(* This file was automatically generated: do not edit *********************)
-set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3_props".
-
-include "ty3/pr3.ma".
+include "LambdaDelta-1/ty3/pr3.ma".
theorem ty3_cred_pr2:
\forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr2 c v1
\def
\lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h:
nat).(\lambda (d: nat).(\lambda (H: (ty3 g c (lift h d t1) x)).(insert_eq T
-(lift h d t1) (\lambda (t: T).(ty3 g c t x)) (\forall (e: C).((drop h d c e)
-\to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) (\lambda (t2: T).(ty3 g
-e t1 t2))))) (\lambda (y: T).(\lambda (H0: (ty3 g c y x)).(unintro nat d
-(\lambda (n: nat).((eq T y (lift h n t1)) \to (\forall (e: C).((drop h n c e)
-\to (ex2 T (\lambda (t2: T).(pc3 c (lift h n t2) x)) (\lambda (t2: T).(ty3 g
-e t1 t2))))))) (unintro T t1 (\lambda (t: T).(\forall (x0: nat).((eq T y
-(lift h x0 t)) \to (\forall (e: C).((drop h x0 c e) \to (ex2 T (\lambda (t2:
-T).(pc3 c (lift h x0 t2) x)) (\lambda (t2: T).(ty3 g e t t2)))))))) (ty3_ind
-g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall
-(x1: nat).((eq T t (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to
-(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) t0)) (\lambda (t2: T).(ty3 g e
-x0 t2))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: T).(\lambda
-(_: (ty3 g c0 t2 t)).(\lambda (_: ((\forall (x0: T).(\forall (x1: nat).((eq T
-t2 (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda
-(t3: T).(pc3 c0 (lift h x1 t3) t)) (\lambda (t3: T).(ty3 g e x0
-t3)))))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u
-t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1
-x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t4: T).(pc3
-c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)))))))))).(\lambda
-(H5: (pc3 c0 t3 t2)).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H6: (eq T
-u (lift h x1 x0))).(\lambda (e: C).(\lambda (H7: (drop h x1 c0 e)).(let H8
-\def (eq_ind T u (\lambda (t0: T).(\forall (x2: T).(\forall (x3: nat).((eq T
-t0 (lift h x3 x2)) \to (\forall (e0: C).((drop h x3 c0 e0) \to (ex2 T
-(\lambda (t4: T).(pc3 c0 (lift h x3 t4) t3)) (\lambda (t4: T).(ty3 g e0 x2
-t4))))))))) H4 (lift h x1 x0) H6) in (let H9 \def (eq_ind T u (\lambda (t0:
-T).(ty3 g c0 t0 t3)) H3 (lift h x1 x0) H6) in (let H10 \def (H8 x0 x1
-(refl_equal T (lift h x1 x0)) e H7) in (ex2_ind T (\lambda (t4: T).(pc3 c0
-(lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)) (ex2 T (\lambda (t4:
-T).(pc3 c0 (lift h x1 t4) t2)) (\lambda (t4: T).(ty3 g e x0 t4))) (\lambda
-(x2: T).(\lambda (H11: (pc3 c0 (lift h x1 x2) t3)).(\lambda (H12: (ty3 g e x0
-x2)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t2)) (\lambda (t4:
-T).(ty3 g e x0 t4)) x2 (pc3_t t3 c0 (lift h x1 x2) H11 t2 H5) H12))))
-H10))))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (x0:
-T).(\lambda (x1: nat).(\lambda (H1: (eq T (TSort m) (lift h x1 x0))).(\lambda
-(e: C).(\lambda (_: (drop h x1 c0 e)).(eq_ind_r T (TSort m) (\lambda (t:
-T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (TSort (next g m))))
-(\lambda (t2: T).(ty3 g e t t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0
-(lift h x1 t2) (TSort (next g m)))) (\lambda (t2: T).(ty3 g e (TSort m) t2))
-(TSort (next g m)) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(pc3 c0 t
-(TSort (next g m)))) (pc3_refl c0 (TSort (next g m))) (lift h x1 (TSort (next
-g m))) (lift_sort (next g m) h x1)) (ty3_sort g e m)) x0 (lift_gen_sort h x1
-m x0 H1))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d0: C).(\lambda
-(u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind Abbr) u))).(\lambda (t:
-T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: ((\forall (x0: T).(\forall
-(x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: C).((drop h x1 d0 e) \to
-(ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e
-x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq T
-(TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda (H5: (drop h x1 c0
-e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6 \def H_x in (or_ind
-(land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1 h) n) (eq T x0 (TLRef
-(minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O
-t))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7: (land (lt n x1) (eq T
-x0 (TLRef n)))).(and_ind (lt n x1) (eq T x0 (TLRef n)) (ex2 T (\lambda (t2:
-T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e x0
-t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0 (TLRef n))).(eq_ind_r T
-(TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2)
-(lift (S n) O t))) (\lambda (t2: T).(ty3 g e t0 t2)))) (let H10 \def (eq_ind
-nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5 (S (plus n (minus x1 (S n))))
-(lt_plus_minus n x1 H8)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
-C).(eq T u (lift h (minus x1 (S n)) v)))) (\lambda (v: T).(\lambda (e0:
-C).(getl n e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h
-x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda
-(x2: T).(\lambda (x3: C).(\lambda (H11: (eq T u (lift h (minus x1 (S n))
-x2))).(\lambda (H12: (getl n e (CHead x3 (Bind Abbr) x2))).(\lambda (H13:
-(drop h (minus x1 (S n)) d0 x3)).(let H14 \def (eq_ind T u (\lambda (t0:
-T).(\forall (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall
-(e0: C).((drop h x5 d0 e0) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x5 t2)
-t)) (\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H3 (lift h (minus x1 (S n)) x2)
-H11) in (let H15 \def (eq_ind T u (\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift
-h (minus x1 (S n)) x2) H11) in (let H16 \def (H14 x2 (minus x1 (S n))
-(refl_equal T (lift h (minus x1 (S n)) x2)) x3 H13) in (ex2_ind T (\lambda
-(t2: T).(pc3 d0 (lift h (minus x1 (S n)) t2) t)) (\lambda (t2: T).(ty3 g x3
-x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t)))
-(\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda (x4: T).(\lambda (H17:
-(pc3 d0 (lift h (minus x1 (S n)) x4) t)).(\lambda (H18: (ty3 g x3 x2
-x4)).(eq_ind_r nat (plus (S n) (minus x1 (S n))) (\lambda (n0: nat).(ex2 T
-(\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift (S n) O t))) (\lambda (t2:
-T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h
-(plus (S n) (minus x1 (S n))) t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g
-e (TLRef n) t2)) (lift (S n) O x4) (eq_ind_r T (lift (S n) O (lift h (minus
-x1 (S n)) x4)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O t))) (pc3_lift c0 d0
-(S n) O (getl_drop Abbr c0 d0 u n H1) (lift h (minus x1 (S n)) x4) t H17)
-(lift h (plus (S n) (minus x1 (S n))) (lift (S n) O x4)) (lift_d x4 h (S n)
-(minus x1 (S n)) O (le_O_n (minus x1 (S n))))) (ty3_abbr g n e x3 x2 H12 x4
-H18)) x1 (le_plus_minus (S n) x1 H8))))) H16))))))))) (getl_drop_conf_lt Abbr
-c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) H7)) (\lambda (H7: (land
-(le (plus x1 h) n) (eq T x0 (TLRef (minus n h))))).(and_ind (le (plus x1 h)
-n) (eq T x0 (TLRef (minus n h))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1
-t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H8: (le
-(plus x1 h) n)).(\lambda (H9: (eq T x0 (TLRef (minus n h)))).(eq_ind_r T
-(TLRef (minus n h)) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h
-x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e t0 t2)))) (ex_intro2 T
+(lift h d t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(\forall (e:
+C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x))
+(\lambda (t2: T).(ty3 g e t1 t2)))))) (\lambda (y: T).(\lambda (H0: (ty3 g c
+y x)).(unintro nat d (\lambda (n: nat).((eq T y (lift h n t1)) \to (\forall
+(e: C).((drop h n c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h n t2) x))
+(\lambda (t2: T).(ty3 g e t1 t2))))))) (unintro T t1 (\lambda (t: T).(\forall
+(x0: nat).((eq T y (lift h x0 t)) \to (\forall (e: C).((drop h x0 c e) \to
+(ex2 T (\lambda (t2: T).(pc3 c (lift h x0 t2) x)) (\lambda (t2: T).(ty3 g e t
+t2)))))))) (ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 x0)) \to (\forall
+(e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2)
+t0)) (\lambda (t2: T).(ty3 g e x0 t2))))))))))) (\lambda (c0: C).(\lambda
+(t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: ((\forall
+(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (\forall (e:
+C).((drop h x1 c0 e) \to (ex2 T (\lambda (t3: T).(pc3 c0 (lift h x1 t3) t))
+(\lambda (t3: T).(ty3 g e x0 t3)))))))))).(\lambda (u: T).(\lambda (t3:
+T).(\lambda (H3: (ty3 g c0 u t3)).(\lambda (H4: ((\forall (x0: T).(\forall
+(x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to
+(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e
+x0 t4)))))))))).(\lambda (H5: (pc3 c0 t3 t2)).(\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H6: (eq T u (lift h x1 x0))).(\lambda (e: C).(\lambda (H7:
+(drop h x1 c0 e)).(let H8 \def (eq_ind T u (\lambda (t0: T).(\forall (x2:
+T).(\forall (x3: nat).((eq T t0 (lift h x3 x2)) \to (\forall (e0: C).((drop h
+x3 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x3 t4) t3)) (\lambda
+(t4: T).(ty3 g e0 x2 t4))))))))) H4 (lift h x1 x0) H6) in (let H9 \def
+(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t3)) H3 (lift h x1 x0) H6) in (let
+H10 \def (H8 x0 x1 (refl_equal T (lift h x1 x0)) e H7) in (ex2_ind T (\lambda
+(t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)) (ex2 T
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) t2)) (\lambda (t4: T).(ty3 g e x0
+t4))) (\lambda (x2: T).(\lambda (H11: (pc3 c0 (lift h x1 x2) t3)).(\lambda
+(H12: (ty3 g e x0 x2)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4)
+t2)) (\lambda (t4: T).(ty3 g e x0 t4)) x2 (pc3_t t3 c0 (lift h x1 x2) H11 t2
+H5) H12)))) H10))))))))))))))))))) (\lambda (c0: C).(\lambda (m:
+nat).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H1: (eq T (TSort m) (lift
+h x1 x0))).(\lambda (e: C).(\lambda (_: (drop h x1 c0 e)).(eq_ind_r T (TSort
+m) (\lambda (t: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (TSort
+(next g m)))) (\lambda (t2: T).(ty3 g e t t2)))) (ex_intro2 T (\lambda (t2:
+T).(pc3 c0 (lift h x1 t2) (TSort (next g m)))) (\lambda (t2: T).(ty3 g e
+(TSort m) t2)) (TSort (next g m)) (eq_ind_r T (TSort (next g m)) (\lambda (t:
+T).(pc3 c0 t (TSort (next g m)))) (pc3_refl c0 (TSort (next g m))) (lift h x1
+(TSort (next g m))) (lift_sort (next g m) h x1)) (ty3_sort g e m)) x0
+(lift_gen_sort h x1 m x0 H1))))))))) (\lambda (n: nat).(\lambda (c0:
+C).(\lambda (d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind
+Abbr) u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3:
+((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall
+(e: C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2)
+t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda
+(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6
+\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1
+h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h
+x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7:
+(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n))
+(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda
+(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0
+(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2:
+T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e t0
+t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5
+(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v))))
+(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abbr) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2:
+T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11:
+(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3
+(Bind Abbr) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14
+\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T
+t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T
+(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4
+t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u
+(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in
+(let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h (minus x1 (S n))
+x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h (minus x1 (S n))
+t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0
+(lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))
+(\lambda (x4: T).(\lambda (H17: (pc3 d0 (lift h (minus x1 (S n)) x4)
+t)).(\lambda (H18: (ty3 g x3 x2 x4)).(eq_ind_r nat (plus (S n) (minus x1 (S
+n))) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift
+(S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda
+(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O t)))
+(\lambda (t2: T).(ty3 g e (TLRef n) t2)) (lift (S n) O x4) (eq_ind_r T (lift
+(S n) O (lift h (minus x1 (S n)) x4)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n)
+O t))) (pc3_lift c0 d0 (S n) O (getl_drop Abbr c0 d0 u n H1) (lift h (minus
+x1 (S n)) x4) t H17) (lift h (plus (S n) (minus x1 (S n))) (lift (S n) O x4))
+(lift_d x4 h (S n) (minus x1 (S n)) O (le_O_n (minus x1 (S n))))) (ty3_abbr g
+n e x3 x2 H12 x4 H18)) x1 (le_plus_minus (S n) x1 H8))))) H16)))))))))
+(getl_drop_conf_lt Abbr c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9)))
+H7)) (\lambda (H7: (land (le (plus x1 h) n) (eq T x0 (TLRef (minus n
+h))))).(land_ind (le (plus x1 h) n) (eq T x0 (TLRef (minus n h))) (ex2 T
(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2:
-T).(ty3 g e (TLRef (minus n h)) t2)) (lift (S (minus n h)) O t) (eq_ind_r T
-(lift (plus h (S (minus n h))) O t) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O
-t))) (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(pc3 c0 (lift n0
-O t) (lift (S n) O t))) (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0)
-O t) (lift (S n) O t))) (pc3_refl c0 (lift (S n) O t)) (plus h (minus n h))
-(le_plus_minus h n (le_trans_plus_r x1 h n H8))) (plus h (S (minus n h)))
-(plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h)) O t)) (lift_free
-t (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus O (S (minus n
-h))) (le_S_minus x1 h n H8) (le_n (plus O (S (minus n h))))) (le_O_n x1)))
-(ty3_abbr g (minus n h) e d0 u (getl_drop_conf_ge n (CHead d0 (Bind Abbr) u)
-c0 H1 e h x1 H5 H8) t H2)) x0 H9))) H7)) H6)))))))))))))))) (\lambda (n:
-nat).(\lambda (c0: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (H1: (getl n
-c0 (CHead d0 (Bind Abst) u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u
-t)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1
-x0)) \to (\forall (e: C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3
-d0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0:
-T).(\lambda (x1: nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda
-(e: C).(\lambda (H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n
-H4) in (let H6 \def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land
-(le (plus x1 h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2:
-T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0
-t2))) (\lambda (H7: (land (lt n x1) (eq T x0 (TLRef n)))).(and_ind (lt n x1)
-(eq T x0 (TLRef n)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S
-n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n
-x1)).(\lambda (H9: (eq T x0 (TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0:
-T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda
-(t2: T).(ty3 g e t0 t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0:
-nat).(drop h n0 c0 e)) H5 (S (plus n (minus x1 (S n)))) (lt_plus_minus n x1
-H8)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus
-x1 (S n)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind
-Abst) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0
-e0))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u)))
-(\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3:
-C).(\lambda (H11: (eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl
-n e (CHead x3 (Bind Abst) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0
-x3)).(let H14 \def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall
-(x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0)
-\to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3
-g e0 x4 t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def
-(eq_ind T u (\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2)
-H11) in (eq_ind_r T (lift h (minus x1 (S n)) x2) (\lambda (t0: T).(ex2 T
-(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t0))) (\lambda (t2:
-T).(ty3 g e (TLRef n) t2)))) (let H16 \def (H14 x2 (minus x1 (S n))
-(refl_equal T (lift h (minus x1 (S n)) x2)) x3 H13) in (ex2_ind T (\lambda
-(t2: T).(pc3 d0 (lift h (minus x1 (S n)) t2) t)) (\lambda (t2: T).(ty3 g x3
-x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O (lift h
-(minus x1 (S n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda
-(x4: T).(\lambda (_: (pc3 d0 (lift h (minus x1 (S n)) x4) t)).(\lambda (H18:
-(ty3 g x3 x2 x4)).(eq_ind_r nat (plus (S n) (minus x1 (S n))) (\lambda (n0:
-nat).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift (S n) O (lift h
-(minus n0 (S n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2
-T (\lambda (t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S
-n) O (lift h (minus (plus (S n) (minus x1 (S n))) (S n)) x2)))) (\lambda (t2:
-T).(ty3 g e (TLRef n) t2)) (lift (S n) O x2) (eq_ind_r T (lift (S n) O (lift
-h (minus x1 (S n)) x2)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O (lift h
-(minus (plus (S n) (minus x1 (S n))) (S n)) x2)))) (eq_ind nat x1 (\lambda
-(n0: nat).(pc3 c0 (lift (S n) O (lift h (minus x1 (S n)) x2)) (lift (S n) O
-(lift h (minus n0 (S n)) x2)))) (pc3_refl c0 (lift (S n) O (lift h (minus x1
-(S n)) x2))) (plus (S n) (minus x1 (S n))) (le_plus_minus (S n) x1 H8)) (lift
-h (plus (S n) (minus x1 (S n))) (lift (S n) O x2)) (lift_d x2 h (S n) (minus
-x1 (S n)) O (le_O_n (minus x1 (S n))))) (ty3_abst g n e x3 x2 H12 x4 H18)) x1
+T).(ty3 g e x0 t2))) (\lambda (H8: (le (plus x1 h) n)).(\lambda (H9: (eq T x0
+(TLRef (minus n h)))).(eq_ind_r T (TLRef (minus n h)) (\lambda (t0: T).(ex2 T
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2:
+T).(ty3 g e t0 t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2)
+(lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef (minus n h)) t2)) (lift
+(S (minus n h)) O t) (eq_ind_r T (lift (plus h (S (minus n h))) O t) (\lambda
+(t0: T).(pc3 c0 t0 (lift (S n) O t))) (eq_ind nat (S (plus h (minus n h)))
+(\lambda (n0: nat).(pc3 c0 (lift n0 O t) (lift (S n) O t))) (eq_ind nat n
+(\lambda (n0: nat).(pc3 c0 (lift (S n0) O t) (lift (S n) O t))) (pc3_refl c0
+(lift (S n) O t)) (plus h (minus n h)) (le_plus_minus h n (le_trans h (plus
+x1 h) n (le_plus_r x1 h) H8))) (plus h (S (minus n h))) (plus_n_Sm h (minus n
+h))) (lift h x1 (lift (S (minus n h)) O t)) (lift_free t (S (minus n h)) h O
+x1 (le_trans x1 (S (minus n h)) (plus O (S (minus n h))) (le_S_minus x1 h n
+H8) (le_n (plus O (S (minus n h))))) (le_O_n x1))) (ty3_abbr g (minus n h) e
+d0 u (getl_drop_conf_ge n (CHead d0 (Bind Abbr) u) c0 H1 e h x1 H5 H8) t H2))
+x0 H9))) H7)) H6)))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda
+(d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind Abst)
+u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: ((\forall
+(x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e:
+C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2) t))
+(\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda
+(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6
+\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1
+h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h
+x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7:
+(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n))
+(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda
+(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0
+(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2:
+T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0
+t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5
+(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v))))
+(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abst) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2:
+T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11:
+(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3
+(Bind Abst) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14
+\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T
+t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T
+(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4
+t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u
+(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in
+(eq_ind_r T (lift h (minus x1 (S n)) x2) (\lambda (t0: T).(ex2 T (\lambda
+(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t0))) (\lambda (t2: T).(ty3 g e
+(TLRef n) t2)))) (let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h
+(minus x1 (S n)) x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h
+(minus x1 (S n)) t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda
+(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O (lift h (minus x1 (S n)) x2))))
+(\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda (x4: T).(\lambda (_: (pc3
+d0 (lift h (minus x1 (S n)) x4) t)).(\lambda (H18: (ty3 g x3 x2
+x4)).(eq_ind_r nat (plus (S n) (minus x1 (S n))) (\lambda (n0: nat).(ex2 T
+(\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift (S n) O (lift h (minus n0 (S
+n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda
+(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O (lift
+h (minus (plus (S n) (minus x1 (S n))) (S n)) x2)))) (\lambda (t2: T).(ty3 g
+e (TLRef n) t2)) (lift (S n) O x2) (eq_ind_r T (lift (S n) O (lift h (minus
+x1 (S n)) x2)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O (lift h (minus (plus
+(S n) (minus x1 (S n))) (S n)) x2)))) (eq_ind nat x1 (\lambda (n0: nat).(pc3
+c0 (lift (S n) O (lift h (minus x1 (S n)) x2)) (lift (S n) O (lift h (minus
+n0 (S n)) x2)))) (pc3_refl c0 (lift (S n) O (lift h (minus x1 (S n)) x2)))
+(plus (S n) (minus x1 (S n))) (le_plus_minus (S n) x1 H8)) (lift h (plus (S
+n) (minus x1 (S n))) (lift (S n) O x2)) (lift_d x2 h (S n) (minus x1 (S n)) O
+(le_O_n (minus x1 (S n))))) (ty3_abst g n e x3 x2 H12 x4 H18)) x1
(le_plus_minus (S n) x1 H8))))) H16)) u H11)))))))) (getl_drop_conf_lt Abst
c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) H7)) (\lambda (H7: (land
-(le (plus x1 h) n) (eq T x0 (TLRef (minus n h))))).(and_ind (le (plus x1 h)
+(le (plus x1 h) n) (eq T x0 (TLRef (minus n h))))).(land_ind (le (plus x1 h)
n) (eq T x0 (TLRef (minus n h))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1
t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H8: (le
(plus x1 h) n)).(\lambda (H9: (eq T x0 (TLRef (minus n h)))).(eq_ind_r T
u))) (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(pc3 c0 (lift n0
O u) (lift (S n) O u))) (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0)
O u) (lift (S n) O u))) (pc3_refl c0 (lift (S n) O u)) (plus h (minus n h))
-(le_plus_minus h n (le_trans_plus_r x1 h n H8))) (plus h (S (minus n h)))
-(plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h)) O u)) (lift_free
-u (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus O (S (minus n
-h))) (le_S_minus x1 h n H8) (le_n (plus O (S (minus n h))))) (le_O_n x1)))
-(ty3_abst g (minus n h) e d0 u (getl_drop_conf_ge n (CHead d0 (Bind Abst) u)
-c0 H1 e h x1 H5 H8) t H2)) x0 H9))) H7)) H6)))))))))))))))) (\lambda (c0:
-C).(\lambda (u: T).(\lambda (t: T).(\lambda (H1: (ty3 g c0 u t)).(\lambda
-(H2: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to
-(\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h
-x1 t2) t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (b: B).(\lambda
-(t2: T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) u) t2
-t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1
-x0)) \to (\forall (e: C).((drop h x1 (CHead c0 (Bind b) u) e) \to (ex2 T
+(le_plus_minus h n (le_trans h (plus x1 h) n (le_plus_r x1 h) H8))) (plus h
+(S (minus n h))) (plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h))
+O u)) (lift_free u (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus
+O (S (minus n h))) (le_S_minus x1 h n H8) (le_n (plus O (S (minus n h)))))
+(le_O_n x1))) (ty3_abst g (minus n h) e d0 u (getl_drop_conf_ge n (CHead d0
+(Bind Abst) u) c0 H1 e h x1 H5 H8) t H2)) x0 H9))) H7)) H6))))))))))))))))
+(\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H1: (ty3 g c0 u
+t)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1
+x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3
+c0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (b:
+B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b)
+u) t2 t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift
+h x1 x0)) \to (\forall (e: C).((drop h x1 (CHead c0 (Bind b) u) e) \to (ex2 T
(\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x1 t4) t3)) (\lambda (t4:
-T).(ty3 g e x0 t4)))))))))).(\lambda (t0: T).(\lambda (H5: (ty3 g (CHead c0
-(Bind b) u) t3 t0)).(\lambda (H6: ((\forall (x0: T).(\forall (x1: nat).((eq T
-t3 (lift h x1 x0)) \to (\forall (e: C).((drop h x1 (CHead c0 (Bind b) u) e)
-\to (ex2 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x1 t4) t0))
-(\lambda (t4: T).(ty3 g e x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1:
-nat).(\lambda (H7: (eq T (THead (Bind b) u t2) (lift h x1 x0))).(\lambda (e:
-C).(\lambda (H8: (drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda
-(z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq
-T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1)
-z)))) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u t3)))
-(\lambda (t4: T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
-(H9: (eq T x0 (THead (Bind b) x2 x3))).(\lambda (H10: (eq T u (lift h x1
-x2))).(\lambda (H11: (eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind
-b) x2 x3) (\lambda (t4: T).(ex2 T (\lambda (t5: T).(pc3 c0 (lift h x1 t5)
-(THead (Bind b) u t3))) (\lambda (t5: T).(ty3 g e t4 t5)))) (let H12 \def
-(eq_ind T t2 (\lambda (t4: T).(\forall (x4: T).(\forall (x5: nat).((eq T t4
-(lift h x5 x4)) \to (\forall (e0: C).((drop h x5 (CHead c0 (Bind b) u) e0)
-\to (ex2 T (\lambda (t5: T).(pc3 (CHead c0 (Bind b) u) (lift h x5 t5) t3))
-(\lambda (t5: T).(ty3 g e0 x4 t5))))))))) H4 (lift h (S x1) x3) H11) in (let
-H13 \def (eq_ind T t2 (\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t4 t3))
-H3 (lift h (S x1) x3) H11) in (let H14 \def (eq_ind T u (\lambda (t4: T).(ty3
-g (CHead c0 (Bind b) t4) (lift h (S x1) x3) t3)) H13 (lift h x1 x2) H10) in
-(let H15 \def (eq_ind T u (\lambda (t4: T).(\forall (x4: T).(\forall (x5:
-nat).((eq T (lift h (S x1) x3) (lift h x5 x4)) \to (\forall (e0: C).((drop h
-x5 (CHead c0 (Bind b) t4) e0) \to (ex2 T (\lambda (t5: T).(pc3 (CHead c0
-(Bind b) t4) (lift h x5 t5) t3)) (\lambda (t5: T).(ty3 g e0 x4 t5)))))))))
-H12 (lift h x1 x2) H10) in (let H16 \def (eq_ind T u (\lambda (t4:
-T).(\forall (x4: T).(\forall (x5: nat).((eq T t3 (lift h x5 x4)) \to (\forall
-(e0: C).((drop h x5 (CHead c0 (Bind b) t4) e0) \to (ex2 T (\lambda (t5:
-T).(pc3 (CHead c0 (Bind b) t4) (lift h x5 t5) t0)) (\lambda (t5: T).(ty3 g e0
-x4 t5))))))))) H6 (lift h x1 x2) H10) in (let H17 \def (eq_ind T u (\lambda
-(t4: T).(ty3 g (CHead c0 (Bind b) t4) t3 t0)) H5 (lift h x1 x2) H10) in (let
-H18 \def (eq_ind T u (\lambda (t4: T).(\forall (x4: T).(\forall (x5:
-nat).((eq T t4 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to
-(ex2 T (\lambda (t5: T).(pc3 c0 (lift h x5 t5) t)) (\lambda (t5: T).(ty3 g e0
-x4 t5))))))))) H2 (lift h x1 x2) H10) in (let H19 \def (eq_ind T u (\lambda
-(t4: T).(ty3 g c0 t4 t)) H1 (lift h x1 x2) H10) in (eq_ind_r T (lift h x1 x2)
-(\lambda (t4: T).(ex2 T (\lambda (t5: T).(pc3 c0 (lift h x1 t5) (THead (Bind
-b) t4 t3))) (\lambda (t5: T).(ty3 g e (THead (Bind b) x2 x3) t5)))) (let H20
-\def (H18 x2 x1 (refl_equal T (lift h x1 x2)) e H8) in (ex2_ind T (\lambda
+T).(ty3 g e x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5:
+(eq T (THead (Bind b) u t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6:
+(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0
+(THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x1
+y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u t3))) (\lambda (t4:
+T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0
+(THead (Bind b) x2 x3))).(\lambda (H8: (eq T u (lift h x1 x2))).(\lambda (H9:
+(eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda
+(t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u
+t3))) (\lambda (t4: T).(ty3 g e t0 t4)))) (let H10 \def (eq_ind T t2 (\lambda
+(t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to
+(\forall (e0: C).((drop h x5 (CHead c0 (Bind b) u) e0) \to (ex2 T (\lambda
+(t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x5 t4) t3)) (\lambda (t4: T).(ty3
+g e0 x4 t4))))))))) H4 (lift h (S x1) x3) H9) in (let H11 \def (eq_ind T t2
+(\lambda (t0: T).(ty3 g (CHead c0 (Bind b) u) t0 t3)) H3 (lift h (S x1) x3)
+H9) in (let H12 \def (eq_ind T u (\lambda (t0: T).(ty3 g (CHead c0 (Bind b)
+t0) (lift h (S x1) x3) t3)) H11 (lift h x1 x2) H8) in (let H13 \def (eq_ind T
+u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T (lift h (S x1)
+x3) (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 (CHead c0 (Bind b) t0)
+e0) \to (ex2 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) t0) (lift h x5 t4)
+t3)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H10 (lift h x1 x2) H8) in (let
+H14 \def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5:
+nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to
+(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t)) (\lambda (t4: T).(ty3 g e0
+x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H15 \def (eq_ind T u (\lambda
+(t0: T).(ty3 g c0 t0 t)) H1 (lift h x1 x2) H8) in (eq_ind_r T (lift h x1 x2)
+(\lambda (t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind
+b) t0 t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)))) (let H16
+\def (H14 x2 x1 (refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda
(t4: T).(pc3 c0 (lift h x1 t4) t)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T
(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3)))
(\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) (\lambda (x4:
-T).(\lambda (_: (pc3 c0 (lift h x1 x4) t)).(\lambda (H22: (ty3 g e x2
-x4)).(let H23 \def (H15 x3 (S x1) (refl_equal T (lift h (S x1) x3)) (CHead e
-(Bind b) x2) (drop_skip_bind h x1 c0 e H8 b x2)) in (ex2_ind T (\lambda (t4:
+T).(\lambda (_: (pc3 c0 (lift h x1 x4) t)).(\lambda (H18: (ty3 g e x2
+x4)).(let H19 \def (H13 x3 (S x1) (refl_equal T (lift h (S x1) x3)) (CHead e
+(Bind b) x2) (drop_skip_bind h x1 c0 e H6 b x2)) in (ex2_ind T (\lambda (t4:
T).(pc3 (CHead c0 (Bind b) (lift h x1 x2)) (lift h (S x1) t4) t3)) (\lambda
(t4: T).(ty3 g (CHead e (Bind b) x2) x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0
(lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e
-(THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda (H24: (pc3 (CHead c0
-(Bind b) (lift h x1 x2)) (lift h (S x1) x5) t3)).(\lambda (H25: (ty3 g (CHead
-e (Bind b) x2) x3 x5)).(ex_ind T (\lambda (t4: T).(ty3 g (CHead e (Bind b)
-x2) x5 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b)
+(THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda (H20: (pc3 (CHead c0
+(Bind b) (lift h x1 x2)) (lift h (S x1) x5) t3)).(\lambda (H21: (ty3 g (CHead
+e (Bind b) x2) x3 x5)).(ex_ind T (\lambda (t0: T).(ty3 g (CHead e (Bind b)
+x2) x5 t0)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b)
(lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)))
-(\lambda (x6: T).(\lambda (H26: (ty3 g (CHead e (Bind b) x2) x5
-x6)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b)
-(lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))
-(THead (Bind b) x2 x5) (eq_ind_r T (THead (Bind b) (lift h x1 x2) (lift h (S
-x1) x5)) (\lambda (t4: T).(pc3 c0 t4 (THead (Bind b) (lift h x1 x2) t3)))
-(pc3_head_2 c0 (lift h x1 x2) (lift h (S x1) x5) t3 (Bind b) H24) (lift h x1
-(THead (Bind b) x2 x5)) (lift_bind b x2 x5 h x1)) (ty3_bind g e x2 x4 H22 b
-x3 x5 H25 x6 H26)))) (ty3_correct g (CHead e (Bind b) x2) x3 x5 H25)))))
-H23))))) H20)) u H10))))))))) x0 H9)))))) (lift_gen_bind b u t2 x0 h x1
-H7)))))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u:
-T).(\lambda (H1: (ty3 g c0 w u)).(\lambda (H2: ((\forall (x0: T).(\forall
-(x1: nat).((eq T w (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to
-(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e
-x0 t2)))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v
-(THead (Bind Abst) u t))).(\lambda (H4: ((\forall (x0: T).(\forall (x1:
-nat).((eq T v (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2
-T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Bind Abst) u t))) (\lambda
-(t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda
-(H5: (eq T (THead (Flat Appl) w v) (lift h x1 x0))).(\lambda (e: C).(\lambda
-(H6: (drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T
-x0 (THead (Flat Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T w (lift
-h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T v (lift h x1 z)))) (ex2 T
-(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) w (THead (Bind
-Abst) u t)))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (x2: T).(\lambda
-(x3: T).(\lambda (H7: (eq T x0 (THead (Flat Appl) x2 x3))).(\lambda (H8: (eq
-T w (lift h x1 x2))).(\lambda (H9: (eq T v (lift h x1 x3))).(eq_ind_r T
-(THead (Flat Appl) x2 x3) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0
-(lift h x1 t2) (THead (Flat Appl) w (THead (Bind Abst) u t)))) (\lambda (t2:
-T).(ty3 g e t0 t2)))) (let H10 \def (eq_ind T v (\lambda (t0: T).(\forall
+(\lambda (x6: T).(\lambda (_: (ty3 g (CHead e (Bind b) x2) x5 x6)).(ex_intro2
+T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2)
+t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)) (THead (Bind b)
+x2 x5) (eq_ind_r T (THead (Bind b) (lift h x1 x2) (lift h (S x1) x5))
+(\lambda (t0: T).(pc3 c0 t0 (THead (Bind b) (lift h x1 x2) t3))) (pc3_head_2
+c0 (lift h x1 x2) (lift h (S x1) x5) t3 (Bind b) H20) (lift h x1 (THead (Bind
+b) x2 x5)) (lift_bind b x2 x5 h x1)) (ty3_bind g e x2 x4 H18 b x3 x5 H21))))
+(ty3_correct g (CHead e (Bind b) x2) x3 x5 H21))))) H19))))) H16)) u
+H8))))))) x0 H7)))))) (lift_gen_bind b u t2 x0 h x1 H5)))))))))))))))))
+(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w
+u)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T w (lift h x1
+x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3
+c0 (lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (v:
+T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v (THead (Bind Abst) u
+t))).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T v (lift h x1
+x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3
+c0 (lift h x1 t2) (THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e x0
+t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead
+(Flat Appl) w v) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: (drop h x1 c0
+e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat
+Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T w (lift h x1 y0))))
+(\lambda (_: T).(\lambda (z: T).(eq T v (lift h x1 z)))) (ex2 T (\lambda (t2:
+T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) w (THead (Bind Abst) u t))))
+(\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
+(H7: (eq T x0 (THead (Flat Appl) x2 x3))).(\lambda (H8: (eq T w (lift h x1
+x2))).(\lambda (H9: (eq T v (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl)
+x2 x3) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead
+(Flat Appl) w (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e t0 t2))))
+(let H10 \def (eq_ind T v (\lambda (t0: T).(\forall (x4: T).(\forall (x5:
+nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to
+(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) (THead (Bind Abst) u t)))
+(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H4 (lift h x1 x3) H9) in (let H11
+\def (eq_ind T v (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3
+(lift h x1 x3) H9) in (let H12 \def (eq_ind T w (\lambda (t0: T).(\forall
(x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0:
-C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2)
-(THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H4 (lift
-h x1 x3) H9) in (let H11 \def (eq_ind T v (\lambda (t0: T).(ty3 g c0 t0
-(THead (Bind Abst) u t))) H3 (lift h x1 x3) H9) in (let H12 \def (eq_ind T w
-(\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5
-x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t2: T).(pc3
-c0 (lift h x5 t2) u)) (\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H2 (lift h x1
-x2) H8) in (let H13 \def (eq_ind T w (\lambda (t0: T).(ty3 g c0 t0 u)) H1
-(lift h x1 x2) H8) in (eq_ind_r T (lift h x1 x2) (\lambda (t0: T).(ex2 T
-(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) t0 (THead (Bind
-Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))) (let
-H14 \def (H12 x2 x1 (refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T
-(\lambda (t2: T).(pc3 c0 (lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x2 t2))
-(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1
-x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl)
-x2 x3) t2))) (\lambda (x4: T).(\lambda (H15: (pc3 c0 (lift h x1 x4)
-u)).(\lambda (H16: (ty3 g e x2 x4)).(let H17 \def (H10 x3 x1 (refl_equal T
-(lift h x1 x3)) e H6) in (ex2_ind T (\lambda (t2: T).(pc3 c0 (lift h x1 t2)
-(THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e x3 t2)) (ex2 T (\lambda
+C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) u))
+(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H2 (lift h x1 x2) H8) in (let H13
+\def (eq_ind T w (\lambda (t0: T).(ty3 g c0 t0 u)) H1 (lift h x1 x2) H8) in
+(eq_ind_r T (lift h x1 x2) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0
+(lift h x1 t2) (THead (Flat Appl) t0 (THead (Bind Abst) u t)))) (\lambda (t2:
+T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))) (let H14 \def (H12 x2 x1
+(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t2: T).(pc3 c0
+(lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x2 t2)) (ex2 T (\lambda (t2:
+T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind
+Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))
+(\lambda (x4: T).(\lambda (H15: (pc3 c0 (lift h x1 x4) u)).(\lambda (H16:
+(ty3 g e x2 x4)).(let H17 \def (H10 x3 x1 (refl_equal T (lift h x1 x3)) e H6)
+in (ex2_ind T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Bind Abst) u
+t))) (\lambda (t2: T).(ty3 g e x3 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift
+h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t))))
+(\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x5:
+T).(\lambda (H18: (pc3 c0 (lift h x1 x5) (THead (Bind Abst) u t))).(\lambda
+(H19: (ty3 g e x3 x5)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t2: T).(pr3
+e x5 (THead (Bind Abst) u1 t2)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c0 u
+(lift h x1 u1)))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) t2)))))) (ex2 T (\lambda
(t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind
Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))
-(\lambda (x5: T).(\lambda (H18: (pc3 c0 (lift h x1 x5) (THead (Bind Abst) u
-t))).(\lambda (H19: (ty3 g e x3 x5)).(ex3_2_ind T T (\lambda (u1: T).(\lambda
-(t2: T).(pr3 e x5 (THead (Bind Abst) u1 t2)))) (\lambda (u1: T).(\lambda (_:
-T).(pr3 c0 u (lift h x1 u1)))) (\lambda (_: T).(\lambda (t2: T).(\forall (b:
-B).(\forall (u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) t2))))))
-(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1
-x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl)
-x2 x3) t2))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (pr3 e x5
-(THead (Bind Abst) x6 x7))).(\lambda (H21: (pr3 c0 u (lift h x1
-x6))).(\lambda (H22: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c0 (Bind
-b) u0) t (lift h (S x1) x7)))))).(ex_ind T (\lambda (t0: T).(ty3 g e x5 t0))
-(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1
-x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl)
-x2 x3) t2))) (\lambda (x8: T).(\lambda (H23: (ty3 g e x5 x8)).(ex4_3_ind T T
-T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 e (THead (Bind Abst)
-x6 t2) x8)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g e x6
-t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead e (Bind
-Abst) x6) x7 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g
-(CHead e (Bind Abst) x6) t2 t3)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1
+(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (pr3 e x5 (THead (Bind Abst)
+x6 x7))).(\lambda (H21: (pr3 c0 u (lift h x1 x6))).(\lambda (H22: ((\forall
+(b: B).(\forall (u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1)
+x7)))))).(ex_ind T (\lambda (t0: T).(ty3 g e x5 t0)) (ex2 T (\lambda (t2:
+T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind
+Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))
+(\lambda (x8: T).(\lambda (H23: (ty3 g e x5 x8)).(let H_y \def (ty3_sred_pr3
+e x5 (THead (Bind Abst) x6 x7) H20 g x8 H23) in (ex3_2_ind T T (\lambda (t2:
+T).(\lambda (_: T).(pc3 e (THead (Bind Abst) x6 t2) x8))) (\lambda (_:
+T).(\lambda (t0: T).(ty3 g e x6 t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
+(CHead e (Bind Abst) x6) x7 t2))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1
t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda
(t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x9: T).(\lambda
-(x10: T).(\lambda (x11: T).(\lambda (_: (pc3 e (THead (Bind Abst) x6 x9)
-x8)).(\lambda (H25: (ty3 g e x6 x10)).(\lambda (H26: (ty3 g (CHead e (Bind
-Abst) x6) x7 x9)).(\lambda (H27: (ty3 g (CHead e (Bind Abst) x6) x9
-x11)).(ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl)
+(x10: T).(\lambda (_: (pc3 e (THead (Bind Abst) x6 x9) x8)).(\lambda (H25:
+(ty3 g e x6 x10)).(\lambda (H26: (ty3 g (CHead e (Bind Abst) x6) x7
+x9)).(ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl)
(lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead
(Flat Appl) x2 x3) t2)) (THead (Flat Appl) x2 (THead (Bind Abst) x6 x7))
(eq_ind_r T (THead (Flat Appl) (lift h x1 x2) (lift h x1 (THead (Bind Abst)
H16 (pc3_gen_lift c0 x4 x6 h x1 (pc3_t u c0 (lift h x1 x4) H15 (lift h x1 x6)
(pc3_pr3_r c0 u (lift h x1 x6) H21)) e H6)) x3 x7 (ty3_conv g e (THead (Bind
Abst) x6 x7) (THead (Bind Abst) x6 x9) (ty3_bind g e x6 x10 H25 Abst x7 x9
-H26 x11 H27) x3 x5 H19 (pc3_pr3_r e x5 (THead (Bind Abst) x6 x7)
-H20))))))))))) (ty3_gen_bind g Abst e x6 x7 x8 (ty3_sred_pr3 e x5 (THead
-(Bind Abst) x6 x7) H20 g x8 H23))))) (ty3_correct g e x3 x5 H19)))))))
+H26) x3 x5 H19 (pc3_pr3_r e x5 (THead (Bind Abst) x6 x7) H20)))))))))
+(ty3_gen_bind g Abst e x6 x7 x8 H_y))))) (ty3_correct g e x3 x5 H19)))))))
(pc3_gen_lift_abst c0 x5 t u h x1 H18 e H6))))) H17))))) H14)) w H8))))) x0
H7)))))) (lift_gen_flat Appl w v x0 h x1 H5)))))))))))))))) (\lambda (c0:
C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t2 t3)).(\lambda
(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0
(THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T t3 (lift h
x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T
-(\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0
-t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead (Flat
-Cast) x2 x3))).(\lambda (H8: (eq T t3 (lift h x1 x2))).(\lambda (H9: (eq T t2
-(lift h x1 x3))).(eq_ind_r T (THead (Flat Cast) x2 x3) (\lambda (t: T).(ex2 T
-(\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e t
-t4)))) (let H10 \def (eq_ind T t3 (\lambda (t: T).(\forall (x4: T).(\forall
-(x5: nat).((eq T t (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0)
-\to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t0)) (\lambda (t4: T).(ty3
-g e0 x4 t4))))))))) H4 (lift h x1 x2) H8) in (let H11 \def (eq_ind T t3
-(\lambda (t: T).(ty3 g c0 t t0)) H3 (lift h x1 x2) H8) in (let H12 \def
-(eq_ind T t3 (\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t2
-(lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda
-(t4: T).(pc3 c0 (lift h x5 t4) t)) (\lambda (t4: T).(ty3 g e0 x4 t4)))))))))
-H2 (lift h x1 x2) H8) in (let H13 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0
-t2 t)) H1 (lift h x1 x2) H8) in (eq_ind_r T (lift h x1 x2) (\lambda (t:
-T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t)) (\lambda (t4: T).(ty3 g
-e (THead (Flat Cast) x2 x3) t4)))) (let H14 \def (eq_ind T t2 (\lambda (t:
-T).(ty3 g c0 t (lift h x1 x2))) H13 (lift h x1 x3) H9) in (let H15 \def
-(eq_ind T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t
-(lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda
-(t4: T).(pc3 c0 (lift h x5 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e0 x4
-t4))))))))) H12 (lift h x1 x3) H9) in (let H16 \def (H15 x3 x1 (refl_equal T
-(lift h x1 x3)) e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 (lift h x1 t4)
-(lift h x1 x2))) (\lambda (t4: T).(ty3 g e x3 t4)) (ex2 T (\lambda (t4:
-T).(pc3 c0 (lift h x1 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e (THead
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 t3))) (\lambda
+(t4: T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq
+T x0 (THead (Flat Cast) x2 x3))).(\lambda (H8: (eq T t3 (lift h x1
+x2))).(\lambda (H9: (eq T t2 (lift h x1 x3))).(eq_ind_r T (THead (Flat Cast)
+x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead
+(Flat Cast) t0 t3))) (\lambda (t4: T).(ty3 g e t t4)))) (let H10 \def (eq_ind
+T t3 (\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5
+x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3
+c0 (lift h x5 t4) t0)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H4 (lift h
+x1 x2) H8) in (let H11 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t0)) H3
+(lift h x1 x2) H8) in (let H12 \def (eq_ind T t3 (\lambda (t: T).(\forall
+(x4: T).(\forall (x5: nat).((eq T t2 (lift h x5 x4)) \to (\forall (e0:
+C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t))
+(\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H13
+\def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t2 t)) H1 (lift h x1 x2) H8) in
+(eq_ind_r T (lift h x1 x2) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0
+(lift h x1 t4) (THead (Flat Cast) t0 t))) (\lambda (t4: T).(ty3 g e (THead
+(Flat Cast) x2 x3) t4)))) (let H14 \def (eq_ind T t2 (\lambda (t: T).(ty3 g
+c0 t (lift h x1 x2))) H13 (lift h x1 x3) H9) in (let H15 \def (eq_ind T t2
+(\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5 x4))
+\to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0
+(lift h x5 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H12
+(lift h x1 x3) H9) in (let H16 \def (H15 x3 x1 (refl_equal T (lift h x1 x3))
+e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (lift h x1 x2)))
+(\lambda (t4: T).(ty3 g e x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1
+t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda (t4: T).(ty3 g e (THead
(Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H17: (pc3 c0 (lift h x1
x4) (lift h x1 x2))).(\lambda (H18: (ty3 g e x3 x4)).(let H19 \def (H10 x2 x1
(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0
(lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T (\lambda (t4:
-T).(pc3 c0 (lift h x1 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e (THead
-(Flat Cast) x2 x3) t4))) (\lambda (x5: T).(\lambda (_: (pc3 c0 (lift h x1 x5)
-t0)).(\lambda (H21: (ty3 g e x2 x5)).(ex_intro2 T (\lambda (t4: T).(pc3 c0
-(lift h x1 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e (THead (Flat Cast)
-x2 x3) t4)) x2 (pc3_refl c0 (lift h x1 x2)) (ty3_cast g e x3 x2 (ty3_conv g e
+T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda
+(t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4))) (\lambda (x5: T).(\lambda
+(H20: (pc3 c0 (lift h x1 x5) t0)).(\lambda (H21: (ty3 g e x2 x5)).(ex_intro2
+T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1
+x2)))) (\lambda (t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4)) (THead (Flat
+Cast) x5 x2) (eq_ind_r T (THead (Flat Cast) (lift h x1 x5) (lift h x1 x2))
+(\lambda (t: T).(pc3 c0 t (THead (Flat Cast) t0 (lift h x1 x2)))) (pc3_head_1
+c0 (lift h x1 x5) t0 H20 (Flat Cast) (lift h x1 x2)) (lift h x1 (THead (Flat
+Cast) x5 x2)) (lift_flat Cast x5 x2 h x1)) (ty3_cast g e x3 x2 (ty3_conv g e
x2 x5 H21 x3 x4 H18 (pc3_gen_lift c0 x4 x2 h x1 H17 e H6)) x5 H21)))))
H19))))) H16)))) t3 H8))))) x0 H7)))))) (lift_gen_flat Cast t3 t2 x0 h x1
H5))))))))))))))) c y x H0))))) H))))))).
\lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H:
(ty3 g c u t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(ex_ind T
(\lambda (t: T).(ty3 g c t1 t)) (ty3 g c u t2) (\lambda (x: T).(\lambda (H1:
-(ty3 g c t1 x)).(ty3_conv g c t2 x (ty3_sred_pr3 c t1 t2 H0 g x H1) u t1 H
-(pc3_pr3_r c t1 t2 H0)))) (ty3_correct g c u t1 H)))))))).
+(ty3 g c t1 x)).(let H_y \def (ty3_sred_pr3 c t1 t2 H0 g x H1) in (ty3_conv g
+c t2 x H_y u t1 H (pc3_pr3_r c t1 t2 H0))))) (ty3_correct g c u t1 H)))))))).
theorem ty3_sconv_pc3:
\forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c
(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c
u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda
(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (pc3 c t1 t2) (\lambda (x:
-T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(ty3_unique g c x
-t1 (ty3_sred_pr3 c u1 x H3 g t1 H) t2 (ty3_sred_pr3 c u2 x H4 g t2 H0)))))
-H2)))))))))).
+T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(let H_y \def
+(ty3_sred_pr3 c u2 x H4 g t2 H0) in (let H_y0 \def (ty3_sred_pr3 c u1 x H3 g
+t1 H) in (ty3_unique g c x t1 H_y0 t2 H_y)))))) H2)))))))))).
theorem ty3_sred_back:
\forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t0: T).((ty3 g c
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