- some improvements in the generation of terms

[helm.git] / matita / matita / contribs / lambdadelta / basic_1 / T / dec.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_1/T/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/T/dec.ma
index e8e7ed885debc48cabb3ecdf08aabc3c85a64921..48a1cd509bec6342b3546d4e721b33ed5c233ee4 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/T/dec.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/T/dec.ma
@@ -20,650 +20,636 @@ theorem terms_props__bind_dec:
  \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) ((eq B b1 b2) \to (\forall 
 (P: Prop).P))))
 \def
- \lambda (b1: B).(let TMP_61 \def (\lambda (b: B).(\forall (b2: B).(let 
-TMP_60 \def (eq B b b2) in (let TMP_59 \def ((eq B b b2) \to (\forall (P: 
-Prop).P)) in (or TMP_60 TMP_59))))) in (let TMP_58 \def (\lambda (b2: B).(let 
-TMP_57 \def (\lambda (b: B).(let TMP_56 \def (eq B Abbr b) in (let TMP_55 
-\def ((eq B Abbr b) \to (\forall (P: Prop).P)) in (or TMP_56 TMP_55)))) in 
-(let TMP_53 \def (eq B Abbr Abbr) in (let TMP_52 \def ((eq B Abbr Abbr) \to 
-(\forall (P: Prop).P)) in (let TMP_51 \def (refl_equal B Abbr) in (let TMP_54 
-\def (or_introl TMP_53 TMP_52 TMP_51) in (let TMP_49 \def (eq B Abbr Abst) in 
-(let TMP_48 \def ((eq B Abbr Abst) \to (\forall (P: Prop).P)) in (let TMP_47 
-\def (\lambda (H: (eq B Abbr Abst)).(\lambda (P: Prop).(let TMP_46 \def 
-(\lambda (ee: B).(match ee in B with [Abbr \Rightarrow True | Abst 
-\Rightarrow False | Void \Rightarrow False])) in (let H0 \def (eq_ind B Abbr 
-TMP_46 I Abst H) in (False_ind P H0))))) in (let TMP_50 \def (or_intror 
-TMP_49 TMP_48 TMP_47) in (let TMP_44 \def (eq B Abbr Void) in (let TMP_43 
-\def ((eq B Abbr Void) \to (\forall (P: Prop).P)) in (let TMP_42 \def 
-(\lambda (H: (eq B Abbr Void)).(\lambda (P: Prop).(let TMP_41 \def (\lambda 
-(ee: B).(match ee in B with [Abbr \Rightarrow True | Abst \Rightarrow False | 
-Void \Rightarrow False])) in (let H0 \def (eq_ind B Abbr TMP_41 I Void H) in 
-(False_ind P H0))))) in (let TMP_45 \def (or_intror TMP_44 TMP_43 TMP_42) in 
-(B_ind TMP_57 TMP_54 TMP_50 TMP_45 b2))))))))))))))) in (let TMP_40 \def 
-(\lambda (b2: B).(let TMP_39 \def (\lambda (b: B).(let TMP_38 \def (eq B Abst 
-b) in (let TMP_37 \def ((eq B Abst b) \to (\forall (P: Prop).P)) in (or 
-TMP_38 TMP_37)))) in (let TMP_35 \def (eq B Abst Abbr) in (let TMP_34 \def 
-((eq B Abst Abbr) \to (\forall (P: Prop).P)) in (let TMP_33 \def (\lambda (H: 
-(eq B Abst Abbr)).(\lambda (P: Prop).(let TMP_32 \def (\lambda (ee: B).(match 
-ee in B with [Abbr \Rightarrow False | Abst \Rightarrow True | Void 
-\Rightarrow False])) in (let H0 \def (eq_ind B Abst TMP_32 I Abbr H) in 
-(False_ind P H0))))) in (let TMP_36 \def (or_intror TMP_35 TMP_34 TMP_33) in 
-(let TMP_30 \def (eq B Abst Abst) in (let TMP_29 \def ((eq B Abst Abst) \to 
-(\forall (P: Prop).P)) in (let TMP_28 \def (refl_equal B Abst) in (let TMP_31 
-\def (or_introl TMP_30 TMP_29 TMP_28) in (let TMP_26 \def (eq B Abst Void) in 
-(let TMP_25 \def ((eq B Abst Void) \to (\forall (P: Prop).P)) in (let TMP_24 
-\def (\lambda (H: (eq B Abst Void)).(\lambda (P: Prop).(let TMP_23 \def 
-(\lambda (ee: B).(match ee in B with [Abbr \Rightarrow False | Abst 
-\Rightarrow True | Void \Rightarrow False])) in (let H0 \def (eq_ind B Abst 
-TMP_23 I Void H) in (False_ind P H0))))) in (let TMP_27 \def (or_intror 
-TMP_26 TMP_25 TMP_24) in (B_ind TMP_39 TMP_36 TMP_31 TMP_27 b2))))))))))))))) 
-in (let TMP_22 \def (\lambda (b2: B).(let TMP_21 \def (\lambda (b: B).(let 
-TMP_20 \def (eq B Void b) in (let TMP_19 \def ((eq B Void b) \to (\forall (P: 
-Prop).P)) in (or TMP_20 TMP_19)))) in (let TMP_17 \def (eq B Void Abbr) in 
-(let TMP_16 \def ((eq B Void Abbr) \to (\forall (P: Prop).P)) in (let TMP_15 
-\def (\lambda (H: (eq B Void Abbr)).(\lambda (P: Prop).(let TMP_14 \def 
-(\lambda (ee: B).(match ee in B with [Abbr \Rightarrow False | Abst 
+ \lambda (b1: B).(let TMP_3 \def (\lambda (b: B).(\forall (b2: B).(let TMP_1 
+\def (eq B b b2) in (let TMP_2 \def ((eq B b b2) \to (\forall (P: Prop).P)) 
+in (or TMP_1 TMP_2))))) in (let TMP_21 \def (\lambda (b2: B).(let TMP_6 \def 
+(\lambda (b: B).(let TMP_4 \def (eq B Abbr b) in (let TMP_5 \def ((eq B Abbr 
+b) \to (\forall (P: Prop).P)) in (or TMP_4 TMP_5)))) in (let TMP_7 \def (eq B 
+Abbr Abbr) in (let TMP_8 \def ((eq B Abbr Abbr) \to (\forall (P: Prop).P)) in 
+(let TMP_9 \def (refl_equal B Abbr) in (let TMP_10 \def (or_introl TMP_7 
+TMP_8 TMP_9) in (let TMP_11 \def (eq B Abbr Abst) in (let TMP_12 \def ((eq B 
+Abbr Abst) \to (\forall (P: Prop).P)) in (let TMP_14 \def (\lambda (H: (eq B 
+Abbr Abst)).(\lambda (P: Prop).(let TMP_13 \def (\lambda (ee: B).(match ee 
+with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow 
+False])) in (let H0 \def (eq_ind B Abbr TMP_13 I Abst H) in (False_ind P 
+H0))))) in (let TMP_15 \def (or_intror TMP_11 TMP_12 TMP_14) in (let TMP_16 
+\def (eq B Abbr Void) in (let TMP_17 \def ((eq B Abbr Void) \to (\forall (P: 
+Prop).P)) in (let TMP_19 \def (\lambda (H: (eq B Abbr Void)).(\lambda (P: 
+Prop).(let TMP_18 \def (\lambda (ee: B).(match ee with [Abbr \Rightarrow True 
+| Abst \Rightarrow False | Void \Rightarrow False])) in (let H0 \def (eq_ind 
+B Abbr TMP_18 I Void H) in (False_ind P H0))))) in (let TMP_20 \def 
+(or_intror TMP_16 TMP_17 TMP_19) in (B_ind TMP_6 TMP_10 TMP_15 TMP_20 
+b2))))))))))))))) in (let TMP_39 \def (\lambda (b2: B).(let TMP_24 \def 
+(\lambda (b: B).(let TMP_22 \def (eq B Abst b) in (let TMP_23 \def ((eq B 
+Abst b) \to (\forall (P: Prop).P)) in (or TMP_22 TMP_23)))) in (let TMP_25 
+\def (eq B Abst Abbr) in (let TMP_26 \def ((eq B Abst Abbr) \to (\forall (P: 
+Prop).P)) in (let TMP_28 \def (\lambda (H: (eq B Abst Abbr)).(\lambda (P: 
+Prop).(let TMP_27 \def (\lambda (ee: B).(match ee with [Abbr \Rightarrow 
+False | Abst \Rightarrow True | Void \Rightarrow False])) in (let H0 \def 
+(eq_ind B Abst TMP_27 I Abbr H) in (False_ind P H0))))) in (let TMP_29 \def 
+(or_intror TMP_25 TMP_26 TMP_28) in (let TMP_30 \def (eq B Abst Abst) in (let 
+TMP_31 \def ((eq B Abst Abst) \to (\forall (P: Prop).P)) in (let TMP_32 \def 
+(refl_equal B Abst) in (let TMP_33 \def (or_introl TMP_30 TMP_31 TMP_32) in 
+(let TMP_34 \def (eq B Abst Void) in (let TMP_35 \def ((eq B Abst Void) \to 
+(\forall (P: Prop).P)) in (let TMP_37 \def (\lambda (H: (eq B Abst 
+Void)).(\lambda (P: Prop).(let TMP_36 \def (\lambda (ee: B).(match ee with 
+[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False])) 
+in (let H0 \def (eq_ind B Abst TMP_36 I Void H) in (False_ind P H0))))) in 
+(let TMP_38 \def (or_intror TMP_34 TMP_35 TMP_37) in (B_ind TMP_24 TMP_29 
+TMP_33 TMP_38 b2))))))))))))))) in (let TMP_57 \def (\lambda (b2: B).(let 
+TMP_42 \def (\lambda (b: B).(let TMP_40 \def (eq B Void b) in (let TMP_41 
+\def ((eq B Void b) \to (\forall (P: Prop).P)) in (or TMP_40 TMP_41)))) in 
+(let TMP_43 \def (eq B Void Abbr) in (let TMP_44 \def ((eq B Void Abbr) \to 
+(\forall (P: Prop).P)) in (let TMP_46 \def (\lambda (H: (eq B Void 
+Abbr)).(\lambda (P: Prop).(let TMP_45 \def (\lambda (ee: B).(match ee with 
+[Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True])) 
+in (let H0 \def (eq_ind B Void TMP_45 I Abbr H) in (False_ind P H0))))) in 
+(let TMP_47 \def (or_intror TMP_43 TMP_44 TMP_46) in (let TMP_48 \def (eq B 
+Void Abst) in (let TMP_49 \def ((eq B Void Abst) \to (\forall (P: Prop).P)) 
+in (let TMP_51 \def (\lambda (H: (eq B Void Abst)).(\lambda (P: Prop).(let 
+TMP_50 \def (\lambda (ee: B).(match ee with [Abbr \Rightarrow False | Abst 
 \Rightarrow False | Void \Rightarrow True])) in (let H0 \def (eq_ind B Void 
-TMP_14 I Abbr H) in (False_ind P H0))))) in (let TMP_18 \def (or_intror 
-TMP_17 TMP_16 TMP_15) in (let TMP_12 \def (eq B Void Abst) in (let TMP_11 
-\def ((eq B Void Abst) \to (\forall (P: Prop).P)) in (let TMP_10 \def 
-(\lambda (H: (eq B Void Abst)).(\lambda (P: Prop).(let TMP_9 \def (\lambda 
-(ee: B).(match ee in B with [Abbr \Rightarrow False | Abst \Rightarrow False 
-| Void \Rightarrow True])) in (let H0 \def (eq_ind B Void TMP_9 I Abst H) in 
-(False_ind P H0))))) in (let TMP_13 \def (or_intror TMP_12 TMP_11 TMP_10) in 
-(let TMP_7 \def (eq B Void Void) in (let TMP_6 \def ((eq B Void Void) \to 
-(\forall (P: Prop).P)) in (let TMP_5 \def (refl_equal B Void) in (let TMP_8 
-\def (or_introl TMP_7 TMP_6 TMP_5) in (B_ind TMP_21 TMP_18 TMP_13 TMP_8 
-b2))))))))))))))) in (B_ind TMP_61 TMP_58 TMP_40 TMP_22 b1))))).
+TMP_50 I Abst H) in (False_ind P H0))))) in (let TMP_52 \def (or_intror 
+TMP_48 TMP_49 TMP_51) in (let TMP_53 \def (eq B Void Void) in (let TMP_54 
+\def ((eq B Void Void) \to (\forall (P: Prop).P)) in (let TMP_55 \def 
+(refl_equal B Void) in (let TMP_56 \def (or_introl TMP_53 TMP_54 TMP_55) in 
+(B_ind TMP_42 TMP_47 TMP_52 TMP_56 b2))))))))))))))) in (B_ind TMP_3 TMP_21 
+TMP_39 TMP_57 b1))))).
 
 theorem bind_dec_not:
  \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) (not (eq B b1 b2))))
 \def
  \lambda (b1: B).(\lambda (b2: B).(let H_x \def (terms_props__bind_dec b1 b2) 
-in (let H \def H_x in (let TMP_73 \def (eq B b1 b2) in (let TMP_72 \def ((eq 
-B b1 b2) \to (\forall (P: Prop).P)) in (let TMP_70 \def (eq B b1 b2) in (let 
-TMP_69 \def ((eq B b1 b2) \to False) in (let TMP_71 \def (or TMP_70 TMP_69) 
-in (let TMP_68 \def (\lambda (H0: (eq B b1 b2)).(let TMP_67 \def (eq B b1 b2) 
-in (let TMP_66 \def ((eq B b1 b2) \to False) in (or_introl TMP_67 TMP_66 
-H0)))) in (let TMP_65 \def (\lambda (H0: (((eq B b1 b2) \to (\forall (P: 
-Prop).P)))).(let TMP_64 \def (eq B b1 b2) in (let TMP_63 \def ((eq B b1 b2) 
-\to False) in (let TMP_62 \def (\lambda (H1: (eq B b1 b2)).(H0 H1 False)) in 
-(or_intror TMP_64 TMP_63 TMP_62))))) in (or_ind TMP_73 TMP_72 TMP_71 TMP_68 
-TMP_65 H))))))))))).
+in (let H \def H_x in (let TMP_1 \def (eq B b1 b2) in (let TMP_2 \def ((eq B 
+b1 b2) \to (\forall (P: Prop).P)) in (let TMP_3 \def (eq B b1 b2) in (let 
+TMP_4 \def ((eq B b1 b2) \to False) in (let TMP_5 \def (or TMP_3 TMP_4) in 
+(let TMP_8 \def (\lambda (H0: (eq B b1 b2)).(let TMP_6 \def (eq B b1 b2) in 
+(let TMP_7 \def ((eq B b1 b2) \to False) in (or_introl TMP_6 TMP_7 H0)))) in 
+(let TMP_12 \def (\lambda (H0: (((eq B b1 b2) \to (\forall (P: 
+Prop).P)))).(let TMP_9 \def (eq B b1 b2) in (let TMP_10 \def ((eq B b1 b2) 
+\to False) in (let TMP_11 \def (\lambda (H1: (eq B b1 b2)).(H0 H1 False)) in 
+(or_intror TMP_9 TMP_10 TMP_11))))) in (or_ind TMP_1 TMP_2 TMP_5 TMP_8 TMP_12 
+H))))))))))).
 
 theorem terms_props__flat_dec:
  \forall (f1: F).(\forall (f2: F).(or (eq F f1 f2) ((eq F f1 f2) \to (\forall 
 (P: Prop).P))))
 \def
- \lambda (f1: F).(let TMP_102 \def (\lambda (f: F).(\forall (f2: F).(let 
-TMP_101 \def (eq F f f2) in (let TMP_100 \def ((eq F f f2) \to (\forall (P: 
-Prop).P)) in (or TMP_101 TMP_100))))) in (let TMP_99 \def (\lambda (f2: 
-F).(let TMP_98 \def (\lambda (f: F).(let TMP_97 \def (eq F Appl f) in (let 
-TMP_96 \def ((eq F Appl f) \to (\forall (P: Prop).P)) in (or TMP_97 
-TMP_96)))) in (let TMP_94 \def (eq F Appl Appl) in (let TMP_93 \def ((eq F 
-Appl Appl) \to (\forall (P: Prop).P)) in (let TMP_92 \def (refl_equal F Appl) 
-in (let TMP_95 \def (or_introl TMP_94 TMP_93 TMP_92) in (let TMP_90 \def (eq 
-F Appl Cast) in (let TMP_89 \def ((eq F Appl Cast) \to (\forall (P: Prop).P)) 
-in (let TMP_88 \def (\lambda (H: (eq F Appl Cast)).(\lambda (P: Prop).(let 
-TMP_87 \def (\lambda (ee: F).(match ee in F with [Appl \Rightarrow True | 
-Cast \Rightarrow False])) in (let H0 \def (eq_ind F Appl TMP_87 I Cast H) in 
-(False_ind P H0))))) in (let TMP_91 \def (or_intror TMP_90 TMP_89 TMP_88) in 
-(F_ind TMP_98 TMP_95 TMP_91 f2))))))))))) in (let TMP_86 \def (\lambda (f2: 
-F).(let TMP_85 \def (\lambda (f: F).(let TMP_84 \def (eq F Cast f) in (let 
-TMP_83 \def ((eq F Cast f) \to (\forall (P: Prop).P)) in (or TMP_84 
-TMP_83)))) in (let TMP_81 \def (eq F Cast Appl) in (let TMP_80 \def ((eq F 
-Cast Appl) \to (\forall (P: Prop).P)) in (let TMP_79 \def (\lambda (H: (eq F 
-Cast Appl)).(\lambda (P: Prop).(let TMP_78 \def (\lambda (ee: F).(match ee in 
-F with [Appl \Rightarrow False | Cast \Rightarrow True])) in (let H0 \def 
-(eq_ind F Cast TMP_78 I Appl H) in (False_ind P H0))))) in (let TMP_82 \def 
-(or_intror TMP_81 TMP_80 TMP_79) in (let TMP_76 \def (eq F Cast Cast) in (let 
-TMP_75 \def ((eq F Cast Cast) \to (\forall (P: Prop).P)) in (let TMP_74 \def 
-(refl_equal F Cast) in (let TMP_77 \def (or_introl TMP_76 TMP_75 TMP_74) in 
-(F_ind TMP_85 TMP_82 TMP_77 f2))))))))))) in (F_ind TMP_102 TMP_99 TMP_86 
-f1)))).
+ \lambda (f1: F).(let TMP_3 \def (\lambda (f: F).(\forall (f2: F).(let TMP_1 
+\def (eq F f f2) in (let TMP_2 \def ((eq F f f2) \to (\forall (P: Prop).P)) 
+in (or TMP_1 TMP_2))))) in (let TMP_16 \def (\lambda (f2: F).(let TMP_6 \def 
+(\lambda (f: F).(let TMP_4 \def (eq F Appl f) in (let TMP_5 \def ((eq F Appl 
+f) \to (\forall (P: Prop).P)) in (or TMP_4 TMP_5)))) in (let TMP_7 \def (eq F 
+Appl Appl) in (let TMP_8 \def ((eq F Appl Appl) \to (\forall (P: Prop).P)) in 
+(let TMP_9 \def (refl_equal F Appl) in (let TMP_10 \def (or_introl TMP_7 
+TMP_8 TMP_9) in (let TMP_11 \def (eq F Appl Cast) in (let TMP_12 \def ((eq F 
+Appl Cast) \to (\forall (P: Prop).P)) in (let TMP_14 \def (\lambda (H: (eq F 
+Appl Cast)).(\lambda (P: Prop).(let TMP_13 \def (\lambda (ee: F).(match ee 
+with [Appl \Rightarrow True | Cast \Rightarrow False])) in (let H0 \def 
+(eq_ind F Appl TMP_13 I Cast H) in (False_ind P H0))))) in (let TMP_15 \def 
+(or_intror TMP_11 TMP_12 TMP_14) in (F_ind TMP_6 TMP_10 TMP_15 f2))))))))))) 
+in (let TMP_29 \def (\lambda (f2: F).(let TMP_19 \def (\lambda (f: F).(let 
+TMP_17 \def (eq F Cast f) in (let TMP_18 \def ((eq F Cast f) \to (\forall (P: 
+Prop).P)) in (or TMP_17 TMP_18)))) in (let TMP_20 \def (eq F Cast Appl) in 
+(let TMP_21 \def ((eq F Cast Appl) \to (\forall (P: Prop).P)) in (let TMP_23 
+\def (\lambda (H: (eq F Cast Appl)).(\lambda (P: Prop).(let TMP_22 \def 
+(\lambda (ee: F).(match ee with [Appl \Rightarrow False | Cast \Rightarrow 
+True])) in (let H0 \def (eq_ind F Cast TMP_22 I Appl H) in (False_ind P 
+H0))))) in (let TMP_24 \def (or_intror TMP_20 TMP_21 TMP_23) in (let TMP_25 
+\def (eq F Cast Cast) in (let TMP_26 \def ((eq F Cast Cast) \to (\forall (P: 
+Prop).P)) in (let TMP_27 \def (refl_equal F Cast) in (let TMP_28 \def 
+(or_introl TMP_25 TMP_26 TMP_27) in (F_ind TMP_19 TMP_24 TMP_28 f2))))))))))) 
+in (F_ind TMP_3 TMP_16 TMP_29 f1)))).
 
 theorem terms_props__kind_dec:
  \forall (k1: K).(\forall (k2: K).(or (eq K k1 k2) ((eq K k1 k2) \to (\forall 
 (P: Prop).P))))
 \def
- \lambda (k1: K).(let TMP_197 \def (\lambda (k: K).(\forall (k2: K).(let 
-TMP_196 \def (eq K k k2) in (let TMP_195 \def ((eq K k k2) \to (\forall (P: 
-Prop).P)) in (or TMP_196 TMP_195))))) in (let TMP_194 \def (\lambda (b: 
-B).(\lambda (k2: K).(let TMP_193 \def (\lambda (k: K).(let TMP_191 \def (Bind 
-b) in (let TMP_192 \def (eq K TMP_191 k) in (let TMP_190 \def ((eq K (Bind b) 
-k) \to (\forall (P: Prop).P)) in (or TMP_192 TMP_190))))) in (let TMP_189 
-\def (\lambda (b0: B).(let H_x \def (terms_props__bind_dec b b0) in (let H 
-\def H_x in (let TMP_188 \def (eq B b b0) in (let TMP_187 \def ((eq B b b0) 
-\to (\forall (P: Prop).P)) in (let TMP_184 \def (Bind b) in (let TMP_183 \def 
-(Bind b0) in (let TMP_185 \def (eq K TMP_184 TMP_183) in (let TMP_182 \def 
-((eq K (Bind b) (Bind b0)) \to (\forall (P: Prop).P)) in (let TMP_186 \def 
-(or TMP_185 TMP_182) in (let TMP_181 \def (\lambda (H0: (eq B b b0)).(let 
-TMP_180 \def (\lambda (b1: B).(let TMP_178 \def (Bind b) in (let TMP_177 \def 
-(Bind b1) in (let TMP_179 \def (eq K TMP_178 TMP_177) in (let TMP_176 \def 
-((eq K (Bind b) (Bind b1)) \to (\forall (P: Prop).P)) in (or TMP_179 
-TMP_176)))))) in (let TMP_173 \def (Bind b) in (let TMP_172 \def (Bind b) in 
-(let TMP_174 \def (eq K TMP_173 TMP_172) in (let TMP_171 \def ((eq K (Bind b) 
-(Bind b)) \to (\forall (P: Prop).P)) in (let TMP_169 \def (Bind b) in (let 
-TMP_170 \def (refl_equal K TMP_169) in (let TMP_175 \def (or_introl TMP_174 
-TMP_171 TMP_170) in (eq_ind B b TMP_180 TMP_175 b0 H0)))))))))) in (let 
-TMP_168 \def (\lambda (H0: (((eq B b b0) \to (\forall (P: Prop).P)))).(let 
-TMP_166 \def (Bind b) in (let TMP_165 \def (Bind b0) in (let TMP_167 \def (eq 
-K TMP_166 TMP_165) in (let TMP_164 \def ((eq K (Bind b) (Bind b0)) \to 
-(\forall (P: Prop).P)) in (let TMP_163 \def (\lambda (H1: (eq K (Bind b) 
-(Bind b0))).(\lambda (P: Prop).(let TMP_160 \def (\lambda (e: K).(match e in 
-K with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])) in (let TMP_159 
-\def (Bind b) in (let TMP_158 \def (Bind b0) in (let H2 \def (f_equal K B 
-TMP_160 TMP_159 TMP_158 H1) in (let TMP_161 \def (\lambda (b1: B).((eq B b 
-b1) \to (\forall (P0: Prop).P0))) in (let H3 \def (eq_ind_r B b0 TMP_161 H0 b 
-H2) in (let TMP_162 \def (refl_equal B b) in (H3 TMP_162 P)))))))))) in 
-(or_intror TMP_167 TMP_164 TMP_163))))))) in (or_ind TMP_188 TMP_187 TMP_186 
-TMP_181 TMP_168 H))))))))))))) in (let TMP_157 \def (\lambda (f: F).(let 
-TMP_155 \def (Bind b) in (let TMP_154 \def (Flat f) in (let TMP_156 \def (eq 
-K TMP_155 TMP_154) in (let TMP_153 \def ((eq K (Bind b) (Flat f)) \to 
-(\forall (P: Prop).P)) in (let TMP_152 \def (\lambda (H: (eq K (Bind b) (Flat 
-f))).(\lambda (P: Prop).(let TMP_151 \def (Bind b) in (let TMP_150 \def 
-(\lambda (ee: K).(match ee in K with [(Bind _) \Rightarrow True | (Flat _) 
-\Rightarrow False])) in (let TMP_149 \def (Flat f) in (let H0 \def (eq_ind K 
-TMP_151 TMP_150 I TMP_149 H) in (False_ind P H0))))))) in (or_intror TMP_156 
-TMP_153 TMP_152))))))) in (K_ind TMP_193 TMP_189 TMP_157 k2)))))) in (let 
-TMP_148 \def (\lambda (f: F).(\lambda (k2: K).(let TMP_147 \def (\lambda (k: 
-K).(let TMP_145 \def (Flat f) in (let TMP_146 \def (eq K TMP_145 k) in (let 
-TMP_144 \def ((eq K (Flat f) k) \to (\forall (P: Prop).P)) in (or TMP_146 
-TMP_144))))) in (let TMP_143 \def (\lambda (b: B).(let TMP_141 \def (Flat f) 
-in (let TMP_140 \def (Bind b) in (let TMP_142 \def (eq K TMP_141 TMP_140) in 
-(let TMP_139 \def ((eq K (Flat f) (Bind b)) \to (\forall (P: Prop).P)) in 
-(let TMP_138 \def (\lambda (H: (eq K (Flat f) (Bind b))).(\lambda (P: 
-Prop).(let TMP_137 \def (Flat f) in (let TMP_136 \def (\lambda (ee: K).(match 
-ee in K with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) in 
-(let TMP_135 \def (Bind b) in (let H0 \def (eq_ind K TMP_137 TMP_136 I 
-TMP_135 H) in (False_ind P H0))))))) in (or_intror TMP_142 TMP_139 
-TMP_138))))))) in (let TMP_134 \def (\lambda (f0: F).(let H_x \def 
-(terms_props__flat_dec f f0) in (let H \def H_x in (let TMP_133 \def (eq F f 
-f0) in (let TMP_132 \def ((eq F f f0) \to (\forall (P: Prop).P)) in (let 
-TMP_129 \def (Flat f) in (let TMP_128 \def (Flat f0) in (let TMP_130 \def (eq 
-K TMP_129 TMP_128) in (let TMP_127 \def ((eq K (Flat f) (Flat f0)) \to 
-(\forall (P: Prop).P)) in (let TMP_131 \def (or TMP_130 TMP_127) in (let 
-TMP_126 \def (\lambda (H0: (eq F f f0)).(let TMP_125 \def (\lambda (f1: 
-F).(let TMP_123 \def (Flat f) in (let TMP_122 \def (Flat f1) in (let TMP_124 
-\def (eq K TMP_123 TMP_122) in (let TMP_121 \def ((eq K (Flat f) (Flat f1)) 
-\to (\forall (P: Prop).P)) in (or TMP_124 TMP_121)))))) in (let TMP_118 \def 
-(Flat f) in (let TMP_117 \def (Flat f) in (let TMP_119 \def (eq K TMP_118 
-TMP_117) in (let TMP_116 \def ((eq K (Flat f) (Flat f)) \to (\forall (P: 
-Prop).P)) in (let TMP_114 \def (Flat f) in (let TMP_115 \def (refl_equal K 
-TMP_114) in (let TMP_120 \def (or_introl TMP_119 TMP_116 TMP_115) in (eq_ind 
-F f TMP_125 TMP_120 f0 H0)))))))))) in (let TMP_113 \def (\lambda (H0: (((eq 
-F f f0) \to (\forall (P: Prop).P)))).(let TMP_111 \def (Flat f) in (let 
-TMP_110 \def (Flat f0) in (let TMP_112 \def (eq K TMP_111 TMP_110) in (let 
-TMP_109 \def ((eq K (Flat f) (Flat f0)) \to (\forall (P: Prop).P)) in (let 
-TMP_108 \def (\lambda (H1: (eq K (Flat f) (Flat f0))).(\lambda (P: Prop).(let 
-TMP_105 \def (\lambda (e: K).(match e in K with [(Bind _) \Rightarrow f | 
-(Flat f1) \Rightarrow f1])) in (let TMP_104 \def (Flat f) in (let TMP_103 
-\def (Flat f0) in (let H2 \def (f_equal K F TMP_105 TMP_104 TMP_103 H1) in 
-(let TMP_106 \def (\lambda (f1: F).((eq F f f1) \to (\forall (P0: Prop).P0))) 
-in (let H3 \def (eq_ind_r F f0 TMP_106 H0 f H2) in (let TMP_107 \def 
-(refl_equal F f) in (H3 TMP_107 P)))))))))) in (or_intror TMP_112 TMP_109 
-TMP_108))))))) in (or_ind TMP_133 TMP_132 TMP_131 TMP_126 TMP_113 
-H))))))))))))) in (K_ind TMP_147 TMP_143 TMP_134 k2)))))) in (K_ind TMP_197 
-TMP_194 TMP_148 k1)))).
+ \lambda (k1: K).(let TMP_3 \def (\lambda (k: K).(\forall (k2: K).(let TMP_1 
+\def (eq K k k2) in (let TMP_2 \def ((eq K k k2) \to (\forall (P: Prop).P)) 
+in (or TMP_1 TMP_2))))) in (let TMP_49 \def (\lambda (b: B).(\lambda (k2: 
+K).(let TMP_7 \def (\lambda (k: K).(let TMP_4 \def (Bind b) in (let TMP_5 
+\def (eq K TMP_4 k) in (let TMP_6 \def ((eq K (Bind b) k) \to (\forall (P: 
+Prop).P)) in (or TMP_5 TMP_6))))) in (let TMP_39 \def (\lambda (b0: B).(let 
+H_x \def (terms_props__bind_dec b b0) in (let H \def H_x in (let TMP_8 \def 
+(eq B b b0) in (let TMP_9 \def ((eq B b b0) \to (\forall (P: Prop).P)) in 
+(let TMP_10 \def (Bind b) in (let TMP_11 \def (Bind b0) in (let TMP_12 \def 
+(eq K TMP_10 TMP_11) in (let TMP_13 \def ((eq K (Bind b) (Bind b0)) \to 
+(\forall (P: Prop).P)) in (let TMP_14 \def (or TMP_12 TMP_13) in (let TMP_27 
+\def (\lambda (H0: (eq B b b0)).(let TMP_19 \def (\lambda (b1: B).(let TMP_15 
+\def (Bind b) in (let TMP_16 \def (Bind b1) in (let TMP_17 \def (eq K TMP_15 
+TMP_16) in (let TMP_18 \def ((eq K (Bind b) (Bind b1)) \to (\forall (P: 
+Prop).P)) in (or TMP_17 TMP_18)))))) in (let TMP_20 \def (Bind b) in (let 
+TMP_21 \def (Bind b) in (let TMP_22 \def (eq K TMP_20 TMP_21) in (let TMP_23 
+\def ((eq K (Bind b) (Bind b)) \to (\forall (P: Prop).P)) in (let TMP_24 \def 
+(Bind b) in (let TMP_25 \def (refl_equal K TMP_24) in (let TMP_26 \def 
+(or_introl TMP_22 TMP_23 TMP_25) in (eq_ind B b TMP_19 TMP_26 b0 H0)))))))))) 
+in (let TMP_38 \def (\lambda (H0: (((eq B b b0) \to (\forall (P: 
+Prop).P)))).(let TMP_28 \def (Bind b) in (let TMP_29 \def (Bind b0) in (let 
+TMP_30 \def (eq K TMP_28 TMP_29) in (let TMP_31 \def ((eq K (Bind b) (Bind 
+b0)) \to (\forall (P: Prop).P)) in (let TMP_37 \def (\lambda (H1: (eq K (Bind 
+b) (Bind b0))).(\lambda (P: Prop).(let TMP_32 \def (\lambda (e: K).(match e 
+with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])) in (let TMP_33 
+\def (Bind b) in (let TMP_34 \def (Bind b0) in (let H2 \def (f_equal K B 
+TMP_32 TMP_33 TMP_34 H1) in (let TMP_35 \def (\lambda (b1: B).((eq B b b1) 
+\to (\forall (P0: Prop).P0))) in (let H3 \def (eq_ind_r B b0 TMP_35 H0 b H2) 
+in (let TMP_36 \def (refl_equal B b) in (H3 TMP_36 P)))))))))) in (or_intror 
+TMP_30 TMP_31 TMP_37))))))) in (or_ind TMP_8 TMP_9 TMP_14 TMP_27 TMP_38 
+H))))))))))))) in (let TMP_48 \def (\lambda (f: F).(let TMP_40 \def (Bind b) 
+in (let TMP_41 \def (Flat f) in (let TMP_42 \def (eq K TMP_40 TMP_41) in (let 
+TMP_43 \def ((eq K (Bind b) (Flat f)) \to (\forall (P: Prop).P)) in (let 
+TMP_47 \def (\lambda (H: (eq K (Bind b) (Flat f))).(\lambda (P: Prop).(let 
+TMP_44 \def (Bind b) in (let TMP_45 \def (\lambda (ee: K).(match ee with 
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])) in (let TMP_46 
+\def (Flat f) in (let H0 \def (eq_ind K TMP_44 TMP_45 I TMP_46 H) in 
+(False_ind P H0))))))) in (or_intror TMP_42 TMP_43 TMP_47))))))) in (K_ind 
+TMP_7 TMP_39 TMP_48 k2)))))) in (let TMP_95 \def (\lambda (f: F).(\lambda 
+(k2: K).(let TMP_53 \def (\lambda (k: K).(let TMP_50 \def (Flat f) in (let 
+TMP_51 \def (eq K TMP_50 k) in (let TMP_52 \def ((eq K (Flat f) k) \to 
+(\forall (P: Prop).P)) in (or TMP_51 TMP_52))))) in (let TMP_62 \def (\lambda 
+(b: B).(let TMP_54 \def (Flat f) in (let TMP_55 \def (Bind b) in (let TMP_56 
+\def (eq K TMP_54 TMP_55) in (let TMP_57 \def ((eq K (Flat f) (Bind b)) \to 
+(\forall (P: Prop).P)) in (let TMP_61 \def (\lambda (H: (eq K (Flat f) (Bind 
+b))).(\lambda (P: Prop).(let TMP_58 \def (Flat f) in (let TMP_59 \def 
+(\lambda (ee: K).(match ee with [(Bind _) \Rightarrow False | (Flat _) 
+\Rightarrow True])) in (let TMP_60 \def (Bind b) in (let H0 \def (eq_ind K 
+TMP_58 TMP_59 I TMP_60 H) in (False_ind P H0))))))) in (or_intror TMP_56 
+TMP_57 TMP_61))))))) in (let TMP_94 \def (\lambda (f0: F).(let H_x \def 
+(terms_props__flat_dec f f0) in (let H \def H_x in (let TMP_63 \def (eq F f 
+f0) in (let TMP_64 \def ((eq F f f0) \to (\forall (P: Prop).P)) in (let 
+TMP_65 \def (Flat f) in (let TMP_66 \def (Flat f0) in (let TMP_67 \def (eq K 
+TMP_65 TMP_66) in (let TMP_68 \def ((eq K (Flat f) (Flat f0)) \to (\forall 
+(P: Prop).P)) in (let TMP_69 \def (or TMP_67 TMP_68) in (let TMP_82 \def 
+(\lambda (H0: (eq F f f0)).(let TMP_74 \def (\lambda (f1: F).(let TMP_70 \def 
+(Flat f) in (let TMP_71 \def (Flat f1) in (let TMP_72 \def (eq K TMP_70 
+TMP_71) in (let TMP_73 \def ((eq K (Flat f) (Flat f1)) \to (\forall (P: 
+Prop).P)) in (or TMP_72 TMP_73)))))) in (let TMP_75 \def (Flat f) in (let 
+TMP_76 \def (Flat f) in (let TMP_77 \def (eq K TMP_75 TMP_76) in (let TMP_78 
+\def ((eq K (Flat f) (Flat f)) \to (\forall (P: Prop).P)) in (let TMP_79 \def 
+(Flat f) in (let TMP_80 \def (refl_equal K TMP_79) in (let TMP_81 \def 
+(or_introl TMP_77 TMP_78 TMP_80) in (eq_ind F f TMP_74 TMP_81 f0 H0)))))))))) 
+in (let TMP_93 \def (\lambda (H0: (((eq F f f0) \to (\forall (P: 
+Prop).P)))).(let TMP_83 \def (Flat f) in (let TMP_84 \def (Flat f0) in (let 
+TMP_85 \def (eq K TMP_83 TMP_84) in (let TMP_86 \def ((eq K (Flat f) (Flat 
+f0)) \to (\forall (P: Prop).P)) in (let TMP_92 \def (\lambda (H1: (eq K (Flat 
+f) (Flat f0))).(\lambda (P: Prop).(let TMP_87 \def (\lambda (e: K).(match e 
+with [(Bind _) \Rightarrow f | (Flat f1) \Rightarrow f1])) in (let TMP_88 
+\def (Flat f) in (let TMP_89 \def (Flat f0) in (let H2 \def (f_equal K F 
+TMP_87 TMP_88 TMP_89 H1) in (let TMP_90 \def (\lambda (f1: F).((eq F f f1) 
+\to (\forall (P0: Prop).P0))) in (let H3 \def (eq_ind_r F f0 TMP_90 H0 f H2) 
+in (let TMP_91 \def (refl_equal F f) in (H3 TMP_91 P)))))))))) in (or_intror 
+TMP_85 TMP_86 TMP_92))))))) in (or_ind TMP_63 TMP_64 TMP_69 TMP_82 TMP_93 
+H))))))))))))) in (K_ind TMP_53 TMP_62 TMP_94 k2)))))) in (K_ind TMP_3 TMP_49 
+TMP_95 k1)))).
 
 theorem term_dec:
  \forall (t1: T).(\forall (t2: T).(or (eq T t1 t2) ((eq T t1 t2) \to (\forall 
 (P: Prop).P))))
 \def
- \lambda (t1: T).(let TMP_447 \def (\lambda (t: T).(\forall (t2: T).(let 
-TMP_446 \def (eq T t t2) in (let TMP_445 \def ((eq T t t2) \to (\forall (P: 
-Prop).P)) in (or TMP_446 TMP_445))))) in (let TMP_444 \def (\lambda (n: 
-nat).(\lambda (t2: T).(let TMP_443 \def (\lambda (t: T).(let TMP_441 \def 
-(TSort n) in (let TMP_442 \def (eq T TMP_441 t) in (let TMP_440 \def ((eq T 
-(TSort n) t) \to (\forall (P: Prop).P)) in (or TMP_442 TMP_440))))) in (let 
-TMP_439 \def (\lambda (n0: nat).(let H_x \def (nat_dec n n0) in (let H \def 
-H_x in (let TMP_438 \def (eq nat n n0) in (let TMP_437 \def ((eq nat n n0) 
-\to (\forall (P: Prop).P)) in (let TMP_434 \def (TSort n) in (let TMP_433 
-\def (TSort n0) in (let TMP_435 \def (eq T TMP_434 TMP_433) in (let TMP_432 
-\def ((eq T (TSort n) (TSort n0)) \to (\forall (P: Prop).P)) in (let TMP_436 
-\def (or TMP_435 TMP_432) in (let TMP_431 \def (\lambda (H0: (eq nat n 
-n0)).(let TMP_430 \def (\lambda (n1: nat).(let TMP_428 \def (TSort n) in (let 
-TMP_427 \def (TSort n1) in (let TMP_429 \def (eq T TMP_428 TMP_427) in (let 
-TMP_426 \def ((eq T (TSort n) (TSort n1)) \to (\forall (P: Prop).P)) in (or 
-TMP_429 TMP_426)))))) in (let TMP_423 \def (TSort n) in (let TMP_422 \def 
-(TSort n) in (let TMP_424 \def (eq T TMP_423 TMP_422) in (let TMP_421 \def 
-((eq T (TSort n) (TSort n)) \to (\forall (P: Prop).P)) in (let TMP_419 \def 
-(TSort n) in (let TMP_420 \def (refl_equal T TMP_419) in (let TMP_425 \def 
-(or_introl TMP_424 TMP_421 TMP_420) in (eq_ind nat n TMP_430 TMP_425 n0 
-H0)))))))))) in (let TMP_418 \def (\lambda (H0: (((eq nat n n0) \to (\forall 
-(P: Prop).P)))).(let TMP_416 \def (TSort n) in (let TMP_415 \def (TSort n0) 
-in (let TMP_417 \def (eq T TMP_416 TMP_415) in (let TMP_414 \def ((eq T 
-(TSort n) (TSort n0)) \to (\forall (P: Prop).P)) in (let TMP_413 \def 
-(\lambda (H1: (eq T (TSort n) (TSort n0))).(\lambda (P: Prop).(let TMP_410 
-\def (\lambda (e: T).(match e in T with [(TSort n1) \Rightarrow n1 | (TLRef 
-_) \Rightarrow n | (THead _ _ _) \Rightarrow n])) in (let TMP_409 \def (TSort 
-n) in (let TMP_408 \def (TSort n0) in (let H2 \def (f_equal T nat TMP_410 
-TMP_409 TMP_408 H1) in (let TMP_411 \def (\lambda (n1: nat).((eq nat n n1) 
-\to (\forall (P0: Prop).P0))) in (let H3 \def (eq_ind_r nat n0 TMP_411 H0 n 
-H2) in (let TMP_412 \def (refl_equal nat n) in (H3 TMP_412 P)))))))))) in 
-(or_intror TMP_417 TMP_414 TMP_413))))))) in (or_ind TMP_438 TMP_437 TMP_436 
-TMP_431 TMP_418 H))))))))))))) in (let TMP_407 \def (\lambda (n0: nat).(let 
-TMP_405 \def (TSort n) in (let TMP_404 \def (TLRef n0) in (let TMP_406 \def 
-(eq T TMP_405 TMP_404) in (let TMP_403 \def ((eq T (TSort n) (TLRef n0)) \to 
-(\forall (P: Prop).P)) in (let TMP_402 \def (\lambda (H: (eq T (TSort n) 
-(TLRef n0))).(\lambda (P: Prop).(let TMP_401 \def (TSort n) in (let TMP_400 
-\def (\lambda (ee: T).(match ee in T with [(TSort _) \Rightarrow True | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) in (let 
-TMP_399 \def (TLRef n0) in (let H0 \def (eq_ind T TMP_401 TMP_400 I TMP_399 
-H) in (False_ind P H0))))))) in (or_intror TMP_406 TMP_403 TMP_402))))))) in 
-(let TMP_398 \def (\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T 
-(TSort n) t) ((eq T (TSort n) t) \to (\forall (P: Prop).P)))).(\lambda (t0: 
-T).(\lambda (_: (or (eq T (TSort n) t0) ((eq T (TSort n) t0) \to (\forall (P: 
-Prop).P)))).(let TMP_396 \def (TSort n) in (let TMP_395 \def (THead k t t0) 
-in (let TMP_397 \def (eq T TMP_396 TMP_395) in (let TMP_394 \def ((eq T 
-(TSort n) (THead k t t0)) \to (\forall (P: Prop).P)) in (let TMP_393 \def 
-(\lambda (H1: (eq T (TSort n) (THead k t t0))).(\lambda (P: Prop).(let 
-TMP_392 \def (TSort n) in (let TMP_391 \def (\lambda (ee: T).(match ee in T 
-with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ 
-_) \Rightarrow False])) in (let TMP_390 \def (THead k t t0) in (let H2 \def 
-(eq_ind T TMP_392 TMP_391 I TMP_390 H1) in (False_ind P H2))))))) in 
-(or_intror TMP_397 TMP_394 TMP_393))))))))))) in (T_ind TMP_443 TMP_439 
-TMP_407 TMP_398 t2))))))) in (let TMP_389 \def (\lambda (n: nat).(\lambda 
-(t2: T).(let TMP_388 \def (\lambda (t: T).(let TMP_386 \def (TLRef n) in (let 
-TMP_387 \def (eq T TMP_386 t) in (let TMP_385 \def ((eq T (TLRef n) t) \to 
-(\forall (P: Prop).P)) in (or TMP_387 TMP_385))))) in (let TMP_384 \def 
-(\lambda (n0: nat).(let TMP_382 \def (TLRef n) in (let TMP_381 \def (TSort 
-n0) in (let TMP_383 \def (eq T TMP_382 TMP_381) in (let TMP_380 \def ((eq T 
-(TLRef n) (TSort n0)) \to (\forall (P: Prop).P)) in (let TMP_379 \def 
-(\lambda (H: (eq T (TLRef n) (TSort n0))).(\lambda (P: Prop).(let TMP_378 
-\def (TLRef n) in (let TMP_377 \def (\lambda (ee: T).(match ee in T with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
-\Rightarrow False])) in (let TMP_376 \def (TSort n0) in (let H0 \def (eq_ind 
-T TMP_378 TMP_377 I TMP_376 H) in (False_ind P H0))))))) in (or_intror 
-TMP_383 TMP_380 TMP_379))))))) in (let TMP_375 \def (\lambda (n0: nat).(let 
-H_x \def (nat_dec n n0) in (let H \def H_x in (let TMP_374 \def (eq nat n n0) 
-in (let TMP_373 \def ((eq nat n n0) \to (\forall (P: Prop).P)) in (let 
-TMP_370 \def (TLRef n) in (let TMP_369 \def (TLRef n0) in (let TMP_371 \def 
-(eq T TMP_370 TMP_369) in (let TMP_368 \def ((eq T (TLRef n) (TLRef n0)) \to 
-(\forall (P: Prop).P)) in (let TMP_372 \def (or TMP_371 TMP_368) in (let 
-TMP_367 \def (\lambda (H0: (eq nat n n0)).(let TMP_366 \def (\lambda (n1: 
-nat).(let TMP_364 \def (TLRef n) in (let TMP_363 \def (TLRef n1) in (let 
-TMP_365 \def (eq T TMP_364 TMP_363) in (let TMP_362 \def ((eq T (TLRef n) 
-(TLRef n1)) \to (\forall (P: Prop).P)) in (or TMP_365 TMP_362)))))) in (let 
-TMP_359 \def (TLRef n) in (let TMP_358 \def (TLRef n) in (let TMP_360 \def 
-(eq T TMP_359 TMP_358) in (let TMP_357 \def ((eq T (TLRef n) (TLRef n)) \to 
-(\forall (P: Prop).P)) in (let TMP_355 \def (TLRef n) in (let TMP_356 \def 
-(refl_equal T TMP_355) in (let TMP_361 \def (or_introl TMP_360 TMP_357 
-TMP_356) in (eq_ind nat n TMP_366 TMP_361 n0 H0)))))))))) in (let TMP_354 
-\def (\lambda (H0: (((eq nat n n0) \to (\forall (P: Prop).P)))).(let TMP_352 
-\def (TLRef n) in (let TMP_351 \def (TLRef n0) in (let TMP_353 \def (eq T 
-TMP_352 TMP_351) in (let TMP_350 \def ((eq T (TLRef n) (TLRef n0)) \to 
-(\forall (P: Prop).P)) in (let TMP_349 \def (\lambda (H1: (eq T (TLRef n) 
-(TLRef n0))).(\lambda (P: Prop).(let TMP_346 \def (\lambda (e: T).(match e in 
-T with [(TSort _) \Rightarrow n | (TLRef n1) \Rightarrow n1 | (THead _ _ _) 
-\Rightarrow n])) in (let TMP_345 \def (TLRef n) in (let TMP_344 \def (TLRef 
-n0) in (let H2 \def (f_equal T nat TMP_346 TMP_345 TMP_344 H1) in (let 
-TMP_347 \def (\lambda (n1: nat).((eq nat n n1) \to (\forall (P0: Prop).P0))) 
-in (let H3 \def (eq_ind_r nat n0 TMP_347 H0 n H2) in (let TMP_348 \def 
-(refl_equal nat n) in (H3 TMP_348 P)))))))))) in (or_intror TMP_353 TMP_350 
-TMP_349))))))) in (or_ind TMP_374 TMP_373 TMP_372 TMP_367 TMP_354 
-H))))))))))))) in (let TMP_343 \def (\lambda (k: K).(\lambda (t: T).(\lambda 
+ \lambda (t1: T).(let TMP_3 \def (\lambda (t: T).(\forall (t2: T).(let TMP_1 
+\def (eq T t t2) in (let TMP_2 \def ((eq T t t2) \to (\forall (P: Prop).P)) 
+in (or TMP_1 TMP_2))))) in (let TMP_58 \def (\lambda (n: nat).(\lambda (t2: 
+T).(let TMP_7 \def (\lambda (t: T).(let TMP_4 \def (TSort n) in (let TMP_5 
+\def (eq T TMP_4 t) in (let TMP_6 \def ((eq T (TSort n) t) \to (\forall (P: 
+Prop).P)) in (or TMP_5 TMP_6))))) in (let TMP_39 \def (\lambda (n0: nat).(let 
+H_x \def (nat_dec n n0) in (let H \def H_x in (let TMP_8 \def (eq nat n n0) 
+in (let TMP_9 \def ((eq nat n n0) \to (\forall (P: Prop).P)) in (let TMP_10 
+\def (TSort n) in (let TMP_11 \def (TSort n0) in (let TMP_12 \def (eq T 
+TMP_10 TMP_11) in (let TMP_13 \def ((eq T (TSort n) (TSort n0)) \to (\forall 
+(P: Prop).P)) in (let TMP_14 \def (or TMP_12 TMP_13) in (let TMP_27 \def 
+(\lambda (H0: (eq nat n n0)).(let TMP_19 \def (\lambda (n1: nat).(let TMP_15 
+\def (TSort n) in (let TMP_16 \def (TSort n1) in (let TMP_17 \def (eq T 
+TMP_15 TMP_16) in (let TMP_18 \def ((eq T (TSort n) (TSort n1)) \to (\forall 
+(P: Prop).P)) in (or TMP_17 TMP_18)))))) in (let TMP_20 \def (TSort n) in 
+(let TMP_21 \def (TSort n) in (let TMP_22 \def (eq T TMP_20 TMP_21) in (let 
+TMP_23 \def ((eq T (TSort n) (TSort n)) \to (\forall (P: Prop).P)) in (let 
+TMP_24 \def (TSort n) in (let TMP_25 \def (refl_equal T TMP_24) in (let 
+TMP_26 \def (or_introl TMP_22 TMP_23 TMP_25) in (eq_ind nat n TMP_19 TMP_26 
+n0 H0)))))))))) in (let TMP_38 \def (\lambda (H0: (((eq nat n n0) \to 
+(\forall (P: Prop).P)))).(let TMP_28 \def (TSort n) in (let TMP_29 \def 
+(TSort n0) in (let TMP_30 \def (eq T TMP_28 TMP_29) in (let TMP_31 \def ((eq 
+T (TSort n) (TSort n0)) \to (\forall (P: Prop).P)) in (let TMP_37 \def 
+(\lambda (H1: (eq T (TSort n) (TSort n0))).(\lambda (P: Prop).(let TMP_32 
+\def (\lambda (e: T).(match e with [(TSort n1) \Rightarrow n1 | (TLRef _) 
+\Rightarrow n | (THead _ _ _) \Rightarrow n])) in (let TMP_33 \def (TSort n) 
+in (let TMP_34 \def (TSort n0) in (let H2 \def (f_equal T nat TMP_32 TMP_33 
+TMP_34 H1) in (let TMP_35 \def (\lambda (n1: nat).((eq nat n n1) \to (\forall 
+(P0: Prop).P0))) in (let H3 \def (eq_ind_r nat n0 TMP_35 H0 n H2) in (let 
+TMP_36 \def (refl_equal nat n) in (H3 TMP_36 P)))))))))) in (or_intror TMP_30 
+TMP_31 TMP_37))))))) in (or_ind TMP_8 TMP_9 TMP_14 TMP_27 TMP_38 
+H))))))))))))) in (let TMP_48 \def (\lambda (n0: nat).(let TMP_40 \def (TSort 
+n) in (let TMP_41 \def (TLRef n0) in (let TMP_42 \def (eq T TMP_40 TMP_41) in 
+(let TMP_43 \def ((eq T (TSort n) (TLRef n0)) \to (\forall (P: Prop).P)) in 
+(let TMP_47 \def (\lambda (H: (eq T (TSort n) (TLRef n0))).(\lambda (P: 
+Prop).(let TMP_44 \def (TSort n) in (let TMP_45 \def (\lambda (ee: T).(match 
+ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ 
+_ _) \Rightarrow False])) in (let TMP_46 \def (TLRef n0) in (let H0 \def 
+(eq_ind T TMP_44 TMP_45 I TMP_46 H) in (False_ind P H0))))))) in (or_intror 
+TMP_42 TMP_43 TMP_47))))))) in (let TMP_57 \def (\lambda (k: K).(\lambda (t: 
+T).(\lambda (_: (or (eq T (TSort n) t) ((eq T (TSort n) t) \to (\forall (P: 
+Prop).P)))).(\lambda (t0: T).(\lambda (_: (or (eq T (TSort n) t0) ((eq T 
+(TSort n) t0) \to (\forall (P: Prop).P)))).(let TMP_49 \def (TSort n) in (let 
+TMP_50 \def (THead k t t0) in (let TMP_51 \def (eq T TMP_49 TMP_50) in (let 
+TMP_52 \def ((eq T (TSort n) (THead k t t0)) \to (\forall (P: Prop).P)) in 
+(let TMP_56 \def (\lambda (H1: (eq T (TSort n) (THead k t t0))).(\lambda (P: 
+Prop).(let TMP_53 \def (TSort n) in (let TMP_54 \def (\lambda (ee: T).(match 
+ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ 
+_ _) \Rightarrow False])) in (let TMP_55 \def (THead k t t0) in (let H2 \def 
+(eq_ind T TMP_53 TMP_54 I TMP_55 H1) in (False_ind P H2))))))) in (or_intror 
+TMP_51 TMP_52 TMP_56))))))))))) in (T_ind TMP_7 TMP_39 TMP_48 TMP_57 
+t2))))))) in (let TMP_113 \def (\lambda (n: nat).(\lambda (t2: T).(let TMP_62 
+\def (\lambda (t: T).(let TMP_59 \def (TLRef n) in (let TMP_60 \def (eq T 
+TMP_59 t) in (let TMP_61 \def ((eq T (TLRef n) t) \to (\forall (P: Prop).P)) 
+in (or TMP_60 TMP_61))))) in (let TMP_71 \def (\lambda (n0: nat).(let TMP_63 
+\def (TLRef n) in (let TMP_64 \def (TSort n0) in (let TMP_65 \def (eq T 
+TMP_63 TMP_64) in (let TMP_66 \def ((eq T (TLRef n) (TSort n0)) \to (\forall 
+(P: Prop).P)) in (let TMP_70 \def (\lambda (H: (eq T (TLRef n) (TSort 
+n0))).(\lambda (P: Prop).(let TMP_67 \def (TLRef n) in (let TMP_68 \def 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) in (let TMP_69 \def 
+(TSort n0) in (let H0 \def (eq_ind T TMP_67 TMP_68 I TMP_69 H) in (False_ind 
+P H0))))))) in (or_intror TMP_65 TMP_66 TMP_70))))))) in (let TMP_103 \def 
+(\lambda (n0: nat).(let H_x \def (nat_dec n n0) in (let H \def H_x in (let 
+TMP_72 \def (eq nat n n0) in (let TMP_73 \def ((eq nat n n0) \to (\forall (P: 
+Prop).P)) in (let TMP_74 \def (TLRef n) in (let TMP_75 \def (TLRef n0) in 
+(let TMP_76 \def (eq T TMP_74 TMP_75) in (let TMP_77 \def ((eq T (TLRef n) 
+(TLRef n0)) \to (\forall (P: Prop).P)) in (let TMP_78 \def (or TMP_76 TMP_77) 
+in (let TMP_91 \def (\lambda (H0: (eq nat n n0)).(let TMP_83 \def (\lambda 
+(n1: nat).(let TMP_79 \def (TLRef n) in (let TMP_80 \def (TLRef n1) in (let 
+TMP_81 \def (eq T TMP_79 TMP_80) in (let TMP_82 \def ((eq T (TLRef n) (TLRef 
+n1)) \to (\forall (P: Prop).P)) in (or TMP_81 TMP_82)))))) in (let TMP_84 
+\def (TLRef n) in (let TMP_85 \def (TLRef n) in (let TMP_86 \def (eq T TMP_84 
+TMP_85) in (let TMP_87 \def ((eq T (TLRef n) (TLRef n)) \to (\forall (P: 
+Prop).P)) in (let TMP_88 \def (TLRef n) in (let TMP_89 \def (refl_equal T 
+TMP_88) in (let TMP_90 \def (or_introl TMP_86 TMP_87 TMP_89) in (eq_ind nat n 
+TMP_83 TMP_90 n0 H0)))))))))) in (let TMP_102 \def (\lambda (H0: (((eq nat n 
+n0) \to (\forall (P: Prop).P)))).(let TMP_92 \def (TLRef n) in (let TMP_93 
+\def (TLRef n0) in (let TMP_94 \def (eq T TMP_92 TMP_93) in (let TMP_95 \def 
+((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P)) in (let TMP_101 \def 
+(\lambda (H1: (eq T (TLRef n) (TLRef n0))).(\lambda (P: Prop).(let TMP_96 
+\def (\lambda (e: T).(match e with [(TSort _) \Rightarrow n | (TLRef n1) 
+\Rightarrow n1 | (THead _ _ _) \Rightarrow n])) in (let TMP_97 \def (TLRef n) 
+in (let TMP_98 \def (TLRef n0) in (let H2 \def (f_equal T nat TMP_96 TMP_97 
+TMP_98 H1) in (let TMP_99 \def (\lambda (n1: nat).((eq nat n n1) \to (\forall 
+(P0: Prop).P0))) in (let H3 \def (eq_ind_r nat n0 TMP_99 H0 n H2) in (let 
+TMP_100 \def (refl_equal nat n) in (H3 TMP_100 P)))))))))) in (or_intror 
+TMP_94 TMP_95 TMP_101))))))) in (or_ind TMP_72 TMP_73 TMP_78 TMP_91 TMP_102 
+H))))))))))))) in (let TMP_112 \def (\lambda (k: K).(\lambda (t: T).(\lambda 
 (_: (or (eq T (TLRef n) t) ((eq T (TLRef n) t) \to (\forall (P: 
 Prop).P)))).(\lambda (t0: T).(\lambda (_: (or (eq T (TLRef n) t0) ((eq T 
-(TLRef n) t0) \to (\forall (P: Prop).P)))).(let TMP_341 \def (TLRef n) in 
-(let TMP_340 \def (THead k t t0) in (let TMP_342 \def (eq T TMP_341 TMP_340) 
-in (let TMP_339 \def ((eq T (TLRef n) (THead k t t0)) \to (\forall (P: 
-Prop).P)) in (let TMP_338 \def (\lambda (H1: (eq T (TLRef n) (THead k t 
-t0))).(\lambda (P: Prop).(let TMP_337 \def (TLRef n) in (let TMP_336 \def 
-(\lambda (ee: T).(match ee in T with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) in (let TMP_335 \def 
-(THead k t t0) in (let H2 \def (eq_ind T TMP_337 TMP_336 I TMP_335 H1) in 
-(False_ind P H2))))))) in (or_intror TMP_342 TMP_339 TMP_338))))))))))) in 
-(T_ind TMP_388 TMP_384 TMP_375 TMP_343 t2))))))) in (let TMP_334 \def 
-(\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).(or (eq T t 
-t2) ((eq T t t2) \to (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda 
-(H0: ((\forall (t2: T).(or (eq T t0 t2) ((eq T t0 t2) \to (\forall (P: 
-Prop).P)))))).(\lambda (t2: T).(let TMP_333 \def (\lambda (t3: T).(let 
-TMP_331 \def (THead k t t0) in (let TMP_332 \def (eq T TMP_331 t3) in (let 
-TMP_330 \def ((eq T (THead k t t0) t3) \to (\forall (P: Prop).P)) in (or 
-TMP_332 TMP_330))))) in (let TMP_329 \def (\lambda (n: nat).(let TMP_327 \def 
-(THead k t t0) in (let TMP_326 \def (TSort n) in (let TMP_328 \def (eq T 
-TMP_327 TMP_326) in (let TMP_325 \def ((eq T (THead k t t0) (TSort n)) \to 
-(\forall (P: Prop).P)) in (let TMP_324 \def (\lambda (H1: (eq T (THead k t 
-t0) (TSort n))).(\lambda (P: Prop).(let TMP_323 \def (THead k t t0) in (let 
-TMP_322 \def (\lambda (ee: T).(match ee in T with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) in 
-(let TMP_321 \def (TSort n) in (let H2 \def (eq_ind T TMP_323 TMP_322 I 
-TMP_321 H1) in (False_ind P H2))))))) in (or_intror TMP_328 TMP_325 
-TMP_324))))))) in (let TMP_320 \def (\lambda (n: nat).(let TMP_318 \def 
-(THead k t t0) in (let TMP_317 \def (TLRef n) in (let TMP_319 \def (eq T 
-TMP_318 TMP_317) in (let TMP_316 \def ((eq T (THead k t t0) (TLRef n)) \to 
-(\forall (P: Prop).P)) in (let TMP_315 \def (\lambda (H1: (eq T (THead k t 
-t0) (TLRef n))).(\lambda (P: Prop).(let TMP_314 \def (THead k t t0) in (let 
-TMP_313 \def (\lambda (ee: T).(match ee in T with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) in 
-(let TMP_312 \def (TLRef n) in (let H2 \def (eq_ind T TMP_314 TMP_313 I 
-TMP_312 H1) in (False_ind P H2))))))) in (or_intror TMP_319 TMP_316 
-TMP_315))))))) in (let TMP_311 \def (\lambda (k0: K).(\lambda (t3: 
-T).(\lambda (H1: (or (eq T (THead k t t0) t3) ((eq T (THead k t t0) t3) \to 
-(\forall (P: Prop).P)))).(\lambda (t4: T).(\lambda (H2: (or (eq T (THead k t 
-t0) t4) ((eq T (THead k t t0) t4) \to (\forall (P: Prop).P)))).(let H_x \def 
-(H t3) in (let H3 \def H_x in (let TMP_310 \def (eq T t t3) in (let TMP_309 
-\def ((eq T t t3) \to (\forall (P: Prop).P)) in (let TMP_306 \def (THead k t 
-t0) in (let TMP_305 \def (THead k0 t3 t4) in (let TMP_307 \def (eq T TMP_306 
-TMP_305) in (let TMP_304 \def ((eq T (THead k t t0) (THead k0 t3 t4)) \to 
-(\forall (P: Prop).P)) in (let TMP_308 \def (or TMP_307 TMP_304) in (let 
-TMP_303 \def (\lambda (H4: (eq T t t3)).(let TMP_228 \def (\lambda (t5: 
-T).(let TMP_226 \def (THead k t t0) in (let TMP_227 \def (eq T TMP_226 t5) in 
-(let TMP_225 \def ((eq T (THead k t t0) t5) \to (\forall (P: Prop).P)) in (or 
-TMP_227 TMP_225))))) in (let H5 \def (eq_ind_r T t3 TMP_228 H1 t H4) in (let 
-TMP_302 \def (\lambda (t5: T).(let TMP_300 \def (THead k t t0) in (let 
-TMP_299 \def (THead k0 t5 t4) in (let TMP_301 \def (eq T TMP_300 TMP_299) in 
-(let TMP_298 \def ((eq T (THead k t t0) (THead k0 t5 t4)) \to (\forall (P: 
-Prop).P)) in (or TMP_301 TMP_298)))))) in (let H_x0 \def (H0 t4) in (let H6 
-\def H_x0 in (let TMP_296 \def (eq T t0 t4) in (let TMP_295 \def ((eq T t0 
-t4) \to (\forall (P: Prop).P)) in (let TMP_292 \def (THead k t t0) in (let 
-TMP_291 \def (THead k0 t t4) in (let TMP_293 \def (eq T TMP_292 TMP_291) in 
-(let TMP_290 \def ((eq T (THead k t t0) (THead k0 t t4)) \to (\forall (P: 
-Prop).P)) in (let TMP_294 \def (or TMP_293 TMP_290) in (let TMP_289 \def 
-(\lambda (H7: (eq T t0 t4)).(let TMP_251 \def (\lambda (t5: T).(let TMP_249 
-\def (THead k t t0) in (let TMP_250 \def (eq T TMP_249 t5) in (let TMP_248 
-\def ((eq T (THead k t t0) t5) \to (\forall (P: Prop).P)) in (or TMP_250 
-TMP_248))))) in (let H8 \def (eq_ind_r T t4 TMP_251 H2 t0 H7) in (let TMP_288 
-\def (\lambda (t5: T).(let TMP_286 \def (THead k t t0) in (let TMP_285 \def 
-(THead k0 t t5) in (let TMP_287 \def (eq T TMP_286 TMP_285) in (let TMP_284 
-\def ((eq T (THead k t t0) (THead k0 t t5)) \to (\forall (P: Prop).P)) in (or 
-TMP_287 TMP_284)))))) in (let H_x1 \def (terms_props__kind_dec k k0) in (let 
-H9 \def H_x1 in (let TMP_282 \def (eq K k k0) in (let TMP_281 \def ((eq K k 
-k0) \to (\forall (P: Prop).P)) in (let TMP_278 \def (THead k t t0) in (let 
-TMP_277 \def (THead k0 t t0) in (let TMP_279 \def (eq T TMP_278 TMP_277) in 
-(let TMP_276 \def ((eq T (THead k t t0) (THead k0 t t0)) \to (\forall (P: 
-Prop).P)) in (let TMP_280 \def (or TMP_279 TMP_276) in (let TMP_275 \def 
-(\lambda (H10: (eq K k k0)).(let TMP_274 \def (\lambda (k1: K).(let TMP_272 
-\def (THead k t t0) in (let TMP_271 \def (THead k1 t t0) in (let TMP_273 \def 
-(eq T TMP_272 TMP_271) in (let TMP_270 \def ((eq T (THead k t t0) (THead k1 t 
-t0)) \to (\forall (P: Prop).P)) in (or TMP_273 TMP_270)))))) in (let TMP_267 
-\def (THead k t t0) in (let TMP_266 \def (THead k t t0) in (let TMP_268 \def 
-(eq T TMP_267 TMP_266) in (let TMP_265 \def ((eq T (THead k t t0) (THead k t 
-t0)) \to (\forall (P: Prop).P)) in (let TMP_263 \def (THead k t t0) in (let 
-TMP_264 \def (refl_equal T TMP_263) in (let TMP_269 \def (or_introl TMP_268 
-TMP_265 TMP_264) in (eq_ind K k TMP_274 TMP_269 k0 H10)))))))))) in (let 
-TMP_262 \def (\lambda (H10: (((eq K k k0) \to (\forall (P: Prop).P)))).(let 
-TMP_260 \def (THead k t t0) in (let TMP_259 \def (THead k0 t t0) in (let 
-TMP_261 \def (eq T TMP_260 TMP_259) in (let TMP_258 \def ((eq T (THead k t 
-t0) (THead k0 t t0)) \to (\forall (P: Prop).P)) in (let TMP_257 \def (\lambda 
-(H11: (eq T (THead k t t0) (THead k0 t t0))).(\lambda (P: Prop).(let TMP_254 
-\def (\lambda (e: T).(match e in T with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) in (let TMP_253 \def (THead 
-k t t0) in (let TMP_252 \def (THead k0 t t0) in (let H12 \def (f_equal T K 
-TMP_254 TMP_253 TMP_252 H11) in (let TMP_255 \def (\lambda (k1: K).((eq K k 
-k1) \to (\forall (P0: Prop).P0))) in (let H13 \def (eq_ind_r K k0 TMP_255 H10 
-k H12) in (let TMP_256 \def (refl_equal K k) in (H13 TMP_256 P)))))))))) in 
-(or_intror TMP_261 TMP_258 TMP_257))))))) in (let TMP_283 \def (or_ind 
-TMP_282 TMP_281 TMP_280 TMP_275 TMP_262 H9) in (eq_ind T t0 TMP_288 TMP_283 
-t4 H7))))))))))))))))) in (let TMP_247 \def (\lambda (H7: (((eq T t0 t4) \to 
-(\forall (P: Prop).P)))).(let TMP_245 \def (THead k t t0) in (let TMP_244 
-\def (THead k0 t t4) in (let TMP_246 \def (eq T TMP_245 TMP_244) in (let 
-TMP_243 \def ((eq T (THead k t t0) (THead k0 t t4)) \to (\forall (P: 
-Prop).P)) in (let TMP_242 \def (\lambda (H8: (eq T (THead k t t0) (THead k0 t 
-t4))).(\lambda (P: Prop).(let TMP_231 \def (\lambda (e: T).(match e in T with 
+(TLRef n) t0) \to (\forall (P: Prop).P)))).(let TMP_104 \def (TLRef n) in 
+(let TMP_105 \def (THead k t t0) in (let TMP_106 \def (eq T TMP_104 TMP_105) 
+in (let TMP_107 \def ((eq T (TLRef n) (THead k t t0)) \to (\forall (P: 
+Prop).P)) in (let TMP_111 \def (\lambda (H1: (eq T (TLRef n) (THead k t 
+t0))).(\lambda (P: Prop).(let TMP_108 \def (TLRef n) in (let TMP_109 \def 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) in (let TMP_110 \def 
+(THead k t t0) in (let H2 \def (eq_ind T TMP_108 TMP_109 I TMP_110 H1) in 
+(False_ind P H2))))))) in (or_intror TMP_106 TMP_107 TMP_111))))))))))) in 
+(T_ind TMP_62 TMP_71 TMP_103 TMP_112 t2))))))) in (let TMP_250 \def (\lambda 
+(k: K).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).(or (eq T t t2) ((eq T 
+t t2) \to (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall 
+(t2: T).(or (eq T t0 t2) ((eq T t0 t2) \to (\forall (P: 
+Prop).P)))))).(\lambda (t2: T).(let TMP_117 \def (\lambda (t3: T).(let 
+TMP_114 \def (THead k t t0) in (let TMP_115 \def (eq T TMP_114 t3) in (let 
+TMP_116 \def ((eq T (THead k t t0) t3) \to (\forall (P: Prop).P)) in (or 
+TMP_115 TMP_116))))) in (let TMP_126 \def (\lambda (n: nat).(let TMP_118 \def 
+(THead k t t0) in (let TMP_119 \def (TSort n) in (let TMP_120 \def (eq T 
+TMP_118 TMP_119) in (let TMP_121 \def ((eq T (THead k t t0) (TSort n)) \to 
+(\forall (P: Prop).P)) in (let TMP_125 \def (\lambda (H1: (eq T (THead k t 
+t0) (TSort n))).(\lambda (P: Prop).(let TMP_122 \def (THead k t t0) in (let 
+TMP_123 \def (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) in (let 
+TMP_124 \def (TSort n) in (let H2 \def (eq_ind T TMP_122 TMP_123 I TMP_124 
+H1) in (False_ind P H2))))))) in (or_intror TMP_120 TMP_121 TMP_125))))))) in 
+(let TMP_135 \def (\lambda (n: nat).(let TMP_127 \def (THead k t t0) in (let 
+TMP_128 \def (TLRef n) in (let TMP_129 \def (eq T TMP_127 TMP_128) in (let 
+TMP_130 \def ((eq T (THead k t t0) (TLRef n)) \to (\forall (P: Prop).P)) in 
+(let TMP_134 \def (\lambda (H1: (eq T (THead k t t0) (TLRef n))).(\lambda (P: 
+Prop).(let TMP_131 \def (THead k t t0) in (let TMP_132 \def (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead _ _ _) \Rightarrow True])) in (let TMP_133 \def (TLRef n) in (let H2 
+\def (eq_ind T TMP_131 TMP_132 I TMP_133 H1) in (False_ind P H2))))))) in 
+(or_intror TMP_129 TMP_130 TMP_134))))))) in (let TMP_249 \def (\lambda (k0: 
+K).(\lambda (t3: T).(\lambda (H1: (or (eq T (THead k t t0) t3) ((eq T (THead 
+k t t0) t3) \to (\forall (P: Prop).P)))).(\lambda (t4: T).(\lambda (H2: (or 
+(eq T (THead k t t0) t4) ((eq T (THead k t t0) t4) \to (\forall (P: 
+Prop).P)))).(let H_x \def (H t3) in (let H3 \def H_x in (let TMP_136 \def (eq 
+T t t3) in (let TMP_137 \def ((eq T t t3) \to (\forall (P: Prop).P)) in (let 
+TMP_138 \def (THead k t t0) in (let TMP_139 \def (THead k0 t3 t4) in (let 
+TMP_140 \def (eq T TMP_138 TMP_139) in (let TMP_141 \def ((eq T (THead k t 
+t0) (THead k0 t3 t4)) \to (\forall (P: Prop).P)) in (let TMP_142 \def (or 
+TMP_140 TMP_141) in (let TMP_221 \def (\lambda (H4: (eq T t t3)).(let TMP_146 
+\def (\lambda (t5: T).(let TMP_143 \def (THead k t t0) in (let TMP_144 \def 
+(eq T TMP_143 t5) in (let TMP_145 \def ((eq T (THead k t t0) t5) \to (\forall 
+(P: Prop).P)) in (or TMP_144 TMP_145))))) in (let H5 \def (eq_ind_r T t3 
+TMP_146 H1 t H4) in (let TMP_151 \def (\lambda (t5: T).(let TMP_147 \def 
+(THead k t t0) in (let TMP_148 \def (THead k0 t5 t4) in (let TMP_149 \def (eq 
+T TMP_147 TMP_148) in (let TMP_150 \def ((eq T (THead k t t0) (THead k0 t5 
+t4)) \to (\forall (P: Prop).P)) in (or TMP_149 TMP_150)))))) in (let H_x0 
+\def (H0 t4) in (let H6 \def H_x0 in (let TMP_152 \def (eq T t0 t4) in (let 
+TMP_153 \def ((eq T t0 t4) \to (\forall (P: Prop).P)) in (let TMP_154 \def 
+(THead k t t0) in (let TMP_155 \def (THead k0 t t4) in (let TMP_156 \def (eq 
+T TMP_154 TMP_155) in (let TMP_157 \def ((eq T (THead k t t0) (THead k0 t 
+t4)) \to (\forall (P: Prop).P)) in (let TMP_158 \def (or TMP_156 TMP_157) in 
+(let TMP_200 \def (\lambda (H7: (eq T t0 t4)).(let TMP_162 \def (\lambda (t5: 
+T).(let TMP_159 \def (THead k t t0) in (let TMP_160 \def (eq T TMP_159 t5) in 
+(let TMP_161 \def ((eq T (THead k t t0) t5) \to (\forall (P: Prop).P)) in (or 
+TMP_160 TMP_161))))) in (let H8 \def (eq_ind_r T t4 TMP_162 H2 t0 H7) in (let 
+TMP_167 \def (\lambda (t5: T).(let TMP_163 \def (THead k t t0) in (let 
+TMP_164 \def (THead k0 t t5) in (let TMP_165 \def (eq T TMP_163 TMP_164) in 
+(let TMP_166 \def ((eq T (THead k t t0) (THead k0 t t5)) \to (\forall (P: 
+Prop).P)) in (or TMP_165 TMP_166)))))) in (let H_x1 \def 
+(terms_props__kind_dec k k0) in (let H9 \def H_x1 in (let TMP_168 \def (eq K 
+k k0) in (let TMP_169 \def ((eq K k k0) \to (\forall (P: Prop).P)) in (let 
+TMP_170 \def (THead k t t0) in (let TMP_171 \def (THead k0 t t0) in (let 
+TMP_172 \def (eq T TMP_170 TMP_171) in (let TMP_173 \def ((eq T (THead k t 
+t0) (THead k0 t t0)) \to (\forall (P: Prop).P)) in (let TMP_174 \def (or 
+TMP_172 TMP_173) in (let TMP_187 \def (\lambda (H10: (eq K k k0)).(let 
+TMP_179 \def (\lambda (k1: K).(let TMP_175 \def (THead k t t0) in (let 
+TMP_176 \def (THead k1 t t0) in (let TMP_177 \def (eq T TMP_175 TMP_176) in 
+(let TMP_178 \def ((eq T (THead k t t0) (THead k1 t t0)) \to (\forall (P: 
+Prop).P)) in (or TMP_177 TMP_178)))))) in (let TMP_180 \def (THead k t t0) in 
+(let TMP_181 \def (THead k t t0) in (let TMP_182 \def (eq T TMP_180 TMP_181) 
+in (let TMP_183 \def ((eq T (THead k t t0) (THead k t t0)) \to (\forall (P: 
+Prop).P)) in (let TMP_184 \def (THead k t t0) in (let TMP_185 \def 
+(refl_equal T TMP_184) in (let TMP_186 \def (or_introl TMP_182 TMP_183 
+TMP_185) in (eq_ind K k TMP_179 TMP_186 k0 H10)))))))))) in (let TMP_198 \def 
+(\lambda (H10: (((eq K k k0) \to (\forall (P: Prop).P)))).(let TMP_188 \def 
+(THead k t t0) in (let TMP_189 \def (THead k0 t t0) in (let TMP_190 \def (eq 
+T TMP_188 TMP_189) in (let TMP_191 \def ((eq T (THead k t t0) (THead k0 t 
+t0)) \to (\forall (P: Prop).P)) in (let TMP_197 \def (\lambda (H11: (eq T 
+(THead k t t0) (THead k0 t t0))).(\lambda (P: Prop).(let TMP_192 \def 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) in (let TMP_193 \def (THead 
+k t t0) in (let TMP_194 \def (THead k0 t t0) in (let H12 \def (f_equal T K 
+TMP_192 TMP_193 TMP_194 H11) in (let TMP_195 \def (\lambda (k1: K).((eq K k 
+k1) \to (\forall (P0: Prop).P0))) in (let H13 \def (eq_ind_r K k0 TMP_195 H10 
+k H12) in (let TMP_196 \def (refl_equal K k) in (H13 TMP_196 P)))))))))) in 
+(or_intror TMP_190 TMP_191 TMP_197))))))) in (let TMP_199 \def (or_ind 
+TMP_168 TMP_169 TMP_174 TMP_187 TMP_198 H9) in (eq_ind T t0 TMP_167 TMP_199 
+t4 H7))))))))))))))))) in (let TMP_219 \def (\lambda (H7: (((eq T t0 t4) \to 
+(\forall (P: Prop).P)))).(let TMP_201 \def (THead k t t0) in (let TMP_202 
+\def (THead k0 t t4) in (let TMP_203 \def (eq T TMP_201 TMP_202) in (let 
+TMP_204 \def ((eq T (THead k t t0) (THead k0 t t4)) \to (\forall (P: 
+Prop).P)) in (let TMP_218 \def (\lambda (H8: (eq T (THead k t t0) (THead k0 t 
+t4))).(\lambda (P: Prop).(let TMP_205 \def (\lambda (e: T).(match e with 
 [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) 
-\Rightarrow k1])) in (let TMP_230 \def (THead k t t0) in (let TMP_229 \def 
-(THead k0 t t4) in (let H9 \def (f_equal T K TMP_231 TMP_230 TMP_229 H8) in 
-(let TMP_234 \def (\lambda (e: T).(match e in T with [(TSort _) \Rightarrow 
-t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t5) \Rightarrow t5])) in (let 
-TMP_233 \def (THead k t t0) in (let TMP_232 \def (THead k0 t t4) in (let H10 
-\def (f_equal T T TMP_234 TMP_233 TMP_232 H8) in (let TMP_241 \def (\lambda 
-(_: (eq K k k0)).(let TMP_235 \def (\lambda (t5: T).((eq T t0 t5) \to 
-(\forall (P0: Prop).P0))) in (let H12 \def (eq_ind_r T t4 TMP_235 H7 t0 H10) 
-in (let TMP_239 \def (\lambda (t5: T).(let TMP_237 \def (THead k t t0) in 
-(let TMP_238 \def (eq T TMP_237 t5) in (let TMP_236 \def ((eq T (THead k t 
-t0) t5) \to (\forall (P0: Prop).P0)) in (or TMP_238 TMP_236))))) in (let H13 
-\def (eq_ind_r T t4 TMP_239 H2 t0 H10) in (let TMP_240 \def (refl_equal T t0) 
-in (H12 TMP_240 P))))))) in (TMP_241 H9)))))))))))) in (or_intror TMP_246 
-TMP_243 TMP_242))))))) in (let TMP_297 \def (or_ind TMP_296 TMP_295 TMP_294 
-TMP_289 TMP_247 H6) in (eq_ind T t TMP_302 TMP_297 t3 H4))))))))))))))))) in 
-(let TMP_224 \def (\lambda (H4: (((eq T t t3) \to (\forall (P: 
-Prop).P)))).(let TMP_222 \def (THead k t t0) in (let TMP_221 \def (THead k0 
-t3 t4) in (let TMP_223 \def (eq T TMP_222 TMP_221) in (let TMP_220 \def ((eq 
-T (THead k t t0) (THead k0 t3 t4)) \to (\forall (P: Prop).P)) in (let TMP_219 
+\Rightarrow k1])) in (let TMP_206 \def (THead k t t0) in (let TMP_207 \def 
+(THead k0 t t4) in (let H9 \def (f_equal T K TMP_205 TMP_206 TMP_207 H8) in 
+(let TMP_208 \def (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | 
+(TLRef _) \Rightarrow t0 | (THead _ _ t5) \Rightarrow t5])) in (let TMP_209 
+\def (THead k t t0) in (let TMP_210 \def (THead k0 t t4) in (let H10 \def 
+(f_equal T T TMP_208 TMP_209 TMP_210 H8) in (let TMP_217 \def (\lambda (_: 
+(eq K k k0)).(let TMP_211 \def (\lambda (t5: T).((eq T t0 t5) \to (\forall 
+(P0: Prop).P0))) in (let H12 \def (eq_ind_r T t4 TMP_211 H7 t0 H10) in (let 
+TMP_215 \def (\lambda (t5: T).(let TMP_212 \def (THead k t t0) in (let 
+TMP_213 \def (eq T TMP_212 t5) in (let TMP_214 \def ((eq T (THead k t t0) t5) 
+\to (\forall (P0: Prop).P0)) in (or TMP_213 TMP_214))))) in (let H13 \def 
+(eq_ind_r T t4 TMP_215 H2 t0 H10) in (let TMP_216 \def (refl_equal T t0) in 
+(H12 TMP_216 P))))))) in (TMP_217 H9)))))))))))) in (or_intror TMP_203 
+TMP_204 TMP_218))))))) in (let TMP_220 \def (or_ind TMP_152 TMP_153 TMP_158 
+TMP_200 TMP_219 H6) in (eq_ind T t TMP_151 TMP_220 t3 H4))))))))))))))))) in 
+(let TMP_248 \def (\lambda (H4: (((eq T t t3) \to (\forall (P: 
+Prop).P)))).(let TMP_222 \def (THead k t t0) in (let TMP_223 \def (THead k0 
+t3 t4) in (let TMP_224 \def (eq T TMP_222 TMP_223) in (let TMP_225 \def ((eq 
+T (THead k t t0) (THead k0 t3 t4)) \to (\forall (P: Prop).P)) in (let TMP_247 
 \def (\lambda (H5: (eq T (THead k t t0) (THead k0 t3 t4))).(\lambda (P: 
-Prop).(let TMP_200 \def (\lambda (e: T).(match e in T with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) in 
-(let TMP_199 \def (THead k t t0) in (let TMP_198 \def (THead k0 t3 t4) in 
-(let H6 \def (f_equal T K TMP_200 TMP_199 TMP_198 H5) in (let TMP_203 \def 
-(\lambda (e: T).(match e in T with [(TSort _) \Rightarrow t | (TLRef _) 
-\Rightarrow t | (THead _ t5 _) \Rightarrow t5])) in (let TMP_202 \def (THead 
-k t t0) in (let TMP_201 \def (THead k0 t3 t4) in (let H7 \def (f_equal T T 
-TMP_203 TMP_202 TMP_201 H5) in (let TMP_206 \def (\lambda (e: T).(match e in 
-T with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t5) 
-\Rightarrow t5])) in (let TMP_205 \def (THead k t t0) in (let TMP_204 \def 
-(THead k0 t3 t4) in (let H8 \def (f_equal T T TMP_206 TMP_205 TMP_204 H5) in 
-(let TMP_217 \def (\lambda (H9: (eq T t t3)).(\lambda (_: (eq K k k0)).(let 
-TMP_210 \def (\lambda (t5: T).(let TMP_208 \def (THead k t t0) in (let 
-TMP_209 \def (eq T TMP_208 t5) in (let TMP_207 \def ((eq T (THead k t t0) t5) 
-\to (\forall (P0: Prop).P0)) in (or TMP_209 TMP_207))))) in (let H11 \def 
-(eq_ind_r T t4 TMP_210 H2 t0 H8) in (let TMP_211 \def (\lambda (t5: T).((eq T 
-t t5) \to (\forall (P0: Prop).P0))) in (let H12 \def (eq_ind_r T t3 TMP_211 
-H4 t H9) in (let TMP_215 \def (\lambda (t5: T).(let TMP_213 \def (THead k t 
-t0) in (let TMP_214 \def (eq T TMP_213 t5) in (let TMP_212 \def ((eq T (THead 
-k t t0) t5) \to (\forall (P0: Prop).P0)) in (or TMP_214 TMP_212))))) in (let 
-H13 \def (eq_ind_r T t3 TMP_215 H1 t H9) in (let TMP_216 \def (refl_equal T 
-t) in (H12 TMP_216 P)))))))))) in (let TMP_218 \def (TMP_217 H7) in (TMP_218 
-H6))))))))))))))))) in (or_intror TMP_223 TMP_220 TMP_219))))))) in (or_ind 
-TMP_310 TMP_309 TMP_308 TMP_303 TMP_224 H3))))))))))))))))) in (T_ind TMP_333 
-TMP_329 TMP_320 TMP_311 t2))))))))))) in (T_ind TMP_447 TMP_444 TMP_389 
-TMP_334 t1))))).
+Prop).(let TMP_226 \def (\lambda (e: T).(match e with [(TSort _) \Rightarrow 
+k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) in (let 
+TMP_227 \def (THead k t t0) in (let TMP_228 \def (THead k0 t3 t4) in (let H6 
+\def (f_equal T K TMP_226 TMP_227 TMP_228 H5) in (let TMP_229 \def (\lambda 
+(e: T).(match e with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | 
+(THead _ t5 _) \Rightarrow t5])) in (let TMP_230 \def (THead k t t0) in (let 
+TMP_231 \def (THead k0 t3 t4) in (let H7 \def (f_equal T T TMP_229 TMP_230 
+TMP_231 H5) in (let TMP_232 \def (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t5) \Rightarrow t5])) 
+in (let TMP_233 \def (THead k t t0) in (let TMP_234 \def (THead k0 t3 t4) in 
+(let H8 \def (f_equal T T TMP_232 TMP_233 TMP_234 H5) in (let TMP_245 \def 
+(\lambda (H9: (eq T t t3)).(\lambda (_: (eq K k k0)).(let TMP_238 \def 
+(\lambda (t5: T).(let TMP_235 \def (THead k t t0) in (let TMP_236 \def (eq T 
+TMP_235 t5) in (let TMP_237 \def ((eq T (THead k t t0) t5) \to (\forall (P0: 
+Prop).P0)) in (or TMP_236 TMP_237))))) in (let H11 \def (eq_ind_r T t4 
+TMP_238 H2 t0 H8) in (let TMP_239 \def (\lambda (t5: T).((eq T t t5) \to 
+(\forall (P0: Prop).P0))) in (let H12 \def (eq_ind_r T t3 TMP_239 H4 t H9) in 
+(let TMP_243 \def (\lambda (t5: T).(let TMP_240 \def (THead k t t0) in (let 
+TMP_241 \def (eq T TMP_240 t5) in (let TMP_242 \def ((eq T (THead k t t0) t5) 
+\to (\forall (P0: Prop).P0)) in (or TMP_241 TMP_242))))) in (let H13 \def 
+(eq_ind_r T t3 TMP_243 H1 t H9) in (let TMP_244 \def (refl_equal T t) in (H12 
+TMP_244 P)))))))))) in (let TMP_246 \def (TMP_245 H7) in (TMP_246 
+H6))))))))))))))))) in (or_intror TMP_224 TMP_225 TMP_247))))))) in (or_ind 
+TMP_136 TMP_137 TMP_142 TMP_221 TMP_248 H3))))))))))))))))) in (T_ind TMP_117 
+TMP_126 TMP_135 TMP_249 t2))))))))))) in (T_ind TMP_3 TMP_58 TMP_113 TMP_250 
+t1))))).
 
 theorem binder_dec:
  \forall (t: T).(or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: 
 T).(eq T t (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall 
 (u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))
 \def
- \lambda (t: T).(let TMP_516 \def (\lambda (t0: T).(let TMP_514 \def (\lambda 
-(b: B).(\lambda (w: T).(\lambda (u: T).(let TMP_512 \def (Bind b) in (let 
-TMP_513 \def (THead TMP_512 w u) in (eq T t0 TMP_513)))))) in (let TMP_515 
-\def (ex_3 B T T TMP_514) in (let TMP_511 \def (\forall (b: B).(\forall (w: 
+ \lambda (t: T).(let TMP_6 \def (\lambda (t0: T).(let TMP_3 \def (\lambda (b: 
+B).(\lambda (w: T).(\lambda (u: T).(let TMP_1 \def (Bind b) in (let TMP_2 
+\def (THead TMP_1 w u) in (eq T t0 TMP_2)))))) in (let TMP_4 \def (ex_3 B T T 
+TMP_3) in (let TMP_5 \def (\forall (b: B).(\forall (w: T).(\forall (u: 
+T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))) in (or TMP_4 
+TMP_5))))) in (let TMP_18 \def (\lambda (n: nat).(let TMP_10 \def (\lambda 
+(b: B).(\lambda (w: T).(\lambda (u: T).(let TMP_7 \def (TSort n) in (let 
+TMP_8 \def (Bind b) in (let TMP_9 \def (THead TMP_8 w u) in (eq T TMP_7 
+TMP_9))))))) in (let TMP_11 \def (ex_3 B T T TMP_10) in (let TMP_12 \def 
+(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (TSort n) (THead (Bind 
+b) w u)) \to (\forall (P: Prop).P))))) in (let TMP_17 \def (\lambda (b: 
+B).(\lambda (w: T).(\lambda (u: T).(\lambda (H: (eq T (TSort n) (THead (Bind 
+b) w u))).(\lambda (P: Prop).(let TMP_13 \def (TSort n) in (let TMP_14 \def 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) in (let TMP_15 \def 
+(Bind b) in (let TMP_16 \def (THead TMP_15 w u) in (let H0 \def (eq_ind T 
+TMP_13 TMP_14 I TMP_16 H) in (False_ind P H0))))))))))) in (or_intror TMP_11 
+TMP_12 TMP_17)))))) in (let TMP_30 \def (\lambda (n: nat).(let TMP_22 \def 
+(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(let TMP_19 \def (TLRef n) in 
+(let TMP_20 \def (Bind b) in (let TMP_21 \def (THead TMP_20 w u) in (eq T 
+TMP_19 TMP_21))))))) in (let TMP_23 \def (ex_3 B T T TMP_22) in (let TMP_24 
+\def (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (TLRef n) (THead 
+(Bind b) w u)) \to (\forall (P: Prop).P))))) in (let TMP_29 \def (\lambda (b: 
+B).(\lambda (w: T).(\lambda (u: T).(\lambda (H: (eq T (TLRef n) (THead (Bind 
+b) w u))).(\lambda (P: Prop).(let TMP_25 \def (TLRef n) in (let TMP_26 \def 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) in (let TMP_27 \def 
+(Bind b) in (let TMP_28 \def (THead TMP_27 w u) in (let H0 \def (eq_ind T 
+TMP_25 TMP_26 I TMP_28 H) in (False_ind P H0))))))))))) in (or_intror TMP_23 
+TMP_24 TMP_29)))))) in (let TMP_69 \def (\lambda (k: K).(let TMP_37 \def 
+(\lambda (k0: K).(\forall (t0: T).((or (ex_3 B T T (\lambda (b: B).(\lambda 
+(w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: 
+B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to 
+(\forall (P: Prop).P)))))) \to (\forall (t1: T).((or (ex_3 B T T (\lambda (b: 
+B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b) w u)))))) 
+(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead (Bind b) w 
+u)) \to (\forall (P: Prop).P)))))) \to (let TMP_34 \def (\lambda (b: 
+B).(\lambda (w: T).(\lambda (u: T).(let TMP_31 \def (THead k0 t0 t1) in (let 
+TMP_32 \def (Bind b) in (let TMP_33 \def (THead TMP_32 w u) in (eq T TMP_31 
+TMP_33))))))) in (let TMP_35 \def (ex_3 B T T TMP_34) in (let TMP_36 \def 
+(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead k0 t0 t1) 
+(THead (Bind b) w u)) \to (\forall (P: Prop).P))))) in (or TMP_35 
+TMP_36))))))))) in (let TMP_54 \def (\lambda (b: B).(\lambda (t0: T).(\lambda 
+(_: (or (ex_3 B T T (\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 
+(THead (Bind b0) w u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: 
+T).((eq T t0 (THead (Bind b0) w u)) \to (\forall (P: Prop).P))))))).(\lambda 
+(t1: T).(\lambda (_: (or (ex_3 B T T (\lambda (b0: B).(\lambda (w: 
+T).(\lambda (u: T).(eq T t1 (THead (Bind b0) w u)))))) (\forall (b0: 
+B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead (Bind b0) w u)) \to 
+(\forall (P: Prop).P))))))).(let TMP_42 \def (\lambda (b0: B).(\lambda (w: 
+T).(\lambda (u: T).(let TMP_38 \def (Bind b) in (let TMP_39 \def (THead 
+TMP_38 t0 t1) in (let TMP_40 \def (Bind b0) in (let TMP_41 \def (THead TMP_40 
+w u) in (eq T TMP_39 TMP_41)))))))) in (let TMP_43 \def (ex_3 B T T TMP_42) 
+in (let TMP_44 \def (\forall (b0: B).(\forall (w: T).(\forall (u: T).((eq T 
+(THead (Bind b) t0 t1) (THead (Bind b0) w u)) \to (\forall (P: Prop).P))))) 
+in (let TMP_49 \def (\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(let 
+TMP_45 \def (Bind b) in (let TMP_46 \def (THead TMP_45 t0 t1) in (let TMP_47 
+\def (Bind b0) in (let TMP_48 \def (THead TMP_47 w u) in (eq T TMP_46 
+TMP_48)))))))) in (let TMP_50 \def (Bind b) in (let TMP_51 \def (THead TMP_50 
+t0 t1) in (let TMP_52 \def (refl_equal T TMP_51) in (let TMP_53 \def 
+(ex_3_intro B T T TMP_49 b t0 t1 TMP_52) in (or_introl TMP_43 TMP_44 
+TMP_53)))))))))))))) in (let TMP_68 \def (\lambda (f: F).(\lambda (t0: 
+T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: 
+T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: 
 T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: 
-Prop).P))))) in (or TMP_515 TMP_511))))) in (let TMP_510 \def (\lambda (n: 
-nat).(let TMP_508 \def (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(let 
-TMP_507 \def (TSort n) in (let TMP_505 \def (Bind b) in (let TMP_506 \def 
-(THead TMP_505 w u) in (eq T TMP_507 TMP_506))))))) in (let TMP_509 \def 
-(ex_3 B T T TMP_508) in (let TMP_504 \def (\forall (b: B).(\forall (w: 
-T).(\forall (u: T).((eq T (TSort n) (THead (Bind b) w u)) \to (\forall (P: 
-Prop).P))))) in (let TMP_503 \def (\lambda (b: B).(\lambda (w: T).(\lambda 
-(u: T).(\lambda (H: (eq T (TSort n) (THead (Bind b) w u))).(\lambda (P: 
-Prop).(let TMP_502 \def (TSort n) in (let TMP_501 \def (\lambda (ee: 
-T).(match ee in T with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow 
-False | (THead _ _ _) \Rightarrow False])) in (let TMP_499 \def (Bind b) in 
-(let TMP_500 \def (THead TMP_499 w u) in (let H0 \def (eq_ind T TMP_502 
-TMP_501 I TMP_500 H) in (False_ind P H0))))))))))) in (or_intror TMP_509 
-TMP_504 TMP_503)))))) in (let TMP_498 \def (\lambda (n: nat).(let TMP_496 
-\def (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(let TMP_495 \def (TLRef 
-n) in (let TMP_493 \def (Bind b) in (let TMP_494 \def (THead TMP_493 w u) in 
-(eq T TMP_495 TMP_494))))))) in (let TMP_497 \def (ex_3 B T T TMP_496) in 
-(let TMP_492 \def (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T 
-(TLRef n) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))) in (let TMP_491 
-\def (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(\lambda (H: (eq T 
-(TLRef n) (THead (Bind b) w u))).(\lambda (P: Prop).(let TMP_490 \def (TLRef 
-n) in (let TMP_489 \def (\lambda (ee: T).(match ee in T with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
-False])) in (let TMP_487 \def (Bind b) in (let TMP_488 \def (THead TMP_487 w 
-u) in (let H0 \def (eq_ind T TMP_490 TMP_489 I TMP_488 H) in (False_ind P 
-H0))))))))))) in (or_intror TMP_497 TMP_492 TMP_491)))))) in (let TMP_486 
-\def (\lambda (k: K).(let TMP_485 \def (\lambda (k0: K).(\forall (t0: T).((or 
-(ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead 
-(Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 
-(THead (Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (\forall (t1: 
-T).((or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 
-(THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: 
-T).((eq T t1 (THead (Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (let 
-TMP_483 \def (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(let TMP_482 
-\def (THead k0 t0 t1) in (let TMP_480 \def (Bind b) in (let TMP_481 \def 
-(THead TMP_480 w u) in (eq T TMP_482 TMP_481))))))) in (let TMP_484 \def 
-(ex_3 B T T TMP_483) in (let TMP_479 \def (\forall (b: B).(\forall (w: 
-T).(\forall (u: T).((eq T (THead k0 t0 t1) (THead (Bind b) w u)) \to (\forall 
-(P: Prop).P))))) in (or TMP_484 TMP_479))))))))) in (let TMP_478 \def 
-(\lambda (b: B).(\lambda (t0: T).(\lambda (_: (or (ex_3 B T T (\lambda (b0: 
-B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b0) w u)))))) 
-(\forall (b0: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b0) w 
-u)) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (_: (or (ex_3 B 
-T T (\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind 
-b0) w u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: T).((eq T t1 
-(THead (Bind b0) w u)) \to (\forall (P: Prop).P))))))).(let TMP_476 \def 
-(\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(let TMP_474 \def (Bind b) 
-in (let TMP_475 \def (THead TMP_474 t0 t1) in (let TMP_472 \def (Bind b0) in 
-(let TMP_473 \def (THead TMP_472 w u) in (eq T TMP_475 TMP_473)))))))) in 
-(let TMP_477 \def (ex_3 B T T TMP_476) in (let TMP_471 \def (\forall (b0: 
-B).(\forall (w: T).(\forall (u: T).((eq T (THead (Bind b) t0 t1) (THead (Bind 
-b0) w u)) \to (\forall (P: Prop).P))))) in (let TMP_469 \def (\lambda (b0: 
-B).(\lambda (w: T).(\lambda (u: T).(let TMP_467 \def (Bind b) in (let TMP_468 
-\def (THead TMP_467 t0 t1) in (let TMP_465 \def (Bind b0) in (let TMP_466 
-\def (THead TMP_465 w u) in (eq T TMP_468 TMP_466)))))))) in (let TMP_462 
-\def (Bind b) in (let TMP_463 \def (THead TMP_462 t0 t1) in (let TMP_464 \def 
-(refl_equal T TMP_463) in (let TMP_470 \def (ex_3_intro B T T TMP_469 b t0 t1 
-TMP_464) in (or_introl TMP_477 TMP_471 TMP_470)))))))))))))) in (let TMP_461 
-\def (\lambda (f: F).(\lambda (t0: T).(\lambda (_: (or (ex_3 B T T (\lambda 
-(b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) 
-(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w 
-u)) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (_: (or (ex_3 B 
-T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b) 
-w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead 
-(Bind b) w u)) \to (\forall (P: Prop).P))))))).(let TMP_459 \def (\lambda (b: 
-B).(\lambda (w: T).(\lambda (u: T).(let TMP_457 \def (Flat f) in (let TMP_458 
-\def (THead TMP_457 t0 t1) in (let TMP_455 \def (Bind b) in (let TMP_456 \def 
-(THead TMP_455 w u) in (eq T TMP_458 TMP_456)))))))) in (let TMP_460 \def 
-(ex_3 B T T TMP_459) in (let TMP_454 \def (\forall (b: B).(\forall (w: 
-T).(\forall (u: T).((eq T (THead (Flat f) t0 t1) (THead (Bind b) w u)) \to 
-(\forall (P: Prop).P))))) in (let TMP_453 \def (\lambda (b: B).(\lambda (w: 
-T).(\lambda (u: T).(\lambda (H1: (eq T (THead (Flat f) t0 t1) (THead (Bind b) 
-w u))).(\lambda (P: Prop).(let TMP_451 \def (Flat f) in (let TMP_452 \def 
-(THead TMP_451 t0 t1) in (let TMP_450 \def (\lambda (ee: T).(match ee in T 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ 
-_) \Rightarrow (match k0 in K with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])])) in (let TMP_448 \def (Bind b) in (let TMP_449 \def 
-(THead TMP_448 w u) in (let H2 \def (eq_ind T TMP_452 TMP_450 I TMP_449 H1) 
-in (False_ind P H2)))))))))))) in (or_intror TMP_460 TMP_454 
-TMP_453)))))))))) in (K_ind TMP_485 TMP_478 TMP_461 k))))) in (T_ind TMP_516 
-TMP_510 TMP_498 TMP_486 t))))).
+Prop).P))))))).(\lambda (t1: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: 
+B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b) w u)))))) 
+(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead (Bind b) w 
+u)) \to (\forall (P: Prop).P))))))).(let TMP_59 \def (\lambda (b: B).(\lambda 
+(w: T).(\lambda (u: T).(let TMP_55 \def (Flat f) in (let TMP_56 \def (THead 
+TMP_55 t0 t1) in (let TMP_57 \def (Bind b) in (let TMP_58 \def (THead TMP_57 
+w u) in (eq T TMP_56 TMP_58)))))))) in (let TMP_60 \def (ex_3 B T T TMP_59) 
+in (let TMP_61 \def (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T 
+(THead (Flat f) t0 t1) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))) in 
+(let TMP_67 \def (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(\lambda 
+(H1: (eq T (THead (Flat f) t0 t1) (THead (Bind b) w u))).(\lambda (P: 
+Prop).(let TMP_62 \def (Flat f) in (let TMP_63 \def (THead TMP_62 t0 t1) in 
+(let TMP_64 \def (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False 
+| (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with 
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) in (let TMP_65 
+\def (Bind b) in (let TMP_66 \def (THead TMP_65 w u) in (let H2 \def (eq_ind 
+T TMP_63 TMP_64 I TMP_66 H1) in (False_ind P H2)))))))))))) in (or_intror 
+TMP_60 TMP_61 TMP_67)))))))))) in (K_ind TMP_37 TMP_54 TMP_68 k))))) in 
+(T_ind TMP_6 TMP_18 TMP_30 TMP_69 t))))).
 
 theorem abst_dec:
  \forall (u: T).(\forall (v: T).(or (ex T (\lambda (t: T).(eq T u (THead 
 (Bind Abst) v t)))) (\forall (t: T).((eq T u (THead (Bind Abst) v t)) \to 
 (\forall (P: Prop).P)))))
 \def
- \lambda (u: T).(let TMP_646 \def (\lambda (t: T).(\forall (v: T).(let 
-TMP_644 \def (\lambda (t0: T).(let TMP_642 \def (Bind Abst) in (let TMP_643 
-\def (THead TMP_642 v t0) in (eq T t TMP_643)))) in (let TMP_645 \def (ex T 
-TMP_644) in (let TMP_641 \def (\forall (t0: T).((eq T t (THead (Bind Abst) v 
-t0)) \to (\forall (P: Prop).P))) in (or TMP_645 TMP_641)))))) in (let TMP_640 
-\def (\lambda (n: nat).(\lambda (v: T).(let TMP_638 \def (\lambda (t: T).(let 
-TMP_637 \def (TSort n) in (let TMP_635 \def (Bind Abst) in (let TMP_636 \def 
-(THead TMP_635 v t) in (eq T TMP_637 TMP_636))))) in (let TMP_639 \def (ex T 
-TMP_638) in (let TMP_634 \def (\forall (t: T).((eq T (TSort n) (THead (Bind 
-Abst) v t)) \to (\forall (P: Prop).P))) in (let TMP_633 \def (\lambda (t: 
-T).(\lambda (H: (eq T (TSort n) (THead (Bind Abst) v t))).(\lambda (P: 
-Prop).(let TMP_632 \def (TSort n) in (let TMP_631 \def (\lambda (ee: 
-T).(match ee in T with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow 
-False | (THead _ _ _) \Rightarrow False])) in (let TMP_629 \def (Bind Abst) 
-in (let TMP_630 \def (THead TMP_629 v t) in (let H0 \def (eq_ind T TMP_632 
-TMP_631 I TMP_630 H) in (False_ind P H0))))))))) in (or_intror TMP_639 
-TMP_634 TMP_633))))))) in (let TMP_628 \def (\lambda (n: nat).(\lambda (v: 
-T).(let TMP_626 \def (\lambda (t: T).(let TMP_625 \def (TLRef n) in (let 
-TMP_623 \def (Bind Abst) in (let TMP_624 \def (THead TMP_623 v t) in (eq T 
-TMP_625 TMP_624))))) in (let TMP_627 \def (ex T TMP_626) in (let TMP_622 \def 
-(\forall (t: T).((eq T (TLRef n) (THead (Bind Abst) v t)) \to (\forall (P: 
-Prop).P))) in (let TMP_621 \def (\lambda (t: T).(\lambda (H: (eq T (TLRef n) 
-(THead (Bind Abst) v t))).(\lambda (P: Prop).(let TMP_620 \def (TLRef n) in 
-(let TMP_619 \def (\lambda (ee: T).(match ee in T with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) in 
-(let TMP_617 \def (Bind Abst) in (let TMP_618 \def (THead TMP_617 v t) in 
-(let H0 \def (eq_ind T TMP_620 TMP_619 I TMP_618 H) in (False_ind P 
-H0))))))))) in (or_intror TMP_627 TMP_622 TMP_621))))))) in (let TMP_616 \def 
-(\lambda (k: K).(\lambda (t: T).(\lambda (_: ((\forall (v: T).(or (ex T 
-(\lambda (t0: T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T 
-t (THead (Bind Abst) v t0)) \to (\forall (P: Prop).P))))))).(\lambda (t0: 
-T).(\lambda (_: ((\forall (v: T).(or (ex T (\lambda (t1: T).(eq T t0 (THead 
-(Bind Abst) v t1)))) (\forall (t1: T).((eq T t0 (THead (Bind Abst) v t1)) \to 
-(\forall (P: Prop).P))))))).(\lambda (v: T).(let TMP_517 \def (Bind Abst) in 
-(let H_x \def (terms_props__kind_dec k TMP_517) in (let H1 \def H_x in (let 
-TMP_614 \def (Bind Abst) in (let TMP_615 \def (eq K k TMP_614) in (let 
-TMP_613 \def ((eq K k (Bind Abst)) \to (\forall (P: Prop).P)) in (let TMP_610 
-\def (\lambda (t1: T).(let TMP_609 \def (THead k t t0) in (let TMP_607 \def 
-(Bind Abst) in (let TMP_608 \def (THead TMP_607 v t1) in (eq T TMP_609 
-TMP_608))))) in (let TMP_611 \def (ex T TMP_610) in (let TMP_606 \def 
-(\forall (t1: T).((eq T (THead k t t0) (THead (Bind Abst) v t1)) \to (\forall 
-(P: Prop).P))) in (let TMP_612 \def (or TMP_611 TMP_606) in (let TMP_605 \def 
-(\lambda (H2: (eq K k (Bind Abst))).(let TMP_604 \def (Bind Abst) in (let 
-TMP_603 \def (\lambda (k0: K).(let TMP_601 \def (\lambda (t1: T).(let TMP_600 
-\def (THead k0 t t0) in (let TMP_598 \def (Bind Abst) in (let TMP_599 \def 
-(THead TMP_598 v t1) in (eq T TMP_600 TMP_599))))) in (let TMP_602 \def (ex T 
-TMP_601) in (let TMP_597 \def (\forall (t1: T).((eq T (THead k0 t t0) (THead 
-(Bind Abst) v t1)) \to (\forall (P: Prop).P))) in (or TMP_602 TMP_597))))) in 
-(let H_x0 \def (term_dec t v) in (let H3 \def H_x0 in (let TMP_595 \def (eq T 
-t v) in (let TMP_594 \def ((eq T t v) \to (\forall (P: Prop).P)) in (let 
-TMP_591 \def (\lambda (t1: T).(let TMP_589 \def (Bind Abst) in (let TMP_590 
-\def (THead TMP_589 t t0) in (let TMP_587 \def (Bind Abst) in (let TMP_588 
-\def (THead TMP_587 v t1) in (eq T TMP_590 TMP_588)))))) in (let TMP_592 \def 
-(ex T TMP_591) in (let TMP_586 \def (\forall (t1: T).((eq T (THead (Bind 
-Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) in (let 
-TMP_593 \def (or TMP_592 TMP_586) in (let TMP_585 \def (\lambda (H4: (eq T t 
-v)).(let TMP_584 \def (\lambda (t1: T).(let TMP_582 \def (\lambda (t2: 
-T).(let TMP_580 \def (Bind Abst) in (let TMP_581 \def (THead TMP_580 t t0) in 
-(let TMP_578 \def (Bind Abst) in (let TMP_579 \def (THead TMP_578 t1 t2) in 
-(eq T TMP_581 TMP_579)))))) in (let TMP_583 \def (ex T TMP_582) in (let 
-TMP_577 \def (\forall (t2: T).((eq T (THead (Bind Abst) t t0) (THead (Bind 
-Abst) t1 t2)) \to (\forall (P: Prop).P))) in (or TMP_583 TMP_577))))) in (let 
-TMP_574 \def (\lambda (t1: T).(let TMP_572 \def (Bind Abst) in (let TMP_573 
-\def (THead TMP_572 t t0) in (let TMP_570 \def (Bind Abst) in (let TMP_571 
-\def (THead TMP_570 t t1) in (eq T TMP_573 TMP_571)))))) in (let TMP_575 \def 
-(ex T TMP_574) in (let TMP_569 \def (\forall (t1: T).((eq T (THead (Bind 
-Abst) t t0) (THead (Bind Abst) t t1)) \to (\forall (P: Prop).P))) in (let 
-TMP_567 \def (\lambda (t1: T).(let TMP_565 \def (Bind Abst) in (let TMP_566 
-\def (THead TMP_565 t t0) in (let TMP_563 \def (Bind Abst) in (let TMP_564 
-\def (THead TMP_563 t t1) in (eq T TMP_566 TMP_564)))))) in (let TMP_560 \def 
-(Bind Abst) in (let TMP_561 \def (THead TMP_560 t t0) in (let TMP_562 \def 
-(refl_equal T TMP_561) in (let TMP_568 \def (ex_intro T TMP_567 t0 TMP_562) 
-in (let TMP_576 \def (or_introl TMP_575 TMP_569 TMP_568) in (eq_ind T t 
-TMP_584 TMP_576 v H4)))))))))))) in (let TMP_559 \def (\lambda (H4: (((eq T t 
-v) \to (\forall (P: Prop).P)))).(let TMP_557 \def (\lambda (t1: T).(let 
-TMP_555 \def (Bind Abst) in (let TMP_556 \def (THead TMP_555 t t0) in (let 
-TMP_553 \def (Bind Abst) in (let TMP_554 \def (THead TMP_553 v t1) in (eq T 
-TMP_556 TMP_554)))))) in (let TMP_558 \def (ex T TMP_557) in (let TMP_552 
-\def (\forall (t1: T).((eq T (THead (Bind Abst) t t0) (THead (Bind Abst) v 
-t1)) \to (\forall (P: Prop).P))) in (let TMP_551 \def (\lambda (t1: 
-T).(\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) v 
-t1))).(\lambda (P: Prop).(let TMP_544 \def (\lambda (e: T).(match e in T with 
+ \lambda (u: T).(let TMP_6 \def (\lambda (t: T).(\forall (v: T).(let TMP_3 
+\def (\lambda (t0: T).(let TMP_1 \def (Bind Abst) in (let TMP_2 \def (THead 
+TMP_1 v t0) in (eq T t TMP_2)))) in (let TMP_4 \def (ex T TMP_3) in (let 
+TMP_5 \def (\forall (t0: T).((eq T t (THead (Bind Abst) v t0)) \to (\forall 
+(P: Prop).P))) in (or TMP_4 TMP_5)))))) in (let TMP_18 \def (\lambda (n: 
+nat).(\lambda (v: T).(let TMP_10 \def (\lambda (t: T).(let TMP_7 \def (TSort 
+n) in (let TMP_8 \def (Bind Abst) in (let TMP_9 \def (THead TMP_8 v t) in (eq 
+T TMP_7 TMP_9))))) in (let TMP_11 \def (ex T TMP_10) in (let TMP_12 \def 
+(\forall (t: T).((eq T (TSort n) (THead (Bind Abst) v t)) \to (\forall (P: 
+Prop).P))) in (let TMP_17 \def (\lambda (t: T).(\lambda (H: (eq T (TSort n) 
+(THead (Bind Abst) v t))).(\lambda (P: Prop).(let TMP_13 \def (TSort n) in 
+(let TMP_14 \def (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow True 
+| (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) in (let 
+TMP_15 \def (Bind Abst) in (let TMP_16 \def (THead TMP_15 v t) in (let H0 
+\def (eq_ind T TMP_13 TMP_14 I TMP_16 H) in (False_ind P H0))))))))) in 
+(or_intror TMP_11 TMP_12 TMP_17))))))) in (let TMP_30 \def (\lambda (n: 
+nat).(\lambda (v: T).(let TMP_22 \def (\lambda (t: T).(let TMP_19 \def (TLRef 
+n) in (let TMP_20 \def (Bind Abst) in (let TMP_21 \def (THead TMP_20 v t) in 
+(eq T TMP_19 TMP_21))))) in (let TMP_23 \def (ex T TMP_22) in (let TMP_24 
+\def (\forall (t: T).((eq T (TLRef n) (THead (Bind Abst) v t)) \to (\forall 
+(P: Prop).P))) in (let TMP_29 \def (\lambda (t: T).(\lambda (H: (eq T (TLRef 
+n) (THead (Bind Abst) v t))).(\lambda (P: Prop).(let TMP_25 \def (TLRef n) in 
+(let TMP_26 \def (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False 
+| (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) in (let 
+TMP_27 \def (Bind Abst) in (let TMP_28 \def (THead TMP_27 v t) in (let H0 
+\def (eq_ind T TMP_25 TMP_26 I TMP_28 H) in (False_ind P H0))))))))) in 
+(or_intror TMP_23 TMP_24 TMP_29))))))) in (let TMP_130 \def (\lambda (k: 
+K).(\lambda (t: T).(\lambda (_: ((\forall (v: T).(or (ex T (\lambda (t0: 
+T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead (Bind 
+Abst) v t0)) \to (\forall (P: Prop).P))))))).(\lambda (t0: T).(\lambda (_: 
+((\forall (v: T).(or (ex T (\lambda (t1: T).(eq T t0 (THead (Bind Abst) v 
+t1)))) (\forall (t1: T).((eq T t0 (THead (Bind Abst) v t1)) \to (\forall (P: 
+Prop).P))))))).(\lambda (v: T).(let TMP_31 \def (Bind Abst) in (let H_x \def 
+(terms_props__kind_dec k TMP_31) in (let H1 \def H_x in (let TMP_32 \def 
+(Bind Abst) in (let TMP_33 \def (eq K k TMP_32) in (let TMP_34 \def ((eq K k 
+(Bind Abst)) \to (\forall (P: Prop).P)) in (let TMP_38 \def (\lambda (t1: 
+T).(let TMP_35 \def (THead k t t0) in (let TMP_36 \def (Bind Abst) in (let 
+TMP_37 \def (THead TMP_36 v t1) in (eq T TMP_35 TMP_37))))) in (let TMP_39 
+\def (ex T TMP_38) in (let TMP_40 \def (\forall (t1: T).((eq T (THead k t t0) 
+(THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) in (let TMP_41 \def (or 
+TMP_39 TMP_40) in (let TMP_107 \def (\lambda (H2: (eq K k (Bind Abst))).(let 
+TMP_42 \def (Bind Abst) in (let TMP_49 \def (\lambda (k0: K).(let TMP_46 \def 
+(\lambda (t1: T).(let TMP_43 \def (THead k0 t t0) in (let TMP_44 \def (Bind 
+Abst) in (let TMP_45 \def (THead TMP_44 v t1) in (eq T TMP_43 TMP_45))))) in 
+(let TMP_47 \def (ex T TMP_46) in (let TMP_48 \def (\forall (t1: T).((eq T 
+(THead k0 t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) in (or 
+TMP_47 TMP_48))))) in (let H_x0 \def (term_dec t v) in (let H3 \def H_x0 in 
+(let TMP_50 \def (eq T t v) in (let TMP_51 \def ((eq T t v) \to (\forall (P: 
+Prop).P)) in (let TMP_56 \def (\lambda (t1: T).(let TMP_52 \def (Bind Abst) 
+in (let TMP_53 \def (THead TMP_52 t t0) in (let TMP_54 \def (Bind Abst) in 
+(let TMP_55 \def (THead TMP_54 v t1) in (eq T TMP_53 TMP_55)))))) in (let 
+TMP_57 \def (ex T TMP_56) in (let TMP_58 \def (\forall (t1: T).((eq T (THead 
+(Bind Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) in 
+(let TMP_59 \def (or TMP_57 TMP_58) in (let TMP_85 \def (\lambda (H4: (eq T t 
+v)).(let TMP_67 \def (\lambda (t1: T).(let TMP_64 \def (\lambda (t2: T).(let 
+TMP_60 \def (Bind Abst) in (let TMP_61 \def (THead TMP_60 t t0) in (let 
+TMP_62 \def (Bind Abst) in (let TMP_63 \def (THead TMP_62 t1 t2) in (eq T 
+TMP_61 TMP_63)))))) in (let TMP_65 \def (ex T TMP_64) in (let TMP_66 \def 
+(\forall (t2: T).((eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)) 
+\to (\forall (P: Prop).P))) in (or TMP_65 TMP_66))))) in (let TMP_72 \def 
+(\lambda (t1: T).(let TMP_68 \def (Bind Abst) in (let TMP_69 \def (THead 
+TMP_68 t t0) in (let TMP_70 \def (Bind Abst) in (let TMP_71 \def (THead 
+TMP_70 t t1) in (eq T TMP_69 TMP_71)))))) in (let TMP_73 \def (ex T TMP_72) 
+in (let TMP_74 \def (\forall (t1: T).((eq T (THead (Bind Abst) t t0) (THead 
+(Bind Abst) t t1)) \to (\forall (P: Prop).P))) in (let TMP_79 \def (\lambda 
+(t1: T).(let TMP_75 \def (Bind Abst) in (let TMP_76 \def (THead TMP_75 t t0) 
+in (let TMP_77 \def (Bind Abst) in (let TMP_78 \def (THead TMP_77 t t1) in 
+(eq T TMP_76 TMP_78)))))) in (let TMP_80 \def (Bind Abst) in (let TMP_81 \def 
+(THead TMP_80 t t0) in (let TMP_82 \def (refl_equal T TMP_81) in (let TMP_83 
+\def (ex_intro T TMP_79 t0 TMP_82) in (let TMP_84 \def (or_introl TMP_73 
+TMP_74 TMP_83) in (eq_ind T t TMP_67 TMP_84 v H4)))))))))))) in (let TMP_105 
+\def (\lambda (H4: (((eq T t v) \to (\forall (P: Prop).P)))).(let TMP_90 \def 
+(\lambda (t1: T).(let TMP_86 \def (Bind Abst) in (let TMP_87 \def (THead 
+TMP_86 t t0) in (let TMP_88 \def (Bind Abst) in (let TMP_89 \def (THead 
+TMP_88 v t1) in (eq T TMP_87 TMP_89)))))) in (let TMP_91 \def (ex T TMP_90) 
+in (let TMP_92 \def (\forall (t1: T).((eq T (THead (Bind Abst) t t0) (THead 
+(Bind Abst) v t1)) \to (\forall (P: Prop).P))) in (let TMP_104 \def (\lambda 
+(t1: T).(\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) v 
+t1))).(\lambda (P: Prop).(let TMP_93 \def (\lambda (e: T).(match e with 
 [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t2 _) 
-\Rightarrow t2])) in (let TMP_542 \def (Bind Abst) in (let TMP_543 \def 
-(THead TMP_542 t t0) in (let TMP_540 \def (Bind Abst) in (let TMP_541 \def 
-(THead TMP_540 v t1) in (let H6 \def (f_equal T T TMP_544 TMP_543 TMP_541 H5) 
-in (let TMP_549 \def (\lambda (e: T).(match e in T with [(TSort _) 
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) 
-in (let TMP_547 \def (Bind Abst) in (let TMP_548 \def (THead TMP_547 t t0) in 
-(let TMP_545 \def (Bind Abst) in (let TMP_546 \def (THead TMP_545 v t1) in 
-(let H7 \def (f_equal T T TMP_549 TMP_548 TMP_546 H5) in (let TMP_550 \def 
-(\lambda (H8: (eq T t v)).(H4 H8 P)) in (TMP_550 H6))))))))))))))))) in 
-(or_intror TMP_558 TMP_552 TMP_551)))))) in (let TMP_596 \def (or_ind TMP_595 
-TMP_594 TMP_593 TMP_585 TMP_559 H3) in (eq_ind_r K TMP_604 TMP_603 TMP_596 k 
-H2))))))))))))))) in (let TMP_539 \def (\lambda (H2: (((eq K k (Bind Abst)) 
-\to (\forall (P: Prop).P)))).(let TMP_537 \def (\lambda (t1: T).(let TMP_536 
-\def (THead k t t0) in (let TMP_534 \def (Bind Abst) in (let TMP_535 \def 
-(THead TMP_534 v t1) in (eq T TMP_536 TMP_535))))) in (let TMP_538 \def (ex T 
-TMP_537) in (let TMP_533 \def (\forall (t1: T).((eq T (THead k t t0) (THead 
-(Bind Abst) v t1)) \to (\forall (P: Prop).P))) in (let TMP_532 \def (\lambda 
-(t1: T).(\lambda (H3: (eq T (THead k t t0) (THead (Bind Abst) v 
-t1))).(\lambda (P: Prop).(let TMP_521 \def (\lambda (e: T).(match e in T with 
-[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
-\Rightarrow k0])) in (let TMP_520 \def (THead k t t0) in (let TMP_518 \def 
-(Bind Abst) in (let TMP_519 \def (THead TMP_518 v t1) in (let H4 \def 
-(f_equal T K TMP_521 TMP_520 TMP_519 H3) in (let TMP_525 \def (\lambda (e: 
-T).(match e in T with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | 
-(THead _ t2 _) \Rightarrow t2])) in (let TMP_524 \def (THead k t t0) in (let 
-TMP_522 \def (Bind Abst) in (let TMP_523 \def (THead TMP_522 v t1) in (let H5 
-\def (f_equal T T TMP_525 TMP_524 TMP_523 H3) in (let TMP_529 \def (\lambda 
-(e: T).(match e in T with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow 
-t0 | (THead _ _ t2) \Rightarrow t2])) in (let TMP_528 \def (THead k t t0) in 
-(let TMP_526 \def (Bind Abst) in (let TMP_527 \def (THead TMP_526 v t1) in 
-(let H6 \def (f_equal T T TMP_529 TMP_528 TMP_527 H3) in (let TMP_530 \def 
-(\lambda (_: (eq T t v)).(\lambda (H8: (eq K k (Bind Abst))).(H2 H8 P))) in 
-(let TMP_531 \def (TMP_530 H5) in (TMP_531 H4))))))))))))))))))))) in 
-(or_intror TMP_538 TMP_533 TMP_532)))))) in (or_ind TMP_615 TMP_613 TMP_612 
-TMP_605 TMP_539 H1))))))))))))))))))) in (T_ind TMP_646 TMP_640 TMP_628 
-TMP_616 u))))).
+\Rightarrow t2])) in (let TMP_94 \def (Bind Abst) in (let TMP_95 \def (THead 
+TMP_94 t t0) in (let TMP_96 \def (Bind Abst) in (let TMP_97 \def (THead 
+TMP_96 v t1) in (let H6 \def (f_equal T T TMP_93 TMP_95 TMP_97 H5) in (let 
+TMP_98 \def (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef 
+_) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) in (let TMP_99 \def 
+(Bind Abst) in (let TMP_100 \def (THead TMP_99 t t0) in (let TMP_101 \def 
+(Bind Abst) in (let TMP_102 \def (THead TMP_101 v t1) in (let H7 \def 
+(f_equal T T TMP_98 TMP_100 TMP_102 H5) in (let TMP_103 \def (\lambda (H8: 
+(eq T t v)).(H4 H8 P)) in (TMP_103 H6))))))))))))))))) in (or_intror TMP_91 
+TMP_92 TMP_104)))))) in (let TMP_106 \def (or_ind TMP_50 TMP_51 TMP_59 TMP_85 
+TMP_105 H3) in (eq_ind_r K TMP_42 TMP_49 TMP_106 k H2))))))))))))))) in (let 
+TMP_129 \def (\lambda (H2: (((eq K k (Bind Abst)) \to (\forall (P: 
+Prop).P)))).(let TMP_111 \def (\lambda (t1: T).(let TMP_108 \def (THead k t 
+t0) in (let TMP_109 \def (Bind Abst) in (let TMP_110 \def (THead TMP_109 v 
+t1) in (eq T TMP_108 TMP_110))))) in (let TMP_112 \def (ex T TMP_111) in (let 
+TMP_113 \def (\forall (t1: T).((eq T (THead k t t0) (THead (Bind Abst) v t1)) 
+\to (\forall (P: Prop).P))) in (let TMP_128 \def (\lambda (t1: T).(\lambda 
+(H3: (eq T (THead k t t0) (THead (Bind Abst) v t1))).(\lambda (P: Prop).(let 
+TMP_114 \def (\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef 
+_) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) in (let TMP_115 \def 
+(THead k t t0) in (let TMP_116 \def (Bind Abst) in (let TMP_117 \def (THead 
+TMP_116 v t1) in (let H4 \def (f_equal T K TMP_114 TMP_115 TMP_117 H3) in 
+(let TMP_118 \def (\lambda (e: T).(match e with [(TSort _) \Rightarrow t | 
+(TLRef _) \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) in (let TMP_119 
+\def (THead k t t0) in (let TMP_120 \def (Bind Abst) in (let TMP_121 \def 
+(THead TMP_120 v t1) in (let H5 \def (f_equal T T TMP_118 TMP_119 TMP_121 H3) 
+in (let TMP_122 \def (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 
+| (TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) in (let TMP_123 
+\def (THead k t t0) in (let TMP_124 \def (Bind Abst) in (let TMP_125 \def 
+(THead TMP_124 v t1) in (let H6 \def (f_equal T T TMP_122 TMP_123 TMP_125 H3) 
+in (let TMP_126 \def (\lambda (_: (eq T t v)).(\lambda (H8: (eq K k (Bind 
+Abst))).(H2 H8 P))) in (let TMP_127 \def (TMP_126 H5) in (TMP_127 
+H4))))))))))))))))))))) in (or_intror TMP_112 TMP_113 TMP_128)))))) in 
+(or_ind TMP_33 TMP_34 TMP_41 TMP_107 TMP_129 H1))))))))))))))))))) in (T_ind 
+TMP_6 TMP_18 TMP_30 TMP_130 u))))).