include "basic_1/pr3/pr3.ma".
-theorem clear_pc3_trans:
+lemma clear_pc3_trans:
\forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pc3 c2 t1 t2) \to
(\forall (c1: C).((clear c1 c2) \to (pc3 c1 t1 t2))))))
\def
\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c2 t1
-t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def H in (let
-TMP_1 \def (\lambda (t: T).(pr3 c2 t1 t)) in (let TMP_2 \def (\lambda (t:
-T).(pr3 c2 t2 t)) in (let TMP_3 \def (pc3 c1 t1 t2) in (let TMP_8 \def
-(\lambda (x: T).(\lambda (H2: (pr3 c2 t1 x)).(\lambda (H3: (pr3 c2 t2
-x)).(let TMP_4 \def (\lambda (t: T).(pr3 c1 t1 t)) in (let TMP_5 \def
-(\lambda (t: T).(pr3 c1 t2 t)) in (let TMP_6 \def (clear_pr3_trans c2 t1 x H2
-c1 H0) in (let TMP_7 \def (clear_pr3_trans c2 t2 x H3 c1 H0) in (ex_intro2 T
-TMP_4 TMP_5 x TMP_6 TMP_7)))))))) in (ex2_ind T TMP_1 TMP_2 TMP_3 TMP_8
-H1))))))))))).
+t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def H in (ex2_ind
+T (\lambda (t: T).(pr3 c2 t1 t)) (\lambda (t: T).(pr3 c2 t2 t)) (pc3 c1 t1
+t2) (\lambda (x: T).(\lambda (H2: (pr3 c2 t1 x)).(\lambda (H3: (pr3 c2 t2
+x)).(ex_intro2 T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2
+t)) x (clear_pr3_trans c2 t1 x H2 c1 H0) (clear_pr3_trans c2 t2 x H3 c1
+H0))))) H1))))))).
-theorem pc3_pr2_r:
+lemma pc3_pr2_r:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pc3 c
t1 t2))))
\def
\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
-t2)).(let TMP_1 \def (\lambda (t: T).(pr3 c t1 t)) in (let TMP_2 \def
-(\lambda (t: T).(pr3 c t2 t)) in (let TMP_3 \def (pr3_pr2 c t1 t2 H) in (let
-TMP_4 \def (pr3_refl c t2) in (ex_intro2 T TMP_1 TMP_2 t2 TMP_3 TMP_4)))))))).
+t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
+t2 (pr3_pr2 c t1 t2 H) (pr3_refl c t2))))).
-theorem pc3_pr2_x:
+lemma pc3_pr2_x:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t2 t1) \to (pc3 c
t1 t2))))
\def
\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t2
-t1)).(let TMP_1 \def (\lambda (t: T).(pr3 c t1 t)) in (let TMP_2 \def
-(\lambda (t: T).(pr3 c t2 t)) in (let TMP_3 \def (pr3_refl c t1) in (let
-TMP_4 \def (pr3_pr2 c t2 t1 H) in (ex_intro2 T TMP_1 TMP_2 t1 TMP_3
-TMP_4)))))))).
+t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
+t1 (pr3_refl c t1) (pr3_pr2 c t2 t1 H))))).
-theorem pc3_pr3_r:
+lemma pc3_pr3_r:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (pc3 c
t1 t2))))
\def
\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
-t2)).(let TMP_1 \def (\lambda (t: T).(pr3 c t1 t)) in (let TMP_2 \def
-(\lambda (t: T).(pr3 c t2 t)) in (let TMP_3 \def (pr3_refl c t2) in
-(ex_intro2 T TMP_1 TMP_2 t2 H TMP_3))))))).
+t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
+t2 H (pr3_refl c t2))))).
-theorem pc3_pr3_x:
+lemma pc3_pr3_x:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t2 t1) \to (pc3 c
t1 t2))))
\def
\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t2
-t1)).(let TMP_1 \def (\lambda (t: T).(pr3 c t1 t)) in (let TMP_2 \def
-(\lambda (t: T).(pr3 c t2 t)) in (let TMP_3 \def (pr3_refl c t1) in
-(ex_intro2 T TMP_1 TMP_2 t1 TMP_3 H))))))).
+t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
+t1 (pr3_refl c t1) H)))).
-theorem pc3_pr3_t:
+lemma pc3_pr3_t:
\forall (c: C).(\forall (t1: T).(\forall (t0: T).((pr3 c t1 t0) \to (\forall
(t2: T).((pr3 c t2 t0) \to (pc3 c t1 t2))))))
\def
\lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H: (pr3 c t1
-t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t2 t0)).(let TMP_1 \def (\lambda
-(t: T).(pr3 c t1 t)) in (let TMP_2 \def (\lambda (t: T).(pr3 c t2 t)) in
-(ex_intro2 T TMP_1 TMP_2 t0 H H0)))))))).
+t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t2 t0)).(ex_intro2 T (\lambda (t:
+T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) t0 H H0)))))).
-theorem pc3_refl:
+lemma pc3_refl:
\forall (c: C).(\forall (t: T).(pc3 c t t))
\def
- \lambda (c: C).(\lambda (t: T).(let TMP_1 \def (\lambda (t0: T).(pr3 c t
-t0)) in (let TMP_2 \def (\lambda (t0: T).(pr3 c t t0)) in (let TMP_3 \def
-(pr3_refl c t) in (let TMP_4 \def (pr3_refl c t) in (ex_intro2 T TMP_1 TMP_2
-t TMP_3 TMP_4)))))).
+ \lambda (c: C).(\lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr3 c t t0))
+(\lambda (t0: T).(pr3 c t t0)) t (pr3_refl c t) (pr3_refl c t))).
-theorem pc3_s:
+lemma pc3_s:
\forall (c: C).(\forall (t2: T).(\forall (t1: T).((pc3 c t1 t2) \to (pc3 c
t2 t1))))
\def
\lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc3 c t1
-t2)).(let H0 \def H in (let TMP_1 \def (\lambda (t: T).(pr3 c t1 t)) in (let
-TMP_2 \def (\lambda (t: T).(pr3 c t2 t)) in (let TMP_3 \def (pc3 c t2 t1) in
-(let TMP_6 \def (\lambda (x: T).(\lambda (H1: (pr3 c t1 x)).(\lambda (H2:
-(pr3 c t2 x)).(let TMP_4 \def (\lambda (t: T).(pr3 c t2 t)) in (let TMP_5
-\def (\lambda (t: T).(pr3 c t1 t)) in (ex_intro2 T TMP_4 TMP_5 x H2 H1))))))
-in (ex2_ind T TMP_1 TMP_2 TMP_3 TMP_6 H0))))))))).
+t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
+T).(pr3 c t2 t)) (pc3 c t2 t1) (\lambda (x: T).(\lambda (H1: (pr3 c t1
+x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t2 t))
+(\lambda (t: T).(pr3 c t1 t)) x H2 H1)))) H0))))).
-theorem pc3_thin_dx:
+lemma pc3_thin_dx:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall
(u: T).(\forall (f: F).(pc3 c (THead (Flat f) u t1) (THead (Flat f) u
t2)))))))
\def
\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
-t2)).(\lambda (u: T).(\lambda (f: F).(let H0 \def H in (let TMP_1 \def
-(\lambda (t: T).(pr3 c t1 t)) in (let TMP_2 \def (\lambda (t: T).(pr3 c t2
-t)) in (let TMP_3 \def (Flat f) in (let TMP_4 \def (THead TMP_3 u t1) in (let
-TMP_5 \def (Flat f) in (let TMP_6 \def (THead TMP_5 u t2) in (let TMP_7 \def
-(pc3 c TMP_4 TMP_6) in (let TMP_18 \def (\lambda (x: T).(\lambda (H1: (pr3 c
-t1 x)).(\lambda (H2: (pr3 c t2 x)).(let TMP_10 \def (\lambda (t: T).(let
-TMP_8 \def (Flat f) in (let TMP_9 \def (THead TMP_8 u t1) in (pr3 c TMP_9
-t)))) in (let TMP_13 \def (\lambda (t: T).(let TMP_11 \def (Flat f) in (let
-TMP_12 \def (THead TMP_11 u t2) in (pr3 c TMP_12 t)))) in (let TMP_14 \def
-(Flat f) in (let TMP_15 \def (THead TMP_14 u x) in (let TMP_16 \def
-(pr3_thin_dx c t1 x H1 u f) in (let TMP_17 \def (pr3_thin_dx c t2 x H2 u f)
-in (ex_intro2 T TMP_10 TMP_13 TMP_15 TMP_16 TMP_17)))))))))) in (ex2_ind T
-TMP_1 TMP_2 TMP_7 TMP_18 H0))))))))))))))).
+t2)).(\lambda (u: T).(\lambda (f: F).(let H0 \def H in (ex2_ind T (\lambda
+(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 c (THead (Flat f) u
+t1) (THead (Flat f) u t2)) (\lambda (x: T).(\lambda (H1: (pr3 c t1
+x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c (THead
+(Flat f) u t1) t)) (\lambda (t: T).(pr3 c (THead (Flat f) u t2) t)) (THead
+(Flat f) u x) (pr3_thin_dx c t1 x H1 u f) (pr3_thin_dx c t2 x H2 u f)))))
+H0))))))).
-theorem pc3_head_1:
+lemma pc3_head_1:
\forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall
(k: K).(\forall (t: T).(pc3 c (THead k u1 t) (THead k u2 t)))))))
\def
\lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1
-u2)).(\lambda (k: K).(\lambda (t: T).(let H0 \def H in (let TMP_1 \def
-(\lambda (t0: T).(pr3 c u1 t0)) in (let TMP_2 \def (\lambda (t0: T).(pr3 c u2
-t0)) in (let TMP_3 \def (THead k u1 t) in (let TMP_4 \def (THead k u2 t) in
-(let TMP_5 \def (pc3 c TMP_3 TMP_4) in (let TMP_17 \def (\lambda (x:
-T).(\lambda (H1: (pr3 c u1 x)).(\lambda (H2: (pr3 c u2 x)).(let TMP_7 \def
-(\lambda (t0: T).(let TMP_6 \def (THead k u1 t) in (pr3 c TMP_6 t0))) in (let
-TMP_9 \def (\lambda (t0: T).(let TMP_8 \def (THead k u2 t) in (pr3 c TMP_8
-t0))) in (let TMP_10 \def (THead k x t) in (let TMP_11 \def (CHead c k x) in
-(let TMP_12 \def (pr3_refl TMP_11 t) in (let TMP_13 \def (pr3_head_12 c u1 x
-H1 k t t TMP_12) in (let TMP_14 \def (CHead c k x) in (let TMP_15 \def
-(pr3_refl TMP_14 t) in (let TMP_16 \def (pr3_head_12 c u2 x H2 k t t TMP_15)
-in (ex_intro2 T TMP_7 TMP_9 TMP_10 TMP_13 TMP_16))))))))))))) in (ex2_ind T
-TMP_1 TMP_2 TMP_5 TMP_17 H0))))))))))))).
+u2)).(\lambda (k: K).(\lambda (t: T).(let H0 \def H in (ex2_ind T (\lambda
+(t0: T).(pr3 c u1 t0)) (\lambda (t0: T).(pr3 c u2 t0)) (pc3 c (THead k u1 t)
+(THead k u2 t)) (\lambda (x: T).(\lambda (H1: (pr3 c u1 x)).(\lambda (H2:
+(pr3 c u2 x)).(ex_intro2 T (\lambda (t0: T).(pr3 c (THead k u1 t) t0))
+(\lambda (t0: T).(pr3 c (THead k u2 t) t0)) (THead k x t) (pr3_head_12 c u1 x
+H1 k t t (pr3_refl (CHead c k x) t)) (pr3_head_12 c u2 x H2 k t t (pr3_refl
+(CHead c k x) t)))))) H0))))))).
-theorem pc3_head_2:
+lemma pc3_head_2:
\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall
(k: K).((pc3 (CHead c k u) t1 t2) \to (pc3 c (THead k u t1) (THead k u
t2)))))))
\def
\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(k: K).(\lambda (H: (pc3 (CHead c k u) t1 t2)).(let H0 \def H in (let TMP_2
-\def (\lambda (t: T).(let TMP_1 \def (CHead c k u) in (pr3 TMP_1 t1 t))) in
-(let TMP_4 \def (\lambda (t: T).(let TMP_3 \def (CHead c k u) in (pr3 TMP_3
-t2 t))) in (let TMP_5 \def (THead k u t1) in (let TMP_6 \def (THead k u t2)
-in (let TMP_7 \def (pc3 c TMP_5 TMP_6) in (let TMP_17 \def (\lambda (x:
-T).(\lambda (H1: (pr3 (CHead c k u) t1 x)).(\lambda (H2: (pr3 (CHead c k u)
-t2 x)).(let TMP_9 \def (\lambda (t: T).(let TMP_8 \def (THead k u t1) in (pr3
-c TMP_8 t))) in (let TMP_11 \def (\lambda (t: T).(let TMP_10 \def (THead k u
-t2) in (pr3 c TMP_10 t))) in (let TMP_12 \def (THead k u x) in (let TMP_13
-\def (pr3_refl c u) in (let TMP_14 \def (pr3_head_12 c u u TMP_13 k t1 x H1)
-in (let TMP_15 \def (pr3_refl c u) in (let TMP_16 \def (pr3_head_12 c u u
-TMP_15 k t2 x H2) in (ex_intro2 T TMP_9 TMP_11 TMP_12 TMP_14
-TMP_16))))))))))) in (ex2_ind T TMP_2 TMP_4 TMP_7 TMP_17 H0))))))))))))).
+(k: K).(\lambda (H: (pc3 (CHead c k u) t1 t2)).(let H0 \def H in (ex2_ind T
+(\lambda (t: T).(pr3 (CHead c k u) t1 t)) (\lambda (t: T).(pr3 (CHead c k u)
+t2 t)) (pc3 c (THead k u t1) (THead k u t2)) (\lambda (x: T).(\lambda (H1:
+(pr3 (CHead c k u) t1 x)).(\lambda (H2: (pr3 (CHead c k u) t2 x)).(ex_intro2
+T (\lambda (t: T).(pr3 c (THead k u t1) t)) (\lambda (t: T).(pr3 c (THead k u
+t2) t)) (THead k u x) (pr3_head_12 c u u (pr3_refl c u) k t1 x H1)
+(pr3_head_12 c u u (pr3_refl c u) k t2 x H2))))) H0))))))).
-theorem pc3_pr2_u:
+lemma pc3_pr2_u:
\forall (c: C).(\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall
(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
\def
\lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr2 c t1
-t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in (let
-TMP_1 \def (\lambda (t: T).(pr3 c t2 t)) in (let TMP_2 \def (\lambda (t:
-T).(pr3 c t3 t)) in (let TMP_3 \def (pc3 c t1 t3) in (let TMP_7 \def (\lambda
-(x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3 x)).(let TMP_4
-\def (\lambda (t: T).(pr3 c t1 t)) in (let TMP_5 \def (\lambda (t: T).(pr3 c
-t3 t)) in (let TMP_6 \def (pr3_sing c t2 t1 H x H2) in (ex_intro2 T TMP_4
-TMP_5 x TMP_6 H3))))))) in (ex2_ind T TMP_1 TMP_2 TMP_3 TMP_7 H1))))))))))).
+t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in
+(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c
+t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3
+x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t3 t))
+x (pr3_sing c t2 t1 H x H2) H3)))) H1))))))).
theorem pc3_t:
\forall (t2: T).(\forall (c: C).(\forall (t1: T).((pc3 c t1 t2) \to (\forall
(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
\def
\lambda (t2: T).(\lambda (c: C).(\lambda (t1: T).(\lambda (H: (pc3 c t1
-t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in (let
-TMP_1 \def (\lambda (t: T).(pr3 c t2 t)) in (let TMP_2 \def (\lambda (t:
-T).(pr3 c t3 t)) in (let TMP_3 \def (pc3 c t1 t3) in (let TMP_15 \def
-(\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3 x)).(let
-H4 \def H in (let TMP_4 \def (\lambda (t: T).(pr3 c t1 t)) in (let TMP_5 \def
-(\lambda (t: T).(pr3 c t2 t)) in (let TMP_6 \def (pc3 c t1 t3) in (let TMP_14
-\def (\lambda (x0: T).(\lambda (H5: (pr3 c t1 x0)).(\lambda (H6: (pr3 c t2
-x0)).(let TMP_7 \def (\lambda (t: T).(pr3 c x0 t)) in (let TMP_8 \def
-(\lambda (t: T).(pr3 c x t)) in (let TMP_9 \def (pc3 c t1 t3) in (let TMP_12
-\def (\lambda (x1: T).(\lambda (H7: (pr3 c x0 x1)).(\lambda (H8: (pr3 c x
-x1)).(let TMP_10 \def (pr3_t x0 t1 c H5 x1 H7) in (let TMP_11 \def (pr3_t x
-t3 c H3 x1 H8) in (pc3_pr3_t c t1 x1 TMP_10 t3 TMP_11)))))) in (let TMP_13
-\def (pr3_confluence c t2 x0 H6 x H2) in (ex2_ind T TMP_7 TMP_8 TMP_9 TMP_12
-TMP_13))))))))) in (ex2_ind T TMP_4 TMP_5 TMP_6 TMP_14 H4))))))))) in
-(ex2_ind T TMP_1 TMP_2 TMP_3 TMP_15 H1))))))))))).
+t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in
+(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c
+t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3
+x)).(let H4 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
+T).(pr3 c t2 t)) (pc3 c t1 t3) (\lambda (x0: T).(\lambda (H5: (pr3 c t1
+x0)).(\lambda (H6: (pr3 c t2 x0)).(ex2_ind T (\lambda (t: T).(pr3 c x0 t))
+(\lambda (t: T).(pr3 c x t)) (pc3 c t1 t3) (\lambda (x1: T).(\lambda (H7:
+(pr3 c x0 x1)).(\lambda (H8: (pr3 c x x1)).(pc3_pr3_t c t1 x1 (pr3_t x0 t1 c
+H5 x1 H7) t3 (pr3_t x t3 c H3 x1 H8))))) (pr3_confluence c t2 x0 H6 x H2)))))
+H4))))) H1))))))).
-theorem pc3_pr2_u2:
+lemma pc3_pr2_u2:
\forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall
(t2: T).((pc3 c t0 t2) \to (pc3 c t1 t2))))))
\def
\lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0
-t1)).(\lambda (t2: T).(\lambda (H0: (pc3 c t0 t2)).(let TMP_1 \def (pc3_pr2_x
-c t1 t0 H) in (pc3_t t0 c t1 TMP_1 t2 H0))))))).
+t1)).(\lambda (t2: T).(\lambda (H0: (pc3 c t0 t2)).(pc3_t t0 c t1 (pc3_pr2_x
+c t1 t0 H) t2 H0)))))).
-theorem pc3_pr3_conf:
+lemma pc3_pr3_conf:
\forall (c: C).(\forall (t: T).(\forall (t1: T).((pc3 c t t1) \to (\forall
(t2: T).((pr3 c t t2) \to (pc3 c t2 t1))))))
\def
\lambda (c: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pc3 c t
-t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(let TMP_1 \def (pc3_pr3_x
-c t2 t H0) in (pc3_t t c t2 TMP_1 t1 H))))))).
+t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(pc3_t t c t2 (pc3_pr3_x c
+t2 t H0) t1 H)))))).
theorem pc3_head_12:
\forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall
\def
\lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1
u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3
-(CHead c k u2) t1 t2)).(let TMP_1 \def (THead k u2 t1) in (let TMP_2 \def
-(THead k u1 t1) in (let TMP_3 \def (pc3_head_1 c u1 u2 H k t1) in (let TMP_4
-\def (THead k u2 t2) in (let TMP_5 \def (pc3_head_2 c u2 t1 t2 k H0) in
-(pc3_t TMP_1 c TMP_2 TMP_3 TMP_4 TMP_5))))))))))))).
+(CHead c k u2) t1 t2)).(pc3_t (THead k u2 t1) c (THead k u1 t1) (pc3_head_1 c
+u1 u2 H k t1) (THead k u2 t2) (pc3_head_2 c u2 t1 t2 k H0))))))))).
theorem pc3_head_21:
\forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall
\def
\lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1
u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3
-(CHead c k u1) t1 t2)).(let TMP_1 \def (THead k u1 t2) in (let TMP_2 \def
-(THead k u1 t1) in (let TMP_3 \def (pc3_head_2 c u1 t1 t2 k H0) in (let TMP_4
-\def (THead k u2 t2) in (let TMP_5 \def (pc3_head_1 c u1 u2 H k t2) in (pc3_t
-TMP_1 c TMP_2 TMP_3 TMP_4 TMP_5))))))))))))).
+(CHead c k u1) t1 t2)).(pc3_t (THead k u1 t2) c (THead k u1 t1) (pc3_head_2 c
+u1 t1 t2 k H0) (THead k u2 t2) (pc3_head_1 c u1 u2 H k t2))))))))).
-theorem pc3_pr0_pr2_t:
+lemma pc3_pr0_pr2_t:
\forall (u1: T).(\forall (u2: T).((pr0 u2 u1) \to (\forall (c: C).(\forall
(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3
(CHead c k u1) t1 t2))))))))
\def
\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u2 u1)).(\lambda (c:
C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2
-(CHead c k u2) t1 t2)).(let TMP_1 \def (CHead c k u2) in (let TMP_2 \def
-(\lambda (c0: C).(pr2 c0 t1 t2)) in (let TMP_4 \def (\lambda (_: C).(let
-TMP_3 \def (CHead c k u1) in (pc3 TMP_3 t1 t2))) in (let TMP_125 \def
-(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(let TMP_6 \def (\lambda (c0:
-C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c k u2)) \to (let TMP_5
-\def (CHead c k u1) in (pc3 TMP_5 t t0)))))) in (let TMP_12 \def (\lambda
-(c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda
-(H3: (eq C c0 (CHead c k u2))).(let TMP_7 \def (\lambda (e: C).e) in (let
-TMP_8 \def (CHead c k u2) in (let H4 \def (f_equal C C TMP_7 c0 TMP_8 H3) in
-(let TMP_9 \def (CHead c k u1) in (let TMP_10 \def (CHead c k u1) in (let
-TMP_11 \def (pr2_free TMP_10 t3 t4 H2) in (pc3_pr2_r TMP_9 t3 t4
-TMP_11)))))))))))) in (let TMP_124 \def (\lambda (c0: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind
-Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3
-t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: (eq C c0
-(CHead c k u2))).(let TMP_13 \def (\lambda (e: C).e) in (let TMP_14 \def
-(CHead c k u2) in (let H6 \def (f_equal C C TMP_13 c0 TMP_14 H5) in (let
-TMP_17 \def (\lambda (c1: C).(let TMP_15 \def (Bind Abbr) in (let TMP_16 \def
-(CHead d TMP_15 u) in (getl i c1 TMP_16)))) in (let TMP_18 \def (CHead c k
-u2) in (let H7 \def (eq_ind C c0 TMP_17 H2 TMP_18 H6) in (let TMP_20 \def
-(\lambda (n: nat).((getl n (CHead c k u2) (CHead d (Bind Abbr) u)) \to
-((subst0 n u t4 t) \to (let TMP_19 \def (CHead c k u1) in (pc3 TMP_19 t3
-t))))) in (let TMP_99 \def (\lambda (H8: (getl O (CHead c k u2) (CHead d
-(Bind Abbr) u))).(\lambda (H9: (subst0 O u t4 t)).(let TMP_22 \def (\lambda
-(k0: K).((clear (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to (let TMP_21 \def
-(CHead c k0 u1) in (pc3 TMP_21 t3 t)))) in (let TMP_76 \def (\lambda (b:
-B).(\lambda (H10: (clear (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(let
-TMP_23 \def (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead
-c1 _ _) \Rightarrow c1])) in (let TMP_24 \def (Bind Abbr) in (let TMP_25 \def
-(CHead d TMP_24 u) in (let TMP_26 \def (Bind b) in (let TMP_27 \def (CHead c
-TMP_26 u2) in (let TMP_28 \def (Bind Abbr) in (let TMP_29 \def (CHead d
-TMP_28 u) in (let TMP_30 \def (clear_gen_bind b c TMP_29 u2 H10) in (let H11
-\def (f_equal C C TMP_23 TMP_25 TMP_27 TMP_30) in (let TMP_31 \def (\lambda
-(e: C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow
-(match k0 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) in
-(let TMP_32 \def (Bind Abbr) in (let TMP_33 \def (CHead d TMP_32 u) in (let
-TMP_34 \def (Bind b) in (let TMP_35 \def (CHead c TMP_34 u2) in (let TMP_36
-\def (Bind Abbr) in (let TMP_37 \def (CHead d TMP_36 u) in (let TMP_38 \def
-(clear_gen_bind b c TMP_37 u2 H10) in (let H12 \def (f_equal C B TMP_31
-TMP_33 TMP_35 TMP_38) in (let TMP_39 \def (\lambda (e: C).(match e with
-[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) in (let TMP_40
-\def (Bind Abbr) in (let TMP_41 \def (CHead d TMP_40 u) in (let TMP_42 \def
-(Bind b) in (let TMP_43 \def (CHead c TMP_42 u2) in (let TMP_44 \def (Bind
-Abbr) in (let TMP_45 \def (CHead d TMP_44 u) in (let TMP_46 \def
-(clear_gen_bind b c TMP_45 u2 H10) in (let H13 \def (f_equal C T TMP_39
-TMP_41 TMP_43 TMP_46) in (let TMP_74 \def (\lambda (H14: (eq B Abbr
-b)).(\lambda (_: (eq C d c)).(let TMP_47 \def (\lambda (t0: T).(subst0 O t0
-t4 t)) in (let H16 \def (eq_ind T u TMP_47 H9 u2 H13) in (let TMP_50 \def
-(\lambda (b0: B).(let TMP_48 \def (Bind b0) in (let TMP_49 \def (CHead c
-TMP_48 u1) in (pc3 TMP_49 t3 t)))) in (let TMP_51 \def (\lambda (t0:
-T).(subst0 O u1 t4 t0)) in (let TMP_52 \def (\lambda (t0: T).(pr0 t t0)) in
-(let TMP_53 \def (Bind Abbr) in (let TMP_54 \def (CHead c TMP_53 u1) in (let
-TMP_55 \def (pc3 TMP_54 t3 t) in (let TMP_71 \def (\lambda (x: T).(\lambda
-(H17: (subst0 O u1 t4 x)).(\lambda (H18: (pr0 t x)).(let TMP_56 \def (Bind
-Abbr) in (let TMP_57 \def (CHead c TMP_56 u1) in (let TMP_58 \def (Bind Abbr)
-in (let TMP_59 \def (CHead c TMP_58 u1) in (let TMP_60 \def (Bind Abbr) in
-(let TMP_61 \def (CHead c TMP_60 u1) in (let TMP_62 \def (getl_refl Abbr c
-u1) in (let TMP_63 \def (pr2_delta TMP_61 c u1 O TMP_62 t3 t4 H3 x H17) in
-(let TMP_64 \def (pr3_pr2 TMP_59 t3 x TMP_63) in (let TMP_65 \def (Bind Abbr)
-in (let TMP_66 \def (CHead c TMP_65 u1) in (let TMP_67 \def (Bind Abbr) in
-(let TMP_68 \def (CHead c TMP_67 u1) in (let TMP_69 \def (pr2_free TMP_68 t x
-H18) in (let TMP_70 \def (pr3_pr2 TMP_66 t x TMP_69) in (pc3_pr3_t TMP_57 t3
-x TMP_64 t TMP_70))))))))))))))))))) in (let TMP_72 \def (pr0_subst0_fwd u2
-t4 t O H16 u1 H) in (let TMP_73 \def (ex2_ind T TMP_51 TMP_52 TMP_55 TMP_71
-TMP_72) in (eq_ind B Abbr TMP_50 TMP_73 b H14)))))))))))))) in (let TMP_75
-\def (TMP_74 H12) in (TMP_75 H11)))))))))))))))))))))))))))))))) in (let
-TMP_94 \def (\lambda (f: F).(\lambda (H10: (clear (CHead c (Flat f) u2)
-(CHead d (Bind Abbr) u))).(let TMP_77 \def (Bind Abbr) in (let TMP_78 \def
-(CHead d TMP_77 u) in (let TMP_79 \def (Bind Abbr) in (let TMP_80 \def (CHead
-d TMP_79 u) in (let TMP_81 \def (Bind Abbr) in (let TMP_82 \def (CHead d
-TMP_81 u) in (let TMP_83 \def (getl_refl Abbr d u) in (let TMP_84 \def
-(pr2_delta TMP_82 d u O TMP_83 t3 t4 H3 t H9) in (let TMP_85 \def (pc3_pr2_r
-TMP_80 t3 t TMP_84) in (let TMP_86 \def (Flat f) in (let TMP_87 \def (CHead c
-TMP_86 u1) in (let TMP_88 \def (Bind Abbr) in (let TMP_89 \def (CHead d
-TMP_88 u) in (let TMP_90 \def (Bind Abbr) in (let TMP_91 \def (CHead d TMP_90
-u) in (let TMP_92 \def (clear_gen_flat f c TMP_91 u2 H10) in (let TMP_93 \def
-(clear_flat c TMP_89 TMP_92 f u1) in (clear_pc3_trans TMP_78 t3 t TMP_85
-TMP_87 TMP_93)))))))))))))))))))) in (let TMP_95 \def (CHead c k u2) in (let
-TMP_96 \def (Bind Abbr) in (let TMP_97 \def (CHead d TMP_96 u) in (let TMP_98
-\def (getl_gen_O TMP_95 TMP_97 H8) in (K_ind TMP_22 TMP_76 TMP_94 k
-TMP_98)))))))))) in (let TMP_123 \def (\lambda (i0: nat).(\lambda (IHi:
-(((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t)
-\to (pc3 (CHead c k u1) t3 t))))).(\lambda (H8: (getl (S i0) (CHead c k u2)
-(CHead d (Bind Abbr) u))).(\lambda (H9: (subst0 (S i0) u t4 t)).(let TMP_101
-\def (\lambda (k0: K).((((getl i0 (CHead c k0 u2) (CHead d (Bind Abbr) u))
-\to ((subst0 i0 u t4 t) \to (pc3 (CHead c k0 u1) t3 t)))) \to ((getl (r k0
-i0) c (CHead d (Bind Abbr) u)) \to (let TMP_100 \def (CHead c k0 u1) in (pc3
-TMP_100 t3 t))))) in (let TMP_112 \def (\lambda (b: B).(\lambda (_: (((getl
-i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) \to
-(pc3 (CHead c (Bind b) u1) t3 t))))).(\lambda (H10: (getl (r (Bind b) i0) c
-(CHead d (Bind Abbr) u))).(let TMP_102 \def (Bind b) in (let TMP_103 \def
-(CHead c TMP_102 u1) in (let TMP_104 \def (Bind b) in (let TMP_105 \def
-(CHead c TMP_104 u1) in (let TMP_106 \def (S i0) in (let TMP_107 \def (Bind
-b) in (let TMP_108 \def (Bind Abbr) in (let TMP_109 \def (CHead d TMP_108 u)
-in (let TMP_110 \def (getl_head TMP_107 i0 c TMP_109 H10 u1) in (let TMP_111
-\def (pr2_delta TMP_105 d u TMP_106 TMP_110 t3 t4 H3 t H9) in (pc3_pr2_r
-TMP_103 t3 t TMP_111)))))))))))))) in (let TMP_119 \def (\lambda (f:
-F).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to
-((subst0 i0 u t4 t) \to (pc3 (CHead c (Flat f) u1) t3 t))))).(\lambda (H10:
-(getl (r (Flat f) i0) c (CHead d (Bind Abbr) u))).(let TMP_113 \def (Flat f)
-in (let TMP_114 \def (CHead c TMP_113 u1) in (let TMP_115 \def (Flat f) in
-(let TMP_116 \def (r TMP_115 i0) in (let TMP_117 \def (pr2_delta c d u
-TMP_116 H10 t3 t4 H3 t H9) in (let TMP_118 \def (pr2_cflat c t3 t TMP_117 f
-u1) in (pc3_pr2_r TMP_114 t3 t TMP_118)))))))))) in (let TMP_120 \def (Bind
-Abbr) in (let TMP_121 \def (CHead d TMP_120 u) in (let TMP_122 \def
-(getl_gen_S k c TMP_121 u2 i0 H8) in (K_ind TMP_101 TMP_112 TMP_119 k IHi
-TMP_122))))))))))) in (nat_ind TMP_20 TMP_99 TMP_123 i H7
-H4))))))))))))))))))))) in (pr2_ind TMP_6 TMP_12 TMP_124 y t1 t2 H1)))))) in
-(insert_eq C TMP_1 TMP_2 TMP_4 TMP_125 H0)))))))))))).
+(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0
+t1 t2)) (\lambda (_: C).(pc3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda
+(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).((eq C c0 (CHead c k u2)) \to (pc3 (CHead c k u1) t t0))))) (\lambda (c0:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3:
+(eq C c0 (CHead c k u2))).(let H4 \def (f_equal C C (\lambda (e: C).e) c0
+(CHead c k u2) H3) in (pc3_pr2_r (CHead c k u1) t3 t4 (pr2_free (CHead c k
+u1) t3 t4 H2)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr)
+u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda
+(t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k
+u2))).(let H6 \def (f_equal C C (\lambda (e: C).e) c0 (CHead c k u2) H5) in
+(let H7 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr)
+u))) H2 (CHead c k u2) H6) in (nat_ind (\lambda (n: nat).((getl n (CHead c k
+u2) (CHead d (Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pc3 (CHead c k u1)
+t3 t)))) (\lambda (H8: (getl O (CHead c k u2) (CHead d (Bind Abbr)
+u))).(\lambda (H9: (subst0 O u t4 t)).(K_ind (\lambda (k0: K).((clear (CHead
+c k0 u2) (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t))) (\lambda
+(b: B).(\lambda (H10: (clear (CHead c (Bind b) u2) (CHead d (Bind Abbr)
+u))).(let H11 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _)
+\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u)
+(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H10)) in
+((let H12 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _)
+\Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u)
+(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H10)) in
+((let H13 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _)
+\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u)
+(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H10)) in
+(\lambda (H14: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H16 \def (eq_ind
+T u (\lambda (t0: T).(subst0 O t0 t4 t)) H9 u2 H13) in (eq_ind B Abbr
+(\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0:
+T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t t0)) (pc3 (CHead c (Bind
+Abbr) u1) t3 t) (\lambda (x: T).(\lambda (H17: (subst0 O u1 t4 x)).(\lambda
+(H18: (pr0 t x)).(pc3_pr3_t (CHead c (Bind Abbr) u1) t3 x (pr3_pr2 (CHead c
+(Bind Abbr) u1) t3 x (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl
+Abbr c u1) t3 t4 H3 x H17)) t (pr3_pr2 (CHead c (Bind Abbr) u1) t x (pr2_free
+(CHead c (Bind Abbr) u1) t x H18)))))) (pr0_subst0_fwd u2 t4 t O H16 u1 H)) b
+H14))))) H12)) H11)))) (\lambda (f: F).(\lambda (H10: (clear (CHead c (Flat
+f) u2) (CHead d (Bind Abbr) u))).(clear_pc3_trans (CHead d (Bind Abbr) u) t3
+t (pc3_pr2_r (CHead d (Bind Abbr) u) t3 t (pr2_delta (CHead d (Bind Abbr) u)
+d u O (getl_refl Abbr d u) t3 t4 H3 t H9)) (CHead c (Flat f) u1) (clear_flat
+c (CHead d (Bind Abbr) u) (clear_gen_flat f c (CHead d (Bind Abbr) u) u2 H10)
+f u1)))) k (getl_gen_O (CHead c k u2) (CHead d (Bind Abbr) u) H8)))) (\lambda
+(i0: nat).(\lambda (IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u))
+\to ((subst0 i0 u t4 t) \to (pc3 (CHead c k u1) t3 t))))).(\lambda (H8: (getl
+(S i0) (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H9: (subst0 (S i0)
+u t4 t)).(K_ind (\lambda (k0: K).((((getl i0 (CHead c k0 u2) (CHead d (Bind
+Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3 (CHead c k0 u1) t3 t)))) \to
+((getl (r k0 i0) c (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t))))
+(\lambda (b: B).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind
+Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3 (CHead c (Bind b) u1) t3
+t))))).(\lambda (H10: (getl (r (Bind b) i0) c (CHead d (Bind Abbr)
+u))).(pc3_pr2_r (CHead c (Bind b) u1) t3 t (pr2_delta (CHead c (Bind b) u1) d
+u (S i0) (getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) H10 u1) t3 t4 H3 t
+H9))))) (\lambda (f: F).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead
+d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3 (CHead c (Flat f) u1) t3
+t))))).(\lambda (H10: (getl (r (Flat f) i0) c (CHead d (Bind Abbr)
+u))).(pc3_pr2_r (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u
+(r (Flat f) i0) H10 t3 t4 H3 t H9) f u1))))) k IHi (getl_gen_S k c (CHead d
+(Bind Abbr) u) u2 i0 H8)))))) i H7 H4)))))))))))))) y t1 t2 H1))) H0)))))))).
-theorem pc3_pr2_pr2_t:
+lemma pc3_pr2_pr2_t:
\forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u2 u1) \to (\forall
(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3
(CHead c k u1) t1 t2))))))))
\def
\lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u2
-u1)).(let TMP_2 \def (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
-T).(\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t) t1
-t2) \to (let TMP_1 \def (CHead c0 k t0) in (pc3 TMP_1 t1 t2))))))))) in (let
-TMP_3 \def (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0:
-(pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H1:
-(pr2 (CHead c0 k t1) t0 t3)).(pc3_pr0_pr2_t t2 t1 H0 c0 t0 t3 k H1)))))))))
-in (let TMP_144 \def (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
-T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr)
-u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda
-(t: T).(\lambda (H2: (subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3:
-T).(\lambda (k: K).(\lambda (H3: (pr2 (CHead c0 k t1) t0 t3)).(let TMP_4 \def
-(CHead c0 k t1) in (let TMP_5 \def (\lambda (c1: C).(pr2 c1 t0 t3)) in (let
-TMP_7 \def (\lambda (_: C).(let TMP_6 \def (CHead c0 k t) in (pc3 TMP_6 t0
-t3))) in (let TMP_143 \def (\lambda (y: C).(\lambda (H4: (pr2 y t0 t3)).(let
-TMP_9 \def (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).((eq C c1
-(CHead c0 k t1)) \to (let TMP_8 \def (CHead c0 k t) in (pc3 TMP_8 t4 t5))))))
-in (let TMP_13 \def (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5:
-T).(\lambda (H5: (pr0 t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t1))).(let
-TMP_10 \def (CHead c0 k t) in (let TMP_11 \def (CHead c0 k t) in (let TMP_12
-\def (pr2_free TMP_11 t4 t5 H5) in (pc3_pr2_r TMP_10 t4 t5 TMP_12))))))))) in
-(let TMP_142 \def (\lambda (c1: C).(\lambda (d0: C).(\lambda (u0: T).(\lambda
-(i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0 (Bind Abbr) u0))).(\lambda (t4:
-T).(\lambda (t5: T).(\lambda (H6: (pr0 t4 t5)).(\lambda (t6: T).(\lambda (H7:
-(subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C c1 (CHead c0 k t1))).(let TMP_16
-\def (\lambda (c2: C).(let TMP_14 \def (Bind Abbr) in (let TMP_15 \def (CHead
-d0 TMP_14 u0) in (getl i0 c2 TMP_15)))) in (let TMP_17 \def (CHead c0 k t1)
-in (let H9 \def (eq_ind C c1 TMP_16 H5 TMP_17 H8) in (let TMP_19 \def
-(\lambda (n: nat).((getl n (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to
-((subst0 n u0 t5 t6) \to (let TMP_18 \def (CHead c0 k t) in (pc3 TMP_18 t4
-t6))))) in (let TMP_117 \def (\lambda (H10: (getl O (CHead c0 k t1) (CHead d0
-(Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(let TMP_21 \def
-(\lambda (k0: K).((clear (CHead c0 k0 t1) (CHead d0 (Bind Abbr) u0)) \to (let
-TMP_20 \def (CHead c0 k0 t) in (pc3 TMP_20 t4 t6)))) in (let TMP_94 \def
-(\lambda (b: B).(\lambda (H12: (clear (CHead c0 (Bind b) t1) (CHead d0 (Bind
-Abbr) u0))).(let TMP_22 \def (\lambda (e: C).(match e with [(CSort _)
-\Rightarrow d0 | (CHead c2 _ _) \Rightarrow c2])) in (let TMP_23 \def (Bind
-Abbr) in (let TMP_24 \def (CHead d0 TMP_23 u0) in (let TMP_25 \def (Bind b)
-in (let TMP_26 \def (CHead c0 TMP_25 t1) in (let TMP_27 \def (Bind Abbr) in
-(let TMP_28 \def (CHead d0 TMP_27 u0) in (let TMP_29 \def (clear_gen_bind b
-c0 TMP_28 t1 H12) in (let H13 \def (f_equal C C TMP_22 TMP_24 TMP_26 TMP_29)
-in (let TMP_30 \def (\lambda (e: C).(match e with [(CSort _) \Rightarrow Abbr
-| (CHead _ k0 _) \Rightarrow (match k0 with [(Bind b0) \Rightarrow b0 | (Flat
-_) \Rightarrow Abbr])])) in (let TMP_31 \def (Bind Abbr) in (let TMP_32 \def
-(CHead d0 TMP_31 u0) in (let TMP_33 \def (Bind b) in (let TMP_34 \def (CHead
-c0 TMP_33 t1) in (let TMP_35 \def (Bind Abbr) in (let TMP_36 \def (CHead d0
-TMP_35 u0) in (let TMP_37 \def (clear_gen_bind b c0 TMP_36 t1 H12) in (let
-H14 \def (f_equal C B TMP_30 TMP_32 TMP_34 TMP_37) in (let TMP_38 \def
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t7)
-\Rightarrow t7])) in (let TMP_39 \def (Bind Abbr) in (let TMP_40 \def (CHead
-d0 TMP_39 u0) in (let TMP_41 \def (Bind b) in (let TMP_42 \def (CHead c0
-TMP_41 t1) in (let TMP_43 \def (Bind Abbr) in (let TMP_44 \def (CHead d0
-TMP_43 u0) in (let TMP_45 \def (clear_gen_bind b c0 TMP_44 t1 H12) in (let
-H15 \def (f_equal C T TMP_38 TMP_40 TMP_42 TMP_45) in (let TMP_92 \def
-(\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let TMP_46 \def
-(\lambda (t7: T).(subst0 O t7 t5 t6)) in (let H18 \def (eq_ind T u0 TMP_46
-H11 t1 H15) in (let TMP_49 \def (\lambda (b0: B).(let TMP_47 \def (Bind b0)
-in (let TMP_48 \def (CHead c0 TMP_47 t) in (pc3 TMP_48 t4 t6)))) in (let
-TMP_50 \def (\lambda (t7: T).(subst0 O t2 t5 t7)) in (let TMP_51 \def
-(\lambda (t7: T).(pr0 t6 t7)) in (let TMP_52 \def (Bind Abbr) in (let TMP_53
-\def (CHead c0 TMP_52 t) in (let TMP_54 \def (pc3 TMP_53 t4 t6) in (let
-TMP_89 \def (\lambda (x: T).(\lambda (H19: (subst0 O t2 t5 x)).(\lambda (H20:
-(pr0 t6 x)).(let TMP_55 \def (\lambda (t7: T).(subst0 O t t5 t7)) in (let
-TMP_58 \def (\lambda (t7: T).(let TMP_56 \def (plus i O) in (let TMP_57 \def
-(S TMP_56) in (subst0 TMP_57 u x t7)))) in (let TMP_59 \def (Bind Abbr) in
-(let TMP_60 \def (CHead c0 TMP_59 t) in (let TMP_61 \def (pc3 TMP_60 t4 t6)
-in (let TMP_87 \def (\lambda (x0: T).(\lambda (H21: (subst0 O t t5
-x0)).(\lambda (H22: (subst0 (S (plus i O)) u x x0)).(let TMP_62 \def (plus i
-O) in (let TMP_63 \def (plus i O) in (let TMP_64 \def (plus_n_O i) in (let
-TMP_65 \def (sym_eq nat i TMP_63 TMP_64) in (let H23 \def (f_equal nat nat S
-TMP_62 i TMP_65) in (let TMP_66 \def (plus i O) in (let TMP_67 \def (S
-TMP_66) in (let TMP_68 \def (\lambda (n: nat).(subst0 n u x x0)) in (let
-TMP_69 \def (S i) in (let H24 \def (eq_ind nat TMP_67 TMP_68 H22 TMP_69 H23)
-in (let TMP_70 \def (Bind Abbr) in (let TMP_71 \def (CHead c0 TMP_70 t) in
-(let TMP_72 \def (Bind Abbr) in (let TMP_73 \def (CHead c0 TMP_72 t) in (let
-TMP_74 \def (getl_refl Abbr c0 t) in (let TMP_75 \def (pr2_delta TMP_73 c0 t
-O TMP_74 t4 t5 H6 x0 H21) in (let TMP_76 \def (Bind Abbr) in (let TMP_77 \def
-(CHead c0 TMP_76 t) in (let TMP_78 \def (Bind Abbr) in (let TMP_79 \def
-(CHead c0 TMP_78 t) in (let TMP_80 \def (S i) in (let TMP_81 \def (Bind Abbr)
-in (let TMP_82 \def (Bind Abbr) in (let TMP_83 \def (CHead d TMP_82 u) in
-(let TMP_84 \def (getl_head TMP_81 i c0 TMP_83 H0 t) in (let TMP_85 \def
-(pr2_delta TMP_79 d u TMP_80 TMP_84 t6 x H20 x0 H24) in (let TMP_86 \def
-(pc3_pr2_x TMP_77 x0 t6 TMP_85) in (pc3_pr2_u TMP_71 x0 t4 TMP_75 t6
-TMP_86))))))))))))))))))))))))))))))) in (let TMP_88 \def (subst0_subst0_back
-t5 x t2 O H19 t u i H2) in (ex2_ind T TMP_55 TMP_58 TMP_61 TMP_87
-TMP_88))))))))))) in (let TMP_90 \def (pr0_subst0_fwd t1 t5 t6 O H18 t2 H1)
-in (let TMP_91 \def (ex2_ind T TMP_50 TMP_51 TMP_54 TMP_89 TMP_90) in (eq_ind
-B Abbr TMP_49 TMP_91 b H16)))))))))))))) in (let TMP_93 \def (TMP_92 H14) in
-(TMP_93 H13)))))))))))))))))))))))))))))))) in (let TMP_112 \def (\lambda (f:
-F).(\lambda (H12: (clear (CHead c0 (Flat f) t1) (CHead d0 (Bind Abbr)
-u0))).(let TMP_95 \def (Bind Abbr) in (let TMP_96 \def (CHead d0 TMP_95 u0)
-in (let TMP_97 \def (Bind Abbr) in (let TMP_98 \def (CHead d0 TMP_97 u0) in
-(let TMP_99 \def (Bind Abbr) in (let TMP_100 \def (CHead d0 TMP_99 u0) in
-(let TMP_101 \def (getl_refl Abbr d0 u0) in (let TMP_102 \def (pr2_delta
-TMP_100 d0 u0 O TMP_101 t4 t5 H6 t6 H11) in (let TMP_103 \def (pc3_pr2_r
-TMP_98 t4 t6 TMP_102) in (let TMP_104 \def (Flat f) in (let TMP_105 \def
-(CHead c0 TMP_104 t) in (let TMP_106 \def (Bind Abbr) in (let TMP_107 \def
-(CHead d0 TMP_106 u0) in (let TMP_108 \def (Bind Abbr) in (let TMP_109 \def
-(CHead d0 TMP_108 u0) in (let TMP_110 \def (clear_gen_flat f c0 TMP_109 t1
-H12) in (let TMP_111 \def (clear_flat c0 TMP_107 TMP_110 f t) in
-(clear_pc3_trans TMP_96 t4 t6 TMP_103 TMP_105 TMP_111)))))))))))))))))))) in
-(let TMP_113 \def (CHead c0 k t1) in (let TMP_114 \def (Bind Abbr) in (let
-TMP_115 \def (CHead d0 TMP_114 u0) in (let TMP_116 \def (getl_gen_O TMP_113
-TMP_115 H10) in (K_ind TMP_21 TMP_94 TMP_112 k TMP_116)))))))))) in (let
-TMP_141 \def (\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c0 k t1)
-(CHead d0 (Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to (pc3 (CHead c0 k t)
-t4 t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t1) (CHead d0 (Bind Abbr)
-u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(let TMP_119 \def (\lambda
-(k0: K).((getl (r k0 i1) c0 (CHead d0 (Bind Abbr) u0)) \to (let TMP_118 \def
-(CHead c0 k0 t) in (pc3 TMP_118 t4 t6)))) in (let TMP_130 \def (\lambda (b:
-B).(\lambda (H12: (getl (r (Bind b) i1) c0 (CHead d0 (Bind Abbr) u0))).(let
-TMP_120 \def (Bind b) in (let TMP_121 \def (CHead c0 TMP_120 t) in (let
-TMP_122 \def (Bind b) in (let TMP_123 \def (CHead c0 TMP_122 t) in (let
-TMP_124 \def (S i1) in (let TMP_125 \def (Bind b) in (let TMP_126 \def (Bind
-Abbr) in (let TMP_127 \def (CHead d0 TMP_126 u0) in (let TMP_128 \def
-(getl_head TMP_125 i1 c0 TMP_127 H12 t) in (let TMP_129 \def (pr2_delta
-TMP_123 d0 u0 TMP_124 TMP_128 t4 t5 H6 t6 H11) in (pc3_pr2_r TMP_121 t4 t6
-TMP_129))))))))))))) in (let TMP_137 \def (\lambda (f: F).(\lambda (H12:
-(getl (r (Flat f) i1) c0 (CHead d0 (Bind Abbr) u0))).(let TMP_131 \def (Flat
-f) in (let TMP_132 \def (CHead c0 TMP_131 t) in (let TMP_133 \def (Flat f) in
-(let TMP_134 \def (r TMP_133 i1) in (let TMP_135 \def (pr2_delta c0 d0 u0
-TMP_134 H12 t4 t5 H6 t6 H11) in (let TMP_136 \def (pr2_cflat c0 t4 t6 TMP_135
-f t) in (pc3_pr2_r TMP_132 t4 t6 TMP_136))))))))) in (let TMP_138 \def (Bind
-Abbr) in (let TMP_139 \def (CHead d0 TMP_138 u0) in (let TMP_140 \def
-(getl_gen_S k c0 TMP_139 t1 i1 H10) in (K_ind TMP_119 TMP_130 TMP_137 k
-TMP_140))))))))))) in (nat_ind TMP_19 TMP_117 TMP_141 i0 H9
-H7)))))))))))))))))) in (pr2_ind TMP_9 TMP_13 TMP_142 y t0 t3 H4)))))) in
-(insert_eq C TMP_4 TMP_5 TMP_7 TMP_143 H3))))))))))))))))))) in (pr2_ind
-TMP_2 TMP_3 TMP_144 c u2 u1 H))))))).
+u1)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1:
+T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t) t1 t2) \to (pc3
+(CHead c0 k t0) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k:
+K).(\lambda (H1: (pr2 (CHead c0 k t1) t0 t3)).(pc3_pr0_pr2_t t2 t1 H0 c0 t0
+t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2:
+(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda
+(H3: (pr2 (CHead c0 k t1) t0 t3)).(insert_eq C (CHead c0 k t1) (\lambda (c1:
+C).(pr2 c1 t0 t3)) (\lambda (_: C).(pc3 (CHead c0 k t) t0 t3)) (\lambda (y:
+C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4:
+T).(\lambda (t5: T).((eq C c1 (CHead c0 k t1)) \to (pc3 (CHead c0 k t) t4
+t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0
+t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t1))).(pc3_pr2_r (CHead c0 k t) t4
+t5 (pr2_free (CHead c0 k t) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0:
+C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0
+(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4
+t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C
+c1 (CHead c0 k t1))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2
+(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t1) H8) in (nat_ind (\lambda (n:
+nat).((getl n (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5
+t6) \to (pc3 (CHead c0 k t) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t1)
+(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind
+(\lambda (k0: K).((clear (CHead c0 k0 t1) (CHead d0 (Bind Abbr) u0)) \to (pc3
+(CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0
+(Bind b) t1) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda
+(e: C).(match e with [(CSort _) \Rightarrow d0 | (CHead c2 _ _) \Rightarrow
+c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0
+(CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H14 \def (f_equal C B (\lambda
+(e: C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow
+(match k0 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])]))
+(CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0 (CHead
+d0 (Bind Abbr) u0) t1 H12)) in ((let H15 \def (f_equal C T (\lambda (e:
+C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7]))
+(CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0 (CHead
+d0 (Bind Abbr) u0) t1 H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq
+C d0 c0)).(let H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6))
+H11 t1 H15) in (eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c0 (Bind b0) t) t4
+t6)) (ex2_ind T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(pr0
+t6 t7)) (pc3 (CHead c0 (Bind Abbr) t) t4 t6) (\lambda (x: T).(\lambda (H19:
+(subst0 O t2 t5 x)).(\lambda (H20: (pr0 t6 x)).(ex2_ind T (\lambda (t7:
+T).(subst0 O t t5 t7)) (\lambda (t7: T).(subst0 (S (plus i O)) u x t7)) (pc3
+(CHead c0 (Bind Abbr) t) t4 t6) (\lambda (x0: T).(\lambda (H21: (subst0 O t
+t5 x0)).(\lambda (H22: (subst0 (S (plus i O)) u x x0)).(let H23 \def (f_equal
+nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H24
+\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n u x x0)) H22 (S
+i) H23) in (pc3_pr2_u (CHead c0 (Bind Abbr) t) x0 t4 (pr2_delta (CHead c0
+(Bind Abbr) t) c0 t O (getl_refl Abbr c0 t) t4 t5 H6 x0 H21) t6 (pc3_pr2_x
+(CHead c0 (Bind Abbr) t) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t) d u (S i)
+(getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H0 t) t6 x H20 x0
+H24)))))))) (subst0_subst0_back t5 x t2 O H19 t u i H2))))) (pr0_subst0_fwd
+t1 t5 t6 O H18 t2 H1)) b H16))))) H14)) H13)))) (\lambda (f: F).(\lambda
+(H12: (clear (CHead c0 (Flat f) t1) (CHead d0 (Bind Abbr)
+u0))).(clear_pc3_trans (CHead d0 (Bind Abbr) u0) t4 t6 (pc3_pr2_r (CHead d0
+(Bind Abbr) u0) t4 t6 (pr2_delta (CHead d0 (Bind Abbr) u0) d0 u0 O (getl_refl
+Abbr d0 u0) t4 t5 H6 t6 H11)) (CHead c0 (Flat f) t) (clear_flat c0 (CHead d0
+(Bind Abbr) u0) (clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t1 H12) f
+t)))) k (getl_gen_O (CHead c0 k t1) (CHead d0 (Bind Abbr) u0) H10))))
+(\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c0 k t1) (CHead d0 (Bind
+Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to (pc3 (CHead c0 k t) t4
+t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t1) (CHead d0 (Bind Abbr)
+u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind (\lambda (k0: K).((getl
+(r k0 i1) c0 (CHead d0 (Bind Abbr) u0)) \to (pc3 (CHead c0 k0 t) t4 t6)))
+(\lambda (b: B).(\lambda (H12: (getl (r (Bind b) i1) c0 (CHead d0 (Bind Abbr)
+u0))).(pc3_pr2_r (CHead c0 (Bind b) t) t4 t6 (pr2_delta (CHead c0 (Bind b) t)
+d0 u0 (S i1) (getl_head (Bind b) i1 c0 (CHead d0 (Bind Abbr) u0) H12 t) t4 t5
+H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (r (Flat f) i1) c0 (CHead
+d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead c0 (Flat f) t) t4 t6 (pr2_cflat c0 t4
+t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) H12 t4 t5 H6 t6 H11) f t)))) k
+(getl_gen_S k c0 (CHead d0 (Bind Abbr) u0) t1 i1 H10)))))) i0 H9
+H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u2 u1 H)))).
-theorem pc3_pr2_pr3_t:
+lemma pc3_pr2_pr3_t:
\forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall
(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u2 u1) \to
(pc3 (CHead c k u1) t1 t2))))))))
\def
\lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(let TMP_1 \def (CHead c k
-u2) in (let TMP_3 \def (\lambda (t: T).(\lambda (t0: T).(\forall (u1:
-T).((pr2 c u2 u1) \to (let TMP_2 \def (CHead c k u1) in (pc3 TMP_2 t t0))))))
-in (let TMP_5 \def (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c u2
-u1)).(let TMP_4 \def (CHead c k u1) in (pc3_refl TMP_4 t))))) in (let TMP_9
-\def (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 (CHead c k u2) t3
-t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead c k u2) t0 t4)).(\lambda (H2:
-((\forall (u1: T).((pr2 c u2 u1) \to (pc3 (CHead c k u1) t0 t4))))).(\lambda
-(u1: T).(\lambda (H3: (pr2 c u2 u1)).(let TMP_6 \def (CHead c k u1) in (let
-TMP_7 \def (pc3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) in (let TMP_8 \def (H2 u1
-H3) in (pc3_t t0 TMP_6 t3 TMP_7 t4 TMP_8)))))))))))) in (pr3_ind TMP_1 TMP_3
-TMP_5 TMP_9 t1 t2 H)))))))))).
+(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2)
+(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u2 u1) \to (pc3
+(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c
+u2 u1)).(pc3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3:
+T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_:
+(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u2 u1)
+\to (pc3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u2
+u1)).(pc3_t t0 (CHead c k u1) t3 (pc3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2
+u1 H3)))))))))) t1 t2 H)))))).
-theorem pc3_pr3_pc3_t:
+lemma pc3_pr3_pc3_t:
\forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u2 u1) \to (\forall
(t1: T).(\forall (t2: T).(\forall (k: K).((pc3 (CHead c k u2) t1 t2) \to (pc3
(CHead c k u1) t1 t2))))))))
\def
\lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u2
-u1)).(let TMP_2 \def (\lambda (t: T).(\lambda (t0: T).(\forall (t1:
-T).(\forall (t2: T).(\forall (k: K).((pc3 (CHead c k t) t1 t2) \to (let TMP_1
-\def (CHead c k t0) in (pc3 TMP_1 t1 t2)))))))) in (let TMP_3 \def (\lambda
-(t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pc3
-(CHead c k t) t1 t2)).H0))))) in (let TMP_17 \def (\lambda (t2: T).(\lambda
+u1)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall
+(t2: T).(\forall (k: K).((pc3 (CHead c k t) t1 t2) \to (pc3 (CHead c k t0) t1
+t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k:
+K).(\lambda (H0: (pc3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda
(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2
t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pc3
(CHead c k t2) t4 t5) \to (pc3 (CHead c k t3) t4 t5))))))).(\lambda (t0:
T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pc3 (CHead c k t1) t0
-t4)).(let H4 \def H3 in (let TMP_5 \def (\lambda (t: T).(let TMP_4 \def
-(CHead c k t1) in (pr3 TMP_4 t0 t))) in (let TMP_7 \def (\lambda (t: T).(let
-TMP_6 \def (CHead c k t1) in (pr3 TMP_6 t4 t))) in (let TMP_8 \def (CHead c k
-t2) in (let TMP_9 \def (pc3 TMP_8 t0 t4) in (let TMP_15 \def (\lambda (x:
-T).(\lambda (H5: (pr3 (CHead c k t1) t0 x)).(\lambda (H6: (pr3 (CHead c k t1)
-t4 x)).(let TMP_10 \def (CHead c k t2) in (let TMP_11 \def (pc3_pr2_pr3_t c
-t1 t0 x k H5 t2 H0) in (let TMP_12 \def (CHead c k t2) in (let TMP_13 \def
-(pc3_pr2_pr3_t c t1 t4 x k H6 t2 H0) in (let TMP_14 \def (pc3_s TMP_12 x t4
-TMP_13) in (pc3_t x TMP_10 t0 TMP_11 t4 TMP_14))))))))) in (let TMP_16 \def
-(ex2_ind T TMP_5 TMP_7 TMP_9 TMP_15 H4) in (H2 t0 t4 k
-TMP_16)))))))))))))))))) in (pr3_ind c TMP_2 TMP_3 TMP_17 u2 u1 H))))))).
+t4)).(H2 t0 t4 k (let H4 \def H3 in (ex2_ind T (\lambda (t: T).(pr3 (CHead c
+k t1) t0 t)) (\lambda (t: T).(pr3 (CHead c k t1) t4 t)) (pc3 (CHead c k t2)
+t0 t4) (\lambda (x: T).(\lambda (H5: (pr3 (CHead c k t1) t0 x)).(\lambda (H6:
+(pr3 (CHead c k t1) t4 x)).(pc3_t x (CHead c k t2) t0 (pc3_pr2_pr3_t c t1 t0
+x k H5 t2 H0) t4 (pc3_s (CHead c k t2) x t4 (pc3_pr2_pr3_t c t1 t4 x k H6 t2
+H0)))))) H4))))))))))))) u2 u1 H)))).
-theorem pc3_lift:
+lemma pc3_lift:
\forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h
d c e) \to (\forall (t1: T).(\forall (t2: T).((pc3 e t1 t2) \to (pc3 c (lift
h d t1) (lift h d t2)))))))))
\def
\lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 e t1
-t2)).(let H1 \def H0 in (let TMP_1 \def (\lambda (t: T).(pr3 e t1 t)) in (let
-TMP_2 \def (\lambda (t: T).(pr3 e t2 t)) in (let TMP_3 \def (lift h d t1) in
-(let TMP_4 \def (lift h d t2) in (let TMP_5 \def (pc3 c TMP_3 TMP_4) in (let
-TMP_11 \def (\lambda (x: T).(\lambda (H2: (pr3 e t1 x)).(\lambda (H3: (pr3 e
-t2 x)).(let TMP_6 \def (lift h d t1) in (let TMP_7 \def (lift h d x) in (let
-TMP_8 \def (pr3_lift c e h d H t1 x H2) in (let TMP_9 \def (lift h d t2) in
-(let TMP_10 \def (pr3_lift c e h d H t2 x H3) in (pc3_pr3_t c TMP_6 TMP_7
-TMP_8 TMP_9 TMP_10))))))))) in (ex2_ind T TMP_1 TMP_2 TMP_5 TMP_11
-H1))))))))))))))).
+t2)).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 e t1 t)) (\lambda (t:
+T).(pr3 e t2 t)) (pc3 c (lift h d t1) (lift h d t2)) (\lambda (x: T).(\lambda
+(H2: (pr3 e t1 x)).(\lambda (H3: (pr3 e t2 x)).(pc3_pr3_t c (lift h d t1)
+(lift h d x) (pr3_lift c e h d H t1 x H2) (lift h d t2) (pr3_lift c e h d H
+t2 x H3))))) H1))))))))).
-theorem pc3_eta:
+lemma pc3_eta:
\forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u: T).((pc3 c t
(THead (Bind Abst) w u)) \to (\forall (v: T).((pc3 c v w) \to (pc3 c (THead
(Bind Abst) v (THead (Flat Appl) (TLRef O) (lift (S O) O t))) t)))))))
\def
\lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u: T).(\lambda (H:
(pc3 c t (THead (Bind Abst) w u))).(\lambda (v: T).(\lambda (H0: (pc3 c v
-w)).(let TMP_1 \def (Bind Abst) in (let TMP_2 \def (Flat Appl) in (let TMP_3
-\def (TLRef O) in (let TMP_4 \def (S O) in (let TMP_5 \def (Bind Abst) in
-(let TMP_6 \def (THead TMP_5 w u) in (let TMP_7 \def (lift TMP_4 O TMP_6) in
-(let TMP_8 \def (THead TMP_2 TMP_3 TMP_7) in (let TMP_9 \def (THead TMP_1 w
-TMP_8) in (let TMP_10 \def (Bind Abst) in (let TMP_11 \def (Flat Appl) in
-(let TMP_12 \def (TLRef O) in (let TMP_13 \def (S O) in (let TMP_14 \def
-(lift TMP_13 O t) in (let TMP_15 \def (THead TMP_11 TMP_12 TMP_14) in (let
-TMP_16 \def (THead TMP_10 v TMP_15) in (let TMP_17 \def (Bind Abst) in (let
-TMP_18 \def (Flat Appl) in (let TMP_19 \def (TLRef O) in (let TMP_20 \def (S
-O) in (let TMP_21 \def (lift TMP_20 O t) in (let TMP_22 \def (THead TMP_18
-TMP_19 TMP_21) in (let TMP_23 \def (Flat Appl) in (let TMP_24 \def (TLRef O)
-in (let TMP_25 \def (S O) in (let TMP_26 \def (Bind Abst) in (let TMP_27 \def
-(THead TMP_26 w u) in (let TMP_28 \def (lift TMP_25 O TMP_27) in (let TMP_29
-\def (THead TMP_23 TMP_24 TMP_28) in (let TMP_30 \def (Bind Abst) in (let
-TMP_31 \def (CHead c TMP_30 v) in (let TMP_32 \def (S O) in (let TMP_33 \def
-(lift TMP_32 O t) in (let TMP_34 \def (S O) in (let TMP_35 \def (Bind Abst)
-in (let TMP_36 \def (THead TMP_35 w u) in (let TMP_37 \def (lift TMP_34 O
-TMP_36) in (let TMP_38 \def (Bind Abst) in (let TMP_39 \def (CHead c TMP_38
-v) in (let TMP_40 \def (S O) in (let TMP_41 \def (Bind Abst) in (let TMP_42
-\def (drop_refl c) in (let TMP_43 \def (drop_drop TMP_41 O c c TMP_42 v) in
-(let TMP_44 \def (Bind Abst) in (let TMP_45 \def (THead TMP_44 w u) in (let
-TMP_46 \def (pc3_lift TMP_39 c TMP_40 O TMP_43 t TMP_45 H) in (let TMP_47
-\def (TLRef O) in (let TMP_48 \def (pc3_thin_dx TMP_31 TMP_33 TMP_37 TMP_46
-TMP_47 Appl) in (let TMP_49 \def (pc3_head_21 c v w H0 TMP_17 TMP_22 TMP_29
-TMP_48) in (let TMP_50 \def (Bind Abst) in (let TMP_51 \def (THead TMP_50 w
-u) in (let TMP_52 \def (Bind Abst) in (let TMP_53 \def (Flat Appl) in (let
-TMP_54 \def (TLRef O) in (let TMP_55 \def (S O) in (let TMP_56 \def (Bind
-Abst) in (let TMP_57 \def (THead TMP_56 w u) in (let TMP_58 \def (lift TMP_55
-O TMP_57) in (let TMP_59 \def (THead TMP_53 TMP_54 TMP_58) in (let TMP_60
-\def (THead TMP_52 w TMP_59) in (let TMP_61 \def (Bind Abst) in (let TMP_62
-\def (Flat Appl) in (let TMP_63 \def (TLRef O) in (let TMP_64 \def (S O) in
-(let TMP_65 \def (Bind Abst) in (let TMP_66 \def (THead TMP_65 w u) in (let
-TMP_67 \def (lift TMP_64 O TMP_66) in (let TMP_68 \def (THead TMP_62 TMP_63
-TMP_67) in (let TMP_69 \def (THead TMP_61 w TMP_68) in (let TMP_70 \def (Bind
-Abst) in (let TMP_71 \def (THead TMP_70 w u) in (let TMP_72 \def (pr3_refl c
-w) in (let TMP_73 \def (pr3_eta c w u w TMP_72) in (let TMP_74 \def
-(pc3_pr3_r c TMP_69 TMP_71 TMP_73) in (let TMP_75 \def (Bind Abst) in (let
-TMP_76 \def (THead TMP_75 w u) in (let TMP_77 \def (pc3_s c TMP_76 t H) in
-(let TMP_78 \def (pc3_t TMP_51 c TMP_60 TMP_74 t TMP_77) in (pc3_t TMP_9 c
-TMP_16 TMP_49 t
-TMP_78))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
-))))))))))))).
+w)).(pc3_t (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O
+(THead (Bind Abst) w u)))) c (THead (Bind Abst) v (THead (Flat Appl) (TLRef
+O) (lift (S O) O t))) (pc3_head_21 c v w H0 (Bind Abst) (THead (Flat Appl)
+(TLRef O) (lift (S O) O t)) (THead (Flat Appl) (TLRef O) (lift (S O) O (THead
+(Bind Abst) w u))) (pc3_thin_dx (CHead c (Bind Abst) v) (lift (S O) O t)
+(lift (S O) O (THead (Bind Abst) w u)) (pc3_lift (CHead c (Bind Abst) v) c (S
+O) O (drop_drop (Bind Abst) O c c (drop_refl c) v) t (THead (Bind Abst) w u)
+H) (TLRef O) Appl)) t (pc3_t (THead (Bind Abst) w u) c (THead (Bind Abst) w
+(THead (Flat Appl) (TLRef O) (lift (S O) O (THead (Bind Abst) w u))))
+(pc3_pr3_r c (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O
+(THead (Bind Abst) w u)))) (THead (Bind Abst) w u) (pr3_eta c w u w (pr3_refl
+c w))) t (pc3_s c (THead (Bind Abst) w u) t H))))))))).
projects / helm.git / blobdiff