(* This file was automatically generated: do not edit *********************)
-include "pc3/defs.ma".
+include "LambdaDelta-1/pc3/defs.ma".
-include "pr3/pr3.ma".
+include "LambdaDelta-1/pr3/pr3.ma".
theorem clear_pc3_trans:
\forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pc3 c2 t1 t2) \to
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_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
-(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_refl:
\forall (c: C).(\forall (t: T).(pc3 c t t))
\def
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:
+ \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
+(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 (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 H1 \def (match H0 in pr2 return (\lambda (c0:
-C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0
-(CHead c k u2)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pc3 (CHead c k u1) t1
-t2)))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0
-(CHead c k u2))).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3
-t2)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2)
-\to ((pr0 t0 t3) \to (pc3 (CHead c k u1) t1 t2))))) (\lambda (H5: (eq T t0
-t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pc3
-(CHead c k u1) t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda
-(t: T).((pr0 t1 t) \to (pc3 (CHead c k u1) t1 t2))) (\lambda (H7: (pr0 t1
-t2)).(pc3_pr2_r (CHead c k u1) t1 t2 (pr2_free (CHead c k u1) t1 t2 H7))) t3
-(sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 (CHead c k u2)
-H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda
-(H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq
-T t t2)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t
-t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i
-u t3 t) \to (pc3 (CHead c k u1) t1 t2))))))) (\lambda (H7: (eq T t0
-t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c k u2)
-(CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pc3
-(CHead c k u1) t1 t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda
-(t4: T).((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to
-((subst0 i u t3 t4) \to (pc3 (CHead c k u1) t1 t2))))) (\lambda (H9: (getl i
-(CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda
-(H11: (subst0 i u t3 t2)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2)
-(CHead d (Bind Abbr) u)) \to ((subst0 n u t3 t2) \to (pc3 (CHead c k u1) t1
-t2)))) (\lambda (H12: (getl O (CHead c k u2) (CHead d (Bind Abbr)
-u))).(\lambda (H13: (subst0 O u t3 t2)).(K_ind (\lambda (k0: K).((clear
-(CHead c k0 u2) (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t1 t2)))
-(\lambda (b: B).(\lambda (H14: (clear (CHead c (Bind b) u2) (CHead d (Bind
-Abbr) u))).(let H15 \def (f_equal C C (\lambda (e: C).(match e in C return
-(\lambda (_: C).C) 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 H14)) in ((let H16 \def (f_equal C B (\lambda (e:
-C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr |
-(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) 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
-H14)) in ((let H17 \def (f_equal C T (\lambda (e: C).(match e in C return
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow
-t4])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c
-(CHead d (Bind Abbr) u) u2 H14)) in (\lambda (H18: (eq B Abbr b)).(\lambda
-(_: (eq C d c)).(let H20 \def (eq_ind T u (\lambda (t4: T).(subst0 O t4 t3
-t2)) H13 u2 H17) in (eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c (Bind b0)
-u1) t1 t2)) (ex2_ind T (\lambda (t4: T).(subst0 O u1 t3 t4)) (\lambda (t4:
-T).(pr0 t2 t4)) (pc3 (CHead c (Bind Abbr) u1) t1 t2) (\lambda (x: T).(\lambda
-(H21: (subst0 O u1 t3 x)).(\lambda (H22: (pr0 t2 x)).(pc3_pr3_t (CHead c
-(Bind Abbr) u1) t1 x (pr3_pr2 (CHead c (Bind Abbr) u1) t1 x (pr2_delta (CHead
-c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 t3 H10 x H21)) t2 (pr3_pr2
-(CHead c (Bind Abbr) u1) t2 x (pr2_free (CHead c (Bind Abbr) u1) t2 x
-H22)))))) (pr0_subst0_fwd u2 t3 t2 O H20 u1 H)) b H18))))) H16)) H15))))
-(\lambda (f: F).(\lambda (H14: (clear (CHead c (Flat f) u2) (CHead d (Bind
-Abbr) u))).(clear_pc3_trans (CHead d (Bind Abbr) u) t1 t2 (pc3_pr2_r (CHead d
-(Bind Abbr) u) t1 t2 (pr2_delta (CHead d (Bind Abbr) u) d u O (getl_refl Abbr
-d u) t1 t3 H10 t2 H13)) (CHead c (Flat f) u1) (clear_flat c (CHead d (Bind
-Abbr) u) (clear_gen_flat f c (CHead d (Bind Abbr) u) u2 H14) f u1)))) k
-(getl_gen_O (CHead c k u2) (CHead d (Bind Abbr) u) H12)))) (\lambda (i0:
-nat).(\lambda (IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to
-((subst0 i0 u t3 t2) \to (pc3 (CHead c k u1) t1 t2))))).(\lambda (H12: (getl
-(S i0) (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H13: (subst0 (S i0)
-u t3 t2)).(K_ind (\lambda (k0: K).((((getl i0 (CHead c k0 u2) (CHead d (Bind
-Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pc3 (CHead c k0 u1) t1 t2)))) \to
-((getl (r k0 i0) c (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t1
-t2)))) (\lambda (b: B).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d
-(Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pc3 (CHead c (Bind b) u1) t1
-t2))))).(\lambda (H14: (getl (r (Bind b) i0) c (CHead d (Bind Abbr)
-u))).(pc3_pr2_r (CHead c (Bind b) u1) t1 t2 (pr2_delta (CHead c (Bind b) u1)
-d u (S i0) (getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) H14 u1) t1 t3 H10
-t2 H13))))) (\lambda (f: F).(\lambda (_: (((getl i0 (CHead c (Flat f) u2)
-(CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pc3 (CHead c (Flat f)
-u1) t1 t2))))).(\lambda (H14: (getl (r (Flat f) i0) c (CHead d (Bind Abbr)
-u))).(pc3_pr2_r (CHead c (Flat f) u1) t1 t2 (pr2_cflat c t1 t2 (pr2_delta c d
-u (r (Flat f) i0) H14 t1 t3 H10 t2 H13) f u1))))) k IHi (getl_gen_S k c
-(CHead d (Bind Abbr) u) u2 i0 H12)))))) i H9 H11)))) t (sym_eq T t t2 H8)))
-t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 (CHead c k u2) H4) H5 H6 H1 H2
-H3))))]) in (H1 (refl_equal C (CHead c k u2)) (refl_equal T t1) (refl_equal T
-t2)))))))))).
+(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 in C return (\lambda
+(_: C).C) 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 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _
+k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) 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 in C return (\lambda (_:
+C).T) 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:
\forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u2 u1) \to (\forall
(CHead c k u1) t1 t2))))))))
\def
\lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u2
-u1)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda (t:
-T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t u2)
-\to ((eq T t0 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)))))))))))) with
-[(pr2_free c0 t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2:
-(eq T t1 u2)).(\lambda (H3: (eq T t2 u1)).(eq_ind C c (\lambda (_: C).((eq T
-t1 u2) \to ((eq T t2 u1) \to ((pr0 t1 t2) \to (\forall (t3: T).(\forall (t4:
-T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3
-t4))))))))) (\lambda (H4: (eq T t1 u2)).(eq_ind T u2 (\lambda (t: T).((eq T
-t2 u1) \to ((pr0 t t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k:
-K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4)))))))) (\lambda
-(H5: (eq T t2 u1)).(eq_ind T u1 (\lambda (t: T).((pr0 u2 t) \to (\forall (t3:
-T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3
-(CHead c k u1) t3 t4))))))) (\lambda (H6: (pr0 u2 u1)).(\lambda (t0:
-T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H7: (pr2 (CHead c k u2) t0
-t3)).(pc3_pr0_pr2_t u1 u2 H6 c t0 t3 k H7)))))) t2 (sym_eq T t2 u1 H5))) t1
-(sym_eq T t1 u2 H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u
-i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq
-T t1 u2)).(\lambda (H5: (eq T t u1)).(eq_ind C c (\lambda (c1: C).((eq T t1
-u2) \to ((eq T t u1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1
-t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k:
-K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4)))))))))))
-(\lambda (H6: (eq T t1 u2)).(eq_ind T u2 (\lambda (t0: T).((eq T t u1) \to
-((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t)
-\to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3
-t4) \to (pc3 (CHead c k u1) t3 t4)))))))))) (\lambda (H7: (eq T t
-u1)).(eq_ind T u1 (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to
-((pr0 u2 t2) \to ((subst0 i u t2 t0) \to (\forall (t3: T).(\forall (t4:
-T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3
-t4))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9:
-(pr0 u2 t2)).(\lambda (H10: (subst0 i u t2 u1)).(\lambda (t0: T).(\lambda
-(t3: T).(\lambda (k: K).(\lambda (H11: (pr2 (CHead c k u2) t0 t3)).(let H12
-\def (match H11 in pr2 return (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5:
-T).(\lambda (_: (pr2 c1 t4 t5)).((eq C c1 (CHead c k u2)) \to ((eq T t4 t0)
-\to ((eq T t5 t3) \to (pc3 (CHead c k u1) t0 t3)))))))) with [(pr2_free c1 t4
-t5 H12) \Rightarrow (\lambda (H13: (eq C c1 (CHead c k u2))).(\lambda (H14:
-(eq T t4 t0)).(\lambda (H15: (eq T t5 t3)).(eq_ind C (CHead c k u2) (\lambda
-(_: C).((eq T t4 t0) \to ((eq T t5 t3) \to ((pr0 t4 t5) \to (pc3 (CHead c k
-u1) t0 t3))))) (\lambda (H16: (eq T t4 t0)).(eq_ind T t0 (\lambda (t6:
-T).((eq T t5 t3) \to ((pr0 t6 t5) \to (pc3 (CHead c k u1) t0 t3)))) (\lambda
-(H17: (eq T t5 t3)).(eq_ind T t3 (\lambda (t6: T).((pr0 t0 t6) \to (pc3
-(CHead c k u1) t0 t3))) (\lambda (H18: (pr0 t0 t3)).(pc3_pr2_r (CHead c k u1)
-t0 t3 (pr2_free (CHead c k u1) t0 t3 H18))) t5 (sym_eq T t5 t3 H17))) t4
-(sym_eq T t4 t0 H16))) c1 (sym_eq C c1 (CHead c k u2) H13) H14 H15 H12)))) |
-(pr2_delta c1 d0 u0 i0 H12 t4 t5 H13 t6 H14) \Rightarrow (\lambda (H15: (eq C
-c1 (CHead c k u2))).(\lambda (H16: (eq T t4 t0)).(\lambda (H17: (eq T t6
-t3)).(eq_ind C (CHead c k u2) (\lambda (c2: C).((eq T t4 t0) \to ((eq T t6
-t3) \to ((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t4 t5) \to ((subst0
-i0 u0 t5 t6) \to (pc3 (CHead c k u1) t0 t3))))))) (\lambda (H18: (eq T t4
-t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t6 t3) \to ((getl i0 (CHead c k u2)
-(CHead d0 (Bind Abbr) u0)) \to ((pr0 t7 t5) \to ((subst0 i0 u0 t5 t6) \to
-(pc3 (CHead c k u1) t0 t3)))))) (\lambda (H19: (eq T t6 t3)).(eq_ind T t3
-(\lambda (t7: T).((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to
-((pr0 t0 t5) \to ((subst0 i0 u0 t5 t7) \to (pc3 (CHead c k u1) t0 t3)))))
-(\lambda (H20: (getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda
-(H21: (pr0 t0 t5)).(\lambda (H22: (subst0 i0 u0 t5 t3)).(nat_ind (\lambda (n:
-nat).((getl n (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5
-t3) \to (pc3 (CHead c k u1) t0 t3)))) (\lambda (H23: (getl O (CHead c k u2)
-(CHead d0 (Bind Abbr) u0))).(\lambda (H24: (subst0 O u0 t5 t3)).(K_ind
-(\lambda (k0: K).((clear (CHead c k0 u2) (CHead d0 (Bind Abbr) u0)) \to (pc3
-(CHead c k0 u1) t0 t3))) (\lambda (b: B).(\lambda (H25: (clear (CHead c (Bind
-b) u2) (CHead d0 (Bind Abbr) u0))).(let H26 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 |
-(CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b)
-u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H25)) in ((let H27 \def
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
+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 in C return (\lambda (_: C).C) 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 in C return (\lambda (_: C).B)
+with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2)
-(clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H25)) in ((let H28 \def
+\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 in C return (\lambda (_: C).T) with
[(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind
-Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0)
-u2 H25)) in (\lambda (H29: (eq B Abbr b)).(\lambda (_: (eq C d0 c)).(let H31
-\def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t3)) H24 u2 H28) in
-(eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t0 t3)) (ex2_ind
-T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(pr0 t3 t7)) (pc3
-(CHead c (Bind Abbr) u1) t0 t3) (\lambda (x: T).(\lambda (H32: (subst0 O t2
-t5 x)).(\lambda (H33: (pr0 t3 x)).(ex2_ind T (\lambda (t7: T).(subst0 O u1 t5
-t7)) (\lambda (t7: T).(subst0 (S (plus i O)) u x t7)) (pc3 (CHead c (Bind
-Abbr) u1) t0 t3) (\lambda (x0: T).(\lambda (H34: (subst0 O u1 t5
-x0)).(\lambda (H35: (subst0 (S (plus i O)) u x x0)).(let H36 \def (f_equal
-nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H37
-\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n u x x0)) H35 (S
-i) H36) in (pc3_pr2_u (CHead c (Bind Abbr) u1) x0 t0 (pr2_delta (CHead c
-(Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t0 t5 H21 x0 H34) t3 (pc3_pr2_x
-(CHead c (Bind Abbr) u1) x0 t3 (pr2_delta (CHead c (Bind Abbr) u1) d u (S i)
-(getl_head (Bind Abbr) i c (CHead d (Bind Abbr) u) H8 u1) t3 x H33 x0
-H37)))))))) (subst0_subst0_back t5 x t2 O H32 u1 u i H10))))) (pr0_subst0_fwd
-u2 t5 t3 O H31 t2 H9)) b H29))))) H27)) H26)))) (\lambda (f: F).(\lambda
-(H25: (clear (CHead c (Flat f) u2) (CHead d0 (Bind Abbr)
-u0))).(clear_pc3_trans (CHead d0 (Bind Abbr) u0) t0 t3 (pc3_pr2_r (CHead d0
-(Bind Abbr) u0) t0 t3 (pr2_delta (CHead d0 (Bind Abbr) u0) d0 u0 O (getl_refl
-Abbr d0 u0) t0 t5 H21 t3 H24)) (CHead c (Flat f) u1) (clear_flat c (CHead d0
-(Bind Abbr) u0) (clear_gen_flat f c (CHead d0 (Bind Abbr) u0) u2 H25) f
-u1)))) k (getl_gen_O (CHead c k u2) (CHead d0 (Bind Abbr) u0) H23))))
-(\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c k u2) (CHead d0 (Bind
-Abbr) u0)) \to ((subst0 i1 u0 t5 t3) \to (pc3 (CHead c k u1) t0
-t3))))).(\lambda (H23: (getl (S i1) (CHead c k u2) (CHead d0 (Bind Abbr)
-u0))).(\lambda (H24: (subst0 (S i1) u0 t5 t3)).(K_ind (\lambda (k0: K).((getl
-(r k0 i1) c (CHead d0 (Bind Abbr) u0)) \to (pc3 (CHead c k0 u1) t0 t3)))
-(\lambda (b: B).(\lambda (H25: (getl (r (Bind b) i1) c (CHead d0 (Bind Abbr)
-u0))).(pc3_pr2_r (CHead c (Bind b) u1) t0 t3 (pr2_delta (CHead c (Bind b) u1)
-d0 u0 (S i1) (getl_head (Bind b) i1 c (CHead d0 (Bind Abbr) u0) H25 u1) t0 t5
-H21 t3 H24)))) (\lambda (f: F).(\lambda (H25: (getl (r (Flat f) i1) c (CHead
-d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead c (Flat f) u1) t0 t3 (pr2_cflat c t0
-t3 (pr2_delta c d0 u0 (r (Flat f) i1) H25 t0 t5 H21 t3 H24) f u1)))) k
-(getl_gen_S k c (CHead d0 (Bind Abbr) u0) u2 i1 H23)))))) i0 H20 H22)))) t6
-(sym_eq T t6 t3 H19))) t4 (sym_eq T t4 t0 H18))) c1 (sym_eq C c1 (CHead c k
-u2) H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C (CHead c k u2))
-(refl_equal T t0) (refl_equal T t3)))))))))) t (sym_eq T t u1 H7))) t1
-(sym_eq T t1 u2 H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0
-(refl_equal C c) (refl_equal T u2) (refl_equal T u1)))))).
+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:
\forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall
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