From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Wed, 4 Mar 2015 22:02:45 +0000 (+0000)
Subject: components: nf2, sn3, ex2
X-Git-Tag: make_still_working~734
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=049d55c73d1746e15a40e89b17fd88b62f002d93;p=helm.git

components: nf2, sn3, ex2
---

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex2/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/ex2/defs.ma
index 35b5df73e..9db1318af 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/ex2/defs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ex2/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/C/defs.ma".
+include "basic_1/C/defs.ma".
 
 definition ex2_c:
  C
@@ -24,5 +24,6 @@ definition ex2_c:
 definition ex2_t:
  T
 \def
- THead (Flat Appl) (TSort O) (TSort O).
+ let TMP_1 \def (Flat Appl) in (let TMP_2 \def (TSort O) in (let TMP_3 \def 
+(TSort O) in (THead TMP_1 TMP_2 TMP_3))).
 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex2/props.ma b/matita/matita/contribs/lambdadelta/basic_1/ex2/props.ma
index 4dfa6a582..a6c71285d 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/ex2/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ex2/props.ma
@@ -14,146 +14,230 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/ex2/defs.ma".
+include "basic_1/ex2/defs.ma".
 
-include "Basic-1/nf2/defs.ma".
+include "basic_1/nf2/defs.ma".
 
-include "Basic-1/pr2/fwd.ma".
+include "basic_1/pr2/fwd.ma".
 
-include "Basic-1/arity/fwd.ma".
+include "basic_1/arity/fwd.ma".
 
 theorem ex2_nf2:
  nf2 ex2_c ex2_t
 \def
  \lambda (t2: T).(\lambda (H: (pr2 (CSort O) (THead (Flat Appl) (TSort O) 
-(TSort O)) t2)).(let H0 \def (pr2_gen_appl (CSort O) (TSort O) (TSort O) t2 
-H) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort 
-O) u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 (CSort O) (TSort O) t3)))) 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O) 
-(Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind b) y1 
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
-T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat 
-Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort 
-O) u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead (CSort O) (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat 
-Appl) (TSort O) (TSort O)) t2) (\lambda (H1: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 (CSort O) (TSort O) t3))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 (CSort O) (TSort O) t3))) (eq T (THead (Flat Appl) (TSort O) 
-(TSort O)) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t2 
-(THead (Flat Appl) x0 x1))).(\lambda (H3: (pr2 (CSort O) (TSort O) 
-x0)).(\lambda (H4: (pr2 (CSort O) (TSort O) x1)).(let H5 \def (eq_ind T x1 
-(\lambda (t: T).(eq T t2 (THead (Flat Appl) x0 t))) H2 (TSort O) 
-(pr2_gen_sort (CSort O) x1 O H4)) in (let H6 \def (eq_ind T x0 (\lambda (t: 
-T).(eq T t2 (THead (Flat Appl) t (TSort O)))) H5 (TSort O) (pr2_gen_sort 
-(CSort O) x0 O H3)) in (eq_ind_r T (THead (Flat Appl) (TSort O) (TSort O)) 
-(\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (refl_equal 
-T (THead (Flat Appl) (TSort O) (TSort O))) t2 H6)))))))) H1)) (\lambda (H1: 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O) 
-(Bind b) u) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))) (\lambda 
+(TSort O)) t2)).(let TMP_1 \def (CSort O) in (let TMP_2 \def (TSort O) in 
+(let TMP_3 \def (TSort O) in (let H0 \def (pr2_gen_appl TMP_1 TMP_2 TMP_3 t2 
+H) in (let TMP_6 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_4 \def (Flat 
+Appl) in (let TMP_5 \def (THead TMP_4 u2 t3) in (eq T t2 TMP_5))))) in (let 
+TMP_9 \def (\lambda (u2: T).(\lambda (_: T).(let TMP_7 \def (CSort O) in (let 
+TMP_8 \def (TSort O) in (pr2 TMP_7 TMP_8 u2))))) in (let TMP_12 \def (\lambda 
+(_: T).(\lambda (t3: T).(let TMP_10 \def (CSort O) in (let TMP_11 \def (TSort 
+O) in (pr2 TMP_10 TMP_11 t3))))) in (let TMP_13 \def (ex3_2 T T TMP_6 TMP_9 
+TMP_12) in (let TMP_17 \def (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(let TMP_14 \def (TSort O) in (let TMP_15 \def (Bind Abst) 
+in (let TMP_16 \def (THead TMP_15 y1 z1) in (eq T TMP_14 TMP_16)))))))) in 
+(let TMP_20 \def (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(let TMP_18 \def (Bind Abbr) in (let TMP_19 \def (THead TMP_18 u2 t3) 
+in (eq T t2 TMP_19))))))) in (let TMP_23 \def (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(let TMP_21 \def (CSort O) in (let TMP_22 
+\def (TSort O) in (pr2 TMP_21 TMP_22 u2))))))) in (let TMP_27 \def (\lambda 
 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: 
-B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) z1 t3))))))) (eq T 
-(THead (Flat Appl) (TSort O) (TSort O)) t2) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H2: (eq T (TSort O) (THead 
-(Bind Abst) x0 x1))).(\lambda (H3: (eq T t2 (THead (Bind Abbr) x2 
-x3))).(\lambda (H4: (pr2 (CSort O) (TSort O) x2)).(\lambda (_: ((\forall (b: 
-B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) x1 x3))))).(let H6 \def 
-(eq_ind T x2 (\lambda (t: T).(eq T t2 (THead (Bind Abbr) t x3))) H3 (TSort O) 
-(pr2_gen_sort (CSort O) x2 O H4)) in (eq_ind_r T (THead (Bind Abbr) (TSort O) 
-x3) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (let H7 
-\def (eq_ind T (TSort O) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 x1) H2) in 
-(False_ind (eq T (THead (Flat Appl) (TSort O) (TSort O)) (THead (Bind Abbr) 
-(TSort O) x3)) H7)) t2 H6)))))))))) H1)) (\lambda (H1: (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(TSort O) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort 
-O) (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: 
+B).(\forall (u: T).(let TMP_24 \def (CSort O) in (let TMP_25 \def (Bind b) in 
+(let TMP_26 \def (CHead TMP_24 TMP_25 u) in (pr2 TMP_26 z1 t3)))))))))) in 
+(let TMP_28 \def (ex4_4 T T T T TMP_17 TMP_20 TMP_23 TMP_27) in (let TMP_30 
+\def (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(let TMP_29 \def (eq B b Abst) in (not TMP_29)))))))) 
+in (let TMP_34 \def (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(let TMP_31 \def (TSort O) 
+in (let TMP_32 \def (Bind b) in (let TMP_33 \def (THead TMP_32 y1 z1) in (eq 
+T TMP_31 TMP_33)))))))))) in (let TMP_41 \def (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(let 
+TMP_35 \def (Bind b) in (let TMP_36 \def (Flat Appl) in (let TMP_37 \def (S 
+O) in (let TMP_38 \def (lift TMP_37 O u2) in (let TMP_39 \def (THead TMP_36 
+TMP_38 z2) in (let TMP_40 \def (THead TMP_35 y2 TMP_39) in (eq T t2 
+TMP_40))))))))))))) in (let TMP_44 \def (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(let 
+TMP_42 \def (CSort O) in (let TMP_43 \def (TSort O) in (pr2 TMP_42 TMP_43 
+u2))))))))) in (let TMP_46 \def (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(let TMP_45 \def (CSort 
+O) in (pr2 TMP_45 y1 y2)))))))) in (let TMP_50 \def (\lambda (b: B).(\lambda 
+(_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: 
+T).(let TMP_47 \def (CSort O) in (let TMP_48 \def (Bind b) in (let TMP_49 
+\def (CHead TMP_47 TMP_48 y2) in (pr2 TMP_49 z1 z2)))))))))) in (let TMP_51 
+\def (ex6_6 B T T T T T TMP_30 TMP_34 TMP_41 TMP_44 TMP_46 TMP_50) in (let 
+TMP_52 \def (Flat Appl) in (let TMP_53 \def (TSort O) in (let TMP_54 \def 
+(TSort O) in (let TMP_55 \def (THead TMP_52 TMP_53 TMP_54) in (let TMP_56 
+\def (eq T TMP_55 t2) in (let TMP_99 \def (\lambda (H1: (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 (CSort O) (TSort O) t3))))).(let TMP_59 \def (\lambda (u2: 
+T).(\lambda (t3: T).(let TMP_57 \def (Flat Appl) in (let TMP_58 \def (THead 
+TMP_57 u2 t3) in (eq T t2 TMP_58))))) in (let TMP_62 \def (\lambda (u2: 
+T).(\lambda (_: T).(let TMP_60 \def (CSort O) in (let TMP_61 \def (TSort O) 
+in (pr2 TMP_60 TMP_61 u2))))) in (let TMP_65 \def (\lambda (_: T).(\lambda 
+(t3: T).(let TMP_63 \def (CSort O) in (let TMP_64 \def (TSort O) in (pr2 
+TMP_63 TMP_64 t3))))) in (let TMP_66 \def (Flat Appl) in (let TMP_67 \def 
+(TSort O) in (let TMP_68 \def (TSort O) in (let TMP_69 \def (THead TMP_66 
+TMP_67 TMP_68) in (let TMP_70 \def (eq T TMP_69 t2) in (let TMP_98 \def 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t2 (THead (Flat Appl) 
+x0 x1))).(\lambda (H3: (pr2 (CSort O) (TSort O) x0)).(\lambda (H4: (pr2 
+(CSort O) (TSort O) x1)).(let TMP_73 \def (\lambda (t: T).(let TMP_71 \def 
+(Flat Appl) in (let TMP_72 \def (THead TMP_71 x0 t) in (eq T t2 TMP_72)))) in 
+(let TMP_74 \def (TSort O) in (let TMP_75 \def (CSort O) in (let TMP_76 \def 
+(pr2_gen_sort TMP_75 x1 O H4) in (let H5 \def (eq_ind T x1 TMP_73 H2 TMP_74 
+TMP_76) in (let TMP_80 \def (\lambda (t: T).(let TMP_77 \def (Flat Appl) in 
+(let TMP_78 \def (TSort O) in (let TMP_79 \def (THead TMP_77 t TMP_78) in (eq 
+T t2 TMP_79))))) in (let TMP_81 \def (TSort O) in (let TMP_82 \def (CSort O) 
+in (let TMP_83 \def (pr2_gen_sort TMP_82 x0 O H3) in (let H6 \def (eq_ind T 
+x0 TMP_80 H5 TMP_81 TMP_83) in (let TMP_84 \def (Flat Appl) in (let TMP_85 
+\def (TSort O) in (let TMP_86 \def (TSort O) in (let TMP_87 \def (THead 
+TMP_84 TMP_85 TMP_86) in (let TMP_92 \def (\lambda (t: T).(let TMP_88 \def 
+(Flat Appl) in (let TMP_89 \def (TSort O) in (let TMP_90 \def (TSort O) in 
+(let TMP_91 \def (THead TMP_88 TMP_89 TMP_90) in (eq T TMP_91 t)))))) in (let 
+TMP_93 \def (Flat Appl) in (let TMP_94 \def (TSort O) in (let TMP_95 \def 
+(TSort O) in (let TMP_96 \def (THead TMP_93 TMP_94 TMP_95) in (let TMP_97 
+\def (refl_equal T TMP_96) in (eq_ind_r T TMP_87 TMP_92 TMP_97 t2 
+H6)))))))))))))))))))))))))) in (ex3_2_ind T T TMP_59 TMP_62 TMP_65 TMP_70 
+TMP_98 H1))))))))))) in (let TMP_147 \def (\lambda (H1: (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort 
+O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O) 
+(Bind b) u) z1 t3))))))))).(let TMP_103 \def (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(let TMP_100 \def (TSort O) in (let 
+TMP_101 \def (Bind Abst) in (let TMP_102 \def (THead TMP_101 y1 z1) in (eq T 
+TMP_100 TMP_102)))))))) in (let TMP_106 \def (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t3: T).(let TMP_104 \def (Bind Abbr) in (let 
+TMP_105 \def (THead TMP_104 u2 t3) in (eq T t2 TMP_105))))))) in (let TMP_109 
+\def (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(let 
+TMP_107 \def (CSort O) in (let TMP_108 \def (TSort O) in (pr2 TMP_107 TMP_108 
+u2))))))) in (let TMP_113 \def (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(let TMP_110 \def (CSort 
+O) in (let TMP_111 \def (Bind b) in (let TMP_112 \def (CHead TMP_110 TMP_111 
+u) in (pr2 TMP_112 z1 t3)))))))))) in (let TMP_114 \def (Flat Appl) in (let 
+TMP_115 \def (TSort O) in (let TMP_116 \def (TSort O) in (let TMP_117 \def 
+(THead TMP_114 TMP_115 TMP_116) in (let TMP_118 \def (eq T TMP_117 t2) in 
+(let TMP_146 \def (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (H2: (eq T (TSort O) (THead (Bind Abst) x0 x1))).(\lambda 
+(H3: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (H4: (pr2 (CSort O) (TSort 
+O) x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O) 
+(Bind b) u) x1 x3))))).(let TMP_121 \def (\lambda (t: T).(let TMP_119 \def 
+(Bind Abbr) in (let TMP_120 \def (THead TMP_119 t x3) in (eq T t2 TMP_120)))) 
+in (let TMP_122 \def (TSort O) in (let TMP_123 \def (CSort O) in (let TMP_124 
+\def (pr2_gen_sort TMP_123 x2 O H4) in (let H6 \def (eq_ind T x2 TMP_121 H3 
+TMP_122 TMP_124) in (let TMP_125 \def (Bind Abbr) in (let TMP_126 \def (TSort 
+O) in (let TMP_127 \def (THead TMP_125 TMP_126 x3) in (let TMP_132 \def 
+(\lambda (t: T).(let TMP_128 \def (Flat Appl) in (let TMP_129 \def (TSort O) 
+in (let TMP_130 \def (TSort O) in (let TMP_131 \def (THead TMP_128 TMP_129 
+TMP_130) in (eq T TMP_131 t)))))) in (let TMP_133 \def (TSort O) in (let 
+TMP_134 \def (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow True | 
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) in (let 
+TMP_135 \def (Bind Abst) in (let TMP_136 \def (THead TMP_135 x0 x1) in (let 
+H7 \def (eq_ind T TMP_133 TMP_134 I TMP_136 H2) in (let TMP_137 \def (Flat 
+Appl) in (let TMP_138 \def (TSort O) in (let TMP_139 \def (TSort O) in (let 
+TMP_140 \def (THead TMP_137 TMP_138 TMP_139) in (let TMP_141 \def (Bind Abbr) 
+in (let TMP_142 \def (TSort O) in (let TMP_143 \def (THead TMP_141 TMP_142 
+x3) in (let TMP_144 \def (eq T TMP_140 TMP_143) in (let TMP_145 \def 
+(False_ind TMP_144 H7) in (eq_ind_r T TMP_127 TMP_132 TMP_145 t2 
+H6)))))))))))))))))))))))))))))))) in (ex4_4_ind T T T T TMP_103 TMP_106 
+TMP_109 TMP_113 TMP_118 TMP_146 H1)))))))))))) in (let TMP_215 \def (\lambda 
+(H1: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind b) y1 z1)))))))) (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S 
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead (CSort O) (Bind b) y2) z1 z2))))))))).(let TMP_149 \def (\lambda (b: 
 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort O) 
-(Bind b) y2) z1 z2))))))) (eq T (THead (Flat Appl) (TSort O) (TSort O)) t2) 
-(\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda 
-(x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H3: (eq 
-T (TSort O) (THead (Bind x0) x1 x2))).(\lambda (H4: (eq T t2 (THead (Bind x0) 
-x5 (THead (Flat Appl) (lift (S O) O x4) x3)))).(\lambda (H5: (pr2 (CSort O) 
-(TSort O) x4)).(\lambda (H6: (pr2 (CSort O) x1 x5)).(\lambda (_: (pr2 (CHead 
-(CSort O) (Bind x0) x5) x2 x3)).(let H_y \def (pr2_gen_csort x1 x5 O H6) in 
-(let H8 \def (eq_ind T x4 (\lambda (t: T).(eq T t2 (THead (Bind x0) x5 (THead 
-(Flat Appl) (lift (S O) O t) x3)))) H4 (TSort O) (pr2_gen_sort (CSort O) x4 O 
-H5)) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O 
-(TSort O)) x3)) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) 
-t)) (let H9 \def (eq_ind T (TSort O) (\lambda (ee: T).(match ee in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind x0) x1 
-x2) H3) in (False_ind (eq T (THead (Flat Appl) (TSort O) (TSort O)) (THead 
-(Bind x0) x5 (THead (Flat Appl) (lift (S O) O (TSort O)) x3))) H9)) t2 
-H8))))))))))))))) H1)) H0))).
-(* COMMENTS
-Initial nodes: 1939
-END *)
+(_: T).(let TMP_148 \def (eq B b Abst) in (not TMP_148)))))))) in (let 
+TMP_153 \def (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(let TMP_150 \def (TSort O) in (let 
+TMP_151 \def (Bind b) in (let TMP_152 \def (THead TMP_151 y1 z1) in (eq T 
+TMP_150 TMP_152)))))))))) in (let TMP_160 \def (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(let 
+TMP_154 \def (Bind b) in (let TMP_155 \def (Flat Appl) in (let TMP_156 \def 
+(S O) in (let TMP_157 \def (lift TMP_156 O u2) in (let TMP_158 \def (THead 
+TMP_155 TMP_157 z2) in (let TMP_159 \def (THead TMP_154 y2 TMP_158) in (eq T 
+t2 TMP_159))))))))))))) in (let TMP_163 \def (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(let 
+TMP_161 \def (CSort O) in (let TMP_162 \def (TSort O) in (pr2 TMP_161 TMP_162 
+u2))))))))) in (let TMP_165 \def (\lambda (_: B).(\lambda (y1: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(let TMP_164 \def 
+(CSort O) in (pr2 TMP_164 y1 y2)))))))) in (let TMP_169 \def (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(let TMP_166 \def (CSort O) in (let TMP_167 \def (Bind b) in (let 
+TMP_168 \def (CHead TMP_166 TMP_167 y2) in (pr2 TMP_168 z1 z2)))))))))) in 
+(let TMP_170 \def (Flat Appl) in (let TMP_171 \def (TSort O) in (let TMP_172 
+\def (TSort O) in (let TMP_173 \def (THead TMP_170 TMP_171 TMP_172) in (let 
+TMP_174 \def (eq T TMP_173 t2) in (let TMP_214 \def (\lambda (x0: B).(\lambda 
+(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H3: (eq T (TSort O) (THead 
+(Bind x0) x1 x2))).(\lambda (H4: (eq T t2 (THead (Bind x0) x5 (THead (Flat 
+Appl) (lift (S O) O x4) x3)))).(\lambda (H5: (pr2 (CSort O) (TSort O) 
+x4)).(\lambda (H6: (pr2 (CSort O) x1 x5)).(\lambda (_: (pr2 (CHead (CSort O) 
+(Bind x0) x5) x2 x3)).(let H_y \def (pr2_gen_csort x1 x5 O H6) in (let 
+TMP_181 \def (\lambda (t: T).(let TMP_175 \def (Bind x0) in (let TMP_176 \def 
+(Flat Appl) in (let TMP_177 \def (S O) in (let TMP_178 \def (lift TMP_177 O 
+t) in (let TMP_179 \def (THead TMP_176 TMP_178 x3) in (let TMP_180 \def 
+(THead TMP_175 x5 TMP_179) in (eq T t2 TMP_180)))))))) in (let TMP_182 \def 
+(TSort O) in (let TMP_183 \def (CSort O) in (let TMP_184 \def (pr2_gen_sort 
+TMP_183 x4 O H5) in (let H8 \def (eq_ind T x4 TMP_181 H4 TMP_182 TMP_184) in 
+(let TMP_185 \def (Bind x0) in (let TMP_186 \def (Flat Appl) in (let TMP_187 
+\def (S O) in (let TMP_188 \def (TSort O) in (let TMP_189 \def (lift TMP_187 
+O TMP_188) in (let TMP_190 \def (THead TMP_186 TMP_189 x3) in (let TMP_191 
+\def (THead TMP_185 x5 TMP_190) in (let TMP_196 \def (\lambda (t: T).(let 
+TMP_192 \def (Flat Appl) in (let TMP_193 \def (TSort O) in (let TMP_194 \def 
+(TSort O) in (let TMP_195 \def (THead TMP_192 TMP_193 TMP_194) in (eq T 
+TMP_195 t)))))) in (let TMP_197 \def (TSort O) in (let TMP_198 \def (\lambda 
+(ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow 
+False | (THead _ _ _) \Rightarrow False])) in (let TMP_199 \def (Bind x0) in 
+(let TMP_200 \def (THead TMP_199 x1 x2) in (let H9 \def (eq_ind T TMP_197 
+TMP_198 I TMP_200 H3) in (let TMP_201 \def (Flat Appl) in (let TMP_202 \def 
+(TSort O) in (let TMP_203 \def (TSort O) in (let TMP_204 \def (THead TMP_201 
+TMP_202 TMP_203) in (let TMP_205 \def (Bind x0) in (let TMP_206 \def (Flat 
+Appl) in (let TMP_207 \def (S O) in (let TMP_208 \def (TSort O) in (let 
+TMP_209 \def (lift TMP_207 O TMP_208) in (let TMP_210 \def (THead TMP_206 
+TMP_209 x3) in (let TMP_211 \def (THead TMP_205 x5 TMP_210) in (let TMP_212 
+\def (eq T TMP_204 TMP_211) in (let TMP_213 \def (False_ind TMP_212 H9) in 
+(eq_ind_r T TMP_191 TMP_196 TMP_213 t2 
+H8))))))))))))))))))))))))))))))))))))))))))))) in (ex6_6_ind B T T T T T 
+TMP_149 TMP_153 TMP_160 TMP_163 TMP_165 TMP_169 TMP_174 TMP_214 
+H1)))))))))))))) in (or3_ind TMP_13 TMP_28 TMP_51 TMP_56 TMP_99 TMP_147 
+TMP_215 H0)))))))))))))))))))))))))))))).
 
 theorem ex2_arity:
  \forall (g: G).(\forall (a: A).((arity g ex2_c ex2_t a) \to (\forall (P: 
 Prop).P)))
 \def
  \lambda (g: G).(\lambda (a: A).(\lambda (H: (arity g (CSort O) (THead (Flat 
-Appl) (TSort O) (TSort O)) a)).(\lambda (P: Prop).(let H0 \def 
-(arity_gen_appl g (CSort O) (TSort O) (TSort O) a H) in (ex2_ind A (\lambda 
-(a1: A).(arity g (CSort O) (TSort O) a1)) (\lambda (a1: A).(arity g (CSort O) 
-(TSort O) (AHead a1 a))) P (\lambda (x: A).(\lambda (_: (arity g (CSort O) 
-(TSort O) x)).(\lambda (H2: (arity g (CSort O) (TSort O) (AHead x a))).(let 
-H_x \def (leq_gen_head1 g x a (ASort O O) (arity_gen_sort g (CSort O) O 
-(AHead x a) H2)) in (let H3 \def H_x in (ex3_2_ind A A (\lambda (a3: 
-A).(\lambda (_: A).(leq g x a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a 
-a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O O) (AHead a3 a4)))) P 
-(\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g x x0)).(\lambda (_: 
-(leq g a x1)).(\lambda (H6: (eq A (ASort O O) (AHead x0 x1))).(let H7 \def 
-(eq_ind A (ASort O O) (\lambda (ee: A).(match ee in A return (\lambda (_: 
-A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow 
-False])) I (AHead x0 x1) H6) in (False_ind P H7))))))) H3)))))) H0))))).
-(* COMMENTS
-Initial nodes: 289
-END *)
+Appl) (TSort O) (TSort O)) a)).(\lambda (P: Prop).(let TMP_1 \def (CSort O) 
+in (let TMP_2 \def (TSort O) in (let TMP_3 \def (TSort O) in (let H0 \def 
+(arity_gen_appl g TMP_1 TMP_2 TMP_3 a H) in (let TMP_6 \def (\lambda (a1: 
+A).(let TMP_4 \def (CSort O) in (let TMP_5 \def (TSort O) in (arity g TMP_4 
+TMP_5 a1)))) in (let TMP_10 \def (\lambda (a1: A).(let TMP_7 \def (CSort O) 
+in (let TMP_8 \def (TSort O) in (let TMP_9 \def (AHead a1 a) in (arity g 
+TMP_7 TMP_8 TMP_9))))) in (let TMP_24 \def (\lambda (x: A).(\lambda (_: 
+(arity g (CSort O) (TSort O) x)).(\lambda (H2: (arity g (CSort O) (TSort O) 
+(AHead x a))).(let TMP_11 \def (ASort O O) in (let TMP_12 \def (CSort O) in 
+(let TMP_13 \def (AHead x a) in (let TMP_14 \def (arity_gen_sort g TMP_12 O 
+TMP_13 H2) in (let H_x \def (leq_gen_head1 g x a TMP_11 TMP_14) in (let H3 
+\def H_x in (let TMP_15 \def (\lambda (a3: A).(\lambda (_: A).(leq g x a3))) 
+in (let TMP_16 \def (\lambda (_: A).(\lambda (a4: A).(leq g a a4))) in (let 
+TMP_19 \def (\lambda (a3: A).(\lambda (a4: A).(let TMP_17 \def (ASort O O) in 
+(let TMP_18 \def (AHead a3 a4) in (eq A TMP_17 TMP_18))))) in (let TMP_23 
+\def (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g x x0)).(\lambda 
+(_: (leq g a x1)).(\lambda (H6: (eq A (ASort O O) (AHead x0 x1))).(let TMP_20 
+\def (ASort O O) in (let TMP_21 \def (\lambda (ee: A).(match ee with [(ASort 
+_ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) in (let TMP_22 \def 
+(AHead x0 x1) in (let H7 \def (eq_ind A TMP_20 TMP_21 I TMP_22 H6) in 
+(False_ind P H7)))))))))) in (ex3_2_ind A A TMP_15 TMP_16 TMP_19 P TMP_23 
+H3)))))))))))))) in (ex2_ind A TMP_6 TMP_10 P TMP_24 H0))))))))))).
 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/arity.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/arity.ma
index 98770d9e9..4ee874d1f 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/arity.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/arity.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/nf2/fwd.ma".
+include "basic_1/nf2/fwd.ma".
 
-include "Basic-1/arity/subst0.ma".
+include "basic_1/arity/subst0.ma".
 
 theorem arity_nf2_inv_all:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
@@ -120,70 +120,63 @@ t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
 i)))))))))) (\lambda (H6: (not (eq B Abst Abst))).(\lambda (_: (arity g 
 (CHead c0 (Bind Abst) u) t0 a2)).(\lambda (_: (nf2 c0 (THead (Bind Abst) u 
-t0))).(let H9 \def (match (H6 (refl_equal B Abst)) in False return (\lambda 
-(_: False).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead 
+t0))).(let H9 \def (match (H6 (refl_equal B Abst)) in False with []) in 
+H9)))) (\lambda (_: (not (eq B Void Abst))).(\lambda (H7: (arity g (CHead c0 
+(Bind Void) u) t0 a2)).(\lambda (H8: (nf2 c0 (THead (Bind Void) u t0))).(let 
+H9 \def (arity_gen_cvoid g (CHead c0 (Bind Void) u) t0 a2 H7 c0 u O 
+(getl_refl Void c0 u)) in (ex_ind T (\lambda (v: T).(eq T t0 (lift (S O) O 
+v))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind 
+Void) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 
+c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) 
+(ex nat (\lambda (n: nat).(eq T (THead (Bind Void) u t0) (TSort n)))) (ex3_2 
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind Void) u 
+t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i)))))) (\lambda (x: T).(\lambda (H10: (eq T t0 (lift (S O) O x))).(let H11 
+\def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead (Bind Void) u t1))) H8 
+(lift (S O) O x) H10) in (eq_ind_r T (lift (S O) O x) (\lambda (t1: T).(or3 
+(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Void) u t1) 
+(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) 
+(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat 
+(\lambda (n: nat).(eq T (THead (Bind Void) u t1) (TSort n)))) (ex3_2 TList 
+nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind Void) u t1) 
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i))))))) (nf2_gen_void c0 u x H11 (or3 (ex3_2 T T (\lambda (w: T).(\lambda 
+(u0: T).(eq T (THead (Bind Void) u (lift (S O) O x)) (THead (Bind Abst) w 
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
+(THead (Bind Void) u (lift (S O) O x)) (TSort n)))) (ex3_2 TList nat (\lambda 
+(ws: TList).(\lambda (i: nat).(eq T (THead (Bind Void) u (lift (S O) O x)) 
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i))))))) t0 H10)))) H9))))) b H0 H3 H5))))))))))))) (\lambda (c0: C).(\lambda 
+(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda 
+(_: (((nf2 c0 u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u 
+(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) 
+(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat 
+(\lambda (n: nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
+TList).(\lambda (i: nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda 
+(a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a2)).(\lambda (_: 
+(((nf2 (CHead c0 (Bind Abst) u) t0) \to (or3 (ex3_2 T T (\lambda (w: 
+T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: 
+T).(\lambda (_: T).(nf2 (CHead c0 (Bind Abst) u) w))) (\lambda (w: 
+T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind Abst) u) (Bind Abst) w) 
+u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat 
+(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef 
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 (CHead c0 (Bind Abst) u) 
+ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead c0 (Bind Abst) u) 
+(TLRef i))))))))).(\lambda (H4: (nf2 c0 (THead (Bind Abst) u t0))).(let H5 
+\def (nf2_gen_abst c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2 (CHead c0 (Bind 
+Abst) u) t0) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead 
 (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: 
 T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) 
 w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) u t0) (TSort 
 n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead 
 (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
 TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
-nat).(nf2 c0 (TLRef i))))))) with []) in H9)))) (\lambda (_: (not (eq B Void 
-Abst))).(\lambda (H7: (arity g (CHead c0 (Bind Void) u) t0 a2)).(\lambda (H8: 
-(nf2 c0 (THead (Bind Void) u t0))).(let H9 \def (arity_gen_cvoid g (CHead c0 
-(Bind Void) u) t0 a2 H7 c0 u O (getl_refl Void c0 u)) in (ex_ind T (\lambda 
-(v: T).(eq T t0 (lift (S O) O v))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda 
-(u0: T).(eq T (THead (Bind Void) u t0) (THead (Bind Abst) w u0)))) (\lambda 
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 
-(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind 
-Void) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
-nat).(eq T (THead (Bind Void) u t0) (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x: T).(\lambda 
-(H10: (eq T t0 (lift (S O) O x))).(let H11 \def (eq_ind T t0 (\lambda (t1: 
-T).(nf2 c0 (THead (Bind Void) u t1))) H8 (lift (S O) O x) H10) in (eq_ind_r T 
-(lift (S O) O x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda 
-(u0: T).(eq T (THead (Bind Void) u t1) (THead (Bind Abst) w u0)))) (\lambda 
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 
-(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind 
-Void) u t1) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
-nat).(eq T (THead (Bind Void) u t1) (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (nf2_gen_void c0 u x H11 
-(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Void) u 
-(lift (S O) O x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: 
-T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) 
-w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Void) u (lift (S O) O 
-x)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq 
-T (THead (Bind Void) u (lift (S O) O x)) (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) t0 H10)))) H9))))) b H0 H3 
-H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda 
-(_: (arity g c0 u (asucc g a1))).(\lambda (_: (((nf2 c0 u) \to (or3 (ex3_2 T 
-T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w u0)))) 
-(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: 
-T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u 
-(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T u 
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 
-(Bind Abst) u) t0 a2)).(\lambda (_: (((nf2 (CHead c0 (Bind Abst) u) t0) \to 
-(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) 
-w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 (Bind Abst) u) w))) 
-(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind Abst) u) (Bind 
-Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList 
-nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws 
-(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 (CHead c0 (Bind 
-Abst) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead c0 (Bind 
-Abst) u) (TLRef i))))))))).(\lambda (H4: (nf2 c0 (THead (Bind Abst) u 
-t0))).(let H5 \def (nf2_gen_abst c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2 
-(CHead c0 (Bind Abst) u) t0) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: 
-T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: 
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead 
-c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) u 
-t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq 
-T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
-TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
 nat).(nf2 c0 (TLRef i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 
 (CHead c0 (Bind Abst) u) t0)).(or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda 
 (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda 
@@ -303,21 +296,21 @@ A).(\lambda (a4: A).(eq A (ASort O x) (AHead a3 a4)))) (or3 (ex3_2 T T
 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
 i)))))) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 
 x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H16: (eq A (ASort O x) (AHead x0 
-x1))).(let H17 \def (eq_ind A (ASort O x) (\lambda (ee: A).(match ee in A 
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) 
-\Rightarrow False])) I (AHead x0 x1) H16) in (False_ind (or3 (ex3_2 T T 
-(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (TSort x)) (THead 
-(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda 
-(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda 
-(n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) (ex3_2 TList nat 
-(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x)) 
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i)))))) H17))))))) H13))) t0 H10))))) H9)) (\lambda (H9: (ex3_2 TList nat 
-(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef 
+x1))).(let H17 \def (eq_ind A (ASort O x) (\lambda (ee: A).(match ee with 
+[(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 
+x1) H16) in (False_ind (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T 
+(THead (Flat Appl) u (TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w: 
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead 
+c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u 
+(TSort x)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
+nat).(eq T (THead (Flat Appl) u (TSort x)) (THeads (Flat Appl) ws (TLRef 
 i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda 
-(ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) 
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) H17))))))) H13))) t0 H10))))) 
+H9)) (\lambda (H9: (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
+nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
+nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda (ws: 
+TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) 
 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
 TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w: 
 T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w 
@@ -490,7 +483,4 @@ x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
 i)))) x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H7 H8)) t0 
 H6)))))) H5)) H4))))))))))) c t a H))))).
-(* COMMENTS
-Initial nodes: 9193
-END *)
 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/dec.ma
index 33b652baf..42826c1f4 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/dec.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/dec.ma
@@ -14,187 +14,339 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/nf2/defs.ma".
+include "basic_1/nf2/defs.ma".
 
-include "Basic-1/pr2/clen.ma".
+include "basic_1/pr2/clen.ma".
 
-include "Basic-1/pr2/fwd.ma".
+include "basic_1/pr0/dec.ma".
 
-include "Basic-1/pr0/dec.ma".
-
-include "Basic-1/C/props.ma".
+include "basic_1/C/props.ma".
 
 theorem nf2_dec:
  \forall (c: C).(\forall (t1: T).(or (nf2 c t1) (ex2 T (\lambda (t2: T).((eq 
 T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)))))
 \def
- \lambda (c: C).(c_tail_ind (\lambda (c0: C).(\forall (t1: T).(or (\forall 
-(t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 
-t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))))) (\lambda 
-(n: nat).(\lambda (t1: T).(let H_x \def (nf0_dec t1) in (let H \def H_x in 
-(or_ind (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: 
-T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2))) 
-(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T 
-(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr2 (CSort n) t1 t2)))) (\lambda (H0: ((\forall (t2: T).((pr0 t1 t2) \to 
-(eq T t1 t2))))).(or_introl (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T 
-t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
-(\lambda (t2: T).(pr2 (CSort n) t1 t2))) (\lambda (t2: T).(\lambda (H1: (pr2 
-(CSort n) t1 t2)).(let H_y \def (pr2_gen_csort t1 t2 n H1) in (H0 t2 
-H_y)))))) (\lambda (H0: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall 
-(P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)))).(ex2_ind T (\lambda (t2: 
-T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)) 
-(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T 
-(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr2 (CSort n) t1 t2)))) (\lambda (x: T).(\lambda (H1: (((eq T t1 x) \to 
-(\forall (P: Prop).P)))).(\lambda (H2: (pr0 t1 x)).(or_intror (\forall (t2: 
-T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T 
-t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2))) 
-(ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
-(\lambda (t2: T).(pr2 (CSort n) t1 t2)) x H1 (pr2_free (CSort n) t1 x 
-H2)))))) H0)) H))))) (\lambda (c0: C).(\lambda (H: ((\forall (t1: T).(or 
+ \lambda (c: C).(let TMP_5 \def (\lambda (c0: C).(\forall (t1: T).(let TMP_1 
+\def (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) in (let TMP_2 \def 
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) in (let TMP_3 \def 
+(\lambda (t2: T).(pr2 c0 t1 t2)) in (let TMP_4 \def (ex2 T TMP_2 TMP_3) in 
+(or TMP_1 TMP_4))))))) in (let TMP_44 \def (\lambda (n: nat).(\lambda (t1: 
+T).(let H_x \def (nf0_dec t1) in (let H \def H_x in (let TMP_6 \def (\forall 
+(t2: T).((pr0 t1 t2) \to (eq T t1 t2))) in (let TMP_7 \def (\lambda (t2: 
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) in (let TMP_8 \def (\lambda (t2: 
+T).(pr0 t1 t2)) in (let TMP_9 \def (ex2 T TMP_7 TMP_8) in (let TMP_10 \def 
+(\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) in (let TMP_11 
+\def (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) in (let 
+TMP_13 \def (\lambda (t2: T).(let TMP_12 \def (CSort n) in (pr2 TMP_12 t1 
+t2))) in (let TMP_14 \def (ex2 T TMP_11 TMP_13) in (let TMP_15 \def (or 
+TMP_10 TMP_14) in (let TMP_22 \def (\lambda (H0: ((\forall (t2: T).((pr0 t1 
+t2) \to (eq T t1 t2))))).(let TMP_16 \def (\forall (t2: T).((pr2 (CSort n) t1 
+t2) \to (eq T t1 t2))) in (let TMP_17 \def (\lambda (t2: T).((eq T t1 t2) \to 
+(\forall (P: Prop).P))) in (let TMP_19 \def (\lambda (t2: T).(let TMP_18 \def 
+(CSort n) in (pr2 TMP_18 t1 t2))) in (let TMP_20 \def (ex2 T TMP_17 TMP_19) 
+in (let TMP_21 \def (\lambda (t2: T).(\lambda (H1: (pr2 (CSort n) t1 
+t2)).(let H_y \def (pr2_gen_csort t1 t2 n H1) in (H0 t2 H_y)))) in (or_introl 
+TMP_16 TMP_20 TMP_21))))))) in (let TMP_43 \def (\lambda (H0: (ex2 T (\lambda 
+(t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 
+t2)))).(let TMP_23 \def (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: 
+Prop).P))) in (let TMP_24 \def (\lambda (t2: T).(pr0 t1 t2)) in (let TMP_25 
+\def (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) in (let 
+TMP_26 \def (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) in 
+(let TMP_28 \def (\lambda (t2: T).(let TMP_27 \def (CSort n) in (pr2 TMP_27 
+t1 t2))) in (let TMP_29 \def (ex2 T TMP_26 TMP_28) in (let TMP_30 \def (or 
+TMP_25 TMP_29) in (let TMP_42 \def (\lambda (x: T).(\lambda (H1: (((eq T t1 
+x) \to (\forall (P: Prop).P)))).(\lambda (H2: (pr0 t1 x)).(let TMP_31 \def 
+(\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) in (let TMP_32 
+\def (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) in (let 
+TMP_34 \def (\lambda (t2: T).(let TMP_33 \def (CSort n) in (pr2 TMP_33 t1 
+t2))) in (let TMP_35 \def (ex2 T TMP_32 TMP_34) in (let TMP_36 \def (\lambda 
+(t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) in (let TMP_38 \def 
+(\lambda (t2: T).(let TMP_37 \def (CSort n) in (pr2 TMP_37 t1 t2))) in (let 
+TMP_39 \def (CSort n) in (let TMP_40 \def (pr2_free TMP_39 t1 x H2) in (let 
+TMP_41 \def (ex_intro2 T TMP_36 TMP_38 x H1 TMP_40) in (or_intror TMP_31 
+TMP_35 TMP_41))))))))))))) in (ex2_ind T TMP_23 TMP_24 TMP_30 TMP_42 
+H0)))))))))) in (or_ind TMP_6 TMP_9 TMP_15 TMP_22 TMP_43 H)))))))))))))))) in 
+(let TMP_404 \def (\lambda (c0: C).(\lambda (H: ((\forall (t1: T).(or 
 (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: 
 T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 
 t2))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (t1: T).(let H_x \def (H 
-t1) in (let H0 \def H_x in (or_ind (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T 
-t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
-(\lambda (t2: T).(pr2 c0 t1 t2))) (or (\forall (t2: T).((pr2 (CTail k t c0) 
-t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall 
-(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (H1: 
-((\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))))).(K_ind (\lambda (k0: 
-K).(or (\forall (t2: T).((pr2 (CTail k0 t c0) t1 t2) \to (eq T t1 t2))) (ex2 
-T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr2 (CTail k0 t c0) t1 t2))))) (\lambda (b: B).(B_ind (\lambda (b0: 
-B).(or (\forall (t2: T).((pr2 (CTail (Bind b0) t c0) t1 t2) \to (eq T t1 
-t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
-(\lambda (t2: T).(pr2 (CTail (Bind b0) t c0) t1 t2))))) (let H_x0 \def 
-(dnf_dec t t1 (clen c0)) in (let H2 \def H_x0 in (ex_ind T (\lambda (v: 
-T).(or (subst0 (clen c0) t t1 (lift (S O) (clen c0) v)) (eq T t1 (lift (S O) 
-(clen c0) v)))) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) 
-\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda 
-(x: T).(\lambda (H3: (or (subst0 (clen c0) t t1 (lift (S O) (clen c0) x)) (eq 
-T t1 (lift (S O) (clen c0) x)))).(or_ind (subst0 (clen c0) t t1 (lift (S O) 
-(clen c0) x)) (eq T t1 (lift (S O) (clen c0) x)) (or (\forall (t2: T).((pr2 
-(CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: 
-T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail 
-(Bind Abbr) t c0) t1 t2)))) (\lambda (H4: (subst0 (clen c0) t t1 (lift (S O) 
-(clen c0) x))).(let H_x1 \def (getl_ctail_clen Abbr t c0) in (let H5 \def 
-H_x1 in (ex_ind nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind Abbr) t 
-c0) (CHead (CSort n) (Bind Abbr) t))) (or (\forall (t2: T).((pr2 (CTail (Bind 
-Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) 
-\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 
-t2)))) (\lambda (x0: nat).(\lambda (H6: (getl (clen c0) (CTail (Bind Abbr) t 
-c0) (CHead (CSort x0) (Bind Abbr) t))).(or_intror (\forall (t2: T).((pr2 
-(CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: 
-T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail 
-(Bind Abbr) t c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to 
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 
-t2)) (lift (S O) (clen c0) x) (\lambda (H7: (eq T t1 (lift (S O) (clen c0) 
-x))).(\lambda (P: Prop).(let H8 \def (eq_ind T t1 (\lambda (t0: T).(subst0 
-(clen c0) t t0 (lift (S O) (clen c0) x))) H4 (lift (S O) (clen c0) x) H7) in 
-(subst0_gen_lift_false x t (lift (S O) (clen c0) x) (S O) (clen c0) (clen c0) 
-(le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) (\lambda (n: nat).(lt 
-(clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen c0) (S O)) (plus_sym 
-(clen c0) (S O))) H8 P)))) (pr2_delta (CTail (Bind Abbr) t c0) (CSort x0) t 
-(clen c0) H6 t1 t1 (pr0_refl t1) (lift (S O) (clen c0) x) H4))))) H5)))) 
-(\lambda (H4: (eq T t1 (lift (S O) (clen c0) x))).(let H5 \def (eq_ind T t1 
-(\lambda (t0: T).(\forall (t2: T).((pr2 c0 t0 t2) \to (eq T t0 t2)))) H1 
-(lift (S O) (clen c0) x) H4) in (eq_ind_r T (lift (S O) (clen c0) x) (\lambda 
-(t0: T).(or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t0 t2) \to (eq T 
-t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) 
-(\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t0 t2))))) (or_introl (\forall 
-(t2: T).((pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2) \to (eq T 
-(lift (S O) (clen c0) x) t2))) (ex2 T (\lambda (t2: T).((eq T (lift (S O) 
-(clen c0) x) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail 
-(Bind Abbr) t c0) (lift (S O) (clen c0) x) t2))) (\lambda (t2: T).(\lambda 
-(H6: (pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2)).(let H_x1 
-\def (pr2_gen_ctail (Bind Abbr) c0 t (lift (S O) (clen c0) x) t2 H6) in (let 
-H7 \def H_x1 in (or_ind (pr2 c0 (lift (S O) (clen c0) x) t2) (ex3 T (\lambda 
-(_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) 
-(clen c0) x) t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T (lift 
-(S O) (clen c0) x) t2) (\lambda (H8: (pr2 c0 (lift (S O) (clen c0) x) 
-t2)).(H5 t2 H8)) (\lambda (H8: (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind 
-Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda (t0: 
-T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Bind Abbr) 
-(Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda 
-(t0: T).(subst0 (clen c0) t t0 t2)) (eq T (lift (S O) (clen c0) x) t2) 
-(\lambda (x0: T).(\lambda (_: (eq K (Bind Abbr) (Bind Abbr))).(\lambda (H10: 
-(pr0 (lift (S O) (clen c0) x) x0)).(\lambda (H11: (subst0 (clen c0) t x0 
-t2)).(ex2_ind T (\lambda (t3: T).(eq T x0 (lift (S O) (clen c0) t3))) 
-(\lambda (t3: T).(pr0 x t3)) (eq T (lift (S O) (clen c0) x) t2) (\lambda (x1: 
-T).(\lambda (H12: (eq T x0 (lift (S O) (clen c0) x1))).(\lambda (_: (pr0 x 
-x1)).(let H14 \def (eq_ind T x0 (\lambda (t0: T).(subst0 (clen c0) t t0 t2)) 
-H11 (lift (S O) (clen c0) x1) H12) in (subst0_gen_lift_false x1 t t2 (S O) 
-(clen c0) (clen c0) (le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) 
-(\lambda (n: nat).(lt (clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen 
-c0) (S O)) (plus_sym (clen c0) (S O))) H14 (eq T (lift (S O) (clen c0) x) 
-t2)))))) (pr0_gen_lift x x0 (S O) (clen c0) H10)))))) H8)) H7)))))) t1 H4))) 
-H3))) H2))) (or_introl (\forall (t2: T).((pr2 (CTail (Bind Abst) t c0) t1 t2) 
-\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abst) t c0) t1 t2))) (\lambda 
-(t2: T).(\lambda (H2: (pr2 (CTail (Bind Abst) t c0) t1 t2)).(let H_x0 \def 
-(pr2_gen_ctail (Bind Abst) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind 
-(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) 
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) 
-(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T 
-(\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq 
-K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: 
-(eq K (Bind Abst) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: 
-(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Abst) (\lambda (ee: 
-K).(match ee in K return (\lambda (_: K).Prop) with [(Bind b0) \Rightarrow 
-(match b0 in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | 
-Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow 
-False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3)))))) 
-(or_introl (\forall (t2: T).((pr2 (CTail (Bind Void) t c0) t1 t2) \to (eq T 
-t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
-(\lambda (t2: T).(pr2 (CTail (Bind Void) t c0) t1 t2))) (\lambda (t2: 
-T).(\lambda (H2: (pr2 (CTail (Bind Void) t c0) t1 t2)).(let H_x0 \def 
-(pr2_gen_ctail (Bind Void) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind 
-(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) 
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) 
-(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T 
-(\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq 
-K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: 
-(eq K (Bind Void) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: 
-(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Void) (\lambda (ee: 
-K).(match ee in K return (\lambda (_: K).Prop) with [(Bind b0) \Rightarrow 
-(match b0 in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | 
-Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow 
-False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3)))))) 
-b)) (\lambda (f: F).(or_introl (\forall (t2: T).((pr2 (CTail (Flat f) t c0) 
-t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall 
-(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Flat f) t c0) t1 t2))) (\lambda 
-(t2: T).(\lambda (H2: (pr2 (CTail (Flat f) t c0) t1 t2)).(let H_x0 \def 
-(pr2_gen_ctail (Flat f) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind (pr2 
-c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: 
-T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T t1 t2) 
-(\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T (\lambda (_: 
-T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Flat f) 
-(Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen 
-c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: (eq K (Flat f) 
-(Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: (subst0 (clen c0) t x0 
-t2)).(let H8 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return 
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3))))))) 
-k)) (\lambda (H1: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))).(ex2_ind T (\lambda (t2: 
-T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)) 
-(or (\forall (t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T 
+t1) in (let H0 \def H_x in (let TMP_45 \def (\forall (t2: T).((pr2 c0 t1 t2) 
+\to (eq T t1 t2))) in (let TMP_46 \def (\lambda (t2: T).((eq T t1 t2) \to 
+(\forall (P: Prop).P))) in (let TMP_47 \def (\lambda (t2: T).(pr2 c0 t1 t2)) 
+in (let TMP_48 \def (ex2 T TMP_46 TMP_47) in (let TMP_49 \def (\forall (t2: 
+T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) in (let TMP_50 \def 
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) in (let TMP_52 
+\def (\lambda (t2: T).(let TMP_51 \def (CTail k t c0) in (pr2 TMP_51 t1 t2))) 
+in (let TMP_53 \def (ex2 T TMP_50 TMP_52) in (let TMP_54 \def (or TMP_49 
+TMP_53) in (let TMP_383 \def (\lambda (H1: ((\forall (t2: T).((pr2 c0 t1 t2) 
+\to (eq T t1 t2))))).(let TMP_60 \def (\lambda (k0: K).(let TMP_55 \def 
+(\forall (t2: T).((pr2 (CTail k0 t c0) t1 t2) \to (eq T t1 t2))) in (let 
+TMP_56 \def (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) in 
+(let TMP_58 \def (\lambda (t2: T).(let TMP_57 \def (CTail k0 t c0) in (pr2 
+TMP_57 t1 t2))) in (let TMP_59 \def (ex2 T TMP_56 TMP_58) in (or TMP_55 
+TMP_59)))))) in (let TMP_350 \def (\lambda (b: B).(let TMP_67 \def (\lambda 
+(b0: B).(let TMP_61 \def (\forall (t2: T).((pr2 (CTail (Bind b0) t c0) t1 t2) 
+\to (eq T t1 t2))) in (let TMP_62 \def (\lambda (t2: T).((eq T t1 t2) \to 
+(\forall (P: Prop).P))) in (let TMP_65 \def (\lambda (t2: T).(let TMP_63 \def 
+(Bind b0) in (let TMP_64 \def (CTail TMP_63 t c0) in (pr2 TMP_64 t1 t2)))) in 
+(let TMP_66 \def (ex2 T TMP_62 TMP_65) in (or TMP_61 TMP_66)))))) in (let 
+TMP_68 \def (clen c0) in (let H_x0 \def (dnf_dec t t1 TMP_68) in (let H2 \def 
+H_x0 in (let TMP_78 \def (\lambda (v: T).(let TMP_69 \def (clen c0) in (let 
+TMP_70 \def (S O) in (let TMP_71 \def (clen c0) in (let TMP_72 \def (lift 
+TMP_70 TMP_71 v) in (let TMP_73 \def (subst0 TMP_69 t t1 TMP_72) in (let 
+TMP_74 \def (S O) in (let TMP_75 \def (clen c0) in (let TMP_76 \def (lift 
+TMP_74 TMP_75 v) in (let TMP_77 \def (eq T t1 TMP_76) in (or TMP_73 
+TMP_77))))))))))) in (let TMP_79 \def (\forall (t2: T).((pr2 (CTail (Bind 
+Abbr) t c0) t1 t2) \to (eq T t1 t2))) in (let TMP_80 \def (\lambda (t2: 
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) in (let TMP_83 \def (\lambda 
+(t2: T).(let TMP_81 \def (Bind Abbr) in (let TMP_82 \def (CTail TMP_81 t c0) 
+in (pr2 TMP_82 t1 t2)))) in (let TMP_84 \def (ex2 T TMP_80 TMP_83) in (let 
+TMP_85 \def (or TMP_79 TMP_84) in (let TMP_284 \def (\lambda (x: T).(\lambda 
+(H3: (or (subst0 (clen c0) t t1 (lift (S O) (clen c0) x)) (eq T t1 (lift (S 
+O) (clen c0) x)))).(let TMP_86 \def (clen c0) in (let TMP_87 \def (S O) in 
+(let TMP_88 \def (clen c0) in (let TMP_89 \def (lift TMP_87 TMP_88 x) in (let 
+TMP_90 \def (subst0 TMP_86 t t1 TMP_89) in (let TMP_91 \def (S O) in (let 
+TMP_92 \def (clen c0) in (let TMP_93 \def (lift TMP_91 TMP_92 x) in (let 
+TMP_94 \def (eq T t1 TMP_93) in (let TMP_95 \def (\forall (t2: T).((pr2 
+(CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) in (let TMP_96 \def 
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) in (let TMP_99 
+\def (\lambda (t2: T).(let TMP_97 \def (Bind Abbr) in (let TMP_98 \def (CTail 
+TMP_97 t c0) in (pr2 TMP_98 t1 t2)))) in (let TMP_100 \def (ex2 T TMP_96 
+TMP_99) in (let TMP_101 \def (or TMP_95 TMP_100) in (let TMP_173 \def 
+(\lambda (H4: (subst0 (clen c0) t t1 (lift (S O) (clen c0) x))).(let H_x1 
+\def (getl_ctail_clen Abbr t c0) in (let H5 \def H_x1 in (let TMP_108 \def 
+(\lambda (n: nat).(let TMP_102 \def (clen c0) in (let TMP_103 \def (Bind 
+Abbr) in (let TMP_104 \def (CTail TMP_103 t c0) in (let TMP_105 \def (CSort 
+n) in (let TMP_106 \def (Bind Abbr) in (let TMP_107 \def (CHead TMP_105 
+TMP_106 t) in (getl TMP_102 TMP_104 TMP_107)))))))) in (let TMP_109 \def 
+(\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) in 
+(let TMP_110 \def (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
+in (let TMP_113 \def (\lambda (t2: T).(let TMP_111 \def (Bind Abbr) in (let 
+TMP_112 \def (CTail TMP_111 t c0) in (pr2 TMP_112 t1 t2)))) in (let TMP_114 
+\def (ex2 T TMP_110 TMP_113) in (let TMP_115 \def (or TMP_109 TMP_114) in 
+(let TMP_172 \def (\lambda (x0: nat).(\lambda (H6: (getl (clen c0) (CTail 
+(Bind Abbr) t c0) (CHead (CSort x0) (Bind Abbr) t))).(let TMP_116 \def 
+(\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) in 
+(let TMP_117 \def (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
+in (let TMP_120 \def (\lambda (t2: T).(let TMP_118 \def (Bind Abbr) in (let 
+TMP_119 \def (CTail TMP_118 t c0) in (pr2 TMP_119 t1 t2)))) in (let TMP_121 
+\def (ex2 T TMP_117 TMP_120) in (let TMP_122 \def (\lambda (t2: T).((eq T t1 
+t2) \to (\forall (P: Prop).P))) in (let TMP_125 \def (\lambda (t2: T).(let 
+TMP_123 \def (Bind Abbr) in (let TMP_124 \def (CTail TMP_123 t c0) in (pr2 
+TMP_124 t1 t2)))) in (let TMP_126 \def (S O) in (let TMP_127 \def (clen c0) 
+in (let TMP_128 \def (lift TMP_126 TMP_127 x) in (let TMP_161 \def (\lambda 
+(H7: (eq T t1 (lift (S O) (clen c0) x))).(\lambda (P: Prop).(let TMP_133 \def 
+(\lambda (t0: T).(let TMP_129 \def (clen c0) in (let TMP_130 \def (S O) in 
+(let TMP_131 \def (clen c0) in (let TMP_132 \def (lift TMP_130 TMP_131 x) in 
+(subst0 TMP_129 t t0 TMP_132)))))) in (let TMP_134 \def (S O) in (let TMP_135 
+\def (clen c0) in (let TMP_136 \def (lift TMP_134 TMP_135 x) in (let H8 \def 
+(eq_ind T t1 TMP_133 H4 TMP_136 H7) in (let TMP_137 \def (S O) in (let 
+TMP_138 \def (clen c0) in (let TMP_139 \def (lift TMP_137 TMP_138 x) in (let 
+TMP_140 \def (S O) in (let TMP_141 \def (clen c0) in (let TMP_142 \def (clen 
+c0) in (let TMP_143 \def (clen c0) in (let TMP_144 \def (le_n TMP_143) in 
+(let TMP_145 \def (S O) in (let TMP_146 \def (clen c0) in (let TMP_147 \def 
+(plus TMP_145 TMP_146) in (let TMP_149 \def (\lambda (n: nat).(let TMP_148 
+\def (clen c0) in (lt TMP_148 n))) in (let TMP_150 \def (S O) in (let TMP_151 
+\def (clen c0) in (let TMP_152 \def (plus TMP_150 TMP_151) in (let TMP_153 
+\def (le_n TMP_152) in (let TMP_154 \def (clen c0) in (let TMP_155 \def (S O) 
+in (let TMP_156 \def (plus TMP_154 TMP_155) in (let TMP_157 \def (clen c0) in 
+(let TMP_158 \def (S O) in (let TMP_159 \def (plus_sym TMP_157 TMP_158) in 
+(let TMP_160 \def (eq_ind_r nat TMP_147 TMP_149 TMP_153 TMP_156 TMP_159) in 
+(subst0_gen_lift_false x t TMP_139 TMP_140 TMP_141 TMP_142 TMP_144 TMP_160 H8 
+P))))))))))))))))))))))))))))))) in (let TMP_162 \def (Bind Abbr) in (let 
+TMP_163 \def (CTail TMP_162 t c0) in (let TMP_164 \def (CSort x0) in (let 
+TMP_165 \def (clen c0) in (let TMP_166 \def (pr0_refl t1) in (let TMP_167 
+\def (S O) in (let TMP_168 \def (clen c0) in (let TMP_169 \def (lift TMP_167 
+TMP_168 x) in (let TMP_170 \def (pr2_delta TMP_163 TMP_164 t TMP_165 H6 t1 t1 
+TMP_166 TMP_169 H4) in (let TMP_171 \def (ex_intro2 T TMP_122 TMP_125 TMP_128 
+TMP_161 TMP_170) in (or_intror TMP_116 TMP_121 TMP_171))))))))))))))))))))))) 
+in (ex_ind nat TMP_108 TMP_115 TMP_172 H5))))))))))) in (let TMP_283 \def 
+(\lambda (H4: (eq T t1 (lift (S O) (clen c0) x))).(let TMP_174 \def (\lambda 
+(t0: T).(\forall (t2: T).((pr2 c0 t0 t2) \to (eq T t0 t2)))) in (let TMP_175 
+\def (S O) in (let TMP_176 \def (clen c0) in (let TMP_177 \def (lift TMP_175 
+TMP_176 x) in (let H5 \def (eq_ind T t1 TMP_174 H1 TMP_177 H4) in (let 
+TMP_178 \def (S O) in (let TMP_179 \def (clen c0) in (let TMP_180 \def (lift 
+TMP_178 TMP_179 x) in (let TMP_187 \def (\lambda (t0: T).(let TMP_181 \def 
+(\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t0 t2) \to (eq T t0 t2))) in 
+(let TMP_182 \def (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) 
+in (let TMP_185 \def (\lambda (t2: T).(let TMP_183 \def (Bind Abbr) in (let 
+TMP_184 \def (CTail TMP_183 t c0) in (pr2 TMP_184 t0 t2)))) in (let TMP_186 
+\def (ex2 T TMP_182 TMP_185) in (or TMP_181 TMP_186)))))) in (let TMP_191 
+\def (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) 
+t2) \to (let TMP_188 \def (S O) in (let TMP_189 \def (clen c0) in (let 
+TMP_190 \def (lift TMP_188 TMP_189 x) in (eq T TMP_190 t2)))))) in (let 
+TMP_192 \def (\lambda (t2: T).((eq T (lift (S O) (clen c0) x) t2) \to 
+(\forall (P: Prop).P))) in (let TMP_198 \def (\lambda (t2: T).(let TMP_193 
+\def (Bind Abbr) in (let TMP_194 \def (CTail TMP_193 t c0) in (let TMP_195 
+\def (S O) in (let TMP_196 \def (clen c0) in (let TMP_197 \def (lift TMP_195 
+TMP_196 x) in (pr2 TMP_194 TMP_197 t2))))))) in (let TMP_199 \def (ex2 T 
+TMP_192 TMP_198) in (let TMP_281 \def (\lambda (t2: T).(\lambda (H6: (pr2 
+(CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2)).(let TMP_200 \def 
+(Bind Abbr) in (let TMP_201 \def (S O) in (let TMP_202 \def (clen c0) in (let 
+TMP_203 \def (lift TMP_201 TMP_202 x) in (let H_x1 \def (pr2_gen_ctail 
+TMP_200 c0 t TMP_203 t2 H6) in (let H7 \def H_x1 in (let TMP_204 \def (S O) 
+in (let TMP_205 \def (clen c0) in (let TMP_206 \def (lift TMP_204 TMP_205 x) 
+in (let TMP_207 \def (pr2 c0 TMP_206 t2) in (let TMP_210 \def (\lambda (_: 
+T).(let TMP_208 \def (Bind Abbr) in (let TMP_209 \def (Bind Abbr) in (eq K 
+TMP_208 TMP_209)))) in (let TMP_214 \def (\lambda (t0: T).(let TMP_211 \def 
+(S O) in (let TMP_212 \def (clen c0) in (let TMP_213 \def (lift TMP_211 
+TMP_212 x) in (pr0 TMP_213 t0))))) in (let TMP_216 \def (\lambda (t0: T).(let 
+TMP_215 \def (clen c0) in (subst0 TMP_215 t t0 t2))) in (let TMP_217 \def 
+(ex3 T TMP_210 TMP_214 TMP_216) in (let TMP_218 \def (S O) in (let TMP_219 
+\def (clen c0) in (let TMP_220 \def (lift TMP_218 TMP_219 x) in (let TMP_221 
+\def (eq T TMP_220 t2) in (let TMP_222 \def (\lambda (H8: (pr2 c0 (lift (S O) 
+(clen c0) x) t2)).(H5 t2 H8)) in (let TMP_280 \def (\lambda (H8: (ex3 T 
+(\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 (lift 
+(S O) (clen c0) x) t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(let 
+TMP_225 \def (\lambda (_: T).(let TMP_223 \def (Bind Abbr) in (let TMP_224 
+\def (Bind Abbr) in (eq K TMP_223 TMP_224)))) in (let TMP_229 \def (\lambda 
+(t0: T).(let TMP_226 \def (S O) in (let TMP_227 \def (clen c0) in (let 
+TMP_228 \def (lift TMP_226 TMP_227 x) in (pr0 TMP_228 t0))))) in (let TMP_231 
+\def (\lambda (t0: T).(let TMP_230 \def (clen c0) in (subst0 TMP_230 t t0 
+t2))) in (let TMP_232 \def (S O) in (let TMP_233 \def (clen c0) in (let 
+TMP_234 \def (lift TMP_232 TMP_233 x) in (let TMP_235 \def (eq T TMP_234 t2) 
+in (let TMP_279 \def (\lambda (x0: T).(\lambda (_: (eq K (Bind Abbr) (Bind 
+Abbr))).(\lambda (H10: (pr0 (lift (S O) (clen c0) x) x0)).(\lambda (H11: 
+(subst0 (clen c0) t x0 t2)).(let TMP_239 \def (\lambda (t3: T).(let TMP_236 
+\def (S O) in (let TMP_237 \def (clen c0) in (let TMP_238 \def (lift TMP_236 
+TMP_237 t3) in (eq T x0 TMP_238))))) in (let TMP_240 \def (\lambda (t3: 
+T).(pr0 x t3)) in (let TMP_241 \def (S O) in (let TMP_242 \def (clen c0) in 
+(let TMP_243 \def (lift TMP_241 TMP_242 x) in (let TMP_244 \def (eq T TMP_243 
+t2) in (let TMP_275 \def (\lambda (x1: T).(\lambda (H12: (eq T x0 (lift (S O) 
+(clen c0) x1))).(\lambda (_: (pr0 x x1)).(let TMP_246 \def (\lambda (t0: 
+T).(let TMP_245 \def (clen c0) in (subst0 TMP_245 t t0 t2))) in (let TMP_247 
+\def (S O) in (let TMP_248 \def (clen c0) in (let TMP_249 \def (lift TMP_247 
+TMP_248 x1) in (let H14 \def (eq_ind T x0 TMP_246 H11 TMP_249 H12) in (let 
+TMP_250 \def (S O) in (let TMP_251 \def (clen c0) in (let TMP_252 \def (clen 
+c0) in (let TMP_253 \def (clen c0) in (let TMP_254 \def (le_n TMP_253) in 
+(let TMP_255 \def (S O) in (let TMP_256 \def (clen c0) in (let TMP_257 \def 
+(plus TMP_255 TMP_256) in (let TMP_259 \def (\lambda (n: nat).(let TMP_258 
+\def (clen c0) in (lt TMP_258 n))) in (let TMP_260 \def (S O) in (let TMP_261 
+\def (clen c0) in (let TMP_262 \def (plus TMP_260 TMP_261) in (let TMP_263 
+\def (le_n TMP_262) in (let TMP_264 \def (clen c0) in (let TMP_265 \def (S O) 
+in (let TMP_266 \def (plus TMP_264 TMP_265) in (let TMP_267 \def (clen c0) in 
+(let TMP_268 \def (S O) in (let TMP_269 \def (plus_sym TMP_267 TMP_268) in 
+(let TMP_270 \def (eq_ind_r nat TMP_257 TMP_259 TMP_263 TMP_266 TMP_269) in 
+(let TMP_271 \def (S O) in (let TMP_272 \def (clen c0) in (let TMP_273 \def 
+(lift TMP_271 TMP_272 x) in (let TMP_274 \def (eq T TMP_273 t2) in 
+(subst0_gen_lift_false x1 t t2 TMP_250 TMP_251 TMP_252 TMP_254 TMP_270 H14 
+TMP_274))))))))))))))))))))))))))))))))) in (let TMP_276 \def (S O) in (let 
+TMP_277 \def (clen c0) in (let TMP_278 \def (pr0_gen_lift x x0 TMP_276 
+TMP_277 H10) in (ex2_ind T TMP_239 TMP_240 TMP_244 TMP_275 
+TMP_278))))))))))))))) in (ex3_ind T TMP_225 TMP_229 TMP_231 TMP_235 TMP_279 
+H8)))))))))) in (or_ind TMP_207 TMP_217 TMP_221 TMP_222 TMP_280 
+H7))))))))))))))))))))))) in (let TMP_282 \def (or_introl TMP_191 TMP_199 
+TMP_281) in (eq_ind_r T TMP_180 TMP_187 TMP_282 t1 H4))))))))))))))))) in 
+(or_ind TMP_90 TMP_94 TMP_101 TMP_173 TMP_283 H3))))))))))))))))))) in (let 
+TMP_285 \def (ex_ind T TMP_78 TMP_85 TMP_284 H2) in (let TMP_286 \def 
+(\forall (t2: T).((pr2 (CTail (Bind Abst) t c0) t1 t2) \to (eq T t1 t2))) in 
+(let TMP_287 \def (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
+in (let TMP_290 \def (\lambda (t2: T).(let TMP_288 \def (Bind Abst) in (let 
+TMP_289 \def (CTail TMP_288 t c0) in (pr2 TMP_289 t1 t2)))) in (let TMP_291 
+\def (ex2 T TMP_287 TMP_290) in (let TMP_316 \def (\lambda (t2: T).(\lambda 
+(H2: (pr2 (CTail (Bind Abst) t c0) t1 t2)).(let TMP_292 \def (Bind Abst) in 
+(let H_x0 \def (pr2_gen_ctail TMP_292 c0 t t1 t2 H2) in (let H3 \def H_x0 in 
+(let TMP_293 \def (pr2 c0 t1 t2) in (let TMP_296 \def (\lambda (_: T).(let 
+TMP_294 \def (Bind Abst) in (let TMP_295 \def (Bind Abbr) in (eq K TMP_294 
+TMP_295)))) in (let TMP_297 \def (\lambda (t0: T).(pr0 t1 t0)) in (let 
+TMP_299 \def (\lambda (t0: T).(let TMP_298 \def (clen c0) in (subst0 TMP_298 
+t t0 t2))) in (let TMP_300 \def (ex3 T TMP_296 TMP_297 TMP_299) in (let 
+TMP_301 \def (eq T t1 t2) in (let TMP_302 \def (\lambda (H4: (pr2 c0 t1 
+t2)).(H1 t2 H4)) in (let TMP_315 \def (\lambda (H4: (ex3 T (\lambda (_: 
+T).(eq K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda 
+(t0: T).(subst0 (clen c0) t t0 t2)))).(let TMP_305 \def (\lambda (_: T).(let 
+TMP_303 \def (Bind Abst) in (let TMP_304 \def (Bind Abbr) in (eq K TMP_303 
+TMP_304)))) in (let TMP_306 \def (\lambda (t0: T).(pr0 t1 t0)) in (let 
+TMP_308 \def (\lambda (t0: T).(let TMP_307 \def (clen c0) in (subst0 TMP_307 
+t t0 t2))) in (let TMP_309 \def (eq T t1 t2) in (let TMP_314 \def (\lambda 
+(x0: T).(\lambda (H5: (eq K (Bind Abst) (Bind Abbr))).(\lambda (_: (pr0 t1 
+x0)).(\lambda (_: (subst0 (clen c0) t x0 t2)).(let TMP_310 \def (Bind Abst) 
+in (let TMP_311 \def (\lambda (ee: K).(match ee with [(Bind b0) \Rightarrow 
+(match b0 with [Abbr \Rightarrow False | Abst \Rightarrow True | Void 
+\Rightarrow False]) | (Flat _) \Rightarrow False])) in (let TMP_312 \def 
+(Bind Abbr) in (let H8 \def (eq_ind K TMP_310 TMP_311 I TMP_312 H5) in (let 
+TMP_313 \def (eq T t1 t2) in (False_ind TMP_313 H8)))))))))) in (ex3_ind T 
+TMP_305 TMP_306 TMP_308 TMP_309 TMP_314 H4))))))) in (or_ind TMP_293 TMP_300 
+TMP_301 TMP_302 TMP_315 H3)))))))))))))) in (let TMP_317 \def (or_introl 
+TMP_286 TMP_291 TMP_316) in (let TMP_318 \def (\forall (t2: T).((pr2 (CTail 
+(Bind Void) t c0) t1 t2) \to (eq T t1 t2))) in (let TMP_319 \def (\lambda 
+(t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) in (let TMP_322 \def 
+(\lambda (t2: T).(let TMP_320 \def (Bind Void) in (let TMP_321 \def (CTail 
+TMP_320 t c0) in (pr2 TMP_321 t1 t2)))) in (let TMP_323 \def (ex2 T TMP_319 
+TMP_322) in (let TMP_348 \def (\lambda (t2: T).(\lambda (H2: (pr2 (CTail 
+(Bind Void) t c0) t1 t2)).(let TMP_324 \def (Bind Void) in (let H_x0 \def 
+(pr2_gen_ctail TMP_324 c0 t t1 t2 H2) in (let H3 \def H_x0 in (let TMP_325 
+\def (pr2 c0 t1 t2) in (let TMP_328 \def (\lambda (_: T).(let TMP_326 \def 
+(Bind Void) in (let TMP_327 \def (Bind Abbr) in (eq K TMP_326 TMP_327)))) in 
+(let TMP_329 \def (\lambda (t0: T).(pr0 t1 t0)) in (let TMP_331 \def (\lambda 
+(t0: T).(let TMP_330 \def (clen c0) in (subst0 TMP_330 t t0 t2))) in (let 
+TMP_332 \def (ex3 T TMP_328 TMP_329 TMP_331) in (let TMP_333 \def (eq T t1 
+t2) in (let TMP_334 \def (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) in (let 
+TMP_347 \def (\lambda (H4: (ex3 T (\lambda (_: T).(eq K (Bind Void) (Bind 
+Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 
+t2)))).(let TMP_337 \def (\lambda (_: T).(let TMP_335 \def (Bind Void) in 
+(let TMP_336 \def (Bind Abbr) in (eq K TMP_335 TMP_336)))) in (let TMP_338 
+\def (\lambda (t0: T).(pr0 t1 t0)) in (let TMP_340 \def (\lambda (t0: T).(let 
+TMP_339 \def (clen c0) in (subst0 TMP_339 t t0 t2))) in (let TMP_341 \def (eq 
+T t1 t2) in (let TMP_346 \def (\lambda (x0: T).(\lambda (H5: (eq K (Bind 
+Void) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: (subst0 (clen c0) 
+t x0 t2)).(let TMP_342 \def (Bind Void) in (let TMP_343 \def (\lambda (ee: 
+K).(match ee with [(Bind b0) \Rightarrow (match b0 with [Abbr \Rightarrow 
+False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) 
+\Rightarrow False])) in (let TMP_344 \def (Bind Abbr) in (let H8 \def (eq_ind 
+K TMP_342 TMP_343 I TMP_344 H5) in (let TMP_345 \def (eq T t1 t2) in 
+(False_ind TMP_345 H8)))))))))) in (ex3_ind T TMP_337 TMP_338 TMP_340 TMP_341 
+TMP_346 H4))))))) in (or_ind TMP_325 TMP_332 TMP_333 TMP_334 TMP_347 
+H3)))))))))))))) in (let TMP_349 \def (or_introl TMP_318 TMP_323 TMP_348) in 
+(B_ind TMP_67 TMP_285 TMP_317 TMP_349 b)))))))))))))))))))))))))) in (let 
+TMP_382 \def (\lambda (f: F).(let TMP_351 \def (\forall (t2: T).((pr2 (CTail 
+(Flat f) t c0) t1 t2) \to (eq T t1 t2))) in (let TMP_352 \def (\lambda (t2: 
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) in (let TMP_355 \def (\lambda 
+(t2: T).(let TMP_353 \def (Flat f) in (let TMP_354 \def (CTail TMP_353 t c0) 
+in (pr2 TMP_354 t1 t2)))) in (let TMP_356 \def (ex2 T TMP_352 TMP_355) in 
+(let TMP_381 \def (\lambda (t2: T).(\lambda (H2: (pr2 (CTail (Flat f) t c0) 
+t1 t2)).(let TMP_357 \def (Flat f) in (let H_x0 \def (pr2_gen_ctail TMP_357 
+c0 t t1 t2 H2) in (let H3 \def H_x0 in (let TMP_358 \def (pr2 c0 t1 t2) in 
+(let TMP_361 \def (\lambda (_: T).(let TMP_359 \def (Flat f) in (let TMP_360 
+\def (Bind Abbr) in (eq K TMP_359 TMP_360)))) in (let TMP_362 \def (\lambda 
+(t0: T).(pr0 t1 t0)) in (let TMP_364 \def (\lambda (t0: T).(let TMP_363 \def 
+(clen c0) in (subst0 TMP_363 t t0 t2))) in (let TMP_365 \def (ex3 T TMP_361 
+TMP_362 TMP_364) in (let TMP_366 \def (eq T t1 t2) in (let TMP_367 \def 
+(\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) in (let TMP_380 \def (\lambda (H4: 
+(ex3 T (\lambda (_: T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 
+t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(let TMP_370 \def 
+(\lambda (_: T).(let TMP_368 \def (Flat f) in (let TMP_369 \def (Bind Abbr) 
+in (eq K TMP_368 TMP_369)))) in (let TMP_371 \def (\lambda (t0: T).(pr0 t1 
+t0)) in (let TMP_373 \def (\lambda (t0: T).(let TMP_372 \def (clen c0) in 
+(subst0 TMP_372 t t0 t2))) in (let TMP_374 \def (eq T t1 t2) in (let TMP_379 
+\def (\lambda (x0: T).(\lambda (H5: (eq K (Flat f) (Bind Abbr))).(\lambda (_: 
+(pr0 t1 x0)).(\lambda (_: (subst0 (clen c0) t x0 t2)).(let TMP_375 \def (Flat 
+f) in (let TMP_376 \def (\lambda (ee: K).(match ee with [(Bind _) \Rightarrow 
+False | (Flat _) \Rightarrow True])) in (let TMP_377 \def (Bind Abbr) in (let 
+H8 \def (eq_ind K TMP_375 TMP_376 I TMP_377 H5) in (let TMP_378 \def (eq T t1 
+t2) in (False_ind TMP_378 H8)))))))))) in (ex3_ind T TMP_370 TMP_371 TMP_373 
+TMP_374 TMP_379 H4))))))) in (or_ind TMP_358 TMP_365 TMP_366 TMP_367 TMP_380 
+H3)))))))))))))) in (or_introl TMP_351 TMP_356 TMP_381))))))) in (K_ind 
+TMP_60 TMP_350 TMP_382 k))))) in (let TMP_403 \def (\lambda (H1: (ex2 T 
 (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (x: T).(\lambda (H2: (((eq T t1 x) 
-\to (\forall (P: Prop).P)))).(\lambda (H3: (pr2 c0 t1 x)).(or_intror (\forall 
-(t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: 
-T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t 
-c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)) x H2 (pr2_ctail c0 t1 
-x H3 k t)))))) H1)) H0)))))))) c).
-(* COMMENTS
-Initial nodes: 3653
-END *)
+T).(pr2 c0 t1 t2)))).(let TMP_384 \def (\lambda (t2: T).((eq T t1 t2) \to 
+(\forall (P: Prop).P))) in (let TMP_385 \def (\lambda (t2: T).(pr2 c0 t1 t2)) 
+in (let TMP_386 \def (\forall (t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T 
+t1 t2))) in (let TMP_387 \def (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: 
+Prop).P))) in (let TMP_389 \def (\lambda (t2: T).(let TMP_388 \def (CTail k t 
+c0) in (pr2 TMP_388 t1 t2))) in (let TMP_390 \def (ex2 T TMP_387 TMP_389) in 
+(let TMP_391 \def (or TMP_386 TMP_390) in (let TMP_402 \def (\lambda (x: 
+T).(\lambda (H2: (((eq T t1 x) \to (\forall (P: Prop).P)))).(\lambda (H3: 
+(pr2 c0 t1 x)).(let TMP_392 \def (\forall (t2: T).((pr2 (CTail k t c0) t1 t2) 
+\to (eq T t1 t2))) in (let TMP_393 \def (\lambda (t2: T).((eq T t1 t2) \to 
+(\forall (P: Prop).P))) in (let TMP_395 \def (\lambda (t2: T).(let TMP_394 
+\def (CTail k t c0) in (pr2 TMP_394 t1 t2))) in (let TMP_396 \def (ex2 T 
+TMP_393 TMP_395) in (let TMP_397 \def (\lambda (t2: T).((eq T t1 t2) \to 
+(\forall (P: Prop).P))) in (let TMP_399 \def (\lambda (t2: T).(let TMP_398 
+\def (CTail k t c0) in (pr2 TMP_398 t1 t2))) in (let TMP_400 \def (pr2_ctail 
+c0 t1 x H3 k t) in (let TMP_401 \def (ex_intro2 T TMP_397 TMP_399 x H2 
+TMP_400) in (or_intror TMP_392 TMP_396 TMP_401)))))))))))) in (ex2_ind T 
+TMP_384 TMP_385 TMP_391 TMP_402 H1)))))))))) in (or_ind TMP_45 TMP_48 TMP_54 
+TMP_383 TMP_403 H0))))))))))))))))))) in (c_tail_ind TMP_5 TMP_44 TMP_404 
+c)))).
 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/defs.ma
index 98e931c0b..d501b8b3c 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/defs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/pr2/defs.ma".
+include "basic_1/pr2/defs.ma".
 
 definition nf2:
  C \to (T \to Prop)
@@ -22,10 +22,7 @@ definition nf2:
  \lambda (c: C).(\lambda (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (eq T t1 
 t2)))).
 
-definition nfs2:
- C \to (TList \to Prop)
-\def
- let rec nfs2 (c: C) (ts: TList) on ts: Prop \def (match ts with [TNil 
-\Rightarrow True | (TCons t ts0) \Rightarrow (land (nf2 c t) (nfs2 c ts0))]) 
-in nfs2.
+let rec nfs2 (c: C) (ts: TList) on ts: Prop \def match ts with [TNil 
+\Rightarrow True | (TCons t ts0) \Rightarrow (let TMP_1 \def (nf2 c t) in 
+(let TMP_2 \def (nfs2 c ts0) in (land TMP_1 TMP_2)))].
 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/fwd.ma
index 9138ff2fa..11d3302e5 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/fwd.ma
@@ -14,13 +14,13 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/nf2/defs.ma".
+include "basic_1/nf2/defs.ma".
 
-include "Basic-1/pr2/clen.ma".
+include "basic_1/pr2/clen.ma".
 
-include "Basic-1/subst0/dec.ma".
+include "basic_1/subst0/dec.ma".
 
-include "Basic-1/T/props.ma".
+include "basic_1/T/props.ma".
 
 theorem nf2_gen_lref:
  \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c 
@@ -28,13 +28,16 @@ theorem nf2_gen_lref:
 \def
  \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
 (H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H0: ((\forall (t2: T).((pr2 
-c (TLRef i) t2) \to (eq T (TLRef i) t2))))).(\lambda (P: 
-Prop).(lift_gen_lref_false (S i) O i (le_O_n i) (le_n (plus O (S i))) u (H0 
-(lift (S i) O u) (pr2_delta c d u i H (TLRef i) (TLRef i) (pr0_refl (TLRef 
-i)) (lift (S i) O u) (subst0_lref u i))) P))))))).
-(* COMMENTS
-Initial nodes: 129
-END *)
+c (TLRef i) t2) \to (eq T (TLRef i) t2))))).(\lambda (P: Prop).(let TMP_1 
+\def (S i) in (let TMP_2 \def (le_O_n i) in (let TMP_3 \def (S i) in (let 
+TMP_4 \def (plus O TMP_3) in (let TMP_5 \def (le_n TMP_4) in (let TMP_6 \def 
+(S i) in (let TMP_7 \def (lift TMP_6 O u) in (let TMP_8 \def (TLRef i) in 
+(let TMP_9 \def (TLRef i) in (let TMP_10 \def (TLRef i) in (let TMP_11 \def 
+(pr0_refl TMP_10) in (let TMP_12 \def (S i) in (let TMP_13 \def (lift TMP_12 
+O u) in (let TMP_14 \def (subst0_lref u i) in (let TMP_15 \def (pr2_delta c d 
+u i H TMP_8 TMP_9 TMP_11 TMP_13 TMP_14) in (let TMP_16 \def (H0 TMP_7 TMP_15) 
+in (lift_gen_lref_false TMP_1 O i TMP_2 TMP_5 u TMP_16 
+P))))))))))))))))))))))).
 
 theorem nf2_gen_abst:
  \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abst) u 
@@ -42,36 +45,43 @@ t)) \to (land (nf2 c u) (nf2 (CHead c (Bind Abst) u) t)))))
 \def
  \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: 
 T).((pr2 c (THead (Bind Abst) u t) t2) \to (eq T (THead (Bind Abst) u t) 
-t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall (t2: 
-T).((pr2 (CHead c (Bind Abst) u) t t2) \to (eq T t t2))) (\lambda (t2: 
-T).(\lambda (H0: (pr2 c u t2)).(let H1 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | 
-(TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst) 
-u t) (THead (Bind Abst) t2 t) (H (THead (Bind Abst) t2 t) (pr2_head_1 c u t2 
-H0 (Bind Abst) t))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 c u 
-t0)) H0 u H1) in (eq_ind T u (\lambda (t0: T).(eq T u t0)) (refl_equal T u) 
-t2 H1))))) (\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind Abst) u) t 
-t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda 
-(_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ 
-_ t0) \Rightarrow t0])) (THead (Bind Abst) u t) (THead (Bind Abst) u t2) (H 
-(THead (Bind Abst) u t2) (let H_y \def (pr2_gen_cbind Abst c u t t2 H0) in 
-H_y))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 (CHead c (Bind 
-Abst) u) t t0)) H0 t H1) in (eq_ind T t (\lambda (t0: T).(eq T t t0)) 
-(refl_equal T t) t2 H1))))))))).
-(* COMMENTS
-Initial nodes: 353
-END *)
+t2))))).(let TMP_1 \def (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) in 
+(let TMP_2 \def (\forall (t2: T).((pr2 (CHead c (Bind Abst) u) t t2) \to (eq 
+T t t2))) in (let TMP_16 \def (\lambda (t2: T).(\lambda (H0: (pr2 c u 
+t2)).(let TMP_3 \def (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | 
+(TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])) in (let TMP_4 \def 
+(Bind Abst) in (let TMP_5 \def (THead TMP_4 u t) in (let TMP_6 \def (Bind 
+Abst) in (let TMP_7 \def (THead TMP_6 t2 t) in (let TMP_8 \def (Bind Abst) in 
+(let TMP_9 \def (THead TMP_8 t2 t) in (let TMP_10 \def (Bind Abst) in (let 
+TMP_11 \def (pr2_head_1 c u t2 H0 TMP_10 t) in (let TMP_12 \def (H TMP_9 
+TMP_11) in (let H1 \def (f_equal T T TMP_3 TMP_5 TMP_7 TMP_12) in (let TMP_13 
+\def (\lambda (t0: T).(pr2 c u t0)) in (let H2 \def (eq_ind_r T t2 TMP_13 H0 
+u H1) in (let TMP_14 \def (\lambda (t0: T).(eq T u t0)) in (let TMP_15 \def 
+(refl_equal T u) in (eq_ind T u TMP_14 TMP_15 t2 H1)))))))))))))))))) in (let 
+TMP_30 \def (\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind Abst) u) t 
+t2)).(let TMP_17 \def (\lambda (e: T).(match e with [(TSort _) \Rightarrow t 
+| (TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0])) in (let TMP_18 
+\def (Bind Abst) in (let TMP_19 \def (THead TMP_18 u t) in (let TMP_20 \def 
+(Bind Abst) in (let TMP_21 \def (THead TMP_20 u t2) in (let TMP_22 \def (Bind 
+Abst) in (let TMP_23 \def (THead TMP_22 u t2) in (let H_y \def (pr2_gen_cbind 
+Abst c u t t2 H0) in (let TMP_24 \def (H TMP_23 H_y) in (let H1 \def (f_equal 
+T T TMP_17 TMP_19 TMP_21 TMP_24) in (let TMP_27 \def (\lambda (t0: T).(let 
+TMP_25 \def (Bind Abst) in (let TMP_26 \def (CHead c TMP_25 u) in (pr2 TMP_26 
+t t0)))) in (let H2 \def (eq_ind_r T t2 TMP_27 H0 t H1) in (let TMP_28 \def 
+(\lambda (t0: T).(eq T t t0)) in (let TMP_29 \def (refl_equal T t) in (eq_ind 
+T t TMP_28 TMP_29 t2 H1))))))))))))))))) in (conj TMP_1 TMP_2 TMP_16 
+TMP_30)))))))).
 
 theorem nf2_gen_cast:
  \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Flat Cast) u 
 t)) \to (\forall (P: Prop).P))))
 \def
  \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (nf2 c (THead 
-(Flat Cast) u t))).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) u t (H t 
-(pr2_free c (THead (Flat Cast) u t) t (pr0_tau t t (pr0_refl t) u))) P))))).
-(* COMMENTS
-Initial nodes: 65
-END *)
+(Flat Cast) u t))).(\lambda (P: Prop).(let TMP_1 \def (Flat Cast) in (let 
+TMP_2 \def (Flat Cast) in (let TMP_3 \def (THead TMP_2 u t) in (let TMP_4 
+\def (pr0_refl t) in (let TMP_5 \def (pr0_tau t t TMP_4 u) in (let TMP_6 \def 
+(pr2_free c TMP_3 t TMP_5) in (let TMP_7 \def (H t TMP_6) in (thead_x_y_y 
+TMP_1 u t TMP_7 P)))))))))))).
 
 theorem nf2_gen_beta:
  \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((nf2 c 
@@ -80,17 +90,20 @@ theorem nf2_gen_beta:
  \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H: 
 ((\forall (t2: T).((pr2 c (THead (Flat Appl) u (THead (Bind Abst) v t)) t2) 
 \to (eq T (THead (Flat Appl) u (THead (Bind Abst) v t)) t2))))).(\lambda (P: 
-Prop).(let H0 \def (eq_ind T (THead (Flat Appl) u (THead (Bind Abst) v t)) 
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
-(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u t) (H (THead (Bind 
-Abbr) u t) (pr2_free c (THead (Flat Appl) u (THead (Bind Abst) v t)) (THead 
-(Bind Abbr) u t) (pr0_beta v u u (pr0_refl u) t t (pr0_refl t))))) in 
-(False_ind P H0))))))).
-(* COMMENTS
-Initial nodes: 183
-END *)
+Prop).(let TMP_1 \def (Flat Appl) in (let TMP_2 \def (Bind Abst) in (let 
+TMP_3 \def (THead TMP_2 v t) in (let TMP_4 \def (THead TMP_1 u TMP_3) in (let 
+TMP_5 \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_6 \def 
+(Bind Abbr) in (let TMP_7 \def (THead TMP_6 u t) in (let TMP_8 \def (Bind 
+Abbr) in (let TMP_9 \def (THead TMP_8 u t) in (let TMP_10 \def (Flat Appl) in 
+(let TMP_11 \def (Bind Abst) in (let TMP_12 \def (THead TMP_11 v t) in (let 
+TMP_13 \def (THead TMP_10 u TMP_12) in (let TMP_14 \def (Bind Abbr) in (let 
+TMP_15 \def (THead TMP_14 u t) in (let TMP_16 \def (pr0_refl u) in (let 
+TMP_17 \def (pr0_refl t) in (let TMP_18 \def (pr0_beta v u u TMP_16 t t 
+TMP_17) in (let TMP_19 \def (pr2_free c TMP_13 TMP_15 TMP_18) in (let TMP_20 
+\def (H TMP_9 TMP_19) in (let H0 \def (eq_ind T TMP_4 TMP_5 I TMP_7 TMP_20) 
+in (False_ind P H0))))))))))))))))))))))))))).
 
 theorem nf2_gen_flat:
  \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c 
@@ -98,83 +111,96 @@ theorem nf2_gen_flat:
 \def
  \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: 
 ((\forall (t2: T).((pr2 c (THead (Flat f) u t) t2) \to (eq T (THead (Flat f) 
-u t) t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall 
-(t2: T).((pr2 c t t2) \to (eq T t t2))) (\lambda (t2: T).(\lambda (H0: (pr2 c 
-u t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | 
-(THead _ t0 _) \Rightarrow t0])) (THead (Flat f) u t) (THead (Flat f) t2 t) 
-(H (THead (Flat f) t2 t) (pr2_head_1 c u t2 H0 (Flat f) t))) in H1))) 
-(\lambda (t2: T).(\lambda (H0: (pr2 c t t2)).(let H1 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0])) 
-(THead (Flat f) u t) (THead (Flat f) u t2) (H (THead (Flat f) u t2) 
-(pr2_head_2 c u t t2 (Flat f) (pr2_cflat c t t2 H0 f u)))) in H1)))))))).
-(* COMMENTS
-Initial nodes: 251
-END *)
+u t) t2))))).(let TMP_1 \def (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) 
+in (let TMP_2 \def (\forall (t2: T).((pr2 c t t2) \to (eq T t t2))) in (let 
+TMP_13 \def (\lambda (t2: T).(\lambda (H0: (pr2 c u t2)).(let TMP_3 \def 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef _) 
+\Rightarrow u | (THead _ t0 _) \Rightarrow t0])) in (let TMP_4 \def (Flat f) 
+in (let TMP_5 \def (THead TMP_4 u t) in (let TMP_6 \def (Flat f) in (let 
+TMP_7 \def (THead TMP_6 t2 t) in (let TMP_8 \def (Flat f) in (let TMP_9 \def 
+(THead TMP_8 t2 t) in (let TMP_10 \def (Flat f) in (let TMP_11 \def 
+(pr2_head_1 c u t2 H0 TMP_10 t) in (let TMP_12 \def (H TMP_9 TMP_11) in (let 
+H1 \def (f_equal T T TMP_3 TMP_5 TMP_7 TMP_12) in H1))))))))))))) in (let 
+TMP_25 \def (\lambda (t2: T).(\lambda (H0: (pr2 c t t2)).(let TMP_14 \def 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow t | (TLRef _) 
+\Rightarrow t | (THead _ _ t0) \Rightarrow t0])) in (let TMP_15 \def (Flat f) 
+in (let TMP_16 \def (THead TMP_15 u t) in (let TMP_17 \def (Flat f) in (let 
+TMP_18 \def (THead TMP_17 u t2) in (let TMP_19 \def (Flat f) in (let TMP_20 
+\def (THead TMP_19 u t2) in (let TMP_21 \def (Flat f) in (let TMP_22 \def 
+(pr2_cflat c t t2 H0 f u) in (let TMP_23 \def (pr2_head_2 c u t t2 TMP_21 
+TMP_22) in (let TMP_24 \def (H TMP_20 TMP_23) in (let H1 \def (f_equal T T 
+TMP_14 TMP_16 TMP_18 TMP_24) in H1)))))))))))))) in (conj TMP_1 TMP_2 TMP_13 
+TMP_25))))))))).
 
 theorem nf2_gen__nf2_gen_aux:
  \forall (b: B).(\forall (x: T).(\forall (u: T).(\forall (d: nat).((eq T 
 (THead (Bind b) u (lift (S O) d x)) x) \to (\forall (P: Prop).P)))))
 \def
- \lambda (b: B).(\lambda (x: T).(T_ind (\lambda (t: T).(\forall (u: 
+ \lambda (b: B).(\lambda (x: T).(let TMP_1 \def (\lambda (t: T).(\forall (u: 
 T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to 
-(\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (d: 
-nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d (TSort n))) (TSort 
-n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead (Bind b) u (lift (S O) 
-d (TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ 
-_) \Rightarrow True])) I (TSort n) H) in (False_ind P H0))))))) (\lambda (n: 
-nat).(\lambda (u: T).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u 
-(lift (S O) d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind 
-T (THead (Bind b) u (lift (S O) d (TLRef n))) (\lambda (ee: T).(match ee in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in 
-(False_ind P H0))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: ((\forall 
-(u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to 
-(\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall (u: 
-T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t0)) t0) \to 
-(\forall (P: Prop).P)))))).(\lambda (u: T).(\lambda (d: nat).(\lambda (H1: 
-(eq T (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t 
-t0))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e 
-in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef 
-_) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u 
-(lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let H3 \def (f_equal T 
-T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t1 _) \Rightarrow t1])) 
-(THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let 
-H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow (THead k ((let rec lref_map (f: ((nat \to nat))) 
-(d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort 
-n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i 
-| false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 
-(lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: 
-nat).(plus x0 (S O))) d t) ((let rec lref_map (f: ((nat \to nat))) (d0: nat) 
-(t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort n) | 
-(TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i | 
-false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 (lref_map 
-f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: nat).(plus 
-x0 (S O))) (s k d) t0)) | (TLRef _) \Rightarrow (THead k ((let rec lref_map 
-(f: ((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort 
-n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) 
-with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2) 
-\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in 
-lref_map) (\lambda (x0: nat).(plus x0 (S O))) d t) ((let rec lref_map (f: 
-((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2) 
-\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in 
-lref_map) (\lambda (x0: nat).(plus x0 (S O))) (s k d) t0)) | (THead _ _ t1) 
-\Rightarrow t1])) (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t 
-t0) H1) in (\lambda (_: (eq T u t)).(\lambda (H6: (eq K (Bind b) k)).(let H7 
-\def (eq_ind_r K k (\lambda (k0: K).(eq T (lift (S O) d (THead k0 t t0)) t0)) 
-H4 (Bind b) H6) in (let H8 \def (eq_ind T (lift (S O) d (THead (Bind b) t 
-t0)) (\lambda (t1: T).(eq T t1 t0)) H7 (THead (Bind b) (lift (S O) d t) (lift 
-(S O) (S d) t0)) (lift_bind b t t0 (S O) d)) in (H0 (lift (S O) d t) (S d) H8 
-P)))))) H3)) H2))))))))))) x)).
-(* COMMENTS
-Initial nodes: 935
-END *)
+(\forall (P: Prop).P))))) in (let TMP_9 \def (\lambda (n: nat).(\lambda (u: 
+T).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d 
+(TSort n))) (TSort n))).(\lambda (P: Prop).(let TMP_2 \def (Bind b) in (let 
+TMP_3 \def (S O) in (let TMP_4 \def (TSort n) in (let TMP_5 \def (lift TMP_3 
+d TMP_4) in (let TMP_6 \def (THead TMP_2 u TMP_5) in (let TMP_7 \def (\lambda 
+(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
+False | (THead _ _ _) \Rightarrow True])) in (let TMP_8 \def (TSort n) in 
+(let H0 \def (eq_ind T TMP_6 TMP_7 I TMP_8 H) in (False_ind P 
+H0)))))))))))))) in (let TMP_17 \def (\lambda (n: nat).(\lambda (u: 
+T).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d 
+(TLRef n))) (TLRef n))).(\lambda (P: Prop).(let TMP_10 \def (Bind b) in (let 
+TMP_11 \def (S O) in (let TMP_12 \def (TLRef n) in (let TMP_13 \def (lift 
+TMP_11 d TMP_12) in (let TMP_14 \def (THead TMP_10 u TMP_13) in (let TMP_15 
+\def (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) in (let TMP_16 \def 
+(TLRef n) in (let H0 \def (eq_ind T TMP_14 TMP_15 I TMP_16 H) in (False_ind P 
+H0)))))))))))))) in (let TMP_97 \def (\lambda (k: K).(\lambda (t: T).(\lambda 
+(_: ((\forall (u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d 
+t)) t) \to (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: 
+((\forall (u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d 
+t0)) t0) \to (\forall (P: Prop).P)))))).(\lambda (u: T).(\lambda (d: 
+nat).(\lambda (H1: (eq T (THead (Bind b) u (lift (S O) d (THead k t t0))) 
+(THead k t t0))).(\lambda (P: Prop).(let TMP_18 \def (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow (Bind b) | (TLRef _) \Rightarrow (Bind b) | 
+(THead k0 _ _) \Rightarrow k0])) in (let TMP_19 \def (Bind b) in (let TMP_20 
+\def (S O) in (let TMP_21 \def (THead k t t0) in (let TMP_22 \def (lift 
+TMP_20 d TMP_21) in (let TMP_23 \def (THead TMP_19 u TMP_22) in (let TMP_24 
+\def (THead k t t0) in (let H2 \def (f_equal T K TMP_18 TMP_23 TMP_24 H1) in 
+(let TMP_25 \def (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | 
+(TLRef _) \Rightarrow u | (THead _ t1 _) \Rightarrow t1])) in (let TMP_26 
+\def (Bind b) in (let TMP_27 \def (S O) in (let TMP_28 \def (THead k t t0) in 
+(let TMP_29 \def (lift TMP_27 d TMP_28) in (let TMP_30 \def (THead TMP_26 u 
+TMP_29) in (let TMP_31 \def (THead k t t0) in (let H3 \def (f_equal T T 
+TMP_25 TMP_30 TMP_31 H1) in (let TMP_66 \def (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow (let TMP_55 \def (\lambda (x0: nat).(let TMP_54 \def 
+(S O) in (plus x0 TMP_54))) in (let TMP_56 \def (lref_map TMP_55 d t) in (let 
+TMP_63 \def (\lambda (x0: nat).(let TMP_62 \def (S O) in (plus x0 TMP_62))) 
+in (let TMP_64 \def (s k d) in (let TMP_65 \def (lref_map TMP_63 TMP_64 t0) 
+in (THead k TMP_56 TMP_65)))))) | (TLRef _) \Rightarrow (let TMP_38 \def 
+(\lambda (x0: nat).(let TMP_37 \def (S O) in (plus x0 TMP_37))) in (let 
+TMP_39 \def (lref_map TMP_38 d t) in (let TMP_46 \def (\lambda (x0: nat).(let 
+TMP_45 \def (S O) in (plus x0 TMP_45))) in (let TMP_47 \def (s k d) in (let 
+TMP_48 \def (lref_map TMP_46 TMP_47 t0) in (THead k TMP_39 TMP_48)))))) | 
+(THead _ _ t1) \Rightarrow t1])) in (let TMP_67 \def (Bind b) in (let TMP_68 
+\def (S O) in (let TMP_69 \def (THead k t t0) in (let TMP_70 \def (lift 
+TMP_68 d TMP_69) in (let TMP_71 \def (THead TMP_67 u TMP_70) in (let TMP_72 
+\def (THead k t t0) in (let H4 \def (f_equal T T TMP_66 TMP_71 TMP_72 H1) in 
+(let TMP_95 \def (\lambda (_: (eq T u t)).(\lambda (H6: (eq K (Bind b) 
+k)).(let TMP_76 \def (\lambda (k0: K).(let TMP_73 \def (S O) in (let TMP_74 
+\def (THead k0 t t0) in (let TMP_75 \def (lift TMP_73 d TMP_74) in (eq T 
+TMP_75 t0))))) in (let TMP_77 \def (Bind b) in (let H7 \def (eq_ind_r K k 
+TMP_76 H4 TMP_77 H6) in (let TMP_78 \def (S O) in (let TMP_79 \def (Bind b) 
+in (let TMP_80 \def (THead TMP_79 t t0) in (let TMP_81 \def (lift TMP_78 d 
+TMP_80) in (let TMP_82 \def (\lambda (t1: T).(eq T t1 t0)) in (let TMP_83 
+\def (Bind b) in (let TMP_84 \def (S O) in (let TMP_85 \def (lift TMP_84 d t) 
+in (let TMP_86 \def (S O) in (let TMP_87 \def (S d) in (let TMP_88 \def (lift 
+TMP_86 TMP_87 t0) in (let TMP_89 \def (THead TMP_83 TMP_85 TMP_88) in (let 
+TMP_90 \def (S O) in (let TMP_91 \def (lift_bind b t t0 TMP_90 d) in (let H8 
+\def (eq_ind T TMP_81 TMP_82 H7 TMP_89 TMP_91) in (let TMP_92 \def (S O) in 
+(let TMP_93 \def (lift TMP_92 d t) in (let TMP_94 \def (S d) in (H0 TMP_93 
+TMP_94 H8 P)))))))))))))))))))))))) in (let TMP_96 \def (TMP_95 H3) in 
+(TMP_96 H2)))))))))))))))))))))))))))))))))))) in (T_ind TMP_1 TMP_9 TMP_17 
+TMP_97 x)))))).
 
 theorem nf2_gen_abbr:
  \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abbr) u 
@@ -183,26 +209,44 @@ t)) \to (\forall (P: Prop).P))))
  \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: 
 T).((pr2 c (THead (Bind Abbr) u t) t2) \to (eq T (THead (Bind Abbr) u t) 
 t2))))).(\lambda (P: Prop).(let H_x \def (dnf_dec u t O) in (let H0 \def H_x 
-in (ex_ind T (\lambda (v: T).(or (subst0 O u t (lift (S O) O v)) (eq T t 
-(lift (S O) O v)))) P (\lambda (x: T).(\lambda (H1: (or (subst0 O u t (lift 
-(S O) O x)) (eq T t (lift (S O) O x)))).(or_ind (subst0 O u t (lift (S O) O 
-x)) (eq T t (lift (S O) O x)) P (\lambda (H2: (subst0 O u t (lift (S O) O 
-x))).(let H3 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda 
-(_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ 
-_ t0) \Rightarrow t0])) (THead (Bind Abbr) u t) (THead (Bind Abbr) u (lift (S 
-O) O x)) (H (THead (Bind Abbr) u (lift (S O) O x)) (pr2_free c (THead (Bind 
-Abbr) u t) (THead (Bind Abbr) u (lift (S O) O x)) (pr0_delta u u (pr0_refl u) 
-t t (pr0_refl t) (lift (S O) O x) H2)))) in (let H4 \def (eq_ind T t (\lambda 
-(t0: T).(subst0 O u t0 (lift (S O) O x))) H2 (lift (S O) O x) H3) in 
-(subst0_refl u (lift (S O) O x) O H4 P)))) (\lambda (H2: (eq T t (lift (S O) 
-O x))).(let H3 \def (eq_ind T t (\lambda (t0: T).(\forall (t2: T).((pr2 c 
-(THead (Bind Abbr) u t0) t2) \to (eq T (THead (Bind Abbr) u t0) t2)))) H 
-(lift (S O) O x) H2) in (nf2_gen__nf2_gen_aux Abbr x u O (H3 x (pr2_free c 
-(THead (Bind Abbr) u (lift (S O) O x)) x (pr0_zeta Abbr not_abbr_abst x x 
-(pr0_refl x) u))) P))) H1))) H0))))))).
-(* COMMENTS
-Initial nodes: 511
-END *)
+in (let TMP_7 \def (\lambda (v: T).(let TMP_1 \def (S O) in (let TMP_2 \def 
+(lift TMP_1 O v) in (let TMP_3 \def (subst0 O u t TMP_2) in (let TMP_4 \def 
+(S O) in (let TMP_5 \def (lift TMP_4 O v) in (let TMP_6 \def (eq T t TMP_5) 
+in (or TMP_3 TMP_6)))))))) in (let TMP_60 \def (\lambda (x: T).(\lambda (H1: 
+(or (subst0 O u t (lift (S O) O x)) (eq T t (lift (S O) O x)))).(let TMP_8 
+\def (S O) in (let TMP_9 \def (lift TMP_8 O x) in (let TMP_10 \def (subst0 O 
+u t TMP_9) in (let TMP_11 \def (S O) in (let TMP_12 \def (lift TMP_11 O x) in 
+(let TMP_13 \def (eq T t TMP_12) in (let TMP_45 \def (\lambda (H2: (subst0 O 
+u t (lift (S O) O x))).(let TMP_14 \def (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0])) 
+in (let TMP_15 \def (Bind Abbr) in (let TMP_16 \def (THead TMP_15 u t) in 
+(let TMP_17 \def (Bind Abbr) in (let TMP_18 \def (S O) in (let TMP_19 \def 
+(lift TMP_18 O x) in (let TMP_20 \def (THead TMP_17 u TMP_19) in (let TMP_21 
+\def (Bind Abbr) in (let TMP_22 \def (S O) in (let TMP_23 \def (lift TMP_22 O 
+x) in (let TMP_24 \def (THead TMP_21 u TMP_23) in (let TMP_25 \def (Bind 
+Abbr) in (let TMP_26 \def (THead TMP_25 u t) in (let TMP_27 \def (Bind Abbr) 
+in (let TMP_28 \def (S O) in (let TMP_29 \def (lift TMP_28 O x) in (let 
+TMP_30 \def (THead TMP_27 u TMP_29) in (let TMP_31 \def (pr0_refl u) in (let 
+TMP_32 \def (pr0_refl t) in (let TMP_33 \def (S O) in (let TMP_34 \def (lift 
+TMP_33 O x) in (let TMP_35 \def (pr0_delta u u TMP_31 t t TMP_32 TMP_34 H2) 
+in (let TMP_36 \def (pr2_free c TMP_26 TMP_30 TMP_35) in (let TMP_37 \def (H 
+TMP_24 TMP_36) in (let H3 \def (f_equal T T TMP_14 TMP_16 TMP_20 TMP_37) in 
+(let TMP_40 \def (\lambda (t0: T).(let TMP_38 \def (S O) in (let TMP_39 \def 
+(lift TMP_38 O x) in (subst0 O u t0 TMP_39)))) in (let TMP_41 \def (S O) in 
+(let TMP_42 \def (lift TMP_41 O x) in (let H4 \def (eq_ind T t TMP_40 H2 
+TMP_42 H3) in (let TMP_43 \def (S O) in (let TMP_44 \def (lift TMP_43 O x) in 
+(subst0_refl u TMP_44 O H4 P))))))))))))))))))))))))))))))))) in (let TMP_59 
+\def (\lambda (H2: (eq T t (lift (S O) O x))).(let TMP_48 \def (\lambda (t0: 
+T).(\forall (t2: T).((pr2 c (THead (Bind Abbr) u t0) t2) \to (let TMP_46 \def 
+(Bind Abbr) in (let TMP_47 \def (THead TMP_46 u t0) in (eq T TMP_47 t2)))))) 
+in (let TMP_49 \def (S O) in (let TMP_50 \def (lift TMP_49 O x) in (let H3 
+\def (eq_ind T t TMP_48 H TMP_50 H2) in (let TMP_51 \def (Bind Abbr) in (let 
+TMP_52 \def (S O) in (let TMP_53 \def (lift TMP_52 O x) in (let TMP_54 \def 
+(THead TMP_51 u TMP_53) in (let TMP_55 \def (pr0_refl x) in (let TMP_56 \def 
+(pr0_zeta Abbr not_abbr_abst x x TMP_55 u) in (let TMP_57 \def (pr2_free c 
+TMP_54 x TMP_56) in (let TMP_58 \def (H3 x TMP_57) in (nf2_gen__nf2_gen_aux 
+Abbr x u O TMP_58 P)))))))))))))) in (or_ind TMP_10 TMP_13 P TMP_45 TMP_59 
+H1))))))))))) in (ex_ind T TMP_7 P TMP_60 H0))))))))).
 
 theorem nf2_gen_void:
  \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Void) u 
@@ -210,11 +254,10 @@ theorem nf2_gen_void:
 \def
  \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: 
 T).((pr2 c (THead (Bind Void) u (lift (S O) O t)) t2) \to (eq T (THead (Bind 
-Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(nf2_gen__nf2_gen_aux 
-Void t u O (H t (pr2_free c (THead (Bind Void) u (lift (S O) O t)) t 
-(pr0_zeta Void (sym_not_eq B Abst Void not_abst_void) t t (pr0_refl t) u))) 
-P))))).
-(* COMMENTS
-Initial nodes: 121
-END *)
+Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(let TMP_1 \def (Bind 
+Void) in (let TMP_2 \def (S O) in (let TMP_3 \def (lift TMP_2 O t) in (let 
+TMP_4 \def (THead TMP_1 u TMP_3) in (let TMP_5 \def (pr0_refl t) in (let 
+TMP_6 \def (pr0_zeta Void not_void_abst t t TMP_5 u) in (let TMP_7 \def 
+(pr2_free c TMP_4 t TMP_6) in (let TMP_8 \def (H t TMP_7) in 
+(nf2_gen__nf2_gen_aux Void t u O TMP_8 P))))))))))))).
 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/iso.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/iso.ma
index 6a2ce00f8..67540f34d 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/iso.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/iso.ma
@@ -14,11 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/nf2/pr3.ma".
+include "basic_1/nf2/pr3.ma".
 
-include "Basic-1/pr3/fwd.ma".
-
-include "Basic-1/iso/props.ma".
+include "basic_1/iso/props.ma".
 
 theorem nf2_iso_appls_lref:
  \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: 
@@ -26,73 +24,119 @@ TList).(\forall (u: T).((pr3 c (THeads (Flat Appl) vs (TLRef i)) u) \to (iso
 (THeads (Flat Appl) vs (TLRef i)) u))))))
 \def
  \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda 
-(vs: TList).(TList_ind (\lambda (t: TList).(\forall (u: T).((pr3 c (THeads 
-(Flat Appl) t (TLRef i)) u) \to (iso (THeads (Flat Appl) t (TLRef i)) u)))) 
-(\lambda (u: T).(\lambda (H0: (pr3 c (TLRef i) u)).(let H_y \def 
-(nf2_pr3_unfold c (TLRef i) u H0 H) in (let H1 \def (eq_ind_r T u (\lambda 
-(t: T).(pr3 c (TLRef i) t)) H0 (TLRef i) H_y) in (eq_ind T (TLRef i) (\lambda 
-(t: T).(iso (TLRef i) t)) (iso_refl (TLRef i)) u H_y))))) (\lambda (t: 
-T).(\lambda (t0: TList).(\lambda (H0: ((\forall (u: T).((pr3 c (THeads (Flat 
-Appl) t0 (TLRef i)) u) \to (iso (THeads (Flat Appl) t0 (TLRef i)) 
-u))))).(\lambda (u: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads 
-(Flat Appl) t0 (TLRef i))) u)).(let H2 \def (pr3_gen_appl c t (THeads (Flat 
-Appl) t0 (TLRef i)) u H1) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T u (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) 
-t0 (TLRef i)) t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat 
-Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: 
-T).(pr3 (CHead c (Bind b) u0) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: 
+(vs: TList).(let TMP_4 \def (\lambda (t: TList).(\forall (u: T).((pr3 c 
+(THeads (Flat Appl) t (TLRef i)) u) \to (let TMP_1 \def (Flat Appl) in (let 
+TMP_2 \def (TLRef i) in (let TMP_3 \def (THeads TMP_1 t TMP_2) in (iso TMP_3 
+u))))))) in (let TMP_14 \def (\lambda (u: T).(\lambda (H0: (pr3 c (TLRef i) 
+u)).(let TMP_5 \def (TLRef i) in (let H_y \def (nf2_pr3_unfold c TMP_5 u H0 
+H) in (let TMP_7 \def (\lambda (t: T).(let TMP_6 \def (TLRef i) in (pr3 c 
+TMP_6 t))) in (let TMP_8 \def (TLRef i) in (let H1 \def (eq_ind_r T u TMP_7 
+H0 TMP_8 H_y) in (let TMP_9 \def (TLRef i) in (let TMP_11 \def (\lambda (t: 
+T).(let TMP_10 \def (TLRef i) in (iso TMP_10 t))) in (let TMP_12 \def (TLRef 
+i) in (let TMP_13 \def (iso_refl TMP_12) in (eq_ind T TMP_9 TMP_11 TMP_13 u 
+H_y)))))))))))) in (let TMP_162 \def (\lambda (t: T).(\lambda (t0: 
+TList).(\lambda (H0: ((\forall (u: T).((pr3 c (THeads (Flat Appl) t0 (TLRef 
+i)) u) \to (iso (THeads (Flat Appl) t0 (TLRef i)) u))))).(\lambda (u: 
+T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef 
+i))) u)).(let TMP_15 \def (Flat Appl) in (let TMP_16 \def (TLRef i) in (let 
+TMP_17 \def (THeads TMP_15 t0 TMP_16) in (let H2 \def (pr3_gen_appl c t 
+TMP_17 u H1) in (let TMP_20 \def (\lambda (u2: T).(\lambda (t2: T).(let 
+TMP_18 \def (Flat Appl) in (let TMP_19 \def (THead TMP_18 u2 t2) in (eq T u 
+TMP_19))))) in (let TMP_21 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c t 
+u2))) in (let TMP_25 \def (\lambda (_: T).(\lambda (t2: T).(let TMP_22 \def 
+(Flat Appl) in (let TMP_23 \def (TLRef i) in (let TMP_24 \def (THeads TMP_22 
+t0 TMP_23) in (pr3 c TMP_24 t2)))))) in (let TMP_26 \def (ex3_2 T T TMP_20 
+TMP_21 TMP_25) in (let TMP_29 \def (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t2: T).(let TMP_27 \def (Bind Abbr) in (let TMP_28 \def 
+(THead TMP_27 u2 t2) in (pr3 c TMP_28 u))))))) in (let TMP_30 \def (\lambda 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) in 
+(let TMP_36 \def (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(let TMP_31 \def (Flat Appl) in (let TMP_32 \def (TLRef i) in (let 
+TMP_33 \def (THeads TMP_31 t0 TMP_32) in (let TMP_34 \def (Bind Abst) in (let 
+TMP_35 \def (THead TMP_34 y1 z1) in (pr3 c TMP_33 TMP_35)))))))))) in (let 
+TMP_39 \def (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: 
+T).(\forall (b: B).(\forall (u0: T).(let TMP_37 \def (Bind b) in (let TMP_38 
+\def (CHead c TMP_37 u0) in (pr3 TMP_38 z1 t2))))))))) in (let TMP_40 \def 
+(ex4_4 T T T T TMP_29 TMP_30 TMP_36 TMP_39) in (let TMP_42 \def (\lambda (b: 
 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat 
-Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: 
-T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
-u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2)))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef 
-i))) u) (\lambda (H3: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T u 
-(THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) 
+(_: T).(let TMP_41 \def (eq B b Abst) in (not TMP_41)))))))) in (let TMP_48 
+\def (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(let TMP_43 \def (Flat Appl) in (let 
+TMP_44 \def (TLRef i) in (let TMP_45 \def (THeads TMP_43 t0 TMP_44) in (let 
+TMP_46 \def (Bind b) in (let TMP_47 \def (THead TMP_46 y1 z1) in (pr3 c 
+TMP_45 TMP_47)))))))))))) in (let TMP_55 \def (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(let 
+TMP_49 \def (Bind b) in (let TMP_50 \def (Flat Appl) in (let TMP_51 \def (S 
+O) in (let TMP_52 \def (lift TMP_51 O u2) in (let TMP_53 \def (THead TMP_50 
+TMP_52 z2) in (let TMP_54 \def (THead TMP_49 y2 TMP_53) in (pr3 c TMP_54 
+u))))))))))))) in (let TMP_56 \def (\lambda (_: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) in 
+(let TMP_57 \def (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) in (let TMP_60 
+\def (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(let TMP_58 \def (Bind b) in (let TMP_59 
+\def (CHead c TMP_58 y2) in (pr3 TMP_59 z1 z2))))))))) in (let TMP_61 \def 
+(ex6_6 B T T T T T TMP_42 TMP_48 TMP_55 TMP_56 TMP_57 TMP_60) in (let TMP_62 
+\def (Flat Appl) in (let TMP_63 \def (Flat Appl) in (let TMP_64 \def (TLRef 
+i) in (let TMP_65 \def (THeads TMP_63 t0 TMP_64) in (let TMP_66 \def (THead 
+TMP_62 t TMP_65) in (let TMP_67 \def (iso TMP_66 u) in (let TMP_96 \def 
+(\lambda (H3: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T u (THead 
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) 
 (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
-t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T u (THead (Flat 
-Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))) (iso 
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H4: (eq T u (THead (Flat Appl) x0 
-x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 
-(TLRef i)) x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(iso 
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (iso_head t x0 
-(THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) u H4)))))) H3)) (\lambda 
-(H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat 
-Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: 
-T).(pr3 (CHead c (Bind b) u0) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: 
+t2))))).(let TMP_70 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_68 \def 
+(Flat Appl) in (let TMP_69 \def (THead TMP_68 u2 t2) in (eq T u TMP_69))))) 
+in (let TMP_71 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) in (let 
+TMP_75 \def (\lambda (_: T).(\lambda (t2: T).(let TMP_72 \def (Flat Appl) in 
+(let TMP_73 \def (TLRef i) in (let TMP_74 \def (THeads TMP_72 t0 TMP_73) in 
+(pr3 c TMP_74 t2)))))) in (let TMP_76 \def (Flat Appl) in (let TMP_77 \def 
+(Flat Appl) in (let TMP_78 \def (TLRef i) in (let TMP_79 \def (THeads TMP_77 
+t0 TMP_78) in (let TMP_80 \def (THead TMP_76 t TMP_79) in (let TMP_81 \def 
+(iso TMP_80 u) in (let TMP_95 \def (\lambda (x0: T).(\lambda (x1: T).(\lambda 
+(H4: (eq T u (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c t x0)).(\lambda 
+(_: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(let TMP_82 \def (Flat 
+Appl) in (let TMP_83 \def (THead TMP_82 x0 x1) in (let TMP_89 \def (\lambda 
+(t1: T).(let TMP_84 \def (Flat Appl) in (let TMP_85 \def (Flat Appl) in (let 
+TMP_86 \def (TLRef i) in (let TMP_87 \def (THeads TMP_85 t0 TMP_86) in (let 
+TMP_88 \def (THead TMP_84 t TMP_87) in (iso TMP_88 t1))))))) in (let TMP_90 
+\def (Flat Appl) in (let TMP_91 \def (TLRef i) in (let TMP_92 \def (THeads 
+TMP_90 t0 TMP_91) in (let TMP_93 \def (Flat Appl) in (let TMP_94 \def 
+(iso_head t x0 TMP_92 x1 TMP_93) in (eq_ind_r T TMP_83 TMP_89 TMP_94 u 
+H4)))))))))))))) in (ex3_2_ind T T TMP_70 TMP_71 TMP_75 TMP_81 TMP_95 
+H3)))))))))))) in (let TMP_125 \def (\lambda (H3: (ex4_4 T T T T (\lambda (_: 
 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind 
 Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
 (_: T).(pr3 c t u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
 T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
 (t2: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) u0) z1 
-t2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda 
-(_: (pr3 c (THead (Bind Abbr) x2 x3) u)).(\lambda (_: (pr3 c t x2)).(\lambda 
-(H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 
-x1))).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) 
-u0) x1 x3))))).(let H_y \def (H0 (THead (Bind Abst) x0 x1) H6) in 
-(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H_y (iso (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (TLRef i))) u))))))))))) H3)) (\lambda (H3: (ex6_6 B T 
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+t2))))))))).(let TMP_99 \def (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t2: T).(let TMP_97 \def (Bind Abbr) in (let TMP_98 \def (THead 
+TMP_97 u2 t2) in (pr3 c TMP_98 u))))))) in (let TMP_100 \def (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) in (let 
+TMP_106 \def (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(let TMP_101 \def (Flat Appl) in (let TMP_102 \def (TLRef i) in (let 
+TMP_103 \def (THeads TMP_101 t0 TMP_102) in (let TMP_104 \def (Bind Abst) in 
+(let TMP_105 \def (THead TMP_104 y1 z1) in (pr3 c TMP_103 TMP_105)))))))))) 
+in (let TMP_109 \def (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: T).(let TMP_107 \def (Bind 
+b) in (let TMP_108 \def (CHead c TMP_107 u0) in (pr3 TMP_108 z1 t2))))))))) 
+in (let TMP_110 \def (Flat Appl) in (let TMP_111 \def (Flat Appl) in (let 
+TMP_112 \def (TLRef i) in (let TMP_113 \def (THeads TMP_111 t0 TMP_112) in 
+(let TMP_114 \def (THead TMP_110 t TMP_113) in (let TMP_115 \def (iso TMP_114 
+u) in (let TMP_124 \def (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (_: (pr3 c (THead (Bind Abbr) x2 x3) 
+u)).(\lambda (_: (pr3 c t x2)).(\lambda (H6: (pr3 c (THeads (Flat Appl) t0 
+(TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (_: ((\forall (b: B).(\forall 
+(u0: T).(pr3 (CHead c (Bind b) u0) x1 x3))))).(let TMP_116 \def (Bind Abst) 
+in (let TMP_117 \def (THead TMP_116 x0 x1) in (let H_y \def (H0 TMP_117 H6) 
+in (let TMP_118 \def (Flat Appl) in (let TMP_119 \def (Flat Appl) in (let 
+TMP_120 \def (TLRef i) in (let TMP_121 \def (THeads TMP_119 t0 TMP_120) in 
+(let TMP_122 \def (THead TMP_118 t TMP_121) in (let TMP_123 \def (iso TMP_122 
+u) in (iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H_y 
+TMP_123)))))))))))))))))) in (ex4_4_ind T T T T TMP_99 TMP_100 TMP_106 
+TMP_109 TMP_115 TMP_124 H3))))))))))))) in (let TMP_161 \def (\lambda (H3: 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) 
 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
@@ -102,29 +146,42 @@ T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2)))))))
 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
-(CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: 
+(CHead c (Bind b) y2) z1 z2))))))))).(let TMP_127 \def (\lambda (b: 
 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat 
-Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
+(_: T).(let TMP_126 \def (eq B b Abst) in (not TMP_126)))))))) in (let 
+TMP_133 \def (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(let TMP_128 \def (Flat Appl) in (let 
+TMP_129 \def (TLRef i) in (let TMP_130 \def (THeads TMP_128 t0 TMP_129) in 
+(let TMP_131 \def (Bind b) in (let TMP_132 \def (THead TMP_131 y1 z1) in (pr3 
+c TMP_130 TMP_132)))))))))))) in (let TMP_140 \def (\lambda (b: B).(\lambda 
 (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: 
-T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
-u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) 
-u) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 
-Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
-x0) x1 x2))).(\lambda (_: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift 
-(S O) O x4) x3)) u)).(\lambda (_: (pr3 c t x4)).(\lambda (_: (pr3 c x1 
-x5)).(\lambda (_: (pr3 (CHead c (Bind x0) x5) x2 x3)).(let H_y \def (H0 
-(THead (Bind x0) x1 x2) H5) in (iso_flats_lref_bind_false Appl x0 i x1 x2 t0 
-H_y (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) 
-u))))))))))))))) H3)) H2))))))) vs)))).
-(* COMMENTS
-Initial nodes: 1817
-END *)
+T).(let TMP_134 \def (Bind b) in (let TMP_135 \def (Flat Appl) in (let 
+TMP_136 \def (S O) in (let TMP_137 \def (lift TMP_136 O u2) in (let TMP_138 
+\def (THead TMP_135 TMP_137 z2) in (let TMP_139 \def (THead TMP_134 y2 
+TMP_138) in (pr3 c TMP_139 u))))))))))))) in (let TMP_141 \def (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c t u2))))))) in (let TMP_142 \def (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
+y2))))))) in (let TMP_145 \def (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(let TMP_143 \def (Bind 
+b) in (let TMP_144 \def (CHead c TMP_143 y2) in (pr3 TMP_144 z1 z2))))))))) 
+in (let TMP_146 \def (Flat Appl) in (let TMP_147 \def (Flat Appl) in (let 
+TMP_148 \def (TLRef i) in (let TMP_149 \def (THeads TMP_147 t0 TMP_148) in 
+(let TMP_150 \def (THead TMP_146 t TMP_149) in (let TMP_151 \def (iso TMP_150 
+u) in (let TMP_160 \def (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B 
+x0 Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead 
+(Bind x0) x1 x2))).(\lambda (_: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) 
+(lift (S O) O x4) x3)) u)).(\lambda (_: (pr3 c t x4)).(\lambda (_: (pr3 c x1 
+x5)).(\lambda (_: (pr3 (CHead c (Bind x0) x5) x2 x3)).(let TMP_152 \def (Bind 
+x0) in (let TMP_153 \def (THead TMP_152 x1 x2) in (let H_y \def (H0 TMP_153 
+H5) in (let TMP_154 \def (Flat Appl) in (let TMP_155 \def (Flat Appl) in (let 
+TMP_156 \def (TLRef i) in (let TMP_157 \def (THeads TMP_155 t0 TMP_156) in 
+(let TMP_158 \def (THead TMP_154 t TMP_157) in (let TMP_159 \def (iso TMP_158 
+u) in (iso_flats_lref_bind_false Appl x0 i x1 x2 t0 H_y 
+TMP_159)))))))))))))))))))))) in (ex6_6_ind B T T T T T TMP_127 TMP_133 
+TMP_140 TMP_141 TMP_142 TMP_145 TMP_151 TMP_160 H3))))))))))))))) in (or3_ind 
+TMP_26 TMP_40 TMP_61 TMP_67 TMP_96 TMP_125 TMP_161 
+H2))))))))))))))))))))))))))))))))))) in (TList_ind TMP_4 TMP_14 TMP_162 
+vs))))))).
 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/lift1.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/lift1.ma
index d50790336..2004f1270 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/lift1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/lift1.ma
@@ -14,28 +14,30 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/nf2/props.ma".
+include "basic_1/nf2/props.ma".
 
-include "Basic-1/drop1/fwd.ma".
+include "basic_1/drop1/fwd.ma".
 
 theorem nf2_lift1:
  \forall (e: C).(\forall (hds: PList).(\forall (c: C).(\forall (t: T).((drop1 
 hds c e) \to ((nf2 e t) \to (nf2 c (lift1 hds t)))))))
 \def
- \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
-(c: C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p 
-t))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c 
-e)).(\lambda (H0: (nf2 e t)).(let H_y \def (drop1_gen_pnil c e H) in 
-(eq_ind_r C e (\lambda (c0: C).(nf2 c0 t)) H0 c H_y)))))) (\lambda (n: 
-nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: 
-C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p 
-t)))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (drop1 (PCons n n0 p) 
-c e)).(\lambda (H1: (nf2 e t)).(let H_x \def (drop1_gen_pcons c e p n n0 H0) 
-in (let H2 \def H_x in (ex2_ind C (\lambda (c2: C).(drop n n0 c c2)) (\lambda 
-(c2: C).(drop1 p c2 e)) (nf2 c (lift n n0 (lift1 p t))) (\lambda (x: 
-C).(\lambda (H3: (drop n n0 c x)).(\lambda (H4: (drop1 p x e)).(nf2_lift x 
-(lift1 p t) (H x t H4 H1) c n n0 H3)))) H2))))))))))) hds)).
-(* COMMENTS
-Initial nodes: 249
-END *)
+ \lambda (e: C).(\lambda (hds: PList).(let TMP_2 \def (\lambda (p: 
+PList).(\forall (c: C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (let 
+TMP_1 \def (lift1 p t) in (nf2 c TMP_1))))))) in (let TMP_4 \def (\lambda (c: 
+C).(\lambda (t: T).(\lambda (H: (drop1 PNil c e)).(\lambda (H0: (nf2 e 
+t)).(let H_y \def (drop1_gen_pnil c e H) in (let TMP_3 \def (\lambda (c0: 
+C).(nf2 c0 t)) in (eq_ind_r C e TMP_3 H0 c H_y))))))) in (let TMP_13 \def 
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: 
+((\forall (c: C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c 
+(lift1 p t)))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (drop1 
+(PCons n n0 p) c e)).(\lambda (H1: (nf2 e t)).(let H_x \def (drop1_gen_pcons 
+c e p n n0 H0) in (let H2 \def H_x in (let TMP_5 \def (\lambda (c2: C).(drop 
+n n0 c c2)) in (let TMP_6 \def (\lambda (c2: C).(drop1 p c2 e)) in (let TMP_7 
+\def (lift1 p t) in (let TMP_8 \def (lift n n0 TMP_7) in (let TMP_9 \def (nf2 
+c TMP_8) in (let TMP_12 \def (\lambda (x: C).(\lambda (H3: (drop n n0 c 
+x)).(\lambda (H4: (drop1 p x e)).(let TMP_10 \def (lift1 p t) in (let TMP_11 
+\def (H x t H4 H1) in (nf2_lift x TMP_10 TMP_11 c n n0 H3)))))) in (ex2_ind C 
+TMP_5 TMP_6 TMP_9 TMP_12 H2))))))))))))))))) in (PList_ind TMP_2 TMP_4 TMP_13 
+hds))))).
 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/pr3.ma
index 3db24223e..6dc1547bd 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/pr3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/pr3.ma
@@ -14,26 +14,28 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/nf2/defs.ma".
+include "basic_1/nf2/defs.ma".
 
-include "Basic-1/pr3/pr3.ma".
+include "basic_1/pr3/pr3.ma".
 
 theorem nf2_pr3_unfold:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to ((nf2 c 
 t1) \to (eq T t1 t2)))))
 \def
  \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
-t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((nf2 c t) \to (eq T t 
-t0)))) (\lambda (t: T).(\lambda (H0: (nf2 c t)).(H0 t (pr2_free c t t 
-(pr0_refl t))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 
-t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: (((nf2 c t0) 
-\to (eq T t0 t4)))).(\lambda (H3: (nf2 c t3)).(let H4 \def H3 in (let H5 \def 
-(eq_ind T t3 (\lambda (t: T).(nf2 c t)) H3 t0 (H4 t0 H0)) in (let H6 \def 
-(eq_ind T t3 (\lambda (t: T).(pr2 c t t0)) H0 t0 (H4 t0 H0)) in (eq_ind_r T 
-t0 (\lambda (t: T).(eq T t t4)) (H2 H5) t3 (H4 t0 H0)))))))))))) t1 t2 H)))).
-(* COMMENTS
-Initial nodes: 187
-END *)
+t2)).(let TMP_1 \def (\lambda (t: T).(\lambda (t0: T).((nf2 c t) \to (eq T t 
+t0)))) in (let TMP_4 \def (\lambda (t: T).(\lambda (H0: (nf2 c t)).(let TMP_2 
+\def (pr0_refl t) in (let TMP_3 \def (pr2_free c t t TMP_2) in (H0 t 
+TMP_3))))) in (let TMP_12 \def (\lambda (t0: T).(\lambda (t3: T).(\lambda 
+(H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda 
+(H2: (((nf2 c t0) \to (eq T t0 t4)))).(\lambda (H3: (nf2 c t3)).(let H4 \def 
+H3 in (let TMP_5 \def (\lambda (t: T).(nf2 c t)) in (let TMP_6 \def (H4 t0 
+H0) in (let H5 \def (eq_ind T t3 TMP_5 H3 t0 TMP_6) in (let TMP_7 \def 
+(\lambda (t: T).(pr2 c t t0)) in (let TMP_8 \def (H4 t0 H0) in (let H6 \def 
+(eq_ind T t3 TMP_7 H0 t0 TMP_8) in (let TMP_9 \def (\lambda (t: T).(eq T t 
+t4)) in (let TMP_10 \def (H2 H5) in (let TMP_11 \def (H4 t0 H0) in (eq_ind_r 
+T t0 TMP_9 TMP_10 t3 TMP_11)))))))))))))))))) in (pr3_ind c TMP_1 TMP_4 
+TMP_12 t1 t2 H))))))).
 
 theorem nf2_pr3_confluence:
  \forall (c: C).(\forall (t1: T).((nf2 c t1) \to (\forall (t2: T).((nf2 c t2) 
@@ -41,16 +43,16 @@ theorem nf2_pr3_confluence:
 \def
  \lambda (c: C).(\lambda (t1: T).(\lambda (H: (nf2 c t1)).(\lambda (t2: 
 T).(\lambda (H0: (nf2 c t2)).(\lambda (t: T).(\lambda (H1: (pr3 c t 
-t1)).(\lambda (H2: (pr3 c t t2)).(ex2_ind T (\lambda (t0: T).(pr3 c t2 t0)) 
-(\lambda (t0: T).(pr3 c t1 t0)) (eq T t1 t2) (\lambda (x: T).(\lambda (H3: 
-(pr3 c t2 x)).(\lambda (H4: (pr3 c t1 x)).(let H_y \def (nf2_pr3_unfold c t1 
-x H4 H) in (let H5 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t1 t0)) H4 t1 
-H_y) in (let H6 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t2 t0)) H3 t1 H_y) 
-in (let H_y0 \def (nf2_pr3_unfold c t2 t1 H6 H0) in (let H7 \def (eq_ind T t2 
-(\lambda (t0: T).(pr3 c t0 t1)) H6 t1 H_y0) in (eq_ind_r T t1 (\lambda (t0: 
-T).(eq T t1 t0)) (refl_equal T t1) t2 H_y0))))))))) (pr3_confluence c t t2 H2 
-t1 H1))))))))).
-(* COMMENTS
-Initial nodes: 215
-END *)
+t1)).(\lambda (H2: (pr3 c t t2)).(let TMP_1 \def (\lambda (t0: T).(pr3 c t2 
+t0)) in (let TMP_2 \def (\lambda (t0: T).(pr3 c t1 t0)) in (let TMP_3 \def 
+(eq T t1 t2) in (let TMP_9 \def (\lambda (x: T).(\lambda (H3: (pr3 c t2 
+x)).(\lambda (H4: (pr3 c t1 x)).(let H_y \def (nf2_pr3_unfold c t1 x H4 H) in 
+(let TMP_4 \def (\lambda (t0: T).(pr3 c t1 t0)) in (let H5 \def (eq_ind_r T x 
+TMP_4 H4 t1 H_y) in (let TMP_5 \def (\lambda (t0: T).(pr3 c t2 t0)) in (let 
+H6 \def (eq_ind_r T x TMP_5 H3 t1 H_y) in (let H_y0 \def (nf2_pr3_unfold c t2 
+t1 H6 H0) in (let TMP_6 \def (\lambda (t0: T).(pr3 c t0 t1)) in (let H7 \def 
+(eq_ind T t2 TMP_6 H6 t1 H_y0) in (let TMP_7 \def (\lambda (t0: T).(eq T t1 
+t0)) in (let TMP_8 \def (refl_equal T t1) in (eq_ind_r T t1 TMP_7 TMP_8 t2 
+H_y0)))))))))))))) in (let TMP_10 \def (pr3_confluence c t t2 H2 t1 H1) in 
+(ex2_ind T TMP_1 TMP_2 TMP_3 TMP_9 TMP_10))))))))))))).
 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/props.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/props.ma
index 2f0f092f2..e1431c4e3 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/props.ma
@@ -14,41 +14,52 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/nf2/defs.ma".
+include "basic_1/nf2/defs.ma".
 
-include "Basic-1/pr2/fwd.ma".
+include "basic_1/pr2/fwd.ma".
 
 theorem nf2_sort:
  \forall (c: C).(\forall (n: nat).(nf2 c (TSort n)))
 \def
  \lambda (c: C).(\lambda (n: nat).(\lambda (t2: T).(\lambda (H: (pr2 c (TSort 
-n) t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal 
-T (TSort n)) t2 (pr2_gen_sort c t2 n H))))).
-(* COMMENTS
-Initial nodes: 55
-END *)
+n) t2)).(let TMP_1 \def (TSort n) in (let TMP_3 \def (\lambda (t: T).(let 
+TMP_2 \def (TSort n) in (eq T TMP_2 t))) in (let TMP_4 \def (TSort n) in (let 
+TMP_5 \def (refl_equal T TMP_4) in (let TMP_6 \def (pr2_gen_sort c t2 n H) in 
+(eq_ind_r T TMP_1 TMP_3 TMP_5 t2 TMP_6))))))))).
 
 theorem nf2_csort_lref:
  \forall (n: nat).(\forall (i: nat).(nf2 (CSort n) (TLRef i)))
 \def
  \lambda (n: nat).(\lambda (i: nat).(\lambda (t2: T).(\lambda (H: (pr2 (CSort 
-n) (TLRef i) t2)).(let H0 \def (pr2_gen_lref (CSort n) t2 i H) in (or_ind (eq 
-T t2 (TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort n) 
-(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S 
-i) O u))))) (eq T (TLRef i) t2) (\lambda (H1: (eq T t2 (TLRef i))).(eq_ind_r 
-T (TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2 
-H1)) (\lambda (H1: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort 
-n) (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift 
-(S i) O u)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort 
-n) (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift 
-(S i) O u)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
-(H2: (getl i (CSort n) (CHead x0 (Bind Abbr) x1))).(\lambda (H3: (eq T t2 
-(lift (S i) O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T 
-(TLRef i) t)) (getl_gen_sort n i (CHead x0 (Bind Abbr) x1) H2 (eq T (TLRef i) 
-(lift (S i) O x1))) t2 H3))))) H1)) H0))))).
-(* COMMENTS
-Initial nodes: 355
-END *)
+n) (TLRef i) t2)).(let TMP_1 \def (CSort n) in (let H0 \def (pr2_gen_lref 
+TMP_1 t2 i H) in (let TMP_2 \def (TLRef i) in (let TMP_3 \def (eq T t2 TMP_2) 
+in (let TMP_7 \def (\lambda (d: C).(\lambda (u: T).(let TMP_4 \def (CSort n) 
+in (let TMP_5 \def (Bind Abbr) in (let TMP_6 \def (CHead d TMP_5 u) in (getl 
+i TMP_4 TMP_6)))))) in (let TMP_10 \def (\lambda (_: C).(\lambda (u: T).(let 
+TMP_8 \def (S i) in (let TMP_9 \def (lift TMP_8 O u) in (eq T t2 TMP_9))))) 
+in (let TMP_11 \def (ex2_2 C T TMP_7 TMP_10) in (let TMP_12 \def (TLRef i) in 
+(let TMP_13 \def (eq T TMP_12 t2) in (let TMP_19 \def (\lambda (H1: (eq T t2 
+(TLRef i))).(let TMP_14 \def (TLRef i) in (let TMP_16 \def (\lambda (t: 
+T).(let TMP_15 \def (TLRef i) in (eq T TMP_15 t))) in (let TMP_17 \def (TLRef 
+i) in (let TMP_18 \def (refl_equal T TMP_17) in (eq_ind_r T TMP_14 TMP_16 
+TMP_18 t2 H1)))))) in (let TMP_41 \def (\lambda (H1: (ex2_2 C T (\lambda (d: 
+C).(\lambda (u: T).(getl i (CSort n) (CHead d (Bind Abbr) u)))) (\lambda (_: 
+C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))))).(let TMP_23 \def (\lambda 
+(d: C).(\lambda (u: T).(let TMP_20 \def (CSort n) in (let TMP_21 \def (Bind 
+Abbr) in (let TMP_22 \def (CHead d TMP_21 u) in (getl i TMP_20 TMP_22)))))) 
+in (let TMP_26 \def (\lambda (_: C).(\lambda (u: T).(let TMP_24 \def (S i) in 
+(let TMP_25 \def (lift TMP_24 O u) in (eq T t2 TMP_25))))) in (let TMP_27 
+\def (TLRef i) in (let TMP_28 \def (eq T TMP_27 t2) in (let TMP_40 \def 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (getl i (CSort n) (CHead x0 
+(Bind Abbr) x1))).(\lambda (H3: (eq T t2 (lift (S i) O x1))).(let TMP_29 \def 
+(S i) in (let TMP_30 \def (lift TMP_29 O x1) in (let TMP_32 \def (\lambda (t: 
+T).(let TMP_31 \def (TLRef i) in (eq T TMP_31 t))) in (let TMP_33 \def (Bind 
+Abbr) in (let TMP_34 \def (CHead x0 TMP_33 x1) in (let TMP_35 \def (TLRef i) 
+in (let TMP_36 \def (S i) in (let TMP_37 \def (lift TMP_36 O x1) in (let 
+TMP_38 \def (eq T TMP_35 TMP_37) in (let TMP_39 \def (getl_gen_sort n i 
+TMP_34 H2 TMP_38) in (eq_ind_r T TMP_30 TMP_32 TMP_39 t2 H3))))))))))))))) in 
+(ex2_2_ind C T TMP_23 TMP_26 TMP_28 TMP_40 H1))))))) in (or_ind TMP_3 TMP_11 
+TMP_13 TMP_19 TMP_41 H0))))))))))))))).
 
 theorem nf2_abst:
  \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (b: B).(\forall (v: 
@@ -59,20 +70,25 @@ Abst) u t))))))))
 \to (eq T u t2))))).(\lambda (b: B).(\lambda (v: T).(\lambda (t: T).(\lambda 
 (H0: ((\forall (t2: T).((pr2 (CHead c (Bind b) v) t t2) \to (eq T t 
 t2))))).(\lambda (t2: T).(\lambda (H1: (pr2 c (THead (Bind Abst) u t) 
-t2)).(let H2 \def (pr2_gen_abst c u t t2 H1) in (ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t t3))))) (eq T (THead 
-(Bind Abst) u t) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T t2 
-(THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 c u x0)).(\lambda (H5: 
-((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t 
-x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T (THead 
-(Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) u x0 t 
-x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 b v))) t2 H3)))))) 
-H2)))))))))).
-(* COMMENTS
-Initial nodes: 299
-END *)
+t2)).(let H2 \def (pr2_gen_abst c u t t2 H1) in (let TMP_3 \def (\lambda (u2: 
+T).(\lambda (t3: T).(let TMP_1 \def (Bind Abst) in (let TMP_2 \def (THead 
+TMP_1 u2 t3) in (eq T t2 TMP_2))))) in (let TMP_4 \def (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c u u2))) in (let TMP_7 \def (\lambda (_: T).(\lambda 
+(t3: T).(\forall (b0: B).(\forall (u0: T).(let TMP_5 \def (Bind b0) in (let 
+TMP_6 \def (CHead c TMP_5 u0) in (pr2 TMP_6 t t3))))))) in (let TMP_8 \def 
+(Bind Abst) in (let TMP_9 \def (THead TMP_8 u t) in (let TMP_10 \def (eq T 
+TMP_9 t2) in (let TMP_24 \def (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: 
+(eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 c u x0)).(\lambda 
+(H5: ((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t 
+x1))))).(let TMP_11 \def (Bind Abst) in (let TMP_12 \def (THead TMP_11 x0 x1) 
+in (let TMP_15 \def (\lambda (t0: T).(let TMP_13 \def (Bind Abst) in (let 
+TMP_14 \def (THead TMP_13 u t) in (eq T TMP_14 t0)))) in (let TMP_16 \def 
+(Bind Abst) in (let TMP_17 \def (Bind Abst) in (let TMP_18 \def (Bind Abst) 
+in (let TMP_19 \def (refl_equal K TMP_18) in (let TMP_20 \def (H x0 H4) in 
+(let TMP_21 \def (H5 b v) in (let TMP_22 \def (H0 x1 TMP_21) in (let TMP_23 
+\def (f_equal3 K T T T THead TMP_16 TMP_17 u x0 t x1 TMP_19 TMP_20 TMP_22) in 
+(eq_ind_r T TMP_12 TMP_15 TMP_23 t2 H3))))))))))))))))) in (ex3_2_ind T T 
+TMP_3 TMP_4 TMP_7 TMP_10 TMP_24 H2))))))))))))))))).
 
 theorem nf2_abst_shift:
  \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (t: T).((nf2 (CHead c 
@@ -82,65 +98,228 @@ theorem nf2_abst_shift:
 \to (eq T u t2))))).(\lambda (t: T).(\lambda (H0: ((\forall (t2: T).((pr2 
 (CHead c (Bind Abst) u) t t2) \to (eq T t t2))))).(\lambda (t2: T).(\lambda 
 (H1: (pr2 c (THead (Bind Abst) u t) t2)).(let H2 \def (pr2_gen_abst c u t t2 
-H1) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
-u0) t t3))))) (eq T (THead (Bind Abst) u t) t2) (\lambda (x0: T).(\lambda 
-(x1: T).(\lambda (H3: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 
-c u x0)).(\lambda (H5: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind 
-b) u0) t x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T 
-(THead (Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) 
-u x0 t x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 Abst u))) t2 
-H3)))))) H2)))))))).
-(* COMMENTS
-Initial nodes: 295
-END *)
+H1) in (let TMP_3 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_1 \def 
+(Bind Abst) in (let TMP_2 \def (THead TMP_1 u2 t3) in (eq T t2 TMP_2))))) in 
+(let TMP_4 \def (\lambda (u2: T).(\lambda (_: T).(pr2 c u u2))) in (let TMP_7 
+\def (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(let 
+TMP_5 \def (Bind b) in (let TMP_6 \def (CHead c TMP_5 u0) in (pr2 TMP_6 t 
+t3))))))) in (let TMP_8 \def (Bind Abst) in (let TMP_9 \def (THead TMP_8 u t) 
+in (let TMP_10 \def (eq T TMP_9 t2) in (let TMP_24 \def (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H3: (eq T t2 (THead (Bind Abst) x0 
+x1))).(\lambda (H4: (pr2 c u x0)).(\lambda (H5: ((\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c (Bind b) u0) t x1))))).(let TMP_11 \def (Bind Abst) in 
+(let TMP_12 \def (THead TMP_11 x0 x1) in (let TMP_15 \def (\lambda (t0: 
+T).(let TMP_13 \def (Bind Abst) in (let TMP_14 \def (THead TMP_13 u t) in (eq 
+T TMP_14 t0)))) in (let TMP_16 \def (Bind Abst) in (let TMP_17 \def (Bind 
+Abst) in (let TMP_18 \def (Bind Abst) in (let TMP_19 \def (refl_equal K 
+TMP_18) in (let TMP_20 \def (H x0 H4) in (let TMP_21 \def (H5 Abst u) in (let 
+TMP_22 \def (H0 x1 TMP_21) in (let TMP_23 \def (f_equal3 K T T T THead TMP_16 
+TMP_17 u x0 t x1 TMP_19 TMP_20 TMP_22) in (eq_ind_r T TMP_12 TMP_15 TMP_23 t2 
+H3))))))))))))))))) in (ex3_2_ind T T TMP_3 TMP_4 TMP_7 TMP_10 TMP_24 
+H2))))))))))))))).
 
 theorem nfs2_tapp:
  \forall (c: C).(\forall (t: T).(\forall (ts: TList).((nfs2 c (TApp ts t)) 
 \to (land (nfs2 c ts) (nf2 c t)))))
 \def
- \lambda (c: C).(\lambda (t: T).(\lambda (ts: TList).(TList_ind (\lambda (t0: 
-TList).((nfs2 c (TApp t0 t)) \to (land (nfs2 c t0) (nf2 c t)))) (\lambda (H: 
-(land (nf2 c t) True)).(let H0 \def H in (land_ind (nf2 c t) True (land True 
-(nf2 c t)) (\lambda (H1: (nf2 c t)).(\lambda (_: True).(conj True (nf2 c t) I 
-H1))) H0))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (((nfs2 c 
+ \lambda (c: C).(\lambda (t: T).(\lambda (ts: TList).(let TMP_3 \def (\lambda 
+(t0: TList).((nfs2 c (TApp t0 t)) \to (let TMP_1 \def (nfs2 c t0) in (let 
+TMP_2 \def (nf2 c t) in (land TMP_1 TMP_2))))) in (let TMP_9 \def (\lambda 
+(H: (land (nf2 c t) True)).(let H0 \def H in (let TMP_4 \def (nf2 c t) in 
+(let TMP_5 \def (nf2 c t) in (let TMP_6 \def (land True TMP_5) in (let TMP_8 
+\def (\lambda (H1: (nf2 c t)).(\lambda (_: True).(let TMP_7 \def (nf2 c t) in 
+(conj True TMP_7 I H1)))) in (land_ind TMP_4 True TMP_6 TMP_8 H0))))))) in 
+(let TMP_34 \def (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (((nfs2 c 
 (TApp t1 t)) \to (land (nfs2 c t1) (nf2 c t))))).(\lambda (H0: (land (nf2 c 
-t0) (nfs2 c (TApp t1 t)))).(let H1 \def H0 in (land_ind (nf2 c t0) (nfs2 c 
-(TApp t1 t)) (land (land (nf2 c t0) (nfs2 c t1)) (nf2 c t)) (\lambda (H2: 
-(nf2 c t0)).(\lambda (H3: (nfs2 c (TApp t1 t))).(let H_x \def (H H3) in (let 
-H4 \def H_x in (land_ind (nfs2 c t1) (nf2 c t) (land (land (nf2 c t0) (nfs2 c 
-t1)) (nf2 c t)) (\lambda (H5: (nfs2 c t1)).(\lambda (H6: (nf2 c t)).(conj 
-(land (nf2 c t0) (nfs2 c t1)) (nf2 c t) (conj (nf2 c t0) (nfs2 c t1) H2 H5) 
-H6))) H4))))) H1)))))) ts))).
-(* COMMENTS
-Initial nodes: 295
-END *)
+t0) (nfs2 c (TApp t1 t)))).(let H1 \def H0 in (let TMP_10 \def (nf2 c t0) in 
+(let TMP_11 \def (TApp t1 t) in (let TMP_12 \def (nfs2 c TMP_11) in (let 
+TMP_13 \def (nf2 c t0) in (let TMP_14 \def (nfs2 c t1) in (let TMP_15 \def 
+(land TMP_13 TMP_14) in (let TMP_16 \def (nf2 c t) in (let TMP_17 \def (land 
+TMP_15 TMP_16) in (let TMP_33 \def (\lambda (H2: (nf2 c t0)).(\lambda (H3: 
+(nfs2 c (TApp t1 t))).(let H_x \def (H H3) in (let H4 \def H_x in (let TMP_18 
+\def (nfs2 c t1) in (let TMP_19 \def (nf2 c t) in (let TMP_20 \def (nf2 c t0) 
+in (let TMP_21 \def (nfs2 c t1) in (let TMP_22 \def (land TMP_20 TMP_21) in 
+(let TMP_23 \def (nf2 c t) in (let TMP_24 \def (land TMP_22 TMP_23) in (let 
+TMP_32 \def (\lambda (H5: (nfs2 c t1)).(\lambda (H6: (nf2 c t)).(let TMP_25 
+\def (nf2 c t0) in (let TMP_26 \def (nfs2 c t1) in (let TMP_27 \def (land 
+TMP_25 TMP_26) in (let TMP_28 \def (nf2 c t) in (let TMP_29 \def (nf2 c t0) 
+in (let TMP_30 \def (nfs2 c t1) in (let TMP_31 \def (conj TMP_29 TMP_30 H2 
+H5) in (conj TMP_27 TMP_28 TMP_31 H6)))))))))) in (land_ind TMP_18 TMP_19 
+TMP_24 TMP_32 H4))))))))))))) in (land_ind TMP_10 TMP_12 TMP_17 TMP_33 
+H1))))))))))))))) in (TList_ind TMP_3 TMP_9 TMP_34 ts)))))).
 
 theorem nf2_appls_lref:
  \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: 
 TList).((nfs2 c vs) \to (nf2 c (THeads (Flat Appl) vs (TLRef i)))))))
 \def
  \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda 
-(vs: TList).(TList_ind (\lambda (t: TList).((nfs2 c t) \to (nf2 c (THeads 
-(Flat Appl) t (TLRef i))))) (\lambda (_: True).H) (\lambda (t: T).(\lambda 
-(t0: TList).(\lambda (H0: (((nfs2 c t0) \to (nf2 c (THeads (Flat Appl) t0 
-(TLRef i)))))).(\lambda (H1: (land (nf2 c t) (nfs2 c t0))).(let H2 \def H1 in 
-(land_ind (nf2 c t) (nfs2 c t0) (nf2 c (THead (Flat Appl) t (THeads (Flat 
-Appl) t0 (TLRef i)))) (\lambda (H3: (nf2 c t)).(\lambda (H4: (nfs2 c 
-t0)).(let H_y \def (H0 H4) in (\lambda (t2: T).(\lambda (H5: (pr2 c (THead 
-(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2)).(let H6 \def 
-(pr2_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) t2 H5) in (or3_ind (ex3_2 
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t3)))) (ex4_4 T T T T (\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat 
-Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 
-t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+(vs: TList).(let TMP_4 \def (\lambda (t: TList).((nfs2 c t) \to (let TMP_1 
+\def (Flat Appl) in (let TMP_2 \def (TLRef i) in (let TMP_3 \def (THeads 
+TMP_1 t TMP_2) in (nf2 c TMP_3)))))) in (let TMP_5 \def (\lambda (_: True).H) 
+in (let TMP_295 \def (\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: 
+(((nfs2 c t0) \to (nf2 c (THeads (Flat Appl) t0 (TLRef i)))))).(\lambda (H1: 
+(land (nf2 c t) (nfs2 c t0))).(let H2 \def H1 in (let TMP_6 \def (nf2 c t) in 
+(let TMP_7 \def (nfs2 c t0) in (let TMP_8 \def (Flat Appl) in (let TMP_9 \def 
+(Flat Appl) in (let TMP_10 \def (TLRef i) in (let TMP_11 \def (THeads TMP_9 
+t0 TMP_10) in (let TMP_12 \def (THead TMP_8 t TMP_11) in (let TMP_13 \def 
+(nf2 c TMP_12) in (let TMP_294 \def (\lambda (H3: (nf2 c t)).(\lambda (H4: 
+(nfs2 c t0)).(let H_y \def (H0 H4) in (\lambda (t2: T).(\lambda (H5: (pr2 c 
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2)).(let TMP_14 \def 
+(Flat Appl) in (let TMP_15 \def (TLRef i) in (let TMP_16 \def (THeads TMP_14 
+t0 TMP_15) in (let H6 \def (pr2_gen_appl c t TMP_16 t2 H5) in (let TMP_19 
+\def (\lambda (u2: T).(\lambda (t3: T).(let TMP_17 \def (Flat Appl) in (let 
+TMP_18 \def (THead TMP_17 u2 t3) in (eq T t2 TMP_18))))) in (let TMP_20 \def 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))) in (let TMP_24 \def (\lambda 
+(_: T).(\lambda (t3: T).(let TMP_21 \def (Flat Appl) in (let TMP_22 \def 
+(TLRef i) in (let TMP_23 \def (THeads TMP_21 t0 TMP_22) in (pr2 c TMP_23 
+t3)))))) in (let TMP_25 \def (ex3_2 T T TMP_19 TMP_20 TMP_24) in (let TMP_31 
+\def (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(let 
+TMP_26 \def (Flat Appl) in (let TMP_27 \def (TLRef i) in (let TMP_28 \def 
+(THeads TMP_26 t0 TMP_27) in (let TMP_29 \def (Bind Abst) in (let TMP_30 \def 
+(THead TMP_29 y1 z1) in (eq T TMP_28 TMP_30)))))))))) in (let TMP_34 \def 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(let TMP_32 
+\def (Bind Abbr) in (let TMP_33 \def (THead TMP_32 u2 t3) in (eq T t2 
+TMP_33))))))) in (let TMP_35 \def (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr2 c t u2))))) in (let TMP_38 \def (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u: T).(let TMP_36 \def (Bind b) in (let TMP_37 \def (CHead c TMP_36 u) in 
+(pr2 TMP_37 z1 t3))))))))) in (let TMP_39 \def (ex4_4 T T T T TMP_31 TMP_34 
+TMP_35 TMP_38) in (let TMP_41 \def (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(let TMP_40 \def (eq B 
+b Abst) in (not TMP_40)))))))) in (let TMP_47 \def (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(let 
+TMP_42 \def (Flat Appl) in (let TMP_43 \def (TLRef i) in (let TMP_44 \def 
+(THeads TMP_42 t0 TMP_43) in (let TMP_45 \def (Bind b) in (let TMP_46 \def 
+(THead TMP_45 y1 z1) in (eq T TMP_44 TMP_46)))))))))))) in (let TMP_54 \def 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(let TMP_48 \def (Bind b) in (let TMP_49 \def (Flat 
+Appl) in (let TMP_50 \def (S O) in (let TMP_51 \def (lift TMP_50 O u2) in 
+(let TMP_52 \def (THead TMP_49 TMP_51 z2) in (let TMP_53 \def (THead TMP_48 
+y2 TMP_52) in (eq T t2 TMP_53))))))))))))) in (let TMP_55 \def (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c t u2))))))) in (let TMP_56 \def (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) in (let TMP_59 \def (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(let TMP_57 \def (Bind 
+b) in (let TMP_58 \def (CHead c TMP_57 y2) in (pr2 TMP_58 z1 z2))))))))) in 
+(let TMP_60 \def (ex6_6 B T T T T T TMP_41 TMP_47 TMP_54 TMP_55 TMP_56 
+TMP_59) in (let TMP_61 \def (Flat Appl) in (let TMP_62 \def (Flat Appl) in 
+(let TMP_63 \def (TLRef i) in (let TMP_64 \def (THeads TMP_62 t0 TMP_63) in 
+(let TMP_65 \def (THead TMP_61 t TMP_64) in (let TMP_66 \def (eq T TMP_65 t2) 
+in (let TMP_132 \def (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THeads (Flat Appl) 
+t0 (TLRef i)) t3))))).(let TMP_69 \def (\lambda (u2: T).(\lambda (t3: T).(let 
+TMP_67 \def (Flat Appl) in (let TMP_68 \def (THead TMP_67 u2 t3) in (eq T t2 
+TMP_68))))) in (let TMP_70 \def (\lambda (u2: T).(\lambda (_: T).(pr2 c t 
+u2))) in (let TMP_74 \def (\lambda (_: T).(\lambda (t3: T).(let TMP_71 \def 
+(Flat Appl) in (let TMP_72 \def (TLRef i) in (let TMP_73 \def (THeads TMP_71 
+t0 TMP_72) in (pr2 c TMP_73 t3)))))) in (let TMP_75 \def (Flat Appl) in (let 
+TMP_76 \def (Flat Appl) in (let TMP_77 \def (TLRef i) in (let TMP_78 \def 
+(THeads TMP_76 t0 TMP_77) in (let TMP_79 \def (THead TMP_75 t TMP_78) in (let 
+TMP_80 \def (eq T TMP_79 t2) in (let TMP_131 \def (\lambda (x0: T).(\lambda 
+(x1: T).(\lambda (H8: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H9: (pr2 
+c t x0)).(\lambda (H10: (pr2 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(let 
+TMP_81 \def (Flat Appl) in (let TMP_82 \def (THead TMP_81 x0 x1) in (let 
+TMP_88 \def (\lambda (t1: T).(let TMP_83 \def (Flat Appl) in (let TMP_84 \def 
+(Flat Appl) in (let TMP_85 \def (TLRef i) in (let TMP_86 \def (THeads TMP_84 
+t0 TMP_85) in (let TMP_87 \def (THead TMP_83 t TMP_86) in (eq T TMP_87 
+t1))))))) in (let TMP_92 \def (\lambda (t1: T).(let TMP_89 \def (Flat Appl) 
+in (let TMP_90 \def (TLRef i) in (let TMP_91 \def (THeads TMP_89 t0 TMP_90) 
+in (pr2 c TMP_91 t1))))) in (let TMP_93 \def (Flat Appl) in (let TMP_94 \def 
+(TLRef i) in (let TMP_95 \def (THeads TMP_93 t0 TMP_94) in (let TMP_96 \def 
+(H_y x1 H10) in (let H11 \def (eq_ind_r T x1 TMP_92 H10 TMP_95 TMP_96) in 
+(let TMP_97 \def (Flat Appl) in (let TMP_98 \def (TLRef i) in (let TMP_99 
+\def (THeads TMP_97 t0 TMP_98) in (let TMP_107 \def (\lambda (t1: T).(let 
+TMP_100 \def (Flat Appl) in (let TMP_101 \def (Flat Appl) in (let TMP_102 
+\def (TLRef i) in (let TMP_103 \def (THeads TMP_101 t0 TMP_102) in (let 
+TMP_104 \def (THead TMP_100 t TMP_103) in (let TMP_105 \def (Flat Appl) in 
+(let TMP_106 \def (THead TMP_105 x0 t1) in (eq T TMP_104 TMP_106))))))))) in 
+(let TMP_108 \def (\lambda (t1: T).(pr2 c t t1)) in (let TMP_109 \def (H3 x0 
+H9) in (let H12 \def (eq_ind_r T x0 TMP_108 H9 t TMP_109) in (let TMP_120 
+\def (\lambda (t1: T).(let TMP_110 \def (Flat Appl) in (let TMP_111 \def 
+(Flat Appl) in (let TMP_112 \def (TLRef i) in (let TMP_113 \def (THeads 
+TMP_111 t0 TMP_112) in (let TMP_114 \def (THead TMP_110 t TMP_113) in (let 
+TMP_115 \def (Flat Appl) in (let TMP_116 \def (Flat Appl) in (let TMP_117 
+\def (TLRef i) in (let TMP_118 \def (THeads TMP_116 t0 TMP_117) in (let 
+TMP_119 \def (THead TMP_115 t1 TMP_118) in (eq T TMP_114 TMP_119)))))))))))) 
+in (let TMP_121 \def (Flat Appl) in (let TMP_122 \def (Flat Appl) in (let 
+TMP_123 \def (TLRef i) in (let TMP_124 \def (THeads TMP_122 t0 TMP_123) in 
+(let TMP_125 \def (THead TMP_121 t TMP_124) in (let TMP_126 \def (refl_equal 
+T TMP_125) in (let TMP_127 \def (H3 x0 H9) in (let TMP_128 \def (eq_ind T t 
+TMP_120 TMP_126 x0 TMP_127) in (let TMP_129 \def (H_y x1 H10) in (let TMP_130 
+\def (eq_ind T TMP_99 TMP_107 TMP_128 x1 TMP_129) in (eq_ind_r T TMP_82 
+TMP_88 TMP_130 t2 H8))))))))))))))))))))))))))))))))) in (ex3_2_ind T T 
+TMP_69 TMP_70 TMP_74 TMP_80 TMP_131 H7)))))))))))) in (let TMP_201 \def 
+(\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) 
+y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 
+(CHead c (Bind b) u) z1 t3))))))))).(let TMP_138 \def (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(let TMP_133 \def (Flat 
+Appl) in (let TMP_134 \def (TLRef i) in (let TMP_135 \def (THeads TMP_133 t0 
+TMP_134) in (let TMP_136 \def (Bind Abst) in (let TMP_137 \def (THead TMP_136 
+y1 z1) in (eq T TMP_135 TMP_137)))))))))) in (let TMP_141 \def (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(let TMP_139 \def (Bind 
+Abbr) in (let TMP_140 \def (THead TMP_139 u2 t3) in (eq T t2 TMP_140))))))) 
+in (let TMP_142 \def (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c t u2))))) in (let TMP_145 \def (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u: T).(let TMP_143 \def (Bind b) in (let TMP_144 \def (CHead c TMP_143 u) in 
+(pr2 TMP_144 z1 t3))))))))) in (let TMP_146 \def (Flat Appl) in (let TMP_147 
+\def (Flat Appl) in (let TMP_148 \def (TLRef i) in (let TMP_149 \def (THeads 
+TMP_147 t0 TMP_148) in (let TMP_150 \def (THead TMP_146 t TMP_149) in (let 
+TMP_151 \def (eq T TMP_150 t2) in (let TMP_200 \def (\lambda (x0: T).(\lambda 
+(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T (THeads (Flat 
+Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (H9: (eq T t2 (THead 
+(Bind Abbr) x2 x3))).(\lambda (_: (pr2 c t x2)).(\lambda (_: ((\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let TMP_152 \def 
+(Bind Abbr) in (let TMP_153 \def (THead TMP_152 x2 x3) in (let TMP_159 \def 
+(\lambda (t1: T).(let TMP_154 \def (Flat Appl) in (let TMP_155 \def (Flat 
+Appl) in (let TMP_156 \def (TLRef i) in (let TMP_157 \def (THeads TMP_155 t0 
+TMP_156) in (let TMP_158 \def (THead TMP_154 t TMP_157) in (eq T TMP_158 
+t1))))))) in (let TMP_167 \def (\lambda (t1: TList).((nf2 c (THeads (Flat 
+Appl) t1 (TLRef i))) \to ((eq T (THeads (Flat Appl) t1 (TLRef i)) (THead 
+(Bind Abst) x0 x1)) \to (let TMP_160 \def (Flat Appl) in (let TMP_161 \def 
+(Flat Appl) in (let TMP_162 \def (TLRef i) in (let TMP_163 \def (THeads 
+TMP_161 t1 TMP_162) in (let TMP_164 \def (THead TMP_160 t TMP_163) in (let 
+TMP_165 \def (Bind Abbr) in (let TMP_166 \def (THead TMP_165 x2 x3) in (eq T 
+TMP_164 TMP_166))))))))))) in (let TMP_180 \def (\lambda (_: (nf2 c (THeads 
+(Flat Appl) TNil (TLRef i)))).(\lambda (H13: (eq T (THeads (Flat Appl) TNil 
+(TLRef i)) (THead (Bind Abst) x0 x1))).(let TMP_168 \def (TLRef i) in (let 
+TMP_169 \def (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) in (let 
+TMP_170 \def (Bind Abst) in (let TMP_171 \def (THead TMP_170 x0 x1) in (let 
+H14 \def (eq_ind T TMP_168 TMP_169 I TMP_171 H13) in (let TMP_172 \def (Flat 
+Appl) in (let TMP_173 \def (Flat Appl) in (let TMP_174 \def (TLRef i) in (let 
+TMP_175 \def (THeads TMP_173 TNil TMP_174) in (let TMP_176 \def (THead 
+TMP_172 t TMP_175) in (let TMP_177 \def (Bind Abbr) in (let TMP_178 \def 
+(THead TMP_177 x2 x3) in (let TMP_179 \def (eq T TMP_176 TMP_178) in 
+(False_ind TMP_179 H14)))))))))))))))) in (let TMP_198 \def (\lambda (t1: 
+T).(\lambda (t3: TList).(\lambda (_: (((nf2 c (THeads (Flat Appl) t3 (TLRef 
+i))) \to ((eq T (THeads (Flat Appl) t3 (TLRef i)) (THead (Bind Abst) x0 x1)) 
+\to (eq T (THead (Flat Appl) t (THeads (Flat Appl) t3 (TLRef i))) (THead 
+(Bind Abbr) x2 x3)))))).(\lambda (_: (nf2 c (THeads (Flat Appl) (TCons t1 t3) 
+(TLRef i)))).(\lambda (H13: (eq T (THeads (Flat Appl) (TCons t1 t3) (TLRef 
+i)) (THead (Bind Abst) x0 x1))).(let TMP_181 \def (Flat Appl) in (let TMP_182 
+\def (Flat Appl) in (let TMP_183 \def (TLRef i) in (let TMP_184 \def (THeads 
+TMP_182 t3 TMP_183) in (let TMP_185 \def (THead TMP_181 t1 TMP_184) in (let 
+TMP_186 \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_187 \def 
+(Bind Abst) in (let TMP_188 \def (THead TMP_187 x0 x1) in (let H14 \def 
+(eq_ind T TMP_185 TMP_186 I TMP_188 H13) in (let TMP_189 \def (Flat Appl) in 
+(let TMP_190 \def (Flat Appl) in (let TMP_191 \def (TCons t1 t3) in (let 
+TMP_192 \def (TLRef i) in (let TMP_193 \def (THeads TMP_190 TMP_191 TMP_192) 
+in (let TMP_194 \def (THead TMP_189 t TMP_193) in (let TMP_195 \def (Bind 
+Abbr) in (let TMP_196 \def (THead TMP_195 x2 x3) in (let TMP_197 \def (eq T 
+TMP_194 TMP_196) in (False_ind TMP_197 H14)))))))))))))))))))))))) in (let 
+TMP_199 \def (TList_ind TMP_167 TMP_180 TMP_198 t0 H_y H8) in (eq_ind_r T 
+TMP_153 TMP_159 TMP_199 t2 H9)))))))))))))))) in (ex4_4_ind T T T T TMP_138 
+TMP_141 TMP_142 TMP_145 TMP_151 TMP_200 H7))))))))))))) in (let TMP_293 \def 
+(\lambda (H7: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
 (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
 (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
@@ -151,141 +330,98 @@ T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t
 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (H7: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(THeads (Flat Appl) t0 (TLRef i)) t3))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(THeads (Flat Appl) t0 (TLRef i)) t3))) (eq T (THead (Flat Appl) t (THeads 
-(Flat Appl) t0 (TLRef i))) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
-(H8: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H9: (pr2 c t 
-x0)).(\lambda (H10: (pr2 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(eq_ind_r T 
-(THead (Flat Appl) x0 x1) (\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads 
-(Flat Appl) t0 (TLRef i))) t1)) (let H11 \def (eq_ind_r T x1 (\lambda (t1: 
-T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t1)) H10 (THeads (Flat Appl) t0 
-(TLRef i)) (H_y x1 H10)) in (eq_ind T (THeads (Flat Appl) t0 (TLRef i)) 
-(\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef 
-i))) (THead (Flat Appl) x0 t1))) (let H12 \def (eq_ind_r T x0 (\lambda (t1: 
-T).(pr2 c t t1)) H9 t (H3 x0 H9)) in (eq_ind T t (\lambda (t1: T).(eq T 
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) (THead (Flat Appl) t1 
-(THeads (Flat Appl) t0 (TLRef i))))) (refl_equal T (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (TLRef i)))) x0 (H3 x0 H9))) x1 (H_y x1 H10))) t2 
-H8)))))) H7)) (\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T T T 
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-z1 t3))))))) (eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) 
-t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H8: (eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) 
-x0 x1))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 
-c t x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) 
-u) x1 x3))))).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t1: T).(eq T 
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind 
-(\lambda (t1: TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T 
-(THeads (Flat Appl) t1 (TLRef i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead 
-(Flat Appl) t (THeads (Flat Appl) t1 (TLRef i))) (THead (Bind Abbr) x2 
-x3))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil (TLRef i)))).(\lambda 
-(H13: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead (Bind Abst) x0 
-x1))).(let H14 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 
-x1) H13) in (False_ind (eq T (THead (Flat Appl) t (THeads (Flat Appl) TNil 
-(TLRef i))) (THead (Bind Abbr) x2 x3)) H14)))) (\lambda (t1: T).(\lambda (t3: 
-TList).(\lambda (_: (((nf2 c (THeads (Flat Appl) t3 (TLRef i))) \to ((eq T 
-(THeads (Flat Appl) t3 (TLRef i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead 
-(Flat Appl) t (THeads (Flat Appl) t3 (TLRef i))) (THead (Bind Abbr) x2 
-x3)))))).(\lambda (_: (nf2 c (THeads (Flat Appl) (TCons t1 t3) (TLRef 
-i)))).(\lambda (H13: (eq T (THeads (Flat Appl) (TCons t1 t3) (TLRef i)) 
-(THead (Bind Abst) x0 x1))).(let H14 \def (eq_ind T (THead (Flat Appl) t1 
-(THeads (Flat Appl) t3 (TLRef i))) (\lambda (ee: T).(match ee in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind Abst) x0 x1) H13) in (False_ind (eq T (THead (Flat 
-Appl) t (THeads (Flat Appl) (TCons t1 t3) (TLRef i))) (THead (Bind Abbr) x2 
-x3)) H14))))))) t0 H_y H8) t2 H9))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T 
-T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq 
-T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S 
-O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) 
-y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(let TMP_203 \def (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(let TMP_202 \def (eq B b Abst) in (not TMP_202)))))))) in 
+(let TMP_209 \def (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(let TMP_204 \def (Flat Appl) in (let 
+TMP_205 \def (TLRef i) in (let TMP_206 \def (THeads TMP_204 t0 TMP_205) in 
+(let TMP_207 \def (Bind b) in (let TMP_208 \def (THead TMP_207 y1 z1) in (eq 
+T TMP_206 TMP_208)))))))))))) in (let TMP_216 \def (\lambda (b: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: 
+T).(let TMP_210 \def (Bind b) in (let TMP_211 \def (Flat Appl) in (let 
+TMP_212 \def (S O) in (let TMP_213 \def (lift TMP_212 O u2) in (let TMP_214 
+\def (THead TMP_211 TMP_213 z2) in (let TMP_215 \def (THead TMP_210 y2 
+TMP_214) in (eq T t2 TMP_215))))))))))))) in (let TMP_217 \def (\lambda (_: 
 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (eq T (THead 
-(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (x0: 
-B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H9: (eq T 
-(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (H10: 
-(eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
-x3)))).(\lambda (_: (pr2 c t x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: 
-(pr2 (CHead c (Bind x0) x5) x2 x3)).(eq_ind_r T (THead (Bind x0) x5 (THead 
-(Flat Appl) (lift (S O) O x4) x3)) (\lambda (t1: T).(eq T (THead (Flat Appl) 
-t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind (\lambda (t1: 
-TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T (THeads (Flat 
-Appl) t1 (TLRef i)) (THead (Bind x0) x1 x2)) \to (eq T (THead (Flat Appl) t 
-(THeads (Flat Appl) t1 (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) 
-(lift (S O) O x4) x3)))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil 
-(TLRef i)))).(\lambda (H15: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead 
-(Bind x0) x1 x2))).(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match 
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead 
-(Bind x0) x1 x2) H15) in (False_ind (eq T (THead (Flat Appl) t (THeads (Flat 
-Appl) TNil (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O 
-x4) x3))) H16)))) (\lambda (t1: T).(\lambda (t3: TList).(\lambda (_: (((nf2 c 
-(THeads (Flat Appl) t3 (TLRef i))) \to ((eq T (THeads (Flat Appl) t3 (TLRef 
-i)) (THead (Bind x0) x1 x2)) \to (eq T (THead (Flat Appl) t (THeads (Flat 
-Appl) t3 (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
-x3))))))).(\lambda (_: (nf2 c (THeads (Flat Appl) (TCons t1 t3) (TLRef 
-i)))).(\lambda (H15: (eq T (THeads (Flat Appl) (TCons t1 t3) (TLRef i)) 
-(THead (Bind x0) x1 x2))).(let H16 \def (eq_ind T (THead (Flat Appl) t1 
-(THeads (Flat Appl) t3 (TLRef i))) (\lambda (ee: T).(match ee in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind x0) x1 x2) H15) in (False_ind (eq T (THead (Flat 
-Appl) t (THeads (Flat Appl) (TCons t1 t3) (TLRef i))) (THead (Bind x0) x5 
-(THead (Flat Appl) (lift (S O) O x4) x3))) H16))))))) t0 H_y H9) t2 
-H10))))))))))))) H7)) H6))))))) H2)))))) vs)))).
-(* COMMENTS
-Initial nodes: 2915
-END *)
+(_: T).(pr2 c t u2))))))) in (let TMP_218 \def (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) in (let TMP_221 \def (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(let TMP_219 \def (Bind 
+b) in (let TMP_220 \def (CHead c TMP_219 y2) in (pr2 TMP_220 z1 z2))))))))) 
+in (let TMP_222 \def (Flat Appl) in (let TMP_223 \def (Flat Appl) in (let 
+TMP_224 \def (TLRef i) in (let TMP_225 \def (THeads TMP_223 t0 TMP_224) in 
+(let TMP_226 \def (THead TMP_222 t TMP_225) in (let TMP_227 \def (eq T 
+TMP_226 t2) in (let TMP_292 \def (\lambda (x0: B).(\lambda (x1: T).(\lambda 
+(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not 
+(eq B x0 Abst))).(\lambda (H9: (eq T (THeads (Flat Appl) t0 (TLRef i)) (THead 
+(Bind x0) x1 x2))).(\lambda (H10: (eq T t2 (THead (Bind x0) x5 (THead (Flat 
+Appl) (lift (S O) O x4) x3)))).(\lambda (_: (pr2 c t x4)).(\lambda (_: (pr2 c 
+x1 x5)).(\lambda (_: (pr2 (CHead c (Bind x0) x5) x2 x3)).(let TMP_228 \def 
+(Bind x0) in (let TMP_229 \def (Flat Appl) in (let TMP_230 \def (S O) in (let 
+TMP_231 \def (lift TMP_230 O x4) in (let TMP_232 \def (THead TMP_229 TMP_231 
+x3) in (let TMP_233 \def (THead TMP_228 x5 TMP_232) in (let TMP_239 \def 
+(\lambda (t1: T).(let TMP_234 \def (Flat Appl) in (let TMP_235 \def (Flat 
+Appl) in (let TMP_236 \def (TLRef i) in (let TMP_237 \def (THeads TMP_235 t0 
+TMP_236) in (let TMP_238 \def (THead TMP_234 t TMP_237) in (eq T TMP_238 
+t1))))))) in (let TMP_251 \def (\lambda (t1: TList).((nf2 c (THeads (Flat 
+Appl) t1 (TLRef i))) \to ((eq T (THeads (Flat Appl) t1 (TLRef i)) (THead 
+(Bind x0) x1 x2)) \to (let TMP_240 \def (Flat Appl) in (let TMP_241 \def 
+(Flat Appl) in (let TMP_242 \def (TLRef i) in (let TMP_243 \def (THeads 
+TMP_241 t1 TMP_242) in (let TMP_244 \def (THead TMP_240 t TMP_243) in (let 
+TMP_245 \def (Bind x0) in (let TMP_246 \def (Flat Appl) in (let TMP_247 \def 
+(S O) in (let TMP_248 \def (lift TMP_247 O x4) in (let TMP_249 \def (THead 
+TMP_246 TMP_248 x3) in (let TMP_250 \def (THead TMP_245 x5 TMP_249) in (eq T 
+TMP_244 TMP_250))))))))))))))) in (let TMP_268 \def (\lambda (_: (nf2 c 
+(THeads (Flat Appl) TNil (TLRef i)))).(\lambda (H15: (eq T (THeads (Flat 
+Appl) TNil (TLRef i)) (THead (Bind x0) x1 x2))).(let TMP_252 \def (TLRef i) 
+in (let TMP_253 \def (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) in 
+(let TMP_254 \def (Bind x0) in (let TMP_255 \def (THead TMP_254 x1 x2) in 
+(let H16 \def (eq_ind T TMP_252 TMP_253 I TMP_255 H15) in (let TMP_256 \def 
+(Flat Appl) in (let TMP_257 \def (Flat Appl) in (let TMP_258 \def (TLRef i) 
+in (let TMP_259 \def (THeads TMP_257 TNil TMP_258) in (let TMP_260 \def 
+(THead TMP_256 t TMP_259) in (let TMP_261 \def (Bind x0) in (let TMP_262 \def 
+(Flat Appl) in (let TMP_263 \def (S O) in (let TMP_264 \def (lift TMP_263 O 
+x4) in (let TMP_265 \def (THead TMP_262 TMP_264 x3) in (let TMP_266 \def 
+(THead TMP_261 x5 TMP_265) in (let TMP_267 \def (eq T TMP_260 TMP_266) in 
+(False_ind TMP_267 H16)))))))))))))))))))) in (let TMP_290 \def (\lambda (t1: 
+T).(\lambda (t3: TList).(\lambda (_: (((nf2 c (THeads (Flat Appl) t3 (TLRef 
+i))) \to ((eq T (THeads (Flat Appl) t3 (TLRef i)) (THead (Bind x0) x1 x2)) 
+\to (eq T (THead (Flat Appl) t (THeads (Flat Appl) t3 (TLRef i))) (THead 
+(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))))))).(\lambda (_: (nf2 
+c (THeads (Flat Appl) (TCons t1 t3) (TLRef i)))).(\lambda (H15: (eq T (THeads 
+(Flat Appl) (TCons t1 t3) (TLRef i)) (THead (Bind x0) x1 x2))).(let TMP_269 
+\def (Flat Appl) in (let TMP_270 \def (Flat Appl) in (let TMP_271 \def (TLRef 
+i) in (let TMP_272 \def (THeads TMP_270 t3 TMP_271) in (let TMP_273 \def 
+(THead TMP_269 t1 TMP_272) in (let TMP_274 \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_275 \def (Bind x0) in (let TMP_276 \def 
+(THead TMP_275 x1 x2) in (let H16 \def (eq_ind T TMP_273 TMP_274 I TMP_276 
+H15) in (let TMP_277 \def (Flat Appl) in (let TMP_278 \def (Flat Appl) in 
+(let TMP_279 \def (TCons t1 t3) in (let TMP_280 \def (TLRef i) in (let 
+TMP_281 \def (THeads TMP_278 TMP_279 TMP_280) in (let TMP_282 \def (THead 
+TMP_277 t TMP_281) in (let TMP_283 \def (Bind x0) in (let TMP_284 \def (Flat 
+Appl) in (let TMP_285 \def (S O) in (let TMP_286 \def (lift TMP_285 O x4) in 
+(let TMP_287 \def (THead TMP_284 TMP_286 x3) in (let TMP_288 \def (THead 
+TMP_283 x5 TMP_287) in (let TMP_289 \def (eq T TMP_282 TMP_288) in (False_ind 
+TMP_289 H16)))))))))))))))))))))))))))) in (let TMP_291 \def (TList_ind 
+TMP_251 TMP_268 TMP_290 t0 H_y H9) in (eq_ind_r T TMP_233 TMP_239 TMP_291 t2 
+H10)))))))))))))))))))))))) in (ex6_6_ind B T T T T T TMP_203 TMP_209 TMP_216 
+TMP_217 TMP_218 TMP_221 TMP_227 TMP_292 H7))))))))))))))) in (or3_ind TMP_25 
+TMP_39 TMP_60 TMP_66 TMP_132 TMP_201 TMP_293 
+H6))))))))))))))))))))))))))))))))))) in (land_ind TMP_6 TMP_7 TMP_13 TMP_294 
+H2))))))))))))))) in (TList_ind TMP_4 TMP_5 TMP_295 vs))))))).
 
 theorem nf2_appl_lref:
  \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (i: nat).((nf2 c 
 (TLRef i)) \to (nf2 c (THead (Flat Appl) u (TLRef i)))))))
 \def
  \lambda (c: C).(\lambda (u: T).(\lambda (H: (nf2 c u)).(\lambda (i: 
-nat).(\lambda (H0: (nf2 c (TLRef i))).(let H_y \def (nf2_appls_lref c i H0 
-(TCons u TNil)) in (H_y (conj (nf2 c u) True H I))))))).
-(* COMMENTS
-Initial nodes: 49
-END *)
+nat).(\lambda (H0: (nf2 c (TLRef i))).(let TMP_1 \def (TCons u TNil) in (let 
+H_y \def (nf2_appls_lref c i H0 TMP_1) in (let TMP_2 \def (nf2 c u) in (let 
+TMP_3 \def (conj TMP_2 True H I) in (H_y TMP_3))))))))).
 
 theorem nf2_lref_abst:
  \forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c 
@@ -293,32 +429,48 @@ theorem nf2_lref_abst:
 \def
  \lambda (c: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
 (H: (getl i c (CHead e (Bind Abst) u))).(\lambda (t2: T).(\lambda (H0: (pr2 c 
-(TLRef i) t2)).(let H1 \def (pr2_gen_lref c t2 i H0) in (or_ind (eq T t2 
-(TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c (CHead d 
-(Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift (S i) O 
-u0))))) (eq T (TLRef i) t2) (\lambda (H2: (eq T t2 (TLRef i))).(eq_ind_r T 
-(TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2 
-H2)) (\lambda (H2: (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c 
-(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift 
-(S i) O u0)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u0: T).(getl i c 
-(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift 
-(S i) O u0)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
-(H3: (getl i c (CHead x0 (Bind Abbr) x1))).(\lambda (H4: (eq T t2 (lift (S i) 
-O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T (TLRef i) t)) 
-(let H5 \def (eq_ind C (CHead e (Bind Abst) u) (\lambda (c0: C).(getl i c 
-c0)) H (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H 
-(CHead x0 (Bind Abbr) x1) H3)) in (let H6 \def (eq_ind C (CHead e (Bind Abst) 
-u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort 
-_) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return 
-(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return 
-(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | 
-Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind 
-Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H (CHead x0 (Bind Abbr) x1) 
-H3)) in (False_ind (eq T (TLRef i) (lift (S i) O x1)) H6))) t2 H4))))) H2)) 
-H1)))))))).
-(* COMMENTS
-Initial nodes: 494
-END *)
+(TLRef i) t2)).(let H1 \def (pr2_gen_lref c t2 i H0) in (let TMP_1 \def 
+(TLRef i) in (let TMP_2 \def (eq T t2 TMP_1) in (let TMP_5 \def (\lambda (d: 
+C).(\lambda (u0: T).(let TMP_3 \def (Bind Abbr) in (let TMP_4 \def (CHead d 
+TMP_3 u0) in (getl i c TMP_4))))) in (let TMP_8 \def (\lambda (_: C).(\lambda 
+(u0: T).(let TMP_6 \def (S i) in (let TMP_7 \def (lift TMP_6 O u0) in (eq T 
+t2 TMP_7))))) in (let TMP_9 \def (ex2_2 C T TMP_5 TMP_8) in (let TMP_10 \def 
+(TLRef i) in (let TMP_11 \def (eq T TMP_10 t2) in (let TMP_17 \def (\lambda 
+(H2: (eq T t2 (TLRef i))).(let TMP_12 \def (TLRef i) in (let TMP_14 \def 
+(\lambda (t: T).(let TMP_13 \def (TLRef i) in (eq T TMP_13 t))) in (let 
+TMP_15 \def (TLRef i) in (let TMP_16 \def (refl_equal T TMP_15) in (eq_ind_r 
+T TMP_12 TMP_14 TMP_16 t2 H2)))))) in (let TMP_56 \def (\lambda (H2: (ex2_2 C 
+T (\lambda (d: C).(\lambda (u0: T).(getl i c (CHead d (Bind Abbr) u0)))) 
+(\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift (S i) O u0)))))).(let TMP_20 
+\def (\lambda (d: C).(\lambda (u0: T).(let TMP_18 \def (Bind Abbr) in (let 
+TMP_19 \def (CHead d TMP_18 u0) in (getl i c TMP_19))))) in (let TMP_23 \def 
+(\lambda (_: C).(\lambda (u0: T).(let TMP_21 \def (S i) in (let TMP_22 \def 
+(lift TMP_21 O u0) in (eq T t2 TMP_22))))) in (let TMP_24 \def (TLRef i) in 
+(let TMP_25 \def (eq T TMP_24 t2) in (let TMP_55 \def (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H3: (getl i c (CHead x0 (Bind Abbr) 
+x1))).(\lambda (H4: (eq T t2 (lift (S i) O x1))).(let TMP_26 \def (S i) in 
+(let TMP_27 \def (lift TMP_26 O x1) in (let TMP_29 \def (\lambda (t: T).(let 
+TMP_28 \def (TLRef i) in (eq T TMP_28 t))) in (let TMP_30 \def (Bind Abst) in 
+(let TMP_31 \def (CHead e TMP_30 u) in (let TMP_32 \def (\lambda (c0: 
+C).(getl i c c0)) in (let TMP_33 \def (Bind Abbr) in (let TMP_34 \def (CHead 
+x0 TMP_33 x1) in (let TMP_35 \def (Bind Abst) in (let TMP_36 \def (CHead e 
+TMP_35 u) in (let TMP_37 \def (Bind Abbr) in (let TMP_38 \def (CHead x0 
+TMP_37 x1) in (let TMP_39 \def (getl_mono c TMP_36 i H TMP_38 H3) in (let H5 
+\def (eq_ind C TMP_31 TMP_32 H TMP_34 TMP_39) in (let TMP_40 \def (Bind Abst) 
+in (let TMP_41 \def (CHead e TMP_40 u) in (let TMP_42 \def (\lambda (ee: 
+C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow 
+(match k with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow False | 
+Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow 
+False])])) in (let TMP_43 \def (Bind Abbr) in (let TMP_44 \def (CHead x0 
+TMP_43 x1) in (let TMP_45 \def (Bind Abst) in (let TMP_46 \def (CHead e 
+TMP_45 u) in (let TMP_47 \def (Bind Abbr) in (let TMP_48 \def (CHead x0 
+TMP_47 x1) in (let TMP_49 \def (getl_mono c TMP_46 i H TMP_48 H3) in (let H6 
+\def (eq_ind C TMP_41 TMP_42 I TMP_44 TMP_49) in (let TMP_50 \def (TLRef i) 
+in (let TMP_51 \def (S i) in (let TMP_52 \def (lift TMP_51 O x1) in (let 
+TMP_53 \def (eq T TMP_50 TMP_52) in (let TMP_54 \def (False_ind TMP_53 H6) in 
+(eq_ind_r T TMP_27 TMP_29 TMP_54 t2 H4))))))))))))))))))))))))))))))))))) in 
+(ex2_2_ind C T TMP_20 TMP_23 TMP_25 TMP_55 H2))))))) in (or_ind TMP_2 TMP_9 
+TMP_11 TMP_17 TMP_56 H1))))))))))))))))).
 
 theorem nf2_lift:
  \forall (d: C).(\forall (t: T).((nf2 d t) \to (\forall (c: C).(\forall (h: 
@@ -327,15 +479,17 @@ nat).(\forall (i: nat).((drop h i c d) \to (nf2 c (lift h i t))))))))
  \lambda (d: C).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).((pr2 d t t2) 
 \to (eq T t t2))))).(\lambda (c: C).(\lambda (h: nat).(\lambda (i: 
 nat).(\lambda (H0: (drop h i c d)).(\lambda (t2: T).(\lambda (H1: (pr2 c 
-(lift h i t) t2)).(let H2 \def (pr2_gen_lift c t t2 h i H1 d H0) in (ex2_ind 
-T (\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t t3)) 
-(eq T (lift h i t) t2) (\lambda (x: T).(\lambda (H3: (eq T t2 (lift h i 
-x))).(\lambda (H4: (pr2 d t x)).(eq_ind_r T (lift h i x) (\lambda (t0: T).(eq 
-T (lift h i t) t0)) (let H_y \def (H x H4) in (let H5 \def (eq_ind_r T x 
-(\lambda (t0: T).(pr2 d t t0)) H4 t H_y) in (eq_ind T t (\lambda (t0: T).(eq 
-T (lift h i t) (lift h i t0))) (refl_equal T (lift h i t)) x H_y))) t2 H3)))) 
-H2)))))))))).
-(* COMMENTS
-Initial nodes: 245
-END *)
+(lift h i t) t2)).(let H2 \def (pr2_gen_lift c t t2 h i H1 d H0) in (let 
+TMP_2 \def (\lambda (t3: T).(let TMP_1 \def (lift h i t3) in (eq T t2 
+TMP_1))) in (let TMP_3 \def (\lambda (t3: T).(pr2 d t t3)) in (let TMP_4 \def 
+(lift h i t) in (let TMP_5 \def (eq T TMP_4 t2) in (let TMP_16 \def (\lambda 
+(x: T).(\lambda (H3: (eq T t2 (lift h i x))).(\lambda (H4: (pr2 d t x)).(let 
+TMP_6 \def (lift h i x) in (let TMP_8 \def (\lambda (t0: T).(let TMP_7 \def 
+(lift h i t) in (eq T TMP_7 t0))) in (let H_y \def (H x H4) in (let TMP_9 
+\def (\lambda (t0: T).(pr2 d t t0)) in (let H5 \def (eq_ind_r T x TMP_9 H4 t 
+H_y) in (let TMP_12 \def (\lambda (t0: T).(let TMP_10 \def (lift h i t) in 
+(let TMP_11 \def (lift h i t0) in (eq T TMP_10 TMP_11)))) in (let TMP_13 \def 
+(lift h i t) in (let TMP_14 \def (refl_equal T TMP_13) in (let TMP_15 \def 
+(eq_ind T t TMP_12 TMP_14 x H_y) in (eq_ind_r T TMP_6 TMP_8 TMP_15 t2 
+H3))))))))))))) in (ex2_ind T TMP_2 TMP_3 TMP_5 TMP_16 H2))))))))))))))).
 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/defs.ma
index 686e6c673..90149df32 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/sn3/defs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/sn3/defs.ma
@@ -14,16 +14,13 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/pr3/defs.ma".
+include "basic_1/pr3/defs.ma".
 
 inductive sn3 (c: C): T \to Prop \def
 | sn3_sing: \forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall 
 (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)).
 
-definition sns3:
- C \to (TList \to Prop)
-\def
- let rec sns3 (c: C) (ts: TList) on ts: Prop \def (match ts with [TNil 
-\Rightarrow True | (TCons t ts0) \Rightarrow (land (sn3 c t) (sns3 c ts0))]) 
-in sns3.
+let rec sns3 (c: C) (ts: TList) on ts: Prop \def match ts with [TNil 
+\Rightarrow True | (TCons t ts0) \Rightarrow (let TMP_1 \def (sn3 c t) in 
+(let TMP_2 \def (sns3 c ts0) in (land TMP_1 TMP_2)))].
 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/fwd.ma
index 68276fe9f..9221e890c 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/sn3/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/sn3/fwd.ma
@@ -14,184 +14,256 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/sn3/defs.ma".
+include "basic_1/sn3/defs.ma".
 
-include "Basic-1/pr3/props.ma".
+include "basic_1/pr3/props.ma".
+
+let rec sn3_ind (c: C) (P: (T \to Prop)) (f: (\forall (t1: T).(((\forall (t2: 
+T).((((eq T t1 t2) \to (\forall (P0: Prop).P0))) \to ((pr3 c t1 t2) \to (sn3 
+c t2))))) \to (((\forall (t2: T).((((eq T t1 t2) \to (\forall (P0: 
+Prop).P0))) \to ((pr3 c t1 t2) \to (P t2))))) \to (P t1))))) (t: T) (s0: sn3 
+c t) on s0: P t \def match s0 with [(sn3_sing t1 s1) \Rightarrow (let TMP_2 
+\def (\lambda (t2: T).(\lambda (p: (((eq T t1 t2) \to (\forall (P0: 
+Prop).P0)))).(\lambda (p0: (pr3 c t1 t2)).(let TMP_1 \def (s1 t2 p p0) in 
+((sn3_ind c P f) t2 TMP_1))))) in (f t1 s1 TMP_2))].
 
 theorem sn3_gen_bind:
  \forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
 (THead (Bind b) u t)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t))))))
 \def
  \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: 
-(sn3 c (THead (Bind b) u t))).(insert_eq T (THead (Bind b) u t) (\lambda (t0: 
-T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 (CHead c (Bind b) u) t))) 
-(\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T 
-y (THead (Bind b) u t0)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t0)))) 
-(unintro T u (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind b) t0 x)) 
-\to (land (sn3 c t0) (sn3 (CHead c (Bind b) t0) x))))) (sn3_ind c (\lambda 
-(t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Bind b) x x0)) \to 
-(land (sn3 c x) (sn3 (CHead c (Bind b) x) x0)))))) (\lambda (t1: T).(\lambda 
-(H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 
-c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) 
-\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall 
-(x0: T).((eq T t2 (THead (Bind b) x x0)) \to (land (sn3 c x) (sn3 (CHead c 
-(Bind b) x) x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T 
-t1 (THead (Bind b) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0: 
+(sn3 c (THead (Bind b) u t))).(let TMP_1 \def (Bind b) in (let TMP_2 \def 
+(THead TMP_1 u t) in (let TMP_3 \def (\lambda (t0: T).(sn3 c t0)) in (let 
+TMP_8 \def (\lambda (_: T).(let TMP_4 \def (sn3 c u) in (let TMP_5 \def (Bind 
+b) in (let TMP_6 \def (CHead c TMP_5 u) in (let TMP_7 \def (sn3 TMP_6 t) in 
+(land TMP_4 TMP_7)))))) in (let TMP_99 \def (\lambda (y: T).(\lambda (H0: 
+(sn3 c y)).(let TMP_13 \def (\lambda (t0: T).((eq T y (THead (Bind b) u t0)) 
+\to (let TMP_9 \def (sn3 c u) in (let TMP_10 \def (Bind b) in (let TMP_11 
+\def (CHead c TMP_10 u) in (let TMP_12 \def (sn3 TMP_11 t0) in (land TMP_9 
+TMP_12))))))) in (let TMP_18 \def (\lambda (t0: T).(\forall (x: T).((eq T y 
+(THead (Bind b) t0 x)) \to (let TMP_14 \def (sn3 c t0) in (let TMP_15 \def 
+(Bind b) in (let TMP_16 \def (CHead c TMP_15 t0) in (let TMP_17 \def (sn3 
+TMP_16 x) in (land TMP_14 TMP_17)))))))) in (let TMP_23 \def (\lambda (t0: 
+T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Bind b) x x0)) \to (let 
+TMP_19 \def (sn3 c x) in (let TMP_20 \def (Bind b) in (let TMP_21 \def (CHead 
+c TMP_20 x) in (let TMP_22 \def (sn3 TMP_21 x0) in (land TMP_19 
+TMP_22))))))))) in (let TMP_96 \def (\lambda (t1: T).(\lambda (H1: ((\forall 
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
+(sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T 
+t2 (THead (Bind b) x x0)) \to (land (sn3 c x) (sn3 (CHead c (Bind b) x) 
+x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 (THead 
+(Bind b) x x0))).(let TMP_28 \def (\lambda (t0: T).(\forall (t2: T).((((eq T 
+t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1: 
+T).(\forall (x2: T).((eq T t2 (THead (Bind b) x1 x2)) \to (let TMP_24 \def 
+(sn3 c x1) in (let TMP_25 \def (Bind b) in (let TMP_26 \def (CHead c TMP_25 
+x1) in (let TMP_27 \def (sn3 TMP_26 x2) in (land TMP_24 TMP_27)))))))))))) in 
+(let TMP_29 \def (Bind b) in (let TMP_30 \def (THead TMP_29 x x0) in (let H4 
+\def (eq_ind T t1 TMP_28 H2 TMP_30 H3) in (let TMP_31 \def (\lambda (t0: 
 T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c 
-t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Bind b) x1 
-x2)) \to (land (sn3 c x1) (sn3 (CHead c (Bind b) x1) x2))))))))) H2 (THead 
-(Bind b) x x0) H3) in (let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall 
-(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to 
-(sn3 c t2))))) H1 (THead (Bind b) x x0) H3) in (conj (sn3 c x) (sn3 (CHead c 
-(Bind b) x) x0) (sn3_sing c x (\lambda (t2: T).(\lambda (H6: (((eq T x t2) 
-\to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x t2)).(let H8 \def (H4 
-(THead (Bind b) t2 x0) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind 
-b) t2 x0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | 
-(TLRef _) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b) x 
-x0) (THead (Bind b) t2 x0) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: 
-T).(pr3 c x t0)) H7 x H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: 
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H6 x H9) in (H11 (refl_equal T 
-x) P)))))) (pr3_head_12 c x t2 H7 (Bind b) x0 x0 (pr3_refl (CHead c (Bind b) 
-t2) x0)) t2 x0 (refl_equal T (THead (Bind b) t2 x0))) in (land_ind (sn3 c t2) 
-(sn3 (CHead c (Bind b) t2) x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda 
-(_: (sn3 (CHead c (Bind b) t2) x0)).H9)) H8)))))) (sn3_sing (CHead c (Bind b) 
-x) x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P: 
-Prop).P)))).(\lambda (H7: (pr3 (CHead c (Bind b) x) x0 t2)).(let H8 \def (H4 
-(THead (Bind b) x t2) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind 
-b) x t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind b) x 
-x0) (THead (Bind b) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: 
-T).(pr3 (CHead c (Bind b) x) x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T 
-t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in 
-(H11 (refl_equal T x0) P)))))) (pr3_head_12 c x x (pr3_refl c x) (Bind b) x0 
-t2 H7) x t2 (refl_equal T (THead (Bind b) x t2))) in (land_ind (sn3 c x) (sn3 
-(CHead c (Bind b) x) t2) (sn3 (CHead c (Bind b) x) t2) (\lambda (_: (sn3 c 
-x)).(\lambda (H10: (sn3 (CHead c (Bind b) x) t2)).H10)) H8))))))))))))))) y 
-H0))))) H))))).
-(* COMMENTS
-Initial nodes: 1055
-END *)
+t0 t2) \to (sn3 c t2))))) in (let TMP_32 \def (Bind b) in (let TMP_33 \def 
+(THead TMP_32 x x0) in (let H5 \def (eq_ind T t1 TMP_31 H1 TMP_33 H3) in (let 
+TMP_34 \def (sn3 c x) in (let TMP_35 \def (Bind b) in (let TMP_36 \def (CHead 
+c TMP_35 x) in (let TMP_37 \def (sn3 TMP_36 x0) in (let TMP_63 \def (\lambda 
+(t2: T).(\lambda (H6: (((eq T x t2) \to (\forall (P: Prop).P)))).(\lambda 
+(H7: (pr3 c x t2)).(let TMP_38 \def (Bind b) in (let TMP_39 \def (THead 
+TMP_38 t2 x0) in (let TMP_48 \def (\lambda (H8: (eq T (THead (Bind b) x x0) 
+(THead (Bind b) t2 x0))).(\lambda (P: Prop).(let TMP_40 \def (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead 
+_ t0 _) \Rightarrow t0])) in (let TMP_41 \def (Bind b) in (let TMP_42 \def 
+(THead TMP_41 x x0) in (let TMP_43 \def (Bind b) in (let TMP_44 \def (THead 
+TMP_43 t2 x0) in (let H9 \def (f_equal T T TMP_40 TMP_42 TMP_44 H8) in (let 
+TMP_45 \def (\lambda (t0: T).(pr3 c x t0)) in (let H10 \def (eq_ind_r T t2 
+TMP_45 H7 x H9) in (let TMP_46 \def (\lambda (t0: T).((eq T x t0) \to 
+(\forall (P0: Prop).P0))) in (let H11 \def (eq_ind_r T t2 TMP_46 H6 x H9) in 
+(let TMP_47 \def (refl_equal T x) in (H11 TMP_47 P)))))))))))))) in (let 
+TMP_49 \def (Bind b) in (let TMP_50 \def (Bind b) in (let TMP_51 \def (CHead 
+c TMP_50 t2) in (let TMP_52 \def (pr3_refl TMP_51 x0) in (let TMP_53 \def 
+(pr3_head_12 c x t2 H7 TMP_49 x0 x0 TMP_52) in (let TMP_54 \def (Bind b) in 
+(let TMP_55 \def (THead TMP_54 t2 x0) in (let TMP_56 \def (refl_equal T 
+TMP_55) in (let H8 \def (H4 TMP_39 TMP_48 TMP_53 t2 x0 TMP_56) in (let TMP_57 
+\def (sn3 c t2) in (let TMP_58 \def (Bind b) in (let TMP_59 \def (CHead c 
+TMP_58 t2) in (let TMP_60 \def (sn3 TMP_59 x0) in (let TMP_61 \def (sn3 c t2) 
+in (let TMP_62 \def (\lambda (H9: (sn3 c t2)).(\lambda (_: (sn3 (CHead c 
+(Bind b) t2) x0)).H9)) in (land_ind TMP_57 TMP_60 TMP_61 TMP_62 
+H8)))))))))))))))))))))) in (let TMP_64 \def (sn3_sing c x TMP_63) in (let 
+TMP_65 \def (Bind b) in (let TMP_66 \def (CHead c TMP_65 x) in (let TMP_94 
+\def (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P: 
+Prop).P)))).(\lambda (H7: (pr3 (CHead c (Bind b) x) x0 t2)).(let TMP_67 \def 
+(Bind b) in (let TMP_68 \def (THead TMP_67 x t2) in (let TMP_79 \def (\lambda 
+(H8: (eq T (THead (Bind b) x x0) (THead (Bind b) x t2))).(\lambda (P: 
+Prop).(let TMP_69 \def (\lambda (e: T).(match e with [(TSort _) \Rightarrow 
+x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) in (let 
+TMP_70 \def (Bind b) in (let TMP_71 \def (THead TMP_70 x x0) in (let TMP_72 
+\def (Bind b) in (let TMP_73 \def (THead TMP_72 x t2) in (let H9 \def 
+(f_equal T T TMP_69 TMP_71 TMP_73 H8) in (let TMP_76 \def (\lambda (t0: 
+T).(let TMP_74 \def (Bind b) in (let TMP_75 \def (CHead c TMP_74 x) in (pr3 
+TMP_75 x0 t0)))) in (let H10 \def (eq_ind_r T t2 TMP_76 H7 x0 H9) in (let 
+TMP_77 \def (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) in 
+(let H11 \def (eq_ind_r T t2 TMP_77 H6 x0 H9) in (let TMP_78 \def (refl_equal 
+T x0) in (H11 TMP_78 P)))))))))))))) in (let TMP_80 \def (pr3_refl c x) in 
+(let TMP_81 \def (Bind b) in (let TMP_82 \def (pr3_head_12 c x x TMP_80 
+TMP_81 x0 t2 H7) in (let TMP_83 \def (Bind b) in (let TMP_84 \def (THead 
+TMP_83 x t2) in (let TMP_85 \def (refl_equal T TMP_84) in (let H8 \def (H4 
+TMP_68 TMP_79 TMP_82 x t2 TMP_85) in (let TMP_86 \def (sn3 c x) in (let 
+TMP_87 \def (Bind b) in (let TMP_88 \def (CHead c TMP_87 x) in (let TMP_89 
+\def (sn3 TMP_88 t2) in (let TMP_90 \def (Bind b) in (let TMP_91 \def (CHead 
+c TMP_90 x) in (let TMP_92 \def (sn3 TMP_91 t2) in (let TMP_93 \def (\lambda 
+(_: (sn3 c x)).(\lambda (H10: (sn3 (CHead c (Bind b) x) t2)).H10)) in 
+(land_ind TMP_86 TMP_89 TMP_92 TMP_93 H8)))))))))))))))))))))) in (let TMP_95 
+\def (sn3_sing TMP_66 x0 TMP_94) in (conj TMP_34 TMP_37 TMP_64 
+TMP_95))))))))))))))))))))))))) in (let TMP_97 \def (sn3_ind c TMP_23 TMP_96 
+y H0) in (let TMP_98 \def (unintro T u TMP_18 TMP_97) in (unintro T t TMP_13 
+TMP_98))))))))) in (insert_eq T TMP_2 TMP_3 TMP_8 TMP_99 H)))))))))).
 
 theorem sn3_gen_flat:
  \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
 (THead (Flat f) u t)) \to (land (sn3 c u) (sn3 c t))))))
 \def
  \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: 
-(sn3 c (THead (Flat f) u t))).(insert_eq T (THead (Flat f) u t) (\lambda (t0: 
-T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 c t))) (\lambda (y: 
-T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T y (THead 
-(Flat f) u t0)) \to (land (sn3 c u) (sn3 c t0)))) (unintro T u (\lambda (t0: 
-T).(\forall (x: T).((eq T y (THead (Flat f) t0 x)) \to (land (sn3 c t0) (sn3 
-c x))))) (sn3_ind c (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T 
-t0 (THead (Flat f) x x0)) \to (land (sn3 c x) (sn3 c x0)))))) (\lambda (t1: 
-T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall 
+(sn3 c (THead (Flat f) u t))).(let TMP_1 \def (Flat f) in (let TMP_2 \def 
+(THead TMP_1 u t) in (let TMP_3 \def (\lambda (t0: T).(sn3 c t0)) in (let 
+TMP_6 \def (\lambda (_: T).(let TMP_4 \def (sn3 c u) in (let TMP_5 \def (sn3 
+c t) in (land TMP_4 TMP_5)))) in (let TMP_75 \def (\lambda (y: T).(\lambda 
+(H0: (sn3 c y)).(let TMP_9 \def (\lambda (t0: T).((eq T y (THead (Flat f) u 
+t0)) \to (let TMP_7 \def (sn3 c u) in (let TMP_8 \def (sn3 c t0) in (land 
+TMP_7 TMP_8))))) in (let TMP_12 \def (\lambda (t0: T).(\forall (x: T).((eq T 
+y (THead (Flat f) t0 x)) \to (let TMP_10 \def (sn3 c t0) in (let TMP_11 \def 
+(sn3 c x) in (land TMP_10 TMP_11)))))) in (let TMP_15 \def (\lambda (t0: 
+T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat f) x x0)) \to (let 
+TMP_13 \def (sn3 c x) in (let TMP_14 \def (sn3 c x0) in (land TMP_13 
+TMP_14))))))) in (let TMP_72 \def (\lambda (t1: T).(\lambda (H1: ((\forall 
 (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
-(\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat f) x x0)) \to (land 
-(sn3 c x) (sn3 c x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: 
-(eq T t1 (THead (Flat f) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0: 
-T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c 
-t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Flat f) x1 
-x2)) \to (land (sn3 c x1) (sn3 c x2))))))))) H2 (THead (Flat f) x x0) H3) in 
-(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) 
-\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead 
-(Flat f) x x0) H3) in (conj (sn3 c x) (sn3 c x0) (sn3_sing c x (\lambda (t2: 
-T).(\lambda (H6: (((eq T x t2) \to (\forall (P: Prop).P)))).(\lambda (H7: 
-(pr3 c x t2)).(let H8 \def (H4 (THead (Flat f) t2 x0) (\lambda (H8: (eq T 
-(THead (Flat f) x x0) (THead (Flat f) t2 x0))).(\lambda (P: Prop).(let H9 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat f) x x0) (THead (Flat f) t2 x0) H8) in (let 
-H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c x t0)) H7 x H9) in (let H11 
-\def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: 
-Prop).P0))) H6 x H9) in (H11 (refl_equal T x) P)))))) (pr3_head_12 c x t2 H7 
-(Flat f) x0 x0 (pr3_refl (CHead c (Flat f) t2) x0)) t2 x0 (refl_equal T 
-(THead (Flat f) t2 x0))) in (land_ind (sn3 c t2) (sn3 c x0) (sn3 c t2) 
-(\lambda (H9: (sn3 c t2)).(\lambda (_: (sn3 c x0)).H9)) H8)))))) (sn3_sing c 
-x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P: 
-Prop).P)))).(\lambda (H7: (pr3 c x0 t2)).(let H8 \def (H4 (THead (Flat f) x 
-t2) (\lambda (H8: (eq T (THead (Flat f) x x0) (THead (Flat f) x 
-t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) 
-\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat f) x x0) 
-(THead (Flat f) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: 
-T).(pr3 c x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: 
-T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in (H11 (refl_equal 
-T x0) P)))))) (pr3_thin_dx c x0 t2 H7 x f) x t2 (refl_equal T (THead (Flat f) 
-x t2))) in (land_ind (sn3 c x) (sn3 c t2) (sn3 c t2) (\lambda (_: (sn3 c 
-x)).(\lambda (H10: (sn3 c t2)).H10)) H8))))))))))))))) y H0))))) H))))).
-(* COMMENTS
-Initial nodes: 925
-END *)
+(sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T 
+t2 (THead (Flat f) x x0)) \to (land (sn3 c x) (sn3 c x0)))))))))).(\lambda 
+(x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 (THead (Flat f) x x0))).(let 
+TMP_18 \def (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall 
+(P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq 
+T t2 (THead (Flat f) x1 x2)) \to (let TMP_16 \def (sn3 c x1) in (let TMP_17 
+\def (sn3 c x2) in (land TMP_16 TMP_17)))))))))) in (let TMP_19 \def (Flat f) 
+in (let TMP_20 \def (THead TMP_19 x x0) in (let H4 \def (eq_ind T t1 TMP_18 
+H2 TMP_20 H3) in (let TMP_21 \def (\lambda (t0: T).(\forall (t2: T).((((eq T 
+t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2))))) in 
+(let TMP_22 \def (Flat f) in (let TMP_23 \def (THead TMP_22 x x0) in (let H5 
+\def (eq_ind T t1 TMP_21 H1 TMP_23 H3) in (let TMP_24 \def (sn3 c x) in (let 
+TMP_25 \def (sn3 c x0) in (let TMP_49 \def (\lambda (t2: T).(\lambda (H6: 
+(((eq T x t2) \to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x t2)).(let 
+TMP_26 \def (Flat f) in (let TMP_27 \def (THead TMP_26 t2 x0) in (let TMP_36 
+\def (\lambda (H8: (eq T (THead (Flat f) x x0) (THead (Flat f) t2 
+x0))).(\lambda (P: Prop).(let TMP_28 \def (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t0 _) 
+\Rightarrow t0])) in (let TMP_29 \def (Flat f) in (let TMP_30 \def (THead 
+TMP_29 x x0) in (let TMP_31 \def (Flat f) in (let TMP_32 \def (THead TMP_31 
+t2 x0) in (let H9 \def (f_equal T T TMP_28 TMP_30 TMP_32 H8) in (let TMP_33 
+\def (\lambda (t0: T).(pr3 c x t0)) in (let H10 \def (eq_ind_r T t2 TMP_33 H7 
+x H9) in (let TMP_34 \def (\lambda (t0: T).((eq T x t0) \to (\forall (P0: 
+Prop).P0))) in (let H11 \def (eq_ind_r T t2 TMP_34 H6 x H9) in (let TMP_35 
+\def (refl_equal T x) in (H11 TMP_35 P)))))))))))))) in (let TMP_37 \def 
+(Flat f) in (let TMP_38 \def (Flat f) in (let TMP_39 \def (CHead c TMP_38 t2) 
+in (let TMP_40 \def (pr3_refl TMP_39 x0) in (let TMP_41 \def (pr3_head_12 c x 
+t2 H7 TMP_37 x0 x0 TMP_40) in (let TMP_42 \def (Flat f) in (let TMP_43 \def 
+(THead TMP_42 t2 x0) in (let TMP_44 \def (refl_equal T TMP_43) in (let H8 
+\def (H4 TMP_27 TMP_36 TMP_41 t2 x0 TMP_44) in (let TMP_45 \def (sn3 c t2) in 
+(let TMP_46 \def (sn3 c x0) in (let TMP_47 \def (sn3 c t2) in (let TMP_48 
+\def (\lambda (H9: (sn3 c t2)).(\lambda (_: (sn3 c x0)).H9)) in (land_ind 
+TMP_45 TMP_46 TMP_47 TMP_48 H8)))))))))))))))))))) in (let TMP_50 \def 
+(sn3_sing c x TMP_49) in (let TMP_70 \def (\lambda (t2: T).(\lambda (H6: 
+(((eq T x0 t2) \to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x0 t2)).(let 
+TMP_51 \def (Flat f) in (let TMP_52 \def (THead TMP_51 x t2) in (let TMP_61 
+\def (\lambda (H8: (eq T (THead (Flat f) x x0) (THead (Flat f) x 
+t2))).(\lambda (P: Prop).(let TMP_53 \def (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) 
+\Rightarrow t0])) in (let TMP_54 \def (Flat f) in (let TMP_55 \def (THead 
+TMP_54 x x0) in (let TMP_56 \def (Flat f) in (let TMP_57 \def (THead TMP_56 x 
+t2) in (let H9 \def (f_equal T T TMP_53 TMP_55 TMP_57 H8) in (let TMP_58 \def 
+(\lambda (t0: T).(pr3 c x0 t0)) in (let H10 \def (eq_ind_r T t2 TMP_58 H7 x0 
+H9) in (let TMP_59 \def (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: 
+Prop).P0))) in (let H11 \def (eq_ind_r T t2 TMP_59 H6 x0 H9) in (let TMP_60 
+\def (refl_equal T x0) in (H11 TMP_60 P)))))))))))))) in (let TMP_62 \def 
+(pr3_thin_dx c x0 t2 H7 x f) in (let TMP_63 \def (Flat f) in (let TMP_64 \def 
+(THead TMP_63 x t2) in (let TMP_65 \def (refl_equal T TMP_64) in (let H8 \def 
+(H4 TMP_52 TMP_61 TMP_62 x t2 TMP_65) in (let TMP_66 \def (sn3 c x) in (let 
+TMP_67 \def (sn3 c t2) in (let TMP_68 \def (sn3 c t2) in (let TMP_69 \def 
+(\lambda (_: (sn3 c x)).(\lambda (H10: (sn3 c t2)).H10)) in (land_ind TMP_66 
+TMP_67 TMP_68 TMP_69 H8)))))))))))))))) in (let TMP_71 \def (sn3_sing c x0 
+TMP_70) in (conj TMP_24 TMP_25 TMP_50 TMP_71))))))))))))))))))))) in (let 
+TMP_73 \def (sn3_ind c TMP_15 TMP_72 y H0) in (let TMP_74 \def (unintro T u 
+TMP_12 TMP_73) in (unintro T t TMP_9 TMP_74))))))))) in (insert_eq T TMP_2 
+TMP_3 TMP_6 TMP_75 H)))))))))).
 
 theorem sn3_gen_head:
  \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
 (THead k u t)) \to (sn3 c u)))))
 \def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (c: C).(\forall (u: 
-T).(\forall (t: T).((sn3 c (THead k0 u t)) \to (sn3 c u)))))) (\lambda (b: 
-B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead 
-(Bind b) u t))).(let H_x \def (sn3_gen_bind b c u t H) in (let H0 \def H_x in 
-(land_ind (sn3 c u) (sn3 (CHead c (Bind b) u) t) (sn3 c u) (\lambda (H1: (sn3 
-c u)).(\lambda (_: (sn3 (CHead c (Bind b) u) t)).H1)) H0)))))))) (\lambda (f: 
-F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead 
-(Flat f) u t))).(let H_x \def (sn3_gen_flat f c u t H) in (let H0 \def H_x in 
-(land_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_: 
-(sn3 c t)).H1)) H0)))))))) k).
-(* COMMENTS
-Initial nodes: 191
-END *)
+ \lambda (k: K).(let TMP_1 \def (\lambda (k0: K).(\forall (c: C).(\forall (u: 
+T).(\forall (t: T).((sn3 c (THead k0 u t)) \to (sn3 c u)))))) in (let TMP_8 
+\def (\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda 
+(H: (sn3 c (THead (Bind b) u t))).(let H_x \def (sn3_gen_bind b c u t H) in 
+(let H0 \def H_x in (let TMP_2 \def (sn3 c u) in (let TMP_3 \def (Bind b) in 
+(let TMP_4 \def (CHead c TMP_3 u) in (let TMP_5 \def (sn3 TMP_4 t) in (let 
+TMP_6 \def (sn3 c u) in (let TMP_7 \def (\lambda (H1: (sn3 c u)).(\lambda (_: 
+(sn3 (CHead c (Bind b) u) t)).H1)) in (land_ind TMP_2 TMP_5 TMP_6 TMP_7 
+H0)))))))))))))) in (let TMP_13 \def (\lambda (f: F).(\lambda (c: C).(\lambda 
+(u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead (Flat f) u t))).(let H_x 
+\def (sn3_gen_flat f c u t H) in (let H0 \def H_x in (let TMP_9 \def (sn3 c 
+u) in (let TMP_10 \def (sn3 c t) in (let TMP_11 \def (sn3 c u) in (let TMP_12 
+\def (\lambda (H1: (sn3 c u)).(\lambda (_: (sn3 c t)).H1)) in (land_ind TMP_9 
+TMP_10 TMP_11 TMP_12 H0)))))))))))) in (K_ind TMP_1 TMP_8 TMP_13 k)))).
 
 theorem sn3_gen_cflat:
  \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 (CHead 
 c (Flat f) u) t) \to (sn3 c t)))))
 \def
  \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: 
-(sn3 (CHead c (Flat f) u) t)).(sn3_ind (CHead c (Flat f) u) (\lambda (t0: 
-T).(sn3 c t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 
-t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to 
-(sn3 (CHead c (Flat f) u) t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T 
-t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to 
-(sn3 c t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) 
-\to (\forall (P: Prop).P)))).(\lambda (H3: (pr3 c t1 t2)).(H1 t2 H2 
-(pr3_cflat c t1 t2 H3 f u))))))))) t H))))).
-(* COMMENTS
-Initial nodes: 175
-END *)
+(sn3 (CHead c (Flat f) u) t)).(let TMP_1 \def (Flat f) in (let TMP_2 \def 
+(CHead c TMP_1 u) in (let TMP_3 \def (\lambda (t0: T).(sn3 c t0)) in (let 
+TMP_6 \def (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) 
+\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to (sn3 
+(CHead c (Flat f) u) t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) 
+\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to (sn3 c 
+t2)))))).(let TMP_5 \def (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) \to 
+(\forall (P: Prop).P)))).(\lambda (H3: (pr3 c t1 t2)).(let TMP_4 \def 
+(pr3_cflat c t1 t2 H3 f u) in (H1 t2 H2 TMP_4))))) in (sn3_sing c t1 
+TMP_5))))) in (sn3_ind TMP_2 TMP_3 TMP_6 t H))))))))).
 
 theorem sn3_gen_lift:
  \forall (c1: C).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((sn3 c1 
 (lift h d t)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t)))))))
 \def
  \lambda (c1: C).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
-(H: (sn3 c1 (lift h d t))).(insert_eq T (lift h d t) (\lambda (t0: T).(sn3 c1 
-t0)) (\lambda (_: T).(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t)))) 
-(\lambda (y: T).(\lambda (H0: (sn3 c1 y)).(unintro T t (\lambda (t0: T).((eq 
-T y (lift h d t0)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t0))))) 
-(sn3_ind c1 (\lambda (t0: T).(\forall (x: T).((eq T t0 (lift h d x)) \to 
-(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 x)))))) (\lambda (t1: 
-T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 c1 t1 t2) \to (sn3 c1 t2)))))).(\lambda (H2: ((\forall 
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t1 t2) \to 
-(\forall (x: T).((eq T t2 (lift h d x)) \to (\forall (c2: C).((drop h d c1 
-c2) \to (sn3 c2 x)))))))))).(\lambda (x: T).(\lambda (H3: (eq T t1 (lift h d 
-x))).(\lambda (c2: C).(\lambda (H4: (drop h d c1 c2)).(let H5 \def (eq_ind T 
-t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 c1 t0 t2) \to (\forall (x0: T).((eq T t2 (lift h d x0)) 
-\to (\forall (c3: C).((drop h d c1 c3) \to (sn3 c3 x0))))))))) H2 (lift h d 
-x) H3) in (let H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq 
-T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t0 t2) \to (sn3 c1 t2))))) 
-H1 (lift h d x) H3) in (sn3_sing c2 x (\lambda (t2: T).(\lambda (H7: (((eq T 
-x t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr3 c2 x t2)).(H5 (lift h d 
-t2) (\lambda (H9: (eq T (lift h d x) (lift h d t2))).(\lambda (P: Prop).(let 
-H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c2 x t0)) H8 x (lift_inj x t2 h 
-d H9)) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to 
-(\forall (P0: Prop).P0))) H7 x (lift_inj x t2 h d H9)) in (H11 (refl_equal T 
-x) P))))) (pr3_lift c1 c2 h d H4 x t2 H8) t2 (refl_equal T (lift h d t2)) c2 
-H4)))))))))))))) y H0)))) H))))).
-(* COMMENTS
-Initial nodes: 565
-END *)
+(H: (sn3 c1 (lift h d t))).(let TMP_1 \def (lift h d t) in (let TMP_2 \def 
+(\lambda (t0: T).(sn3 c1 t0)) in (let TMP_3 \def (\lambda (_: T).(\forall 
+(c2: C).((drop h d c1 c2) \to (sn3 c2 t)))) in (let TMP_23 \def (\lambda (y: 
+T).(\lambda (H0: (sn3 c1 y)).(let TMP_4 \def (\lambda (t0: T).((eq T y (lift 
+h d t0)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t0))))) in (let 
+TMP_5 \def (\lambda (t0: T).(\forall (x: T).((eq T t0 (lift h d x)) \to 
+(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 x)))))) in (let TMP_21 \def 
+(\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
+(P: Prop).P))) \to ((pr3 c1 t1 t2) \to (sn3 c1 t2)))))).(\lambda (H2: 
+((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t1 
+t2) \to (\forall (x: T).((eq T t2 (lift h d x)) \to (\forall (c2: C).((drop h 
+d c1 c2) \to (sn3 c2 x)))))))))).(\lambda (x: T).(\lambda (H3: (eq T t1 (lift 
+h d x))).(\lambda (c2: C).(\lambda (H4: (drop h d c1 c2)).(let TMP_6 \def 
+(\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) 
+\to ((pr3 c1 t0 t2) \to (\forall (x0: T).((eq T t2 (lift h d x0)) \to 
+(\forall (c3: C).((drop h d c1 c3) \to (sn3 c3 x0))))))))) in (let TMP_7 \def 
+(lift h d x) in (let H5 \def (eq_ind T t1 TMP_6 H2 TMP_7 H3) in (let TMP_8 
+\def (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: 
+Prop).P))) \to ((pr3 c1 t0 t2) \to (sn3 c1 t2))))) in (let TMP_9 \def (lift h 
+d x) in (let H6 \def (eq_ind T t1 TMP_8 H1 TMP_9 H3) in (let TMP_20 \def 
+(\lambda (t2: T).(\lambda (H7: (((eq T x t2) \to (\forall (P: 
+Prop).P)))).(\lambda (H8: (pr3 c2 x t2)).(let TMP_10 \def (lift h d t2) in 
+(let TMP_16 \def (\lambda (H9: (eq T (lift h d x) (lift h d t2))).(\lambda 
+(P: Prop).(let TMP_11 \def (\lambda (t0: T).(pr3 c2 x t0)) in (let TMP_12 
+\def (lift_inj x t2 h d H9) in (let H10 \def (eq_ind_r T t2 TMP_11 H8 x 
+TMP_12) in (let TMP_13 \def (\lambda (t0: T).((eq T x t0) \to (\forall (P0: 
+Prop).P0))) in (let TMP_14 \def (lift_inj x t2 h d H9) in (let H11 \def 
+(eq_ind_r T t2 TMP_13 H7 x TMP_14) in (let TMP_15 \def (refl_equal T x) in 
+(H11 TMP_15 P)))))))))) in (let TMP_17 \def (pr3_lift c1 c2 h d H4 x t2 H8) 
+in (let TMP_18 \def (lift h d t2) in (let TMP_19 \def (refl_equal T TMP_18) 
+in (H5 TMP_10 TMP_16 TMP_17 t2 TMP_19 c2 H4))))))))) in (sn3_sing c2 x 
+TMP_20))))))))))))))) in (let TMP_22 \def (sn3_ind c1 TMP_5 TMP_21 y H0) in 
+(unintro T t TMP_4 TMP_22))))))) in (insert_eq T TMP_1 TMP_2 TMP_3 TMP_23 
+H))))))))).
 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/lift1.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/lift1.ma
index 1b64c22bb..51bc3426c 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/sn3/lift1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/sn3/lift1.ma
@@ -14,33 +14,40 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/sn3/props.ma".
+include "basic_1/sn3/props.ma".
 
-include "Basic-1/drop1/fwd.ma".
+include "basic_1/drop1/fwd.ma".
 
-include "Basic-1/lift1/fwd.ma".
+include "basic_1/lift1/props.ma".
 
 theorem sns3_lifts1:
  \forall (e: C).(\forall (hds: PList).(\forall (c: C).((drop1 hds c e) \to 
 (\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 hds ts)))))))
 \def
- \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
-(c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to (sns3 c 
-(lifts1 p ts))))))) (\lambda (c: C).(\lambda (H: (drop1 PNil c e)).(\lambda 
-(ts: TList).(\lambda (H0: (sns3 e ts)).(let H_y \def (drop1_gen_pnil c e H) 
-in (eq_ind_r C e (\lambda (c0: C).(sns3 c0 (lifts1 PNil ts))) (eq_ind_r TList 
-ts (\lambda (t: TList).(sns3 e t)) H0 (lifts1 PNil ts) (lifts1_nil ts)) c 
-H_y)))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda 
-(H: ((\forall (c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to 
-(sns3 c (lifts1 p ts)))))))).(\lambda (c: C).(\lambda (H0: (drop1 (PCons n n0 
-p) c e)).(\lambda (ts: TList).(\lambda (H1: (sns3 e ts)).(let H_x \def 
-(drop1_gen_pcons c e p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda 
-(c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 e)) (sns3 c (lifts1 
-(PCons n n0 p) ts)) (\lambda (x: C).(\lambda (H3: (drop n n0 c x)).(\lambda 
-(H4: (drop1 p x e)).(eq_ind_r TList (lifts n n0 (lifts1 p ts)) (\lambda (t: 
-TList).(sns3 c t)) (sns3_lifts c x n n0 H3 (lifts1 p ts) (H x H4 ts H1)) 
-(lifts1 (PCons n n0 p) ts) (lifts1_cons n n0 p ts))))) H2))))))))))) hds)).
-(* COMMENTS
-Initial nodes: 323
-END *)
+ \lambda (e: C).(\lambda (hds: PList).(let TMP_2 \def (\lambda (p: 
+PList).(\forall (c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) 
+\to (let TMP_1 \def (lifts1 p ts) in (sns3 c TMP_1))))))) in (let TMP_9 \def 
+(\lambda (c: C).(\lambda (H: (drop1 PNil c e)).(\lambda (ts: TList).(\lambda 
+(H0: (sns3 e ts)).(let H_y \def (drop1_gen_pnil c e H) in (let TMP_4 \def 
+(\lambda (c0: C).(let TMP_3 \def (lifts1 PNil ts) in (sns3 c0 TMP_3))) in 
+(let TMP_5 \def (\lambda (t: TList).(sns3 e t)) in (let TMP_6 \def (lifts1 
+PNil ts) in (let TMP_7 \def (lifts1_nil ts) in (let TMP_8 \def (eq_ind_r 
+TList ts TMP_5 H0 TMP_6 TMP_7) in (eq_ind_r C e TMP_4 TMP_8 c H_y))))))))))) 
+in (let TMP_25 \def (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: 
+PList).(\lambda (H: ((\forall (c: C).((drop1 p c e) \to (\forall (ts: 
+TList).((sns3 e ts) \to (sns3 c (lifts1 p ts)))))))).(\lambda (c: C).(\lambda 
+(H0: (drop1 (PCons n n0 p) c e)).(\lambda (ts: TList).(\lambda (H1: (sns3 e 
+ts)).(let H_x \def (drop1_gen_pcons c e p n n0 H0) in (let H2 \def H_x in 
+(let TMP_10 \def (\lambda (c2: C).(drop n n0 c c2)) in (let TMP_11 \def 
+(\lambda (c2: C).(drop1 p c2 e)) in (let TMP_12 \def (PCons n n0 p) in (let 
+TMP_13 \def (lifts1 TMP_12 ts) in (let TMP_14 \def (sns3 c TMP_13) in (let 
+TMP_24 \def (\lambda (x: C).(\lambda (H3: (drop n n0 c x)).(\lambda (H4: 
+(drop1 p x e)).(let TMP_15 \def (lifts1 p ts) in (let TMP_16 \def (lifts n n0 
+TMP_15) in (let TMP_17 \def (\lambda (t: TList).(sns3 c t)) in (let TMP_18 
+\def (lifts1 p ts) in (let TMP_19 \def (H x H4 ts H1) in (let TMP_20 \def 
+(sns3_lifts c x n n0 H3 TMP_18 TMP_19) in (let TMP_21 \def (PCons n n0 p) in 
+(let TMP_22 \def (lifts1 TMP_21 ts) in (let TMP_23 \def (lifts1_cons n n0 p 
+ts) in (eq_ind_r TList TMP_16 TMP_17 TMP_20 TMP_22 TMP_23))))))))))))) in 
+(ex2_ind C TMP_10 TMP_11 TMP_14 TMP_24 H2))))))))))))))))) in (PList_ind 
+TMP_2 TMP_9 TMP_25 hds))))).
 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/nf2.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/nf2.ma
index 824428084..05744e7d0 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/sn3/nf2.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/sn3/nf2.ma
@@ -14,53 +14,61 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/sn3/defs.ma".
+include "basic_1/sn3/fwd.ma".
 
-include "Basic-1/nf2/dec.ma".
+include "basic_1/nf2/dec.ma".
 
-include "Basic-1/nf2/pr3.ma".
+include "basic_1/nf2/pr3.ma".
 
 theorem sn3_nf2:
  \forall (c: C).(\forall (t: T).((nf2 c t) \to (sn3 c t)))
 \def
- \lambda (c: C).(\lambda (t: T).(\lambda (H: (nf2 c t)).(sn3_sing c t 
+ \lambda (c: C).(\lambda (t: T).(\lambda (H: (nf2 c t)).(let TMP_7 \def 
 (\lambda (t2: T).(\lambda (H0: (((eq T t t2) \to (\forall (P: 
 Prop).P)))).(\lambda (H1: (pr3 c t t2)).(let H_y \def (nf2_pr3_unfold c t t2 
-H1 H) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c t t0)) H1 t H_y) 
-in (let H3 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T t t0) \to (\forall (P: 
-Prop).P))) H0 t H_y) in (eq_ind T t (\lambda (t0: T).(sn3 c t0)) (H3 
-(refl_equal T t) (sn3 c t)) t2 H_y)))))))))).
-(* COMMENTS
-Initial nodes: 129
-END *)
+H1 H) in (let TMP_1 \def (\lambda (t0: T).(pr3 c t t0)) in (let H2 \def 
+(eq_ind_r T t2 TMP_1 H1 t H_y) in (let TMP_2 \def (\lambda (t0: T).((eq T t 
+t0) \to (\forall (P: Prop).P))) in (let H3 \def (eq_ind_r T t2 TMP_2 H0 t 
+H_y) in (let TMP_3 \def (\lambda (t0: T).(sn3 c t0)) in (let TMP_4 \def 
+(refl_equal T t) in (let TMP_5 \def (sn3 c t) in (let TMP_6 \def (H3 TMP_4 
+TMP_5) in (eq_ind T t TMP_3 TMP_6 t2 H_y))))))))))))) in (sn3_sing c t 
+TMP_7)))).
 
 theorem nf2_sn3:
  \forall (c: C).(\forall (t: T).((sn3 c t) \to (ex2 T (\lambda (u: T).(pr3 c 
 t u)) (\lambda (u: T).(nf2 c u)))))
 \def
- \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(sn3_ind c (\lambda 
-(t0: T).(ex2 T (\lambda (u: T).(pr3 c t0 u)) (\lambda (u: T).(nf2 c u)))) 
+ \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(let TMP_3 \def 
+(\lambda (t0: T).(let TMP_1 \def (\lambda (u: T).(pr3 c t0 u)) in (let TMP_2 
+\def (\lambda (u: T).(nf2 c u)) in (ex2 T TMP_1 TMP_2)))) in (let TMP_32 \def 
 (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
 (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall 
 (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
 (ex2 T (\lambda (u: T).(pr3 c t2 u)) (\lambda (u: T).(nf2 c u)))))))).(let 
-H_x \def (nf2_dec c t1) in (let H2 \def H_x in (or_ind (nf2 c t1) (ex2 T 
-(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr2 c t1 t2))) (ex2 T (\lambda (u: T).(pr3 c t1 u)) (\lambda (u: T).(nf2 
-c u))) (\lambda (H3: (nf2 c t1)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u)) 
-(\lambda (u: T).(nf2 c u)) t1 (pr3_refl c t1) H3)) (\lambda (H3: (ex2 T 
-(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr2 c t1 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t1 t2) \to (\forall 
-(P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)) (ex2 T (\lambda (u: T).(pr3 c 
-t1 u)) (\lambda (u: T).(nf2 c u))) (\lambda (x: T).(\lambda (H4: (((eq T t1 
-x) \to (\forall (P: Prop).P)))).(\lambda (H5: (pr2 c t1 x)).(let H_y \def (H1 
-x H4) in (let H6 \def (H_y (pr3_pr2 c t1 x H5)) in (ex2_ind T (\lambda (u: 
-T).(pr3 c x u)) (\lambda (u: T).(nf2 c u)) (ex2 T (\lambda (u: T).(pr3 c t1 
-u)) (\lambda (u: T).(nf2 c u))) (\lambda (x0: T).(\lambda (H7: (pr3 c x 
-x0)).(\lambda (H8: (nf2 c x0)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u)) 
-(\lambda (u: T).(nf2 c u)) x0 (pr3_sing c x t1 H5 x0 H7) H8)))) H6)))))) H3)) 
-H2)))))) t H))).
-(* COMMENTS
-Initial nodes: 443
-END *)
+H_x \def (nf2_dec c t1) in (let H2 \def H_x in (let TMP_4 \def (nf2 c t1) in 
+(let TMP_5 \def (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) in 
+(let TMP_6 \def (\lambda (t2: T).(pr2 c t1 t2)) in (let TMP_7 \def (ex2 T 
+TMP_5 TMP_6) in (let TMP_8 \def (\lambda (u: T).(pr3 c t1 u)) in (let TMP_9 
+\def (\lambda (u: T).(nf2 c u)) in (let TMP_10 \def (ex2 T TMP_8 TMP_9) in 
+(let TMP_14 \def (\lambda (H3: (nf2 c t1)).(let TMP_11 \def (\lambda (u: 
+T).(pr3 c t1 u)) in (let TMP_12 \def (\lambda (u: T).(nf2 c u)) in (let 
+TMP_13 \def (pr3_refl c t1) in (ex_intro2 T TMP_11 TMP_12 t1 TMP_13 H3))))) 
+in (let TMP_31 \def (\lambda (H3: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)))).(let TMP_15 \def 
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) in (let TMP_16 
+\def (\lambda (t2: T).(pr2 c t1 t2)) in (let TMP_17 \def (\lambda (u: T).(pr3 
+c t1 u)) in (let TMP_18 \def (\lambda (u: T).(nf2 c u)) in (let TMP_19 \def 
+(ex2 T TMP_17 TMP_18) in (let TMP_30 \def (\lambda (x: T).(\lambda (H4: (((eq 
+T t1 x) \to (\forall (P: Prop).P)))).(\lambda (H5: (pr2 c t1 x)).(let H_y 
+\def (H1 x H4) in (let TMP_20 \def (pr3_pr2 c t1 x H5) in (let H6 \def (H_y 
+TMP_20) in (let TMP_21 \def (\lambda (u: T).(pr3 c x u)) in (let TMP_22 \def 
+(\lambda (u: T).(nf2 c u)) in (let TMP_23 \def (\lambda (u: T).(pr3 c t1 u)) 
+in (let TMP_24 \def (\lambda (u: T).(nf2 c u)) in (let TMP_25 \def (ex2 T 
+TMP_23 TMP_24) in (let TMP_29 \def (\lambda (x0: T).(\lambda (H7: (pr3 c x 
+x0)).(\lambda (H8: (nf2 c x0)).(let TMP_26 \def (\lambda (u: T).(pr3 c t1 u)) 
+in (let TMP_27 \def (\lambda (u: T).(nf2 c u)) in (let TMP_28 \def (pr3_sing 
+c x t1 H5 x0 H7) in (ex_intro2 T TMP_26 TMP_27 x0 TMP_28 H8))))))) in 
+(ex2_ind T TMP_21 TMP_22 TMP_25 TMP_29 H6))))))))))))) in (ex2_ind T TMP_15 
+TMP_16 TMP_19 TMP_30 H3)))))))) in (or_ind TMP_4 TMP_7 TMP_10 TMP_14 TMP_31 
+H2))))))))))))))) in (sn3_ind c TMP_3 TMP_32 t H))))).
 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/props.ma
index ea72c8869..e5d7cf28b 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/sn3/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/sn3/props.ma
@@ -14,13 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/sn3/nf2.ma".
+include "basic_1/sn3/nf2.ma".
 
-include "Basic-1/sn3/fwd.ma".
+include "basic_1/nf2/iso.ma".
 
-include "Basic-1/nf2/iso.ma".
-
-include "Basic-1/pr3/iso.ma".
+include "basic_1/pr3/iso.ma".
 
 theorem sn3_pr3_trans:
  \forall (c: C).(\forall (t1: T).((sn3 c t1) \to (\forall (t2: T).((pr3 c t1 
@@ -41,9 +39,6 @@ H7 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t t0)) H4 t2 H6) in (let H8
 H3 t2 H6) in (let H9 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t2 t)) H2 t2 
 H6) in (H0 t0 H8 H7))))) (\lambda (H6: (((eq T t2 t3) \to (\forall (P: 
 Prop).P)))).(H1 t3 H6 H2 t0 H4)) H5)))))))))))) t1 H))).
-(* COMMENTS
-Initial nodes: 289
-END *)
 
 theorem sn3_pr2_intro:
  \forall (c: C).(\forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to 
@@ -74,9 +69,6 @@ t6) \to (sn3 c t6))))) H7 t3 H10) in (let H13 \def (eq_ind T t4 (\lambda (t:
 T).(pr2 c t t3)) H4 t3 H10) in (H6 H12 H11))))) (\lambda (H10: (((eq T t4 t3) 
 \to (\forall (P: Prop).P)))).(sn3_pr3_trans c t3 (H7 t3 H10 H4) t5 H5)) 
 H9))))))))))) t1 t2 H1 H3)) H2)))))))).
-(* COMMENTS
-Initial nodes: 467
-END *)
 
 theorem sn3_cast:
  \forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: T).((sn3 c t) \to 
@@ -142,9 +134,6 @@ Cast) x0 t3)) \to (\forall (P: Prop).P))) H12 t0 H16) in (let H18 \def
 H11))) H15))))) H13))) t2 H9))))))) H8)) (\lambda (H8: (pr2 c t0 
 t2)).(sn3_pr3_trans c t0 (sn3_sing c t0 H3) t2 (pr3_pr2 c t0 t2 H8))) 
 H7))))))))) t H2)))))) u H))).
-(* COMMENTS
-Initial nodes: 1239
-END *)
 
 theorem sn3_cflat:
  \forall (c: C).(\forall (t: T).((sn3 c t) \to (\forall (f: F).(\forall (u: 
@@ -159,9 +148,6 @@ F).(\lambda (u: T).(sn3_ind c (\lambda (t0: T).(sn3 (CHead c (Flat f) u) t0))
 (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) \to (\forall (P: 
 Prop).P)))).(\lambda (H3: (pr2 (CHead c (Flat f) u) t1 t2)).(H1 t2 H2 
 (pr3_pr2 c t1 t2 (pr2_gen_cflat f c u t1 t2 H3)))))))))) t H))))).
-(* COMMENTS
-Initial nodes: 175
-END *)
 
 theorem sn3_shift:
  \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c 
@@ -172,9 +158,6 @@ theorem sn3_shift:
 H0 \def H_x in (land_ind (sn3 c v) (sn3 (CHead c (Bind b) v) t) (sn3 (CHead c 
 (Bind b) v) t) (\lambda (_: (sn3 c v)).(\lambda (H2: (sn3 (CHead c (Bind b) 
 v) t)).H2)) H0))))))).
-(* COMMENTS
-Initial nodes: 95
-END *)
 
 theorem sn3_change:
  \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: 
@@ -193,9 +176,6 @@ t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to
 Prop).P)))).(\lambda (H4: (pr2 (CHead c (Bind b) v2) t1 t2)).(H2 t2 H3 
 (pr3_pr2 (CHead c (Bind b) v1) t1 t2 (pr2_change b H c v2 t1 t2 H4 
 v1)))))))))) t H0))))))).
-(* COMMENTS
-Initial nodes: 239
-END *)
 
 theorem sn3_gen_def:
  \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
@@ -207,9 +187,6 @@ i))).(sn3_gen_lift c v (S i) O (sn3_pr3_trans c (TLRef i) H0 (lift (S i) O v)
 (pr3_pr2 c (TLRef i) (lift (S i) O v) (pr2_delta c d v i H (TLRef i) (TLRef 
 i) (pr0_refl (TLRef i)) (lift (S i) O v) (subst0_lref v i)))) d (getl_drop 
 Abbr c d v i H))))))).
-(* COMMENTS
-Initial nodes: 139
-END *)
 
 theorem sn3_cdelta:
  \forall (v: T).(\forall (t: T).(\forall (i: nat).(((\forall (w: T).(ex T 
@@ -264,9 +241,6 @@ v0))))))).(\lambda (c: C).(\lambda (d: C).(\lambda (H6: (getl i0 c (CHead d
 (Bind Abbr) v0))).(\lambda (H7: (sn3 c (THead k u1 t1))).(let H_y \def 
 (sn3_gen_head k c u1 t1 H7) in (H3 c d H6 H_y))))))))))))))))) i v t x H1))) 
 H0)))))).
-(* COMMENTS
-Initial nodes: 949
-END *)
 
 theorem sn3_cpr3_trans:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
@@ -283,9 +257,6 @@ Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u2)
 t2)))))).(sn3_sing (CHead c k u2) t1 (\lambda (t2: T).(\lambda (H3: (((eq T 
 t1 t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 (CHead c k u2) t1 
 t2)).(H2 t2 H3 (pr3_pr3_pr3_t c u1 u2 H t1 t2 k H4))))))))) t H0))))))).
-(* COMMENTS
-Initial nodes: 203
-END *)
 
 theorem sn3_bind:
  \forall (b: B).(\forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: 
@@ -400,9 +371,6 @@ t3))).(sn3_gen_lift (CHead c (Bind b) t1) t3 (S O) O (sn3_pr3_trans (CHead c
 (Bind b) t1) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3) (lift (S O) O t3) H10) 
 c (drop_drop (Bind b) O c c (drop_refl c) t1))) H9)))) H7)))))))))) t 
 H2)))))) u H)))).
-(* COMMENTS
-Initial nodes: 2401
-END *)
 
 theorem sn3_beta:
  \forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c (THead (Bind Abbr) v 
@@ -514,48 +482,46 @@ Abst) t2 x0))) (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)))) x4
 H31)))) (\lambda (H31: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead 
 (Bind Abbr) x x4) (\lambda (H32: (eq T (THead (Bind Abbr) x x0) (THead (Bind 
 Abbr) x x4))).(\lambda (P: Prop).(let H33 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind 
-Abbr) x x0) (THead (Bind Abbr) x x4) H32) in (let H34 \def (eq_ind_r T x4 
-(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H31 x0 H33) in 
-(let H35 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H33) in (H34 (refl_equal T x0) 
-P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) 
-(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead 
-(Bind Abbr) x x4)) t2 (sn3_sing c t2 H7))) H30))) x1 H27)))) (\lambda (H27: 
-(((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind Abbr) x1 x4) 
-(\lambda (H28: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x1 
-x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) 
-\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
+T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
+x x4) H32) in (let H34 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to 
+(\forall (P0: Prop).P0))) H31 x0 H33) in (let H35 \def (eq_ind_r T x4 
+(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 
+t0)))) H20 x0 H33) in (H34 (refl_equal T x0) P)))))) (pr3_pr2 c (THead (Bind 
+Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4 (Bind Abbr) (H20 
+Abbr x))) x x4 (refl_equal T (THead (Bind Abbr) x x4)) t2 (sn3_sing c t2 
+H7))) H30))) x1 H27)))) (\lambda (H27: (((eq T x x1) \to (\forall (P: 
+Prop).P)))).(H5 (THead (Bind Abbr) x1 x4) (\lambda (H28: (eq T (THead (Bind 
+Abbr) x x0) (THead (Bind Abbr) x1 x4))).(\lambda (P: Prop).(let H29 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x | (TLRef 
+_) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
 (THead (Bind Abbr) x1 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind 
-Abbr) x x0) (THead (Bind Abbr) x1 x4) H28) in (\lambda (H31: (eq T x 
-x1)).(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (let H33 \def 
-(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) 
-H27 x H31) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 
-x H31) in (H33 (refl_equal T x) P)))))) H29)))) (pr3_head_12 c x x1 (pr3_pr2 
-c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20 
-Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) t2 (sn3_sing c t2 
-H7))) H26))) x3 H23)))) (\lambda (H23: (((eq T t2 x3) \to (\forall (P: 
-Prop).P)))).(let H_x0 \def (term_dec x x1) in (let H24 \def H_x0 in (or_ind 
-(eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) 
-x1 (THead (Bind Abst) x3 x4))) (\lambda (H25: (eq T x x1)).(let H26 \def 
-(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H25) in (eq_ind T x 
-(\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4)))) 
-(let H_x1 \def (term_dec x0 x4) in (let H27 \def H_x1 in (or_ind (eq T x0 x4) 
-((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x (THead 
-(Bind Abst) x3 x4))) (\lambda (H28: (eq T x0 x4)).(let H29 \def (eq_ind_r T 
-x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-x0 t0)))) H20 x0 H28) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat 
-Appl) x (THead (Bind Abst) x3 t0)))) (H8 x3 H23 (pr3_pr2 c t2 x3 H19)) x4 
-H28))) (\lambda (H28: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead 
-(Bind Abbr) x x4) (\lambda (H29: (eq T (THead (Bind Abbr) x x0) (THead (Bind 
-Abbr) x x4))).(\lambda (P: Prop).(let H30 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind 
+T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
+x1 x4) H28) in (\lambda (H31: (eq T x x1)).(let H32 \def (eq_ind_r T x4 
+(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 
+t0)))) H20 x0 H30) in (let H33 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T x 
+t0) \to (\forall (P0: Prop).P0))) H27 x H31) in (let H34 \def (eq_ind_r T x1 
+(\lambda (t0: T).(pr2 c x t0)) H14 x H31) in (H33 (refl_equal T x) P)))))) 
+H29)))) (pr3_head_12 c x x1 (pr3_pr2 c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 
+(CHead c (Bind Abbr) x1) x0 x4 (H20 Abbr x1))) x1 x4 (refl_equal T (THead 
+(Bind Abbr) x1 x4)) t2 (sn3_sing c t2 H7))) H26))) x3 H23)))) (\lambda (H23: 
+(((eq T t2 x3) \to (\forall (P: Prop).P)))).(let H_x0 \def (term_dec x x1) in 
+(let H24 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: 
+Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) x3 x4))) (\lambda 
+(H25: (eq T x x1)).(let H26 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x 
+t0)) H14 x H25) in (eq_ind T x (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 
+(THead (Bind Abst) x3 x4)))) (let H_x1 \def (term_dec x0 x4) in (let H27 \def 
+H_x1 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c 
+(THead (Flat Appl) x (THead (Bind Abst) x3 x4))) (\lambda (H28: (eq T x0 
+x4)).(let H29 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (eq_ind T x0 
+(\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) x3 t0)))) (H8 
+x3 H23 (pr3_pr2 c t2 x3 H19)) x4 H28))) (\lambda (H28: (((eq T x0 x4) \to 
+(\forall (P: Prop).P)))).(H5 (THead (Bind Abbr) x x4) (\lambda (H29: (eq T 
+(THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4))).(\lambda (P: Prop).(let 
+H30 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 
+| (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind 
 Abbr) x x0) (THead (Bind Abbr) x x4) H29) in (let H31 \def (eq_ind_r T x4 
 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H28 x0 H30) in 
 (let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: 
@@ -566,21 +532,20 @@ P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4)
 (\lambda (H25: (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind 
 Abbr) x1 x4) (\lambda (H26: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
 x1 x4))).(\lambda (P: Prop).(let H27 \def (f_equal T T (\lambda (e: T).(match 
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) 
-\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
-(THead (Bind Abbr) x1 x4) H26) in ((let H28 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind 
-Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in (\lambda (H29: (eq T x 
-x1)).(let H30 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (let H31 \def 
-(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) 
-H25 x H29) in (let H32 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 
-x H29) in (H31 (refl_equal T x) P)))))) H27)))) (pr3_head_12 c x x1 (pr3_pr2 
-c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20 
-Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) x3 (H7 x3 H23 
-(pr3_pr2 c t2 x3 H19)))) H24)))) H22))) x2 H18))))))) H17)) t3 H13))))))) 
-H12)) (\lambda (H12: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+e with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t0 _) 
+\Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in 
+((let H28 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) 
+(THead (Bind Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in (\lambda (H29: (eq 
+T x x1)).(let H30 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (let 
+H31 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: 
+Prop).P0))) H25 x H29) in (let H32 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 
+c x t0)) H14 x H29) in (H31 (refl_equal T x) P)))))) H27)))) (pr3_head_12 c x 
+x1 (pr3_pr2 c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) 
+x0 x4 (H20 Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) x3 (H7 
+x3 H23 (pr3_pr2 c t2 x3 H19)))) H24)))) H22))) x2 H18))))))) H17)) t3 
+H13))))))) H12)) (\lambda (H12: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead 
 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
 T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
@@ -601,65 +566,63 @@ x4))).(\lambda (H15: (pr2 c x x3)).(\lambda (H16: ((\forall (b: B).(\forall
 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) 
 \to (\forall (P: Prop).P))) H9 (THead (Bind Abbr) x3 x4) H14) in (eq_ind_r T 
 (THead (Bind Abbr) x3 x4) (\lambda (t0: T).(sn3 c t0)) (let H18 \def (f_equal 
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) \Rightarrow t0])) 
-(THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in ((let H19 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) 
-\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in 
-(\lambda (_: (eq T t2 x1)).(let H21 \def (eq_ind_r T x2 (\lambda (t0: 
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t0 x4)))) H16 x0 
-H19) in (let H_x \def (term_dec x x3) in (let H22 \def H_x in (or_ind (eq T x 
-x3) ((eq T x x3) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abbr) x3 x4)) 
-(\lambda (H23: (eq T x x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: 
-T).(pr2 c x t0)) H15 x H23) in (eq_ind T x (\lambda (t0: T).(sn3 c (THead 
-(Bind Abbr) t0 x4))) (let H_x0 \def (term_dec x0 x4) in (let H25 \def H_x0 in 
-(or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead 
-(Bind Abbr) x x4)) (\lambda (H26: (eq T x0 x4)).(let H27 \def (eq_ind_r T x4 
+T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t2 | (TLRef _) 
+\Rightarrow t2 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst) t2 x0) 
+(THead (Bind Abst) x1 x2) H13) in ((let H19 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind Abst) 
+x1 x2) H13) in (\lambda (_: (eq T t2 x1)).(let H21 \def (eq_ind_r T x2 
+(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t0 
+x4)))) H16 x0 H19) in (let H_x \def (term_dec x x3) in (let H22 \def H_x in 
+(or_ind (eq T x x3) ((eq T x x3) \to (\forall (P: Prop).P)) (sn3 c (THead 
+(Bind Abbr) x3 x4)) (\lambda (H23: (eq T x x3)).(let H24 \def (eq_ind_r T x3 
+(\lambda (t0: T).(pr2 c x t0)) H15 x H23) in (eq_ind T x (\lambda (t0: 
+T).(sn3 c (THead (Bind Abbr) t0 x4))) (let H_x0 \def (term_dec x0 x4) in (let 
+H25 \def H_x0 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: 
+Prop).P)) (sn3 c (THead (Bind Abbr) x x4)) (\lambda (H26: (eq T x0 x4)).(let 
+H27 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 
+(CHead c (Bind b) u) x0 t0)))) H21 x0 H26) in (eq_ind T x0 (\lambda (t0: 
+T).(sn3 c (THead (Bind Abbr) x t0))) (sn3_sing c (THead (Bind Abbr) x x0) H6) 
+x4 H26))) (\lambda (H26: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H6 
+(THead (Bind Abbr) x x4) (\lambda (H27: (eq T (THead (Bind Abbr) x x0) (THead 
+(Bind Abbr) x x4))).(\lambda (P: Prop).(let H28 \def (f_equal T T (\lambda 
+(e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
+x x4) H27) in (let H29 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to 
+(\forall (P0: Prop).P0))) H26 x0 H28) in (let H30 \def (eq_ind_r T x4 
 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 
-t0)))) H21 x0 H26) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Bind Abbr) 
-x t0))) (sn3_sing c (THead (Bind Abbr) x x0) H6) x4 H26))) (\lambda (H26: 
-(((eq T x0 x4) \to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x x4) 
-(\lambda (H27: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x 
-x4))).(\lambda (P: Prop).(let H28 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) 
-\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
-(THead (Bind Abbr) x x4) H27) in (let H29 \def (eq_ind_r T x4 (\lambda (t0: 
-T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H26 x0 H28) in (let H30 \def 
-(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c 
-(Bind b) u) x0 t0)))) H21 x0 H28) in (H29 (refl_equal T x0) P)))))) (pr3_pr2 
-c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4 
-(Bind Abbr) (H21 Abbr x))))) H25))) x3 H23))) (\lambda (H23: (((eq T x x3) 
-\to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x3 x4) (\lambda (H24: (eq 
-T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x3 x4))).(\lambda (P: 
-Prop).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | 
-(THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
-x3 x4) H24) in ((let H26 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) 
-\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
-(THead (Bind Abbr) x3 x4) H24) in (\lambda (H27: (eq T x x3)).(let H28 \def 
-(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c 
-(Bind b) u) x0 t0)))) H21 x0 H26) in (let H29 \def (eq_ind_r T x3 (\lambda 
-(t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H23 x H27) in (let H30 
-\def (eq_ind_r T x3 (\lambda (t0: T).(pr2 c x t0)) H15 x H27) in (H29 
-(refl_equal T x) P)))))) H25)))) (pr3_head_12 c x x3 (pr3_pr2 c x x3 H15) 
-(Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x3) x0 x4 (H21 Abbr x3))))) 
-H22)))))) H18)) t3 H14)))))))))) H12)) (\lambda (H12: (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) 
-y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+t0)))) H21 x0 H28) in (H29 (refl_equal T x0) P)))))) (pr3_pr2 c (THead (Bind 
+Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4 (Bind Abbr) (H21 
+Abbr x))))) H25))) x3 H23))) (\lambda (H23: (((eq T x x3) \to (\forall (P: 
+Prop).P)))).(H6 (THead (Bind Abbr) x3 x4) (\lambda (H24: (eq T (THead (Bind 
+Abbr) x x0) (THead (Bind Abbr) x3 x4))).(\lambda (P: Prop).(let H25 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x | (TLRef 
+_) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
+(THead (Bind Abbr) x3 x4) H24) in ((let H26 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
+x3 x4) H24) in (\lambda (H27: (eq T x x3)).(let H28 \def (eq_ind_r T x4 
+(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 
+t0)))) H21 x0 H26) in (let H29 \def (eq_ind_r T x3 (\lambda (t0: T).((eq T x 
+t0) \to (\forall (P0: Prop).P0))) H23 x H27) in (let H30 \def (eq_ind_r T x3 
+(\lambda (t0: T).(pr2 c x t0)) H15 x H27) in (H29 (refl_equal T x) P)))))) 
+H25)))) (pr3_head_12 c x x3 (pr3_pr2 c x x3 H15) (Bind Abbr) x0 x4 (pr3_pr2 
+(CHead c (Bind Abbr) x3) x0 x4 (H21 Abbr x3))))) H22)))))) H18)) t3 
+H14)))))))))) H12)) (\lambda (H12: (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
+Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead 
 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
@@ -679,30 +642,24 @@ T t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
 Prop).P))) H9 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) 
 H15) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) 
 x4)) (\lambda (t0: T).(sn3 c t0)) (let H20 \def (f_equal T B (\lambda (e: 
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | 
-(TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return 
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
-Abst])])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in ((let H21 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in 
-((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ 
-t0) \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) 
-in (\lambda (H23: (eq T t2 x2)).(\lambda (H24: (eq B Abst x1)).(let H25 \def 
-(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H18 x0 
-H22) in (let H26 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H17 t2 
-H23) in (let H27 \def (eq_ind_r B x1 (\lambda (b: B).(pr2 (CHead c (Bind b) 
-x6) x0 x4)) H25 Abst H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b: 
-B).(not (eq B b Abst))) H13 Abst H24) in (eq_ind B Abst (\lambda (b: B).(sn3 
-c (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (let H29 
-\def (match (H28 (refl_equal B Abst)) in False return (\lambda (_: 
-False).(sn3 c (THead (Bind Abst) x6 (THead (Flat Appl) (lift (S O) O x5) 
-x4)))) with []) in H29) x1 H24)))))))) H21)) H20)) t3 H15)))))))))))))) H12)) 
+T).(match e with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | 
+(THead k _ _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) 
+\Rightarrow Abst])])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) 
+in ((let H21 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) \Rightarrow t0])) 
+(THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in ((let H22 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef 
+_) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abst) t2 
+x0) (THead (Bind x1) x2 x3) H14) in (\lambda (H23: (eq T t2 x2)).(\lambda 
+(H24: (eq B Abst x1)).(let H25 \def (eq_ind_r T x3 (\lambda (t0: T).(pr2 
+(CHead c (Bind x1) x6) t0 x4)) H18 x0 H22) in (let H26 \def (eq_ind_r T x2 
+(\lambda (t0: T).(pr2 c t0 x6)) H17 t2 H23) in (let H27 \def (eq_ind_r B x1 
+(\lambda (b: B).(pr2 (CHead c (Bind b) x6) x0 x4)) H25 Abst H24) in (let H28 
+\def (eq_ind_r B x1 (\lambda (b: B).(not (eq B b Abst))) H13 Abst H24) in 
+(eq_ind B Abst (\lambda (b: B).(sn3 c (THead (Bind b) x6 (THead (Flat Appl) 
+(lift (S O) O x5) x4)))) (let H29 \def (match (H28 (refl_equal B Abst)) in 
+False with []) in H29) x1 H24)))))))) H21)) H20)) t3 H15)))))))))))))) H12)) 
 H11))))))))) w H4))))))))))) y H0))))) H)))).
-(* COMMENTS
-Initial nodes: 5699
-END *)
 
 theorem sn3_appl_lref:
  \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (v: 
@@ -782,11 +739,22 @@ T).(\lambda (H7: (eq T (TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H8:
 H11 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) 
 t) \to (\forall (P: Prop).P))) H3 (THead (Bind Abbr) x2 x3) H8) in (eq_ind_r 
 T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) (let H12 \def (eq_ind 
-T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
-\Rightarrow False])) I (THead (Bind Abst) x0 x1) H7) in (False_ind (sn3 c 
-(THead (Bind Abbr) x2 x3)) H12)) t2 H8)))))))))) H6)) (\lambda (H6: (ex6_6 B 
-T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T (TLRef i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead 
+(Bind Abst) x0 x1) H7) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H12)) 
+t2 H8)))))))))) H6)) (\lambda (H6: (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
+B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
@@ -796,36 +764,21 @@ T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
 T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1: 
 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind b) y1 
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
-T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat 
-Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2) (\lambda (x0: B).(\lambda (x1: 
-T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: 
-T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H8: (eq T (TLRef i) (THead 
-(Bind x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x5 (THead (Flat 
-Appl) (lift (S O) O x4) x3)))).(\lambda (_: (pr2 c t1 x4)).(\lambda (_: (pr2 
-c x1 x5)).(\lambda (_: (pr2 (CHead c (Bind x0) x5) x2 x3)).(let H13 \def 
-(eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to 
-(\forall (P: Prop).P))) H3 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) 
-O x4) x3)) H9) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S 
-O) O x4) x3)) (\lambda (t: T).(sn3 c t)) (let H14 \def (eq_ind T (TLRef i) 
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
-False])) I (THead (Bind x0) x1 x2) H8) in (False_ind (sn3 c (THead (Bind x0) 
-x5 (THead (Flat Appl) (lift (S O) O x4) x3))) H14)) t2 H9)))))))))))))) H6)) 
-H5))))))))) v H0))))).
-(* COMMENTS
-Initial nodes: 2125
-END *)
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) 
+(sn3 c t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 
+Abst))).(\lambda (H8: (eq T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda (H9: 
+(eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
+x3)))).(\lambda (_: (pr2 c t1 x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: 
+(pr2 (CHead c (Bind x0) x5) x2 x3)).(let H13 \def (eq_ind T t2 (\lambda (t: 
+T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to (\forall (P: Prop).P))) H3 
+(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H9) in 
+(eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) 
+(\lambda (t: T).(sn3 c t)) (let H14 \def (eq_ind T (TLRef i) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(THead _ _ _) \Rightarrow False])) I (THead (Bind x0) x1 x2) H8) in 
+(False_ind (sn3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
+x3))) H14)) t2 H9)))))))))))))) H6)) H5))))))))) v H0))))).
 
 theorem sn3_appl_abbr:
  \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
@@ -904,38 +857,37 @@ in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c (THead
 H19)))) (\lambda (H19: (((eq T x x0) \to (\forall (P: Prop).P)))).(H5 (THead 
 (Flat Appl) x0 (lift (S i) O w)) (\lambda (H20: (eq T (THead (Flat Appl) x 
 (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)))).(\lambda (P: 
-Prop).(let H21 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | 
-(THead _ t _) \Rightarrow t])) (THead (Flat Appl) x (lift (S i) O w)) (THead 
-(Flat Appl) x0 (lift (S i) O w)) H20) in (let H22 \def (eq_ind_r T x0 
-(\lambda (t: T).((eq T x t) \to (\forall (P0: Prop).P0))) H19 x H21) in (let 
-H23 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H21) in (H22 
-(refl_equal T x) P)))))) (pr3_pr2 c (THead (Flat Appl) x (lift (S i) O w)) 
-(THead (Flat Appl) x0 (lift (S i) O w)) (pr2_head_1 c x x0 H12 (Flat Appl) 
-(lift (S i) O w))) x0 (refl_equal T (THead (Flat Appl) x0 (lift (S i) O 
-w))))) H18))) x1 H16))) (\lambda (H16: (ex2_2 C T (\lambda (d0: C).(\lambda 
-(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
-T).(eq T x1 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda 
-(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
-T).(eq T x1 (lift (S i) O u)))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda 
-(x2: C).(\lambda (x3: T).(\lambda (H17: (getl i c (CHead x2 (Bind Abbr) 
-x3))).(\lambda (H18: (eq T x1 (lift (S i) O x3))).(let H19 \def (eq_ind T x1 
-(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 
-t)) \to (\forall (P: Prop).P))) H14 (lift (S i) O x3) H18) in (eq_ind_r T 
-(lift (S i) O x3) (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H20 
-\def (eq_ind C (CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H 
+Prop).(let H21 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t])) 
+(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O 
+w)) H20) in (let H22 \def (eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to 
+(\forall (P0: Prop).P0))) H19 x H21) in (let H23 \def (eq_ind_r T x0 (\lambda 
+(t: T).(pr2 c x t)) H12 x H21) in (H22 (refl_equal T x) P)))))) (pr3_pr2 c 
+(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O 
+w)) (pr2_head_1 c x x0 H12 (Flat Appl) (lift (S i) O w))) x0 (refl_equal T 
+(THead (Flat Appl) x0 (lift (S i) O w))))) H18))) x1 H16))) (\lambda (H16: 
+(ex2_2 C T (\lambda (d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) 
+u)))) (\lambda (_: C).(\lambda (u: T).(eq T x1 (lift (S i) O 
+u)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u: T).(getl i c (CHead d0 
+(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x1 (lift (S i) O 
+u)))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda (x2: C).(\lambda (x3: 
+T).(\lambda (H17: (getl i c (CHead x2 (Bind Abbr) x3))).(\lambda (H18: (eq T 
+x1 (lift (S i) O x3))).(let H19 \def (eq_ind T x1 (\lambda (t: T).((eq T 
+(THead (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P: 
+Prop).P))) H14 (lift (S i) O x3) H18) in (eq_ind_r T (lift (S i) O x3) 
+(\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H20 \def (eq_ind C 
+(CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H (CHead x2 (Bind 
+Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 (Bind Abbr) x3) 
+H17)) in (let H21 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) w) 
 (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 
-(Bind Abbr) x3) H17)) in (let H21 \def (f_equal C C (\lambda (e: C).(match e 
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) 
-\Rightarrow c0])) (CHead d (Bind Abbr) w) (CHead x2 (Bind Abbr) x3) 
-(getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 (Bind Abbr) x3) H17)) in 
-((let H22 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: 
-C).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d 
-(Bind Abbr) w) (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) 
-i H (CHead x2 (Bind Abbr) x3) H17)) in (\lambda (H23: (eq C d x2)).(let H24 
-\def (eq_ind_r T x3 (\lambda (t: T).(getl i c (CHead x2 (Bind Abbr) t))) H20 
-w H22) in (eq_ind T w (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 (lift (S 
-i) O t)))) (let H25 \def (eq_ind_r C x2 (\lambda (c0: C).(getl i c (CHead c0 
+(Bind Abbr) x3) H17)) in ((let H22 \def (f_equal C T (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind 
+Abbr) w) (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H 
+(CHead x2 (Bind Abbr) x3) H17)) in (\lambda (H23: (eq C d x2)).(let H24 \def 
+(eq_ind_r T x3 (\lambda (t: T).(getl i c (CHead x2 (Bind Abbr) t))) H20 w 
+H22) in (eq_ind T w (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 (lift (S i) 
+O t)))) (let H25 \def (eq_ind_r C x2 (\lambda (c0: C).(getl i c (CHead c0 
 (Bind Abbr) w))) H24 d H23) in (let H_x \def (term_dec x x0) in (let H26 \def 
 H_x in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c 
 (THead (Flat Appl) x0 (lift (S i) O w))) (\lambda (H27: (eq T x x0)).(let H28 
@@ -945,39 +897,38 @@ H_x in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c
 x0) \to (\forall (P: Prop).P)))).(H6 (THead (Flat Appl) x0 (lift (S i) O w)) 
 (\lambda (H28: (eq T (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat 
 Appl) x0 (lift (S i) O w)))).(\lambda (P: Prop).(let H29 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t])) 
-(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O 
-w)) H28) in (let H30 \def (eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to 
-(\forall (P0: Prop).P0))) H27 x H29) in (let H31 \def (eq_ind_r T x0 (\lambda 
-(t: T).(pr2 c x t)) H12 x H29) in (H30 (refl_equal T x) P)))))) (pr3_pr2 c 
-(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O 
-w)) (pr2_head_1 c x x0 H12 (Flat Appl) (lift (S i) O w))))) H26)))) x3 
-H22)))) H21))) x1 H18)))))) H16)) H15)) t2 H11))))))) H10)) (\lambda (H10: 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 
-t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind 
-b) u) z1 t3))))))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: 
-T).(\lambda (x3: T).(\lambda (H11: (eq T (TLRef i) (THead (Bind Abst) x0 
-x1))).(\lambda (H12: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c 
-x x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) 
-u) x1 x3))))).(let H15 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat 
-Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 (THead (Bind Abbr) x2 
-x3) H12) in (eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) 
-(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 
-x1) H11) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H16)) t2 
-H12)))))))))) H10)) (\lambda (H10: (ex6_6 B T T T T T (\lambda (b: 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow x | (TLRef _) 
+\Rightarrow x | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) x (lift (S 
+i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)) H28) in (let H30 \def 
+(eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to (\forall (P0: Prop).P0))) H27 
+x H29) in (let H31 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x 
+H29) in (H30 (refl_equal T x) P)))))) (pr3_pr2 c (THead (Flat Appl) x (lift 
+(S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)) (pr2_head_1 c x x0 H12 
+(Flat Appl) (lift (S i) O w))))) H26)))) x3 H22)))) H21))) x1 H18)))))) H16)) 
+H15)) t2 H11))))))) H10)) (\lambda (H10: (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T T T 
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))) 
+(sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H11: (eq T (TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H12: 
+(eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c x x2)).(\lambda (_: 
+((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let 
+H15 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) 
+t) \to (\forall (P: Prop).P))) H7 (THead (Bind Abbr) x2 x3) H12) in (eq_ind_r 
+T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) (let H16 \def (eq_ind 
+T (TLRef i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead 
+(Bind Abst) x0 x1) H11) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H16)) 
+t2 H12)))))))))) H10)) (\lambda (H10: (ex6_6 B T T T T T (\lambda (b: 
 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) 
@@ -1010,14 +961,10 @@ T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7
 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H13) in 
 (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) 
 (\lambda (t: T).(sn3 c t)) (let H18 \def (eq_ind T (TLRef i) (\lambda (ee: 
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
-(THead (Bind x0) x1 x2) H12) in (False_ind (sn3 c (THead (Bind x0) x5 (THead 
-(Flat Appl) (lift (S O) O x4) x3))) H18)) t2 H13)))))))))))))) H10)) 
-H9))))))))))))) y H1)))) H0))))))).
-(* COMMENTS
-Initial nodes: 3727
-END *)
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(THead _ _ _) \Rightarrow False])) I (THead (Bind x0) x1 x2) H12) in 
+(False_ind (sn3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
+x3))) H18)) t2 H13)))))))))))))) H10)) H9))))))))))))) y H1)))) H0))))))).
 
 theorem sn3_appl_cast:
  \forall (c: C).(\forall (v: T).(\forall (u: T).((sn3 c (THead (Flat Appl) v 
@@ -1126,12 +1073,11 @@ H27 \def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2
 x4)) ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) \to (\forall 
 (P: Prop).P)) (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x4 x5))) 
 (\lambda (H28: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 
-x4))).(let H29 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | 
-(THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x0) (THead (Flat Appl) 
-x2 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) 
-\Rightarrow x0 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x0) 
+x4))).(let H29 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t3 _) \Rightarrow t3])) 
+(THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4) H28) in ((let H30 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef 
+_) \Rightarrow x0 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x0) 
 (THead (Flat Appl) x2 x4) H28) in (\lambda (H31: (eq T x x2)).(let H32 \def 
 (eq_ind_r T x4 (\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat 
 Cast) x0 x1)) (THead (Flat Appl) x2 (THead (Flat Cast) t3 x5))) \to (\forall 
@@ -1148,43 +1094,42 @@ Appl) x x1) (THead (Flat Appl) x x5)) ((eq T (THead (Flat Appl) x x1) (THead
 (Flat Appl) x x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x 
 (THead (Flat Cast) x0 x5))) (\lambda (H37: (eq T (THead (Flat Appl) x x1) 
 (THead (Flat Appl) x x5))).(let H38 \def (f_equal T T (\lambda (e: T).(match 
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) 
-\Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x1) 
-(THead (Flat Appl) x x5) H37) in (let H39 \def (eq_ind_r T x5 (\lambda (t3: 
-T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) 
-x (THead (Flat Cast) x0 t3))) \to (\forall (P: Prop).P))) H34 x1 H38) in (let 
-H40 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H38) in 
-(eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) 
-x0 t3)))) (H39 (refl_equal T (THead (Flat Appl) x (THead (Flat Cast) x0 x1))) 
-(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)))) x5 H38))))) (\lambda 
-(H37: (((eq T (THead (Flat Appl) x x1) (THead (Flat Appl) x x5)) \to (\forall 
-(P: Prop).P)))).(H9 (THead (Flat Appl) x x5) H37 (pr3_pr2 c (THead (Flat 
-Appl) x x1) (THead (Flat Appl) x x5) (pr2_thin_dx c x1 x5 H25 x Appl)) x5 
-(refl_equal T (THead (Flat Appl) x x5)))) H36))) x2 H31))) x4 H30))))) H29))) 
-(\lambda (H28: (((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) 
-\to (\forall (P: Prop).P)))).(let H_x0 \def (term_dec (THead (Flat Appl) x 
-x1) (THead (Flat Appl) x2 x5)) in (let H29 \def H_x0 in (or_ind (eq T (THead 
-(Flat Appl) x x1) (THead (Flat Appl) x2 x5)) ((eq T (THead (Flat Appl) x x1) 
-(THead (Flat Appl) x2 x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat 
-Appl) x2 (THead (Flat Cast) x4 x5))) (\lambda (H30: (eq T (THead (Flat Appl) 
-x x1) (THead (Flat Appl) x2 x5))).(let H31 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | 
-(TLRef _) \Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) 
-x x1) (THead (Flat Appl) x2 x5) H30) in ((let H32 \def (f_equal T T (\lambda 
-(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 
-| (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat 
-Appl) x x1) (THead (Flat Appl) x2 x5) H30) in (\lambda (H33: (eq T x 
-x2)).(let H34 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H32) 
-in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 (THead (Flat 
-Cast) x4 t3)))) (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead 
-(Flat Appl) x x0) (THead (Flat Appl) t3 x4)) \to (\forall (P: Prop).P))) H28 
-x H33) in (let H36 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18 x 
-H33) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead 
-(Flat Cast) x4 x1)))) (H11 (THead (Flat Appl) x x4) H35 (pr3_pr2 c (THead 
-(Flat Appl) x x0) (THead (Flat Appl) x x4) (pr2_thin_dx c x0 x4 H24 x Appl)) 
-x x4 (refl_equal T (THead (Flat Appl) x x4)) x1 (sn3_sing c (THead (Flat 
-Appl) x x1) H10)) x2 H33))) x5 H32)))) H31))) (\lambda (H30: (((eq T (THead 
-(Flat Appl) x x1) (THead (Flat Appl) x2 x5)) \to (\forall (P: 
+e with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t3) 
+\Rightarrow t3])) (THead (Flat Appl) x x1) (THead (Flat Appl) x x5) H37) in 
+(let H39 \def (eq_ind_r T x5 (\lambda (t3: T).((eq T (THead (Flat Appl) x 
+(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x (THead (Flat Cast) x0 t3))) 
+\to (\forall (P: Prop).P))) H34 x1 H38) in (let H40 \def (eq_ind_r T x5 
+(\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H38) in (eq_ind T x1 (\lambda (t3: 
+T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 t3)))) (H39 (refl_equal 
+T (THead (Flat Appl) x (THead (Flat Cast) x0 x1))) (sn3 c (THead (Flat Appl) 
+x (THead (Flat Cast) x0 x1)))) x5 H38))))) (\lambda (H37: (((eq T (THead 
+(Flat Appl) x x1) (THead (Flat Appl) x x5)) \to (\forall (P: Prop).P)))).(H9 
+(THead (Flat Appl) x x5) H37 (pr3_pr2 c (THead (Flat Appl) x x1) (THead (Flat 
+Appl) x x5) (pr2_thin_dx c x1 x5 H25 x Appl)) x5 (refl_equal T (THead (Flat 
+Appl) x x5)))) H36))) x2 H31))) x4 H30))))) H29))) (\lambda (H28: (((eq T 
+(THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) \to (\forall (P: 
+Prop).P)))).(let H_x0 \def (term_dec (THead (Flat Appl) x x1) (THead (Flat 
+Appl) x2 x5)) in (let H29 \def H_x0 in (or_ind (eq T (THead (Flat Appl) x x1) 
+(THead (Flat Appl) x2 x5)) ((eq T (THead (Flat Appl) x x1) (THead (Flat Appl) 
+x2 x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x2 (THead (Flat 
+Cast) x4 x5))) (\lambda (H30: (eq T (THead (Flat Appl) x x1) (THead (Flat 
+Appl) x2 x5))).(let H31 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t3 _) 
+\Rightarrow t3])) (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x5) H30) in 
+((let H32 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) 
+(THead (Flat Appl) x x1) (THead (Flat Appl) x2 x5) H30) in (\lambda (H33: (eq 
+T x x2)).(let H34 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 
+H32) in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 (THead 
+(Flat Cast) x4 t3)))) (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T 
+(THead (Flat Appl) x x0) (THead (Flat Appl) t3 x4)) \to (\forall (P: 
+Prop).P))) H28 x H33) in (let H36 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c 
+x t3)) H18 x H33) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) 
+t3 (THead (Flat Cast) x4 x1)))) (H11 (THead (Flat Appl) x x4) H35 (pr3_pr2 c 
+(THead (Flat Appl) x x0) (THead (Flat Appl) x x4) (pr2_thin_dx c x0 x4 H24 x 
+Appl)) x x4 (refl_equal T (THead (Flat Appl) x x4)) x1 (sn3_sing c (THead 
+(Flat Appl) x x1) H10)) x2 H33))) x5 H32)))) H31))) (\lambda (H30: (((eq T 
+(THead (Flat Appl) x x1) (THead (Flat Appl) x2 x5)) \to (\forall (P: 
 Prop).P)))).(H11 (THead (Flat Appl) x2 x4) H28 (pr3_flat c x x2 (pr3_pr2 c x 
 x2 H18) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x2 x4 (refl_equal T (THead (Flat 
 Appl) x2 x4)) x5 (H10 (THead (Flat Appl) x2 x5) H30 (pr3_flat c x x2 (pr3_pr2 
@@ -1194,24 +1139,23 @@ Appl) x x1) (THead (Flat Appl) x2 x3)) in (let H23 \def H_x in (or_ind (eq T
 (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3)) ((eq T (THead (Flat Appl) 
 x x1) (THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)) (sn3 c (THead 
 (Flat Appl) x2 x3)) (\lambda (H24: (eq T (THead (Flat Appl) x x1) (THead 
-(Flat Appl) x2 x3))).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) 
-\Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x1) 
-(THead (Flat Appl) x2 x3) H24) in ((let H26 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | 
-(TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat 
-Appl) x x1) (THead (Flat Appl) x2 x3) H24) in (\lambda (H27: (eq T x 
-x2)).(let H28 \def (eq_ind_r T x3 (\lambda (t3: T).(pr2 c x1 t3)) H22 x1 H26) 
-in (let H29 \def (eq_ind_r T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x 
-(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P: 
-Prop).P))) H20 x1 H26) in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat 
-Appl) x2 t3))) (let H30 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead 
-(Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) t3 x1)) \to 
-(\forall (P: Prop).P))) H29 x H27) in (let H31 \def (eq_ind_r T x2 (\lambda 
-(t3: T).(pr2 c x t3)) H18 x H27) in (eq_ind T x (\lambda (t3: T).(sn3 c 
-(THead (Flat Appl) t3 x1))) (sn3_sing c (THead (Flat Appl) x x1) H10) x2 
-H27))) x3 H26))))) H25))) (\lambda (H24: (((eq T (THead (Flat Appl) x x1) 
-(THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat 
+(Flat Appl) x2 x3))).(let H25 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t3 _) 
+\Rightarrow t3])) (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3) H24) in 
+((let H26 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) 
+(THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3) H24) in (\lambda (H27: (eq 
+T x x2)).(let H28 \def (eq_ind_r T x3 (\lambda (t3: T).(pr2 c x1 t3)) H22 x1 
+H26) in (let H29 \def (eq_ind_r T x3 (\lambda (t3: T).((eq T (THead (Flat 
+Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall 
+(P: Prop).P))) H20 x1 H26) in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead 
+(Flat Appl) x2 t3))) (let H30 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T 
+(THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) t3 x1)) 
+\to (\forall (P: Prop).P))) H29 x H27) in (let H31 \def (eq_ind_r T x2 
+(\lambda (t3: T).(pr2 c x t3)) H18 x H27) in (eq_ind T x (\lambda (t3: 
+T).(sn3 c (THead (Flat Appl) t3 x1))) (sn3_sing c (THead (Flat Appl) x x1) 
+H10) x2 H27))) x3 H26))))) H25))) (\lambda (H24: (((eq T (THead (Flat Appl) x 
+x1) (THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat 
 Appl) x2 x3) H24 (pr3_flat c x x2 (pr3_pr2 c x x2 H18) x1 x3 (pr3_pr2 c x1 x3 
 H22) Appl))) H23)))) H21)) t2 H17))))))) H16)) (\lambda (H16: (ex4_4 T T T T 
 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
@@ -1235,9 +1179,8 @@ u0) x3 x5))))).(let H21 \def (eq_ind T t2 (\lambda (t3: T).((eq T (THead
 (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: Prop).P))) H13 
 (THead (Bind Abbr) x4 x5) H18) in (eq_ind_r T (THead (Bind Abbr) x4 x5) 
 (\lambda (t3: T).(sn3 c t3)) (let H22 \def (eq_ind T (THead (Flat Cast) x0 
-x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
+x1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) x2 
 x3) H17) in (False_ind (sn3 c (THead (Bind Abbr) x4 x5)) H22)) t2 
 H18)))))))))) H16)) (\lambda (H16: (ex6_6 B T T T T T (\lambda (b: 
@@ -1274,16 +1217,12 @@ T t2 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda
 Prop).P))) H13 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) 
 H19) in (eq_ind_r T (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) 
 x5)) (\lambda (t3: T).(sn3 c t3)) (let H24 \def (eq_ind T (THead (Flat Cast) 
-x0 x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
+x0 x1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef 
+_) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x2) x3 x4) 
 H18) in (False_ind (sn3 c (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) 
 O x6) x5))) H24)) t2 H19)))))))))))))) H16)) H15))))))))))))))) y0 H5)))) 
 H4))))))))) y H0))))) H)))).
-(* COMMENTS
-Initial nodes: 5149
-END *)
 
 theorem sn3_appl_bind:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: 
@@ -1407,167 +1346,121 @@ x (THead (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to
 Prop).P)))).(H8 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H34: (eq T 
 (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
 x0))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
-(lift (S O) O x1) x0) H34) in (let H36 \def (eq_ind_r T x1 (\lambda (t0: 
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H33 x (lift_inj x x1 (S O) O 
-H35)) in (let H37 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x 
-(lift_inj x x1 (S O) O H35)) in (H36 (refl_equal T x) P)))))) (pr3_flat 
-(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c 
-(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 
-(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl) x1 x0 
-(refl_equal T (THead (Flat Appl) (lift (S O) O x1) x0)))) H32))) x4 H29)))) 
-(\lambda (H29: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H8 (THead (Flat 
-Appl) (lift (S O) O x1) x4) (\lambda (H30: (eq T (THead (Flat Appl) (lift (S 
-O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P: 
-Prop).(let H31 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat 
-\to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
-(lift (S O) O x1) x4) H30) in ((let H32 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat 
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H30) in 
-(\lambda (H33: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H34 \def 
-(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) 
-H29 x0 H32) in (let H35 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T (THead 
-(Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 (THead (Bind b) 
-t1 t0))) \to (\forall (P0: Prop).P0))) H26 x0 H32) in (let H36 \def (eq_ind_r 
-T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) H22 x0 H32) in (let 
-H37 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead 
-(Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to (\forall 
-(P0: Prop).P0))) H35 x (lift_inj x x1 (S O) O H33)) in (let H38 \def 
+with [(TSort _) \Rightarrow (lref_map (\lambda (x5: nat).(plus x5 (S O))) O 
+x) | (TLRef _) \Rightarrow (lref_map (\lambda (x5: nat).(plus x5 (S O))) O x) 
+| (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) 
+(THead (Flat Appl) (lift (S O) O x1) x0) H34) in (let H36 \def (eq_ind_r T x1 
+(\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H33 x (lift_inj x 
+x1 (S O) O H35)) in (let H37 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x 
+t0)) H15 x (lift_inj x x1 (S O) O H35)) in (H36 (refl_equal T x) P)))))) 
+(pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift 
+(CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x 
+x1 (pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl) x1 
+x0 (refl_equal T (THead (Flat Appl) (lift (S O) O x1) x0)))) H32))) x4 
+H29)))) (\lambda (H29: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H8 (THead 
+(Flat Appl) (lift (S O) O x1) x4) (\lambda (H30: (eq T (THead (Flat Appl) 
+(lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P: 
+Prop).(let H31 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow (lref_map (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) 
+\Rightarrow (lref_map (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 
+_) \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat 
+Appl) (lift (S O) O x1) x4) H30) in ((let H32 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) 
+(THead (Flat Appl) (lift (S O) O x1) x4) H30) in (\lambda (H33: (eq T (lift 
+(S O) O x) (lift (S O) O x1))).(let H34 \def (eq_ind_r T x4 (\lambda (t0: 
+T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H29 x0 H32) in (let H35 \def 
+(eq_ind_r T x4 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) 
+t1 x0)) (THead (Flat Appl) x1 (THead (Bind b) t1 t0))) \to (\forall (P0: 
+Prop).P0))) H26 x0 H32) in (let H36 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 
+(CHead c (Bind b) t1) x0 t0)) H22 x0 H32) in (let H37 \def (eq_ind_r T x1 
+(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead 
+(Flat Appl) t0 (THead (Bind b) t1 x0))) \to (\forall (P0: Prop).P0))) H35 x 
+(lift_inj x x1 (S O) O H33)) in (let H38 \def (eq_ind_r T x1 (\lambda (t0: 
+T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H33)) in (H34 (refl_equal T x0) 
+P)))))))) H31)))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift (S 
+O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c c 
+(drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x4 H22 Appl) x1 x4 
+(refl_equal T (THead (Flat Appl) (lift (S O) O x1) x4)))) H28))) x3 H25)))) 
+(\lambda (H25: (((eq T t1 x3) \to (\forall (P: Prop).P)))).(H2 x3 H25 H21 x4 
+x1 (sn3_cpr3_trans c t1 x3 H21 (Bind b) (THead (Flat Appl) (lift (S O) O x1) 
+x4) (let H_x1 \def (term_dec x0 x4) in (let H26 \def H_x1 in (or_ind (eq T x0 
+x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) 
+(THead (Flat Appl) (lift (S O) O x1) x4)) (\lambda (H27: (eq T x0 x4)).(let 
+H28 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) 
+H22 x0 H27) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) 
+(THead (Flat Appl) (lift (S O) O x1) t0))) (let H_x2 \def (term_dec x x1) in 
+(let H29 \def H_x2 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: 
+Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) 
+x0)) (\lambda (H30: (eq T x x1)).(let H31 \def (eq_ind_r T x1 (\lambda (t0: 
+T).(pr2 c x t0)) H15 x H30) in (eq_ind T x (\lambda (t0: T).(sn3 (CHead c 
+(Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0))) (sn3_sing (CHead c 
+(Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9) x1 H30))) (\lambda 
+(H30: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift 
+(S O) O x1) x0) (\lambda (H31: (eq T (THead (Flat Appl) (lift (S O) O x) x0) 
+(THead (Flat Appl) (lift (S O) O x1) x0))).(\lambda (P: Prop).(let H32 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (lref_map 
+(\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow (lref_map 
+(\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) \Rightarrow t0])) 
+(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
+x0) H31) in (let H33 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to 
+(\forall (P0: Prop).P0))) H30 x (lift_inj x x1 (S O) O H32)) in (let H34 \def 
 (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O 
-H33)) in (H34 (refl_equal T x0) P)))))))) H31)))) (pr3_flat (CHead c (Bind b) 
+H32)) in (H33 (refl_equal T x) P)))))) (pr3_flat (CHead c (Bind b) t1) (lift 
+(S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O 
+(drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x0 
+(pr3_refl (CHead c (Bind b) t1) x0) Appl))) H29))) x4 H27))) (\lambda (H27: 
+(((eq T x0 x4) \to (\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S 
+O) O x1) x4) (\lambda (H28: (eq T (THead (Flat Appl) (lift (S O) O x) x0) 
+(THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P: Prop).(let H29 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (lref_map 
+(\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow (lref_map 
+(\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) \Rightarrow t0])) 
+(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
+x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow 
+t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) 
+O x1) x4) H28) in (\lambda (H31: (eq T (lift (S O) O x) (lift (S O) O 
+x1))).(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to 
+(\forall (P0: Prop).P0))) H27 x0 H30) in (let H33 \def (eq_ind_r T x4 
+(\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) H22 x0 H30) in (let H34 
+\def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) 
+O H31)) in (H32 (refl_equal T x0) P)))))) H29)))) (pr3_flat (CHead c (Bind b) 
 t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S 
 O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) 
-x0 x4 H22 Appl) x1 x4 (refl_equal T (THead (Flat Appl) (lift (S O) O x1) 
-x4)))) H28))) x3 H25)))) (\lambda (H25: (((eq T t1 x3) \to (\forall (P: 
-Prop).P)))).(H2 x3 H25 H21 x4 x1 (sn3_cpr3_trans c t1 x3 H21 (Bind b) (THead 
-(Flat Appl) (lift (S O) O x1) x4) (let H_x1 \def (term_dec x0 x4) in (let H26 
-\def H_x1 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) 
-(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x4)) (\lambda 
-(H27: (eq T x0 x4)).(let H28 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead 
-c (Bind b) t1) x0 t0)) H22 x0 H27) in (eq_ind T x0 (\lambda (t0: T).(sn3 
-(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) t0))) (let H_x2 
-\def (term_dec x x1) in (let H29 \def H_x2 in (or_ind (eq T x x1) ((eq T x 
-x1) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) 
-(lift (S O) O x1) x0)) (\lambda (H30: (eq T x x1)).(let H31 \def (eq_ind_r T 
-x1 (\lambda (t0: T).(pr2 c x t0)) H15 x H30) in (eq_ind T x (\lambda (t0: 
-T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0))) 
-(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9) 
-x1 H30))) (\lambda (H30: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9 
-(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H31: (eq T (THead (Flat 
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
-x0))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
-(lift (S O) O x1) x0) H31) in (let H33 \def (eq_ind_r T x1 (\lambda (t0: 
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H30 x (lift_inj x x1 (S O) O 
-H32)) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x 
-(lift_inj x x1 (S O) O H32)) in (H33 (refl_equal T x) P)))))) (pr3_flat 
-(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c 
-(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 
-(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl))) 
-H29))) x4 H27))) (\lambda (H27: (((eq T x0 x4) \to (\forall (P: 
-Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x1) x4) (\lambda (H28: (eq T 
-(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
-x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
-(lift (S O) O x1) x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat 
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H28) in 
-(\lambda (H31: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H32 \def 
-(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) 
-H27 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c 
-(Bind b) t1) x0 t0)) H22 x0 H30) in (let H34 \def (eq_ind_r T x1 (\lambda 
-(t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H31)) in (H32 (refl_equal 
-T x0) P)))))) H29)))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift 
-(S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c 
-c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x4 H22 Appl))) H26)))))) 
-H24))) x2 H20))))))) H19)) (\lambda (H19: (pr3 (CHead c (Bind b) t1) x0 (lift 
-(S O) O x2))).(sn3_gen_lift (CHead c (Bind b) t1) (THead (Flat Appl) x1 x2) 
-(S O) O (eq_ind_r T (THead (Flat Appl) (lift (S O) O x1) (lift (S O) (s (Flat 
-Appl) O) x2)) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) t0)) (sn3_pr3_trans 
-(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x0) (let H_x0 \def 
-(term_dec x x1) in (let H20 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to 
-(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S 
-O) O x1) x0)) (\lambda (H21: (eq T x x1)).(let H22 \def (eq_ind_r T x1 
-(\lambda (t0: T).(pr2 c x t0)) H15 x H21) in (eq_ind T x (\lambda (t0: 
-T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0))) 
-(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9) 
-x1 H21))) (\lambda (H21: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9 
-(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H22: (eq T (THead (Flat 
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
-x0))).(\lambda (P: Prop).(let H23 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
-lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t4))]) in lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
-(lift (S O) O x1) x0) H22) in (let H24 \def (eq_ind_r T x1 (\lambda (t0: 
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H21 x (lift_inj x x1 (S O) O 
-H23)) in (let H25 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x 
-(lift_inj x x1 (S O) O H23)) in (H24 (refl_equal T x) P)))))) (pr3_flat 
-(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c 
-(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 
-(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl))) 
-H20))) (THead (Flat Appl) (lift (S O) O x1) (lift (S O) O x2)) (pr3_thin_dx 
-(CHead c (Bind b) t1) x0 (lift (S O) O x2) H19 (lift (S O) O x1) Appl)) (lift 
-(S O) O (THead (Flat Appl) x1 x2)) (lift_head (Flat Appl) x1 x2 (S O) O)) c 
-(drop_drop (Bind b) O c c (drop_refl c) t1))) H18))) t3 H14))))))) H13)) 
-(\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: 
-T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
+x0 x4 H22 Appl))) H26)))))) H24))) x2 H20))))))) H19)) (\lambda (H19: (pr3 
+(CHead c (Bind b) t1) x0 (lift (S O) O x2))).(sn3_gen_lift (CHead c (Bind b) 
+t1) (THead (Flat Appl) x1 x2) (S O) O (eq_ind_r T (THead (Flat Appl) (lift (S 
+O) O x1) (lift (S O) (s (Flat Appl) O) x2)) (\lambda (t0: T).(sn3 (CHead c 
+(Bind b) t1) t0)) (sn3_pr3_trans (CHead c (Bind b) t1) (THead (Flat Appl) 
+(lift (S O) O x1) x0) (let H_x0 \def (term_dec x x1) in (let H20 \def H_x0 in 
+(or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 (CHead c 
+(Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x0)) (\lambda (H21: (eq T x 
+x1)).(let H22 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x H21) 
+in (eq_ind T x (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) 
+(lift (S O) O t0) x0))) (sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) 
+(lift (S O) O x) x0) H9) x1 H21))) (\lambda (H21: (((eq T x x1) \to (\forall 
+(P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H22: 
+(eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) 
+O x1) x0))).(\lambda (P: Prop).(let H23 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow (lref_map (\lambda (x3: nat).(plus x3 
+(S O))) O x) | (TLRef _) \Rightarrow (lref_map (\lambda (x3: nat).(plus x3 (S 
+O))) O x) | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O 
+x) x0) (THead (Flat Appl) (lift (S O) O x1) x0) H22) in (let H24 \def 
+(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) 
+H21 x (lift_inj x x1 (S O) O H23)) in (let H25 \def (eq_ind_r T x1 (\lambda 
+(t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H23)) in (H24 (refl_equal 
+T x) P)))))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O 
+x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c c 
+(drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind 
+b) t1) x0) Appl))) H20))) (THead (Flat Appl) (lift (S O) O x1) (lift (S O) O 
+x2)) (pr3_thin_dx (CHead c (Bind b) t1) x0 (lift (S O) O x2) H19 (lift (S O) 
+O x1) Appl)) (lift (S O) O (THead (Flat Appl) x1 x2)) (lift_head (Flat Appl) 
+x1 x2 (S O) O)) c (drop_drop (Bind b) O c c (drop_refl c) t1))) H18))) t3 
+H14))))))) H13)) (\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda 
 (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: 
 T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))))).(ex4_4_ind T T T T (\lambda 
@@ -1585,110 +1478,105 @@ b0) u0) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T (THead
 (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) H10 
 (THead (Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4) 
 (\lambda (t0: T).(sn3 c t0)) (let H19 \def (f_equal T B (\lambda (e: 
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | 
-(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return 
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
-b])])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in ((let H20 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in 
-((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ 
-t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) 
-in (\lambda (_: (eq T t1 x1)).(\lambda (H23: (eq B b Abst)).(let H24 \def 
-(eq_ind_r T x2 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead 
-c (Bind b0) u0) t0 x4)))) H17 x0 H21) in (let H25 \def (eq_ind B b (\lambda 
-(b0: B).((eq T (THead (Flat Appl) x (THead (Bind b0) t1 x0)) (THead (Bind 
-Abbr) x3 x4)) \to (\forall (P: Prop).P))) H18 Abst H23) in (let H26 \def 
-(eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T (THead (Flat Appl) 
-(lift (S O) O x) x0) t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind 
-b0) t1) (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (sn3 (CHead c (Bind 
-b0) t1) t4))))) H9 Abst H23) in (let H27 \def (eq_ind B b (\lambda (b0: 
-B).(\forall (t4: T).((((eq T (THead (Flat Appl) (lift (S O) O x) x0) t4) \to 
-(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) (THead (Flat Appl) 
-(lift (S O) O x) x0) t4) \to (\forall (x5: T).(\forall (x6: T).((eq T t4 
-(THead (Flat Appl) (lift (S O) O x5) x6)) \to (sn3 c (THead (Flat Appl) x5 
-(THead (Bind b0) t1 x6)))))))))) H8 Abst H23) in (let H28 \def (eq_ind B b 
-(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P))) 
-\to ((pr3 c t1 t4) \to (\forall (t0: T).(\forall (v0: T).((sn3 (CHead c (Bind 
-b0) t4) (THead (Flat Appl) (lift (S O) O v0) t0)) \to (sn3 c (THead (Flat 
-Appl) v0 (THead (Bind b0) t4 t0)))))))))) H2 Abst H23) in (let H29 \def 
-(eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H Abst H23) in (let H30 
-\def (match (H29 (refl_equal B Abst)) in False return (\lambda (_: 
-False).(sn3 c (THead (Bind Abbr) x3 x4))) with []) in H30)))))))))) H20)) 
-H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 B T T T T T (\lambda (b0: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) 
-t1 x0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-t3 (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
-y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 
-Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind 
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b0) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b0: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2))))))) (sn3 c t3) (\lambda (x1: 
-B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: 
-T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: (eq T 
-(THead (Bind b) t1 x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T t3 
-(THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda 
+T).(match e with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead 
+k _ _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) 
+\Rightarrow b])])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in 
+((let H20 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) \Rightarrow t0])) 
+(THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in ((let H21 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef 
+_) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind b) t1 x0) 
+(THead (Bind Abst) x1 x2) H14) in (\lambda (_: (eq T t1 x1)).(\lambda (H23: 
+(eq B b Abst)).(let H24 \def (eq_ind_r T x2 (\lambda (t0: T).(\forall (b0: 
+B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t0 x4)))) H17 x0 H21) in (let 
+H25 \def (eq_ind B b (\lambda (b0: B).((eq T (THead (Flat Appl) x (THead 
+(Bind b0) t1 x0)) (THead (Bind Abbr) x3 x4)) \to (\forall (P: Prop).P))) H18 
+Abst H23) in (let H26 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: 
+T).((((eq T (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (\forall (P: 
+Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) (THead (Flat Appl) (lift (S O) O 
+x) x0) t4) \to (sn3 (CHead c (Bind b0) t1) t4))))) H9 Abst H23) in (let H27 
+\def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T (THead (Flat 
+Appl) (lift (S O) O x) x0) t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c 
+(Bind b0) t1) (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (\forall (x5: 
+T).(\forall (x6: T).((eq T t4 (THead (Flat Appl) (lift (S O) O x5) x6)) \to 
+(sn3 c (THead (Flat Appl) x5 (THead (Bind b0) t1 x6)))))))))) H8 Abst H23) in 
+(let H28 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) 
+\to (\forall (P: Prop).P))) \to ((pr3 c t1 t4) \to (\forall (t0: T).(\forall 
+(v0: T).((sn3 (CHead c (Bind b0) t4) (THead (Flat Appl) (lift (S O) O v0) 
+t0)) \to (sn3 c (THead (Flat Appl) v0 (THead (Bind b0) t4 t0)))))))))) H2 
+Abst H23) in (let H29 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) 
+H Abst H23) in (let H30 \def (match (H29 (refl_equal B Abst)) in False with 
+[]) in H30)))))))))) H20)) H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 
+B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind b0) y1 z1)))))))) (\lambda 
+(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T t3 (THead (Bind b0) y2 (THead (Flat Appl) (lift (S 
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) 
+y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead 
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
+b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2))))))) (sn3 c t3) 
+(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda 
+(x5: T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: 
+(eq T (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T 
+t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda 
 (H17: (pr2 c x x5)).(\lambda (H18: (pr2 c x2 x6)).(\lambda (H19: (pr2 (CHead 
 c (Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t0: T).((eq T 
 (THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) 
 H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) H16) in 
 (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) 
 (\lambda (t0: T).(sn3 c t0)) (let H21 \def (f_equal T B (\lambda (e: 
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | 
-(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return 
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
-b])])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in ((let H22 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in 
-((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ 
-t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in 
-(\lambda (H24: (eq T t1 x2)).(\lambda (H25: (eq B b x1)).(let H26 \def 
-(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H19 x0 
-H23) in (let H27 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H18 t1 
-H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b0: B).(pr2 (CHead c (Bind b0) 
-x6) x0 x4)) H26 b H25) in (eq_ind B b (\lambda (b0: B).(sn3 c (THead (Bind 
-b0) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (sn3_pr3_trans c (THead 
-(Bind b) t1 (THead (Flat Appl) (lift (S O) O x5) x4)) (sn3_bind b c t1 
-(sn3_sing c t1 H1) (THead (Flat Appl) (lift (S O) O x5) x4) (let H_x \def 
-(term_dec x x5) in (let H29 \def H_x in (or_ind (eq T x x5) ((eq T x x5) \to 
-(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S 
-O) O x5) x4)) (\lambda (H30: (eq T x x5)).(let H31 \def (eq_ind_r T x5 
-(\lambda (t0: T).(pr2 c x t0)) H17 x H30) in (eq_ind T x (\lambda (t0: 
-T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x4))) (let 
-H_x0 \def (term_dec x0 x4) in (let H32 \def H_x0 in (or_ind (eq T x0 x4) ((eq 
-T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat 
-Appl) (lift (S O) O x) x4)) (\lambda (H33: (eq T x0 x4)).(let H34 \def 
-(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0 
-H33) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat 
-Appl) (lift (S O) O x) t0))) (sn3_sing (CHead c (Bind b) t1) (THead (Flat 
-Appl) (lift (S O) O x) x0) H9) x4 H33))) (\lambda (H33: (((eq T x0 x4) \to 
-(\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x) x4) (\lambda 
-(H34: (eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift 
-(S O) O x) x4))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat 
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x) x4) H34) in 
-(let H36 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: 
-Prop).P0))) H33 x0 H35) in (let H37 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2 
-(CHead c (Bind b) x6) x0 t0)) H28 x0 H35) in (H36 (refl_equal T x0) P)))))) 
+T).(match e with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead 
+k _ _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) 
+\Rightarrow b])])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in 
+((let H22 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) \Rightarrow t0])) 
+(THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in ((let H23 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef 
+_) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind b) t1 x0) 
+(THead (Bind x1) x2 x3) H15) in (\lambda (H24: (eq T t1 x2)).(\lambda (H25: 
+(eq B b x1)).(let H26 \def (eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c 
+(Bind x1) x6) t0 x4)) H19 x0 H23) in (let H27 \def (eq_ind_r T x2 (\lambda 
+(t0: T).(pr2 c t0 x6)) H18 t1 H24) in (let H28 \def (eq_ind_r B x1 (\lambda 
+(b0: B).(pr2 (CHead c (Bind b0) x6) x0 x4)) H26 b H25) in (eq_ind B b 
+(\lambda (b0: B).(sn3 c (THead (Bind b0) x6 (THead (Flat Appl) (lift (S O) O 
+x5) x4)))) (sn3_pr3_trans c (THead (Bind b) t1 (THead (Flat Appl) (lift (S O) 
+O x5) x4)) (sn3_bind b c t1 (sn3_sing c t1 H1) (THead (Flat Appl) (lift (S O) 
+O x5) x4) (let H_x \def (term_dec x x5) in (let H29 \def H_x in (or_ind (eq T 
+x x5) ((eq T x x5) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) 
+(THead (Flat Appl) (lift (S O) O x5) x4)) (\lambda (H30: (eq T x x5)).(let 
+H31 \def (eq_ind_r T x5 (\lambda (t0: T).(pr2 c x t0)) H17 x H30) in (eq_ind 
+T x (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S 
+O) O t0) x4))) (let H_x0 \def (term_dec x0 x4) in (let H32 \def H_x0 in 
+(or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c 
+(Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x4)) (\lambda (H33: (eq T x0 
+x4)).(let H34 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) 
+x0 t0)) H28 x0 H33) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) 
+t1) (THead (Flat Appl) (lift (S O) O x) t0))) (sn3_sing (CHead c (Bind b) t1) 
+(THead (Flat Appl) (lift (S O) O x) x0) H9) x4 H33))) (\lambda (H33: (((eq T 
+x0 x4) \to (\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x) 
+x4) (\lambda (H34: (eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat 
+Appl) (lift (S O) O x) x4))).(\lambda (P: Prop).(let H35 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) 
+\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat Appl) (lift (S 
+O) O x) x0) (THead (Flat Appl) (lift (S O) O x) x4) H34) in (let H36 \def 
+(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) 
+H33 x0 H35) in (let H37 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c 
+(Bind b) x6) x0 t0)) H28 x0 H35) in (H36 (refl_equal T x0) P)))))) 
 (pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O 
 x) x0) (THead (Flat Appl) (lift (S O) O x) x4) (Bind b) (pr3_pr2 (CHead c 
 (Bind b) x6) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift 
@@ -1697,41 +1585,29 @@ Appl))))) H32))) x5 H30))) (\lambda (H30: (((eq T x x5) \to (\forall (P:
 Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x5) x4) (\lambda (H31: (eq T 
 (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) 
 x4))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
-lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t4))]) in lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
-(lift (S O) O x5) x4) H31) in ((let H33 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat 
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) x4) H31) in 
-(\lambda (H34: (eq T (lift (S O) O x) (lift (S O) O x5))).(let H35 \def 
-(eq_ind_r T x5 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) 
-H30 x (lift_inj x x5 (S O) O H34)) in (let H36 \def (eq_ind_r T x5 (\lambda 
-(t0: T).(pr2 c x t0)) H17 x (lift_inj x x5 (S O) O H34)) in (let H37 \def 
-(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0 
-H33) in (H35 (refl_equal T x) P)))))) H32)))) (pr3_pr3_pr3_t c t1 x6 (pr3_pr2 
-c t1 x6 H27) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift 
-(S O) O x5) x4) (Bind b) (pr3_flat (CHead c (Bind b) x6) (lift (S O) O x) 
-(lift (S O) O x5) (pr3_lift (CHead c (Bind b) x6) c (S O) O (drop_drop (Bind 
-b) O c c (drop_refl c) x6) x x5 (pr3_pr2 c x x5 H17)) x0 x4 (pr3_pr2 (CHead c 
-(Bind b) x6) x0 x4 H28) Appl)))) H29)))) (THead (Bind b) x6 (THead (Flat 
-Appl) (lift (S O) O x5) x4)) (pr3_pr2 c (THead (Bind b) t1 (THead (Flat Appl) 
-(lift (S O) O x5) x4)) (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O 
-x5) x4)) (pr2_head_1 c t1 x6 H27 (Bind b) (THead (Flat Appl) (lift (S O) O 
-x5) x4)))) x1 H25))))))) H22)) H21)) t3 H16)))))))))))))) H13)) 
-H12)))))))))))))) y H4))))) H3))))))) u H0))))).
-(* COMMENTS
-Initial nodes: 9191
-END *)
+with [(TSort _) \Rightarrow (lref_map (\lambda (x7: nat).(plus x7 (S O))) O 
+x) | (TLRef _) \Rightarrow (lref_map (\lambda (x7: nat).(plus x7 (S O))) O x) 
+| (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) 
+(THead (Flat Appl) (lift (S O) O x5) x4) H31) in ((let H33 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) 
+\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat Appl) (lift (S 
+O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) x4) H31) in (\lambda (H34: 
+(eq T (lift (S O) O x) (lift (S O) O x5))).(let H35 \def (eq_ind_r T x5 
+(\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H30 x (lift_inj x 
+x5 (S O) O H34)) in (let H36 \def (eq_ind_r T x5 (\lambda (t0: T).(pr2 c x 
+t0)) H17 x (lift_inj x x5 (S O) O H34)) in (let H37 \def (eq_ind_r T x4 
+(\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0 H33) in (H35 
+(refl_equal T x) P)))))) H32)))) (pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) 
+(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) 
+x4) (Bind b) (pr3_flat (CHead c (Bind b) x6) (lift (S O) O x) (lift (S O) O 
+x5) (pr3_lift (CHead c (Bind b) x6) c (S O) O (drop_drop (Bind b) O c c 
+(drop_refl c) x6) x x5 (pr3_pr2 c x x5 H17)) x0 x4 (pr3_pr2 (CHead c (Bind b) 
+x6) x0 x4 H28) Appl)))) H29)))) (THead (Bind b) x6 (THead (Flat Appl) (lift 
+(S O) O x5) x4)) (pr3_pr2 c (THead (Bind b) t1 (THead (Flat Appl) (lift (S O) 
+O x5) x4)) (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) 
+(pr2_head_1 c t1 x6 H27 (Bind b) (THead (Flat Appl) (lift (S O) O x5) x4)))) 
+x1 H25))))))) H22)) H21)) t3 H16)))))))))))))) H13)) H12)))))))))))))) y 
+H4))))) H3))))))) u H0))))).
 
 theorem sn3_appl_appl:
  \forall (v1: T).(\forall (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in 
@@ -1866,20 +1742,19 @@ Appl) x3 x4) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H_x \def
 ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) \to (\forall (P: 
 Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 x4))) (\lambda 
 (H27: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))).(let H28 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) 
-\Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27) in 
-((let H29 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ 
-t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27) 
-in (\lambda (H30: (eq T x x3)).(let H31 \def (eq_ind_r T x4 (\lambda (t: 
-T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) 
-x1 (THead (Flat Appl) x3 t))) \to (\forall (P: Prop).P))) H25 x0 H29) in (let 
-H32 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H24 x0 H29) in (eq_ind 
-T x0 (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 t)))) 
-(let H33 \def (eq_ind_r T x3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 
-(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) t x0))) 
-\to (\forall (P: Prop).P))) H31 x H30) in (let H34 \def (eq_ind_r T x3 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x | 
+(TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) x 
+x0) (THead (Flat Appl) x3 x4) H27) in ((let H29 \def (f_equal T T (\lambda 
+(e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
+(THead _ _ t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 
+x4) H27) in (\lambda (H30: (eq T x x3)).(let H31 \def (eq_ind_r T x4 (\lambda 
+(t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat 
+Appl) x1 (THead (Flat Appl) x3 t))) \to (\forall (P: Prop).P))) H25 x0 H29) 
+in (let H32 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H24 x0 H29) in 
+(eq_ind T x0 (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) 
+x3 t)))) (let H33 \def (eq_ind_r T x3 (\lambda (t: T).((eq T (THead (Flat 
+Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) t 
+x0))) \to (\forall (P: Prop).P))) H31 x H30) in (let H34 \def (eq_ind_r T x3 
 (\lambda (t: T).(pr2 c x t)) H23 x H30) in (eq_ind T x (\lambda (t: T).(sn3 c 
 (THead (Flat Appl) x1 (THead (Flat Appl) t x0)))) (let H_x0 \def (term_dec t0 
 x1) in (let H35 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to (\forall 
@@ -1960,68 +1835,51 @@ Abbr) x x4) (pr0_beta x3 x x (pr0_refl x) x4 x4 (pr0_refl x4))) (THead (Bind
 Abbr) x5 x6) (pr3_head_12 c x x5 (pr3_pr2 c x x5 H24) (Bind Abbr) x4 x6 
 (pr3_pr2 (CHead c (Bind Abbr) x5) x4 x6 (H25 Abbr x5)))) (\lambda (H32: (iso 
 (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind Abbr) x5 
-x6))).(\lambda (P: Prop).(let H33 \def (match H32 in iso return (\lambda (t: 
-T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x 
-(THead (Bind Abst) x3 x4))) \to ((eq T t4 (THead (Bind Abbr) x5 x6)) \to 
-P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H33: (eq T (TSort n1) 
-(THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H34: (eq T (TSort 
-n2) (THead (Bind Abbr) x5 x6))).((let H35 \def (eq_ind T (TSort n1) (\lambda 
-(e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I 
-(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T 
-(TSort n2) (THead (Bind Abbr) x5 x6)) \to P) H35)) H34))) | (iso_lref i1 i2) 
-\Rightarrow (\lambda (H33: (eq T (TLRef i1) (THead (Flat Appl) x (THead (Bind 
-Abst) x3 x4)))).(\lambda (H34: (eq T (TLRef i2) (THead (Bind Abbr) x5 
-x6))).((let H35 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x 
-(THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T (TLRef i2) (THead (Bind 
-Abbr) x5 x6)) \to P) H35)) H34))) | (iso_head v4 v5 t4 t5 k) \Rightarrow 
-(\lambda (H33: (eq T (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) 
-x3 x4)))).(\lambda (H34: (eq T (THead k v5 t5) (THead (Bind Abbr) x5 
-x6))).((let H35 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 
-| (THead _ _ t) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead 
-(Bind Abst) x3 x4)) H33) in ((let H36 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v4 | 
-(TLRef _) \Rightarrow v4 | (THead _ t _) \Rightarrow t])) (THead k v4 t4) 
-(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in ((let H37 \def 
-(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with 
-[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
-\Rightarrow k0])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) x3 
-x4)) H33) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) \to ((eq T 
-t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead k0 v5 t5) (THead (Bind Abbr) 
-x5 x6)) \to P)))) (\lambda (H38: (eq T v4 x)).(eq_ind T x (\lambda (_: 
-T).((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead (Flat Appl) v5 t5) 
-(THead (Bind Abbr) x5 x6)) \to P))) (\lambda (H39: (eq T t4 (THead (Bind 
-Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 x4) (\lambda (_: T).((eq T 
-(THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6)) \to P)) (\lambda (H40: 
-(eq T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6))).(let H41 \def 
-(eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind Abbr) x5 x6) H40) in (False_ind P H41))) t4 (sym_eq 
-T t4 (THead (Bind Abst) x3 x4) H39))) v4 (sym_eq T v4 x H38))) k (sym_eq K k 
-(Flat Appl) H37))) H36)) H35)) H34)))]) in (H33 (refl_equal T (THead (Flat 
-Appl) x (THead (Bind Abst) x3 x4))) (refl_equal T (THead (Bind Abbr) x5 
-x6))))))) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6)) (pr3_pr2 c (THead 
-(Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (THead (Flat Appl) x1 (THead (Bind 
-Abbr) x5 x6)) (pr2_head_1 c t0 x1 H17 (Flat Appl) (THead (Bind Abbr) x5 
-x6))))))))) x2 H23)))))))))) H21)) (\lambda (H21: (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
-B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+x6))).(\lambda (P: Prop).(let H33 \def (match H32 with [(iso_sort n1 n2) 
+\Rightarrow (\lambda (H33: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind 
+Abst) x3 x4)))).(\lambda (H34: (eq T (TSort n2) (THead (Bind Abbr) x5 
+x6))).((let H35 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow False])) I (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) 
+in (False_ind ((eq T (TSort n2) (THead (Bind Abbr) x5 x6)) \to P) H35)) 
+H34))) | (iso_lref i1 i2) \Rightarrow (\lambda (H33: (eq T (TLRef i1) (THead 
+(Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H34: (eq T (TLRef i2) 
+(THead (Bind Abbr) x5 x6))).((let H35 \def (eq_ind T (TLRef i1) (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x (THead (Bind Abst) 
+x3 x4)) H33) in (False_ind ((eq T (TLRef i2) (THead (Bind Abbr) x5 x6)) \to 
+P) H35)) H34))) | (iso_head v4 v5 t4 t5 k) \Rightarrow (\lambda (H33: (eq T 
+(THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda 
+(H34: (eq T (THead k v5 t5) (THead (Bind Abbr) x5 x6))).((let H35 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t4 | (TLRef 
+_) \Rightarrow t4 | (THead _ _ t) \Rightarrow t])) (THead k v4 t4) (THead 
+(Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in ((let H36 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow v4 | (TLRef _) 
+\Rightarrow v4 | (THead _ t _) \Rightarrow t])) (THead k v4 t4) (THead (Flat 
+Appl) x (THead (Bind Abst) x3 x4)) H33) in ((let H37 \def (f_equal T K 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k v4 t4) (THead (Flat 
+Appl) x (THead (Bind Abst) x3 x4)) H33) in (eq_ind K (Flat Appl) (\lambda 
+(k0: K).((eq T v4 x) \to ((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq T 
+(THead k0 v5 t5) (THead (Bind Abbr) x5 x6)) \to P)))) (\lambda (H38: (eq T v4 
+x)).(eq_ind T x (\lambda (_: T).((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq 
+T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6)) \to P))) (\lambda 
+(H39: (eq T t4 (THead (Bind Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 
+x4) (\lambda (_: T).((eq T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 
+x6)) \to P)) (\lambda (H40: (eq T (THead (Flat Appl) v5 t5) (THead (Bind 
+Abbr) x5 x6))).(let H41 \def (eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead k0 _ _) \Rightarrow (match k0 with [(Bind _) \Rightarrow False | (Flat 
+_) \Rightarrow True])])) I (THead (Bind Abbr) x5 x6) H40) in (False_ind P 
+H41))) t4 (sym_eq T t4 (THead (Bind Abst) x3 x4) H39))) v4 (sym_eq T v4 x 
+H38))) k (sym_eq K k (Flat Appl) H37))) H36)) H35)) H34)))]) in (H33 
+(refl_equal T (THead (Flat Appl) x (THead (Bind Abst) x3 x4))) (refl_equal T 
+(THead (Bind Abbr) x5 x6))))))) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 
+x6)) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (THead (Flat 
+Appl) x1 (THead (Bind Abbr) x5 x6)) (pr2_head_1 c t0 x1 H17 (Flat Appl) 
+(THead (Bind Abbr) x5 x6))))))))) x2 H23)))))))))) H21)) (\lambda (H21: 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
@@ -2030,38 +1888,49 @@ x2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
 T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) 
-(sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: B).(\lambda (x4: T).(\lambda 
-(x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H22: 
-(not (eq B x3 Abst))).(\lambda (H23: (eq T x0 (THead (Bind x3) x4 
-x5))).(\lambda (H24: (eq T x2 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S 
-O) O x7) x6)))).(\lambda (H25: (pr2 c x x7)).(\lambda (H26: (pr2 c x4 
-x8)).(\lambda (H27: (pr2 (CHead c (Bind x3) x8) x5 x6)).(let H28 \def (eq_ind 
-T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) 
-(THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H19 (THead (Bind x3) x8 
-(THead (Flat Appl) (lift (S O) O x7) x6)) H24) in (eq_ind_r T (THead (Bind 
-x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (\lambda (t: T).(sn3 c 
-(THead (Flat Appl) x1 t))) (let H29 \def (eq_ind T x0 (\lambda (t: T).((eq T 
-(THead (Flat Appl) t0 (THead (Flat Appl) x t)) (THead (Flat Appl) x1 (THead 
-(Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))) \to (\forall (P: 
-Prop).P))) H28 (THead (Bind x3) x4 x5) H23) in (let H30 \def (eq_ind T x0 
-(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to 
-(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c 
-t4))))) H11 (THead (Bind x3) x4 x5) H23) in (let H31 \def (eq_ind T x0 
-(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to 
-(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall 
-(x9: T).(\forall (x10: T).((eq T t4 (THead (Flat Appl) x9 x10)) \to (\forall 
-(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2) 
-\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to 
-(sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind x3) x4 x5) H23) 
-in (let H32 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c (THead 
-(Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P: 
-Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead (Bind x3) x4 
-x5) H23) in (let H33 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4: 
-T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t4) \to 
-(((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead 
-(Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat 
-Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c (Bind b) y2) z1 z2))))))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda 
+(x3: B).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: 
+T).(\lambda (x8: T).(\lambda (H22: (not (eq B x3 Abst))).(\lambda (H23: (eq T 
+x0 (THead (Bind x3) x4 x5))).(\lambda (H24: (eq T x2 (THead (Bind x3) x8 
+(THead (Flat Appl) (lift (S O) O x7) x6)))).(\lambda (H25: (pr2 c x 
+x7)).(\lambda (H26: (pr2 c x4 x8)).(\lambda (H27: (pr2 (CHead c (Bind x3) x8) 
+x5 x6)).(let H28 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) 
+t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: 
+Prop).P))) H19 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) 
+H24) in (eq_ind_r T (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) 
+x6)) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H29 \def (eq_ind 
+T x0 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x t)) 
+(THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O 
+x7) x6)))) \to (\forall (P: Prop).P))) H28 (THead (Bind x3) x4 x5) H23) in 
+(let H30 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4: T).((((eq T (THead 
+(Flat Appl) x t) t4) \to (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat 
+Appl) x t) t4) \to (sn3 c t4))))) H11 (THead (Bind x3) x4 x5) H23) in (let 
+H31 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat 
+Appl) x t) t4) \to (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x 
+t) t4) \to (\forall (x9: T).(\forall (x10: T).((eq T t4 (THead (Flat Appl) x9 
+x10)) \to (\forall (v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) 
+\to ((((iso t4 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) 
+v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind 
+x3) x4 x5) H23) in (let H32 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: 
+T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) 
+u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 
+(THead (Bind x3) x4 x5) H23) in (let H33 \def (eq_ind T x0 (\lambda (t: 
+T).(\forall (t4: T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c 
+t0 t4) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso 
+(THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead 
+(Flat Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x 
 t)))))))) H9 (THead (Bind x3) x4 x5) H23) in (sn3_pr3_trans c (THead (Flat 
 Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (H32 
 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (pr3_sing c 
@@ -2078,102 +1947,96 @@ x7 H25)) (Flat Appl) x5 x6 (pr3_pr2 (CHead (CHead c (Bind x3) x8) (Flat Appl)
 (lift (S O) O x7)) x5 x6 (pr2_cflat (CHead c (Bind x3) x8) x5 x6 H27 Appl 
 (lift (S O) O x7)))))) (\lambda (H34: (iso (THead (Flat Appl) x (THead (Bind 
 x3) x4 x5)) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) 
-x6)))).(\lambda (P: Prop).(let H35 \def (match H34 in iso return (\lambda (t: 
-T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x 
-(THead (Bind x3) x4 x5))) \to ((eq T t4 (THead (Bind x3) x8 (THead (Flat 
-Appl) (lift (S O) O x7) x6))) \to P))))) with [(iso_sort n1 n2) \Rightarrow 
-(\lambda (H35: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind x3) x4 
-x5)))).(\lambda (H36: (eq T (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) 
-(lift (S O) O x7) x6)))).((let H37 \def (eq_ind T (TSort n1) (\lambda (e: 
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I 
-(THead (Flat Appl) x (THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T 
-(TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to 
-P) H37)) H36))) | (iso_lref i1 i2) \Rightarrow (\lambda (H35: (eq T (TLRef 
-i1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T 
-(TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) 
-x6)))).((let H37 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x 
-(THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T (TLRef i2) (THead (Bind 
+x6)))).(\lambda (P: Prop).(let H35 \def (match H34 with [(iso_sort n1 n2) 
+\Rightarrow (\lambda (H35: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind 
+x3) x4 x5)))).(\lambda (H36: (eq T (TSort n2) (THead (Bind x3) x8 (THead 
+(Flat Appl) (lift (S O) O x7) x6)))).((let H37 \def (eq_ind T (TSort n1) 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow True | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x 
+(THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T (TSort n2) (THead (Bind 
 x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H37)) H36))) | 
-(iso_head v4 v5 t4 t5 k) \Rightarrow (\lambda (H35: (eq T (THead k v4 t4) 
-(THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T (THead k 
-v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let 
-H37 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) 
+(iso_lref i1 i2) \Rightarrow (\lambda (H35: (eq T (TLRef i1) (THead (Flat 
+Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T (TLRef i2) (THead 
+(Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let H37 \def 
+(eq_ind T (TLRef i1) (\lambda (e: T).(match e with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
+(THead (Flat Appl) x (THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T 
+(TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to 
+P) H37)) H36))) | (iso_head v4 v5 t4 t5 k) \Rightarrow (\lambda (H35: (eq T 
+(THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda 
+(H36: (eq T (THead k v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S 
+O) O x7) x6)))).((let H37 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) 
 \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4 
-x5)) H35) in ((let H38 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4 
-| (THead _ t _) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead 
-(Bind x3) x4 x5)) H35) in ((let H39 \def (f_equal T K (\lambda (e: T).(match 
-e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k v4 t4) (THead (Flat 
-Appl) x (THead (Bind x3) x4 x5)) H35) in (eq_ind K (Flat Appl) (\lambda (k0: 
-K).((eq T v4 x) \to ((eq T t4 (THead (Bind x3) x4 x5)) \to ((eq T (THead k0 
-v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to 
-P)))) (\lambda (H40: (eq T v4 x)).(eq_ind T x (\lambda (_: T).((eq T t4 
-(THead (Bind x3) x4 x5)) \to ((eq T (THead (Flat Appl) v5 t5) (THead (Bind 
-x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P))) (\lambda (H41: (eq 
-T t4 (THead (Bind x3) x4 x5))).(eq_ind T (THead (Bind x3) x4 x5) (\lambda (_: 
-T).((eq T (THead (Flat Appl) v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) 
-(lift (S O) O x7) x6))) \to P)) (\lambda (H42: (eq T (THead (Flat Appl) v5 
-t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(let H43 
-\def (eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) 
-H42) in (False_ind P H43))) t4 (sym_eq T t4 (THead (Bind x3) x4 x5) H41))) v4 
-(sym_eq T v4 x H40))) k (sym_eq K k (Flat Appl) H39))) H38)) H37)) H36)))]) 
-in (H35 (refl_equal T (THead (Flat Appl) x (THead (Bind x3) x4 x5))) 
-(refl_equal T (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) 
-x6)))))))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift 
-(S O) O x7) x6))) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x3) x8 (THead 
-(Flat Appl) (lift (S O) O x7) x6))) (THead (Flat Appl) x1 (THead (Bind x3) x8 
-(THead (Flat Appl) (lift (S O) O x7) x6))) (pr2_head_1 c t0 x1 H17 (Flat 
-Appl) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))))))))) 
-x2 H24)))))))))))))) H21)) H20)) t3 H16))))))) H15)) (\lambda (H15: (ex4_4 T 
+x5)) H35) in ((let H38 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4 | (THead _ t _) 
+\Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4 
+x5)) H35) in ((let H39 \def (f_equal T K (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
+\Rightarrow k0])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4 
+x5)) H35) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) \to ((eq T 
+t4 (THead (Bind x3) x4 x5)) \to ((eq T (THead k0 v5 t5) (THead (Bind x3) x8 
+(THead (Flat Appl) (lift (S O) O x7) x6))) \to P)))) (\lambda (H40: (eq T v4 
+x)).(eq_ind T x (\lambda (_: T).((eq T t4 (THead (Bind x3) x4 x5)) \to ((eq T 
+(THead (Flat Appl) v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) 
+O x7) x6))) \to P))) (\lambda (H41: (eq T t4 (THead (Bind x3) x4 
+x5))).(eq_ind T (THead (Bind x3) x4 x5) (\lambda (_: T).((eq T (THead (Flat 
+Appl) v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) 
+\to P)) (\lambda (H42: (eq T (THead (Flat Appl) v5 t5) (THead (Bind x3) x8 
+(THead (Flat Appl) (lift (S O) O x7) x6)))).(let H43 \def (eq_ind T (THead 
+(Flat Appl) v5 t5) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False 
+| (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with 
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
+x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) H42) in (False_ind P H43))) 
+t4 (sym_eq T t4 (THead (Bind x3) x4 x5) H41))) v4 (sym_eq T v4 x H40))) k 
+(sym_eq K k (Flat Appl) H39))) H38)) H37)) H36)))]) in (H35 (refl_equal T 
+(THead (Flat Appl) x (THead (Bind x3) x4 x5))) (refl_equal T (THead (Bind x3) 
+x8 (THead (Flat Appl) (lift (S O) O x7) x6)))))))) (THead (Flat Appl) x1 
+(THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (pr3_pr2 c 
+(THead (Flat Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O 
+x7) x6))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift 
+(S O) O x7) x6))) (pr2_head_1 c t0 x1 H17 (Flat Appl) (THead (Bind x3) x8 
+(THead (Flat Appl) (lift (S O) O x7) x6)))))))))) x2 H24)))))))))))))) H21)) 
+H20)) t3 H16))))))) H15)) (\lambda (H15: (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) 
+x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T 
 T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
 (THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
 (_: T).(pr2 c t0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
 T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-z1 t4))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: 
-T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c t3) (\lambda (x1: 
-T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H16: (eq T 
-(THead (Flat Appl) x x0) (THead (Bind Abst) x1 x2))).(\lambda (H17: (eq T t3 
-(THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c t0 x3)).(\lambda (_: 
-((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let 
-H20 \def (eq_ind T t3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead 
-(Flat Appl) x x0)) t) \to (\forall (P: Prop).P))) H12 (THead (Bind Abbr) x3 
-x4) H17) in (eq_ind_r T (THead (Bind Abbr) x3 x4) (\lambda (t: T).(sn3 c t)) 
-(let H21 \def (eq_ind T (THead (Flat Appl) x x0) (\lambda (ee: T).(match ee 
-in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef 
-_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return 
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind Abst) x1 x2) H16) in (False_ind (sn3 c (THead (Bind 
-Abbr) x3 x4)) H21)) t3 H17)))))))))) H15)) (\lambda (H15: (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) 
-y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (x4: T).(\lambda (H16: (eq T (THead (Flat Appl) x x0) (THead 
+(Bind Abst) x1 x2))).(\lambda (H17: (eq T t3 (THead (Bind Abbr) x3 
+x4))).(\lambda (_: (pr2 c t0 x3)).(\lambda (_: ((\forall (b: B).(\forall (u: 
+T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H20 \def (eq_ind T t3 (\lambda 
+(t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall 
+(P: Prop).P))) H12 (THead (Bind Abbr) x3 x4) H17) in (eq_ind_r T (THead (Bind 
+Abbr) x3 x4) (\lambda (t: T).(sn3 c t)) (let H21 \def (eq_ind T (THead (Flat 
+Appl) x x0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind 
+_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) x1 
+x2) H16) in (False_ind (sn3 c (THead (Bind Abbr) x3 x4)) H21)) t3 
+H17)))))))))) H15)) (\lambda (H15: (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat 
+Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead 
 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
@@ -2193,16 +2056,12 @@ t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
 Prop).P))) H12 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) 
 H18) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) 
 x4)) (\lambda (t: T).(sn3 c t)) (let H23 \def (eq_ind T (THead (Flat Appl) x 
-x0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
+x0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x1) x2 x3) 
 H17) in (False_ind (sn3 c (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) 
 O x5) x4))) H23)) t3 H18)))))))))))))) H15)) H14)))))) t2 H3))))))))) v2 
 H4))))))))) y H0))))) H))))).
-(* COMMENTS
-Initial nodes: 9317
-END *)
 
 theorem sn3_appl_beta:
  \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((sn3 c 
@@ -2223,9 +2082,6 @@ Abst) w t)) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat
 Appl) u (THead (Bind Abbr) v t)) H (THead (Flat Appl) u u2) (pr3_thin_dx c 
 (THead (Bind Abbr) v t) u2 (pr3_iso_beta v w t c u2 H4 H5) u Appl)))))))) 
 H1))))))))).
-(* COMMENTS
-Initial nodes: 289
-END *)
 
 theorem sn3_appl_appls:
  \forall (v1: T).(\forall (t1: T).(\forall (vs: TList).(let u1 \def (THeads 
@@ -2242,9 +2098,6 @@ theorem sn3_appl_appls:
 (Flat Appl) vs t1)) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat 
 Appl) v2 u2))))))).(sn3_appl_appl v1 (THeads (Flat Appl) vs t1) c H v2 H0 
 H1))))))))).
-(* COMMENTS
-Initial nodes: 141
-END *)
 
 theorem sn3_appls_lref:
  \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (us: 
@@ -2276,9 +2129,6 @@ Appl) (TCons t1 t2) (TLRef i)) u2)).(\lambda (H9: (((iso (THeads (Flat Appl)
 (TCons t1 t2) (TLRef i)) u2) \to (\forall (P: Prop).P)))).(H9 
 (nf2_iso_appls_lref c i H (TCons t1 t2) u2 H8) (sn3 c (THead (Flat Appl) t 
 u2))))))))) H5))) H3))))))) t0))) us)))).
-(* COMMENTS
-Initial nodes: 577
-END *)
 
 theorem sn3_appls_cast:
  \forall (c: C).(\forall (vs: TList).(\forall (u: T).((sn3 c (THeads (Flat 
@@ -2332,9 +2182,6 @@ Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t (THeads (Flat Appl) (TCons
 t1 t2) t3)) H2 (THead (Flat Appl) t u2) (pr3_thin_dx c (THeads (Flat Appl) 
 (TCons t1 t2) t3) u2 (pr3_iso_appls_cast c u t3 (TCons t1 t2) u2 H11 H12) t 
 Appl))))))))) H7)))))) H3))))))))))) t0))) vs)).
-(* COMMENTS
-Initial nodes: 1025
-END *)
 
 theorem sn3_appls_bind:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: 
@@ -2387,9 +2234,6 @@ H (TCons t t0) u t1 c u2 H7 H8) in (sn3_pr3_trans c (THead (Flat Appl) v
 t1) v H3) (THead (Flat Appl) v u2) (pr3_flat c v v (pr3_refl c v) (THead 
 (Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) u2 H9 
 Appl)))))))))) H4))))))))) vs0))) vs)))))).
-(* COMMENTS
-Initial nodes: 1143
-END *)
 
 theorem sn3_appls_beta:
  \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (us: TList).((sn3 c 
@@ -2439,9 +2283,6 @@ Appl) v (THead (Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8
 (THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v 
 t))) H1 (THead (Flat Appl) u u2) (pr3_thin_dx c (THeads (Flat Appl) (TCons t0 
 t1) (THead (Bind Abbr) v t)) u2 H8 u Appl))))))))) H3)))))))))) us0))) us)))).
-(* COMMENTS
-Initial nodes: 987
-END *)
 
 theorem sn3_lift:
  \forall (d: C).(\forall (t: T).((sn3 d t) \to (\forall (c: C).(\forall (h: 
@@ -2468,9 +2309,6 @@ x)).(\lambda (P: Prop).(let H10 \def (eq_ind_r T x (\lambda (t0: T).((eq T
 H11 \def (eq_ind_r T x (\lambda (t0: T).(pr2 d t1 t0)) H7 t1 H9) in (H10 
 (refl_equal T (lift h i t1)) P))))) (pr3_pr2 d t1 x H7) c h i H2) t2 H6))))) 
 H5))))))))))))) t H))).
-(* COMMENTS
-Initial nodes: 439
-END *)
 
 theorem sn3_abbr:
  \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
@@ -2497,22 +2335,18 @@ x1))).(\lambda (H6: (eq T t2 (lift (S i) O x1))).(let H7 \def (eq_ind T t2
 i) O x1) H6) in (eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(sn3 c t)) (let 
 H8 \def (eq_ind C (CHead d (Bind Abbr) v) (\lambda (c0: C).(getl i c c0)) H 
 (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 
-(Bind Abbr) x1) H5)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in 
-C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) 
-\Rightarrow c0])) (CHead d (Bind Abbr) v) (CHead x0 (Bind Abbr) x1) 
-(getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 (Bind Abbr) x1) H5)) in 
-((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: 
-C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead d 
+(Bind Abbr) x1) H5)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d 
 (Bind Abbr) v) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) 
-i H (CHead x0 (Bind Abbr) x1) H5)) in (\lambda (H11: (eq C d x0)).(let H12 
-\def (eq_ind_r T x1 (\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 v 
-H10) in (eq_ind T v (\lambda (t: T).(sn3 c (lift (S i) O t))) (let H13 \def 
-(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) v))) H12 d 
-H11) in (sn3_lift d v H0 c (S i) O (getl_drop Abbr c d v i H13))) x1 H10)))) 
-H9))) t2 H6)))))) H4)) H3))))))))))).
-(* COMMENTS
-Initial nodes: 743
-END *)
+i H (CHead x0 (Bind Abbr) x1) H5)) in ((let H10 \def (f_equal C T (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow 
+t])) (CHead d (Bind Abbr) v) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d 
+(Bind Abbr) v) i H (CHead x0 (Bind Abbr) x1) H5)) in (\lambda (H11: (eq C d 
+x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t: T).(getl i c (CHead x0 (Bind 
+Abbr) t))) H8 v H10) in (eq_ind T v (\lambda (t: T).(sn3 c (lift (S i) O t))) 
+(let H13 \def (eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) 
+v))) H12 d H11) in (sn3_lift d v H0 c (S i) O (getl_drop Abbr c d v i H13))) 
+x1 H10)))) H9))) t2 H6)))))) H4)) H3))))))))))).
 
 theorem sn3_appls_abbr:
  \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
@@ -2552,9 +2386,6 @@ Appl) (TCons t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2)
 (pr3_thin_dx c (THeads (Flat Appl) (TCons t t0) (lift (S i) O w)) u2 
 (pr3_iso_appls_abbr c d w i H (TCons t t0) u2 H6 H7) v Appl)))))))) 
 H3)))))))) vs0))) vs)))))).
-(* COMMENTS
-Initial nodes: 797
-END *)
 
 theorem sns3_lifts:
  \forall (c: C).(\forall (d: C).(\forall (h: nat).(\forall (i: nat).((drop h 
@@ -2569,7 +2400,4 @@ H1 in (land_ind (sn3 d t) (sns3 d t0) (land (sn3 c (lift h i t)) (sns3 c
 (lifts h i t0))) (\lambda (H3: (sn3 d t)).(\lambda (H4: (sns3 d t0)).(conj 
 (sn3 c (lift h i t)) (sns3 c (lifts h i t0)) (sn3_lift d t H3 c h i H) (H0 
 H4)))) H2)))))) ts)))))).
-(* COMMENTS
-Initial nodes: 185
-END *)