include "basic_1/T/props.ma".
-theorem pr1_pr0:
+lemma pr1_pr0:
\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr1 t1 t2)))
\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(let TMP_1 \def
-(pr1_refl t2) in (pr1_sing t2 t1 H t2 TMP_1)))).
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr1_sing t2 t1 H
+t2 (pr1_refl t2)))).
theorem pr1_t:
\forall (t2: T).(\forall (t1: T).((pr1 t1 t2) \to (\forall (t3: T).((pr1 t2
t3) \to (pr1 t1 t3)))))
\def
- \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr1 t1 t2)).(let TMP_1 \def
-(\lambda (t: T).(\lambda (t0: T).(\forall (t3: T).((pr1 t0 t3) \to (pr1 t
-t3))))) in (let TMP_2 \def (\lambda (t: T).(\lambda (t3: T).(\lambda (H0:
-(pr1 t t3)).H0))) in (let TMP_4 \def (\lambda (t0: T).(\lambda (t3:
-T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: (pr1 t0
-t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t4 t5) \to (pr1 t0
-t5))))).(\lambda (t5: T).(\lambda (H3: (pr1 t4 t5)).(let TMP_3 \def (H2 t5
-H3) in (pr1_sing t0 t3 H0 t5 TMP_3)))))))))) in (pr1_ind TMP_1 TMP_2 TMP_4 t1
-t2 H)))))).
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda
+(t: T).(\lambda (t0: T).(\forall (t3: T).((pr1 t0 t3) \to (pr1 t t3)))))
+(\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr1 t t3)).H0))) (\lambda
+(t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda
+(_: (pr1 t0 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t4 t5) \to (pr1 t0
+t5))))).(\lambda (t5: T).(\lambda (H3: (pr1 t4 t5)).(pr1_sing t0 t3 H0 t5 (H2
+t5 H3)))))))))) t1 t2 H))).
-theorem pr1_head_1:
+lemma pr1_head_1:
\forall (u1: T).(\forall (u2: T).((pr1 u1 u2) \to (\forall (t: T).(\forall
(k: K).(pr1 (THead k u1 t) (THead k u2 t))))))
\def
\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr1 u1 u2)).(\lambda (t:
-T).(\lambda (k: K).(let TMP_3 \def (\lambda (t0: T).(\lambda (t1: T).(let
-TMP_1 \def (THead k t0 t) in (let TMP_2 \def (THead k t1 t) in (pr1 TMP_1
-TMP_2))))) in (let TMP_5 \def (\lambda (t0: T).(let TMP_4 \def (THead k t0 t)
-in (pr1_refl TMP_4))) in (let TMP_11 \def (\lambda (t2: T).(\lambda (t1:
-T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t3: T).(\lambda (_: (pr1 t2
-t3)).(\lambda (H2: (pr1 (THead k t2 t) (THead k t3 t))).(let TMP_6 \def
-(THead k t2 t) in (let TMP_7 \def (THead k t1 t) in (let TMP_8 \def (pr0_refl
-t) in (let TMP_9 \def (pr0_comp t1 t2 H0 t t TMP_8 k) in (let TMP_10 \def
-(THead k t3 t) in (pr1_sing TMP_6 TMP_7 TMP_9 TMP_10 H2)))))))))))) in
-(pr1_ind TMP_3 TMP_5 TMP_11 u1 u2 H)))))))).
+T).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t1: T).(pr1 (THead k
+t0 t) (THead k t1 t)))) (\lambda (t0: T).(pr1_refl (THead k t0 t))) (\lambda
+(t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t3: T).(\lambda
+(_: (pr1 t2 t3)).(\lambda (H2: (pr1 (THead k t2 t) (THead k t3 t))).(pr1_sing
+(THead k t2 t) (THead k t1 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k) (THead k
+t3 t) H2))))))) u1 u2 H))))).
-theorem pr1_head_2:
+lemma pr1_head_2:
\forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (u: T).(\forall
(k: K).(pr1 (THead k u t1) (THead k u t2))))))
\def
\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(\lambda (u:
-T).(\lambda (k: K).(let TMP_3 \def (\lambda (t: T).(\lambda (t0: T).(let
-TMP_1 \def (THead k u t) in (let TMP_2 \def (THead k u t0) in (pr1 TMP_1
-TMP_2))))) in (let TMP_5 \def (\lambda (t: T).(let TMP_4 \def (THead k u t)
-in (pr1_refl TMP_4))) in (let TMP_11 \def (\lambda (t0: T).(\lambda (t3:
-T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: (pr1 t0
-t4)).(\lambda (H2: (pr1 (THead k u t0) (THead k u t4))).(let TMP_6 \def
-(THead k u t0) in (let TMP_7 \def (THead k u t3) in (let TMP_8 \def (pr0_refl
-u) in (let TMP_9 \def (pr0_comp u u TMP_8 t3 t0 H0 k) in (let TMP_10 \def
-(THead k u t4) in (pr1_sing TMP_6 TMP_7 TMP_9 TMP_10 H2)))))))))))) in
-(pr1_ind TMP_3 TMP_5 TMP_11 t1 t2 H)))))))).
+T).(\lambda (k: K).(pr1_ind (\lambda (t: T).(\lambda (t0: T).(pr1 (THead k u
+t) (THead k u t0)))) (\lambda (t: T).(pr1_refl (THead k u t))) (\lambda (t0:
+T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_:
+(pr1 t0 t4)).(\lambda (H2: (pr1 (THead k u t0) (THead k u t4))).(pr1_sing
+(THead k u t0) (THead k u t3) (pr0_comp u u (pr0_refl u) t3 t0 H0 k) (THead k
+u t4) H2))))))) t1 t2 H))))).
theorem pr1_comp:
\forall (v: T).(\forall (w: T).((pr1 v w) \to (\forall (t: T).(\forall (u:
T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k v t) (THead k w u))))))))
\def
- \lambda (v: T).(\lambda (w: T).(\lambda (H: (pr1 v w)).(let TMP_3 \def
-(\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall (u: T).((pr1 t1 u)
-\to (\forall (k: K).(let TMP_1 \def (THead k t t1) in (let TMP_2 \def (THead
-k t0 u) in (pr1 TMP_1 TMP_2))))))))) in (let TMP_4 \def (\lambda (t:
-T).(\lambda (t0: T).(\lambda (u: T).(\lambda (H0: (pr1 t0 u)).(\lambda (k:
-K).(pr1_head_2 t0 u H0 t k)))))) in (let TMP_16 \def (\lambda (t2:
-T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t3: T).(\lambda (H1:
-(pr1 t2 t3)).(\lambda (_: ((\forall (t: T).(\forall (u: T).((pr1 t u) \to
-(\forall (k: K).(pr1 (THead k t2 t) (THead k t3 u)))))))).(\lambda (t:
-T).(\lambda (u: T).(\lambda (H3: (pr1 t u)).(\lambda (k: K).(let TMP_7 \def
-(\lambda (t0: T).(\lambda (t4: T).(let TMP_5 \def (THead k t1 t0) in (let
-TMP_6 \def (THead k t3 t4) in (pr1 TMP_5 TMP_6))))) in (let TMP_9 \def
-(\lambda (t0: T).(let TMP_8 \def (pr1_sing t2 t1 H0 t3 H1) in (pr1_head_1 t1
-t3 TMP_8 t0 k))) in (let TMP_15 \def (\lambda (t0: T).(\lambda (t4:
-T).(\lambda (H4: (pr0 t4 t0)).(\lambda (t5: T).(\lambda (_: (pr1 t0
-t5)).(\lambda (H6: (pr1 (THead k t1 t0) (THead k t3 t5))).(let TMP_10 \def
-(THead k t1 t0) in (let TMP_11 \def (THead k t1 t4) in (let TMP_12 \def
-(pr0_refl t1) in (let TMP_13 \def (pr0_comp t1 t1 TMP_12 t4 t0 H4 k) in (let
-TMP_14 \def (THead k t3 t5) in (pr1_sing TMP_10 TMP_11 TMP_13 TMP_14
-H6)))))))))))) in (pr1_ind TMP_7 TMP_9 TMP_15 t u H3)))))))))))))) in
-(pr1_ind TMP_3 TMP_4 TMP_16 v w H)))))).
+ \lambda (v: T).(\lambda (w: T).(\lambda (H: (pr1 v w)).(pr1_ind (\lambda (t:
+T).(\lambda (t0: T).(\forall (t1: T).(\forall (u: T).((pr1 t1 u) \to (\forall
+(k: K).(pr1 (THead k t t1) (THead k t0 u)))))))) (\lambda (t: T).(\lambda
+(t0: T).(\lambda (u: T).(\lambda (H0: (pr1 t0 u)).(\lambda (k: K).(pr1_head_2
+t0 u H0 t k)))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1
+t2)).(\lambda (t3: T).(\lambda (H1: (pr1 t2 t3)).(\lambda (_: ((\forall (t:
+T).(\forall (u: T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k t2 t) (THead
+k t3 u)))))))).(\lambda (t: T).(\lambda (u: T).(\lambda (H3: (pr1 t
+u)).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t4: T).(pr1 (THead k
+t1 t0) (THead k t3 t4)))) (\lambda (t0: T).(pr1_head_1 t1 t3 (pr1_sing t2 t1
+H0 t3 H1) t0 k)) (\lambda (t0: T).(\lambda (t4: T).(\lambda (H4: (pr0 t4
+t0)).(\lambda (t5: T).(\lambda (_: (pr1 t0 t5)).(\lambda (H6: (pr1 (THead k
+t1 t0) (THead k t3 t5))).(pr1_sing (THead k t1 t0) (THead k t1 t4) (pr0_comp
+t1 t1 (pr0_refl t1) t4 t0 H4 k) (THead k t3 t5) H6))))))) t u H3))))))))))) v
+w H))).
-theorem pr1_eta:
+lemma pr1_eta:
\forall (w: T).(\forall (u: T).(let t \def (THead (Bind Abst) w u) in
(\forall (v: T).((pr1 v w) \to (pr1 (THead (Bind Abst) v (THead (Flat Appl)
(TLRef O) (lift (S O) O t))) t)))))
\def
- \lambda (w: T).(\lambda (u: T).(let TMP_1 \def (Bind Abst) in (let t \def
-(THead TMP_1 w u) in (\lambda (v: T).(\lambda (H: (pr1 v w)).(let TMP_2 \def
-(Bind Abst) in (let TMP_3 \def (S O) in (let TMP_4 \def (lift TMP_3 O w) in
-(let TMP_5 \def (S O) in (let TMP_6 \def (S O) in (let TMP_7 \def (lift TMP_5
-TMP_6 u) in (let TMP_8 \def (THead TMP_2 TMP_4 TMP_7) in (let TMP_16 \def
-(\lambda (t0: T).(let TMP_9 \def (Bind Abst) in (let TMP_10 \def (Flat Appl)
-in (let TMP_11 \def (TLRef O) in (let TMP_12 \def (THead TMP_10 TMP_11 t0) in
-(let TMP_13 \def (THead TMP_9 v TMP_12) in (let TMP_14 \def (Bind Abst) in
-(let TMP_15 \def (THead TMP_14 w u) in (pr1 TMP_13 TMP_15))))))))) in (let
-TMP_17 \def (Flat Appl) in (let TMP_18 \def (TLRef O) in (let TMP_19 \def
-(Bind Abst) in (let TMP_20 \def (S O) in (let TMP_21 \def (lift TMP_20 O w)
-in (let TMP_22 \def (S O) in (let TMP_23 \def (S O) in (let TMP_24 \def (lift
-TMP_22 TMP_23 u) in (let TMP_25 \def (THead TMP_19 TMP_21 TMP_24) in (let
-TMP_26 \def (THead TMP_17 TMP_18 TMP_25) in (let TMP_27 \def (Bind Abbr) in
-(let TMP_28 \def (TLRef O) in (let TMP_29 \def (S O) in (let TMP_30 \def (S
-O) in (let TMP_31 \def (lift TMP_29 TMP_30 u) in (let TMP_32 \def (THead
-TMP_27 TMP_28 TMP_31) in (let TMP_33 \def (Flat Appl) in (let TMP_34 \def
-(TLRef O) in (let TMP_35 \def (Bind Abst) in (let TMP_36 \def (S O) in (let
-TMP_37 \def (lift TMP_36 O w) in (let TMP_38 \def (S O) in (let TMP_39 \def
-(S O) in (let TMP_40 \def (lift TMP_38 TMP_39 u) in (let TMP_41 \def (THead
-TMP_35 TMP_37 TMP_40) in (let TMP_42 \def (THead TMP_33 TMP_34 TMP_41) in
-(let TMP_43 \def (S O) in (let TMP_44 \def (lift TMP_43 O w) in (let TMP_45
-\def (TLRef O) in (let TMP_46 \def (TLRef O) in (let TMP_47 \def (TLRef O) in
-(let TMP_48 \def (pr0_refl TMP_47) in (let TMP_49 \def (S O) in (let TMP_50
-\def (S O) in (let TMP_51 \def (lift TMP_49 TMP_50 u) in (let TMP_52 \def (S
-O) in (let TMP_53 \def (S O) in (let TMP_54 \def (lift TMP_52 TMP_53 u) in
-(let TMP_55 \def (S O) in (let TMP_56 \def (S O) in (let TMP_57 \def (lift
-TMP_55 TMP_56 u) in (let TMP_58 \def (pr0_refl TMP_57) in (let TMP_59 \def
-(pr0_beta TMP_44 TMP_45 TMP_46 TMP_48 TMP_51 TMP_54 TMP_58) in (let TMP_60
-\def (Bind Abbr) in (let TMP_61 \def (TLRef O) in (let TMP_62 \def (S O) in
-(let TMP_63 \def (lift TMP_62 O u) in (let TMP_64 \def (THead TMP_60 TMP_61
-TMP_63) in (let TMP_65 \def (Bind Abbr) in (let TMP_66 \def (TLRef O) in (let
-TMP_67 \def (S O) in (let TMP_68 \def (S O) in (let TMP_69 \def (lift TMP_67
-TMP_68 u) in (let TMP_70 \def (THead TMP_65 TMP_66 TMP_69) in (let TMP_71
-\def (TLRef O) in (let TMP_72 \def (TLRef O) in (let TMP_73 \def (TLRef O) in
-(let TMP_74 \def (pr0_refl TMP_73) in (let TMP_75 \def (S O) in (let TMP_76
-\def (S O) in (let TMP_77 \def (lift TMP_75 TMP_76 u) in (let TMP_78 \def (S
-O) in (let TMP_79 \def (S O) in (let TMP_80 \def (lift TMP_78 TMP_79 u) in
-(let TMP_81 \def (S O) in (let TMP_82 \def (S O) in (let TMP_83 \def (lift
-TMP_81 TMP_82 u) in (let TMP_84 \def (pr0_refl TMP_83) in (let TMP_85 \def (S
-O) in (let TMP_86 \def (lift TMP_85 O u) in (let TMP_87 \def (le_O_n O) in
-(let TMP_88 \def (subst1_lift_S u O O TMP_87) in (let TMP_89 \def (pr0_delta1
-TMP_71 TMP_72 TMP_74 TMP_77 TMP_80 TMP_84 TMP_86 TMP_88) in (let TMP_90 \def
-(Bind Abbr) in (let TMP_91 \def (TLRef O) in (let TMP_92 \def (S O) in (let
-TMP_93 \def (lift TMP_92 O u) in (let TMP_94 \def (THead TMP_90 TMP_91
-TMP_93) in (let TMP_95 \def (pr0_refl u) in (let TMP_96 \def (TLRef O) in
-(let TMP_97 \def (pr0_zeta Abbr not_abbr_abst u u TMP_95 TMP_96) in (let
-TMP_98 \def (pr1_pr0 TMP_94 u TMP_97) in (let TMP_99 \def (pr1_sing TMP_64
-TMP_70 TMP_89 u TMP_98) in (let TMP_100 \def (pr1_sing TMP_32 TMP_42 TMP_59 u
-TMP_99) in (let TMP_101 \def (Bind Abst) in (let TMP_102 \def (pr1_comp v w H
-TMP_26 u TMP_100 TMP_101) in (let TMP_103 \def (S O) in (let TMP_104 \def
-(Bind Abst) in (let TMP_105 \def (THead TMP_104 w u) in (let TMP_106 \def
-(lift TMP_103 O TMP_105) in (let TMP_107 \def (S O) in (let TMP_108 \def
-(lift_bind Abst w u TMP_107 O) in (eq_ind_r T TMP_8 TMP_16 TMP_102 TMP_106
-TMP_108)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
-))))))))))))))))))))))))))))))))))).
+ \lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind Abst) w u) in
+(\lambda (v: T).(\lambda (H: (pr1 v w)).(eq_ind_r T (THead (Bind Abst) (lift
+(S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr1 (THead (Bind Abst) v
+(THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w u))) (pr1_comp v w H
+(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O)
+(S O) u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u))
+(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O)
+(S O) u))) (pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef
+O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O)
+u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind
+Abbr) (TLRef O) (lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O)
+(pr0_refl (TLRef O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl
+(lift (S O) (S O) u)) (lift (S O) O u) (subst1_lift_S u O O (le_O_n O))) u
+(pr1_pr0 (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr
+not_abbr_abst u u (pr0_refl u) (TLRef O))))) (Bind Abst)) (lift (S O) O
+(THead (Bind Abst) w u)) (lift_bind Abst w u (S O) O)))))).
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