+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Basic-1/ty3/props.ma".
-
-include "Basic-1/pc3/fsubst0.ma".
-
-include "Basic-1/getl/getl.ma".
-
-theorem ty3_fsubst0:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1
-t1 t) \to (\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2:
-T).((fsubst0 i u c1 t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind
-Abbr) u)) \to (ty3 g c2 t2 t))))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda
-(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda
-(t2: T).(\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3:
-T).((fsubst0 i u c t0 c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind
-Abbr) u)) \to (ty3 g c2 t3 t2))))))))))) (\lambda (c: C).(\lambda (t2:
-T).(\lambda (t0: T).(\lambda (H0: (ty3 g c t2 t0)).(\lambda (H1: ((\forall
-(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2
-c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (ty3 g c2
-t3 t0)))))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 g c u
-t3)).(\lambda (H3: ((\forall (i: nat).(\forall (u0: T).(\forall (c2:
-C).(\forall (t4: T).((fsubst0 i u0 c u c2 t4) \to (\forall (e: C).((getl i c
-(CHead e (Bind Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (H4: (pc3 c
-t3 t2)).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4:
-T).(\lambda (H5: (fsubst0 i u0 c u c2 t4)).(fsubst0_ind i u0 c u (\lambda
-(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0))
-\to (ty3 g c0 t5 t2))))) (\lambda (t5: T).(\lambda (H6: (subst0 i u0 u
-t5)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr)
-u0))).(ty3_conv g c t2 t0 H0 t5 t3 (H3 i u0 c t5 (fsubst0_snd i u0 c u t5 H6)
-e H7) H4))))) (\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c c3)).(\lambda
-(e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) u0))).(ty3_conv g c3 t2
-t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H6) e H7) u t3 (H3 i u0 c3 u
-(fsubst0_fst i u0 c u c3 H6) e H7) (pc3_fsubst0 c t3 t2 H4 i u0 c3 t3
-(fsubst0_fst i u0 c t3 c3 H6) e H7)))))) (\lambda (t5: T).(\lambda (H6:
-(subst0 i u0 u t5)).(\lambda (c3: C).(\lambda (H7: (csubst0 i u0 c
-c3)).(\lambda (e: C).(\lambda (H8: (getl i c (CHead e (Bind Abbr)
-u0))).(ty3_conv g c3 t2 t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H7) e H8)
-t5 t3 (H3 i u0 c3 t5 (fsubst0_both i u0 c u t5 H6 c3 H7) e H8) (pc3_fsubst0 c
-t3 t2 H4 i u0 c3 t3 (fsubst0_fst i u0 c t3 c3 H7) e H8)))))))) c2 t4
-H5)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (i:
-nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H0: (fsubst0
-i u c (TSort m) c2 t2)).(fsubst0_ind i u c (TSort m) (\lambda (c0:
-C).(\lambda (t0: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to
-(ty3 g c0 t0 (TSort (next g m))))))) (\lambda (t3: T).(\lambda (H1: (subst0 i
-u (TSort m) t3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind Abbr)
-u))).(subst0_gen_sort u t3 i m H1 (ty3 g c t3 (TSort (next g m))))))))
-(\lambda (c3: C).(\lambda (_: (csubst0 i u c c3)).(\lambda (e: C).(\lambda
-(_: (getl i c (CHead e (Bind Abbr) u))).(ty3_sort g c3 m))))) (\lambda (t3:
-T).(\lambda (H1: (subst0 i u (TSort m) t3)).(\lambda (c3: C).(\lambda (_:
-(csubst0 i u c c3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind
-Abbr) u))).(subst0_gen_sort u t3 i m H1 (ty3 g c3 t3 (TSort (next g
-m)))))))))) c2 t2 H0)))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda
-(t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2: ((\forall (i:
-nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 d u c2
-t2) \to (\forall (e: C).((getl i d (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2
-t0)))))))))).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda
-(t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n) c2 t2)).(fsubst0_ind i u0 c
-(TLRef n) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c (CHead
-e (Bind Abbr) u0)) \to (ty3 g c0 t3 (lift (S n) O t0)))))) (\lambda (t3:
-T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (e: C).(\lambda (H5:
-(getl i c (CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) (eq T t3 (lift (S
-n) O u0)) (ty3 g c t3 (lift (S n) O t0)) (\lambda (H6: (eq nat n i)).(\lambda
-(H7: (eq T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4:
-T).(ty3 g c t4 (lift (S n) O t0))) (let H8 \def (eq_ind_r nat i (\lambda (n0:
-nat).(getl n0 c (CHead e (Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C
-(CHead d (Bind Abbr) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind
-Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0)
-H8)) in (let H10 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow
-c0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d
-(Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H8)) in ((let H11 \def (f_equal
-C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d (Bind Abbr) u)
-(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e
-(Bind Abbr) u0) H8)) in (\lambda (H12: (eq C d e)).(let H13 \def (eq_ind_r C
-e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H9 d H12) in (let
-H14 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d (Bind Abbr) t4)))
-H13 u H11) in (eq_ind T u (\lambda (t4: T).(ty3 g c (lift (S n) O t4) (lift
-(S n) O t0))) (ty3_lift g d u t0 H1 c O (S n) (getl_drop Abbr c d u n H14))
-u0 H11))))) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n H4)))))) (\lambda
-(c3: C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H5:
-(getl i c (CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3 (TLRef n) (lift
-(S n) O t0)) (\lambda (H6: (lt n i)).(let H7 \def (csubst0_getl_lt i n H6 c
-c3 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind (getl n c3 (CHead d (Bind
-Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C
-(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b)
-u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C
-(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(minus i (S n)) u0 e1 e2))))))) (ty3 g c3 (TLRef n) (lift (S n) O t0))
-(\lambda (H8: (getl n c3 (CHead d (Bind Abbr) u))).(ty3_abbr g n c3 d u H8 t0
-H1)) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
-(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b)
-u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1
-w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))) (ty3 g c3 (TLRef n)
-(lift (S n) O t0)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
-T).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1
-(Bind x0) x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind x0) x3))).(\lambda
-(H11: (subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda
-(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow
-d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind
-x0) x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C
-return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match
-e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
-t3) \Rightarrow t3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in
-(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def
-(eq_ind_r T x2 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x3)) H11 u H14)
-in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(getl n c3 (CHead c0 (Bind
-x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n
-c3 (CHead d (Bind b) x3))) H18 Abbr H15) in (let H20 \def (eq_ind nat (minus
-i n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abbr) x3) (CHead e (Bind
-Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i
-(le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead d (Bind Abbr) x3) n
-H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6))
-in (ty3_abbr g n c3 d x3 H19 t0 (H2 (minus i (S n)) u0 d x3 (fsubst0_snd
-(minus i (S n)) u0 d u x3 H17) e (getl_gen_S (Bind Abbr) d (CHead e (Bind
-Abbr) u0) x3 (minus i (S n)) H20)))))))))) H13)) H12))))))))) H8)) (\lambda
-(H8: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2
-(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda
-(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
-Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-u0 e1 e2))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) (\lambda (x0:
-B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H9: (eq C
-(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H10: (getl n c3
-(CHead x2 (Bind x0) x3))).(\lambda (H11: (csubst0 (minus i (S n)) u0 x1
-x2)).(let H12 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow
-c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def
-(f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
-Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14
-\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d
-(Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H15: (eq B Abbr
-x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x3 (\lambda (t3:
-T).(getl n c3 (CHead x2 (Bind x0) t3))) H10 u H14) in (let H18 \def (eq_ind_r
-C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H11 d H16) in (let
-H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) u)))
-H17 Abbr H15) in (let H20 \def (eq_ind nat (minus i n) (\lambda (n0:
-nat).(getl n0 (CHead x2 (Bind Abbr) u) (CHead e (Bind Abbr) u0)))
-(getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c
-c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abbr) u) n H19 (le_S_n
-n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr
-g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n))
-u0 d u x2 H18) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n
-(minus i (S n))) d x2 u0 H18 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abbr)
-x2 (CHead e (Bind Abbr) u0) u (minus i (S n)) H20))))))))))) H13))
-H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind
-Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C
-(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S n) O t0))
-(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda
-(x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
-x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H11:
-(subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S n)) u0
-x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow
-c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def
-(f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
-Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H15
-\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d
-(Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B Abbr
-x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda (t3:
-T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) in (let H19 \def (eq_ind_r C
-x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H12 d H17) in (let
-H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) x4)))
-H10 Abbr H16) in (let H21 \def (eq_ind nat (minus i n) (\lambda (n0:
-nat).(getl n0 (CHead x2 (Bind Abbr) x4) (CHead e (Bind Abbr) u0)))
-(getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c
-c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abbr) x4) n H20 (le_S_n
-n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr
-g n c3 x2 x4 H20 t0 (H2 (minus i (S n)) u0 x2 x4 (fsubst0_both (minus i (S
-n)) u0 d u x4 H18 x2 H19) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S
-n)) (le_n (minus i (S n))) d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S
-(Bind Abbr) x2 (CHead e (Bind Abbr) u0) x4 (minus i (S n)) H21)))))))))))
-H14)) H13))))))))))) H8)) H7))) (\lambda (H6: (le i n)).(ty3_abbr g n c3 d u
-(csubst0_getl_ge i n H6 c c3 u0 H4 (CHead d (Bind Abbr) u) H0) t0 H1)))))))
-(\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3:
-C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H6: (getl i c
-(CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0))
-(ty3 g c3 t3 (lift (S n) O t0)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq
-T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3
-g c3 t4 (lift (S n) O t0))) (let H9 \def (eq_ind_r nat i (\lambda (n0:
-nat).(getl n0 c (CHead e (Bind Abbr) u0))) H6 n H7) in (let H10 \def
-(eq_ind_r nat i (\lambda (n0: nat).(csubst0 n0 u0 c c3)) H5 n H7) in (let H11
-\def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl n c c0)) H0
-(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e
-(Bind Abbr) u0) H9)) in (let H12 \def (f_equal C C (\lambda (e0: C).(match e0
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
-\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono
-c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in ((let H13
-\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d
-(Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u)
-n H0 (CHead e (Bind Abbr) u0) H9)) in (\lambda (H14: (eq C d e)).(let H15
-\def (eq_ind_r C e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H11
-d H14) in (let H16 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d
-(Bind Abbr) t4))) H15 u H13) in (let H17 \def (eq_ind_r T u0 (\lambda (t4:
-T).(csubst0 n t4 c c3)) H10 u H13) in (eq_ind T u (\lambda (t4: T).(ty3 g c3
-(lift (S n) O t4) (lift (S n) O t0))) (ty3_lift g d u t0 H1 c3 O (S n)
-(getl_drop Abbr c3 d u n (csubst0_getl_ge n n (le_n n) c c3 u H17 (CHead d
-(Bind Abbr) u) H16))) u0 H13)))))) H12))))) t3 H8))) (subst0_gen_lref u0 t3 i
-n H4)))))))) c2 t2 H3)))))))))))))) (\lambda (n: nat).(\lambda (c:
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind
-Abst) u))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2:
-((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2:
-T).((fsubst0 i u0 d u c2 t2) \to (\forall (e: C).((getl i d (CHead e (Bind
-Abbr) u0)) \to (ty3 g c2 t2 t0)))))))))).(\lambda (i: nat).(\lambda (u0:
-T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n)
-c2 t2)).(fsubst0_ind i u0 c (TLRef n) (\lambda (c0: C).(\lambda (t3:
-T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3
-(lift (S n) O u)))))) (\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n)
-t3)).(\lambda (e: C).(\lambda (H5: (getl i c (CHead e (Bind Abbr)
-u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c t3 (lift (S
-n) O u)) (\lambda (H6: (eq nat n i)).(\lambda (H7: (eq T t3 (lift (S n) O
-u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c t4 (lift (S n)
-O u))) (let H8 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e
-(Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u)
-(\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0) (getl_mono c
-(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H8)) in (let H10 \def
-(eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind
-Abst) u) n H0 (CHead e (Bind Abbr) u0) H8)) in (False_ind (ty3 g c (lift (S
-n) O u0) (lift (S n) O u)) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n
-H4)))))) (\lambda (c3: C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e:
-C).(\lambda (H5: (getl i c (CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3
-(TLRef n) (lift (S n) O u)) (\lambda (H6: (lt n i)).(let H7 \def
-(csubst0_getl_lt i n H6 c c3 u0 H4 (CHead d (Bind Abst) u) H0) in (or4_ind
-(getl n c3 (CHead d (Bind Abst) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n))
-u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3
-(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
-n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))) (ty3 g c3 (TLRef n) (lift (S
-n) O u)) (\lambda (H8: (getl n c3 (CHead d (Bind Abst) u))).(ty3_abst g n c3
-d u H8 t0 H1)) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1
-w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))) (ty3 g c3 (TLRef n)
-(lift (S n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda
-(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
-x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind x0) x3))).(\lambda (H11:
-(subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
-(CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
-x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return
-(\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead
-x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match
-e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
-t3) \Rightarrow t3])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) H9) in
-(\lambda (H15: (eq B Abst x0)).(\lambda (H16: (eq C d x1)).(let H17 \def
-(eq_ind_r T x2 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x3)) H11 u H14)
-in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(getl n c3 (CHead c0 (Bind
-x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n
-c3 (CHead d (Bind b) x3))) H18 Abst H15) in (let H20 \def (eq_ind nat (minus
-i n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abst) x3) (CHead e (Bind
-Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i
-(le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead d (Bind Abst) x3) n
-H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6))
-in (ty3_conv g c3 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g d u t0 H1 c3
-O (S n) (getl_drop Abst c3 d x3 n H19)) (TLRef n) (lift (S n) O x3) (ty3_abst
-g n c3 d x3 H19 t0 (H2 (minus i (S n)) u0 d x3 (fsubst0_snd (minus i (S n))
-u0 d u x3 H17) e (getl_gen_S (Bind Abst) d (CHead e (Bind Abbr) u0) x3 (minus
-i (S n)) H20))) (pc3_lift c3 d (S n) O (getl_drop Abst c3 d x3 n H19) x3 u
-(pc3_pr2_x d x3 u (pr2_delta d e u0 (r (Bind Abst) (minus i (S n)))
-(getl_gen_S (Bind Abst) d (CHead e (Bind Abbr) u0) x3 (minus i (S n)) H20) u
-u (pr0_refl u) x3 H17))))))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq
-C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2
-(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda
-(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
-Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-u0 e1 e2))))) (ty3 g c3 (TLRef n) (lift (S n) O u)) (\lambda (x0: B).(\lambda
-(x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind
-Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind
-x0) x3))).(\lambda (H11: (csubst0 (minus i (S n)) u0 x1 x2)).(let H12 \def
-(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind
-Abst) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda
-(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
-Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
-[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst)
-u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
-(CHead _ _ t3) \Rightarrow t3])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
-x3) H9) in (\lambda (H15: (eq B Abst x0)).(\lambda (H16: (eq C d x1)).(let
-H17 \def (eq_ind_r T x3 (\lambda (t3: T).(getl n c3 (CHead x2 (Bind x0) t3)))
-H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i
-(S n)) u0 c0 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b:
-B).(getl n c3 (CHead x2 (Bind b) u))) H17 Abst H15) in (let H20 \def (eq_ind
-nat (minus i n) (\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abst) u) (CHead e
-(Bind Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3
-(csubst0_getl_ge i i (le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead
-x2 (Bind Abst) u) n H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n)))
-(minus_x_Sy i n H6)) in (ty3_abst g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0
-x2 u (fsubst0_fst (minus i (S n)) u0 d u x2 H18) e (csubst0_getl_ge_back
-(minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) d x2 u0 H18 (CHead e
-(Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e (Bind Abbr) u0) u (minus
-i (S n)) H20))))))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T
-T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
-n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
-n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S
-n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3:
-T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1
-(Bind x0) x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda
-(H11: (subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S
-n)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
-\Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in
-((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda
-(_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match
-k in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _)
-\Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in
-((let H15 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3]))
-(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B
-Abst x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda
-(t3: T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) in (let H19 \def
-(eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H12 d
-H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2
-(Bind b) x4))) H10 Abst H16) in (let H21 \def (eq_ind nat (minus i n)
-(\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abst) x4) (CHead e (Bind Abbr)
-u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n
-i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abst) x4) n H20
-(le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in
-(ty3_conv g c3 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x2 u t0 (H2
-(minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n)) u0 d u x2 H19) e
-(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n)))
-d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e
-(Bind Abbr) u0) x4 (minus i (S n)) H21))) c3 O (S n) (getl_drop Abst c3 x2 x4
-n H20)) (TLRef n) (lift (S n) O x4) (ty3_abst g n c3 x2 x4 H20 t0 (H2 (minus
-i (S n)) u0 x2 x4 (fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) e
-(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n)))
-d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e
-(Bind Abbr) u0) x4 (minus i (S n)) H21)))) (pc3_lift c3 x2 (S n) O (getl_drop
-Abst c3 x2 x4 n H20) x4 u (pc3_fsubst0 d u u (pc3_refl d u) (minus i (S n))
-u0 x2 x4 (fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) e
-(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n)))
-d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e
-(Bind Abbr) u0) x4 (minus i (S n)) H21)))))))))))) H14)) H13))))))))))) H8))
-H7))) (\lambda (H6: (le i n)).(ty3_abst g n c3 d u (csubst0_getl_ge i n H6 c
-c3 u0 H4 (CHead d (Bind Abst) u) H0) t0 H1))))))) (\lambda (t3: T).(\lambda
-(H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3: C).(\lambda (H5: (csubst0 i u0
-c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr)
-u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c3 t3 (lift
-(S n) O u)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t3 (lift (S n) O
-u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c3 t4 (lift (S n)
-O u))) (let H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e
-(Bind Abbr) u0))) H6 n H7) in (let H10 \def (eq_ind_r nat i (\lambda (n0:
-nat).(csubst0 n0 u0 c c3)) H5 n H7) in (let H11 \def (eq_ind C (CHead d (Bind
-Abst) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0)
-(getl_mono c (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in
-(let H12 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in
-C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k
-_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind
-Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind (ty3 g c3 (lift (S
-n) O u0) (lift (S n) O u)) H12))))) t3 H8))) (subst0_gen_lref u0 t3 i n
-H4)))))))) c2 t2 H3)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda
-(t0: T).(\lambda (H0: (ty3 g c u t0)).(\lambda (H1: ((\forall (i:
-nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2
-t2) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2
-t0)))))))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2:
-(ty3 g (CHead c (Bind b) u) t2 t3)).(\lambda (H3: ((\forall (i: nat).(\forall
-(u0: T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u0 (CHead c (Bind b) u)
-t2 c2 t4) \to (\forall (e: C).((getl i (CHead c (Bind b) u) (CHead e (Bind
-Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u0:
-T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H4: (fsubst0 i u0 c (THead
-(Bind b) u t2) c2 t4)).(fsubst0_ind i u0 c (THead (Bind b) u t2) (\lambda
-(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0))
-\to (ty3 g c0 t5 (THead (Bind b) u t3)))))) (\lambda (t5: T).(\lambda (H5:
-(subst0 i u0 (THead (Bind b) u t2) t5)).(\lambda (e: C).(\lambda (H6: (getl i
-c (CHead e (Bind Abbr) u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead
-(Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6:
-T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i)
-u0 t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead
-(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
-(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c
-t5 (THead (Bind b) u t3)) (\lambda (H7: (ex2 T (\lambda (u2: T).(eq T t5
-(THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T
-(\lambda (u2: T).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i
-u0 u u2)) (ty3 g c t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H8:
-(eq T t5 (THead (Bind b) x t2))).(\lambda (H9: (subst0 i u0 u x)).(eq_ind_r T
-(THead (Bind b) x t2) (\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3)))
-(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c
-(THead (Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H10:
-(ty3 g (CHead c (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead
-c (Bind b) x) t3 t6)) (ty3 g c (THead (Bind b) x t2) (THead (Bind b) u t3))
-(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c (Bind b) x) t3 x1)).(ty3_conv g
-c (THead (Bind b) u t3) (THead (Bind b) u x0) (ty3_bind g c u t0 H0 b t3 x0
-H10) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c x t0 (H1 i u0
-c x (fsubst0_snd i u0 c u x H9) e H6) b t2 t3 (H3 (S i) u0 (CHead c (Bind b)
-x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c (Bind b) x)
-(csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c (CHead e (Bind
-Abbr) u0) H6 u))) (pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3)
-(pc3_refl c (THead (Bind b) u t3)) i u0 c (THead (Bind b) x t3) (fsubst0_snd
-i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3) (subst0_fst u0 x u i H9 t3
-(Bind b))) e H6)))) (ty3_correct g (CHead c (Bind b) x) t2 t3 (H3 (S i) u0
-(CHead c (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead
-c (Bind b) x) (csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c
-(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3
-H2)) t5 H8)))) H7)) (\lambda (H7: (ex2 T (\lambda (t6: T).(eq T t5 (THead
-(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2
-t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6))) (\lambda
-(t6: T).(subst0 (s (Bind b) i) u0 t2 t6)) (ty3 g c t5 (THead (Bind b) u t3))
-(\lambda (x: T).(\lambda (H8: (eq T t5 (THead (Bind b) u x))).(\lambda (H9:
-(subst0 (s (Bind b) i) u0 t2 x)).(eq_ind_r T (THead (Bind b) u x) (\lambda
-(t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g
-(CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) u x) (THead (Bind b) u
-t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t3
-x0)).(ty3_bind g c u t0 H0 b x t3 (H3 (S i) u0 (CHead c (Bind b) u) x
-(fsubst0_snd (S i) u0 (CHead c (Bind b) u) t2 x H9) e (getl_head (Bind b) i c
-(CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g (CHead c (Bind b) u) x t3
-(H3 (S i) u0 (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c (Bind b)
-u) t2 x H9) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u)))) t5
-H8)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T
-t5 (THead (Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u
-u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2
-t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead
-(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
-(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c
-t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq
-T t5 (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i u0 u x0)).(\lambda
-(H10: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1)
-(\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6:
-T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) x0 x1) (THead
-(Bind b) u t3)) (\lambda (x: T).(\lambda (H11: (ty3 g (CHead c (Bind b) u) t3
-x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) x0) t3 t6)) (ty3 g c
-(THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_:
-(ty3 g (CHead c (Bind b) x0) t3 x2)).(ty3_conv g c (THead (Bind b) u t3)
-(THead (Bind b) u x) (ty3_bind g c u t0 H0 b t3 x H11) (THead (Bind b) x0 x1)
-(THead (Bind b) x0 t3) (ty3_bind g c x0 t0 (H1 i u0 c x0 (fsubst0_snd i u0 c
-u x0 H9) e H6) b x1 t3 (H3 (S i) u0 (CHead c (Bind b) x0) x1 (fsubst0_both (S
-i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead c (Bind b) x0) (csubst0_snd_bind
-b i u0 u x0 H9 c)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u)))
-(pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead
-(Bind b) u t3)) i u0 c (THead (Bind b) x0 t3) (fsubst0_snd i u0 c (THead
-(Bind b) u t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H9 t3 (Bind b)))
-e H6)))) (ty3_correct g (CHead c (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c
-(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead
-c (Bind b) x0) (csubst0_snd_bind b i u0 u x0 H9 c)) e (getl_head (Bind b) i c
-(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3
-H2)) t5 H8)))))) H7)) (subst0_gen_head (Bind b) u0 u t2 t5 i H5))))))
-(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda
-(H6: (getl i c (CHead e (Bind Abbr) u0))).(ex_ind T (\lambda (t5: T).(ty3 g
-(CHead c3 (Bind b) u) t3 t5)) (ty3 g c3 (THead (Bind b) u t2) (THead (Bind b)
-u t3)) (\lambda (x: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3
-x)).(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H5) e H6) b t2
-t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind
-b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H5 u)) e
-(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g
-(CHead c3 (Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2
-(fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) u)
-(csubst0_fst_bind b i c c3 u0 H5 u)) e (getl_head (Bind b) i c (CHead e (Bind
-Abbr) u0) H6 u)))))))) (\lambda (t5: T).(\lambda (H5: (subst0 i u0 (THead
-(Bind b) u t2) t5)).(\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c
-c3)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr)
-u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind b) u2 t2)))
-(\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6: T).(eq T t5 (THead
-(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ex3_2 T
-T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Bind b) u2 t6))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
-T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c3 t5 (THead
-(Bind b) u t3)) (\lambda (H8: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind
-b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2:
-T).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2))
-(ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9: (eq T t5
-(THead (Bind b) x t2))).(\lambda (H10: (subst0 i u0 u x)).(eq_ind_r T (THead
-(Bind b) x t2) (\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind
-T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead
-(Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g
-(CHead c3 (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3
-(Bind b) u) x0 t6)) (ty3 g c3 (THead (Bind b) x t2) (THead (Bind b) u t3))
-(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x0 x1)).(ex_ind T
-(\lambda (t6: T).(ty3 g (CHead c3 (Bind b) x) t3 t6)) (ty3 g c3 (THead (Bind
-b) x t2) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_: (ty3 g (CHead
-c3 (Bind b) x) t3 x2)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u
-x0) (ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3
-x0 H11) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c3 x t0 (H1 i
-u0 c3 x (fsubst0_both i u0 c u x H10 c3 H6) e H7) b t2 t3 (H3 (S i) u0 (CHead
-c3 (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3
-(Bind b) x) (csubst0_both_bind b i u0 u x H10 c c3 H6)) e (getl_head (Bind b)
-i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c (THead (Bind b) u t3)
-(THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u t3)) i u0 c3 (THead (Bind
-b) x t3) (fsubst0_both i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3)
-(subst0_fst u0 x u i H10 t3 (Bind b)) c3 H6) e H7)))) (ty3_correct g (CHead
-c3 (Bind b) x) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) x) t2 (fsubst0_fst (S i)
-u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) x) (csubst0_both_bind b i u0 u
-x H10 c c3 H6)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u))))))
-(ty3_correct g (CHead c3 (Bind b) u) t3 x0 H11)))) (ty3_correct g (CHead c3
-(Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0
-(CHead c (Bind b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0
-H6 u)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9))))
-H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6)))
-(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))).(ex2_ind T (\lambda (t6:
-T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i)
-u0 t2 t6)) (ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9:
-(eq T t5 (THead (Bind b) u x))).(\lambda (H10: (subst0 (s (Bind b) i) u0 t2
-x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t6: T).(ty3 g c3 t6 (THead
-(Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3
-t6)) (ty3 g c3 (THead (Bind b) u x) (THead (Bind b) u t3)) (\lambda (x0:
-T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3 x0)).(ty3_bind g c3 u t0 (H1
-i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b x t3 (H3 (S i) u0 (CHead c3
-(Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x H10 (CHead c3
-(Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind b) i c
-(CHead e (Bind Abbr) u0) H7 u))))) (ty3_correct g (CHead c3 (Bind b) u) x t3
-(H3 (S i) u0 (CHead c3 (Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b)
-u) t2 x H10 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e
-(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))) H8))
-(\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead
-(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
-(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2
-t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead
-(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
-(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c3
-t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq
-T t5 (THead (Bind b) x0 x1))).(\lambda (H10: (subst0 i u0 u x0)).(\lambda
-(H11: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1)
-(\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6:
-T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1)
-(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H12: (ty3 g (CHead c3 (Bind
-b) u) t3 x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) x t6))
-(ty3 g c3 (THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2:
-T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x x2)).(ex_ind T (\lambda (t6:
-T).(ty3 g (CHead c3 (Bind b) x0) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1)
-(THead (Bind b) u t3)) (\lambda (x3: T).(\lambda (_: (ty3 g (CHead c3 (Bind
-b) x0) t3 x3)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u x)
-(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3 x
-H12) (THead (Bind b) x0 x1) (THead (Bind b) x0 t3) (ty3_bind g c3 x0 t0 (H1 i
-u0 c3 x0 (fsubst0_both i u0 c u x0 H10 c3 H6) e H7) b x1 t3 (H3 (S i) u0
-(CHead c3 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1
-H11 (CHead c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e
-(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c
-(THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u
-t3)) i u0 c3 (THead (Bind b) x0 t3) (fsubst0_both i u0 c (THead (Bind b) u
-t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H10 t3 (Bind b)) c3 H6) e
-H7)))) (ty3_correct g (CHead c3 (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c3
-(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H11 (CHead
-c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e (getl_head
-(Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))))) (ty3_correct g (CHead c3
-(Bind b) u) t3 x H12)))) (ty3_correct g (CHead c3 (Bind b) u) t2 t3 (H3 (S i)
-u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2
-(CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind
-b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))))) H8)) (subst0_gen_head
-(Bind b) u0 u t2 t5 i H5)))))))) c2 t4 H4)))))))))))))))) (\lambda (c:
-C).(\lambda (w: T).(\lambda (u: T).(\lambda (H0: (ty3 g c w u)).(\lambda (H1:
-((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2:
-T).((fsubst0 i u0 c w c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind
-Abbr) u0)) \to (ty3 g c2 t2 u)))))))))).(\lambda (v: T).(\lambda (t0:
-T).(\lambda (H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3:
-((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2:
-T).((fsubst0 i u0 c v c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind
-Abbr) u0)) \to (ty3 g c2 t2 (THead (Bind Abst) u t0))))))))))).(\lambda (i:
-nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4:
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-H9) e H6) t2 t3 H0 (pc3_s c t3 x (pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c x
-(fsubst0_snd i u c t3 x H9) e H6))) t0 (H3 i u c x (fsubst0_snd i u c t3 x
-H9) e H6)) (pc3_fsubst0 c (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 t3)
-(pc3_refl c (THead (Flat Cast) t0 t3)) i u c (THead (Flat Cast) t0 x)
-(fsubst0_snd i u c (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 x)
-(subst0_snd (Flat Cast) u x t3 i H9 t0)) e H6)))) (ty3_correct g c x t0 (H3 i
-u c x (fsubst0_snd i u c t3 x H9) e H6))) t5 H8)))) H7)) (\lambda (H7: (ex2 T
-(\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6:
-T).(subst0 (s (Flat Cast) i) u t2 t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5
-(THead (Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2
-t6)) (ty3 g c t5 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H8: (eq
-T t5 (THead (Flat Cast) t3 x))).(\lambda (H9: (subst0 (s (Flat Cast) i) u t2
-x)).(eq_ind_r T (THead (Flat Cast) t3 x) (\lambda (t6: T).(ty3 g c t6 (THead
-(Flat Cast) t0 t3))) (ty3_cast g c x t3 (H1 (s (Flat Cast) i) u c x
-(fsubst0_snd (s (Flat Cast) i) u c t2 x H9) e H6) t0 H2) t5 H8)))) H7))
-(\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead
-(Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2)))
-(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2
-t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead
-(Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2)))
-(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g
-c t5 (THead (Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda
-(H8: (eq T t5 (THead (Flat Cast) x0 x1))).(\lambda (H9: (subst0 i u t3
-x0)).(\lambda (H10: (subst0 (s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead
-(Flat Cast) x0 x1) (\lambda (t6: T).(ty3 g c t6 (THead (Flat Cast) t0 t3)))
-(ex_ind T (\lambda (t6: T).(ty3 g c t0 t6)) (ty3 g c (THead (Flat Cast) x0
-x1) (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H11: (ty3 g c t0
-x)).(ty3_conv g c (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0)
-(ty3_cast g c t3 t0 H2 x H11) (THead (Flat Cast) x0 x1) (THead (Flat Cast) t0
-x0) (ty3_cast g c x1 x0 (ty3_conv g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c
-t3 x0 H9) e H6) x1 t3 (H1 (s (Flat Cast) i) u c x1 (fsubst0_snd (s (Flat
-Cast) i) u c t2 x1 H10) e H6) (pc3_s c t3 x0 (pc3_fsubst0 c t3 t3 (pc3_refl c
-t3) i u c x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t0 (H3 i u c x0
-(fsubst0_snd i u c t3 x0 H9) e H6)) (pc3_fsubst0 c (THead (Flat Cast) t0 t3)
-(THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat Cast) t0 t3)) i u c (THead
-(Flat Cast) t0 x0) (fsubst0_snd i u c (THead (Flat Cast) t0 t3) (THead (Flat
-Cast) t0 x0) (subst0_snd (Flat Cast) u x0 t3 i H9 t0)) e H6)))) (ty3_correct
-g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t5 H8)))))) H7))
-(subst0_gen_head (Flat Cast) u t3 t2 t5 i H5)))))) (\lambda (c3: C).(\lambda
-(H5: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e
-(Bind Abbr) u))).(ty3_cast g c3 t2 t3 (H1 i u c3 t2 (fsubst0_fst i u c t2 c3
-H5) e H6) t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H5) e H6)))))) (\lambda
-(t5: T).(\lambda (H5: (subst0 i u (THead (Flat Cast) t3 t2) t5)).(\lambda
-(c3: C).(\lambda (H6: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H7: (getl
-i c (CHead e (Bind Abbr) u))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5
-(THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2))) (ex2 T
-(\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6:
-T).(subst0 (s (Flat Cast) i) u t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda
-(t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat
-Cast) i) u t2 t6)))) (ty3 g c3 t5 (THead (Flat Cast) t0 t3)) (\lambda (H8:
-(ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2:
-T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t5 (THead (Flat
-Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)) (ty3 g c3 t5 (THead (Flat
-Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) x
-t2))).(\lambda (H10: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat Cast) x t2)
-(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda
-(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x t2) (THead (Flat
-Cast) t0 t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g c3 t0 x0)).(ty3_conv g
-c3 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x0 t0) (ty3_cast g c3 t3 t0
-(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x0 H11) (THead (Flat Cast) x
-t2) (THead (Flat Cast) t0 x) (ty3_cast g c3 t2 x (ty3_conv g c3 x t0 (H3 i u
-c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7) t2 t3 (H1 i u c3 t2
-(fsubst0_fst i u c t2 c3 H6) e H7) (pc3_s c3 t3 x (pc3_fsubst0 c t3 t3
-(pc3_refl c t3) i u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7))) t0 (H3 i
-u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7)) (pc3_fsubst0 c (THead (Flat
-Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat Cast) t0 t3))
-i u c3 (THead (Flat Cast) t0 x) (fsubst0_both i u c (THead (Flat Cast) t0 t3)
-(THead (Flat Cast) t0 x) (subst0_snd (Flat Cast) u x t3 i H10 t0) c3 H6) e
-H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e
-H7))) t5 H9)))) H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead
-(Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2
-t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6)))
-(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6)) (ty3 g c3 t5 (THead
-(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast)
-t3 x))).(\lambda (H10: (subst0 (s (Flat Cast) i) u t2 x)).(eq_ind_r T (THead
-(Flat Cast) t3 x) (\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3)))
-(ty3_cast g c3 x t3 (H1 i u c3 x (fsubst0_both i u c t2 x H10 c3 H6) e H7) t0
-(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7)) t5 H9)))) H8)) (\lambda
-(H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast)
-u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_:
-T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_:
-T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g c3 t5 (THead
-(Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t5
-(THead (Flat Cast) x0 x1))).(\lambda (H10: (subst0 i u t3 x0)).(\lambda (H11:
-(subst0 (s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1)
-(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda
-(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x0 x1) (THead (Flat
-Cast) t0 t3)) (\lambda (x: T).(\lambda (H12: (ty3 g c3 t0 x)).(ty3_conv g c3
-(THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c3 t3 t0 (H3 i
-u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x H12) (THead (Flat Cast) x0 x1)
-(THead (Flat Cast) t0 x0) (ty3_cast g c3 x1 x0 (ty3_conv g c3 x0 t0 (H3 i u
-c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7) x1 t3 (H1 i u c3 x1
-(fsubst0_both i u c t2 x1 H11 c3 H6) e H7) (pc3_s c3 t3 x0 (pc3_fsubst0 c t3
-t3 (pc3_refl c t3) i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7))) t0
-(H3 i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7)) (pc3_fsubst0 c
-(THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat
-Cast) t0 t3)) i u c3 (THead (Flat Cast) t0 x0) (fsubst0_both i u c (THead
-(Flat Cast) t0 t3) (THead (Flat Cast) t0 x0) (subst0_snd (Flat Cast) u x0 t3
-i H10 t0) c3 H6) e H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst
-i u c t3 c3 H6) e H7))) t5 H9)))))) H8)) (subst0_gen_head (Flat Cast) u t3 t2
-t5 i H5)))))))) c2 t4 H4)))))))))))))) c1 t1 t H))))).
-(* COMMENTS
-Initial nodes: 23439
-END *)
-
-theorem ty3_csubst0:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1
-t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c1
-(CHead e (Bind Abbr) u)) \to (\forall (c2: C).((csubst0 i u c1 c2) \to (ty3 g
-c2 t1 t2)))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (ty3 g c1 t1 t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (i:
-nat).(\lambda (H0: (getl i c1 (CHead e (Bind Abbr) u))).(\lambda (c2:
-C).(\lambda (H1: (csubst0 i u c1 c2)).(ty3_fsubst0 g c1 t1 t2 H i u c2 t1
-(fsubst0_fst i u c1 t1 c2 H1) e H0))))))))))).
-(* COMMENTS
-Initial nodes: 89
-END *)
-
-theorem ty3_subst0:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((ty3 g c t1
-t) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead e
-(Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (ty3 g c t2
-t)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
-(ty3 g c t1 t)).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H0: (getl i c (CHead e (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1:
-(subst0 i u t1 t2)).(ty3_fsubst0 g c t1 t H i u c t2 (fsubst0_snd i u c t1 t2
-H1) e H0))))))))))).
-(* COMMENTS
-Initial nodes: 89
-END *)
-
projects / helm.git / blobdiff