component: pr0

[helm.git] / matita / matita / contribs / lambdadelta / basic_1 / pr0 / props.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma
index d7f69d691abb71849d095ce10739f316e6dd9c78..a99bdf288154c419401c6c4fb59921443c5c4c98 100644 (file)
--- 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 
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-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 
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-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 
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-(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))))))))).