include "basic_1/sc3/arity.ma".
-theorem ty3_predicative:
+lemma ty3_predicative:
\forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (u:
T).((ty3 g c (THead (Bind Abst) v t) u) \to ((pc3 c u v) \to (\forall (P:
Prop).P)))))))
\def
\lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (u:
T).(\lambda (H: (ty3 g c (THead (Bind Abst) v t) u)).(\lambda (H0: (pc3 c u
-v)).(\lambda (P: Prop).(let H1 \def H in (let TMP_3 \def (\lambda (t2:
-T).(\lambda (_: T).(let TMP_1 \def (Bind Abst) in (let TMP_2 \def (THead
-TMP_1 v t2) in (pc3 c TMP_2 u))))) in (let TMP_4 \def (\lambda (_:
-T).(\lambda (t0: T).(ty3 g c v t0))) in (let TMP_7 \def (\lambda (t2:
-T).(\lambda (_: T).(let TMP_5 \def (Bind Abst) in (let TMP_6 \def (CHead c
-TMP_5 v) in (ty3 g TMP_6 t t2))))) in (let TMP_34 \def (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (_: (pc3 c (THead (Bind Abst) v x0) u)).(\lambda
-(H3: (ty3 g c v x1)).(\lambda (_: (ty3 g (CHead c (Bind Abst) v) t x0)).(let
-TMP_8 \def (Bind Abst) in (let TMP_9 \def (THead TMP_8 v t) in (let H_y \def
-(ty3_conv g c v x1 H3 TMP_9 u H H0) in (let TMP_10 \def (Bind Abst) in (let
-TMP_11 \def (THead TMP_10 v t) in (let H_x \def (ty3_arity g c TMP_11 v H_y)
-in (let H5 \def H_x in (let TMP_14 \def (\lambda (a1: A).(let TMP_12 \def
-(Bind Abst) in (let TMP_13 \def (THead TMP_12 v t) in (arity g c TMP_13
-a1)))) in (let TMP_16 \def (\lambda (a1: A).(let TMP_15 \def (asucc g a1) in
-(arity g c v TMP_15))) in (let TMP_33 \def (\lambda (x: A).(\lambda (H6:
-(arity g c (THead (Bind Abst) v t) x)).(\lambda (H7: (arity g c v (asucc g
-x))).(let H8 \def (arity_gen_abst g c v t x H6) in (let TMP_18 \def (\lambda
-(a1: A).(\lambda (a2: A).(let TMP_17 \def (AHead a1 a2) in (eq A x TMP_17))))
-in (let TMP_20 \def (\lambda (a1: A).(\lambda (_: A).(let TMP_19 \def (asucc
-g a1) in (arity g c v TMP_19)))) in (let TMP_23 \def (\lambda (_: A).(\lambda
-(a2: A).(let TMP_21 \def (Bind Abst) in (let TMP_22 \def (CHead c TMP_21 v)
-in (arity g TMP_22 t a2))))) in (let TMP_32 \def (\lambda (x2: A).(\lambda
-(x3: A).(\lambda (H9: (eq A x (AHead x2 x3))).(\lambda (H10: (arity g c v
-(asucc g x2))).(\lambda (_: (arity g (CHead c (Bind Abst) v) t x3)).(let
-TMP_25 \def (\lambda (a: A).(let TMP_24 \def (asucc g a) in (arity g c v
-TMP_24))) in (let TMP_26 \def (AHead x2 x3) in (let H12 \def (eq_ind A x
-TMP_25 H7 TMP_26 H9) in (let TMP_27 \def (asucc g x3) in (let TMP_28 \def
-(AHead x2 x3) in (let TMP_29 \def (asucc g TMP_28) in (let TMP_30 \def (asucc
-g x2) in (let TMP_31 \def (arity_mono g c v TMP_29 H12 TMP_30 H10) in
-(leq_ahead_asucc_false g x2 TMP_27 TMP_31 P)))))))))))))) in (ex3_2_ind A A
-TMP_18 TMP_20 TMP_23 P TMP_32 H8))))))))) in (ex2_ind A TMP_14 TMP_16 P
-TMP_33 H5)))))))))))))))) in (let TMP_35 \def (ty3_gen_bind g Abst c v t u
-H1) in (ex3_2_ind T T TMP_3 TMP_4 TMP_7 P TMP_34 TMP_35)))))))))))))).
+v)).(\lambda (P: Prop).(let H1 \def H in (ex3_2_ind T T (\lambda (t2:
+T).(\lambda (_: T).(pc3 c (THead (Bind Abst) v t2) u))) (\lambda (_:
+T).(\lambda (t0: T).(ty3 g c v t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind Abst) v) t t2))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(_: (pc3 c (THead (Bind Abst) v x0) u)).(\lambda (H3: (ty3 g c v
+x1)).(\lambda (_: (ty3 g (CHead c (Bind Abst) v) t x0)).(let H_y \def
+(ty3_conv g c v x1 H3 (THead (Bind Abst) v t) u H H0) in (let H_x \def
+(ty3_arity g c (THead (Bind Abst) v t) v H_y) in (let H5 \def H_x in (ex2_ind
+A (\lambda (a1: A).(arity g c (THead (Bind Abst) v t) a1)) (\lambda (a1:
+A).(arity g c v (asucc g a1))) P (\lambda (x: A).(\lambda (H6: (arity g c
+(THead (Bind Abst) v t) x)).(\lambda (H7: (arity g c v (asucc g x))).(let H8
+\def (arity_gen_abst g c v t x H6) in (ex3_2_ind A A (\lambda (a1:
+A).(\lambda (a2: A).(eq A x (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g c v (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead c (Bind Abst) v) t a2))) P (\lambda (x2: A).(\lambda (x3: A).(\lambda
+(H9: (eq A x (AHead x2 x3))).(\lambda (H10: (arity g c v (asucc g
+x2))).(\lambda (_: (arity g (CHead c (Bind Abst) v) t x3)).(let H12 \def
+(eq_ind A x (\lambda (a: A).(arity g c v (asucc g a))) H7 (AHead x2 x3) H9)
+in (leq_ahead_asucc_false g x2 (asucc g x3) (arity_mono g c v (asucc g (AHead
+x2 x3)) H12 (asucc g x2) H10) P))))))) H8))))) H5))))))))) (ty3_gen_bind g
+Abst c v t u H1)))))))))).
theorem ty3_repellent:
\forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (u1:
\lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (u1:
T).(\lambda (H: (ty3 g c (THead (Bind Abst) w t) u1)).(\lambda (u2:
T).(\lambda (H0: (ty3 g (CHead c (Bind Abst) w) t (lift (S O) O
-u2))).(\lambda (H1: (pc3 c u1 u2)).(\lambda (P: Prop).(let TMP_5 \def
-(\lambda (t0: T).(let TMP_1 \def (Bind Abst) in (let TMP_2 \def (CHead c
-TMP_1 w) in (let TMP_3 \def (S O) in (let TMP_4 \def (lift TMP_3 O u2) in
-(ty3 g TMP_2 TMP_4 t0)))))) in (let TMP_55 \def (\lambda (x: T).(\lambda (H2:
-(ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) x)).(let TMP_6 \def (Bind
-Abst) in (let TMP_7 \def (CHead c TMP_6 w) in (let TMP_8 \def (S O) in (let
-TMP_9 \def (Bind Abst) in (let TMP_10 \def (drop_refl c) in (let TMP_11 \def
-(drop_drop TMP_9 O c c TMP_10 w) in (let H3 \def (ty3_gen_lift g TMP_7 u2 x
-TMP_8 O H2 c TMP_11) in (let TMP_16 \def (\lambda (t2: T).(let TMP_12 \def
-(Bind Abst) in (let TMP_13 \def (CHead c TMP_12 w) in (let TMP_14 \def (S O)
-in (let TMP_15 \def (lift TMP_14 O t2) in (pc3 TMP_13 TMP_15 x)))))) in (let
-TMP_17 \def (\lambda (t2: T).(ty3 g c u2 t2)) in (let TMP_54 \def (\lambda
-(x0: T).(\lambda (_: (pc3 (CHead c (Bind Abst) w) (lift (S O) O x0)
-x)).(\lambda (H5: (ty3 g c u2 x0)).(let TMP_18 \def (Bind Abst) in (let
-TMP_19 \def (THead TMP_18 w t) in (let H_y \def (ty3_conv g c u2 x0 H5 TMP_19
-u1 H H1) in (let TMP_20 \def (Bind Abst) in (let TMP_21 \def (CHead c TMP_20
-w) in (let TMP_22 \def (S O) in (let TMP_23 \def (lift TMP_22 O u2) in (let
-H_x \def (ty3_arity g TMP_21 t TMP_23 H0) in (let H6 \def H_x in (let TMP_26
-\def (\lambda (a1: A).(let TMP_24 \def (Bind Abst) in (let TMP_25 \def (CHead
-c TMP_24 w) in (arity g TMP_25 t a1)))) in (let TMP_32 \def (\lambda (a1:
-A).(let TMP_27 \def (Bind Abst) in (let TMP_28 \def (CHead c TMP_27 w) in
-(let TMP_29 \def (S O) in (let TMP_30 \def (lift TMP_29 O u2) in (let TMP_31
-\def (asucc g a1) in (arity g TMP_28 TMP_30 TMP_31))))))) in (let TMP_53 \def
-(\lambda (x1: A).(\lambda (H7: (arity g (CHead c (Bind Abst) w) t
-x1)).(\lambda (H8: (arity g (CHead c (Bind Abst) w) (lift (S O) O u2) (asucc
-g x1))).(let TMP_33 \def (Bind Abst) in (let TMP_34 \def (THead TMP_33 w t)
-in (let H_x0 \def (ty3_arity g c TMP_34 u2 H_y) in (let H9 \def H_x0 in (let
-TMP_37 \def (\lambda (a1: A).(let TMP_35 \def (Bind Abst) in (let TMP_36 \def
-(THead TMP_35 w t) in (arity g c TMP_36 a1)))) in (let TMP_39 \def (\lambda
-(a1: A).(let TMP_38 \def (asucc g a1) in (arity g c u2 TMP_38))) in (let
-TMP_52 \def (\lambda (x2: A).(\lambda (H10: (arity g c (THead (Bind Abst) w
-t) x2)).(\lambda (H11: (arity g c u2 (asucc g x2))).(let TMP_40 \def (asucc g
-x1) in (let TMP_41 \def (Bind Abst) in (let TMP_42 \def (CHead c TMP_41 w) in
-(let TMP_43 \def (asucc g x1) in (let TMP_44 \def (S O) in (let TMP_45 \def
-(Bind Abst) in (let TMP_46 \def (drop_refl c) in (let TMP_47 \def (drop_drop
-TMP_45 O c c TMP_46 w) in (let TMP_48 \def (arity_gen_lift g TMP_42 u2 TMP_43
-TMP_44 O H8 c TMP_47) in (let TMP_49 \def (asucc g x2) in (let TMP_50 \def
-(arity_mono g c u2 TMP_40 TMP_48 TMP_49 H11) in (let TMP_51 \def (asucc_inj g
-x1 x2 TMP_50) in (arity_repellent g c w t x1 H7 x2 H10 TMP_51
-P)))))))))))))))) in (ex2_ind A TMP_37 TMP_39 P TMP_52 H9))))))))))) in
-(ex2_ind A TMP_26 TMP_32 P TMP_53 H6)))))))))))))))) in (ex2_ind T TMP_16
-TMP_17 P TMP_54 H3))))))))))))) in (let TMP_56 \def (Bind Abst) in (let
-TMP_57 \def (CHead c TMP_56 w) in (let TMP_58 \def (S O) in (let TMP_59 \def
-(lift TMP_58 O u2) in (let TMP_60 \def (ty3_correct g TMP_57 t TMP_59 H0) in
-(ex_ind T TMP_5 P TMP_55 TMP_60))))))))))))))))).
+u2))).(\lambda (H1: (pc3 c u1 u2)).(\lambda (P: Prop).(ex_ind T (\lambda (t0:
+T).(ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) t0)) P (\lambda (x:
+T).(\lambda (H2: (ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) x)).(let H3
+\def (ty3_gen_lift g (CHead c (Bind Abst) w) u2 x (S O) O H2 c (drop_drop
+(Bind Abst) O c c (drop_refl c) w)) in (ex2_ind T (\lambda (t2: T).(pc3
+(CHead c (Bind Abst) w) (lift (S O) O t2) x)) (\lambda (t2: T).(ty3 g c u2
+t2)) P (\lambda (x0: T).(\lambda (_: (pc3 (CHead c (Bind Abst) w) (lift (S O)
+O x0) x)).(\lambda (H5: (ty3 g c u2 x0)).(let H_y \def (ty3_conv g c u2 x0 H5
+(THead (Bind Abst) w t) u1 H H1) in (let H_x \def (ty3_arity g (CHead c (Bind
+Abst) w) t (lift (S O) O u2) H0) in (let H6 \def H_x in (ex2_ind A (\lambda
+(a1: A).(arity g (CHead c (Bind Abst) w) t a1)) (\lambda (a1: A).(arity g
+(CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g a1))) P (\lambda (x1:
+A).(\lambda (H7: (arity g (CHead c (Bind Abst) w) t x1)).(\lambda (H8: (arity
+g (CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g x1))).(let H_x0 \def
+(ty3_arity g c (THead (Bind Abst) w t) u2 H_y) in (let H9 \def H_x0 in
+(ex2_ind A (\lambda (a1: A).(arity g c (THead (Bind Abst) w t) a1)) (\lambda
+(a1: A).(arity g c u2 (asucc g a1))) P (\lambda (x2: A).(\lambda (H10: (arity
+g c (THead (Bind Abst) w t) x2)).(\lambda (H11: (arity g c u2 (asucc g
+x2))).(arity_repellent g c w t x1 H7 x2 H10 (asucc_inj g x1 x2 (arity_mono g
+c u2 (asucc g x1) (arity_gen_lift g (CHead c (Bind Abst) w) u2 (asucc g x1)
+(S O) O H8 c (drop_drop (Bind Abst) O c c (drop_refl c) w)) (asucc g x2)
+H11)) P)))) H9)))))) H6))))))) H3)))) (ty3_correct g (CHead c (Bind Abst) w)
+t (lift (S O) O u2) H0))))))))))).
-theorem ty3_acyclic:
+lemma ty3_acyclic:
\forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
u) \to ((pc3 c u t) \to (\forall (P: Prop).P))))))
\def
\lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
(ty3 g c t u)).(\lambda (H0: (pc3 c u t)).(\lambda (P: Prop).(let H_y \def
(ty3_conv g c t u H t u H H0) in (let H_x \def (ty3_arity g c t t H_y) in
-(let H1 \def H_x in (let TMP_1 \def (\lambda (a1: A).(arity g c t a1)) in
-(let TMP_3 \def (\lambda (a1: A).(let TMP_2 \def (asucc g a1) in (arity g c t
-TMP_2))) in (let TMP_6 \def (\lambda (x: A).(\lambda (H2: (arity g c t
-x)).(\lambda (H3: (arity g c t (asucc g x))).(let TMP_4 \def (asucc g x) in
-(let TMP_5 \def (arity_mono g c t TMP_4 H3 x H2) in (leq_asucc_false g x
-TMP_5 P)))))) in (ex2_ind A TMP_1 TMP_3 P TMP_6 H1))))))))))))).
+(let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda
+(a1: A).(arity g c t (asucc g a1))) P (\lambda (x: A).(\lambda (H2: (arity g
+c t x)).(\lambda (H3: (arity g c t (asucc g x))).(leq_asucc_false g x
+(arity_mono g c t (asucc g x) H3 x H2) P)))) H1)))))))))).
-theorem ty3_sn3:
+lemma ty3_sn3:
\forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
u) \to (sn3 c t)))))
\def
\lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
(ty3 g c t u)).(let H_x \def (ty3_arity g c t u H) in (let H0 \def H_x in
-(let TMP_1 \def (\lambda (a1: A).(arity g c t a1)) in (let TMP_3 \def
-(\lambda (a1: A).(let TMP_2 \def (asucc g a1) in (arity g c u TMP_2))) in
-(let TMP_4 \def (sn3 c t) in (let TMP_6 \def (\lambda (x: A).(\lambda (H1:
-(arity g c t x)).(\lambda (_: (arity g c u (asucc g x))).(let TMP_5 \def
-(sc3_arity g c t x H1) in (sc3_sn3 g x c t TMP_5))))) in (ex2_ind A TMP_1
-TMP_3 TMP_4 TMP_6 H0))))))))))).
+(ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1: A).(arity g c u
+(asucc g a1))) (sn3 c t) (\lambda (x: A).(\lambda (H1: (arity g c t
+x)).(\lambda (_: (arity g c u (asucc g x))).(sc3_sn3 g x c t (sc3_arity g c t
+x H1))))) H0))))))).
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