From 049d55c73d1746e15a40e89b17fd88b62f002d93 Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Wed, 4 Mar 2015 22:02:45 +0000 Subject: [PATCH] components: nf2, sn3, ex2 --- .../contribs/lambdadelta/basic_1/ex2/defs.ma | 5 +- .../contribs/lambdadelta/basic_1/ex2/props.ma | 336 ++-- .../contribs/lambdadelta/basic_1/nf2/arity.ma | 144 +- .../contribs/lambdadelta/basic_1/nf2/dec.ma | 494 +++-- .../contribs/lambdadelta/basic_1/nf2/defs.ma | 11 +- .../contribs/lambdadelta/basic_1/nf2/fwd.ma | 331 ++-- .../contribs/lambdadelta/basic_1/nf2/iso.ma | 229 ++- .../contribs/lambdadelta/basic_1/nf2/lift1.ma | 38 +- .../contribs/lambdadelta/basic_1/nf2/pr3.ma | 52 +- .../contribs/lambdadelta/basic_1/nf2/props.ma | 654 ++++--- .../contribs/lambdadelta/basic_1/sn3/defs.ma | 11 +- .../contribs/lambdadelta/basic_1/sn3/fwd.ma | 368 ++-- .../contribs/lambdadelta/basic_1/sn3/lift1.ma | 51 +- .../contribs/lambdadelta/basic_1/sn3/nf2.ma | 72 +- .../contribs/lambdadelta/basic_1/sn3/props.ma | 1596 ++++++++--------- 15 files changed, 2393 insertions(+), 1999 deletions(-) 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 *) -- 2.39.2