TMP_1 \def (lift h d t) in (let TMP_2 \def (lift h d t0) in (pr0 TMP_1 +TMP_2))))))) in (let TMP_5 \def (\lambda (t: T).(\lambda (h: nat).(\lambda +(d: nat).(let TMP_4 \def (lift h d t) in (pr0_refl TMP_4))))) in (let TMP_39 +\def (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda +(H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d u1) (lift h d +u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda +(H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) (lift h d +t4)))))).(\lambda (k: K).(\lambda (h: nat).(\lambda (d: nat).(let TMP_6 \def +(lift h d u1) in (let TMP_7 \def (s k d) in (let TMP_8 \def (lift h TMP_7 t3) +in (let TMP_9 \def (THead k TMP_6 TMP_8) in (let TMP_12 \def (\lambda (t: +T).(let TMP_10 \def (THead k u2 t4) in (let TMP_11 \def (lift h d TMP_10) in +(pr0 t TMP_11)))) in (let TMP_13 \def (lift h d u2) in (let TMP_14 \def (s k +d) in (let TMP_15 \def (lift h TMP_14 t4) in (let TMP_16 \def (THead k TMP_13 +TMP_15) in (let TMP_21 \def (\lambda (t: T).(let TMP_17 \def (lift h d u1) in +(let TMP_18 \def (s k d) in (let TMP_19 \def (lift h TMP_18 t3) in (let +TMP_20 \def (THead k TMP_17 TMP_19) in (pr0 TMP_20 t)))))) in (let TMP_22 +\def (lift h d u1) in (let TMP_23 \def (lift h d u2) in (let TMP_24 \def (H1 +h d) in (let TMP_25 \def (s k d) in (let TMP_26 \def (lift h TMP_25 t3) in +(let TMP_27 \def (s k d) in (let TMP_28 \def (lift h TMP_27 t4) in (let +TMP_29 \def (s k d) in (let TMP_30 \def (H3 h TMP_29) in (let TMP_31 \def +(pr0_comp TMP_22 TMP_23 TMP_24 TMP_26 TMP_28 TMP_30 k) in (let TMP_32 \def +(THead k u2 t4) in (let TMP_33 \def (lift h d TMP_32) in (let TMP_34 \def +(lift_head k u2 t4 h d) in (let TMP_35 \def (eq_ind_r T TMP_16 TMP_21 TMP_31 +TMP_33 TMP_34) in (let TMP_36 \def (THead k u1 t3) in (let TMP_37 \def (lift +h d TMP_36) in (let TMP_38 \def (lift_head k u1 t3 h d) in (eq_ind_r T TMP_9 +TMP_12 TMP_35 TMP_37 TMP_38))))))))))))))))))))))))))))))))))))))) in (let +TMP_132 \def (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d v1) (lift h d v2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) -(lift h d t4)))))).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead -(Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) (THead (Bind Abst) u -t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r -T (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s -(Flat Appl) d)) t3)) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) -(lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r T (THead (Bind Abbr) (lift h -d v2) (lift h (s (Bind Abbr) d) t4)) (\lambda (t: T).(pr0 (THead (Flat Appl) -(lift h d v1) (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s -(Bind Abst) (s (Flat Appl) d)) t3))) t)) (pr0_beta (lift h (s (Flat Appl) d) -u) (lift h d v1) (lift h d v2) (H1 h d) (lift h (s (Bind Abst) (s (Flat Appl) -d)) t3) (lift h (s (Bind Abbr) d) t4) (H3 h (s (Bind Abbr) d))) (lift h d -(THead (Bind Abbr) v2 t4)) (lift_head (Bind Abbr) v2 t4 h d)) (lift h (s -(Flat Appl) d) (THead (Bind Abst) u t3)) (lift_head (Bind Abst) u t3 h (s -(Flat Appl) d))) (lift h d (THead (Flat Appl) v1 (THead (Bind Abst) u t3))) -(lift_head (Flat Appl) v1 (THead (Bind Abst) u t3) h d))))))))))))) (\lambda -(b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (pr0 v1 v2)).(\lambda (H2: ((\forall (h: nat).(\forall (d: -nat).(pr0 (lift h d v1) (lift h d v2)))))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (_: (pr0 u1 u2)).(\lambda (H4: ((\forall (h: nat).(\forall (d: -nat).(pr0 (lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (pr0 t3 t4)).(\lambda (H6: ((\forall (h: nat).(\forall (d: -nat).(pr0 (lift h d t3) (lift h d t4)))))).(\lambda (h: nat).(\lambda (d: -nat).(eq_ind_r T (THead (Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) -(THead (Bind b) u1 t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead (Bind b) -(lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) t3)) -(\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) (lift h d (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead -(Bind b) (lift h d u2) (lift h (s (Bind b) d) (THead (Flat Appl) (lift (S O) -O v2) t4))) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) (THead -(Bind b) (lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) -t3))) t)) (eq_ind_r T (THead (Flat Appl) (lift h (s (Bind b) d) (lift (S O) O -v2)) (lift h (s (Flat Appl) (s (Bind b) d)) t4)) (\lambda (t: T).(pr0 (THead -(Flat Appl) (lift h d v1) (THead (Bind b) (lift h (s (Flat Appl) d) u1) (lift -h (s (Bind b) (s (Flat Appl) d)) t3))) (THead (Bind b) (lift h d u2) t))) -(eq_ind nat (plus (S O) d) (\lambda (n: nat).(pr0 (THead (Flat Appl) (lift h -d v1) (THead (Bind b) (lift h d u1) (lift h n t3))) (THead (Bind b) (lift h d -u2) (THead (Flat Appl) (lift h n (lift (S O) O v2)) (lift h n t4))))) -(eq_ind_r T (lift (S O) O (lift h d v2)) (\lambda (t: T).(pr0 (THead (Flat -Appl) (lift h d v1) (THead (Bind b) (lift h d u1) (lift h (plus (S O) d) -t3))) (THead (Bind b) (lift h d u2) (THead (Flat Appl) t (lift h (plus (S O) -d) t4))))) (pr0_upsilon b H0 (lift h d v1) (lift h d v2) (H2 h d) (lift h d -u1) (lift h d u2) (H4 h d) (lift h (plus (S O) d) t3) (lift h (plus (S O) d) -t4) (H6 h (plus (S O) d))) (lift h (plus (S O) d) (lift (S O) O v2)) (lift_d -v2 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) (lift h (s (Bind b) -d) (THead (Flat Appl) (lift (S O) O v2) t4)) (lift_head (Flat Appl) (lift (S -O) O v2) t4 h (s (Bind b) d))) (lift h d (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4))) (lift_head (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4) h d)) (lift h (s (Flat Appl) d) (THead (Bind b) u1 t3)) -(lift_head (Bind b) u1 t3 h (s (Flat Appl) d))) (lift h d (THead (Flat Appl) -v1 (THead (Bind b) u1 t3))) (lift_head (Flat Appl) v1 (THead (Bind b) u1 t3) -h d)))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 +(lift h d t4)))))).(\lambda (h: nat).(\lambda (d: nat).(let TMP_40 \def (Flat +Appl) in (let TMP_41 \def (lift h d v1) in (let TMP_42 \def (Flat Appl) in +(let TMP_43 \def (s TMP_42 d) in (let TMP_44 \def (Bind Abst) in (let TMP_45 +\def (THead TMP_44 u t3) in (let TMP_46 \def (lift h TMP_43 TMP_45) in (let +TMP_47 \def (THead TMP_40 TMP_41 TMP_46) in (let TMP_51 \def (\lambda (t: +T).(let TMP_48 \def (Bind Abbr) in (let TMP_49 \def (THead TMP_48 v2 t4) in +(let TMP_50 \def (lift h d TMP_49) in (pr0 t TMP_50))))) in (let TMP_52 \def +(Bind Abst) in (let TMP_53 \def (Flat Appl) in (let TMP_54 \def (s TMP_53 d) +in (let TMP_55 \def (lift h TMP_54 u) in (let TMP_56 \def (Bind Abst) in (let +TMP_57 \def (Flat Appl) in (let TMP_58 \def (s TMP_57 d) in (let TMP_59 \def +(s TMP_56 TMP_58) in (let TMP_60 \def (lift h TMP_59 t3) in (let TMP_61 \def +(THead TMP_52 TMP_55 TMP_60) in (let TMP_68 \def (\lambda (t: T).(let TMP_62 +\def (Flat Appl) in (let TMP_63 \def (lift h d v1) in (let TMP_64 \def (THead +TMP_62 TMP_63 t) in (let TMP_65 \def (Bind Abbr) in (let TMP_66 \def (THead +TMP_65 v2 t4) in (let TMP_67 \def (lift h d TMP_66) in (pr0 TMP_64 +TMP_67)))))))) in (let TMP_69 \def (Bind Abbr) in (let TMP_70 \def (lift h d +v2) in (let TMP_71 \def (Bind Abbr) in (let TMP_72 \def (s TMP_71 d) in (let +TMP_73 \def (lift h TMP_72 t4) in (let TMP_74 \def (THead TMP_69 TMP_70 +TMP_73) in (let TMP_88 \def (\lambda (t: T).(let TMP_75 \def (Flat Appl) in +(let TMP_76 \def (lift h d v1) in (let TMP_77 \def (Bind Abst) in (let TMP_78 +\def (Flat Appl) in (let TMP_79 \def (s TMP_78 d) in (let TMP_80 \def (lift h +TMP_79 u) in (let TMP_81 \def (Bind Abst) in (let TMP_82 \def (Flat Appl) in +(let TMP_83 \def (s TMP_82 d) in (let TMP_84 \def (s TMP_81 TMP_83) in (let +TMP_85 \def (lift h TMP_84 t3) in (let TMP_86 \def (THead TMP_77 TMP_80 +TMP_85) in (let TMP_87 \def (THead TMP_75 TMP_76 TMP_86) in (pr0 TMP_87 +t))))))))))))))) in (let TMP_89 \def (Flat Appl) in (let TMP_90 \def (s +TMP_89 d) in (let TMP_91 \def (lift h TMP_90 u) in (let TMP_92 \def (lift h d +v1) in (let TMP_93 \def (lift h d v2) in (let TMP_94 \def (H1 h d) in (let +TMP_95 \def (Bind Abst) in (let TMP_96 \def (Flat Appl) in (let TMP_97 \def +(s TMP_96 d) in (let TMP_98 \def (s TMP_95 TMP_97) in (let TMP_99 \def (lift +h TMP_98 t3) in (let TMP_100 \def (Bind Abbr) in (let TMP_101 \def (s TMP_100 +d) in (let TMP_102 \def (lift h TMP_101 t4) in (let TMP_103 \def (Bind Abbr) +in (let TMP_104 \def (s TMP_103 d) in (let TMP_105 \def (H3 h TMP_104) in +(let TMP_106 \def (pr0_beta TMP_91 TMP_92 TMP_93 TMP_94 TMP_99 TMP_102 +TMP_105) in (let TMP_107 \def (Bind Abbr) in (let TMP_108 \def (THead TMP_107 +v2 t4) in (let TMP_109 \def (lift h d TMP_108) in (let TMP_110 \def (Bind +Abbr) in (let TMP_111 \def (lift_head TMP_110 v2 t4 h d) in (let TMP_112 \def +(eq_ind_r T TMP_74 TMP_88 TMP_106 TMP_109 TMP_111) in (let TMP_113 \def (Flat +Appl) in (let TMP_114 \def (s TMP_113 d) in (let TMP_115 \def (Bind Abst) in +(let TMP_116 \def (THead TMP_115 u t3) in (let TMP_117 \def (lift h TMP_114 +TMP_116) in (let TMP_118 \def (Bind Abst) in (let TMP_119 \def (Flat Appl) in +(let TMP_120 \def (s TMP_119 d) in (let TMP_121 \def (lift_head TMP_118 u t3 +h TMP_120) in (let TMP_122 \def (eq_ind_r T TMP_61 TMP_68 TMP_112 TMP_117 +TMP_121) in (let TMP_123 \def (Flat Appl) in (let TMP_124 \def (Bind Abst) in +(let TMP_125 \def (THead TMP_124 u t3) in (let TMP_126 \def (THead TMP_123 v1 +TMP_125) in (let TMP_127 \def (lift h d TMP_126) in (let TMP_128 \def (Flat +Appl) in (let TMP_129 \def (Bind Abst) in (let TMP_130 \def (THead TMP_129 u +t3) in (let TMP_131 \def (lift_head TMP_128 v1 TMP_130 h d) in (eq_ind_r T +TMP_47 TMP_51 TMP_122 TMP_127 +TMP_131))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) +))))))))))) in (let TMP_339 \def (\lambda (b: B).(\lambda (H0: (not (eq B b +Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda +(H2: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d v1) (lift h d +v2)))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda +(H4: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d u1) (lift h d +u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda +(H6: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) (lift h d +t4)))))).(\lambda (h: nat).(\lambda (d: nat).(let TMP_133 \def (Flat Appl) in +(let TMP_134 \def (lift h d v1) in (let TMP_135 \def (Flat Appl) in (let +TMP_136 \def (s TMP_135 d) in (let TMP_137 \def (Bind b) in (let TMP_138 \def +(THead TMP_137 u1 t3) in (let TMP_139 \def (lift h TMP_136 TMP_138) in (let +TMP_140 \def (THead TMP_133 TMP_134 TMP_139) in (let TMP_148 \def (\lambda +(t: T).(let TMP_141 \def (Bind b) in (let TMP_142 \def (Flat Appl) in (let +TMP_143 \def (S O) in (let TMP_144 \def (lift TMP_143 O v2) in (let TMP_145 +\def (THead TMP_142 TMP_144 t4) in (let TMP_146 \def (THead TMP_141 u2 +TMP_145) in (let TMP_147 \def (lift h d TMP_146) in (pr0 t TMP_147))))))))) +in (let TMP_149 \def (Bind b) in (let TMP_150 \def (Flat Appl) in (let +TMP_151 \def (s TMP_150 d) in (let TMP_152 \def (lift h TMP_151 u1) in (let +TMP_153 \def (Bind b) in (let TMP_154 \def (Flat Appl) in (let TMP_155 \def +(s TMP_154 d) in (let TMP_156 \def (s TMP_153 TMP_155) in (let TMP_157 \def +(lift h TMP_156 t3) in (let TMP_158 \def (THead TMP_149 TMP_152 TMP_157) in +(let TMP_169 \def (\lambda (t: T).(let TMP_159 \def (Flat Appl) in (let +TMP_160 \def (lift h d v1) in (let TMP_161 \def (THead TMP_159 TMP_160 t) in +(let TMP_162 \def (Bind b) in (let TMP_163 \def (Flat Appl) in (let TMP_164 +\def (S O) in (let TMP_165 \def (lift TMP_164 O v2) in (let TMP_166 \def +(THead TMP_163 TMP_165 t4) in (let TMP_167 \def (THead TMP_162 u2 TMP_166) in +(let TMP_168 \def (lift h d TMP_167) in (pr0 TMP_161 TMP_168)))))))))))) in +(let TMP_170 \def (Bind b) in (let TMP_171 \def (lift h d u2) in (let TMP_172 +\def (Bind b) in (let TMP_173 \def (s TMP_172 d) in (let TMP_174 \def (Flat +Appl) in (let TMP_175 \def (S O) in (let TMP_176 \def (lift TMP_175 O v2) in +(let TMP_177 \def (THead TMP_174 TMP_176 t4) in (let TMP_178 \def (lift h +TMP_173 TMP_177) in (let TMP_179 \def (THead TMP_170 TMP_171 TMP_178) in (let +TMP_193 \def (\lambda (t: T).(let TMP_180 \def (Flat Appl) in (let TMP_181 +\def (lift h d v1) in (let TMP_182 \def (Bind b) in (let TMP_183 \def (Flat +Appl) in (let TMP_184 \def (s TMP_183 d) in (let TMP_185 \def (lift h TMP_184 +u1) in (let TMP_186 \def (Bind b) in (let TMP_187 \def (Flat Appl) in (let +TMP_188 \def (s TMP_187 d) in (let TMP_189 \def (s TMP_186 TMP_188) in (let +TMP_190 \def (lift h TMP_189 t3) in (let TMP_191 \def (THead TMP_182 TMP_185 +TMP_190) in (let TMP_192 \def (THead TMP_180 TMP_181 TMP_191) in (pr0 TMP_192 +t))))))))))))))) in (let TMP_194 \def (Flat Appl) in (let TMP_195 \def (Bind +b) in (let TMP_196 \def (s TMP_195 d) in (let TMP_197 \def (S O) in (let +TMP_198 \def (lift TMP_197 O v2) in (let TMP_199 \def (lift h TMP_196 +TMP_198) in (let TMP_200 \def (Flat Appl) in (let TMP_201 \def (Bind b) in +(let TMP_202 \def (s TMP_201 d) in (let TMP_203 \def (s TMP_200 TMP_202) in +(let TMP_204 \def (lift h TMP_203 t4) in (let TMP_205 \def (THead TMP_194 +TMP_199 TMP_204) in (let TMP_222 \def (\lambda (t: T).(let TMP_206 \def (Flat +Appl) in (let TMP_207 \def (lift h d v1) in (let TMP_208 \def (Bind b) in +(let TMP_209 \def (Flat Appl) in (let TMP_210 \def (s TMP_209 d) in (let +TMP_211 \def (lift h TMP_210 u1) in (let TMP_212 \def (Bind b) in (let +TMP_213 \def (Flat Appl) in (let TMP_214 \def (s TMP_213 d) in (let TMP_215 +\def (s TMP_212 TMP_214) in (let TMP_216 \def (lift h TMP_215 t3) in (let +TMP_217 \def (THead TMP_208 TMP_211 TMP_216) in (let TMP_218 \def (THead +TMP_206 TMP_207 TMP_217) in (let TMP_219 \def (Bind b) in (let TMP_220 \def +(lift h d u2) in (let TMP_221 \def (THead TMP_219 TMP_220 t) in (pr0 TMP_218 +TMP_221)))))))))))))))))) in (let TMP_223 \def (S O) in (let TMP_224 \def +(plus TMP_223 d) in (let TMP_241 \def (\lambda (n: nat).(let TMP_225 \def +(Flat Appl) in (let TMP_226 \def (lift h d v1) in (let TMP_227 \def (Bind b) +in (let TMP_228 \def (lift h d u1) in (let TMP_229 \def (lift h n t3) in (let +TMP_230 \def (THead TMP_227 TMP_228 TMP_229) in (let TMP_231 \def (THead +TMP_225 TMP_226 TMP_230) in (let TMP_232 \def (Bind b) in (let TMP_233 \def +(lift h d u2) in (let TMP_234 \def (Flat Appl) in (let TMP_235 \def (S O) in +(let TMP_236 \def (lift TMP_235 O v2) in (let TMP_237 \def (lift h n TMP_236) +in (let TMP_238 \def (lift h n t4) in (let TMP_239 \def (THead TMP_234 +TMP_237 TMP_238) in (let TMP_240 \def (THead TMP_232 TMP_233 TMP_239) in (pr0 +TMP_231 TMP_240)))))))))))))))))) in (let TMP_242 \def (S O) in (let TMP_243 +\def (lift h d v2) in (let TMP_244 \def (lift TMP_242 O TMP_243) in (let +TMP_262 \def (\lambda (t: T).(let TMP_245 \def (Flat Appl) in (let TMP_246 +\def (lift h d v1) in (let TMP_247 \def (Bind b) in (let TMP_248 \def (lift h +d u1) in (let TMP_249 \def (S O) in (let TMP_250 \def (plus TMP_249 d) in +(let TMP_251 \def (lift h TMP_250 t3) in (let TMP_252 \def (THead TMP_247 +TMP_248 TMP_251) in (let TMP_253 \def (THead TMP_245 TMP_246 TMP_252) in (let +TMP_254 \def (Bind b) in (let TMP_255 \def (lift h d u2) in (let TMP_256 \def +(Flat Appl) in (let TMP_257 \def (S O) in (let TMP_258 \def (plus TMP_257 d) +in (let TMP_259 \def (lift h TMP_258 t4) in (let TMP_260 \def (THead TMP_256 +t TMP_259) in (let TMP_261 \def (THead TMP_254 TMP_255 TMP_260) in (pr0 +TMP_253 TMP_261))))))))))))))))))) in (let TMP_263 \def (lift h d v1) in (let +TMP_264 \def (lift h d v2) in (let TMP_265 \def (H2 h d) in (let TMP_266 \def +(lift h d u1) in (let TMP_267 \def (lift h d u2) in (let TMP_268 \def (H4 h +d) in (let TMP_269 \def (S O) in (let TMP_270 \def (plus TMP_269 d) in (let +TMP_271 \def (lift h TMP_270 t3) in (let TMP_272 \def (S O) in (let TMP_273 +\def (plus TMP_272 d) in (let TMP_274 \def (lift h TMP_273 t4) in (let +TMP_275 \def (S O) in (let TMP_276 \def (plus TMP_275 d) in (let TMP_277 \def +(H6 h TMP_276) in (let TMP_278 \def (pr0_upsilon b H0 TMP_263 TMP_264 TMP_265 +TMP_266 TMP_267 TMP_268 TMP_271 TMP_274 TMP_277) in (let TMP_279 \def (S O) +in (let TMP_280 \def (plus TMP_279 d) in (let TMP_281 \def (S O) in (let +TMP_282 \def (lift TMP_281 O v2) in (let TMP_283 \def (lift h TMP_280 +TMP_282) in (let TMP_284 \def (S O) in (let TMP_285 \def (le_O_n d) in (let +TMP_286 \def (lift_d v2 h TMP_284 d O TMP_285) in (let TMP_287 \def (eq_ind_r +T TMP_244 TMP_262 TMP_278 TMP_283 TMP_286) in (let TMP_288 \def (S d) in (let +TMP_289 \def (S d) in (let TMP_290 \def (refl_equal nat TMP_289) in (let +TMP_291 \def (eq_ind nat TMP_224 TMP_241 TMP_287 TMP_288 TMP_290) in (let +TMP_292 \def (Bind b) in (let TMP_293 \def (s TMP_292 d) in (let TMP_294 \def +(Flat Appl) in (let TMP_295 \def (S O) in (let TMP_296 \def (lift TMP_295 O +v2) in (let TMP_297 \def (THead TMP_294 TMP_296 t4) in (let TMP_298 \def +(lift h TMP_293 TMP_297) in (let TMP_299 \def (Flat Appl) in (let TMP_300 +\def (S O) in (let TMP_301 \def (lift TMP_300 O v2) in (let TMP_302 \def +(Bind b) in (let TMP_303 \def (s TMP_302 d) in (let TMP_304 \def (lift_head +TMP_299 TMP_301 t4 h TMP_303) in (let TMP_305 \def (eq_ind_r T TMP_205 +TMP_222 TMP_291 TMP_298 TMP_304) in (let TMP_306 \def (Bind b) in (let +TMP_307 \def (Flat Appl) in (let TMP_308 \def (S O) in (let TMP_309 \def +(lift TMP_308 O v2) in (let TMP_310 \def (THead TMP_307 TMP_309 t4) in (let +TMP_311 \def (THead TMP_306 u2 TMP_310) in (let TMP_312 \def (lift h d +TMP_311) in (let TMP_313 \def (Bind b) in (let TMP_314 \def (Flat Appl) in +(let TMP_315 \def (S O) in (let TMP_316 \def (lift TMP_315 O v2) in (let +TMP_317 \def (THead TMP_314 TMP_316 t4) in (let TMP_318 \def (lift_head +TMP_313 u2 TMP_317 h d) in (let TMP_319 \def (eq_ind_r T TMP_179 TMP_193 +TMP_305 TMP_312 TMP_318) in (let TMP_320 \def (Flat Appl) in (let TMP_321 +\def (s TMP_320 d) in (let TMP_322 \def (Bind b) in (let TMP_323 \def (THead +TMP_322 u1 t3) in (let TMP_324 \def (lift h TMP_321 TMP_323) in (let TMP_325 +\def (Bind b) in (let TMP_326 \def (Flat Appl) in (let TMP_327 \def (s +TMP_326 d) in (let TMP_328 \def (lift_head TMP_325 u1 t3 h TMP_327) in (let +TMP_329 \def (eq_ind_r T TMP_158 TMP_169 TMP_319 TMP_324 TMP_328) in (let +TMP_330 \def (Flat Appl) in (let TMP_331 \def (Bind b) in (let TMP_332 \def +(THead TMP_331 u1 t3) in (let TMP_333 \def (THead TMP_330 v1 TMP_332) in (let +TMP_334 \def (lift h d TMP_333) in (let TMP_335 \def (Flat Appl) in (let +TMP_336 \def (Bind b) in (let TMP_337 \def (THead TMP_336 u1 t3) in (let +TMP_338 \def (lift_head TMP_335 v1 TMP_337 h d) in (eq_ind_r T TMP_140 +TMP_148 TMP_329 TMP_334 +TMP_338))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) +))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) in +(let TMP_405 \def (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) (lift h d t4)))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t4 w)).(\lambda -(h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind Abbr) (lift h d u1) (lift -h (s (Bind Abbr) d) t3)) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) -u2 w)))) (eq_ind_r T (THead (Bind Abbr) (lift h d u2) (lift h (s (Bind Abbr) -d) w)) (\lambda (t: T).(pr0 (THead (Bind Abbr) (lift h d u1) (lift h (s (Bind -Abbr) d) t3)) t)) (pr0_delta (lift h d u1) (lift h d u2) (H1 h d) (lift h (S -d) t3) (lift h (S d) t4) (H3 h (S d)) (lift h (S d) w) (let d' \def (S d) in -(eq_ind nat (minus (S d) (S O)) (\lambda (n: nat).(subst0 O (lift h n u2) -(lift h d' t4) (lift h d' w))) (subst0_lift_lt t4 w u2 O H4 (S d) (le_n_S O d -(le_O_n d)) h) d (eq_ind nat d (\lambda (n: nat).(eq nat n d)) (refl_equal -nat d) (minus d O) (minus_n_O d))))) (lift h d (THead (Bind Abbr) u2 w)) -(lift_head (Bind Abbr) u2 w h d)) (lift h d (THead (Bind Abbr) u1 t3)) -(lift_head (Bind Abbr) u1 t3 h d)))))))))))))) (\lambda (b: B).(\lambda (H0: -(not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 -t4)).(\lambda (H2: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) -(lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: -nat).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s (Bind b) d) (lift (S -O) O t3))) (\lambda (t: T).(pr0 t (lift h d t4))) (eq_ind nat (plus (S O) d) -(\lambda (n: nat).(pr0 (THead (Bind b) (lift h d u) (lift h n (lift (S O) O -t3))) (lift h d t4))) (eq_ind_r T (lift (S O) O (lift h d t3)) (\lambda (t: -T).(pr0 (THead (Bind b) (lift h d u) t) (lift h d t4))) (pr0_zeta b H0 (lift -h d t3) (lift h d t4) (H2 h d) (lift h d u)) (lift h (plus (S O) d) (lift (S -O) O t3)) (lift_d t3 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) -(lift h d (THead (Bind b) u (lift (S O) O t3))) (lift_head (Bind b) u (lift -(S O) O t3) h d))))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda (_: -(pr0 t3 t4)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h -d t3) (lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: -nat).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d) -t3)) (\lambda (t: T).(pr0 t (lift h d t4))) (pr0_tau (lift h (s (Flat Cast) -d) t3) (lift h d t4) (H1 h d) (lift h d u)) (lift h d (THead (Flat Cast) u -t3)) (lift_head (Flat Cast) u t3 h d))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 2845 -END *) +(h: nat).(\lambda (d: nat).(let TMP_340 \def (Bind Abbr) in (let TMP_341 \def +(lift h d u1) in (let TMP_342 \def (Bind Abbr) in (let TMP_343 \def (s +TMP_342 d) in (let TMP_344 \def (lift h TMP_343 t3) in (let TMP_345 \def +(THead TMP_340 TMP_341 TMP_344) in (let TMP_349 \def (\lambda (t: T).(let +TMP_346 \def (Bind Abbr) in (let TMP_347 \def (THead TMP_346 u2 w) in (let +TMP_348 \def (lift h d TMP_347) in (pr0 t TMP_348))))) in (let TMP_350 \def +(Bind Abbr) in (let TMP_351 \def (lift h d u2) in (let TMP_352 \def (Bind +Abbr) in (let TMP_353 \def (s TMP_352 d) in (let TMP_354 \def (lift h TMP_353 +w) in (let TMP_355 \def (THead TMP_350 TMP_351 TMP_354) in (let TMP_362 \def +(\lambda (t: T).(let TMP_356 \def (Bind Abbr) in (let TMP_357 \def (lift h d +u1) in (let TMP_358 \def (Bind Abbr) in (let TMP_359 \def (s TMP_358 d) in +(let TMP_360 \def (lift h TMP_359 t3) in (let TMP_361 \def (THead TMP_356 +TMP_357 TMP_360) in (pr0 TMP_361 t)))))))) in (let TMP_363 \def (lift h d u1) +in (let TMP_364 \def (lift h d u2) in (let TMP_365 \def (H1 h d) in (let +TMP_366 \def (S d) in (let TMP_367 \def (lift h TMP_366 t3) in (let TMP_368 +\def (S d) in (let TMP_369 \def (lift h TMP_368 t4) in (let TMP_370 \def (S +d) in (let TMP_371 \def (H3 h TMP_370) in (let TMP_372 \def (S d) in (let +TMP_373 \def (lift h TMP_372 w) in (let d' \def (S d) in (let TMP_374 \def (S +d) in (let TMP_375 \def (S O) in (let TMP_376 \def (minus TMP_374 TMP_375) in +(let TMP_380 \def (\lambda (n: nat).(let TMP_377 \def (lift h n u2) in (let +TMP_378 \def (lift h d' t4) in (let TMP_379 \def (lift h d' w) in (subst0 O +TMP_377 TMP_378 TMP_379))))) in (let TMP_381 \def (S d) in (let TMP_382 \def +(le_O_n d) in (let TMP_383 \def (le_n_S O d TMP_382) in (let TMP_384 \def +(subst0_lift_lt t4 w u2 O H4 TMP_381 TMP_383 h) in (let TMP_385 \def (\lambda +(n: nat).(eq nat n d)) in (let TMP_386 \def (le_n d) in (let TMP_387 \def +(le_n d) in (let TMP_388 \def (le_antisym d d TMP_386 TMP_387) in (let +TMP_389 \def (minus d O) in (let TMP_390 \def (minus_n_O d) in (let TMP_391 +\def (eq_ind nat d TMP_385 TMP_388 TMP_389 TMP_390) in (let TMP_392 \def +(eq_ind nat TMP_376 TMP_380 TMP_384 d TMP_391) in (let TMP_393 \def +(pr0_delta TMP_363 TMP_364 TMP_365 TMP_367 TMP_369 TMP_371 TMP_373 TMP_392) +in (let TMP_394 \def (Bind Abbr) in (let TMP_395 \def (THead TMP_394 u2 w) in +(let TMP_396 \def (lift h d TMP_395) in (let TMP_397 \def (Bind Abbr) in (let +TMP_398 \def (lift_head TMP_397 u2 w h d) in (let TMP_399 \def (eq_ind_r T +TMP_355 TMP_362 TMP_393 TMP_396 TMP_398) in (let TMP_400 \def (Bind Abbr) in +(let TMP_401 \def (THead TMP_400 u1 t3) in (let TMP_402 \def (lift h d +TMP_401) in (let TMP_403 \def (Bind Abbr) in (let TMP_404 \def (lift_head +TMP_403 u1 t3 h d) in (eq_ind_r T TMP_345 TMP_349 TMP_399 TMP_402 +TMP_404))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) in +(let TMP_461 \def (\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda +(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (H2: ((\forall +(h: nat).(\forall (d: nat).(pr0 (lift h d t3) (lift h d t4)))))).(\lambda (u: +T).(\lambda (h: nat).(\lambda (d: nat).(let TMP_406 \def (Bind b) in (let +TMP_407 \def (lift h d u) in (let TMP_408 \def (Bind b) in (let TMP_409 \def +(s TMP_408 d) in (let TMP_410 \def (S O) in (let TMP_411 \def (lift TMP_410 O +t3) in (let TMP_412 \def (lift h TMP_409 TMP_411) in (let TMP_413 \def (THead +TMP_406 TMP_407 TMP_412) in (let TMP_415 \def (\lambda (t: T).(let TMP_414 +\def (lift h d t4) in (pr0 t TMP_414))) in (let TMP_416 \def (S O) in (let +TMP_417 \def (plus TMP_416 d) in (let TMP_425 \def (\lambda (n: nat).(let +TMP_418 \def (Bind b) in (let TMP_419 \def (lift h d u) in (let TMP_420 \def +(S O) in (let TMP_421 \def (lift TMP_420 O t3) in (let TMP_422 \def (lift h n +TMP_421) in (let TMP_423 \def (THead TMP_418 TMP_419 TMP_422) in (let TMP_424 +\def (lift h d t4) in (pr0 TMP_423 TMP_424))))))))) in (let TMP_426 \def (S +O) in (let TMP_427 \def (lift h d t3) in (let TMP_428 \def (lift TMP_426 O +TMP_427) in (let TMP_433 \def (\lambda (t: T).(let TMP_429 \def (Bind b) in +(let TMP_430 \def (lift h d u) in (let TMP_431 \def (THead TMP_429 TMP_430 t) +in (let TMP_432 \def (lift h d t4) in (pr0 TMP_431 TMP_432)))))) in (let +TMP_434 \def (lift h d t3) in (let TMP_435 \def (lift h d t4) in (let TMP_436 +\def (H2 h d) in (let TMP_437 \def (lift h d u) in (let TMP_438 \def +(pr0_zeta b H0 TMP_434 TMP_435 TMP_436 TMP_437) in (let TMP_439 \def (S O) in +(let TMP_440 \def (plus TMP_439 d) in (let TMP_441 \def (S O) in (let TMP_442 +\def (lift TMP_441 O t3) in (let TMP_443 \def (lift h TMP_440 TMP_442) in +(let TMP_444 \def (S O) in (let TMP_445 \def (le_O_n d) in (let TMP_446 \def +(lift_d t3 h TMP_444 d O TMP_445) in (let TMP_447 \def (eq_ind_r T TMP_428 +TMP_433 TMP_438 TMP_443 TMP_446) in (let TMP_448 \def (S d) in (let TMP_449 +\def (S d) in (let TMP_450 \def (refl_equal nat TMP_449) in (let TMP_451 \def +(eq_ind nat TMP_417 TMP_425 TMP_447 TMP_448 TMP_450) in (let TMP_452 \def +(Bind b) in (let TMP_453 \def (S O) in (let TMP_454 \def (lift TMP_453 O t3) +in (let TMP_455 \def (THead TMP_452 u TMP_454) in (let TMP_456 \def (lift h d +TMP_455) in (let TMP_457 \def (Bind b) in (let TMP_458 \def (S O) in (let +TMP_459 \def (lift TMP_458 O t3) in (let TMP_460 \def (lift_head TMP_457 u +TMP_459 h d) in (eq_ind_r T TMP_413 TMP_415 TMP_451 TMP_456 +TMP_460))))))))))))))))))))))))))))))))))))))))))))))))))))) in (let TMP_482 +\def (\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda +(H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) (lift h d +t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: nat).(let TMP_462 +\def (Flat Cast) in (let TMP_463 \def (lift h d u) in (let TMP_464 \def (Flat +Cast) in (let TMP_465 \def (s TMP_464 d) in (let TMP_466 \def (lift h TMP_465 +t3) in (let TMP_467 \def (THead TMP_462 TMP_463 TMP_466) in (let TMP_469 \def +(\lambda (t: T).(let TMP_468 \def (lift h d t4) in (pr0 t TMP_468))) in (let +TMP_470 \def (Flat Cast) in (let TMP_471 \def (s TMP_470 d) in (let TMP_472 +\def (lift h TMP_471 t3) in (let TMP_473 \def (lift h d t4) in (let TMP_474 +\def (H1 h d) in (let TMP_475 \def (lift h d u) in (let TMP_476 \def (pr0_tau +TMP_472 TMP_473 TMP_474 TMP_475) in (let TMP_477 \def (Flat Cast) in (let +TMP_478 \def (THead TMP_477 u t3) in (let TMP_479 \def (lift h d TMP_478) in +(let TMP_480 \def (Flat Cast) in (let TMP_481 \def (lift_head TMP_480 u t3 h +d) in (eq_ind_r T TMP_467 TMP_469 TMP_476 TMP_479 +TMP_481))))))))))))))))))))))))))) in (pr0_ind TMP_3 TMP_5 TMP_39 TMP_132 +TMP_339 TMP_405 TMP_461 TMP_482 t1 t2 H))))))))))). -theorem pr0_subst0_back: - \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 -i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: -T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t2))))))))) +theorem pr0_gen_abbr: + \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abbr) u1 +t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead +(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) +(\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x)))))) \def - \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 u1 t) \to (ex2 T -(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t4 t3))))))))) -(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 u1 -v)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: -T).(pr0 t (lift (S i0) O v))) (lift (S i0) O u1) (subst0_lref u1 i0) -(pr0_lift u1 v H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda -(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: -((\forall (u4: T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) -(\lambda (t: T).(pr0 t u3))))))).(\lambda (t: T).(\lambda (k: K).(\lambda -(u0: T).(\lambda (H2: (pr0 u0 v)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 -u1 t0)) (\lambda (t0: T).(pr0 t0 u3)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 -(THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 (THead k u3 t)))) (\lambda (x: -T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 x u3)).(ex_intro2 T -(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 -(THead k u3 t))) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp x u3 -H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda (v: -T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: (subst0 -(s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 u1 v) \to (ex2 T -(\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t -t3))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 u1 v)).(ex2_ind -T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t t3)) (ex2 -T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 t -(THead k u t3)))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 -x)).(\lambda (H4: (pr0 x t3)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 -(THead k u t4) t)) (\lambda (t: T).(pr0 t (THead k u t3))) (THead k u x) -(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) x t3 H4 k))))) (H1 -u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda -(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: -T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: -T).(pr0 t u3))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: -T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda -(t: T).(pr0 t t4))))))).(\lambda (u0: T).(\lambda (H4: (pr0 u0 v)).(ex2_ind T -(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t t4)) (ex2 T -(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t -(THead k u3 t4)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 -x)).(\lambda (H6: (pr0 x t4)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) -(\lambda (t: T).(pr0 t u3)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 -t3) t)) (\lambda (t: T).(pr0 t (THead k u3 t4)))) (\lambda (x0: T).(\lambda -(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 x0 u3)).(ex_intro2 T (\lambda -(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t (THead k u3 -t4))) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp x0 u3 -H8 x t4 H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))). -(* COMMENTS -Initial nodes: 979 -END *) + \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead +(Bind Abbr) u1 t1) x)).(let TMP_1 \def (Bind Abbr) in (let TMP_2 \def (THead +TMP_1 u1 t1) in (let TMP_3 \def (\lambda (t: T).(pr0 t x)) in (let TMP_17 +\def (\lambda (_: T).(let TMP_6 \def (\lambda (u2: T).(\lambda (t2: T).(let +TMP_4 \def (Bind Abbr) in (let TMP_5 \def (THead TMP_4 u2 t2) in (eq T x +TMP_5))))) in (let TMP_7 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) +in (let TMP_12 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_8 \def (pr0 t1 +t2) in (let TMP_9 \def (\lambda (y: T).(pr0 t1 y)) in (let TMP_10 \def +(\lambda (y: T).(subst0 O u2 y t2)) in (let TMP_11 \def (ex2 T TMP_9 TMP_10) +in (or TMP_8 TMP_11))))))) in (let TMP_13 \def (ex3_2 T T TMP_6 TMP_7 TMP_12) +in (let TMP_14 \def (S O) in (let TMP_15 \def (lift TMP_14 O x) in (let +TMP_16 \def (pr0 t1 TMP_15) in (or TMP_13 TMP_16))))))))) in (let TMP_448 +\def (\lambda (y: T).(\lambda (H0: (pr0 y x)).(let TMP_31 \def (\lambda (t: +T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (let TMP_20 \def +(\lambda (u2: T).(\lambda (t2: T).(let TMP_18 \def (Bind Abbr) in (let TMP_19 +\def (THead TMP_18 u2 t2) in (eq T t0 TMP_19))))) in (let TMP_21 \def +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) in (let TMP_26 \def (\lambda +(u2: T).(\lambda (t2: T).(let TMP_22 \def (pr0 t1 t2) in (let TMP_23 \def +(\lambda (y0: T).(pr0 t1 y0)) in (let TMP_24 \def (\lambda (y0: T).(subst0 O +u2 y0 t2)) in (let TMP_25 \def (ex2 T TMP_23 TMP_24) in (or TMP_22 +TMP_25))))))) in (let TMP_27 \def (ex3_2 T T TMP_20 TMP_21 TMP_26) in (let +TMP_28 \def (S O) in (let TMP_29 \def (lift TMP_28 O t0) in (let TMP_30 \def +(pr0 t1 TMP_29) in (or TMP_27 TMP_30))))))))))) in (let TMP_91 \def (\lambda +(t: T).(\lambda (H1: (eq T t (THead (Bind Abbr) u1 t1))).(let TMP_32 \def +(\lambda (e: T).e) in (let TMP_33 \def (Bind Abbr) in (let TMP_34 \def (THead +TMP_33 u1 t1) in (let H2 \def (f_equal T T TMP_32 t TMP_34 H1) in (let TMP_35 +\def (Bind Abbr) in (let TMP_36 \def (THead TMP_35 u1 t1) in (let TMP_50 \def +(\lambda (t0: T).(let TMP_39 \def (\lambda (u2: T).(\lambda (t2: T).(let +TMP_37 \def (Bind Abbr) in (let TMP_38 \def (THead TMP_37 u2 t2) in (eq T t0 +TMP_38))))) in (let TMP_40 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 +u2))) in (let TMP_45 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_41 \def +(pr0 t1 t2) in (let TMP_42 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_43 +\def (\lambda (y0: T).(subst0 O u2 y0 t2)) in (let TMP_44 \def (ex2 T TMP_42 +TMP_43) in (or TMP_41 TMP_44))))))) in (let TMP_46 \def (ex3_2 T T TMP_39 +TMP_40 TMP_45) in (let TMP_47 \def (S O) in (let TMP_48 \def (lift TMP_47 O +t0) in (let TMP_49 \def (pr0 t1 TMP_48) in (or TMP_46 TMP_49))))))))) in (let +TMP_55 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_51 \def (Bind Abbr) in +(let TMP_52 \def (THead TMP_51 u1 t1) in (let TMP_53 \def (Bind Abbr) in (let +TMP_54 \def (THead TMP_53 u2 t2) in (eq T TMP_52 TMP_54))))))) in (let TMP_56 +\def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) in (let TMP_61 \def +(\lambda (u2: T).(\lambda (t2: T).(let TMP_57 \def (pr0 t1 t2) in (let TMP_58 +\def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_59 \def (\lambda (y0: +T).(subst0 O u2 y0 t2)) in (let TMP_60 \def (ex2 T TMP_58 TMP_59) in (or +TMP_57 TMP_60))))))) in (let TMP_62 \def (ex3_2 T T TMP_55 TMP_56 TMP_61) in +(let TMP_63 \def (S O) in (let TMP_64 \def (Bind Abbr) in (let TMP_65 \def +(THead TMP_64 u1 t1) in (let TMP_66 \def (lift TMP_63 O TMP_65) in (let +TMP_67 \def (pr0 t1 TMP_66) in (let TMP_72 \def (\lambda (u2: T).(\lambda +(t2: T).(let TMP_68 \def (Bind Abbr) in (let TMP_69 \def (THead TMP_68 u1 t1) +in (let TMP_70 \def (Bind Abbr) in (let TMP_71 \def (THead TMP_70 u2 t2) in +(eq T TMP_69 TMP_71))))))) in (let TMP_73 \def (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) in (let TMP_78 \def (\lambda (u2: T).(\lambda (t2: T).(let +TMP_74 \def (pr0 t1 t2) in (let TMP_75 \def (\lambda (y0: T).(pr0 t1 y0)) in +(let TMP_76 \def (\lambda (y0: T).(subst0 O u2 y0 t2)) in (let TMP_77 \def +(ex2 T TMP_75 TMP_76) in (or TMP_74 TMP_77))))))) in (let TMP_79 \def (Bind +Abbr) in (let TMP_80 \def (THead TMP_79 u1 t1) in (let TMP_81 \def +(refl_equal T TMP_80) in (let TMP_82 \def (pr0_refl u1) in (let TMP_83 \def +(pr0 t1 t1) in (let TMP_84 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_85 +\def (\lambda (y0: T).(subst0 O u1 y0 t1)) in (let TMP_86 \def (ex2 T TMP_84 +TMP_85) in (let TMP_87 \def (pr0_refl t1) in (let TMP_88 \def (or_introl +TMP_83 TMP_86 TMP_87) in (let TMP_89 \def (ex3_2_intro T T TMP_72 TMP_73 +TMP_78 u1 t1 TMP_81 TMP_82 TMP_88) in (let TMP_90 \def (or_introl TMP_62 +TMP_67 TMP_89) in (eq_ind_r T TMP_36 TMP_50 TMP_90 t +H2)))))))))))))))))))))))))))))))))) in (let TMP_191 \def (\lambda (u0: +T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 +(THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: +T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: +T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T +(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 +t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: +(pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or +(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 +t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: +T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O +t2)))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) (THead (Bind +Abbr) u1 t1))).(let TMP_92 \def (\lambda (e: T).(match e with [(TSort _) +\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) in +(let TMP_93 \def (THead k u0 t0) in (let TMP_94 \def (Bind Abbr) in (let +TMP_95 \def (THead TMP_94 u1 t1) in (let H6 \def (f_equal T K TMP_92 TMP_93 +TMP_95 H5) in (let TMP_96 \def (\lambda (e: T).(match e with [(TSort _) +\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) in +(let TMP_97 \def (THead k u0 t0) in (let TMP_98 \def (Bind Abbr) in (let +TMP_99 \def (THead TMP_98 u1 t1) in (let H7 \def (f_equal T T TMP_96 TMP_97 +TMP_99 H5) in (let TMP_100 \def (\lambda (e: T).(match e with [(TSort _) +\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) in +(let TMP_101 \def (THead k u0 t0) in (let TMP_102 \def (Bind Abbr) in (let +TMP_103 \def (THead TMP_102 u1 t1) in (let H8 \def (f_equal T T TMP_100 +TMP_101 TMP_103 H5) in (let TMP_189 \def (\lambda (H9: (eq T u0 u1)).(\lambda +(H10: (eq K k (Bind Abbr))).(let TMP_104 \def (Bind Abbr) in (let TMP_120 +\def (\lambda (k0: K).(let TMP_108 \def (\lambda (u3: T).(\lambda (t3: +T).(let TMP_105 \def (THead k0 u2 t2) in (let TMP_106 \def (Bind Abbr) in +(let TMP_107 \def (THead TMP_106 u3 t3) in (eq T TMP_105 TMP_107)))))) in +(let TMP_109 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let +TMP_114 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_110 \def (pr0 t1 t3) +in (let TMP_111 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_112 \def +(\lambda (y0: T).(subst0 O u3 y0 t3)) in (let TMP_113 \def (ex2 T TMP_111 +TMP_112) in (or TMP_110 TMP_113))))))) in (let TMP_115 \def (ex3_2 T T +TMP_108 TMP_109 TMP_114) in (let TMP_116 \def (S O) in (let TMP_117 \def +(THead k0 u2 t2) in (let TMP_118 \def (lift TMP_116 O TMP_117) in (let +TMP_119 \def (pr0 t1 TMP_118) in (or TMP_115 TMP_119)))))))))) in (let +TMP_134 \def (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (let +TMP_123 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_121 \def (Bind Abbr) +in (let TMP_122 \def (THead TMP_121 u3 t3) in (eq T t2 TMP_122))))) in (let +TMP_124 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let TMP_129 +\def (\lambda (u3: T).(\lambda (t3: T).(let TMP_125 \def (pr0 t1 t3) in (let +TMP_126 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_127 \def (\lambda (y0: +T).(subst0 O u3 y0 t3)) in (let TMP_128 \def (ex2 T TMP_126 TMP_127) in (or +TMP_125 TMP_128))))))) in (let TMP_130 \def (ex3_2 T T TMP_123 TMP_124 +TMP_129) in (let TMP_131 \def (S O) in (let TMP_132 \def (lift TMP_131 O t2) +in (let TMP_133 \def (pr0 t1 TMP_132) in (or TMP_130 TMP_133)))))))))) in +(let H11 \def (eq_ind T t0 TMP_134 H4 t1 H8) in (let TMP_135 \def (\lambda +(t: T).(pr0 t t2)) in (let H12 \def (eq_ind T t0 TMP_135 H3 t1 H8) in (let +TMP_149 \def (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (let +TMP_138 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_136 \def (Bind Abbr) +in (let TMP_137 \def (THead TMP_136 u3 t3) in (eq T u2 TMP_137))))) in (let +TMP_139 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let TMP_144 +\def (\lambda (u3: T).(\lambda (t3: T).(let TMP_140 \def (pr0 t1 t3) in (let +TMP_141 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_142 \def (\lambda (y0: +T).(subst0 O u3 y0 t3)) in (let TMP_143 \def (ex2 T TMP_141 TMP_142) in (or +TMP_140 TMP_143))))))) in (let TMP_145 \def (ex3_2 T T TMP_138 TMP_139 +TMP_144) in (let TMP_146 \def (S O) in (let TMP_147 \def (lift TMP_146 O u2) +in (let TMP_148 \def (pr0 t1 TMP_147) in (or TMP_145 TMP_148)))))))))) in +(let H13 \def (eq_ind T u0 TMP_149 H2 u1 H9) in (let TMP_150 \def (\lambda +(t: T).(pr0 t u2)) in (let H14 \def (eq_ind T u0 TMP_150 H1 u1 H9) in (let +TMP_155 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_151 \def (Bind Abbr) +in (let TMP_152 \def (THead TMP_151 u2 t2) in (let TMP_153 \def (Bind Abbr) +in (let TMP_154 \def (THead TMP_153 u3 t3) in (eq T TMP_152 TMP_154))))))) in +(let TMP_156 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let +TMP_161 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_157 \def (pr0 t1 t3) +in (let TMP_158 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_159 \def +(\lambda (y0: T).(subst0 O u3 y0 t3)) in (let TMP_160 \def (ex2 T TMP_158 +TMP_159) in (or TMP_157 TMP_160))))))) in (let TMP_162 \def (ex3_2 T T +TMP_155 TMP_156 TMP_161) in (let TMP_163 \def (S O) in (let TMP_164 \def +(Bind Abbr) in (let TMP_165 \def (THead TMP_164 u2 t2) in (let TMP_166 \def +(lift TMP_163 O TMP_165) in (let TMP_167 \def (pr0 t1 TMP_166) in (let +TMP_172 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_168 \def (Bind Abbr) +in (let TMP_169 \def (THead TMP_168 u2 t2) in (let TMP_170 \def (Bind Abbr) +in (let TMP_171 \def (THead TMP_170 u3 t3) in (eq T TMP_169 TMP_171))))))) in +(let TMP_173 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let +TMP_178 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_174 \def (pr0 t1 t3) +in (let TMP_175 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_176 \def +(\lambda (y0: T).(subst0 O u3 y0 t3)) in (let TMP_177 \def (ex2 T TMP_175 +TMP_176) in (or TMP_174 TMP_177))))))) in (let TMP_179 \def (Bind Abbr) in +(let TMP_180 \def (THead TMP_179 u2 t2) in (let TMP_181 \def (refl_equal T +TMP_180) in (let TMP_182 \def (pr0 t1 t2) in (let TMP_183 \def (\lambda (y0: +T).(pr0 t1 y0)) in (let TMP_184 \def (\lambda (y0: T).(subst0 O u2 y0 t2)) in +(let TMP_185 \def (ex2 T TMP_183 TMP_184) in (let TMP_186 \def (or_introl +TMP_182 TMP_185 H12) in (let TMP_187 \def (ex3_2_intro T T TMP_172 TMP_173 +TMP_178 u2 t2 TMP_181 H14 TMP_186) in (let TMP_188 \def (or_introl TMP_162 +TMP_167 TMP_187) in (eq_ind_r K TMP_104 TMP_120 TMP_188 k +H10))))))))))))))))))))))))))))))))))) in (let TMP_190 \def (TMP_189 H7) in +(TMP_190 H6)))))))))))))))))))))))))))) in (let TMP_217 \def (\lambda (u: +T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: +(((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 +t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 +t2))))))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (t0: T).(\lambda (t2: +T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 +t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O +t2)))))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) +(THead (Bind Abbr) u1 t1))).(let TMP_192 \def (Flat Appl) in (let TMP_193 +\def (Bind Abst) in (let TMP_194 \def (THead TMP_193 u t0) in (let TMP_195 +\def (THead TMP_192 v1 TMP_194) in (let TMP_196 \def (\lambda (ee: T).(match +ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k +_ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) in (let TMP_197 \def (Bind Abbr) in (let TMP_198 \def +(THead TMP_197 u1 t1) in (let H6 \def (eq_ind T TMP_195 TMP_196 I TMP_198 H5) +in (let TMP_203 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_199 \def +(Bind Abbr) in (let TMP_200 \def (THead TMP_199 v2 t2) in (let TMP_201 \def +(Bind Abbr) in (let TMP_202 \def (THead TMP_201 u2 t3) in (eq T TMP_200 +TMP_202))))))) in (let TMP_204 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 +u2))) in (let TMP_209 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_205 +\def (pr0 t1 t3) in (let TMP_206 \def (\lambda (y0: T).(pr0 t1 y0)) in (let +TMP_207 \def (\lambda (y0: T).(subst0 O u2 y0 t3)) in (let TMP_208 \def (ex2 +T TMP_206 TMP_207) in (or TMP_205 TMP_208))))))) in (let TMP_210 \def (ex3_2 +T T TMP_203 TMP_204 TMP_209) in (let TMP_211 \def (S O) in (let TMP_212 \def +(Bind Abbr) in (let TMP_213 \def (THead TMP_212 v2 t2) in (let TMP_214 \def +(lift TMP_211 O TMP_213) in (let TMP_215 \def (pr0 t1 TMP_214) in (let +TMP_216 \def (or TMP_210 TMP_215) in (False_ind TMP_216 +H6))))))))))))))))))))))))))))) in (let TMP_251 \def (\lambda (b: B).(\lambda +(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 +v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: +T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: +T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: +T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead +(Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq +T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 +u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: +T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 t1 (lift (S +O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 +t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T +(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: +T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: +T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq +T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abbr) u1 +t1))).(let TMP_218 \def (Flat Appl) in (let TMP_219 \def (Bind b) in (let +TMP_220 \def (THead TMP_219 u0 t0) in (let TMP_221 \def (THead TMP_218 v1 +TMP_220) in (let TMP_222 \def (\lambda (ee: T).(match ee with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) in +(let TMP_223 \def (Bind Abbr) in (let TMP_224 \def (THead TMP_223 u1 t1) in +(let H9 \def (eq_ind T TMP_221 TMP_222 I TMP_224 H8) in (let TMP_233 \def +(\lambda (u3: T).(\lambda (t3: T).(let TMP_225 \def (Bind b) in (let TMP_226 +\def (Flat Appl) in (let TMP_227 \def (S O) in (let TMP_228 \def (lift +TMP_227 O v2) in (let TMP_229 \def (THead TMP_226 TMP_228 t2) in (let TMP_230 +\def (THead TMP_225 u2 TMP_229) in (let TMP_231 \def (Bind Abbr) in (let +TMP_232 \def (THead TMP_231 u3 t3) in (eq T TMP_230 TMP_232))))))))))) in +(let TMP_234 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let +TMP_239 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_235 \def (pr0 t1 t3) +in (let TMP_236 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_237 \def +(\lambda (y0: T).(subst0 O u3 y0 t3)) in (let TMP_238 \def (ex2 T TMP_236 +TMP_237) in (or TMP_235 TMP_238))))))) in (let TMP_240 \def (ex3_2 T T +TMP_233 TMP_234 TMP_239) in (let TMP_241 \def (S O) in (let TMP_242 \def +(Bind b) in (let TMP_243 \def (Flat Appl) in (let TMP_244 \def (S O) in (let +TMP_245 \def (lift TMP_244 O v2) in (let TMP_246 \def (THead TMP_243 TMP_245 +t2) in (let TMP_247 \def (THead TMP_242 u2 TMP_246) in (let TMP_248 \def +(lift TMP_241 O TMP_247) in (let TMP_249 \def (pr0 t1 TMP_248) in (let +TMP_250 \def (or TMP_240 TMP_249) in (False_ind TMP_250 +H9)))))))))))))))))))))))))))))))))))))) in (let TMP_333 \def (\lambda (u0: +T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 +(THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: +T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: +T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T +(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 +t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: +(pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or +(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 +t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: +T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O +t2)))))).(\lambda (w: T).(\lambda (H5: (subst0 O u2 t2 w)).(\lambda (H6: (eq +T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1))).(let TMP_252 \def +(\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | (TLRef _) +\Rightarrow u0 | (THead _ t _) \Rightarrow t])) in (let TMP_253 \def (Bind +Abbr) in (let TMP_254 \def (THead TMP_253 u0 t0) in (let TMP_255 \def (Bind +Abbr) in (let TMP_256 \def (THead TMP_255 u1 t1) in (let H7 \def (f_equal T T +TMP_252 TMP_254 TMP_256 H6) in (let TMP_257 \def (\lambda (e: T).(match e +with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) +\Rightarrow t])) in (let TMP_258 \def (Bind Abbr) in (let TMP_259 \def (THead +TMP_258 u0 t0) in (let TMP_260 \def (Bind Abbr) in (let TMP_261 \def (THead +TMP_260 u1 t1) in (let H8 \def (f_equal T T TMP_257 TMP_259 TMP_261 H6) in +(let TMP_332 \def (\lambda (H9: (eq T u0 u1)).(let TMP_275 \def (\lambda (t: +T).((eq T t (THead (Bind Abbr) u1 t1)) \to (let TMP_264 \def (\lambda (u3: +T).(\lambda (t3: T).(let TMP_262 \def (Bind Abbr) in (let TMP_263 \def (THead +TMP_262 u3 t3) in (eq T t2 TMP_263))))) in (let TMP_265 \def (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) in (let TMP_270 \def (\lambda (u3: +T).(\lambda (t3: T).(let TMP_266 \def (pr0 t1 t3) in (let TMP_267 \def +(\lambda (y0: T).(pr0 t1 y0)) in (let TMP_268 \def (\lambda (y0: T).(subst0 O +u3 y0 t3)) in (let TMP_269 \def (ex2 T TMP_267 TMP_268) in (or TMP_266 +TMP_269))))))) in (let TMP_271 \def (ex3_2 T T TMP_264 TMP_265 TMP_270) in +(let TMP_272 \def (S O) in (let TMP_273 \def (lift TMP_272 O t2) in (let +TMP_274 \def (pr0 t1 TMP_273) in (or TMP_271 TMP_274)))))))))) in (let H10 +\def (eq_ind T t0 TMP_275 H4 t1 H8) in (let TMP_276 \def (\lambda (t: T).(pr0 +t t2)) in (let H11 \def (eq_ind T t0 TMP_276 H3 t1 H8) in (let TMP_290 \def +(\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (let TMP_279 \def +(\lambda (u3: T).(\lambda (t3: T).(let TMP_277 \def (Bind Abbr) in (let +TMP_278 \def (THead TMP_277 u3 t3) in (eq T u2 TMP_278))))) in (let TMP_280 +\def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let TMP_285 \def +(\lambda (u3: T).(\lambda (t3: T).(let TMP_281 \def (pr0 t1 t3) in (let +TMP_282 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_283 \def (\lambda (y0: +T).(subst0 O u3 y0 t3)) in (let TMP_284 \def (ex2 T TMP_282 TMP_283) in (or +TMP_281 TMP_284))))))) in (let TMP_286 \def (ex3_2 T T TMP_279 TMP_280 +TMP_285) in (let TMP_287 \def (S O) in (let TMP_288 \def (lift TMP_287 O u2) +in (let TMP_289 \def (pr0 t1 TMP_288) in (or TMP_286 TMP_289)))))))))) in +(let H12 \def (eq_ind T u0 TMP_290 H2 u1 H9) in (let TMP_291 \def (\lambda +(t: T).(pr0 t u2)) in (let H13 \def (eq_ind T u0 TMP_291 H1 u1 H9) in (let +TMP_296 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_292 \def (Bind Abbr) +in (let TMP_293 \def (THead TMP_292 u2 w) in (let TMP_294 \def (Bind Abbr) in +(let TMP_295 \def (THead TMP_294 u3 t3) in (eq T TMP_293 TMP_295))))))) in +(let TMP_297 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let +TMP_302 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_298 \def (pr0 t1 t3) +in (let TMP_299 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_300 \def +(\lambda (y0: T).(subst0 O u3 y0 t3)) in (let TMP_301 \def (ex2 T TMP_299 +TMP_300) in (or TMP_298 TMP_301))))))) in (let TMP_303 \def (ex3_2 T T +TMP_296 TMP_297 TMP_302) in (let TMP_304 \def (S O) in (let TMP_305 \def +(Bind Abbr) in (let TMP_306 \def (THead TMP_305 u2 w) in (let TMP_307 \def +(lift TMP_304 O TMP_306) in (let TMP_308 \def (pr0 t1 TMP_307) in (let +TMP_313 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_309 \def (Bind Abbr) +in (let TMP_310 \def (THead TMP_309 u2 w) in (let TMP_311 \def (Bind Abbr) in +(let TMP_312 \def (THead TMP_311 u3 t3) in (eq T TMP_310 TMP_312))))))) in +(let TMP_314 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let +TMP_319 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_315 \def (pr0 t1 t3) +in (let TMP_316 \def (\lambda (y0: T).(pr0 t1 y0)) in (let TMP_317 \def +(\lambda (y0: T).(subst0 O u3 y0 t3)) in (let TMP_318 \def (ex2 T TMP_316 +TMP_317) in (or TMP_315 TMP_318))))))) in (let TMP_320 \def (Bind Abbr) in +(let TMP_321 \def (THead TMP_320 u2 w) in (let TMP_322 \def (refl_equal T +TMP_321) in (let TMP_323 \def (pr0 t1 w) in (let TMP_324 \def (\lambda (y0: +T).(pr0 t1 y0)) in (let TMP_325 \def (\lambda (y0: T).(subst0 O u2 y0 w)) in +(let TMP_326 \def (ex2 T TMP_324 TMP_325) in (let TMP_327 \def (\lambda (y0: +T).(pr0 t1 y0)) in (let TMP_328 \def (\lambda (y0: T).(subst0 O u2 y0 w)) in +(let TMP_329 \def (ex_intro2 T TMP_327 TMP_328 t2 H11 H5) in (let TMP_330 +\def (or_intror TMP_323 TMP_326 TMP_329) in (let TMP_331 \def (ex3_2_intro T +T TMP_313 TMP_314 TMP_319 u2 w TMP_322 H13 TMP_330) in (or_introl TMP_303 +TMP_308 TMP_331)))))))))))))))))))))))))))))))))) in (TMP_332 +H7))))))))))))))))))))))))) in (let TMP_427 \def (\lambda (b: B).(\lambda +(H1: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: +(pr0 t0 t2)).(\lambda (H3: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: +T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O +t2)))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O +t0)) (THead (Bind Abbr) u1 t1))).(let TMP_334 \def (\lambda (e: T).(match e +with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) +\Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow +b])])) in (let TMP_335 \def (Bind b) in (let TMP_336 \def (S O) in (let +TMP_337 \def (lift TMP_336 O t0) in (let TMP_338 \def (THead TMP_335 u +TMP_337) in (let TMP_339 \def (Bind Abbr) in (let TMP_340 \def (THead TMP_339 +u1 t1) in (let H5 \def (f_equal T B TMP_334 TMP_338 TMP_340 H4) in (let +TMP_341 \def (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef +_) \Rightarrow u | (THead _ t _) \Rightarrow t])) in (let TMP_342 \def (Bind +b) in (let TMP_343 \def (S O) in (let TMP_344 \def (lift TMP_343 O t0) in +(let TMP_345 \def (THead TMP_342 u TMP_344) in (let TMP_346 \def (Bind Abbr) +in (let TMP_347 \def (THead TMP_346 u1 t1) in (let H6 \def (f_equal T T +TMP_341 TMP_345 TMP_347 H4) in (let TMP_362 \def (\lambda (e: T).(match e +with [(TSort _) \Rightarrow (let TMP_361 \def (\lambda (x0: nat).(let TMP_360 +\def (S O) in (plus x0 TMP_360))) in (lref_map TMP_361 O t0)) | (TLRef _) +\Rightarrow (let TMP_354 \def (\lambda (x0: nat).(let TMP_353 \def (S O) in +(plus x0 TMP_353))) in (lref_map TMP_354 O t0)) | (THead _ _ t) \Rightarrow +t])) in (let TMP_363 \def (Bind b) in (let TMP_364 \def (S O) in (let TMP_365 +\def (lift TMP_364 O t0) in (let TMP_366 \def (THead TMP_363 u TMP_365) in +(let TMP_367 \def (Bind Abbr) in (let TMP_368 \def (THead TMP_367 u1 t1) in +(let H7 \def (f_equal T T TMP_362 TMP_366 TMP_368 H4) in (let TMP_425 \def +(\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abbr)).(let TMP_370 \def +(\lambda (b0: B).(let TMP_369 \def (eq B b0 Abst) in (not TMP_369))) in (let +H10 \def (eq_ind B b TMP_370 H1 Abbr H9) in (let TMP_384 \def (\lambda (t: +T).((eq T t0 (THead (Bind Abbr) u1 t)) \to (let TMP_373 \def (\lambda (u2: +T).(\lambda (t3: T).(let TMP_371 \def (Bind Abbr) in (let TMP_372 \def (THead +TMP_371 u2 t3) in (eq T t2 TMP_372))))) in (let TMP_374 \def (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) in (let TMP_379 \def (\lambda (u2: +T).(\lambda (t3: T).(let TMP_375 \def (pr0 t t3) in (let TMP_376 \def +(\lambda (y0: T).(pr0 t y0)) in (let TMP_377 \def (\lambda (y0: T).(subst0 O +u2 y0 t3)) in (let TMP_378 \def (ex2 T TMP_376 TMP_377) in (or TMP_375 +TMP_378))))))) in (let TMP_380 \def (ex3_2 T T TMP_373 TMP_374 TMP_379) in +(let TMP_381 \def (S O) in (let TMP_382 \def (lift TMP_381 O t2) in (let +TMP_383 \def (pr0 t TMP_382) in (or TMP_380 TMP_383)))))))))) in (let TMP_385 +\def (S O) in (let TMP_386 \def (lift TMP_385 O t0) in (let H11 \def +(eq_ind_r T t1 TMP_384 H3 TMP_386 H7) in (let TMP_387 \def (S O) in (let +TMP_388 \def (lift TMP_387 O t0) in (let TMP_402 \def (\lambda (t: T).(let +TMP_391 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_389 \def (Bind Abbr) +in (let TMP_390 \def (THead TMP_389 u2 t3) in (eq T t2 TMP_390))))) in (let +TMP_392 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) in (let TMP_397 +\def (\lambda (u2: T).(\lambda (t3: T).(let TMP_393 \def (pr0 t t3) in (let +TMP_394 \def (\lambda (y0: T).(pr0 t y0)) in (let TMP_395 \def (\lambda (y0: +T).(subst0 O u2 y0 t3)) in (let TMP_396 \def (ex2 T TMP_394 TMP_395) in (or +TMP_393 TMP_396))))))) in (let TMP_398 \def (ex3_2 T T TMP_391 TMP_392 +TMP_397) in (let TMP_399 \def (S O) in (let TMP_400 \def (lift TMP_399 O t2) +in (let TMP_401 \def (pr0 t TMP_400) in (or TMP_398 TMP_401))))))))) in (let +TMP_405 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_403 \def (Bind Abbr) +in (let TMP_404 \def (THead TMP_403 u2 t3) in (eq T t2 TMP_404))))) in (let +TMP_406 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) in (let TMP_415 +\def (\lambda (u2: T).(\lambda (t3: T).(let TMP_407 \def (S O) in (let +TMP_408 \def (lift TMP_407 O t0) in (let TMP_409 \def (pr0 TMP_408 t3) in +(let TMP_412 \def (\lambda (y0: T).(let TMP_410 \def (S O) in (let TMP_411 +\def (lift TMP_410 O t0) in (pr0 TMP_411 y0)))) in (let TMP_413 \def (\lambda +(y0: T).(subst0 O u2 y0 t3)) in (let TMP_414 \def (ex2 T TMP_412 TMP_413) in +(or TMP_409 TMP_414))))))))) in (let TMP_416 \def (ex3_2 T T TMP_405 TMP_406 +TMP_415) in (let TMP_417 \def (S O) in (let TMP_418 \def (lift TMP_417 O t0) +in (let TMP_419 \def (S O) in (let TMP_420 \def (lift TMP_419 O t2) in (let +TMP_421 \def (pr0 TMP_418 TMP_420) in (let TMP_422 \def (S O) in (let TMP_423 +\def (pr0_lift t0 t2 H2 TMP_422 O) in (let TMP_424 \def (or_intror TMP_416 +TMP_421 TMP_423) in (eq_ind T TMP_388 TMP_402 TMP_424 t1 +H7)))))))))))))))))))))))) in (let TMP_426 \def (TMP_425 H6) in (TMP_426 +H5))))))))))))))))))))))))))))))))))) in (let TMP_447 \def (\lambda (t0: +T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead +(Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq +T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 +u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: +T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S +O) O t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) +(THead (Bind Abbr) u1 t1))).(let TMP_428 \def (Flat Cast) in (let TMP_429 +\def (THead TMP_428 u t0) in (let TMP_430 \def (\lambda (ee: T).(match ee +with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ +_) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) in (let TMP_431 \def (Bind Abbr) in (let TMP_432 \def +(THead TMP_431 u1 t1) in (let H4 \def (eq_ind T TMP_429 TMP_430 I TMP_432 H3) +in (let TMP_435 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_433 \def +(Bind Abbr) in (let TMP_434 \def (THead TMP_433 u2 t3) in (eq T t2 +TMP_434))))) in (let TMP_436 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 +u2))) in (let TMP_441 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_437 +\def (pr0 t1 t3) in (let TMP_438 \def (\lambda (y0: T).(pr0 t1 y0)) in (let +TMP_439 \def (\lambda (y0: T).(subst0 O u2 y0 t3)) in (let TMP_440 \def (ex2 +T TMP_438 TMP_439) in (or TMP_437 TMP_440))))))) in (let TMP_442 \def (ex3_2 +T T TMP_435 TMP_436 TMP_441) in (let TMP_443 \def (S O) in (let TMP_444 \def +(lift TMP_443 O t2) in (let TMP_445 \def (pr0 t1 TMP_444) in (let TMP_446 +\def (or TMP_442 TMP_445) in (False_ind TMP_446 H4))))))))))))))))))))) in +(pr0_ind TMP_31 TMP_91 TMP_191 TMP_217 TMP_251 TMP_333 TMP_427 TMP_447 y x +H0))))))))))) in (insert_eq T TMP_2 TMP_3 TMP_17 TMP_448 H))))))))). -theorem pr0_subst0_fwd: - \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 -i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: -T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t))))))))) +theorem pr0_gen_void: + \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Void) u1 +t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead +(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x)))))) \def - \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 t u1) \to (ex2 T -(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t3 t4))))))))) -(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 v -u1)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: -T).(pr0 (lift (S i0) O v) t)) (lift (S i0) O u1) (subst0_lref u1 i0) -(pr0_lift v u1 H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda -(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: -((\forall (u4: T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) -(\lambda (t: T).(pr0 u3 t))))))).(\lambda (t: T).(\lambda (k: K).(\lambda -(u0: T).(\lambda (H2: (pr0 v u0)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 -u1 t0)) (\lambda (t0: T).(pr0 u3 t0)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 -(THead k u1 t) t0)) (\lambda (t0: T).(pr0 (THead k u3 t) t0))) (\lambda (x: -T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 u3 x)).(ex_intro2 T -(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 -(THead k u3 t) t0)) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp u3 -x H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda -(v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: -(subst0 (s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 v u1) \to -(ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 -t))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 v u1)).(ex2_ind -T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 -T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 -(THead k u t3) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 -x)).(\lambda (H4: (pr0 t3 x)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 -(THead k u t4) t)) (\lambda (t: T).(pr0 (THead k u t3) t)) (THead k u x) -(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) t3 x H4 k))))) (H1 -u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda -(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: -T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: -T).(pr0 u3 t))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: -T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda -(t: T).(pr0 t4 t))))))).(\lambda (u0: T).(\lambda (H4: (pr0 v u0)).(ex2_ind T -(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t4 t)) (ex2 T -(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead -k u3 t4) t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 -x)).(\lambda (H6: (pr0 t4 x)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) -(\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 -t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) t))) (\lambda (x0: T).(\lambda -(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 u3 x0)).(ex_intro2 T (\lambda -(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) -t)) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp u3 x0 H8 -t4 x H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))). -(* COMMENTS -Initial nodes: 979 -END *) - -theorem pr0_subst0: - \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall -(w1: T).(\forall (i: nat).((subst0 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 -v2) \to (or (pr0 w1 t2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t2 w2)))))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (v1: T).(\forall (w1: T).(\forall (i: -nat).((subst0 i v1 t w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 -t0) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t0 -w2)))))))))))) (\lambda (t: T).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: -nat).(\lambda (H0: (subst0 i v1 t w1)).(\lambda (v2: T).(\lambda (H1: (pr0 v1 -v2)).(or_intror (pr0 w1 t) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t w2))) (ex2_sym T (subst0 i v2 t) (pr0 w1) (pr0_subst0_fwd -v1 t w1 i H0 v2 H1)))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: -(pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: T).(\forall (i: -nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 -u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 u2 -w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 -t4)).(\lambda (H3: ((\forall (v1: T).(\forall (w1: T).(\forall (i: -nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 -t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2)))))))))))).(\lambda (k: K).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: -nat).(\lambda (H4: (subst0 i v1 (THead k u1 t3) w1)).(\lambda (v2: -T).(\lambda (H5: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T w1 -(THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3))) (ex2 T (\lambda (t5: -T).(eq T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5))) -(ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5)))) -(\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))) (or (pr0 w1 (THead k u2 t4)) -(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 (THead k -u2 t4) w2)))) (\lambda (H6: (ex2 T (\lambda (u3: T).(eq T w1 (THead k u3 -t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq -T w1 (THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)) (or (pr0 w1 -(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1 -(THead k x t3))).(\lambda (H8: (subst0 i v1 u1 x)).(eq_ind_r T (THead k x t3) -(\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 t -w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind (pr0 x u2) -(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) -(or (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead -k x t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda -(H9: (pr0 x u2)).(or_introl (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2 -(THead k u2 t4) w2))) (pr0_comp x u2 H9 t3 t4 H2 k))) (\lambda (H9: (ex2 T -(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))).(ex2_ind -T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 -(THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x t3) -w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0: -T).(\lambda (H10: (pr0 x x0)).(\lambda (H11: (subst0 i v2 u2 x0)).(or_intror -(pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x -t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2 -(THead k u2 t4) w2)) (THead k x0 t4) (pr0_comp x x0 H10 t3 t4 H2 k) -(subst0_fst v2 x0 u2 i H11 t4 k)))))) H9)) (H1 v1 x i H8 v2 H5)) w1 H7)))) -H6)) (\lambda (H6: (ex2 T (\lambda (t5: T).(eq T w1 (THead k u1 t5))) -(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq -T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5)) (or (pr0 -w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1 -(THead k u1 x))).(\lambda (H8: (subst0 (s k i) v1 t3 x)).(eq_ind_r T (THead k -u1 x) (\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind -(pr0 x t4) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s k -i) v2 t4 w2))) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) -w2)))) (\lambda (H9: (pr0 x t4)).(or_introl (pr0 (THead k u1 x) (THead k u2 -t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2))) (pr0_comp u1 u2 H0 x t4 H9 k))) -(\lambda (H9: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s -k i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: -T).(subst0 (s k i) v2 t4 w2)) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 -(THead k u2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (pr0 x x0)).(\lambda -(H11: (subst0 (s k i) v2 t4 x0)).(or_intror (pr0 (THead k u1 x) (THead k u2 -t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 -(THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)) (THead -k u2 x0) (pr0_comp u1 u2 H0 x x0 H10 k) (subst0_snd k v2 x0 t4 i H11 u2)))))) -H9)) (H3 v1 x (s k i) H8 v2 H5)) w1 H7)))) H6)) (\lambda (H6: (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s k i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda -(t5: T).(eq T w1 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 -i v1 u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5))) -(or (pr0 w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda -(w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H7: (eq T w1 (THead k x0 x1))).(\lambda (H8: (subst0 i v1 u1 -x0)).(\lambda (H9: (subst0 (s k i) v1 t3 x1)).(eq_ind_r T (THead k x0 x1) -(\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 t -w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind (pr0 x1 -t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 -t4 w2))) (or (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) -w2)))) (\lambda (H10: (pr0 x1 t4)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead k x0 -x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (H11: (pr0 x0 -u2)).(or_introl (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) -w2))) (pr0_comp x0 u2 H11 x1 t4 H10 k))) (\lambda (H11: (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead k -x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda -(H12: (pr0 x0 x)).(\lambda (H13: (subst0 i v2 u2 x)).(or_intror (pr0 (THead k -x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda -(w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 -t4) w2)) (THead k x t4) (pr0_comp x0 x H12 x1 t4 H10 k) (subst0_fst v2 x u2 i -H13 t4 k)))))) H11)) (H1 v1 x0 i H8 v2 H5))) (\lambda (H10: (ex2 T (\lambda -(w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 t4 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 t4 w2)) (or -(pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k -x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: -T).(\lambda (H11: (pr0 x1 x)).(\lambda (H12: (subst0 (s k i) v2 t4 -x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v2 u2 w2))) (or (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 -(THead k u2 t4) w2)))) (\lambda (H13: (pr0 x0 u2)).(or_intror (pr0 (THead k -x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda -(w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 -t4) w2)) (THead k u2 x) (pr0_comp x0 u2 H13 x1 x H11 k) (subst0_snd k v2 x t4 -i H12 u2)))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) -(\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead k x0 x1) (THead k u2 -t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x2: T).(\lambda (H14: (pr0 -x0 x2)).(\lambda (H15: (subst0 i v2 u2 x2)).(or_intror (pr0 (THead k x0 x1) -(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda -(w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 -(THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)) -(THead k x2 x) (pr0_comp x0 x2 H14 x1 x H11 k) (subst0_both v2 u2 x2 i H15 k -t4 x H12)))))) H13)) (H1 v1 x0 i H8 v2 H5))))) H10)) (H3 v1 x1 (s k i) H9 v2 -H5)) w1 H7)))))) H6)) (subst0_gen_head k v1 u1 t3 w1 i H4))))))))))))))))) -(\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (pr0 v1 -v2)).(\lambda (H1: ((\forall (v3: T).(\forall (w1: T).(\forall (i: -nat).((subst0 i v3 v1 w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 -v2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 v2 -w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 -t4)).(\lambda (H3: ((\forall (v3: T).(\forall (w1: T).(\forall (i: -nat).((subst0 i v3 t3 w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 -t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 t4 -w2)))))))))))).(\lambda (v0: T).(\lambda (w1: T).(\lambda (i: nat).(\lambda -(H4: (subst0 i v0 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) -w1)).(\lambda (v3: T).(\lambda (H5: (pr0 v0 v3)).(or3_ind (ex2 T (\lambda -(u2: T).(eq T w1 (THead (Flat Appl) u2 (THead (Bind Abst) u t3)))) (\lambda -(u2: T).(subst0 i v0 v1 u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat -Appl) v1 t5))) (\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind -Abst) u t3) t5))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 -(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 v1 -u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead -(Bind Abst) u t3) t5)))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2)))) (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T w1 (THead -(Flat Appl) u2 (THead (Bind Abst) u t3)))) (\lambda (u2: T).(subst0 i v0 v1 -u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat Appl) u2 (THead -(Bind Abst) u t3)))) (\lambda (u2: T).(subst0 i v0 v1 u2)) (or (pr0 w1 (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda -(H7: (eq T w1 (THead (Flat Appl) x (THead (Bind Abst) u t3)))).(\lambda (H8: -(subst0 i v0 v1 x)).(eq_ind_r T (THead (Flat Appl) x (THead (Bind Abst) u -t3)) (\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda -(w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2))))) (or_ind (pr0 x v2) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: -T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x (THead (Bind Abst) u -t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2)))) (\lambda (H9: (pr0 x v2)).(or_introl (pr0 (THead -(Flat Appl) x (THead (Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta u x -v2 H9 t3 t4 H2))) (\lambda (H9: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda -(w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) -(\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x (THead -(Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: T).(subst0 -i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (pr0 x -x0)).(\lambda (H11: (subst0 i v3 v2 x0)).(or_intror (pr0 (THead (Flat Appl) x -(THead (Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x0 t4) -(pr0_beta u x x0 H10 t3 t4 H2) (subst0_fst v3 x0 v2 i H11 t4 (Bind -Abbr))))))) H9)) (H1 v0 x i H8 v3 H5)) w1 H7)))) H6)) (\lambda (H6: (ex2 T -(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: -T).(subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) t5)))).(ex2_ind T -(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: -T).(subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) t5)) (or (pr0 w1 -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda -(H7: (eq T w1 (THead (Flat Appl) v1 x))).(\lambda (H8: (subst0 (s (Flat Appl) -i) v0 (THead (Bind Abst) u t3) x)).(or3_ind (ex2 T (\lambda (u2: T).(eq T x -(THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u -u2))) (ex2 T (\lambda (t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda -(t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 -t3 t5)))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) -(\lambda (H9: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 t3))) -(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda -(u2: T).(eq T x (THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat -Appl) i) v0 u u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead (Bind Abst) x0 -t3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(let H12 \def (eq_ind -T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) -x0 t3) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) -(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2))))) (or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 t3 t4 H2)) w1 H12))))) H9)) (\lambda -(H9: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda -(t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind -Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead -(Bind Abst) u x0))).(\lambda (H11: (subst0 (s (Bind Abst) (s (Flat Appl) i)) -v0 t3 x0)).(let H12 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat -Appl) v1 t))) H7 (THead (Bind Abst) u x0) H10) in (eq_ind_r T (THead (Flat -Appl) v1 (THead (Bind Abst) u x0)) (\lambda (t: T).(or (pr0 t (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 -t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead -(Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)))) (\lambda (H13: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta u v1 v2 H0 x0 t4 -H13))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) -i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2)))) (\lambda (x1: T).(\lambda (H14: (pr0 x0 x1)).(\lambda -(H15: (subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 -(THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 -T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind -Abbr) v2 x1) (pr0_beta u v1 v2 H0 x0 x1 H14) (subst0_snd (Bind Abbr) v3 x1 t4 -i H15 v2)))))) H13)) (H3 v0 x0 (s (Bind Abst) (s (Flat Appl) i)) H11 v3 H5)) -w1 H12))))) H9)) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T x (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 -t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T x (THead (Bind Abst) -x0 x1))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(\lambda (H12: -(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x1)).(let H13 \def (eq_ind T -x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) -x0 x1) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) -(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda -(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead -(Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H14: -(pr0 x1 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 x1 t4 H14))) (\lambda (H14: (ex2 T -(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s -(Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) -(\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)) (or -(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 -t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 -x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) -(\lambda (x2: T).(\lambda (H15: (pr0 x1 x2)).(\lambda (H16: (subst0 (s (Bind -Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_intror (pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) -(pr0_beta x0 v1 v2 H0 x1 x2 H15) (subst0_snd (Bind Abbr) v3 x2 t4 i H16 -v2)))))) H14)) (H3 v0 x1 (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 -H13))))))) H9)) (subst0_gen_head (Bind Abst) v0 u t3 x (s (Flat Appl) i) -H8))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 -v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead -(Bind Abst) u t3) t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: -T).(eq T w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i v0 v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat -Appl) i) v0 (THead (Bind Abst) u t3) t5))) (or (pr0 w1 (THead (Bind Abbr) v2 -t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(H7: (eq T w1 (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i v0 v1 -x0)).(\lambda (H9: (subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) -x1)).(or3_ind (ex2 T (\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) -(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2))) (ex2 T (\lambda (t5: -T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind -Abst) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))) (or (pr0 w1 (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H10: (ex2 T -(\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) (\lambda (u2: -T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T x1 -(THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u -u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda -(x: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x t3))).(\lambda (_: -(subst0 (s (Flat Appl) i) v0 u x)).(let H13 \def (eq_ind T x1 (\lambda (t: -T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) x t3) H11) in -(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (\lambda (t: -T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 -x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2)))) (\lambda (H14: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat Appl) x0 -(THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta x x0 v2 H14 t3 t4 -H2))) (\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind -Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead -(Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 -x2)).(\lambda (H16: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) -x0 (THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x2 t4) -(pr0_beta x x0 x2 H15 t3 t4 H2) (subst0_fst v3 x2 v2 i H16 t4 (Bind -Abbr))))))) H14)) (H1 v0 x0 i H8 v3 H5)) w1 H13))))) H10)) (\lambda (H10: -(ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda -(t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind -Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda (H11: (eq T x1 (THead -(Bind Abst) u x))).(\lambda (H12: (subst0 (s (Bind Abst) (s (Flat Appl) i)) -v0 t3 x)).(let H13 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat -Appl) x0 t))) H7 (THead (Bind Abst) u x) H11) in (eq_ind_r T (THead (Flat -Appl) x0 (THead (Bind Abst) u x)) (\lambda (t: T).(or (pr0 t (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x t4) (ex2 T (\lambda (w2: -T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 -t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2)))) (\lambda (H14: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat -Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda -(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) (\lambda -(w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: (pr0 x0 -v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2))) (pr0_beta u x0 v2 H15 x t4 H14))) (\lambda (H15: (ex2 T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 -(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda -(x2: T).(\lambda (H16: (pr0 x0 x2)).(\lambda (H17: (subst0 i v3 v2 -x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)) (THead (Bind Abbr) x2 t4) (pr0_beta u x0 x2 H16 x t4 H14) -(subst0_fst v3 x2 v2 i H17 t4 (Bind Abbr))))))) H15)) (H1 v0 x0 i H8 v3 H5))) -(\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 -(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: -T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 -t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x x2)).(\lambda (H16: (subst0 (s -(Bind Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead -(Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H17: -(pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 -(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 -(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) (pr0_beta u x0 v2 H17 x x2 H15) -(subst0_snd (Bind Abbr) v3 x2 t4 i H16 v2)))) (\lambda (H17: (ex2 T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 -(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda -(x3: T).(\lambda (H18: (pr0 x0 x3)).(\lambda (H19: (subst0 i v3 v2 -x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)) (THead (Bind Abbr) x3 x2) (pr0_beta u x0 x3 H18 x x2 H15) -(subst0_both v3 v2 x3 i H19 (Bind Abbr) t4 x2 H16)))))) H17)) (H1 v0 x0 i H8 -v3 H5))))) H14)) (H3 v0 x (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 -H13))))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 -t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) -(\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) -x2 x3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x2)).(\lambda (H13: -(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x3)).(let H14 \def (eq_ind T -x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) -x2 x3) H11) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) -(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2))))) (or_ind (pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda -(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead -(Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: -(pr0 x3 t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) -(\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H16: (pr0 x0 -v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2))) (pr0_beta x2 x0 v2 H16 x3 t4 H15))) (\lambda (H16: (ex2 T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 -(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda -(x: T).(\lambda (H17: (pr0 x0 x)).(\lambda (H18: (subst0 i v3 v2 -x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)) (THead (Bind Abbr) x t4) (pr0_beta x2 x0 x H17 x3 t4 H15) -(subst0_fst v3 x v2 i H18 t4 (Bind Abbr))))))) H16)) (H1 v0 x0 i H8 v3 H5))) -(\lambda (H15: (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 -(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: -T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 -t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)))) (\lambda (x: T).(\lambda (H16: (pr0 x3 x)).(\lambda (H17: (subst0 -(s (Bind Abst) (s (Flat Appl) i)) v3 t4 x)).(or_ind (pr0 x0 v2) (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 -(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda -(H18: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) -x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x) (pr0_beta x2 x0 v2 H18 x3 x -H16) (subst0_snd (Bind Abbr) v3 x t4 i H17 v2)))) (\lambda (H18: (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)))) (\lambda (x4: T).(\lambda (H19: (pr0 x0 x4)).(\lambda (H20: -(subst0 i v3 v2 x4)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) -x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x4 x) (pr0_beta x2 x0 x4 H19 x3 x -H16) (subst0_both v3 v2 x4 i H20 (Bind Abbr) t4 x H17)))))) H18)) (H1 v0 x0 i -H8 v3 H5))))) H15)) (H3 v0 x3 (s (Bind Abst) (s (Flat Appl) i)) H13 v3 H5)) -w1 H14))))))) H10)) (subst0_gen_head (Bind Abst) v0 u t3 x1 (s (Flat Appl) i) -H9))))))) H6)) (subst0_gen_head (Flat Appl) v0 v1 (THead (Bind Abst) u t3) w1 -i H4))))))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b -Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H1: (pr0 v1 v2)).(\lambda -(H2: ((\forall (v3: T).(\forall (w1: T).(\forall (i: nat).((subst0 i v3 v1 -w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 v2) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 v2 w2)))))))))))).(\lambda -(u1: T).(\lambda (u2: T).(\lambda (H3: (pr0 u1 u2)).(\lambda (H4: ((\forall -(v3: T).(\forall (w1: T).(\forall (i: nat).((subst0 i v3 u1 w1) \to (\forall -(v4: T).((pr0 v3 v4) \to (or (pr0 w1 u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) -(\lambda (w2: T).(subst0 i v4 u2 w2)))))))))))).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H5: (pr0 t3 t4)).(\lambda (H6: ((\forall (v3: T).(\forall -(w1: T).(\forall (i: nat).((subst0 i v3 t3 w1) \to (\forall (v4: T).((pr0 v3 -v4) \to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v4 t4 w2)))))))))))).(\lambda (v0: T).(\lambda (w1: T).(\lambda -(i: nat).(\lambda (H7: (subst0 i v0 (THead (Flat Appl) v1 (THead (Bind b) u1 -t3)) w1)).(\lambda (v3: T).(\lambda (H8: (pr0 v0 v3)).(or3_ind (ex2 T -(\lambda (u3: T).(eq T w1 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)))) -(\lambda (u3: T).(subst0 i v0 v1 u3))) (ex2 T (\lambda (t5: T).(eq T w1 -(THead (Flat Appl) v1 t5))) (\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 -(THead (Bind b) u1 t3) t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq -T w1 (THead (Flat Appl) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i -v0 v1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 -(THead (Bind b) u1 t3) t5)))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda -(w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4)) w2)))) (\lambda (H9: (ex2 T (\lambda (u3: T).(eq T w1 (THead (Flat Appl) -u3 (THead (Bind b) u1 t3)))) (\lambda (u3: T).(subst0 i v0 v1 u3)))).(ex2_ind -T (\lambda (u3: T).(eq T w1 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)))) -(\lambda (u3: T).(subst0 i v0 v1 u3)) (or (pr0 w1 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H10: (eq T w1 (THead (Flat -Appl) x (THead (Bind b) u1 t3)))).(\lambda (H11: (subst0 i v0 v1 -x)).(eq_ind_r T (THead (Flat Appl) x (THead (Bind b) u1 t3)) (\lambda (t: -T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) -(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x -v2) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2))) (or (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (H12: (pr0 x v2)).(or_introl (pr0 (THead (Flat Appl) x (THead (Bind -b) u1 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 -T (\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2))) (pr0_upsilon b H0 x v2 H12 u1 u2 H3 t3 t4 H5))) (\lambda -(H12: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x0: T).(\lambda (H13: (pr0 x x0)).(\lambda (H14: (subst0 i v3 v2 -x0)).(or_intror (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind b) -u1 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O x0) t4)) (pr0_upsilon b H0 x x0 H13 u1 u2 H3 t3 t4 H5) (subst0_snd -(Bind b) v3 (THead (Flat Appl) (lift (S O) O x0) t4) (THead (Flat Appl) (lift -(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x0) (lift (S O) O v2) (s (Bind -b) i) (subst0_lift_ge_s v2 x0 v3 i H14 O (le_O_n i) b) t4 (Flat Appl)) -u2)))))) H12)) (H2 v0 x i H11 v3 H8)) w1 H10)))) H9)) (\lambda (H9: (ex2 T -(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: -T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) t5)))).(ex2_ind T -(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: -T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) t5)) (or (pr0 w1 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H10: (eq -T w1 (THead (Flat Appl) v1 x))).(\lambda (H11: (subst0 (s (Flat Appl) i) v0 -(THead (Bind b) u1 t3) x)).(or3_ind (ex2 T (\lambda (u3: T).(eq T x (THead -(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (ex2 -T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 -(s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u3: -T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or -(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H12: (ex2 T -(\lambda (u3: T).(eq T x (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s -(Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x (THead (Bind -b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or (pr0 w1 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq -T x (THead (Bind b) x0 t3))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 -x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 -t))) H10 (THead (Bind b) x0 t3) H13) in (eq_ind_r T (THead (Flat Appl) v1 -(THead (Bind b) x0 t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2))))) (or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 -w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or (pr0 (THead -(Flat Appl) v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H16: (pr0 x0 -u2)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H16 t3 t4 H5))) (\lambda (H16: (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 -u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 -(s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) -x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda -(H18: (subst0 (s (Flat Appl) i) v3 u2 x1)).(or_intror (pr0 (THead (Flat Appl) -v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) -O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind -b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead -(Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead -(Bind b) x1 (THead (Flat Appl) (lift (S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 -H1 x0 x1 H17 t3 t4 H5) (subst0_fst v3 x1 u2 i H18 (THead (Flat Appl) (lift (S -O) O v2) t4) (Bind b))))))) H16)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8)) w1 -H15))))) H12)) (\lambda (H12: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind b) -u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 -t5)))).(ex2_ind T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda -(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq T x -(THead (Bind b) u1 x0))).(\lambda (H14: (subst0 (s (Bind b) (s (Flat Appl) -i)) v0 t3 x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead -(Flat Appl) v1 t))) H10 (THead (Bind b) u1 x0) H13) in (eq_ind_r T (THead -(Flat Appl) v1 (THead (Bind b) u1 x0)) (\lambda (t: T).(or (pr0 t (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 -w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (H16: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead -(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 -x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 x0 t4 H16))) -(\lambda (H16: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 -(s (Bind b) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 -x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 w2)) -(or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda (H18: (subst0 (s (Bind -b) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 (THead (Flat Appl) v1 (THead -(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 -x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) x1)) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 -x0 x1 H17) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O v2) x1) -(THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) v3 x1 t4 -(s (Bind b) i) H18 (lift (S O) O v2)) u2)))))) H16)) (H6 v0 x0 (s (Bind b) (s -(Flat Appl) i)) H14 v3 H8)) w1 H15))))) H12)) (\lambda (H12: (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda -(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 -t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind -b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 -u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) -v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda -(x0: T).(\lambda (x1: T).(\lambda (H13: (eq T x (THead (Bind b) x0 -x1))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 x0)).(\lambda (H15: -(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x1)).(let H16 \def (eq_ind T x -(\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H10 (THead (Bind b) x0 -x1) H13) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) -(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) -O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind -(pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s -(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead -(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 -x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (H17: (pr0 x1 t4)).(or_ind (pr0 x0 u2) -(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) -i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)))) (\lambda (H18: (pr0 x0 u2)).(or_introl (pr0 (THead (Flat Appl) v1 -(THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) -x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H18 x1 t4 -H17))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 (s (Flat Appl) i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 -w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead -(Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda -(H19: (pr0 x0 x2)).(\lambda (H20: (subst0 (s (Flat Appl) i) v3 u2 -x2)).(or_intror (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind -b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift -(S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 H1 x0 x2 H19 x1 t4 H17) (subst0_fst -v3 x2 u2 i H20 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))))) H18)) -(H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: -T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s -(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead -(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 -x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x1 -x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind -(pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s -(Flat Appl) i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 -x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 u2)).(or_intror (pr0 (THead (Flat -Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead -(Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 v1 v2 -H1 x0 u2 H20 x1 x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S -O) O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat -Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: -(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) -i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead -(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 -x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (x3: T).(\lambda (H21: (pr0 x0 -x3)).(\lambda (H22: (subst0 (s (Flat Appl) i) v3 u2 x3)).(or_intror (pr0 -(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) -v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)) (THead (Bind b) x3 (THead (Flat Appl) (lift (S O) O v2) x2)) -(pr0_upsilon b H0 v1 v2 H1 x0 x3 H21 x1 x2 H18) (subst0_both v3 u2 x3 i H22 -(Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S -O) O v2) x2) (subst0_snd (Flat Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) -O v2)))))))) H20)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))))) H17)) (H6 v0 x1 -(s (Bind b) (s (Flat Appl) i)) H15 v3 H8)) w1 H16))))))) H12)) -(subst0_gen_head (Bind b) v0 u1 t3 x (s (Flat Appl) i) H11))))) H9)) (\lambda -(H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead (Flat Appl) -u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) -t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead -(Flat Appl) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) -u1 t3) t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda -(x0: T).(\lambda (x1: T).(\lambda (H10: (eq T w1 (THead (Flat Appl) x0 -x1))).(\lambda (H11: (subst0 i v0 v1 x0)).(\lambda (H12: (subst0 (s (Flat -Appl) i) v0 (THead (Bind b) u1 t3) x1)).(or3_ind (ex2 T (\lambda (u3: T).(eq -T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 -u1 u3))) (ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind b) u1 t5))) (\lambda -(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda -(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or -(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H13: (ex2 T -(\lambda (u3: T).(eq T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 -(s (Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x1 (THead -(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or -(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda -(H14: (eq T x1 (THead (Bind b) x t3))).(\lambda (H15: (subst0 (s (Flat Appl) -i) v0 u1 x)).(let H16 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat -Appl) x0 t))) H10 (THead (Bind b) x t3) H14) in (eq_ind_r T (THead (Flat -Appl) x0 (THead (Bind b) x t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x u2) (ex2 T (\lambda (w2: -T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or -(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H17: -(pr0 x u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) -x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat -Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 x0 v2 H18 x u2 H17 -t3 t4 H5))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x -t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda -(H20: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind -b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 -T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 -(THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 x u2 H17 t3 t4 -H5) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead -(Flat Appl) (lift (S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S -O) O v2) (s (Bind b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 -(Flat Appl)) u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T -(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s -(Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x -t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x x2)).(\lambda -(H19: (subst0 (s (Flat Appl) i) v3 u2 x2)).(or_ind (pr0 x0 v2) (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 -(THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) -x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 -v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift -(S O) O v2) t4)) (pr0_upsilon b H0 x0 v2 H20 x x2 H18 t3 t4 H5) (subst0_fst -v3 x2 u2 i H19 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))) (\lambda -(H20: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 -x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift -(S O) O x3) t4)) (pr0_upsilon b H0 x0 x3 H21 x x2 H18 t3 t4 H5) (subst0_both -v3 u2 x2 i H19 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat -Appl) (lift (S O) O x3) t4) (subst0_fst v3 (lift (S O) O x3) (lift (S O) O -v2) (s (Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) t4 (Flat -Appl)))))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H4 v0 x (s (Flat Appl) -i) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex2 T (\lambda (t5: T).(eq T -x1 (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat -Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T x1 (THead (Bind b) -u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) -(or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) -(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: -T).(\lambda (H14: (eq T x1 (THead (Bind b) u1 x))).(\lambda (H15: (subst0 (s -(Bind b) (s (Flat Appl) i)) v0 t3 x)).(let H16 \def (eq_ind T x1 (\lambda (t: -T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) u1 x) H14) in -(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (\lambda (t: T).(or -(pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x t4) (ex2 T -(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat -Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda -(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)))) (\lambda (H17: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat -Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 -v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (pr0_upsilon b H0 x0 v2 H18 u1 u2 H3 x t4 H17))) (\lambda (H18: (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda (H20: (subst0 i v3 v2 -x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 u1 u2 H3 x t4 H17) (subst0_snd -(Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead (Flat Appl) (lift -(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S O) O v2) (s (Bind -b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 (Flat Appl)) -u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: -T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s -(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead -(Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x -x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind -(pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i -v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)))) (\lambda (H20: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 -(THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) -u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead -(Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 x0 v2 -H20 u1 u2 H3 x x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) -O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) -v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 -x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O x3) x2)) (pr0_upsilon b H0 x0 x3 H21 u1 u2 H3 x x2 H18) (subst0_snd -(Bind b) v3 (THead (Flat Appl) (lift (S O) O x3) x2) (THead (Flat Appl) (lift -(S O) O v2) t4) i (subst0_both v3 (lift (S O) O v2) (lift (S O) O x3) (s -(Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) (Flat Appl) t4 -x2 H19) u2)))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H6 v0 x (s (Bind b) -(s (Flat Appl) i)) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda -(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 -t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead -(Bind b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) -v0 u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat -Appl) i)) v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T x1 (THead (Bind -b) x2 x3))).(\lambda (H15: (subst0 (s (Flat Appl) i) v0 u1 x2)).(\lambda -(H16: (subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x3)).(let H17 \def (eq_ind -T x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) -x2 x3) H14) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) -(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) -O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind -(pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s -(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead -(Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 -x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x3 t4)).(or_ind (pr0 x2 u2) -(ex2 T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) -i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)))) (\lambda (H19: (pr0 x2 u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat -Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 -v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (pr0_upsilon b H0 x0 v2 H20 x2 u2 H19 x3 t4 H18))) (\lambda (H20: (ex2 -T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x: T).(\lambda (H21: (pr0 x0 x)).(\lambda (H22: (subst0 i v3 v2 -x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O x) t4)) (pr0_upsilon b H0 x0 x H21 x2 u2 H19 x3 t4 H18) (subst0_snd -(Bind b) v3 (THead (Flat Appl) (lift (S O) O x) t4) (THead (Flat Appl) (lift -(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x) (lift (S O) O v2) (s (Bind -b) i) (subst0_lift_ge_s v2 x v3 i H22 O (le_O_n i) b) t4 (Flat Appl)) -u2)))))) H20)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H19: (ex2 T (\lambda (w2: -T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s -(Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 -x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H20: (pr0 x2 x)).(\lambda -(H21: (subst0 (s (Flat Appl) i) v3 u2 x)).(or_ind (pr0 x0 v2) (ex2 T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead -(Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 -(THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H22: (pr0 x0 -v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x (THead (Flat Appl) (lift -(S O) O v2) t4)) (pr0_upsilon b H0 x0 v2 H22 x2 x H20 x3 t4 H18) (subst0_fst -v3 x u2 i H21 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))) (\lambda -(H22: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x4: T).(\lambda (H23: (pr0 x0 x4)).(\lambda (H24: (subst0 i v3 v2 -x4)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x (THead (Flat Appl) (lift -(S O) O x4) t4)) (pr0_upsilon b H0 x0 x4 H23 x2 x H20 x3 t4 H18) (subst0_both -v3 u2 x i H21 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat -Appl) (lift (S O) O x4) t4) (subst0_fst v3 (lift (S O) O x4) (lift (S O) O -v2) (s (Bind b) i) (subst0_lift_ge_s v2 x4 v3 i H24 O (le_O_n i) b) t4 (Flat -Appl)))))))) H22)) (H2 v0 x0 i H11 v3 H8))))) H19)) (H4 v0 x2 (s (Flat Appl) -i) H15 v3 H8))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda -(w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat -Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda -(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)))) (\lambda (x: T).(\lambda (H19: (pr0 x3 x)).(\lambda (H20: (subst0 (s -(Bind b) (s (Flat Appl) i)) v3 t4 x)).(or_ind (pr0 x2 u2) (ex2 T (\lambda -(w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) -(or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (H21: (pr0 x2 u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 -x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 -(THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) -x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H22: (pr0 x0 v2)).(or_intror -(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -x)) (pr0_upsilon b H0 x0 v2 H22 x2 u2 H21 x3 x H19) (subst0_snd (Bind b) v3 -(THead (Flat Appl) (lift (S O) O v2) x) (THead (Flat Appl) (lift (S O) O v2) -t4) i (subst0_snd (Flat Appl) v3 x t4 (s (Bind b) i) H20 (lift (S O) O v2)) -u2)))) (\lambda (H22: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) -x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (x4: T).(\lambda (H23: (pr0 x0 x4)).(\lambda -(H24: (subst0 i v3 v2 x4)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind -b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 -T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 -(THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O x4) x)) (pr0_upsilon b H0 x0 x4 H23 x2 u2 H21 x3 x -H19) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O x4) x) (THead -(Flat Appl) (lift (S O) O v2) t4) i (subst0_both v3 (lift (S O) O v2) (lift -(S O) O x4) (s (Bind b) i) (subst0_lift_ge_s v2 x4 v3 i H24 O (le_O_n i) b) -(Flat Appl) t4 x H20) u2)))))) H22)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H21: -(ex2 T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) -i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: -T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead -(Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 -x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (x4: T).(\lambda (H22: (pr0 x2 -x4)).(\lambda (H23: (subst0 (s (Flat Appl) i) v3 u2 x4)).(or_ind (pr0 x0 v2) -(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) -(or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (H24: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead -(Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 -x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x4 -(THead (Flat Appl) (lift (S O) O v2) x)) (pr0_upsilon b H0 x0 v2 H24 x2 x4 -H22 x3 x H19) (subst0_both v3 u2 x4 i H23 (Bind b) (THead (Flat Appl) (lift -(S O) O v2) t4) (THead (Flat Appl) (lift (S O) O v2) x) (subst0_snd (Flat -Appl) v3 x t4 (s (Bind b) i) H20 (lift (S O) O v2)))))) (\lambda (H24: (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x5: T).(\lambda (H25: (pr0 x0 x5)).(\lambda (H26: (subst0 i v3 v2 -x5)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x4 (THead (Flat Appl) (lift -(S O) O x5) x)) (pr0_upsilon b H0 x0 x5 H25 x2 x4 H22 x3 x H19) (subst0_both -v3 u2 x4 i H23 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat -Appl) (lift (S O) O x5) x) (subst0_both v3 (lift (S O) O v2) (lift (S O) O -x5) (s (Bind b) i) (subst0_lift_ge_s v2 x5 v3 i H26 O (le_O_n i) b) (Flat -Appl) t4 x H20))))))) H24)) (H2 v0 x0 i H11 v3 H8))))) H21)) (H4 v0 x2 (s -(Flat Appl) i) H15 v3 H8))))) H18)) (H6 v0 x3 (s (Bind b) (s (Flat Appl) i)) -H16 v3 H8)) w1 H17))))))) H13)) (subst0_gen_head (Bind b) v0 u1 t3 x1 (s -(Flat Appl) i) H12))))))) H9)) (subst0_gen_head (Flat Appl) v0 v1 (THead -(Bind b) u1 t3) w1 i H7)))))))))))))))))))))) (\lambda (u1: T).(\lambda (u2: -T).(\lambda (H0: (pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: -T).(\forall (i: nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) -\to (or (pr0 w1 u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 u2 w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda -(H2: (pr0 t3 t4)).(\lambda (H3: ((\forall (v1: T).(\forall (w1: T).(\forall -(i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 -w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2)))))))))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t4 w)).(\lambda -(v1: T).(\lambda (w1: T).(\lambda (i: nat).(\lambda (H5: (subst0 i v1 (THead -(Bind Abbr) u1 t3) w1)).(\lambda (v2: T).(\lambda (H6: (pr0 v1 v2)).(or3_ind -(ex2 T (\lambda (u3: T).(eq T w1 (THead (Bind Abbr) u3 t3))) (\lambda (u3: -T).(subst0 i v1 u1 u3))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind Abbr) -u1 t5))) (\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead (Bind Abbr) u3 t5)))) -(\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5)))) (or (pr0 w1 (THead -(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H7: (ex2 T (\lambda -(u3: T).(eq T w1 (THead (Bind Abbr) u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 -u3)))).(ex2_ind T (\lambda (u3: T).(eq T w1 (THead (Bind Abbr) u3 t3))) -(\lambda (u3: T).(subst0 i v1 u1 u3)) (or (pr0 w1 (THead (Bind Abbr) u2 w)) -(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 (THead -(Bind Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H8: (eq T w1 (THead (Bind -Abbr) x t3))).(\lambda (H9: (subst0 i v1 u1 x)).(eq_ind_r T (THead (Bind -Abbr) x t3) (\lambda (t: T).(or (pr0 t (THead (Bind Abbr) u2 w)) (ex2 T -(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) -u2 w) w2))))) (or_ind (pr0 x u2) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda -(w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind Abbr) x t3) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H10: -(pr0 x u2)).(or_introl (pr0 (THead (Bind Abbr) x t3) (THead (Bind Abbr) u2 -w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) (\lambda (w2: -T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (pr0_delta x u2 H10 t3 t4 H2 w -H4))) (\lambda (H10: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: -T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda -(w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead (Bind Abbr) x t3) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x0: -T).(\lambda (H11: (pr0 x x0)).(\lambda (H12: (subst0 i v2 u2 x0)).(ex2_ind T -(\lambda (t: T).(subst0 O x0 t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 -w t)) (or (pr0 (THead (Bind Abbr) x t3) (THead (Bind Abbr) u2 w)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x1: T).(\lambda (H13: (subst0 -O x0 t4 x1)).(\lambda (H14: (subst0 (S (plus i O)) v2 w x1)).(let H15 \def -(f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in -(let H16 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w -x1)) H14 (S i) H15) in (or_intror (pr0 (THead (Bind Abbr) x t3) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x0 x1) (pr0_delta x x0 -H11 t3 t4 H2 x1 H13) (subst0_both v2 u2 x0 i H12 (Bind Abbr) w x1 H16)))))))) -(subst0_subst0_back t4 w u2 O H4 x0 v2 i H12))))) H10)) (H1 v1 x i H9 v2 H6)) -w1 H8)))) H7)) (\lambda (H7: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind -Abbr) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 -t5)))).(ex2_ind T (\lambda (t5: T).(eq T w1 (THead (Bind Abbr) u1 t5))) -(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5)) (or (pr0 w1 (THead -(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H8: -(eq T w1 (THead (Bind Abbr) u1 x))).(\lambda (H9: (subst0 (s (Bind Abbr) i) -v1 t3 x)).(eq_ind_r T (THead (Bind Abbr) u1 x) (\lambda (t: T).(or (pr0 t -(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: -T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))))) (or_ind (pr0 x t4) (ex2 T -(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 -w2))) (or (pr0 (THead (Bind Abbr) u1 x) (THead (Bind Abbr) u2 w)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H10: (pr0 x t4)).(or_introl -(pr0 (THead (Bind Abbr) u1 x) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead -(Bind Abbr) u2 w) w2))) (pr0_delta u1 u2 H0 x t4 H10 w H4))) (\lambda (H10: -(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) -i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: -T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead (Bind Abbr) u1 x) -(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 -x) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) -(\lambda (x0: T).(\lambda (H11: (pr0 x x0)).(\lambda (H12: (subst0 (s (Bind -Abbr) i) v2 t4 x0)).(ex2_ind T (\lambda (t: T).(subst0 O u2 x0 t)) (\lambda -(t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead (Bind Abbr) u1 x) -(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 -x) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) -(\lambda (x1: T).(\lambda (H13: (subst0 O u2 x0 x1)).(\lambda (H14: (subst0 -(s (Bind Abbr) i) v2 w x1)).(or_intror (pr0 (THead (Bind Abbr) u1 x) (THead -(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) u2 x1) (pr0_delta u1 u2 -H0 x x0 H11 x1 H13) (subst0_snd (Bind Abbr) v2 x1 w i H14 u2)))))) -(subst0_confluence_neq t4 x0 v2 (s (Bind Abbr) i) H12 w u2 O H4 (sym_not_eq -nat O (S i) (O_S i))))))) H10)) (H3 v1 x (s (Bind Abbr) i) H9 v2 H6)) w1 -H8)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T -w1 (THead (Bind Abbr) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v1 -u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 -t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead -(Bind Abbr) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5))) (or -(pr0 w1 (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H8: (eq T w1 (THead (Bind Abbr) x0 -x1))).(\lambda (H9: (subst0 i v1 u1 x0)).(\lambda (H10: (subst0 (s (Bind -Abbr) i) v1 t3 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or -(pr0 t (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda -(w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))))) (or_ind (pr0 x1 t4) -(ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) -i) v2 t4 w2))) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: -T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H11: (pr0 x1 -t4)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H12: -(pr0 x0 u2)).(or_introl (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 -w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: -T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (pr0_delta x0 u2 H12 x1 t4 H11 -w H4))) (\lambda (H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x: -T).(\lambda (H13: (pr0 x0 x)).(\lambda (H14: (subst0 i v2 u2 x)).(ex2_ind T -(\lambda (t: T).(subst0 O x t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 -w t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 -O x t4 x2)).(\lambda (H16: (subst0 (S (plus i O)) v2 w x2)).(let H17 \def -(f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in -(let H18 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w -x2)) H16 (S i) H17) in (or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x x2) (pr0_delta x0 x -H13 x1 t4 H11 x2 H15) (subst0_both v2 u2 x i H14 (Bind Abbr) w x2 H18)))))))) -(subst0_subst0_back t4 w u2 O H4 x v2 i H14))))) H12)) (H1 v1 x0 i H9 v2 -H6))) (\lambda (H11: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: -T).(subst0 (s (Bind Abbr) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 -w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead -(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind -Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H12: (pr0 x1 x)).(\lambda (H13: -(subst0 (s (Bind Abbr) i) v2 t4 x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind -Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead -(Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 -w) w2)))) (\lambda (H14: (pr0 x0 u2)).(ex2_ind T (\lambda (t: T).(subst0 O u2 -x t)) (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead -(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind -Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 O u2 x -x2)).(\lambda (H16: (subst0 (s (Bind Abbr) i) v2 w x2)).(or_intror (pr0 -(THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead -(Bind Abbr) u2 w) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) -x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)) -(THead (Bind Abbr) u2 x2) (pr0_delta x0 u2 H14 x1 x H12 x2 H15) (subst0_snd -(Bind Abbr) v2 x2 w i H16 u2)))))) (subst0_confluence_neq t4 x v2 (s (Bind -Abbr) i) H13 w u2 O H4 (sym_not_eq nat O (S i) (O_S i))))) (\lambda (H14: -(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 -u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 -x2)).(\lambda (H16: (subst0 i v2 u2 x2)).(ex2_ind T (\lambda (t: T).(subst0 O -x2 t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 w t)) (or (pr0 (THead -(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind -Abbr) u2 w) w2)))) (\lambda (x3: T).(\lambda (H17: (subst0 O x2 t4 -x3)).(\lambda (H18: (subst0 (S (plus i O)) v2 w x3)).(let H19 \def (f_equal -nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H20 -\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w x3)) H18 (S -i) H19) in (ex2_ind T (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 x3 t)) -(\lambda (t: T).(subst0 O x2 x t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead -(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) -w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda -(x4: T).(\lambda (H21: (subst0 (s (Bind Abbr) i) v2 x3 x4)).(\lambda (H22: -(subst0 O x2 x x4)).(or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x2 x4) (pr0_delta x0 x2 -H15 x1 x H12 x4 H22) (subst0_both v2 u2 x2 i H16 (Bind Abbr) w x4 -(subst0_trans x3 w v2 (s (Bind Abbr) i) H20 x4 H21))))))) -(subst0_confluence_neq t4 x3 x2 O H17 x v2 (s (Bind Abbr) i) H13 (O_S -i)))))))) (subst0_subst0_back t4 w u2 O H4 x2 v2 i H16))))) H14)) (H1 v1 x0 i -H9 v2 H6))))) H11)) (H3 v1 x1 (s (Bind Abbr) i) H10 v2 H6)) w1 H8)))))) H7)) -(subst0_gen_head (Bind Abbr) v1 u1 t3 w1 i H5)))))))))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H1: (pr0 t3 t4)).(\lambda (H2: ((\forall (v1: T).(\forall (w1: -T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) -\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda -(w1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v1 (THead (Bind b) u (lift -(S O) O t3)) w1)).(\lambda (v2: T).(\lambda (H4: (pr0 v1 v2)).(or3_ind (ex2 T -(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda -(u2: T).(subst0 i v1 u u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) -u t5))) (\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))) (or -(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i -v2 t4 w2)))) (\lambda (H5: (ex2 T (\lambda (u2: T).(eq T w1 (THead (Bind b) -u2 (lift (S O) O t3)))) (\lambda (u2: T).(subst0 i v1 u u2)))).(ex2_ind T -(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda -(u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 -w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: -(eq T w1 (THead (Bind b) x (lift (S O) O t3)))).(\lambda (_: (subst0 i v1 u -x)).(eq_ind_r T (THead (Bind b) x (lift (S O) O t3)) (\lambda (t: T).(or (pr0 -t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2))))) (or_introl (pr0 (THead (Bind b) x (lift (S O) O t3)) t4) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind b) x (lift (S O) O t3)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 t3 t4 H1 x)) w1 H6)))) H5)) (\lambda -(H5: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: -T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))).(ex2_ind T (\lambda -(t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: T).(subst0 (s (Bind b) -i) v1 (lift (S O) O t3) t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 -w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: -(eq T w1 (THead (Bind b) u x))).(\lambda (H7: (subst0 (s (Bind b) i) v1 (lift -(S O) O t3) x)).(ex2_ind T (\lambda (t5: T).(eq T x (lift (S O) O t5))) -(\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or (pr0 w1 -t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2)))) (\lambda (x0: T).(\lambda (H8: (eq T x (lift (S O) O x0))).(\lambda -(H9: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x0)).(let H10 \def (eq_ind T -x (\lambda (t: T).(eq T w1 (THead (Bind b) u t))) H6 (lift (S O) O x0) H8) in -(eq_ind_r T (THead (Bind b) u (lift (S O) O x0)) (\lambda (t: T).(or (pr0 t -t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2))))) (let H11 \def (eq_ind_r nat (minus i O) (\lambda (n: nat).(subst0 n -v1 t3 x0)) H9 i (minus_n_O i)) in (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or (pr0 (THead (Bind -b) u (lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u -(lift (S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda -(H12: (pr0 x0 t4)).(or_introl (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) -(ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) -(\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x0 t4 H12 u))) (\lambda -(H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 -t4 w2)) (or (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T (\lambda -(w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))) (\lambda (x1: T).(\lambda (H13: (pr0 x0 -x1)).(\lambda (H14: (subst0 i v2 t4 x1)).(or_intror (pr0 (THead (Bind b) u -(lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift -(S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)) x1 (pr0_zeta b H0 x0 x1 H13 u) H14))))) H12)) (H2 v1 -x0 i H11 v2 H4))) w1 H10))))) (subst0_gen_lift_ge v1 t3 x (s (Bind b) i) (S -O) O H7 (le_n_S O i (le_O_n i))))))) H5)) (\lambda (H5: (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda -(u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) (or (pr0 w1 t4) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T w1 (THead (Bind b) x0 -x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H8: (subst0 (s (Bind b) i) -v1 (lift (S O) O t3) x1)).(ex2_ind T (\lambda (t5: T).(eq T x1 (lift (S O) O -t5))) (\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or -(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i -v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (eq T x1 (lift (S O) O -x))).(\lambda (H10: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x)).(let H11 -\def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Bind b) x0 t))) H6 (lift -(S O) O x) H9) in (eq_ind_r T (THead (Bind b) x0 (lift (S O) O x)) (\lambda -(t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2))))) (let H12 \def (eq_ind_r nat (minus i O) (\lambda -(n: nat).(subst0 n v1 t3 x)) H10 i (minus_n_O i)) in (or_ind (pr0 x t4) (ex2 -T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or -(pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 -(THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2)))) (\lambda (H13: (pr0 x t4)).(or_introl (pr0 (THead (Bind b) x0 (lift (S -O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O -x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x t4 H13 x0))) -(\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i -v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 -i v2 t4 w2)) (or (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))) (\lambda (x2: T).(\lambda (H14: (pr0 x -x2)).(\lambda (H15: (subst0 i v2 t4 x2)).(or_intror (pr0 (THead (Bind b) x0 -(lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift -(S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda -(w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)) x2 (pr0_zeta b H0 x x2 H14 x0) H15))))) H13)) (H2 v1 -x i H12 v2 H4))) w1 H11))))) (subst0_gen_lift_ge v1 t3 x1 (s (Bind b) i) (S -O) O H8 (le_n_S O i (le_O_n i))))))))) H5)) (subst0_gen_head (Bind b) v1 u -(lift (S O) O t3) w1 i H3))))))))))))))) (\lambda (t3: T).(\lambda (t4: -T).(\lambda (H0: (pr0 t3 t4)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: -T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) -\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda -(w1: T).(\lambda (i: nat).(\lambda (H2: (subst0 i v1 (THead (Flat Cast) u t3) -w1)).(\lambda (v2: T).(\lambda (H3: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda -(u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u -u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda -(t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Flat Cast) i) v1 t3 t5)))) (or (pr0 w1 t4) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H4: -(ex2 T (\lambda (u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: -T).(subst0 i v1 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat -Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) -(\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x t3))).(\lambda -(_: (subst0 i v1 u x)).(eq_ind_r T (THead (Flat Cast) x t3) (\lambda (t: -T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2))))) (or_introl (pr0 (THead (Flat Cast) x t3) t4) (ex2 -T (\lambda (w2: T).(pr0 (THead (Flat Cast) x t3) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2))) (pr0_tau t3 t4 H0 x)) w1 H5)))) H4)) (\lambda (H4: -(ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5: -T).(subst0 (s (Flat Cast) i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T -w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 (s (Flat Cast) i) v1 -t3 t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat -Cast) u x))).(\lambda (H6: (subst0 (s (Flat Cast) i) v1 t3 x)).(eq_ind_r T -(THead (Flat Cast) u x) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (or_ind (pr0 x t4) -(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) -i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) -(\lambda (H7: (pr0 x t4)).(or_introl (pr0 (THead (Flat Cast) u x) t4) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i -v2 t4 w2))) (pr0_tau x t4 H7 u))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 -x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 -w2)) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead -(Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: -T).(\lambda (H8: (pr0 x x0)).(\lambda (H9: (subst0 (s (Flat Cast) i) v2 t4 -x0)).(or_intror (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) -(ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)) x0 (pr0_tau x x0 H8 u) H9))))) H7)) (H1 v1 x (s (Flat -Cast) i) H6 v2 H3)) w1 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Flat Cast) i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (or (pr0 w1 t4) (ex2 T (\lambda (w2: -T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x0 -x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H7: (subst0 (s (Flat Cast) -i) v1 t3 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (pr0 -t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda -(w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) x0 -x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda -(w2: T).(subst0 i v2 t4 w2)))) (\lambda (H8: (pr0 x1 t4)).(or_introl (pr0 -(THead (Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) -x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_tau x1 t4 H8 x0))) -(\lambda (H8: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 -(s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) -(\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)) (or (pr0 (THead (Flat -Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (pr0 x1 -x)).(\lambda (H10: (subst0 (s (Flat Cast) i) v2 t4 x)).(or_intror (pr0 (THead -(Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) -w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)) -x (pr0_tau x1 x H9 x0) H10))))) H8)) (H1 v1 x1 (s (Flat Cast) i) H7 v2 H3)) -w1 H5)))))) H4)) (subst0_gen_head (Flat Cast) v1 u t3 w1 i H2))))))))))))) t1 -t2 H))). -(* COMMENTS -Initial nodes: 38857 -END *) + \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead +(Bind Void) u1 t1) x)).(let TMP_1 \def (Bind Void) in (let TMP_2 \def (THead +TMP_1 u1 t1) in (let TMP_3 \def (\lambda (t: T).(pr0 t x)) in (let TMP_13 +\def (\lambda (_: T).(let TMP_6 \def (\lambda (u2: T).(\lambda (t2: T).(let +TMP_4 \def (Bind Void) in (let TMP_5 \def (THead TMP_4 u2 t2) in (eq T x +TMP_5))))) in (let TMP_7 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) +in (let TMP_8 \def (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) in (let +TMP_9 \def (ex3_2 T T TMP_6 TMP_7 TMP_8) in (let TMP_10 \def (S O) in (let +TMP_11 \def (lift TMP_10 O x) in (let TMP_12 \def (pr0 t1 TMP_11) in (or +TMP_9 TMP_12))))))))) in (let TMP_310 \def (\lambda (y: T).(\lambda (H0: (pr0 +y x)).(let TMP_23 \def (\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Bind +Void) u1 t1)) \to (let TMP_16 \def (\lambda (u2: T).(\lambda (t2: T).(let +TMP_14 \def (Bind Void) in (let TMP_15 \def (THead TMP_14 u2 t2) in (eq T t0 +TMP_15))))) in (let TMP_17 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 +u2))) in (let TMP_18 \def (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) in +(let TMP_19 \def (ex3_2 T T TMP_16 TMP_17 TMP_18) in (let TMP_20 \def (S O) +in (let TMP_21 \def (lift TMP_20 O t0) in (let TMP_22 \def (pr0 t1 TMP_21) in +(or TMP_19 TMP_22))))))))))) in (let TMP_66 \def (\lambda (t: T).(\lambda +(H1: (eq T t (THead (Bind Void) u1 t1))).(let TMP_24 \def (\lambda (e: T).e) +in (let TMP_25 \def (Bind Void) in (let TMP_26 \def (THead TMP_25 u1 t1) in +(let H2 \def (f_equal T T TMP_24 t TMP_26 H1) in (let TMP_27 \def (Bind Void) +in (let TMP_28 \def (THead TMP_27 u1 t1) in (let TMP_38 \def (\lambda (t0: +T).(let TMP_31 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_29 \def (Bind +Void) in (let TMP_30 \def (THead TMP_29 u2 t2) in (eq T t0 TMP_30))))) in +(let TMP_32 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) in (let +TMP_33 \def (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) in (let TMP_34 +\def (ex3_2 T T TMP_31 TMP_32 TMP_33) in (let TMP_35 \def (S O) in (let +TMP_36 \def (lift TMP_35 O t0) in (let TMP_37 \def (pr0 t1 TMP_36) in (or +TMP_34 TMP_37))))))))) in (let TMP_43 \def (\lambda (u2: T).(\lambda (t2: +T).(let TMP_39 \def (Bind Void) in (let TMP_40 \def (THead TMP_39 u1 t1) in +(let TMP_41 \def (Bind Void) in (let TMP_42 \def (THead TMP_41 u2 t2) in (eq +T TMP_40 TMP_42))))))) in (let TMP_44 \def (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) in (let TMP_45 \def (\lambda (_: T).(\lambda (t2: T).(pr0 t1 +t2))) in (let TMP_46 \def (ex3_2 T T TMP_43 TMP_44 TMP_45) in (let TMP_47 +\def (S O) in (let TMP_48 \def (Bind Void) in (let TMP_49 \def (THead TMP_48 +u1 t1) in (let TMP_50 \def (lift TMP_47 O TMP_49) in (let TMP_51 \def (pr0 t1 +TMP_50) in (let TMP_56 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_52 +\def (Bind Void) in (let TMP_53 \def (THead TMP_52 u1 t1) in (let TMP_54 \def +(Bind Void) in (let TMP_55 \def (THead TMP_54 u2 t2) in (eq T TMP_53 +TMP_55))))))) in (let TMP_57 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 +u2))) in (let TMP_58 \def (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) in +(let TMP_59 \def (Bind Void) in (let TMP_60 \def (THead TMP_59 u1 t1) in (let +TMP_61 \def (refl_equal T TMP_60) in (let TMP_62 \def (pr0_refl u1) in (let +TMP_63 \def (pr0_refl t1) in (let TMP_64 \def (ex3_2_intro T T TMP_56 TMP_57 +TMP_58 u1 t1 TMP_61 TMP_62 TMP_63) in (let TMP_65 \def (or_introl TMP_46 +TMP_51 TMP_64) in (eq_ind_r T TMP_28 TMP_38 TMP_65 t +H2))))))))))))))))))))))))))))) in (let TMP_141 \def (\lambda (u0: +T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 +(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: +T).(eq T u2 (THead (Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: +T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 +(lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 +t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T +T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (k: K).(\lambda +(H5: (eq T (THead k u0 t0) (THead (Bind Void) u1 t1))).(let TMP_67 \def +(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) +\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) in (let TMP_68 \def (THead k +u0 t0) in (let TMP_69 \def (Bind Void) in (let TMP_70 \def (THead TMP_69 u1 +t1) in (let H6 \def (f_equal T K TMP_67 TMP_68 TMP_70 H5) in (let TMP_71 \def +(\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | (TLRef _) +\Rightarrow u0 | (THead _ t _) \Rightarrow t])) in (let TMP_72 \def (THead k +u0 t0) in (let TMP_73 \def (Bind Void) in (let TMP_74 \def (THead TMP_73 u1 +t1) in (let H7 \def (f_equal T T TMP_71 TMP_72 TMP_74 H5) in (let TMP_75 \def +(\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) +\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) in (let TMP_76 \def (THead k +u0 t0) in (let TMP_77 \def (Bind Void) in (let TMP_78 \def (THead TMP_77 u1 +t1) in (let H8 \def (f_equal T T TMP_75 TMP_76 TMP_78 H5) in (let TMP_139 +\def (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind Void))).(let +TMP_79 \def (Bind Void) in (let TMP_91 \def (\lambda (k0: K).(let TMP_83 \def +(\lambda (u3: T).(\lambda (t3: T).(let TMP_80 \def (THead k0 u2 t2) in (let +TMP_81 \def (Bind Void) in (let TMP_82 \def (THead TMP_81 u3 t3) in (eq T +TMP_80 TMP_82)))))) in (let TMP_84 \def (\lambda (u3: T).(\lambda (_: T).(pr0 +u1 u3))) in (let TMP_85 \def (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) +in (let TMP_86 \def (ex3_2 T T TMP_83 TMP_84 TMP_85) in (let TMP_87 \def (S +O) in (let TMP_88 \def (THead k0 u2 t2) in (let TMP_89 \def (lift TMP_87 O +TMP_88) in (let TMP_90 \def (pr0 t1 TMP_89) in (or TMP_86 TMP_90)))))))))) in +(let TMP_101 \def (\lambda (t: T).((eq T t (THead (Bind Void) u1 t1)) \to +(let TMP_94 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_92 \def (Bind +Void) in (let TMP_93 \def (THead TMP_92 u3 t3) in (eq T t2 TMP_93))))) in +(let TMP_95 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let +TMP_96 \def (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) in (let TMP_97 +\def (ex3_2 T T TMP_94 TMP_95 TMP_96) in (let TMP_98 \def (S O) in (let +TMP_99 \def (lift TMP_98 O t2) in (let TMP_100 \def (pr0 t1 TMP_99) in (or +TMP_97 TMP_100)))))))))) in (let H11 \def (eq_ind T t0 TMP_101 H4 t1 H8) in +(let TMP_102 \def (\lambda (t: T).(pr0 t t2)) in (let H12 \def (eq_ind T t0 +TMP_102 H3 t1 H8) in (let TMP_112 \def (\lambda (t: T).((eq T t (THead (Bind +Void) u1 t1)) \to (let TMP_105 \def (\lambda (u3: T).(\lambda (t3: T).(let +TMP_103 \def (Bind Void) in (let TMP_104 \def (THead TMP_103 u3 t3) in (eq T +u2 TMP_104))))) in (let TMP_106 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 +u3))) in (let TMP_107 \def (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) in +(let TMP_108 \def (ex3_2 T T TMP_105 TMP_106 TMP_107) in (let TMP_109 \def (S +O) in (let TMP_110 \def (lift TMP_109 O u2) in (let TMP_111 \def (pr0 t1 +TMP_110) in (or TMP_108 TMP_111)))))))))) in (let H13 \def (eq_ind T u0 +TMP_112 H2 u1 H9) in (let TMP_113 \def (\lambda (t: T).(pr0 t u2)) in (let +H14 \def (eq_ind T u0 TMP_113 H1 u1 H9) in (let TMP_118 \def (\lambda (u3: +T).(\lambda (t3: T).(let TMP_114 \def (Bind Void) in (let TMP_115 \def (THead +TMP_114 u2 t2) in (let TMP_116 \def (Bind Void) in (let TMP_117 \def (THead +TMP_116 u3 t3) in (eq T TMP_115 TMP_117))))))) in (let TMP_119 \def (\lambda +(u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let TMP_120 \def (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3))) in (let TMP_121 \def (ex3_2 T T TMP_118 +TMP_119 TMP_120) in (let TMP_122 \def (S O) in (let TMP_123 \def (Bind Void) +in (let TMP_124 \def (THead TMP_123 u2 t2) in (let TMP_125 \def (lift TMP_122 +O TMP_124) in (let TMP_126 \def (pr0 t1 TMP_125) in (let TMP_131 \def +(\lambda (u3: T).(\lambda (t3: T).(let TMP_127 \def (Bind Void) in (let +TMP_128 \def (THead TMP_127 u2 t2) in (let TMP_129 \def (Bind Void) in (let +TMP_130 \def (THead TMP_129 u3 t3) in (eq T TMP_128 TMP_130))))))) in (let +TMP_132 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) in (let TMP_133 +\def (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) in (let TMP_134 \def +(Bind Void) in (let TMP_135 \def (THead TMP_134 u2 t2) in (let TMP_136 \def +(refl_equal T TMP_135) in (let TMP_137 \def (ex3_2_intro T T TMP_131 TMP_132 +TMP_133 u2 t2 TMP_136 H14 H12) in (let TMP_138 \def (or_introl TMP_121 +TMP_126 TMP_137) in (eq_ind_r K TMP_79 TMP_91 TMP_138 k +H10)))))))))))))))))))))))))))))) in (let TMP_140 \def (TMP_139 H7) in +(TMP_140 H6)))))))))))))))))))))))))))) in (let TMP_163 \def (\lambda (u: +T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: +(((eq T v1 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T v2 (THead (Bind Void) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 +t2)))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (t0: T).(\lambda (t2: +T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 +t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O +t2)))))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) +(THead (Bind Void) u1 t1))).(let TMP_142 \def (Flat Appl) in (let TMP_143 +\def (Bind Abst) in (let TMP_144 \def (THead TMP_143 u t0) in (let TMP_145 +\def (THead TMP_142 v1 TMP_144) in (let TMP_146 \def (\lambda (ee: T).(match +ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k +_ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) in (let TMP_147 \def (Bind Void) in (let TMP_148 \def +(THead TMP_147 u1 t1) in (let H6 \def (eq_ind T TMP_145 TMP_146 I TMP_148 H5) +in (let TMP_153 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_149 \def +(Bind Abbr) in (let TMP_150 \def (THead TMP_149 v2 t2) in (let TMP_151 \def +(Bind Void) in (let TMP_152 \def (THead TMP_151 u2 t3) in (eq T TMP_150 +TMP_152))))))) in (let TMP_154 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 +u2))) in (let TMP_155 \def (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) in +(let TMP_156 \def (ex3_2 T T TMP_153 TMP_154 TMP_155) in (let TMP_157 \def (S +O) in (let TMP_158 \def (Bind Abbr) in (let TMP_159 \def (THead TMP_158 v2 +t2) in (let TMP_160 \def (lift TMP_157 O TMP_159) in (let TMP_161 \def (pr0 +t1 TMP_160) in (let TMP_162 \def (or TMP_156 TMP_161) in (False_ind TMP_162 +H6))))))))))))))))))))))))))))) in (let TMP_193 \def (\lambda (b: B).(\lambda +(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 +v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Void) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: +T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: T).(\lambda +(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) +u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead +(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda +(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O +u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda +(_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq T (THead (Flat Appl) +v1 (THead (Bind b) u0 t0)) (THead (Bind Void) u1 t1))).(let TMP_164 \def +(Flat Appl) in (let TMP_165 \def (Bind b) in (let TMP_166 \def (THead TMP_165 +u0 t0) in (let TMP_167 \def (THead TMP_164 v1 TMP_166) in (let TMP_168 \def +(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) in (let TMP_169 \def (Bind +Void) in (let TMP_170 \def (THead TMP_169 u1 t1) in (let H9 \def (eq_ind T +TMP_167 TMP_168 I TMP_170 H8) in (let TMP_179 \def (\lambda (u3: T).(\lambda +(t3: T).(let TMP_171 \def (Bind b) in (let TMP_172 \def (Flat Appl) in (let +TMP_173 \def (S O) in (let TMP_174 \def (lift TMP_173 O v2) in (let TMP_175 +\def (THead TMP_172 TMP_174 t2) in (let TMP_176 \def (THead TMP_171 u2 +TMP_175) in (let TMP_177 \def (Bind Void) in (let TMP_178 \def (THead TMP_177 +u3 t3) in (eq T TMP_176 TMP_178))))))))))) in (let TMP_180 \def (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) in (let TMP_181 \def (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3))) in (let TMP_182 \def (ex3_2 T T TMP_179 +TMP_180 TMP_181) in (let TMP_183 \def (S O) in (let TMP_184 \def (Bind b) in +(let TMP_185 \def (Flat Appl) in (let TMP_186 \def (S O) in (let TMP_187 \def +(lift TMP_186 O v2) in (let TMP_188 \def (THead TMP_185 TMP_187 t2) in (let +TMP_189 \def (THead TMP_184 u2 TMP_188) in (let TMP_190 \def (lift TMP_183 O +TMP_189) in (let TMP_191 \def (pr0 t1 TMP_190) in (let TMP_192 \def (or +TMP_182 TMP_191) in (False_ind TMP_192 +H9)))))))))))))))))))))))))))))))))))))) in (let TMP_213 \def (\lambda (u0: +T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead +(Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq +T u2 (THead (Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 +u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O +u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda +(_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (w: T).(\lambda (_: (subst0 O +u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Void) u1 +t1))).(let TMP_194 \def (Bind Abbr) in (let TMP_195 \def (THead TMP_194 u0 +t0) in (let TMP_196 \def (\lambda (ee: T).(match ee with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow True | +Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow +False])])) in (let TMP_197 \def (Bind Void) in (let TMP_198 \def (THead +TMP_197 u1 t1) in (let H7 \def (eq_ind T TMP_195 TMP_196 I TMP_198 H6) in +(let TMP_203 \def (\lambda (u3: T).(\lambda (t3: T).(let TMP_199 \def (Bind +Abbr) in (let TMP_200 \def (THead TMP_199 u2 w) in (let TMP_201 \def (Bind +Void) in (let TMP_202 \def (THead TMP_201 u3 t3) in (eq T TMP_200 +TMP_202))))))) in (let TMP_204 \def (\lambda (u3: T).(\lambda (_: T).(pr0 u1 +u3))) in (let TMP_205 \def (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) in +(let TMP_206 \def (ex3_2 T T TMP_203 TMP_204 TMP_205) in (let TMP_207 \def (S +O) in (let TMP_208 \def (Bind Abbr) in (let TMP_209 \def (THead TMP_208 u2 w) +in (let TMP_210 \def (lift TMP_207 O TMP_209) in (let TMP_211 \def (pr0 t1 +TMP_210) in (let TMP_212 \def (or TMP_206 TMP_211) in (False_ind TMP_212 +H7)))))))))))))))))))))))))))) in (let TMP_293 \def (\lambda (b: B).(\lambda +(H1: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: +(pr0 t0 t2)).(\lambda (H3: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda +(u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind +Void) u1 t1))).(let TMP_214 \def (\lambda (e: T).(match e with [(TSort _) +\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k +with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) in (let TMP_215 +\def (Bind b) in (let TMP_216 \def (S O) in (let TMP_217 \def (lift TMP_216 O +t0) in (let TMP_218 \def (THead TMP_215 u TMP_217) in (let TMP_219 \def (Bind +Void) in (let TMP_220 \def (THead TMP_219 u1 t1) in (let H5 \def (f_equal T B +TMP_214 TMP_218 TMP_220 H4) in (let TMP_221 \def (\lambda (e: T).(match e +with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) +\Rightarrow t])) in (let TMP_222 \def (Bind b) in (let TMP_223 \def (S O) in +(let TMP_224 \def (lift TMP_223 O t0) in (let TMP_225 \def (THead TMP_222 u +TMP_224) in (let TMP_226 \def (Bind Void) in (let TMP_227 \def (THead TMP_226 +u1 t1) in (let H6 \def (f_equal T T TMP_221 TMP_225 TMP_227 H4) in (let +TMP_242 \def (\lambda (e: T).(match e with [(TSort _) \Rightarrow (let +TMP_241 \def (\lambda (x0: nat).(let TMP_240 \def (S O) in (plus x0 +TMP_240))) in (lref_map TMP_241 O t0)) | (TLRef _) \Rightarrow (let TMP_234 +\def (\lambda (x0: nat).(let TMP_233 \def (S O) in (plus x0 TMP_233))) in +(lref_map TMP_234 O t0)) | (THead _ _ t) \Rightarrow t])) in (let TMP_243 +\def (Bind b) in (let TMP_244 \def (S O) in (let TMP_245 \def (lift TMP_244 O +t0) in (let TMP_246 \def (THead TMP_243 u TMP_245) in (let TMP_247 \def (Bind +Void) in (let TMP_248 \def (THead TMP_247 u1 t1) in (let H7 \def (f_equal T T +TMP_242 TMP_246 TMP_248 H4) in (let TMP_291 \def (\lambda (_: (eq T u +u1)).(\lambda (H9: (eq B b Void)).(let TMP_250 \def (\lambda (b0: B).(let +TMP_249 \def (eq B b0 Abst) in (not TMP_249))) in (let H10 \def (eq_ind B b +TMP_250 H1 Void H9) in (let TMP_260 \def (\lambda (t: T).((eq T t0 (THead +(Bind Void) u1 t)) \to (let TMP_253 \def (\lambda (u2: T).(\lambda (t3: +T).(let TMP_251 \def (Bind Void) in (let TMP_252 \def (THead TMP_251 u2 t3) +in (eq T t2 TMP_252))))) in (let TMP_254 \def (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) in (let TMP_255 \def (\lambda (_: T).(\lambda (t3: T).(pr0 t +t3))) in (let TMP_256 \def (ex3_2 T T TMP_253 TMP_254 TMP_255) in (let +TMP_257 \def (S O) in (let TMP_258 \def (lift TMP_257 O t2) in (let TMP_259 +\def (pr0 t TMP_258) in (or TMP_256 TMP_259)))))))))) in (let TMP_261 \def (S +O) in (let TMP_262 \def (lift TMP_261 O t0) in (let H11 \def (eq_ind_r T t1 +TMP_260 H3 TMP_262 H7) in (let TMP_263 \def (S O) in (let TMP_264 \def (lift +TMP_263 O t0) in (let TMP_274 \def (\lambda (t: T).(let TMP_267 \def (\lambda +(u2: T).(\lambda (t3: T).(let TMP_265 \def (Bind Void) in (let TMP_266 \def +(THead TMP_265 u2 t3) in (eq T t2 TMP_266))))) in (let TMP_268 \def (\lambda +(u2: T).(\lambda (_: T).(pr0 u1 u2))) in (let TMP_269 \def (\lambda (_: +T).(\lambda (t3: T).(pr0 t t3))) in (let TMP_270 \def (ex3_2 T T TMP_267 +TMP_268 TMP_269) in (let TMP_271 \def (S O) in (let TMP_272 \def (lift +TMP_271 O t2) in (let TMP_273 \def (pr0 t TMP_272) in (or TMP_270 +TMP_273))))))))) in (let TMP_277 \def (\lambda (u2: T).(\lambda (t3: T).(let +TMP_275 \def (Bind Void) in (let TMP_276 \def (THead TMP_275 u2 t3) in (eq T +t2 TMP_276))))) in (let TMP_278 \def (\lambda (u2: T).(\lambda (_: T).(pr0 u1 +u2))) in (let TMP_281 \def (\lambda (_: T).(\lambda (t3: T).(let TMP_279 \def +(S O) in (let TMP_280 \def (lift TMP_279 O t0) in (pr0 TMP_280 t3))))) in +(let TMP_282 \def (ex3_2 T T TMP_277 TMP_278 TMP_281) in (let TMP_283 \def (S +O) in (let TMP_284 \def (lift TMP_283 O t0) in (let TMP_285 \def (S O) in +(let TMP_286 \def (lift TMP_285 O t2) in (let TMP_287 \def (pr0 TMP_284 +TMP_286) in (let TMP_288 \def (S O) in (let TMP_289 \def (pr0_lift t0 t2 H2 +TMP_288 O) in (let TMP_290 \def (or_intror TMP_282 TMP_287 TMP_289) in +(eq_ind T TMP_264 TMP_274 TMP_290 t1 H7)))))))))))))))))))))))) in (let +TMP_292 \def (TMP_291 H6) in (TMP_292 H5))))))))))))))))))))))))))))))))))) +in (let TMP_309 \def (\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 +t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (u: T).(\lambda +(H3: (eq T (THead (Flat Cast) u t0) (THead (Bind Void) u1 t1))).(let TMP_294 +\def (Flat Cast) in (let TMP_295 \def (THead TMP_294 u t0) in (let TMP_296 +\def (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) in (let TMP_297 \def (Bind +Void) in (let TMP_298 \def (THead TMP_297 u1 t1) in (let H4 \def (eq_ind T +TMP_295 TMP_296 I TMP_298 H3) in (let TMP_301 \def (\lambda (u2: T).(\lambda +(t3: T).(let TMP_299 \def (Bind Void) in (let TMP_300 \def (THead TMP_299 u2 +t3) in (eq T t2 TMP_300))))) in (let TMP_302 \def (\lambda (u2: T).(\lambda +(_: T).(pr0 u1 u2))) in (let TMP_303 \def (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3))) in (let TMP_304 \def (ex3_2 T T TMP_301 TMP_302 TMP_303) in +(let TMP_305 \def (S O) in (let TMP_306 \def (lift TMP_305 O t2) in (let +TMP_307 \def (pr0 t1 TMP_306) in (let TMP_308 \def (or TMP_304 TMP_307) in +(False_ind TMP_308 H4))))))))))))))))))))) in (pr0_ind TMP_23 TMP_66 TMP_141 +TMP_163 TMP_193 TMP_213 TMP_293 TMP_309 y x H0))))))))))) in (insert_eq T +TMP_2 TMP_3 TMP_13 TMP_310 H))))))))).