--- /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 "LambdaDelta-1/ty3/fwd.ma".
+
+include "LambdaDelta-1/pc3/fwd.ma".
+
+theorem ty3_lift:
+ \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((ty3 g e
+t1 t2) \to (\forall (c: C).(\forall (d: nat).(\forall (h: nat).((drop h d c
+e) \to (ty3 g c (lift h d t1) (lift h d t2))))))))))
+\def
+ \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (ty3 g e t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda (t0:
+T).(\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to
+(ty3 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda (t0:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda (H1: ((\forall (c0:
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h
+d t0) (lift h d t)))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3
+g c u t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d: nat).(\forall (h:
+nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d
+t3)))))))).(\lambda (H4: (pc3 c t3 t0)).(\lambda (c0: C).(\lambda (d:
+nat).(\lambda (h: nat).(\lambda (H5: (drop h d c0 c)).(ty3_conv g c0 (lift h
+d t0) (lift h d t) (H1 c0 d h H5) (lift h d u) (lift h d t3) (H3 c0 d h H5)
+(pc3_lift c0 c h d H5 t3 t0 H4)))))))))))))))) (\lambda (c: C).(\lambda (m:
+nat).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (_: (drop
+h d c0 c)).(eq_ind_r T (TSort m) (\lambda (t: T).(ty3 g c0 t (lift h d (TSort
+(next g m))))) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0
+(TSort m) t)) (ty3_sort g c0 m) (lift h d (TSort (next g m))) (lift_sort
+(next g m) h d)) (lift h d (TSort m)) (lift_sort m h d)))))))) (\lambda (n:
+nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c
+(CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u
+t)).(\lambda (H2: ((\forall (c0: C).(\forall (d0: nat).(\forall (h:
+nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u) (lift h d0
+t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: nat).(\lambda (H3:
+(drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0
+(lift (S n) O t))) (\lambda (H4: (lt n d0)).(let H5 \def (drop_getl_trans_le
+n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) u) H0)
+in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop n O c0 e0)))
+(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n) e0 e1))) (\lambda (_:
+C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (ty3 g c0 (lift h d0
+(TLRef n)) (lift h d0 (lift (S n) O t))) (\lambda (x0: C).(\lambda (x1:
+C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop h (minus d0 n) x0
+x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) u))).(let H9 \def (eq_ind
+nat (minus d0 n) (\lambda (n0: nat).(drop h n0 x0 x1)) H7 (S (minus d0 (S
+n))) (minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0
+(S n)) H9 Abbr d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1
+(Bind Abbr) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h (minus d0
+(S n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O t)))
+(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus
+d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x d)).(eq_ind_r T
+(TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t))))
+(eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3 g c0 (TLRef
+n) (lift h n0 (lift (S n) O t)))) (eq_ind_r T (lift (S n) O (lift h (minus d0
+(S n)) t)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat d0 (\lambda
+(_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S n)) t))))
+(ty3_abbr g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0 (CHead x
+(Bind Abbr) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus d0 (S n))
+t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n)))
+(le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S
+n) O t)) (lift_d t h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0
+(le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0
+H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n
+h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t)))) (eq_ind nat
+(S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O
+t)))) (eq_ind_r T (lift (plus h (S n)) O t) (\lambda (t0: T).(ty3 g c0 (TLRef
+(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0
+(TLRef (plus n h)) (lift n0 O t))) (ty3_abbr g (plus n h) c0 d u
+(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) t H1) (plus
+h (S n)) (plus_sym h (S n))) (lift h d0 (lift (S n) O t)) (lift_free t (S n)
+h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O)
+n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n))
+(plus n (S O)) (plus_sym n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h
+d0 H4)))))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda
+(t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c0: C).(\forall
+(d0: nat).(\forall (h: nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u)
+(lift h d0 t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h:
+nat).(\lambda (H3: (drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0
+(TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda (H4: (lt n d0)).(let H5
+\def (drop_getl_trans_le n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3
+(CHead d (Bind Abst) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_:
+C).(drop n O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n)
+e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst)
+u)))) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda
+(x0: C).(\lambda (x1: C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop
+h (minus d0 n) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) u))).(let
+H9 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(drop h n0 x0 x1)) H7 (S
+(minus d0 (S n))) (minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0
+h (minus d0 (S n)) H9 Abst d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0
+(CHead c1 (Bind Abst) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h
+(minus d0 (S n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S
+n) O u))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift
+h (minus d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x
+d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S
+n) O u)))) (eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3
+g c0 (TLRef n) (lift h n0 (lift (S n) O u)))) (eq_ind_r T (lift (S n) O (lift
+h (minus d0 (S n)) u)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat
+d0 (\lambda (_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S
+n)) u)))) (ty3_abst g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0
+(CHead x (Bind Abst) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus
+d0 (S n)) t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n)))
+(le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S
+n) O u)) (lift_d u h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0
+(le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0
+H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n
+h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O u)))) (eq_ind nat
+(S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O
+u)))) (eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t0: T).(ty3 g c0 (TLRef
+(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0
+(TLRef (plus n h)) (lift n0 O u))) (ty3_abst g (plus n h) c0 d u
+(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) t H1) (plus
+h (S n)) (plus_sym h (S n))) (lift h d0 (lift (S n) O u)) (lift_free u (S n)
+h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O)
+n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n))
+(plus n (S O)) (plus_sym n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h
+d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c0: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d
+t)))))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3
+g (CHead c (Bind b) u) t0 t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c0 (CHead c (Bind b) u)) \to (ty3 g c0
+(lift h d t0) (lift h d t3)))))))).(\lambda (c0: C).(\lambda (d:
+nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead
+(Bind b) (lift h d u) (lift h (s (Bind b) d) t0)) (\lambda (t4: T).(ty3 g c0
+t4 (lift h d (THead (Bind b) u t3)))) (eq_ind_r T (THead (Bind b) (lift h d
+u) (lift h (s (Bind b) d) t3)) (\lambda (t4: T).(ty3 g c0 (THead (Bind b)
+(lift h d u) (lift h (s (Bind b) d) t0)) t4)) (ty3_bind g c0 (lift h d u)
+(lift h d t) (H1 c0 d h H4) b (lift h (S d) t0) (lift h (S d) t3) (H3 (CHead
+c0 (Bind b) (lift h d u)) (S d) h (drop_skip_bind h d c0 c H4 b u))) (lift h
+d (THead (Bind b) u t3)) (lift_head (Bind b) u t3 h d)) (lift h d (THead
+(Bind b) u t0)) (lift_head (Bind b) u t0 h d)))))))))))))))) (\lambda (c:
+C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w u)).(\lambda (H1:
+((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to
+(ty3 g c0 (lift h d w) (lift h d u)))))))).(\lambda (v: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall
+(c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0
+(lift h d v) (lift h d (THead (Bind Abst) u t))))))))).(\lambda (c0:
+C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0
+c)).(eq_ind_r T (THead (Flat Appl) (lift h d w) (lift h (s (Flat Appl) d) v))
+(\lambda (t0: T).(ty3 g c0 t0 (lift h d (THead (Flat Appl) w (THead (Bind
+Abst) u t))))) (eq_ind_r T (THead (Flat Appl) (lift h d w) (lift h (s (Flat
+Appl) d) (THead (Bind Abst) u t))) (\lambda (t0: T).(ty3 g c0 (THead (Flat
+Appl) (lift h d w) (lift h (s (Flat Appl) d) v)) t0)) (eq_ind_r T (THead
+(Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s (Flat
+Appl) d)) t)) (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) (lift h d w)
+(lift h (s (Flat Appl) d) v)) (THead (Flat Appl) (lift h d w) t0))) (ty3_appl
+g c0 (lift h d w) (lift h d u) (H1 c0 d h H4) (lift h d v) (lift h (S d) t)
+(eq_ind T (lift h d (THead (Bind Abst) u t)) (\lambda (t0: T).(ty3 g c0 (lift
+h d v) t0)) (H3 c0 d h H4) (THead (Bind Abst) (lift h d u) (lift h (S d) t))
+(lift_bind Abst u t h d))) (lift h (s (Flat Appl) d) (THead (Bind Abst) u t))
+(lift_head (Bind Abst) u t h (s (Flat Appl) d))) (lift h d (THead (Flat Appl)
+w (THead (Bind Abst) u t))) (lift_head (Flat Appl) w (THead (Bind Abst) u t)
+h d)) (lift h d (THead (Flat Appl) w v)) (lift_head (Flat Appl) w v h
+d))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda
+(_: (ty3 g c t0 t3)).(\lambda (H1: ((\forall (c0: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d t0) (lift h d
+t3)))))))).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3:
+((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to
+(ty3 g c0 (lift h d t3) (lift h d t4)))))))).(\lambda (c0: C).(\lambda (d:
+nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead
+(Flat Cast) (lift h d t3) (lift h (s (Flat Cast) d) t0)) (\lambda (t: T).(ty3
+g c0 t (lift h d (THead (Flat Cast) t4 t3)))) (eq_ind_r T (THead (Flat Cast)
+(lift h d t4) (lift h (s (Flat Cast) d) t3)) (\lambda (t: T).(ty3 g c0 (THead
+(Flat Cast) (lift h d t3) (lift h (s (Flat Cast) d) t0)) t)) (ty3_cast g c0
+(lift h (s (Flat Cast) d) t0) (lift h (s (Flat Cast) d) t3) (H1 c0 (s (Flat
+Cast) d) h H4) (lift h d t4) (H3 c0 d h H4)) (lift h d (THead (Flat Cast) t4
+t3)) (lift_head (Flat Cast) t4 t3 h d)) (lift h d (THead (Flat Cast) t3 t0))
+(lift_head (Flat Cast) t3 t0 h d)))))))))))))) e t1 t2 H))))).
+
+theorem ty3_correct:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
+t1 t2) \to (ex T (\lambda (t: T).(ty3 g c t2 t)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda
+(t0: T).(ex T (\lambda (t3: T).(ty3 g c0 t0 t3)))))) (\lambda (c0:
+C).(\lambda (t0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t0 t)).(\lambda
+(_: (ex T (\lambda (t3: T).(ty3 g c0 t t3)))).(\lambda (u: T).(\lambda (t3:
+T).(\lambda (_: (ty3 g c0 u t3)).(\lambda (_: (ex T (\lambda (t4: T).(ty3 g
+c0 t3 t4)))).(\lambda (_: (pc3 c0 t3 t0)).(ex_intro T (\lambda (t4: T).(ty3 g
+c0 t0 t4)) t H0))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro T
+(\lambda (t: T).(ty3 g c0 (TSort (next g m)) t)) (TSort (next g (next g m)))
+(ty3_sort g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr)
+u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex T (\lambda
+(t0: T).(ty3 g d t t0)))).(let H3 \def H2 in (ex_ind T (\lambda (t0: T).(ty3
+g d t t0)) (ex T (\lambda (t0: T).(ty3 g c0 (lift (S n) O t) t0))) (\lambda
+(x: T).(\lambda (H4: (ty3 g d t x)).(ex_intro T (\lambda (t0: T).(ty3 g c0
+(lift (S n) O t) t0)) (lift (S n) O x) (ty3_lift g d t x H4 c0 O (S n)
+(getl_drop Abbr c0 d u n H0))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind
+Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (_: (ex T
+(\lambda (t0: T).(ty3 g d t t0)))).(ex_intro T (\lambda (t0: T).(ty3 g c0
+(lift (S n) O u) t0)) (lift (S n) O t) (ty3_lift g d u t H1 c0 O (S n)
+(getl_drop Abst c0 d u n H0))))))))))) (\lambda (c0: C).(\lambda (u:
+T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda (_: (ex T (\lambda
+(t0: T).(ty3 g c0 t t0)))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3:
+T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (H3: (ex T
+(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)))).(let H4 \def H3 in
+(ex_ind T (\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)) (ex T
+(\lambda (t4: T).(ty3 g c0 (THead (Bind b) u t3) t4))) (\lambda (x:
+T).(\lambda (H5: (ty3 g (CHead c0 (Bind b) u) t3 x)).(ex_intro T (\lambda
+(t4: T).(ty3 g c0 (THead (Bind b) u t3) t4)) (THead (Bind b) u x) (ty3_bind g
+c0 u t H0 b t3 x H5)))) H4)))))))))))) (\lambda (c0: C).(\lambda (w:
+T).(\lambda (u: T).(\lambda (H0: (ty3 g c0 w u)).(\lambda (H1: (ex T (\lambda
+(t: T).(ty3 g c0 u t)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g
+c0 v (THead (Bind Abst) u t))).(\lambda (H3: (ex T (\lambda (t0: T).(ty3 g c0
+(THead (Bind Abst) u t) t0)))).(let H4 \def H1 in (ex_ind T (\lambda (t0:
+T).(ty3 g c0 u t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w
+(THead (Bind Abst) u t)) t0))) (\lambda (x: T).(\lambda (_: (ty3 g c0 u
+x)).(let H6 \def H3 in (ex_ind T (\lambda (t0: T).(ty3 g c0 (THead (Bind
+Abst) u t) t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead
+(Bind Abst) u t)) t0))) (\lambda (x0: T).(\lambda (H7: (ty3 g c0 (THead (Bind
+Abst) u t) x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0
+(THead (Bind Abst) u t3) x0))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u
+t0))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind Abst) u) t
+t3))) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead (Bind
+Abst) u t)) t0))) (\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c0
+(THead (Bind Abst) u x1) x0)).(\lambda (H9: (ty3 g c0 u x2)).(\lambda (H10:
+(ty3 g (CHead c0 (Bind Abst) u) t x1)).(ex_intro T (\lambda (t0: T).(ty3 g c0
+(THead (Flat Appl) w (THead (Bind Abst) u t)) t0)) (THead (Flat Appl) w
+(THead (Bind Abst) u x1)) (ty3_appl g c0 w u H0 (THead (Bind Abst) u t) x1
+(ty3_bind g c0 u x2 H9 Abst t x1 H10)))))))) (ty3_gen_bind g Abst c0 u t x0
+H7)))) H6)))) H4))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t3:
+T).(\lambda (_: (ty3 g c0 t0 t3)).(\lambda (_: (ex T (\lambda (t: T).(ty3 g
+c0 t3 t)))).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 t3 t4)).(\lambda (H3:
+(ex T (\lambda (t: T).(ty3 g c0 t4 t)))).(let H4 \def H3 in (ex_ind T
+(\lambda (t: T).(ty3 g c0 t4 t)) (ex T (\lambda (t: T).(ty3 g c0 (THead (Flat
+Cast) t4 t3) t))) (\lambda (x: T).(\lambda (H5: (ty3 g c0 t4 x)).(ex_intro T
+(\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t4 t3) t)) (THead (Flat Cast) x
+t4) (ty3_cast g c0 t3 t4 H2 x H5)))) H4)))))))))) c t1 t2 H))))).
+
+theorem ty3_unique:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u
+t1) \to (\forall (t2: T).((ty3 g c u t2) \to (pc3 c t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H:
+(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).(\forall (t2: T).((ty3 g c0 t t2) \to (pc3 c0 t0 t2)))))) (\lambda (c0:
+C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda
+(_: ((\forall (t3: T).((ty3 g c0 t2 t3) \to (pc3 c0 t t3))))).(\lambda (u0:
+T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H3: ((\forall
+(t3: T).((ty3 g c0 u0 t3) \to (pc3 c0 t0 t3))))).(\lambda (H4: (pc3 c0 t0
+t2)).(\lambda (t3: T).(\lambda (H5: (ty3 g c0 u0 t3)).(pc3_t t0 c0 t2 (pc3_s
+c0 t2 t0 H4) t3 (H3 t3 H5)))))))))))))) (\lambda (c0: C).(\lambda (m:
+nat).(\lambda (t2: T).(\lambda (H0: (ty3 g c0 (TSort m) t2)).(ty3_gen_sort g
+c0 t2 m H0))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda
+(u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u0))).(\lambda (t:
+T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall (t2: T).((ty3 g d u0
+t2) \to (pc3 d t t2))))).(\lambda (t2: T).(\lambda (H3: (ty3 g c0 (TLRef n)
+t2)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
+T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1:
+T).(\lambda (t0: T).(ty3 g e u1 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1)))))
+(\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))) (pc3 c0
+(lift (S n) O t) t2) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (_:
+T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda
+(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e:
+C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0)
+t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g
+e u1 t0)))) (pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O x2) t2)).(\lambda
+(H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g x0 x1
+x2)).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n
+c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n
+H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
+(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind
+Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr)
+x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t0)
+\Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1)
+(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in
+(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0:
+T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H8 u0 H10) in (let H13 \def
+(eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def
+(eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0))) H12 d
+H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3 g c1 u0 x2)) H13 d
+H11) in (pc3_t (lift (S n) O x2) c0 (lift (S n) O t) (pc3_lift c0 d (S n) O
+(getl_drop Abbr c0 d u0 n H14) t x2 (H2 x2 H15)) t2 H5))))))) H9)))))))))
+H4)) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_:
+T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1:
+T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda
+(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst)
+u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))
+(pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2:
+T).(\lambda (_: (pc3 c0 (lift (S n) O x1) t2)).(\lambda (H6: (getl n c0
+(CHead x0 (Bind Abst) x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def
+(eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead
+x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0
+(Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abbr) u0)
+(\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 True | Abst \Rightarrow False | Void
+\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abst)
+x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1)
+H6)) in (False_ind (pc3 c0 (lift (S n) O t) t2) H9))))))))) H4))
+(ty3_gen_lref g c0 t2 n H3)))))))))))) (\lambda (n: nat).(\lambda (c0:
+C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind
+Abst) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_:
+((\forall (t2: T).((ty3 g d u0 t2) \to (pc3 d t t2))))).(\lambda (t2:
+T).(\lambda (H3: (ty3 g c0 (TLRef n) t2)).(or_ind (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda
+(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
+u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))))
+(ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift
+(S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda
+(t0: T).(ty3 g e u1 t0))))) (pc3 c0 (lift (S n) O u0) t2) (\lambda (H4:
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift
+(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda
+(t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_:
+T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda
+(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e:
+C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O
+u0) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3
+c0 (lift (S n) O x2) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abbr)
+x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind
+Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1)
+(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in
+(let H9 \def (eq_ind C (CHead d (Bind Abst) u0) (\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 x0 (Bind Abbr) x1) (getl_mono c0 (CHead d
+(Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind (pc3 c0
+(lift (S n) O u0) t2) H9))))))))) H4)) (\lambda (H4: (ex3_3 C T T (\lambda
+(_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2))))
+(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1
+t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_:
+T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1:
+T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O u0) t2) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O
+x1) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7:
+(ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda
+(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d
+(Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (f_equal
+C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abst) u0)
+(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead
+x0 (Bind Abst) x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match
+e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _
+t0) \Rightarrow t0])) (CHead d (Bind Abst) u0) (CHead x0 (Bind Abst) x1)
+(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in
+(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0:
+T).(getl n c0 (CHead x0 (Bind Abst) t0))) H8 u0 H10) in (let H13 \def
+(eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def
+(eq_ind_r T x1 (\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)) H5 u0 H10) in
+(let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind
+Abst) u0))) H12 d H11) in (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3
+g c1 u0 x2)) H13 d H11) in H14))))))) H9))))))))) H4)) (ty3_gen_lref g c0 t2
+n H3)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda
+(_: (ty3 g c0 u0 t)).(\lambda (_: ((\forall (t2: T).((ty3 g c0 u0 t2) \to
+(pc3 c0 t t2))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t2: T).(\lambda
+(_: (ty3 g (CHead c0 (Bind b) u0) t0 t2)).(\lambda (H3: ((\forall (t3:
+T).((ty3 g (CHead c0 (Bind b) u0) t0 t3) \to (pc3 (CHead c0 (Bind b) u0) t2
+t3))))).(\lambda (t3: T).(\lambda (H4: (ty3 g c0 (THead (Bind b) u0 t0)
+t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b)
+u0 t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u0 t5))) (\lambda
+(t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u0) t0 t4))) (pc3 c0 (THead
+(Bind b) u0 t2) t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (pc3 c0
+(THead (Bind b) u0 x0) t3)).(\lambda (_: (ty3 g c0 u0 x1)).(\lambda (H7: (ty3
+g (CHead c0 (Bind b) u0) t0 x0)).(pc3_t (THead (Bind b) u0 x0) c0 (THead
+(Bind b) u0 t2) (pc3_head_2 c0 u0 t2 x0 (Bind b) (H3 x0 H7)) t3 H5))))))
+(ty3_gen_bind g b c0 u0 t0 t3 H4)))))))))))))) (\lambda (c0: C).(\lambda (w:
+T).(\lambda (u0: T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2:
+T).((ty3 g c0 w t2) \to (pc3 c0 u0 t2))))).(\lambda (v: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 t))).(\lambda (H3:
+((\forall (t2: T).((ty3 g c0 v t2) \to (pc3 c0 (THead (Bind Abst) u0 t)
+t2))))).(\lambda (t2: T).(\lambda (H4: (ty3 g c0 (THead (Flat Appl) w v)
+t2)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t0: T).(pc3 c0 (THead (Flat
+Appl) w (THead (Bind Abst) u1 t0)) t2))) (\lambda (u1: T).(\lambda (t0:
+T).(ty3 g c0 v (THead (Bind Abst) u1 t0)))) (\lambda (u1: T).(\lambda (_:
+T).(ty3 g c0 w u1))) (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t))
+t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (pc3 c0 (THead (Flat
+Appl) w (THead (Bind Abst) x0 x1)) t2)).(\lambda (H6: (ty3 g c0 v (THead
+(Bind Abst) x0 x1))).(\lambda (_: (ty3 g c0 w x0)).(pc3_t (THead (Flat Appl)
+w (THead (Bind Abst) x0 x1)) c0 (THead (Flat Appl) w (THead (Bind Abst) u0
+t)) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) (THead (Bind Abst) x0 x1) (H3
+(THead (Bind Abst) x0 x1) H6) w Appl) t2 H5)))))) (ty3_gen_appl g c0 w v t2
+H4))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda
+(_: (ty3 g c0 t0 t2)).(\lambda (_: ((\forall (t3: T).((ty3 g c0 t0 t3) \to
+(pc3 c0 t2 t3))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2 t3)).(\lambda
+(H3: ((\forall (t4: T).((ty3 g c0 t2 t4) \to (pc3 c0 t3 t4))))).(\lambda (t4:
+T).(\lambda (H4: (ty3 g c0 (THead (Flat Cast) t2 t0) t4)).(ex3_ind T (\lambda
+(t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t0
+t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)) (pc3 c0 (THead (Flat Cast) t3 t2) t4)
+(\lambda (x0: T).(\lambda (H5: (pc3 c0 (THead (Flat Cast) x0 t2)
+t4)).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (H7: (ty3 g c0 t2 x0)).(pc3_t
+(THead (Flat Cast) x0 t2) c0 (THead (Flat Cast) t3 t2) (pc3_head_1 c0 t3 x0
+(H3 x0 H7) (Flat Cast) t2) t4 H5))))) (ty3_gen_cast g c0 t0 t2 t4
+H4)))))))))))) c u t1 H))))).
+
+theorem ty3_gen_abst_abst:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall
+(t2: T).((ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (ex2
+T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst)
+u) t1 t2))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (H: (ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u
+t2))).(ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) u t2) t)) (ex2 T
+(\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u)
+t1 t2))) (\lambda (x: T).(\lambda (H0: (ty3 g c (THead (Bind Abst) u t2)
+x)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c (THead (Bind Abst)
+u t3) x))) (\lambda (_: T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t3:
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t2 t3))) (ex2 T (\lambda
+(w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c (THead (Bind Abst) u
+x0) x)).(\lambda (_: (ty3 g c u x1)).(\lambda (H3: (ty3 g (CHead c (Bind
+Abst) u) t2 x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c
+(THead (Bind Abst) u t3) (THead (Bind Abst) u t2)))) (\lambda (_: T).(\lambda
+(t: T).(ty3 g c u t))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c (Bind
+Abst) u) t1 t3))) (ex2 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3
+g (CHead c (Bind Abst) u) t1 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
+(H4: (pc3 c (THead (Bind Abst) u x2) (THead (Bind Abst) u t2))).(\lambda (H5:
+(ty3 g c u x3)).(\lambda (H6: (ty3 g (CHead c (Bind Abst) u) t1 x2)).(let H_y
+\def (pc3_gen_abst_shift c u x2 t2 H4) in (ex_intro2 T (\lambda (w: T).(ty3 g
+c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)) x3 H5
+(ty3_conv g (CHead c (Bind Abst) u) t2 x0 H3 t1 x2 H6 H_y))))))))
+(ty3_gen_bind g Abst c u t1 (THead (Bind Abst) u t2) H))))))) (ty3_gen_bind g
+Abst c u t2 x H0)))) (ty3_correct g c (THead (Bind Abst) u t1) (THead (Bind
+Abst) u t2) H))))))).
+
+theorem ty3_typecheck:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (v: T).((ty3 g c t
+v) \to (ex T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (v: T).(\lambda (H:
+(ty3 g c t v)).(ex_ind T (\lambda (t0: T).(ty3 g c v t0)) (ex T (\lambda (u:
+T).(ty3 g c (THead (Flat Cast) v t) u))) (\lambda (x: T).(\lambda (H0: (ty3 g
+c v x)).(ex_intro T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u))
+(THead (Flat Cast) x v) (ty3_cast g c t v H x H0)))) (ty3_correct g c t v
+H)))))).
+
+theorem ty3_getl_subst0:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
+u) \to (\forall (v0: T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 t
+t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c (CHead d
+(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
+(ty3 g c t u)).(ty3_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
+T).(\forall (v0: T).(\forall (t2: T).(\forall (i: nat).((subst0 i v0 t0 t2)
+\to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d
+(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda
+(c0: C).(\lambda (t2: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2
+t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
+nat).((subst0 i v0 t2 t1) \to (\forall (b: B).(\forall (d: C).(\forall (v:
+T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
+w))))))))))))).(\lambda (u0: T).(\lambda (t1: T).(\lambda (_: (ty3 g c0 u0
+t1)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i:
+nat).((subst0 i v0 u0 t3) \to (\forall (b: B).(\forall (d: C).(\forall (v:
+T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
+w))))))))))))).(\lambda (_: (pc3 c0 t1 t2)).(\lambda (v0: T).(\lambda (t3:
+T).(\lambda (i: nat).(\lambda (H5: (subst0 i v0 u0 t3)).(\lambda (b:
+B).(\lambda (d: C).(\lambda (v: T).(\lambda (H6: (getl i c0 (CHead d (Bind b)
+v))).(H3 v0 t3 i H5 b d v H6))))))))))))))))))) (\lambda (c0: C).(\lambda (m:
+nat).(\lambda (v0: T).(\lambda (t0: T).(\lambda (i: nat).(\lambda (H0:
+(subst0 i v0 (TSort m) t0)).(\lambda (b: B).(\lambda (d: C).(\lambda (v:
+T).(\lambda (_: (getl i c0 (CHead d (Bind b) v))).(subst0_gen_sort v0 t0 i m
+H0 (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda (n:
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n
+c0 (CHead d (Bind Abbr) u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0
+t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
+nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d0: C).(\forall (v:
+T).((getl i d (CHead d0 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v
+w))))))))))))).(\lambda (v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
+(H3: (subst0 i v0 (TLRef n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda
+(v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(land_ind (eq nat n
+i) (eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w)))
+(\lambda (H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7
+\def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v)))
+H4 n H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1:
+C).(getl n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
+Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (let H9 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0)
+(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0
+(Bind b) v) H7)) in ((let H10 \def (f_equal C B (\lambda (e: C).(match e in C
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0)
+(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0
+(Bind b) v) H7)) in ((let H11 \def (f_equal C T (\lambda (e: C).(match e in C
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t2)
+\Rightarrow t2])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v) (getl_mono
+c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12:
+(eq B Abbr b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v
+(\lambda (t2: T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T
+u0 (\lambda (t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def
+(eq_ind_r C d0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d
+H13) in (eq_ind C d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w))))
+(let H16 \def (eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0)
+u0))) H15 Abbr H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0
+H13)) v H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n
+H3)))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst)
+u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0 t0)).(\lambda (_: ((\forall
+(v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) \to (\forall
+(b: B).(\forall (d0: C).(\forall (v: T).((getl i d (CHead d0 (Bind b) v)) \to
+(ex T (\lambda (w: T).(ty3 g d0 v w))))))))))))).(\lambda (v0: T).(\lambda
+(t1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v0 (TLRef n) t1)).(\lambda
+(b: B).(\lambda (d0: C).(\lambda (v: T).(\lambda (H4: (getl i c0 (CHead d0
+(Bind b) v))).(land_ind (eq nat n i) (eq T t1 (lift (S n) O v0)) (ex T
+(\lambda (w: T).(ty3 g d0 v w))) (\lambda (H5: (eq nat n i)).(\lambda (_: (eq
+T t1 (lift (S n) O v0))).(let H7 \def (eq_ind_r nat i (\lambda (n0:
+nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n H5) in (let H8 \def (eq_ind C
+(CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead d0 (Bind
+b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7))
+in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1]))
+(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
+Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H10 \def (f_equal C B
+(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
+\Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda
+(_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])]))
+(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
+Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H11 \def (f_equal C T
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2])) (CHead d (Bind Abst) u0)
+(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0
+(Bind b) v) H7)) in (\lambda (H12: (eq B Abst b)).(\lambda (H13: (eq C d
+d0)).(let H14 \def (eq_ind_r T v (\lambda (t2: T).(getl n c0 (CHead d0 (Bind
+b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda (t2: T).(ex T (\lambda (w:
+T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0 (\lambda (c1: C).(getl n
+c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C d (\lambda (c1: C).(ex T
+(\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def (eq_ind_r B b (\lambda (b0:
+B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abst H12) in (ex_intro T (\lambda
+(w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v H11))))) H10)) H9))))))
+(subst0_gen_lref v0 t1 i n H3)))))))))))))))))) (\lambda (c0: C).(\lambda
+(u0: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H1:
+((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) \to
+(\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b)
+v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (b:
+B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b)
+u0) t1 t2)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i:
+nat).((subst0 i v0 t1 t3) \to (\forall (b0: B).(\forall (d: C).(\forall (v:
+T).((getl i (CHead c0 (Bind b) u0) (CHead d (Bind b0) v)) \to (ex T (\lambda
+(w: T).(ty3 g d v w))))))))))))).(\lambda (v0: T).(\lambda (t3: T).(\lambda
+(i: nat).(\lambda (H4: (subst0 i v0 (THead (Bind b) u0 t1) t3)).(\lambda (b0:
+B).(\lambda (d: C).(\lambda (v: T).(\lambda (H5: (getl i c0 (CHead d (Bind
+b0) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1)))
+(\lambda (u2: T).(subst0 i v0 u0 u2))) (ex2 T (\lambda (t4: T).(eq T t3
+(THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2
+t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_:
+T).(\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))) (ex T (\lambda (w:
+T).(ty3 g d v w))) (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T t3 (THead
+(Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0 u2)))).(ex2_ind T (\lambda
+(u2: T).(eq T t3 (THead (Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0
+u2)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T
+t3 (THead (Bind b) x t1))).(\lambda (H8: (subst0 i v0 u0 x)).(H1 v0 x i H8 b0
+d v H5)))) H6)) (\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Bind
+b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))).(ex2_ind T
+(\lambda (t4: T).(eq T t3 (THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0
+(s (Bind b) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x:
+T).(\lambda (_: (eq T t3 (THead (Bind b) u0 x))).(\lambda (H8: (subst0 (s
+(Bind b) i) v0 t1 x)).(H3 v0 x (S i) H8 b0 d v (getl_head (Bind b) i c0
+(CHead d (Bind b0) v) H5 u0))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4:
+T).(subst0 (s (Bind b) i) v0 t1 t4))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4:
+T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex T (\lambda (w: T).(ty3 g d v w)))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t3 (THead (Bind b) x0
+x1))).(\lambda (H8: (subst0 i v0 u0 x0)).(\lambda (_: (subst0 (s (Bind b) i)
+v0 t1 x1)).(H1 v0 x0 i H8 b0 d v H5)))))) H6)) (subst0_gen_head (Bind b) v0
+u0 t1 t3 i H4)))))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda
+(u0: T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (H1: ((\forall (v0:
+T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 w t0) \to (\forall (b:
+B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) v)) \to (ex
+T (\lambda (w0: T).(ty3 g d v w0))))))))))))).(\lambda (v: T).(\lambda (t0:
+T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 t0))).(\lambda (H3:
+((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 v t1) \to
+(\forall (b: B).(\forall (d: C).(\forall (v1: T).((getl i c0 (CHead d (Bind
+b) v1)) \to (ex T (\lambda (w0: T).(ty3 g d v1 w0))))))))))))).(\lambda (v0:
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat
+Appl) w v) t1)).(\lambda (b: B).(\lambda (d: C).(\lambda (v1: T).(\lambda
+(H5: (getl i c0 (CHead d (Bind b) v1))).(or3_ind (ex2 T (\lambda (u2: T).(eq
+T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w u2))) (ex2 T
+(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0
+(s (Flat Appl) i) v0 v t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq
+T t1 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i
+v0 w u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v
+t2)))) (ex T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (H6: (ex2 T (\lambda
+(u2: T).(eq T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T t1 (THead (Flat Appl) u2 v)))
+(\lambda (u2: T).(subst0 i v0 w u2)) (ex T (\lambda (w0: T).(ty3 g d v1 w0)))
+(\lambda (x: T).(\lambda (_: (eq T t1 (THead (Flat Appl) x v))).(\lambda (H8:
+(subst0 i v0 w x)).(H1 v0 x i H8 b d v1 H5)))) H6)) (\lambda (H6: (ex2 T
+(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0
+(s (Flat Appl) i) v0 v t2)))).(ex2_ind T (\lambda (t2: T).(eq T t1 (THead
+(Flat Appl) w t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2)) (ex
+T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (x: T).(\lambda (_: (eq T t1
+(THead (Flat Appl) w x))).(\lambda (H8: (subst0 (s (Flat Appl) i) v0 v
+x)).(H3 v0 x (s (Flat Appl) i) H8 b d v1 H5)))) H6)) (\lambda (H6: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))) (ex T (\lambda (w0:
+T).(ty3 g d v1 w0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t1
+(THead (Flat Appl) x0 x1))).(\lambda (_: (subst0 i v0 w x0)).(\lambda (H9:
+(subst0 (s (Flat Appl) i) v0 v x1)).(H3 v0 x1 (s (Flat Appl) i) H9 b d v1
+H5)))))) H6)) (subst0_gen_head (Flat Appl) v0 w v t1 i H4)))))))))))))))))))
+(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1
+t2)).(\lambda (H1: ((\forall (v0: T).(\forall (t0: T).(\forall (i:
+nat).((subst0 i v0 t1 t0) \to (\forall (b: B).(\forall (d: C).(\forall (v:
+T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
+w))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (H3:
+((\forall (v0: T).(\forall (t3: T).(\forall (i: nat).((subst0 i v0 t2 t3) \to
+(\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b)
+v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (v0:
+T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat
+Cast) t2 t1) t3)).(\lambda (b: B).(\lambda (d: C).(\lambda (v: T).(\lambda
+(H5: (getl i c0 (CHead d (Bind b) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T
+t3 (THead (Flat Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2))) (ex2 T
+(\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4:
+T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex3_2 T T (\lambda (u2: T).(\lambda
+(t4: T).(eq T t3 (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i v0 t2 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat
+Cast) i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (H6:
+(ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Cast) u2 t1))) (\lambda (u2:
+T).(subst0 i v0 t2 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat
+Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2)) (ex T (\lambda (w:
+T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Flat Cast) x
+t1))).(\lambda (H8: (subst0 i v0 t2 x)).(H3 v0 x i H8 b d v H5)))) H6))
+(\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4)))
+(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4)))).(ex2_ind T (\lambda
+(t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4: T).(subst0 (s
+(Flat Cast) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x:
+T).(\lambda (_: (eq T t3 (THead (Flat Cast) t2 x))).(\lambda (H8: (subst0 (s
+(Flat Cast) i) v0 t1 x)).(H1 v0 x (s (Flat Cast) i) H8 b d v H5)))) H6))
+(\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
+(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2)))
+(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1
+t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
+(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2)))
+(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex T
+(\lambda (w: T).(ty3 g d v w))) (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(_: (eq T t3 (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i v0 t2
+x0)).(\lambda (_: (subst0 (s (Flat Cast) i) v0 t1 x1)).(H3 v0 x0 i H8 b d v
+H5)))))) H6)) (subst0_gen_head (Flat Cast) v0 t2 t1 t3 i H4))))))))))))))))))
+c t u H))))).
+
projects / helm.git / blobdiff