include "LambdaDelta-1/ty3/pr3_props.ma".
-include "LambdaDelta-1/tau0/defs.ma".
+include "LambdaDelta-1/tau0/fwd.ma".
theorem ty3_tau0:
\forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u
((\forall (t4: T).((tau0 g c0 u0 t4) \to (ty3 g c0 u0 t4))))).(\lambda (_:
(pc3 c0 t3 t2)).(\lambda (t0: T).(\lambda (H5: (tau0 g c0 u0 t0)).(H3 t0
H5))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (t2: T).(\lambda
-(H0: (tau0 g c0 (TSort m) t2)).(let H1 \def (match H0 in tau0 return (\lambda
-(c1: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (tau0 ? c1 t t0)).((eq
-C c1 c0) \to ((eq T t (TSort m)) \to ((eq T t0 t2) \to (ty3 g c0 (TSort m)
-t2)))))))) with [(tau0_sort c1 n) \Rightarrow (\lambda (H1: (eq C c1
-c0)).(\lambda (H2: (eq T (TSort n) (TSort m))).(\lambda (H3: (eq T (TSort
-(next g n)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n) (TSort m)) \to
-((eq T (TSort (next g n)) t2) \to (ty3 g c0 (TSort m) t2)))) (\lambda (H4:
-(eq T (TSort n) (TSort m))).(let H5 \def (f_equal T nat (\lambda (e:
-T).(match e in T return (\lambda (_: T).nat) with [(TSort n0) \Rightarrow n0
-| (TLRef _) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TSort n) (TSort
-m) H4) in (eq_ind nat m (\lambda (n0: nat).((eq T (TSort (next g n0)) t2) \to
-(ty3 g c0 (TSort m) t2))) (\lambda (H6: (eq T (TSort (next g m)) t2)).(eq_ind
-T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 (TSort m) t)) (ty3_sort g c0
-m) t2 H6)) n (sym_eq nat n m H5)))) c1 (sym_eq C c1 c0 H1) H2 H3)))) |
-(tau0_abbr c1 d v i H1 w H2) \Rightarrow (\lambda (H3: (eq C c1 c0)).(\lambda
-(H4: (eq T (TLRef i) (TSort m))).(\lambda (H5: (eq T (lift (S i) O w)
-t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i) (TSort m)) \to ((eq T
-(lift (S i) O w) t2) \to ((getl i c2 (CHead d (Bind Abbr) v)) \to ((tau0 g d
-v w) \to (ty3 g c0 (TSort m) t2)))))) (\lambda (H6: (eq T (TLRef i) (TSort
-m))).(let H7 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort m) H6) in
-(False_ind ((eq T (lift (S i) O w) t2) \to ((getl i c0 (CHead d (Bind Abbr)
-v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))) H7))) c1 (sym_eq C c1
-c0 H3) H4 H5 H1 H2)))) | (tau0_abst c1 d v i H1 w H2) \Rightarrow (\lambda
-(H3: (eq C c1 c0)).(\lambda (H4: (eq T (TLRef i) (TSort m))).(\lambda (H5:
-(eq T (lift (S i) O v) t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i)
-(TSort m)) \to ((eq T (lift (S i) O v) t2) \to ((getl i c2 (CHead d (Bind
-Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))))) (\lambda (H6:
-(eq T (TLRef i) (TSort m))).(let H7 \def (eq_ind T (TLRef i) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
-(TSort m) H6) in (False_ind ((eq T (lift (S i) O v) t2) \to ((getl i c0
-(CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2))))
-H7))) c1 (sym_eq C c1 c0 H3) H4 H5 H1 H2)))) | (tau0_bind b c1 v t0 t3 H1)
-\Rightarrow (\lambda (H2: (eq C c1 c0)).(\lambda (H3: (eq T (THead (Bind b) v
-t0) (TSort m))).(\lambda (H4: (eq T (THead (Bind b) v t3) t2)).(eq_ind C c0
-(\lambda (c2: C).((eq T (THead (Bind b) v t0) (TSort m)) \to ((eq T (THead
-(Bind b) v t3) t2) \to ((tau0 g (CHead c2 (Bind b) v) t0 t3) \to (ty3 g c0
-(TSort m) t2))))) (\lambda (H5: (eq T (THead (Bind b) v t0) (TSort m))).(let
-H6 \def (eq_ind T (THead (Bind b) v t0) (\lambda (e: T).(match e in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort m) H5) in
-(False_ind ((eq T (THead (Bind b) v t3) t2) \to ((tau0 g (CHead c0 (Bind b)
-v) t0 t3) \to (ty3 g c0 (TSort m) t2))) H6))) c1 (sym_eq C c1 c0 H2) H3 H4
-H1)))) | (tau0_appl c1 v t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c1
-c0)).(\lambda (H3: (eq T (THead (Flat Appl) v t0) (TSort m))).(\lambda (H4:
-(eq T (THead (Flat Appl) v t3) t2)).(eq_ind C c0 (\lambda (c2: C).((eq T
-(THead (Flat Appl) v t0) (TSort m)) \to ((eq T (THead (Flat Appl) v t3) t2)
-\to ((tau0 g c2 t0 t3) \to (ty3 g c0 (TSort m) t2))))) (\lambda (H5: (eq T
-(THead (Flat Appl) v t0) (TSort m))).(let H6 \def (eq_ind T (THead (Flat
-Appl) v t0) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TSort m) H5) in (False_ind ((eq T (THead (Flat Appl) v
-t3) t2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (TSort m) t2))) H6))) c1 (sym_eq
-C c1 c0 H2) H3 H4 H1)))) | (tau0_cast c1 v1 v2 H1 t0 t3 H2) \Rightarrow
-(\lambda (H3: (eq C c1 c0)).(\lambda (H4: (eq T (THead (Flat Cast) v1 t0)
-(TSort m))).(\lambda (H5: (eq T (THead (Flat Cast) v2 t3) t2)).(eq_ind C c0
-(\lambda (c2: C).((eq T (THead (Flat Cast) v1 t0) (TSort m)) \to ((eq T
-(THead (Flat Cast) v2 t3) t2) \to ((tau0 g c2 v1 v2) \to ((tau0 g c2 t0 t3)
-\to (ty3 g c0 (TSort m) t2)))))) (\lambda (H6: (eq T (THead (Flat Cast) v1
-t0) (TSort m))).(let H7 \def (eq_ind T (THead (Flat Cast) v1 t0) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
-(TSort m) H6) in (False_ind ((eq T (THead (Flat Cast) v2 t3) t2) \to ((tau0 g
-c0 v1 v2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (TSort m) t2)))) H7))) c1
-(sym_eq C c1 c0 H3) H4 H5 H1 H2))))]) in (H1 (refl_equal C c0) (refl_equal T
-(TSort m)) (refl_equal T t2))))))) (\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).((tau0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2:
-T).(\lambda (H3: (tau0 g c0 (TLRef n) t2)).(let H4 \def (match H3 in tau0
-return (\lambda (c1: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (tau0
-? c1 t0 t3)).((eq C c1 c0) \to ((eq T t0 (TLRef n)) \to ((eq T t3 t2) \to
-(ty3 g c0 (TLRef n) t2)))))))) with [(tau0_sort c1 n0) \Rightarrow (\lambda
-(H4: (eq C c1 c0)).(\lambda (H5: (eq T (TSort n0) (TLRef n))).(\lambda (H6:
-(eq T (TSort (next g n0)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n0)
-(TLRef n)) \to ((eq T (TSort (next g n0)) t2) \to (ty3 g c0 (TLRef n) t2))))
-(\lambda (H7: (eq T (TSort n0) (TLRef n))).(let H8 \def (eq_ind T (TSort n0)
-(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-False])) I (TLRef n) H7) in (False_ind ((eq T (TSort (next g n0)) t2) \to
-(ty3 g c0 (TLRef n) t2)) H8))) c1 (sym_eq C c1 c0 H4) H5 H6)))) | (tau0_abbr
-c1 d0 v i H4 w H5) \Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq
-T (TLRef i) (TLRef n))).(\lambda (H8: (eq T (lift (S i) O w) t2)).(eq_ind C
-c0 (\lambda (c2: C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O w)
-t2) \to ((getl i c2 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g
-c0 (TLRef n) t2)))))) (\lambda (H9: (eq T (TLRef i) (TLRef n))).(let H10 \def
-(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
-[(TSort _) \Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _)
-\Rightarrow i])) (TLRef i) (TLRef n) H9) in (eq_ind nat n (\lambda (n0:
-nat).((eq T (lift (S n0) O w) t2) \to ((getl n0 c0 (CHead d0 (Bind Abbr) v))
-\to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H11: (eq T
-(lift (S n) O w) t2)).(eq_ind T (lift (S n) O w) (\lambda (t0: T).((getl n c0
-(CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t0))))
-(\lambda (H12: (getl n c0 (CHead d0 (Bind Abbr) v))).(\lambda (H13: (tau0 g
-d0 v w)).(let H14 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c2:
-C).(getl n c0 c2)) H0 (CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind
-Abbr) u0) n H0 (CHead d0 (Bind Abbr) v) H12)) in (let H15 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow d | (CHead c2 _ _) \Rightarrow c2])) (CHead d (Bind Abbr) u0)
-(CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead
-d0 (Bind Abbr) v) H12)) in ((let H16 \def (f_equal C T (\lambda (e: C).(match
+(H0: (tau0 g c0 (TSort m) t2)).(let H_y \def (tau0_gen_sort g c0 t2 m H0) in
+(let H1 \def (f_equal T T (\lambda (e: T).e) t2 (TSort (next g m)) H_y) in
+(eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 (TSort m) t))
+(ty3_sort g c0 m) t2 H1))))))) (\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).((tau0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: T).(\lambda
+(H3: (tau0 g c0 (TLRef n) t2)).(let H_x \def (tau0_gen_lref g c0 t2 n H3) in
+(let H4 \def H_x in (or_ind (ex3_3 C T T (\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).(tau0 g e u1 t0)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0)))))) (ex3_3 C
+T T (\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).(tau0 g
+e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift
+(S n) O u1)))))) (ty3 g c0 (TLRef n) t2) (\lambda (H5: (ex3_3 C T T (\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).(tau0 g e u1 t0))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O
+t0))))))).(ex3_3_ind C T T (\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).(tau0 g e u1 t0)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))) (ty3 g c0 (TLRef n) t2)
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0
+(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (tau0 g x0 x1 x2)).(\lambda (H8:
+(eq T t2 (lift (S n) O x2))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2
+(lift (S n) O x2) H8) in (eq_ind_r T (lift (S n) O x2) (\lambda (t0: T).(ty3
+g c0 (TLRef n) t0)) (let H10 \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 H11 \def (f_equal
+C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0)
+(CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead
+x0 (Bind Abbr) x1) H6)) in ((let H12 \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 d0 (Bind Abbr) v)
-(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abbr) v) H12)) in
-(\lambda (H17: (eq C d d0)).(let H18 \def (eq_ind_r T v (\lambda (t0:
-T).(getl n c0 (CHead d0 (Bind Abbr) t0))) H14 u0 H16) in (let H19 \def
-(eq_ind_r T v (\lambda (t0: T).(tau0 g d0 t0 w)) H13 u0 H16) in (let H20 \def
-(eq_ind_r C d0 (\lambda (c2: C).(getl n c0 (CHead c2 (Bind Abbr) u0))) H18 d
-H17) in (let H21 \def (eq_ind_r C d0 (\lambda (c2: C).(tau0 g c2 u0 w)) H19 d
-H17) in (ty3_abbr g n c0 d u0 H20 w (H2 w H21)))))))) H15))))) t2 H11)) i
-(sym_eq nat i n H10)))) c1 (sym_eq C c1 c0 H6) H7 H8 H4 H5)))) | (tau0_abst
-c1 d0 v i H4 w H5) \Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq
-T (TLRef i) (TLRef n))).(\lambda (H8: (eq T (lift (S i) O v) t2)).(eq_ind C
-c0 (\lambda (c2: C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O v)
-t2) \to ((getl i c2 (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g
-c0 (TLRef n) t2)))))) (\lambda (H9: (eq T (TLRef i) (TLRef n))).(let H10 \def
-(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
-[(TSort _) \Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _)
-\Rightarrow i])) (TLRef i) (TLRef n) H9) in (eq_ind nat n (\lambda (n0:
-nat).((eq T (lift (S n0) O v) t2) \to ((getl n0 c0 (CHead d0 (Bind Abst) v))
-\to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H11: (eq T
-(lift (S n) O v) t2)).(eq_ind T (lift (S n) O v) (\lambda (t0: T).((getl n c0
-(CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t0))))
-(\lambda (H12: (getl n c0 (CHead d0 (Bind Abst) v))).(\lambda (_: (tau0 g d0
-v w)).(let H14 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c2: C).(getl
-n c0 c2)) H0 (CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abbr) u0)
-n H0 (CHead d0 (Bind Abst) v) H12)) in (let H15 \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 d0 (Bind
-Abst) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abst) v)
-H12)) in (False_ind (ty3 g c0 (TLRef n) (lift (S n) O v)) H15))))) t2 H11)) i
-(sym_eq nat i n H10)))) c1 (sym_eq C c1 c0 H6) H7 H8 H4 H5)))) | (tau0_bind b
-c1 v t0 t3 H4) \Rightarrow (\lambda (H5: (eq C c1 c0)).(\lambda (H6: (eq T
-(THead (Bind b) v t0) (TLRef n))).(\lambda (H7: (eq T (THead (Bind b) v t3)
-t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Bind b) v t0) (TLRef n))
-\to ((eq T (THead (Bind b) v t3) t2) \to ((tau0 g (CHead c2 (Bind b) v) t0
-t3) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H8: (eq T (THead (Bind b) v t0)
-(TLRef n))).(let H9 \def (eq_ind T (THead (Bind b) v t0) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
-(TLRef n) H8) in (False_ind ((eq T (THead (Bind b) v t3) t2) \to ((tau0 g
-(CHead c0 (Bind b) v) t0 t3) \to (ty3 g c0 (TLRef n) t2))) H9))) c1 (sym_eq C
-c1 c0 H5) H6 H7 H4)))) | (tau0_appl c1 v t0 t3 H4) \Rightarrow (\lambda (H5:
-(eq C c1 c0)).(\lambda (H6: (eq T (THead (Flat Appl) v t0) (TLRef
-n))).(\lambda (H7: (eq T (THead (Flat Appl) v t3) t2)).(eq_ind C c0 (\lambda
-(c2: C).((eq T (THead (Flat Appl) v t0) (TLRef n)) \to ((eq T (THead (Flat
-Appl) v t3) t2) \to ((tau0 g c2 t0 t3) \to (ty3 g c0 (TLRef n) t2)))))
-(\lambda (H8: (eq T (THead (Flat Appl) v t0) (TLRef n))).(let H9 \def (eq_ind
-T (THead (Flat Appl) v t0) (\lambda (e: T).(match e in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow True])) I (TLRef n) H8) in (False_ind ((eq T (THead
-(Flat Appl) v t3) t2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (TLRef n) t2)))
-H9))) c1 (sym_eq C c1 c0 H5) H6 H7 H4)))) | (tau0_cast c1 v1 v2 H4 t0 t3 H5)
-\Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq T (THead (Flat
-Cast) v1 t0) (TLRef n))).(\lambda (H8: (eq T (THead (Flat Cast) v2 t3)
-t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Flat Cast) v1 t0) (TLRef
-n)) \to ((eq T (THead (Flat Cast) v2 t3) t2) \to ((tau0 g c2 v1 v2) \to
-((tau0 g c2 t0 t3) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H9: (eq T
-(THead (Flat Cast) v1 t0) (TLRef n))).(let H10 \def (eq_ind T (THead (Flat
-Cast) v1 t0) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TLRef n) H9) in (False_ind ((eq T (THead (Flat Cast)
-v2 t3) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (TLRef
-n) t2)))) H10))) c1 (sym_eq C c1 c0 H6) H7 H8 H4 H5))))]) in (H4 (refl_equal
-C c0) (refl_equal T (TLRef n)) (refl_equal T t2))))))))))))) (\lambda (n:
+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 (H13: (eq C d x0)).(let H14 \def (eq_ind_r T x1 (\lambda (t0:
+T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H10 u0 H12) in (let H15 \def
+(eq_ind_r T x1 (\lambda (t0: T).(tau0 g x0 t0 x2)) H7 u0 H12) in (let H16
+\def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0)))
+H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1: C).(tau0 g c1 u0
+x2)) H15 d H13) in (ty3_abbr g n c0 d u0 H16 x2 (H2 x2 H17)))))))) H11))) t2
+H9)))))))) H5)) (\lambda (H5: (ex3_3 C T T (\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).(tau0 g e u1 t0)))) (\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O
+u1))))))).(ex3_3_ind C T T (\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).(tau0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0 (TLRef n) t2)
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0
+(CHead x0 (Bind Abst) x1))).(\lambda (_: (tau0 g x0 x1 x2)).(\lambda (H8: (eq
+T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2
+(lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1) (\lambda (t0: T).(ty3
+g c0 (TLRef n) t0)) (let H10 \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 H11 \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 (ty3 g c0 (TLRef n) (lift
+(S n) O x1)) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\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 (H1: (ty3 g d u0
t)).(\lambda (_: ((\forall (t2: T).((tau0 g d u0 t2) \to (ty3 g d u0
-t2))))).(\lambda (t2: T).(\lambda (H3: (tau0 g c0 (TLRef n) t2)).(let H4 \def
-(match H3 in tau0 return (\lambda (c1: C).(\lambda (t0: T).(\lambda (t3:
-T).(\lambda (_: (tau0 ? c1 t0 t3)).((eq C c1 c0) \to ((eq T t0 (TLRef n)) \to
-((eq T t3 t2) \to (ty3 g c0 (TLRef n) t2)))))))) with [(tau0_sort c1 n0)
-\Rightarrow (\lambda (H4: (eq C c1 c0)).(\lambda (H5: (eq T (TSort n0) (TLRef
-n))).(\lambda (H6: (eq T (TSort (next g n0)) t2)).(eq_ind C c0 (\lambda (_:
-C).((eq T (TSort n0) (TLRef n)) \to ((eq T (TSort (next g n0)) t2) \to (ty3 g
-c0 (TLRef n) t2)))) (\lambda (H7: (eq T (TSort n0) (TLRef n))).(let H8 \def
-(eq_ind T (TSort n0) (\lambda (e: T).(match e in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow False])) I (TLRef n) H7) in (False_ind ((eq T
-(TSort (next g n0)) t2) \to (ty3 g c0 (TLRef n) t2)) H8))) c1 (sym_eq C c1 c0
-H4) H5 H6)))) | (tau0_abbr c1 d0 v i H4 w H5) \Rightarrow (\lambda (H6: (eq C
-c1 c0)).(\lambda (H7: (eq T (TLRef i) (TLRef n))).(\lambda (H8: (eq T (lift
-(S i) O w) t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i) (TLRef n)) \to
-((eq T (lift (S i) O w) t2) \to ((getl i c2 (CHead d0 (Bind Abbr) v)) \to
-((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H9: (eq T (TLRef
-i) (TLRef n))).(let H10 \def (f_equal T nat (\lambda (e: T).(match e in T
-return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | (TLRef n0)
-\Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H9) in
-(eq_ind nat n (\lambda (n0: nat).((eq T (lift (S n0) O w) t2) \to ((getl n0
-c0 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n)
-t2))))) (\lambda (H11: (eq T (lift (S n) O w) t2)).(eq_ind T (lift (S n) O w)
-(\lambda (t0: T).((getl n c0 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w)
-\to (ty3 g c0 (TLRef n) t0)))) (\lambda (H12: (getl n c0 (CHead d0 (Bind
-Abbr) v))).(\lambda (_: (tau0 g d0 v w)).(let H14 \def (eq_ind C (CHead d
-(Bind Abst) u0) (\lambda (c2: C).(getl n c0 c2)) H0 (CHead d0 (Bind Abbr) v)
-(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abbr) v) H12)) in
-(let H15 \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 d0 (Bind Abbr) v) (getl_mono c0 (CHead d
-(Bind Abst) u0) n H0 (CHead d0 (Bind Abbr) v) H12)) in (False_ind (ty3 g c0
-(TLRef n) (lift (S n) O w)) H15))))) t2 H11)) i (sym_eq nat i n H10)))) c1
-(sym_eq C c1 c0 H6) H7 H8 H4 H5)))) | (tau0_abst c1 d0 v i H4 w H5)
-\Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq T (TLRef i) (TLRef
-n))).(\lambda (H8: (eq T (lift (S i) O v) t2)).(eq_ind C c0 (\lambda (c2:
-C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O v) t2) \to ((getl i
-c2 (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n)
-t2)))))) (\lambda (H9: (eq T (TLRef i) (TLRef n))).(let H10 \def (f_equal T
-nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _)
-\Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i]))
-(TLRef i) (TLRef n) H9) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S
-n0) O v) t2) \to ((getl n0 c0 (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w)
-\to (ty3 g c0 (TLRef n) t2))))) (\lambda (H11: (eq T (lift (S n) O v)
-t2)).(eq_ind T (lift (S n) O v) (\lambda (t0: T).((getl n c0 (CHead d0 (Bind
-Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t0)))) (\lambda (H12:
-(getl n c0 (CHead d0 (Bind Abst) v))).(\lambda (H13: (tau0 g d0 v w)).(let
-H14 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c2: C).(getl n c0 c2))
-H0 (CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0
-(CHead d0 (Bind Abst) v) H12)) in (let H15 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
-(CHead c2 _ _) \Rightarrow c2])) (CHead d (Bind Abst) u0) (CHead d0 (Bind
-Abst) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abst) v)
-H12)) in ((let H16 \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 d0 (Bind Abst) v)
-(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abst) v) H12)) in
-(\lambda (H17: (eq C d d0)).(let H18 \def (eq_ind_r T v (\lambda (t0:
-T).(getl n c0 (CHead d0 (Bind Abst) t0))) H14 u0 H16) in (let H19 \def
-(eq_ind_r T v (\lambda (t0: T).(tau0 g d0 t0 w)) H13 u0 H16) in (eq_ind T u0
-(\lambda (t0: T).(ty3 g c0 (TLRef n) (lift (S n) O t0))) (let H20 \def
-(eq_ind_r C d0 (\lambda (c2: C).(getl n c0 (CHead c2 (Bind Abst) u0))) H18 d
-H17) in (let H21 \def (eq_ind_r C d0 (\lambda (c2: C).(tau0 g c2 u0 w)) H19 d
-H17) in (ty3_abst g n c0 d u0 H20 t H1))) v H16))))) H15))))) t2 H11)) i
-(sym_eq nat i n H10)))) c1 (sym_eq C c1 c0 H6) H7 H8 H4 H5)))) | (tau0_bind b
-c1 v t0 t3 H4) \Rightarrow (\lambda (H5: (eq C c1 c0)).(\lambda (H6: (eq T
-(THead (Bind b) v t0) (TLRef n))).(\lambda (H7: (eq T (THead (Bind b) v t3)
-t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Bind b) v t0) (TLRef n))
-\to ((eq T (THead (Bind b) v t3) t2) \to ((tau0 g (CHead c2 (Bind b) v) t0
-t3) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H8: (eq T (THead (Bind b) v t0)
-(TLRef n))).(let H9 \def (eq_ind T (THead (Bind b) v t0) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
-(TLRef n) H8) in (False_ind ((eq T (THead (Bind b) v t3) t2) \to ((tau0 g
-(CHead c0 (Bind b) v) t0 t3) \to (ty3 g c0 (TLRef n) t2))) H9))) c1 (sym_eq C
-c1 c0 H5) H6 H7 H4)))) | (tau0_appl c1 v t0 t3 H4) \Rightarrow (\lambda (H5:
-(eq C c1 c0)).(\lambda (H6: (eq T (THead (Flat Appl) v t0) (TLRef
-n))).(\lambda (H7: (eq T (THead (Flat Appl) v t3) t2)).(eq_ind C c0 (\lambda
-(c2: C).((eq T (THead (Flat Appl) v t0) (TLRef n)) \to ((eq T (THead (Flat
-Appl) v t3) t2) \to ((tau0 g c2 t0 t3) \to (ty3 g c0 (TLRef n) t2)))))
-(\lambda (H8: (eq T (THead (Flat Appl) v t0) (TLRef n))).(let H9 \def (eq_ind
-T (THead (Flat Appl) v t0) (\lambda (e: T).(match e in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow True])) I (TLRef n) H8) in (False_ind ((eq T (THead
-(Flat Appl) v t3) t2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (TLRef n) t2)))
-H9))) c1 (sym_eq C c1 c0 H5) H6 H7 H4)))) | (tau0_cast c1 v1 v2 H4 t0 t3 H5)
-\Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq T (THead (Flat
-Cast) v1 t0) (TLRef n))).(\lambda (H8: (eq T (THead (Flat Cast) v2 t3)
-t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Flat Cast) v1 t0) (TLRef
-n)) \to ((eq T (THead (Flat Cast) v2 t3) t2) \to ((tau0 g c2 v1 v2) \to
-((tau0 g c2 t0 t3) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H9: (eq T
-(THead (Flat Cast) v1 t0) (TLRef n))).(let H10 \def (eq_ind T (THead (Flat
-Cast) v1 t0) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TLRef n) H9) in (False_ind ((eq T (THead (Flat Cast)
-v2 t3) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (TLRef
-n) t2)))) H10))) c1 (sym_eq C c1 c0 H6) H7 H8 H4 H5))))]) in (H4 (refl_equal
-C c0) (refl_equal T (TLRef n)) (refl_equal T t2))))))))))))) (\lambda (c0:
-C).(\lambda (u0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u0 t)).(\lambda
-(_: ((\forall (t2: T).((tau0 g c0 u0 t2) \to (ty3 g c0 u0 t2))))).(\lambda
-(b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind
-b) u0) t2 t3)).(\lambda (H3: ((\forall (t4: T).((tau0 g (CHead c0 (Bind b)
-u0) t2 t4) \to (ty3 g (CHead c0 (Bind b) u0) t2 t4))))).(\lambda (t0:
-T).(\lambda (H4: (tau0 g c0 (THead (Bind b) u0 t2) t0)).(let H5 \def (match
-H4 in tau0 return (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda
-(_: (tau0 ? c1 t4 t5)).((eq C c1 c0) \to ((eq T t4 (THead (Bind b) u0 t2))
-\to ((eq T t5 t0) \to (ty3 g c0 (THead (Bind b) u0 t2) t0)))))))) with
-[(tau0_sort c1 n) \Rightarrow (\lambda (H5: (eq C c1 c0)).(\lambda (H6: (eq T
-(TSort n) (THead (Bind b) u0 t2))).(\lambda (H7: (eq T (TSort (next g n))
-t0)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n) (THead (Bind b) u0 t2))
-\to ((eq T (TSort (next g n)) t0) \to (ty3 g c0 (THead (Bind b) u0 t2) t0))))
-(\lambda (H8: (eq T (TSort n) (THead (Bind b) u0 t2))).(let H9 \def (eq_ind T
-(TSort n) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow False])) I (THead (Bind b) u0 t2) H8) in (False_ind ((eq T (TSort
-(next g n)) t0) \to (ty3 g c0 (THead (Bind b) u0 t2) t0)) H9))) c1 (sym_eq C
-c1 c0 H5) H6 H7)))) | (tau0_abbr c1 d v i H5 w H6) \Rightarrow (\lambda (H7:
-(eq C c1 c0)).(\lambda (H8: (eq T (TLRef i) (THead (Bind b) u0 t2))).(\lambda
-(H9: (eq T (lift (S i) O w) t0)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef
-i) (THead (Bind b) u0 t2)) \to ((eq T (lift (S i) O w) t0) \to ((getl i c2
-(CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0
-t2) t0)))))) (\lambda (H10: (eq T (TLRef i) (THead (Bind b) u0 t2))).(let H11
-\def (eq_ind T (TLRef i) (\lambda (e: T).(match e in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u0 t2) H10) in
-(False_ind ((eq T (lift (S i) O w) t0) \to ((getl i c0 (CHead d (Bind Abbr)
-v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 t2) t0)))) H11))) c1
-(sym_eq C c1 c0 H7) H8 H9 H5 H6)))) | (tau0_abst c1 d v i H5 w H6)
-\Rightarrow (\lambda (H7: (eq C c1 c0)).(\lambda (H8: (eq T (TLRef i) (THead
-(Bind b) u0 t2))).(\lambda (H9: (eq T (lift (S i) O v) t0)).(eq_ind C c0
-(\lambda (c2: C).((eq T (TLRef i) (THead (Bind b) u0 t2)) \to ((eq T (lift (S
-i) O v) t0) \to ((getl i c2 (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to
-(ty3 g c0 (THead (Bind b) u0 t2) t0)))))) (\lambda (H10: (eq T (TLRef i)
-(THead (Bind b) u0 t2))).(let H11 \def (eq_ind T (TLRef i) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
-(THead (Bind b) u0 t2) H10) in (False_ind ((eq T (lift (S i) O v) t0) \to
-((getl i c0 (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead
-(Bind b) u0 t2) t0)))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 H6)))) |
-(tau0_bind b0 c1 v t4 t5 H5) \Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda
-(H7: (eq T (THead (Bind b0) v t4) (THead (Bind b) u0 t2))).(\lambda (H8: (eq
-T (THead (Bind b0) v t5) t0)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead
-(Bind b0) v t4) (THead (Bind b) u0 t2)) \to ((eq T (THead (Bind b0) v t5) t0)
-\to ((tau0 g (CHead c2 (Bind b0) v) t4 t5) \to (ty3 g c0 (THead (Bind b) u0
-t2) t0))))) (\lambda (H9: (eq T (THead (Bind b0) v t4) (THead (Bind b) u0
-t2))).(let H10 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4
-| (THead _ _ t6) \Rightarrow t6])) (THead (Bind b0) v t4) (THead (Bind b) u0
-t2) H9) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v |
-(THead _ t6 _) \Rightarrow t6])) (THead (Bind b0) v t4) (THead (Bind b) u0
-t2) H9) in ((let H12 \def (f_equal T B (\lambda (e: T).(match e in T return
-(\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0
-| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with
-[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) v
-t4) (THead (Bind b) u0 t2) H9) in (eq_ind B b (\lambda (b1: B).((eq T v u0)
-\to ((eq T t4 t2) \to ((eq T (THead (Bind b1) v t5) t0) \to ((tau0 g (CHead
-c0 (Bind b1) v) t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t0)))))) (\lambda
-(H13: (eq T v u0)).(eq_ind T u0 (\lambda (t6: T).((eq T t4 t2) \to ((eq T
-(THead (Bind b) t6 t5) t0) \to ((tau0 g (CHead c0 (Bind b) t6) t4 t5) \to
-(ty3 g c0 (THead (Bind b) u0 t2) t0))))) (\lambda (H14: (eq T t4 t2)).(eq_ind
-T t2 (\lambda (t6: T).((eq T (THead (Bind b) u0 t5) t0) \to ((tau0 g (CHead
-c0 (Bind b) u0) t6 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t0)))) (\lambda
-(H15: (eq T (THead (Bind b) u0 t5) t0)).(eq_ind T (THead (Bind b) u0 t5)
-(\lambda (t6: T).((tau0 g (CHead c0 (Bind b) u0) t2 t5) \to (ty3 g c0 (THead
-(Bind b) u0 t2) t6))) (\lambda (H16: (tau0 g (CHead c0 (Bind b) u0) t2
-t5)).(ty3_bind g c0 u0 t H0 b t2 t5 (H3 t5 H16))) t0 H15)) t4 (sym_eq T t4 t2
-H14))) v (sym_eq T v u0 H13))) b0 (sym_eq B b0 b H12))) H11)) H10))) c1
-(sym_eq C c1 c0 H6) H7 H8 H5)))) | (tau0_appl c1 v t4 t5 H5) \Rightarrow
-(\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq T (THead (Flat Appl) v t4)
-(THead (Bind b) u0 t2))).(\lambda (H8: (eq T (THead (Flat Appl) v t5)
-t0)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Flat Appl) v t4) (THead
-(Bind b) u0 t2)) \to ((eq T (THead (Flat Appl) v t5) t0) \to ((tau0 g c2 t4
-t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t0))))) (\lambda (H9: (eq T (THead
-(Flat Appl) v t4) (THead (Bind b) u0 t2))).(let H10 \def (eq_ind T (THead
-(Flat Appl) v t4) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
-_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 t2)
-H9) in (False_ind ((eq T (THead (Flat Appl) v t5) t0) \to ((tau0 g c0 t4 t5)
-\to (ty3 g c0 (THead (Bind b) u0 t2) t0))) H10))) c1 (sym_eq C c1 c0 H6) H7
-H8 H5)))) | (tau0_cast c1 v1 v2 H5 t4 t5 H6) \Rightarrow (\lambda (H7: (eq C
-c1 c0)).(\lambda (H8: (eq T (THead (Flat Cast) v1 t4) (THead (Bind b) u0
-t2))).(\lambda (H9: (eq T (THead (Flat Cast) v2 t5) t0)).(eq_ind C c0
-(\lambda (c2: C).((eq T (THead (Flat Cast) v1 t4) (THead (Bind b) u0 t2)) \to
-((eq T (THead (Flat Cast) v2 t5) t0) \to ((tau0 g c2 v1 v2) \to ((tau0 g c2
-t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t0)))))) (\lambda (H10: (eq T
-(THead (Flat Cast) v1 t4) (THead (Bind b) u0 t2))).(let H11 \def (eq_ind T
-(THead (Flat Cast) v1 t4) (\lambda (e: T).(match e in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
-b) u0 t2) H10) in (False_ind ((eq T (THead (Flat Cast) v2 t5) t0) \to ((tau0
-g c0 v1 v2) \to ((tau0 g c0 t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2)
-t0)))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 H6))))]) in (H5 (refl_equal C
-c0) (refl_equal T (THead (Bind b) u0 t2)) (refl_equal T t0)))))))))))))))
-(\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda (H0: (ty3 g c0 w
-u0)).(\lambda (_: ((\forall (t2: T).((tau0 g c0 w t2) \to (ty3 g c0 w
-t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v (THead
-(Bind Abst) u0 t))).(\lambda (H3: ((\forall (t2: T).((tau0 g c0 v t2) \to
-(ty3 g c0 v t2))))).(\lambda (t2: T).(\lambda (H4: (tau0 g c0 (THead (Flat
-Appl) w v) t2)).(let H5 \def (match H4 in tau0 return (\lambda (c1:
-C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (tau0 ? c1 t0 t3)).((eq C
-c1 c0) \to ((eq T t0 (THead (Flat Appl) w v)) \to ((eq T t3 t2) \to (ty3 g c0
-(THead (Flat Appl) w v) t2)))))))) with [(tau0_sort c1 n) \Rightarrow
-(\lambda (H5: (eq C c1 c0)).(\lambda (H6: (eq T (TSort n) (THead (Flat Appl)
-w v))).(\lambda (H7: (eq T (TSort (next g n)) t2)).(eq_ind C c0 (\lambda (_:
-C).((eq T (TSort n) (THead (Flat Appl) w v)) \to ((eq T (TSort (next g n))
-t2) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) (\lambda (H8: (eq T (TSort
-n) (THead (Flat Appl) w v))).(let H9 \def (eq_ind T (TSort n) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
-(THead (Flat Appl) w v) H8) in (False_ind ((eq T (TSort (next g n)) t2) \to
-(ty3 g c0 (THead (Flat Appl) w v) t2)) H9))) c1 (sym_eq C c1 c0 H5) H6 H7))))
-| (tau0_abbr c1 d v0 i H5 w0 H6) \Rightarrow (\lambda (H7: (eq C c1
-c0)).(\lambda (H8: (eq T (TLRef i) (THead (Flat Appl) w v))).(\lambda (H9:
-(eq T (lift (S i) O w0) t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i)
-(THead (Flat Appl) w v)) \to ((eq T (lift (S i) O w0) t2) \to ((getl i c2
-(CHead d (Bind Abbr) v0)) \to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat
-Appl) w v) t2)))))) (\lambda (H10: (eq T (TLRef i) (THead (Flat Appl) w
-v))).(let H11 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w
-v) H10) in (False_ind ((eq T (lift (S i) O w0) t2) \to ((getl i c0 (CHead d
-(Bind Abbr) v0)) \to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v)
-t2)))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 H6)))) | (tau0_abst c1 d v0 i
-H5 w0 H6) \Rightarrow (\lambda (H7: (eq C c1 c0)).(\lambda (H8: (eq T (TLRef
-i) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (lift (S i) O v0)
-t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i) (THead (Flat Appl) w v))
-\to ((eq T (lift (S i) O v0) t2) \to ((getl i c2 (CHead d (Bind Abst) v0))
-\to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))))) (\lambda
-(H10: (eq T (TLRef i) (THead (Flat Appl) w v))).(let H11 \def (eq_ind T
-(TLRef i) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
-\Rightarrow False])) I (THead (Flat Appl) w v) H10) in (False_ind ((eq T
-(lift (S i) O v0) t2) \to ((getl i c0 (CHead d (Bind Abst) v0)) \to ((tau0 g
-d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) H11))) c1 (sym_eq C c1
-c0 H7) H8 H9 H5 H6)))) | (tau0_bind b c1 v0 t0 t3 H5) \Rightarrow (\lambda
-(H6: (eq C c1 c0)).(\lambda (H7: (eq T (THead (Bind b) v0 t0) (THead (Flat
-Appl) w v))).(\lambda (H8: (eq T (THead (Bind b) v0 t3) t2)).(eq_ind C c0
-(\lambda (c2: C).((eq T (THead (Bind b) v0 t0) (THead (Flat Appl) w v)) \to
-((eq T (THead (Bind b) v0 t3) t2) \to ((tau0 g (CHead c2 (Bind b) v0) t0 t3)
-\to (ty3 g c0 (THead (Flat Appl) w v) t2))))) (\lambda (H9: (eq T (THead
-(Bind b) v0 t0) (THead (Flat Appl) w v))).(let H10 \def (eq_ind T (THead
-(Bind b) v0 t0) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
-_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v)
-H9) in (False_ind ((eq T (THead (Bind b) v0 t3) t2) \to ((tau0 g (CHead c0
-(Bind b) v0) t0 t3) \to (ty3 g c0 (THead (Flat Appl) w v) t2))) H10))) c1
-(sym_eq C c1 c0 H6) H7 H8 H5)))) | (tau0_appl c1 v0 t0 t3 H5) \Rightarrow
-(\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq T (THead (Flat Appl) v0 t0)
-(THead (Flat Appl) w v))).(\lambda (H8: (eq T (THead (Flat Appl) v0 t3)
-t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Flat Appl) v0 t0) (THead
-(Flat Appl) w v)) \to ((eq T (THead (Flat Appl) v0 t3) t2) \to ((tau0 g c2 t0
-t3) \to (ty3 g c0 (THead (Flat Appl) w v) t2))))) (\lambda (H9: (eq T (THead
-(Flat Appl) v0 t0) (THead (Flat Appl) w v))).(let H10 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t4) \Rightarrow t4]))
-(THead (Flat Appl) v0 t0) (THead (Flat Appl) w v) H9) in ((let H11 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t4 _)
-\Rightarrow t4])) (THead (Flat Appl) v0 t0) (THead (Flat Appl) w v) H9) in
-(eq_ind T w (\lambda (t4: T).((eq T t0 v) \to ((eq T (THead (Flat Appl) t4
-t3) t2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))))
-(\lambda (H12: (eq T t0 v)).(eq_ind T v (\lambda (t4: T).((eq T (THead (Flat
-Appl) w t3) t2) \to ((tau0 g c0 t4 t3) \to (ty3 g c0 (THead (Flat Appl) w v)
-t2)))) (\lambda (H13: (eq T (THead (Flat Appl) w t3) t2)).(eq_ind T (THead
-(Flat Appl) w t3) (\lambda (t4: T).((tau0 g c0 v t3) \to (ty3 g c0 (THead
-(Flat Appl) w v) t4))) (\lambda (H14: (tau0 g c0 v t3)).(let H_y \def (H3 t3
-H14) in (let H15 \def (ty3_unique g c0 v t3 H_y (THead (Bind Abst) u0 t) H2)
-in (ex_ind T (\lambda (t4: T).(ty3 g c0 t3 t4)) (ty3 g c0 (THead (Flat Appl)
-w v) (THead (Flat Appl) w t3)) (\lambda (x: T).(\lambda (H16: (ty3 g c0 t3
-x)).(ex_ind T (\lambda (t4: T).(ty3 g c0 u0 t4)) (ty3 g c0 (THead (Flat Appl)
-w v) (THead (Flat Appl) w t3)) (\lambda (x0: T).(\lambda (_: (ty3 g c0 u0
-x0)).(ex_ind T (\lambda (t4: T).(ty3 g c0 (THead (Bind Abst) u0 t) t4)) (ty3
-g c0 (THead (Flat Appl) w v) (THead (Flat Appl) w t3)) (\lambda (x1:
-T).(\lambda (H18: (ty3 g c0 (THead (Bind Abst) u0 t) x1)).(ex3_2_ind T T
-(\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u0 t4) x1)))
-(\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u0 t5))) (\lambda (t4: T).(\lambda
-(_: T).(ty3 g (CHead c0 (Bind Abst) u0) t t4))) (ty3 g c0 (THead (Flat Appl)
-w v) (THead (Flat Appl) w t3)) (\lambda (x2: T).(\lambda (x3: T).(\lambda (_:
-(pc3 c0 (THead (Bind Abst) u0 x2) x1)).(\lambda (H20: (ty3 g c0 u0
-x3)).(\lambda (H21: (ty3 g (CHead c0 (Bind Abst) u0) t x2)).(ty3_conv g c0
-(THead (Flat Appl) w t3) (THead (Flat Appl) w (THead (Bind Abst) u0 x2))
-(ty3_appl g c0 w u0 H0 t3 x2 (ty3_sconv g c0 t3 x H16 (THead (Bind Abst) u0
-t) (THead (Bind Abst) u0 x2) (ty3_bind g c0 u0 x3 H20 Abst t x2 H21) H15))
-(THead (Flat Appl) w v) (THead (Flat Appl) w (THead (Bind Abst) u0 t))
-(ty3_appl g c0 w u0 H0 v t H2) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) t3
-(ty3_unique g c0 v (THead (Bind Abst) u0 t) H2 t3 H_y) w Appl)))))))
-(ty3_gen_bind g Abst c0 u0 t x1 H18)))) (ty3_correct g c0 v (THead (Bind
-Abst) u0 t) H2)))) (ty3_correct g c0 w u0 H0)))) (ty3_correct g c0 v t3
-H_y))))) t2 H13)) t0 (sym_eq T t0 v H12))) v0 (sym_eq T v0 w H11))) H10))) c1
-(sym_eq C c1 c0 H6) H7 H8 H5)))) | (tau0_cast c1 v1 v2 H5 t0 t3 H6)
-\Rightarrow (\lambda (H7: (eq C c1 c0)).(\lambda (H8: (eq T (THead (Flat
-Cast) v1 t0) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (THead (Flat Cast)
-v2 t3) t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Flat Cast) v1 t0)
-(THead (Flat Appl) w v)) \to ((eq T (THead (Flat Cast) v2 t3) t2) \to ((tau0
-g c2 v1 v2) \to ((tau0 g c2 t0 t3) \to (ty3 g c0 (THead (Flat Appl) w v)
-t2)))))) (\lambda (H10: (eq T (THead (Flat Cast) v1 t0) (THead (Flat Appl) w
-v))).(let H11 \def (eq_ind T (THead (Flat Cast) v1 t0) (\lambda (e: T).(match
-e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow
-(match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow False |
-Cast \Rightarrow True])])])) I (THead (Flat Appl) w v) H10) in (False_ind
-((eq T (THead (Flat Cast) v2 t3) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0
-t0 t3) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) H11))) c1 (sym_eq C c1 c0
-H7) H8 H9 H5 H6))))]) in (H5 (refl_equal C c0) (refl_equal T (THead (Flat
-Appl) w v)) (refl_equal T t2)))))))))))))) (\lambda (c0: C).(\lambda (t2:
-T).(\lambda (t3: T).(\lambda (H0: (ty3 g c0 t2 t3)).(\lambda (H1: ((\forall
-(t4: T).((tau0 g c0 t2 t4) \to (ty3 g c0 t2 t4))))).(\lambda (t0: T).(\lambda
-(_: (ty3 g c0 t3 t0)).(\lambda (H3: ((\forall (t4: T).((tau0 g c0 t3 t4) \to
-(ty3 g c0 t3 t4))))).(\lambda (t4: T).(\lambda (H4: (tau0 g c0 (THead (Flat
-Cast) t3 t2) t4)).(let H5 \def (match H4 in tau0 return (\lambda (c1:
-C).(\lambda (t: T).(\lambda (t5: T).(\lambda (_: (tau0 ? c1 t t5)).((eq C c1
-c0) \to ((eq T t (THead (Flat Cast) t3 t2)) \to ((eq T t5 t4) \to (ty3 g c0
-(THead (Flat Cast) t3 t2) t4)))))))) with [(tau0_sort c1 n) \Rightarrow
-(\lambda (H5: (eq C c1 c0)).(\lambda (H6: (eq T (TSort n) (THead (Flat Cast)
-t3 t2))).(\lambda (H7: (eq T (TSort (next g n)) t4)).(eq_ind C c0 (\lambda
-(_: C).((eq T (TSort n) (THead (Flat Cast) t3 t2)) \to ((eq T (TSort (next g
-n)) t4) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))) (\lambda (H8: (eq T
-(TSort n) (THead (Flat Cast) t3 t2))).(let H9 \def (eq_ind T (TSort n)
-(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-False])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T (TSort (next g
-n)) t4) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)) H9))) c1 (sym_eq C c1 c0
-H5) H6 H7)))) | (tau0_abbr c1 d v i H5 w H6) \Rightarrow (\lambda (H7: (eq C
-c1 c0)).(\lambda (H8: (eq T (TLRef i) (THead (Flat Cast) t3 t2))).(\lambda
-(H9: (eq T (lift (S i) O w) t4)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef
-i) (THead (Flat Cast) t3 t2)) \to ((eq T (lift (S i) O w) t4) \to ((getl i c2
-(CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast)
-t3 t2) t4)))))) (\lambda (H10: (eq T (TLRef i) (THead (Flat Cast) t3
-t2))).(let H11 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t3
-t2) H10) in (False_ind ((eq T (lift (S i) O w) t4) \to ((getl i c0 (CHead d
-(Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2)
-t4)))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 H6)))) | (tau0_abst c1 d v i H5
-w H6) \Rightarrow (\lambda (H7: (eq C c1 c0)).(\lambda (H8: (eq T (TLRef i)
-(THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T (lift (S i) O v) t4)).(eq_ind
-C c0 (\lambda (c2: C).((eq T (TLRef i) (THead (Flat Cast) t3 t2)) \to ((eq T
-(lift (S i) O v) t4) \to ((getl i c2 (CHead d (Bind Abst) v)) \to ((tau0 g d
-v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H10: (eq T
-(TLRef i) (THead (Flat Cast) t3 t2))).(let H11 \def (eq_ind T (TLRef i)
-(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
-False])) I (THead (Flat Cast) t3 t2) H10) in (False_ind ((eq T (lift (S i) O
-v) t4) \to ((getl i c0 (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3
-g c0 (THead (Flat Cast) t3 t2) t4)))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5
-H6)))) | (tau0_bind b c1 v t5 t6 H5) \Rightarrow (\lambda (H6: (eq C c1
-c0)).(\lambda (H7: (eq T (THead (Bind b) v t5) (THead (Flat Cast) t3
-t2))).(\lambda (H8: (eq T (THead (Bind b) v t6) t4)).(eq_ind C c0 (\lambda
-(c2: C).((eq T (THead (Bind b) v t5) (THead (Flat Cast) t3 t2)) \to ((eq T
-(THead (Bind b) v t6) t4) \to ((tau0 g (CHead c2 (Bind b) v) t5 t6) \to (ty3
-g c0 (THead (Flat Cast) t3 t2) t4))))) (\lambda (H9: (eq T (THead (Bind b) v
-t5) (THead (Flat Cast) t3 t2))).(let H10 \def (eq_ind T (THead (Bind b) v t5)
-(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True |
-(Flat _) \Rightarrow False])])) I (THead (Flat Cast) t3 t2) H9) in (False_ind
-((eq T (THead (Bind b) v t6) t4) \to ((tau0 g (CHead c0 (Bind b) v) t5 t6)
-\to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))) H10))) c1 (sym_eq C c1 c0 H6)
-H7 H8 H5)))) | (tau0_appl c1 v t5 t6 H5) \Rightarrow (\lambda (H6: (eq C c1
-c0)).(\lambda (H7: (eq T (THead (Flat Appl) v t5) (THead (Flat Cast) t3
-t2))).(\lambda (H8: (eq T (THead (Flat Appl) v t6) t4)).(eq_ind C c0 (\lambda
-(c2: C).((eq T (THead (Flat Appl) v t5) (THead (Flat Cast) t3 t2)) \to ((eq T
-(THead (Flat Appl) v t6) t4) \to ((tau0 g c2 t5 t6) \to (ty3 g c0 (THead
-(Flat Cast) t3 t2) t4))))) (\lambda (H9: (eq T (THead (Flat Appl) v t5)
-(THead (Flat Cast) t3 t2))).(let H10 \def (eq_ind T (THead (Flat Appl) v t5)
-(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
-(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl
-\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2)
-H9) in (False_ind ((eq T (THead (Flat Appl) v t6) t4) \to ((tau0 g c0 t5 t6)
-\to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))) H10))) c1 (sym_eq C c1 c0 H6)
-H7 H8 H5)))) | (tau0_cast c1 v1 v2 H5 t5 t6 H6) \Rightarrow (\lambda (H7: (eq
-C c1 c0)).(\lambda (H8: (eq T (THead (Flat Cast) v1 t5) (THead (Flat Cast) t3
-t2))).(\lambda (H9: (eq T (THead (Flat Cast) v2 t6) t4)).(eq_ind C c0
-(\lambda (c2: C).((eq T (THead (Flat Cast) v1 t5) (THead (Flat Cast) t3 t2))
-\to ((eq T (THead (Flat Cast) v2 t6) t4) \to ((tau0 g c2 v1 v2) \to ((tau0 g
-c2 t5 t6) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H10: (eq
-T (THead (Flat Cast) v1 t5) (THead (Flat Cast) t3 t2))).(let H11 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t)
-\Rightarrow t])) (THead (Flat Cast) v1 t5) (THead (Flat Cast) t3 t2) H10) in
-((let H12 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t
-_) \Rightarrow t])) (THead (Flat Cast) v1 t5) (THead (Flat Cast) t3 t2) H10)
-in (eq_ind T t3 (\lambda (t: T).((eq T t5 t2) \to ((eq T (THead (Flat Cast)
-v2 t6) t4) \to ((tau0 g c0 t v2) \to ((tau0 g c0 t5 t6) \to (ty3 g c0 (THead
-(Flat Cast) t3 t2) t4)))))) (\lambda (H13: (eq T t5 t2)).(eq_ind T t2
-(\lambda (t: T).((eq T (THead (Flat Cast) v2 t6) t4) \to ((tau0 g c0 t3 v2)
-\to ((tau0 g c0 t t6) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))
-(\lambda (H14: (eq T (THead (Flat Cast) v2 t6) t4)).(eq_ind T (THead (Flat
-Cast) v2 t6) (\lambda (t: T).((tau0 g c0 t3 v2) \to ((tau0 g c0 t2 t6) \to
-(ty3 g c0 (THead (Flat Cast) t3 t2) t)))) (\lambda (H15: (tau0 g c0 t3
-v2)).(\lambda (H16: (tau0 g c0 t2 t6)).(let H_y \def (H1 t6 H16) in (let H_y0
-\def (H3 v2 H15) in (let H17 \def (ty3_unique g c0 t2 t6 H_y t3 H0) in
-(ex_ind T (\lambda (t: T).(ty3 g c0 v2 t)) (ty3 g c0 (THead (Flat Cast) t3
-t2) (THead (Flat Cast) v2 t6)) (\lambda (x: T).(\lambda (H18: (ty3 g c0 v2
-x)).(ex_ind T (\lambda (t: T).(ty3 g c0 t6 t)) (ty3 g c0 (THead (Flat Cast)
-t3 t2) (THead (Flat Cast) v2 t6)) (\lambda (x0: T).(\lambda (H19: (ty3 g c0
-t6 x0)).(ty3_conv g c0 (THead (Flat Cast) v2 t6) (THead (Flat Cast) x v2)
-(ty3_cast g c0 t6 v2 (ty3_sconv g c0 t6 x0 H19 t3 v2 H_y0 H17) x H18) (THead
-(Flat Cast) t3 t2) (THead (Flat Cast) v2 t3) (ty3_cast g c0 t2 t3 H0 v2 H_y0)
-(pc3_thin_dx c0 t3 t6 (ty3_unique g c0 t2 t3 H0 t6 H_y) v2 Cast))))
-(ty3_correct g c0 t2 t6 H_y)))) (ty3_correct g c0 t3 v2 H_y0))))))) t4 H14))
-t5 (sym_eq T t5 t2 H13))) v1 (sym_eq T v1 t3 H12))) H11))) c1 (sym_eq C c1 c0
-H7) H8 H9 H5 H6))))]) in (H5 (refl_equal C c0) (refl_equal T (THead (Flat
-Cast) t3 t2)) (refl_equal T t4))))))))))))) c u t1 H))))).
+t2))))).(\lambda (t2: T).(\lambda (H3: (tau0 g c0 (TLRef n) t2)).(let H_x
+\def (tau0_gen_lref g c0 t2 n H3) in (let H4 \def H_x in (or_ind (ex3_3 C T T
+(\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).(tau0 g e u1
+t0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n)
+O t0)))))) (ex3_3 C T T (\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).(tau0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(eq T t2 (lift (S n) O u1)))))) (ty3 g c0 (TLRef n) t2)
+(\lambda (H5: (ex3_3 C T T (\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).(tau0 g e u1 t0)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))))).(ex3_3_ind C T T
+(\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).(tau0 g e u1
+t0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n)
+O t0))))) (ty3 g c0 (TLRef n) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda
+(x2: T).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (_:
+(tau0 g x0 x1 x2)).(\lambda (H8: (eq T t2 (lift (S n) O x2))).(let H9 \def
+(f_equal T T (\lambda (e: T).e) t2 (lift (S n) O x2) H8) in (eq_ind_r T (lift
+(S n) O x2) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (let H10 \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 H11 \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
+(ty3 g c0 (TLRef n) (lift (S n) O x2)) H11))) t2 H9)))))))) H5)) (\lambda
+(H5: (ex3_3 C T T (\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).(tau0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T
+t2 (lift (S n) O u1))))))).(ex3_3_ind C T T (\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).(tau0 g e u1 t0)))) (\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0
+(TLRef n) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda
+(H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7: (tau0 g x0 x1
+x2)).(\lambda (H8: (eq T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T
+(\lambda (e: T).e) t2 (lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1)
+(\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (let H10 \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 H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_:
+C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead
+d (Bind 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 H12 \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 (H13: (eq C d x0)).(let H14 \def
+(eq_ind_r T x1 (\lambda (t0: T).(getl n c0 (CHead x0 (Bind Abst) t0))) H10 u0
+H12) in (let H15 \def (eq_ind_r T x1 (\lambda (t0: T).(tau0 g x0 t0 x2)) H7
+u0 H12) in (eq_ind T u0 (\lambda (t0: T).(ty3 g c0 (TLRef n) (lift (S n) O
+t0))) (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1
+(Bind Abst) u0))) H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1:
+C).(tau0 g c1 u0 x2)) H15 d H13) in (ty3_abst g n c0 d u0 H16 t H1))) x1
+H12))))) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\lambda (c0: C).(\lambda
+(u0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u0 t)).(\lambda (_: ((\forall
+(t2: T).((tau0 g c0 u0 t2) \to (ty3 g c0 u0 t2))))).(\lambda (b: B).(\lambda
+(t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t2
+t3)).(\lambda (H3: ((\forall (t4: T).((tau0 g (CHead c0 (Bind b) u0) t2 t4)
+\to (ty3 g (CHead c0 (Bind b) u0) t2 t4))))).(\lambda (t0: T).(\lambda (H4:
+(tau0 g c0 (THead (Bind b) u0 t2) t0)).(let H_x \def (tau0_gen_bind g b c0 u0
+t2 t0 H4) in (let H5 \def H_x in (ex2_ind T (\lambda (t4: T).(tau0 g (CHead
+c0 (Bind b) u0) t2 t4)) (\lambda (t4: T).(eq T t0 (THead (Bind b) u0 t4)))
+(ty3 g c0 (THead (Bind b) u0 t2) t0) (\lambda (x: T).(\lambda (H6: (tau0 g
+(CHead c0 (Bind b) u0) t2 x)).(\lambda (H7: (eq T t0 (THead (Bind b) u0
+x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u0 x)
+H7) in (eq_ind_r T (THead (Bind b) u0 x) (\lambda (t4: T).(ty3 g c0 (THead
+(Bind b) u0 t2) t4)) (ty3_bind g c0 u0 t H0 b t2 x (H3 x H6)) t0 H8)))))
+H5))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda
+(H0: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: T).((tau0 g c0 w t2) \to
+(ty3 g c0 w t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v
+(THead (Bind Abst) u0 t))).(\lambda (H3: ((\forall (t2: T).((tau0 g c0 v t2)
+\to (ty3 g c0 v t2))))).(\lambda (t2: T).(\lambda (H4: (tau0 g c0 (THead
+(Flat Appl) w v) t2)).(let H_x \def (tau0_gen_appl g c0 w v t2 H4) in (let H5
+\def H_x in (ex2_ind T (\lambda (t3: T).(tau0 g c0 v t3)) (\lambda (t3:
+T).(eq T t2 (THead (Flat Appl) w t3))) (ty3 g c0 (THead (Flat Appl) w v) t2)
+(\lambda (x: T).(\lambda (H6: (tau0 g c0 v x)).(\lambda (H7: (eq T t2 (THead
+(Flat Appl) w x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t2 (THead
+(Flat Appl) w x) H7) in (eq_ind_r T (THead (Flat Appl) w x) (\lambda (t0:
+T).(ty3 g c0 (THead (Flat Appl) w v) t0)) (let H_y \def (H3 x H6) in (let H9
+\def (ty3_unique g c0 v x H_y (THead (Bind Abst) u0 t) H2) in (ex_ind T
+(\lambda (t0: T).(ty3 g c0 x t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead
+(Flat Appl) w x)) (\lambda (x0: T).(\lambda (H10: (ty3 g c0 x x0)).(ex_ind T
+(\lambda (t0: T).(ty3 g c0 u0 t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead
+(Flat Appl) w x)) (\lambda (x1: T).(\lambda (_: (ty3 g c0 u0 x1)).(ex_ind T
+(\lambda (t0: T).(ty3 g c0 (THead (Bind Abst) u0 t) t0)) (ty3 g c0 (THead
+(Flat Appl) w v) (THead (Flat Appl) w x)) (\lambda (x2: T).(\lambda (H12:
+(ty3 g c0 (THead (Bind Abst) u0 t) x2)).(ex3_2_ind T T (\lambda (t3:
+T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u0 t3) x2))) (\lambda (_:
+T).(\lambda (t0: T).(ty3 g c0 u0 t0))) (\lambda (t3: T).(\lambda (_: T).(ty3
+g (CHead c0 (Bind Abst) u0) t t3))) (ty3 g c0 (THead (Flat Appl) w v) (THead
+(Flat Appl) w x)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (_: (pc3 c0
+(THead (Bind Abst) u0 x3) x2)).(\lambda (H14: (ty3 g c0 u0 x4)).(\lambda
+(H15: (ty3 g (CHead c0 (Bind Abst) u0) t x3)).(ty3_conv g c0 (THead (Flat
+Appl) w x) (THead (Flat Appl) w (THead (Bind Abst) u0 x3)) (ty3_appl g c0 w
+u0 H0 x x3 (ty3_sconv g c0 x x0 H10 (THead (Bind Abst) u0 t) (THead (Bind
+Abst) u0 x3) (ty3_bind g c0 u0 x4 H14 Abst t x3 H15) H9)) (THead (Flat Appl)
+w v) (THead (Flat Appl) w (THead (Bind Abst) u0 t)) (ty3_appl g c0 w u0 H0 v
+t H2) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) x (ty3_unique g c0 v (THead
+(Bind Abst) u0 t) H2 x H_y) w Appl))))))) (ty3_gen_bind g Abst c0 u0 t x2
+H12)))) (ty3_correct g c0 v (THead (Bind Abst) u0 t) H2)))) (ty3_correct g c0
+w u0 H0)))) (ty3_correct g c0 v x H_y)))) t2 H8))))) H5))))))))))))))
+(\lambda (c0: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (ty3 g c0 t2
+t3)).(\lambda (H1: ((\forall (t4: T).((tau0 g c0 t2 t4) \to (ty3 g c0 t2
+t4))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t3 t0)).(\lambda (H3:
+((\forall (t4: T).((tau0 g c0 t3 t4) \to (ty3 g c0 t3 t4))))).(\lambda (t4:
+T).(\lambda (H4: (tau0 g c0 (THead (Flat Cast) t3 t2) t4)).(let H_x \def
+(tau0_gen_cast g c0 t3 t2 t4 H4) in (let H5 \def H_x in (ex3_2_ind T T
+(\lambda (v2: T).(\lambda (_: T).(tau0 g c0 t3 v2))) (\lambda (_: T).(\lambda
+(t5: T).(tau0 g c0 t2 t5))) (\lambda (v2: T).(\lambda (t5: T).(eq T t4 (THead
+(Flat Cast) v2 t5)))) (ty3 g c0 (THead (Flat Cast) t3 t2) t4) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H6: (tau0 g c0 t3 x0)).(\lambda (H7: (tau0 g c0
+t2 x1)).(\lambda (H8: (eq T t4 (THead (Flat Cast) x0 x1))).(let H9 \def
+(f_equal T T (\lambda (e: T).e) t4 (THead (Flat Cast) x0 x1) H8) in (eq_ind_r
+T (THead (Flat Cast) x0 x1) (\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t3
+t2) t)) (let H_y \def (H1 x1 H7) in (let H_y0 \def (H3 x0 H6) in (let H10
+\def (ty3_unique g c0 t2 x1 H_y t3 H0) in (ex_ind T (\lambda (t: T).(ty3 g c0
+x0 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0 x1))
+(\lambda (x: T).(\lambda (H11: (ty3 g c0 x0 x)).(ex_ind T (\lambda (t:
+T).(ty3 g c0 x1 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0
+x1)) (\lambda (x2: T).(\lambda (H12: (ty3 g c0 x1 x2)).(ty3_conv g c0 (THead
+(Flat Cast) x0 x1) (THead (Flat Cast) x x0) (ty3_cast g c0 x1 x0 (ty3_sconv g
+c0 x1 x2 H12 t3 x0 H_y0 H10) x H11) (THead (Flat Cast) t3 t2) (THead (Flat
+Cast) x0 t3) (ty3_cast g c0 t2 t3 H0 x0 H_y0) (pc3_thin_dx c0 t3 x1
+(ty3_unique g c0 t2 t3 H0 x1 H_y) x0 Cast)))) (ty3_correct g c0 t2 x1 H_y))))
+(ty3_correct g c0 t3 x0 H_y0))))) t4 H9))))))) H5))))))))))))) c u t1 H))))).
projects / helm.git / blobdiff