(* This file was automatically generated: do not edit *********************)
-set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/fwd".
+include "LambdaDelta-1/ty3/defs.ma".
-include "ty3/defs.ma".
-
-include "pc3/props.ma".
+include "LambdaDelta-1/pc3/props.ma".
theorem ty3_gen_sort:
\forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c
\def
\lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda
(H: (ty3 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(ty3 g c t
-x)) (pc3 c (TSort (next g n)) x) (\lambda (y: T).(\lambda (H0: (ty3 g c y
-x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t
-(TSort n)) \to (pc3 c0 (TSort (next g n)) t0))))) (\lambda (c0: C).(\lambda
-(t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2
-(TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (u: T).(\lambda (t1:
-T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (TSort n)) \to (pc3
-c0 (TSort (next g n)) t1)))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: (eq
-T u (TSort n))).(let H7 \def (f_equal T T (\lambda (e: T).e) u (TSort n) H6)
-in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0 (TSort n)) \to (pc3 c0
-(TSort (next g n)) t1))) H4 (TSort n) H7) in (let H9 \def (eq_ind T u
-(\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TSort n) H7) in (pc3_t t1 c0 (TSort
-(next g n)) (H8 (refl_equal T (TSort n))) t2 H5))))))))))))))) (\lambda (c0:
-C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (TSort n))).(let H2 \def
-(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
-[(TSort n0) \Rightarrow n0 | (TLRef _) \Rightarrow m | (THead _ _ _)
-\Rightarrow m])) (TSort m) (TSort n) H1) in (eq_ind_r nat n (\lambda (n0:
-nat).(pc3 c0 (TSort (next g n)) (TSort (next g n0)))) (pc3_refl c0 (TSort
-(next g n))) m H2))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (_: (getl n0 c0 (CHead d (Bind Abbr)
-u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u
-(TSort n)) \to (pc3 d (TSort (next g n)) t)))).(\lambda (H4: (eq T (TLRef n0)
-(TSort n))).(let H5 \def (eq_ind T (TLRef n0) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) H4) in
-(False_ind (pc3 c0 (TSort (next g n)) (lift (S n0) O t)) H5)))))))))))
-(\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
-(_: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g
-d u t)).(\lambda (_: (((eq T u (TSort n)) \to (pc3 d (TSort (next g n))
-t)))).(\lambda (H4: (eq T (TLRef n0) (TSort n))).(let H5 \def (eq_ind T
-(TLRef n0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
-\Rightarrow False])) I (TSort n) H4) in (False_ind (pc3 c0 (TSort (next g n))
-(lift (S n0) O u)) H5))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda
-(t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TSort n)) \to
-(pc3 c0 (TSort (next g n)) t)))).(\lambda (b: B).(\lambda (t1: T).(\lambda
-(t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq
-T t1 (TSort n)) \to (pc3 (CHead c0 (Bind b) u) (TSort (next g n))
-t2)))).(\lambda (t0: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t2
-t0)).(\lambda (_: (((eq T t2 (TSort n)) \to (pc3 (CHead c0 (Bind b) u) (TSort
-(next g n)) t0)))).(\lambda (H7: (eq T (THead (Bind b) u t1) (TSort n))).(let
-H8 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee in T
+x)) (\lambda (_: T).(pc3 c (TSort (next g n)) x)) (\lambda (y: T).(\lambda
+(H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).((eq T t (TSort n)) \to (pc3 c0 (TSort (next g n)) t0))))) (\lambda (c0:
+C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda
+(_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (u:
+T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u
+(TSort n)) \to (pc3 c0 (TSort (next g n)) t1)))).(\lambda (H5: (pc3 c0 t1
+t2)).(\lambda (H6: (eq T u (TSort n))).(let H7 \def (f_equal T T (\lambda (e:
+T).e) u (TSort n) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0
+(TSort n)) \to (pc3 c0 (TSort (next g n)) t1))) H4 (TSort n) H7) in (let H9
+\def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TSort n) H7) in
+(pc3_t t1 c0 (TSort (next g n)) (H8 (refl_equal T (TSort n))) t2
+H5))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T
+(TSort m) (TSort n))).(let H2 \def (f_equal T nat (\lambda (e: T).(match e in
+T return (\lambda (_: T).nat) with [(TSort n0) \Rightarrow n0 | (TLRef _)
+\Rightarrow m | (THead _ _ _) \Rightarrow m])) (TSort m) (TSort n) H1) in
+(eq_ind_r nat n (\lambda (n0: nat).(pc3 c0 (TSort (next g n)) (TSort (next g
+n0)))) (pc3_refl c0 (TSort (next g n))) m H2))))) (\lambda (n0: nat).(\lambda
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n0 c0 (CHead d
+(Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_:
+(((eq T u (TSort n)) \to (pc3 d (TSort (next g n)) t)))).(\lambda (H4: (eq T
+(TLRef n0) (TSort n))).(let H5 \def (eq_ind T (TLRef n0) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
+(TSort n) H4) in (False_ind (pc3 c0 (TSort (next g n)) (lift (S n0) O t))
+H5))))))))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda
+(u: T).(\lambda (_: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t:
+T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u (TSort n)) \to (pc3 d
+(TSort (next g n)) t)))).(\lambda (H4: (eq T (TLRef n0) (TSort n))).(let H5
+\def (eq_ind T (TLRef n0) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (TSort n) H4) in (False_ind (pc3 c0
+(TSort (next g n)) (lift (S n0) O u)) H5))))))))))) (\lambda (c0: C).(\lambda
+(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u
+(TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (b: B).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1
+t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (pc3 (CHead c0 (Bind b) u) (TSort
+(next g n)) t2)))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TSort n))).(let
+H6 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee in T
return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H7) in
-(False_ind (pc3 c0 (TSort (next g n)) (THead (Bind b) u t2))
-H8)))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda
-(_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (TSort n)) \to (pc3 c0 (TSort
-(next g n)) u)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v
-(THead (Bind Abst) u t))).(\lambda (_: (((eq T v (TSort n)) \to (pc3 c0
-(TSort (next g n)) (THead (Bind Abst) u t))))).(\lambda (H5: (eq T (THead
-(Flat Appl) w v) (TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) w v)
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-True])) I (TSort n) H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat
-Appl) w (THead (Bind Abst) u t))) H6)))))))))))) (\lambda (c0: C).(\lambda
-(t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T
-t1 (TSort n)) \to (pc3 c0 (TSort (next g n)) t2)))).(\lambda (t0: T).(\lambda
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in
+(False_ind (pc3 c0 (TSort (next g n)) (THead (Bind b) u t2)) H6)))))))))))))
+(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w
+u)).(\lambda (_: (((eq T w (TSort n)) \to (pc3 c0 (TSort (next g n))
+u)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind
+Abst) u t))).(\lambda (_: (((eq T v (TSort n)) \to (pc3 c0 (TSort (next g n))
+(THead (Bind Abst) u t))))).(\lambda (H5: (eq T (THead (Flat Appl) w v)
+(TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TSort n) H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Appl) w
+(THead (Bind Abst) u t))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1
+(TSort n)) \to (pc3 c0 (TSort (next g n)) t2)))).(\lambda (t0: T).(\lambda
(_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort
(next g n)) t0)))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (TSort
n))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match
\def
\lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda
(H: (ty3 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(ty3 g c t
-x)) (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c
-(lift (S n) O t) x)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl
-n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
-T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(pc3 c (lift (S n) O u) x)))) (\lambda (e: C).(\lambda (u: T).(\lambda
-(_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t: T).(ty3 g e u t)))))) (\lambda (y: T).(\lambda (H0: (ty3 g c
-y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t
-(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t1:
-T).(pc3 c0 (lift (S n) O t1) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda
+x)) (\lambda (_: T).(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda
+(t0: T).(pc3 c (lift (S n) O t0) x)))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda
+(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0)))))))
+(\lambda (y: T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0:
+C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (ex3_3 C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(pc3 c0 (lift (S n) O t1)
+t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(ty3 g e
+u t1))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3
+c0 (lift (S n) O u) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t1: T).(ty3 g e u t1)))))))))) (\lambda (c0: C).(\lambda (t2:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2
+(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
+T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda (e: C).(\lambda (u: T).(\lambda
(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t1: T).(ty3 g e u t1))))) (ex3_3 C T T (\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) t0)))) (\lambda (e:
+T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) t)))) (\lambda (e:
C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u)))))
-(\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(ty3 g e u t1))))))))))
-(\lambda (c0: C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2
-t)).(\lambda (_: (((eq T t2 (TLRef n)) \to (or (ex3_3 C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u)))))
-(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C
-T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u)
-t)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e
-(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e
-u t0))))))))).(\lambda (u: T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u
+(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u
+t0))))))))).(\lambda (u: T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u
t1)).(\lambda (H4: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_:
C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda
(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead c0 (Bind
b) u) (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abst) u0))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (t0:
-T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t2 t0)).(\lambda (_: (((eq T t2
-(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t3:
-T).(pc3 (CHead c0 (Bind b) u) (lift (S n) O t3) t0)))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e
-(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t3: T).(ty3 g
-e u0 t3))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
-T).(pc3 (CHead c0 (Bind b) u) (lift (S n) O u0) t0)))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e
-(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t3: T).(ty3 g
-e u0 t3))))))))).(\lambda (H7: (eq T (THead (Bind b) u t1) (TLRef n))).(let
-H8 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in
-(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t3:
-T).(pc3 c0 (lift (S n) O t3) (THead (Bind b) u t2))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0)))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t3: T).(ty3 g e u0 t3))))) (ex3_3
-C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O
-u0) (THead (Bind b) u t2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (t3: T).(ty3 g e u0 t3)))))) H8)))))))))))))))) (\lambda (c0:
-C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_:
-(((eq T w (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_:
-T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) u)))) (\lambda (e: C).(\lambda
-(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda
-(_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) u))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0
-t))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead
-(Bind Abst) u t))).(\lambda (_: (((eq T v (TLRef n)) \to (or (ex3_3 C T T
-(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0)
-(THead (Bind Abst) u t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
-(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind Abst) u t)))))
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (H5:
+(eq T (THead (Bind b) u t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Bind
+b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda
+(_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead
+(Bind b) u t2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0:
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind b) u t2)))))
(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
+t0)))))) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u:
+T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (TLRef n)) \to (or
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S
+n) O t) u)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0
+(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t:
+T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda
+(_: T).(pc3 c0 (lift (S n) O u0) u)))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))))))).(\lambda (v:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u
+t))).(\lambda (_: (((eq T v (TLRef n)) \to (or (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead (Bind
+Abst) u t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0
+(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0:
+T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda
+(_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind Abst) u t))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
t0))))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (TLRef n))).(let H6
\def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match ee in T return
(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
theorem ty3_gen_bind:
\forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1:
-T).(\forall (x: T).((ty3 g c (THead (Bind b) u t1) x) \to (ex4_3 T T T
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2)
-x)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c u t))))
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind b) u)
-t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c
-(Bind b) u) t2 t0)))))))))))
+T).(\forall (x: T).((ty3 g c (THead (Bind b) u t1) x) \to (ex3_2 T T (\lambda
+(t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))) (\lambda (_:
+T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind b) u) t1 t2))))))))))
\def
\lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1:
T).(\lambda (x: T).(\lambda (H: (ty3 g c (THead (Bind b) u t1) x)).(insert_eq
-T (THead (Bind b) u t1) (\lambda (t: T).(ty3 g c t x)) (ex4_3 T T T (\lambda
-(t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))))
-(\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c u t)))) (\lambda
-(t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind b) u) t1 t2))))
-(\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c (Bind b) u)
-t2 t0))))) (\lambda (y: T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda
-(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Bind b) u t1)) \to
-(ex4_3 T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead
-(Bind b) u t2) t0)))) (\lambda (_: T).(\lambda (t3: T).(\lambda (_: T).(ty3 g
-c0 u t3)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0
-(Bind b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t4: T).(ty3
-g (CHead c0 (Bind b) u) t2 t4))))))))) (\lambda (c0: C).(\lambda (t2:
-T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2
-(THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t3: T).(\lambda (_:
-T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) t)))) (\lambda (_:
-T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c0 u t0)))) (\lambda (t3:
-T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))
-(\lambda (t3: T).(\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind b)
-u) t3 t4)))))))).(\lambda (u0: T).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 u0
-t0)).(\lambda (H4: (((eq T u0 (THead (Bind b) u t1)) \to (ex4_3 T T T
-(\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u
-t3) t0)))) (\lambda (_: T).(\lambda (t4: T).(\lambda (_: T).(ty3 g c0 u
-t4)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0
-(Bind b) u) t1 t3)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (t5: T).(ty3
-g (CHead c0 (Bind b) u) t3 t5)))))))).(\lambda (H5: (pc3 c0 t0 t2)).(\lambda
-(H6: (eq T u0 (THead (Bind b) u t1))).(let H7 \def (f_equal T T (\lambda (e:
-T).e) u0 (THead (Bind b) u t1) H6) in (let H8 \def (eq_ind T u0 (\lambda (t3:
-T).((eq T t3 (THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t4:
-T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t0))))
-(\lambda (_: T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c0 u t5)))) (\lambda
-(t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1
-t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead c0
-(Bind b) u) t4 t6))))))) H4 (THead (Bind b) u t1) H7) in (let H9 \def (eq_ind
-T u0 (\lambda (t3: T).(ty3 g c0 t3 t0)) H3 (THead (Bind b) u t1) H7) in (let
-H10 \def (H8 (refl_equal T (THead (Bind b) u t1))) in (ex4_3_ind T T T
-(\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u
-t3) t0)))) (\lambda (_: T).(\lambda (t4: T).(\lambda (_: T).(ty3 g c0 u
-t4)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0
-(Bind b) u) t1 t3)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (t5: T).(ty3
-g (CHead c0 (Bind b) u) t3 t5)))) (ex4_3 T T T (\lambda (t3: T).(\lambda (_:
-T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) t2)))) (\lambda (_:
-T).(\lambda (t4: T).(\lambda (_: T).(ty3 g c0 u t4)))) (\lambda (t3:
-T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))
-(\lambda (t3: T).(\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b)
-u) t3 t5))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda
-(H11: (pc3 c0 (THead (Bind b) u x0) t0)).(\lambda (H12: (ty3 g c0 u
-x1)).(\lambda (H13: (ty3 g (CHead c0 (Bind b) u) t1 x0)).(\lambda (H14: (ty3
-g (CHead c0 (Bind b) u) x0 x2)).(ex4_3_intro T T T (\lambda (t3: T).(\lambda
-(_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) t2)))) (\lambda (_:
-T).(\lambda (t4: T).(\lambda (_: T).(ty3 g c0 u t4)))) (\lambda (t3:
-T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))
-(\lambda (t3: T).(\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b)
-u) t3 t5)))) x0 x1 x2 (pc3_t t0 c0 (THead (Bind b) u x0) H11 t2 H5) H12 H13
-H14)))))))) H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda
-(H1: (eq T (TSort m) (THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort m)
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+T (THead (Bind b) u t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2
+T T (\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x)))
+(\lambda (_: T).(\lambda (t0: T).(ty3 g c u t0))) (\lambda (t2: T).(\lambda
+(_: T).(ty3 g (CHead c (Bind b) u) t1 t2))))) (\lambda (y: T).(\lambda (H0:
+(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).((eq T t (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda
+(_: T).(pc3 c0 (THead (Bind b) u t2) t0))) (\lambda (_: T).(\lambda (t3:
+T).(ty3 g c0 u t3))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind
+b) u) t1 t2)))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (THead (Bind b) u
+t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b)
+u t3) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t3:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))))).(\lambda (u0:
+T).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 u0 t0)).(\lambda (H4: (((eq T u0
+(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3
+c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u
+t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1
+t3))))))).(\lambda (H5: (pc3 c0 t0 t2)).(\lambda (H6: (eq T u0 (THead (Bind
+b) u t1))).(let H7 \def (f_equal T T (\lambda (e: T).e) u0 (THead (Bind b) u
+t1) H6) in (let H8 \def (eq_ind T u0 (\lambda (t3: T).((eq T t3 (THead (Bind
+b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead
+(Bind b) u t4) t0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5)))
+(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))))) H4
+(THead (Bind b) u t1) H7) in (let H9 \def (eq_ind T u0 (\lambda (t3: T).(ty3
+g c0 t3 t0)) H3 (THead (Bind b) u t1) H7) in (let H10 \def (H8 (refl_equal T
+(THead (Bind b) u t1))) in (ex3_2_ind T T (\lambda (t3: T).(\lambda (_:
+T).(pc3 c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3
+g c0 u t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1
+t3))) (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u
+t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H11: (pc3 c0 (THead (Bind b) u x0)
+t0)).(\lambda (H12: (ty3 g c0 u x1)).(\lambda (H13: (ty3 g (CHead c0 (Bind b)
+u) t1 x0)).(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead
+(Bind b) u t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4)))
+(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))) x0 x1
+(pc3_t t0 c0 (THead (Bind b) u x0) H11 t2 H5) H12 H13))))))
+H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T
+(TSort m) (THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort m) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-False])) I (THead (Bind b) u t1) H1) in (False_ind (ex4_3 T T T (\lambda (t2:
-T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (TSort (next
-g m)))))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u t))))
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u)
-t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c0
-(Bind b) u) t2 t0))))) H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda
-(d: C).(\lambda (u0: T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr)
+False])) I (THead (Bind b) u t1) H1) in (False_ind (ex3_2 T T (\lambda (t2:
+T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (TSort (next g m)))))
+(\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2: T).(\lambda
+(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) H2))))) (\lambda (n:
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (_: (getl n
+c0 (CHead d (Bind Abbr) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0
+t)).(\lambda (_: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
+(t2: T).(\lambda (_: T).(pc3 d (THead (Bind b) u t2) t))) (\lambda (_:
+T).(\lambda (t0: T).(ty3 g d u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
+(CHead d (Bind b) u) t1 t2))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind
+b) u t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1)
+H4) in (False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead
+(Bind b) u t2) (lift (S n) O t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0
+u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1
+t2)))) H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (_: (getl n c0 (CHead d (Bind Abst)
u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: (((eq T u0
-(THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t2: T).(\lambda (_:
-T).(\lambda (_: T).(pc3 d (THead (Bind b) u t2) t)))) (\lambda (_:
-T).(\lambda (t0: T).(\lambda (_: T).(ty3 g d u t0)))) (\lambda (t2:
-T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1 t2))))
-(\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead d (Bind b) u)
-t2 t3)))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind b) u t1))).(let H5
-\def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_:
+(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 d
+(THead (Bind b) u t2) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g d u t0)))
+(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1
+t2))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind b) u t1))).(let H5 \def
+(eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_:
T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H4) in (False_ind
-(ex4_3 T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead
-(Bind b) u t2) (lift (S n) O t))))) (\lambda (_: T).(\lambda (t0: T).(\lambda
-(_: T).(ty3 g c0 u t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_:
-T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_:
-T).(\lambda (t3: T).(ty3 g (CHead c0 (Bind b) u) t2 t3))))) H5)))))))))))
-(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda
-(_: (getl n c0 (CHead d (Bind Abst) u0))).(\lambda (t: T).(\lambda (_: (ty3 g
-d u0 t)).(\lambda (_: (((eq T u0 (THead (Bind b) u t1)) \to (ex4_3 T T T
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 d (THead (Bind b) u t2)
-t)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g d u t0))))
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u)
-t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead d
-(Bind b) u) t2 t3)))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind b) u
-t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1)
-H4) in (False_ind (ex4_3 T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_:
-T).(pc3 c0 (THead (Bind b) u t2) (lift (S n) O u0))))) (\lambda (_:
-T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c0 u t0)))) (\lambda (t2:
-T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))
-(\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c0 (Bind b)
-u) t2 t3))))) H5))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t:
-T).(\lambda (H1: (ty3 g c0 u0 t)).(\lambda (H2: (((eq T u0 (THead (Bind b) u
-t1)) \to (ex4_3 T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3
-c0 (THead (Bind b) u t2) t)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_:
-T).(ty3 g c0 u t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g
-(CHead c0 (Bind b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda
-(t3: T).(ty3 g (CHead c0 (Bind b) u) t2 t3)))))))).(\lambda (b0: B).(\lambda
+(ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2)
+(lift (S n) O u0)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0)))
+(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))
+H5))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda (H1:
+(ty3 g c0 u0 t)).(\lambda (H2: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T
+T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) t)))
+(\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda
+(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (b0: B).(\lambda
(t0: T).(\lambda (t2: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b0) u0) t0
-t2)).(\lambda (H4: (((eq T t0 (THead (Bind b) u t1)) \to (ex4_3 T T T
-(\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0)
-(THead (Bind b) u t3) t2)))) (\lambda (_: T).(\lambda (t4: T).(\lambda (_:
-T).(ty3 g (CHead c0 (Bind b0) u0) u t4)))) (\lambda (t3: T).(\lambda (_:
-T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1
-t3)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead (CHead
-c0 (Bind b0) u0) (Bind b) u) t3 t5)))))))).(\lambda (t3: T).(\lambda (H5:
-(ty3 g (CHead c0 (Bind b0) u0) t2 t3)).(\lambda (H6: (((eq T t2 (THead (Bind
-b) u t1)) \to (ex4_3 T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_:
-T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t4) t3)))) (\lambda (_:
-T).(\lambda (t5: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b0) u0) u t5))))
-(\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead (CHead c0
-(Bind b0) u0) (Bind b) u) t1 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda
-(t6: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t4
-t6)))))))).(\lambda (H7: (eq T (THead (Bind b0) u0 t0) (THead (Bind b) u
-t1))).(let H8 \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) u0 t0) (THead
-(Bind b) u t1) H7) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e in
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
-\Rightarrow u0 | (THead _ t4 _) \Rightarrow t4])) (THead (Bind b0) u0 t0)
-(THead (Bind b) u t1) H7) in ((let H10 \def (f_equal T T (\lambda (e:
+t2)).(\lambda (H4: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
+(t3: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t3)
+t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind b0) u0) u t4)))
+(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind
+b) u) t1 t3))))))).(\lambda (H5: (eq T (THead (Bind b0) u0 t0) (THead (Bind
+b) u t1))).(let H6 \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) u0
+t0) (THead (Bind b) u t1) H5) in ((let H7 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 |
+(TLRef _) \Rightarrow u0 | (THead _ t3 _) \Rightarrow t3])) (THead (Bind b0)
+u0 t0) (THead (Bind b) u t1) H5) in ((let H8 \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 (Bind b0)
-u0 t0) (THead (Bind b) u t1) H7) in (\lambda (H11: (eq T u0 u)).(\lambda
-(H12: (eq B b0 b)).(let H13 \def (eq_ind T t0 (\lambda (t4: T).((eq T t4
-(THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t5: T).(\lambda (_:
-T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t5) t2))))
-(\lambda (_: T).(\lambda (t6: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b0)
-u0) u t6)))) (\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead
-(CHead c0 (Bind b0) u0) (Bind b) u) t1 t5)))) (\lambda (t5: T).(\lambda (_:
-T).(\lambda (t7: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t5
-t7))))))) H4 t1 H10) in (let H14 \def (eq_ind T t0 (\lambda (t4: T).(ty3 g
-(CHead c0 (Bind b0) u0) t4 t2)) H3 t1 H10) in (let H15 \def (eq_ind B b0
-(\lambda (b1: B).((eq T t2 (THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda
-(t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b1) u0) (THead
-(Bind b) u t4) t3)))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_: T).(ty3 g
-(CHead c0 (Bind b1) u0) u t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda
-(_: T).(ty3 g (CHead (CHead c0 (Bind b1) u0) (Bind b) u) t1 t4)))) (\lambda
-(t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead (CHead c0 (Bind b1)
-u0) (Bind b) u) t4 t6))))))) H6 b H12) in (let H16 \def (eq_ind B b0 (\lambda
-(b1: B).(ty3 g (CHead c0 (Bind b1) u0) t2 t3)) H5 b H12) in (let H17 \def
-(eq_ind B b0 (\lambda (b1: B).((eq T t1 (THead (Bind b) u t1)) \to (ex4_3 T T
-T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b1)
-u0) (THead (Bind b) u t4) t2)))) (\lambda (_: T).(\lambda (t5: T).(\lambda
-(_: T).(ty3 g (CHead c0 (Bind b1) u0) u t5)))) (\lambda (t4: T).(\lambda (_:
-T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b1) u0) (Bind b) u) t1
-t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead (CHead
-c0 (Bind b1) u0) (Bind b) u) t4 t6))))))) H13 b H12) in (let H18 \def (eq_ind
-B b0 (\lambda (b1: B).(ty3 g (CHead c0 (Bind b1) u0) t1 t2)) H14 b H12) in
-(eq_ind_r B b (\lambda (b1: B).(ex4_3 T T T (\lambda (t4: T).(\lambda (_:
-T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) (THead (Bind b1) u0 t2)))))
-(\lambda (_: T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c0 u t5)))) (\lambda
-(t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1
-t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead c0
-(Bind b) u) t4 t6)))))) (let H19 \def (eq_ind T u0 (\lambda (t4: T).((eq T t2
-(THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t5: T).(\lambda (_:
-T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) t4) (THead (Bind b) u t5) t3))))
-(\lambda (_: T).(\lambda (t6: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b)
-t4) u t6)))) (\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead
-(CHead c0 (Bind b) t4) (Bind b) u) t1 t5)))) (\lambda (t5: T).(\lambda (_:
-T).(\lambda (t7: T).(ty3 g (CHead (CHead c0 (Bind b) t4) (Bind b) u) t5
-t7))))))) H15 u H11) in (let H20 \def (eq_ind T u0 (\lambda (t4: T).(ty3 g
-(CHead c0 (Bind b) t4) t2 t3)) H16 u H11) in (let H21 \def (eq_ind T u0
-(\lambda (t4: T).((eq T t1 (THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda
-(t5: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) t4) (THead
-(Bind b) u t5) t2)))) (\lambda (_: T).(\lambda (t6: T).(\lambda (_: T).(ty3 g
-(CHead c0 (Bind b) t4) u t6)))) (\lambda (t5: T).(\lambda (_: T).(\lambda (_:
-T).(ty3 g (CHead (CHead c0 (Bind b) t4) (Bind b) u) t1 t5)))) (\lambda (t5:
-T).(\lambda (_: T).(\lambda (t7: T).(ty3 g (CHead (CHead c0 (Bind b) t4)
-(Bind b) u) t5 t7))))))) H17 u H11) in (let H22 \def (eq_ind T u0 (\lambda
-(t4: T).(ty3 g (CHead c0 (Bind b) t4) t1 t2)) H18 u H11) in (let H23 \def
-(eq_ind T u0 (\lambda (t4: T).((eq T t4 (THead (Bind b) u t1)) \to (ex4_3 T T
-T (\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u
-t5) t)))) (\lambda (_: T).(\lambda (t6: T).(\lambda (_: T).(ty3 g c0 u t6))))
-(\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u)
-t1 t5)))) (\lambda (t5: T).(\lambda (_: T).(\lambda (t7: T).(ty3 g (CHead c0
-(Bind b) u) t5 t7))))))) H2 u H11) in (let H24 \def (eq_ind T u0 (\lambda
-(t4: T).(ty3 g c0 t4 t)) H1 u H11) in (eq_ind_r T u (\lambda (t4: T).(ex4_3 T
-T T (\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b)
-u t5) (THead (Bind b) t4 t2))))) (\lambda (_: T).(\lambda (t6: T).(\lambda
-(_: T).(ty3 g c0 u t6)))) (\lambda (t5: T).(\lambda (_: T).(\lambda (_:
-T).(ty3 g (CHead c0 (Bind b) u) t1 t5)))) (\lambda (t5: T).(\lambda (_:
-T).(\lambda (t7: T).(ty3 g (CHead c0 (Bind b) u) t5 t7)))))) (ex4_3_intro T T
-T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u
-t4) (THead (Bind b) u t2))))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_:
-T).(ty3 g c0 u t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g
-(CHead c0 (Bind b) u) t1 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda
-(t6: T).(ty3 g (CHead c0 (Bind b) u) t4 t6)))) t2 t t3 (pc3_refl c0 (THead
-(Bind b) u t2)) H24 H22 H20) u0 H11))))))) b0 H12)))))))))) H9))
-H8)))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda
-(_: (ty3 g c0 w u0)).(\lambda (_: (((eq T w (THead (Bind b) u t1)) \to (ex4_3
-T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind
-b) u t2) u0)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u
-t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind
-b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g
-(CHead c0 (Bind b) u) t2 t0)))))))).(\lambda (v: T).(\lambda (t: T).(\lambda
-(_: (ty3 g c0 v (THead (Bind Abst) u0 t))).(\lambda (_: (((eq T v (THead
-(Bind b) u t1)) \to (ex4_3 T T T (\lambda (t2: T).(\lambda (_: T).(\lambda
-(_: T).(pc3 c0 (THead (Bind b) u t2) (THead (Bind Abst) u0 t))))) (\lambda
-(_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c0 u t0)))) (\lambda (t2:
-T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))
-(\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c0 (Bind b)
-u) t2 t3)))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (THead (Bind b)
-u t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee:
-T).(match ee 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) u t1) H5) in (False_ind (ex4_3 T T T
-(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u
-t2) (THead (Flat Appl) w (THead (Bind Abst) u0 t)))))) (\lambda (_:
-T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c0 u t0)))) (\lambda (t2:
-T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))
-(\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c0 (Bind b)
-u) t2 t3))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2:
-T).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u
-t1)) \to (ex4_3 T T T (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3
-c0 (THead (Bind b) u t3) t2)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_:
-T).(ty3 g c0 u t)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(ty3 g
-(CHead c0 (Bind b) u) t1 t3)))) (\lambda (t3: T).(\lambda (_: T).(\lambda
-(t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)))))))).(\lambda (t3: T).(\lambda
-(_: (ty3 g c0 t2 t3)).(\lambda (_: (((eq T t2 (THead (Bind b) u t1)) \to
-(ex4_3 T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead
-(Bind b) u t4) t3)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g
-c0 u t)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0
-(Bind b) u) t1 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t5: T).(ty3
-g (CHead c0 (Bind b) u) t4 t5)))))))).(\lambda (H5: (eq T (THead (Flat Cast)
-t2 t0) (THead (Bind b) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) t2
-t0) (\lambda (ee: T).(match ee 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) u t1)
-H5) in (False_ind (ex4_3 T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_:
-T).(pc3 c0 (THead (Bind b) u t4) (THead (Flat Cast) t3 t2))))) (\lambda (_:
-T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u t)))) (\lambda (t4:
-T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4))))
-(\lambda (t4: T).(\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b)
-u) t4 t5))))) H6))))))))))) c y x H0))) H))))))).
+(TLRef _) \Rightarrow t0 | (THead _ _ t3) \Rightarrow t3])) (THead (Bind b0)
+u0 t0) (THead (Bind b) u t1) H5) in (\lambda (H9: (eq T u0 u)).(\lambda (H10:
+(eq B b0 b)).(let H11 \def (eq_ind T t0 (\lambda (t3: T).((eq T t3 (THead
+(Bind b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 (CHead
+c0 (Bind b0) u0) (THead (Bind b) u t4) t2))) (\lambda (_: T).(\lambda (t5:
+T).(ty3 g (CHead c0 (Bind b0) u0) u t5))) (\lambda (t4: T).(\lambda (_:
+T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1 t4)))))) H4 t1 H8) in
+(let H12 \def (eq_ind T t0 (\lambda (t3: T).(ty3 g (CHead c0 (Bind b0) u0) t3
+t2)) H3 t1 H8) in (let H13 \def (eq_ind B b0 (\lambda (b1: B).((eq T t1
+(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3
+(CHead c0 (Bind b1) u0) (THead (Bind b) u t3) t2))) (\lambda (_: T).(\lambda
+(t4: T).(ty3 g (CHead c0 (Bind b1) u0) u t4))) (\lambda (t3: T).(\lambda (_:
+T).(ty3 g (CHead (CHead c0 (Bind b1) u0) (Bind b) u) t1 t3)))))) H11 b H10)
+in (let H14 \def (eq_ind B b0 (\lambda (b1: B).(ty3 g (CHead c0 (Bind b1) u0)
+t1 t2)) H12 b H10) in (eq_ind_r B b (\lambda (b1: B).(ex3_2 T T (\lambda (t3:
+T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) (THead (Bind b1) u0 t2))))
+(\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3: T).(\lambda
+(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))) (let H15 \def (eq_ind T u0
+(\lambda (t3: T).((eq T t1 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
+(t4: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) t3) (THead (Bind b) u t4)
+t2))) (\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) t3) u t5)))
+(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b) t3) (Bind
+b) u) t1 t4)))))) H13 u H9) in (let H16 \def (eq_ind T u0 (\lambda (t3:
+T).(ty3 g (CHead c0 (Bind b) t3) t1 t2)) H14 u H9) in (let H17 \def (eq_ind T
+u0 (\lambda (t3: T).((eq T t3 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
+(t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t))) (\lambda (_:
+T).(\lambda (t5: T).(ty3 g c0 u t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g
+(CHead c0 (Bind b) u) t1 t4)))))) H2 u H9) in (let H18 \def (eq_ind T u0
+(\lambda (t3: T).(ty3 g c0 t3 t)) H1 u H9) in (eq_ind_r T u (\lambda (t3:
+T).(ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4)
+(THead (Bind b) t3 t2)))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5)))
+(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))))
+(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u
+t3) (THead (Bind b) u t2)))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u
+t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))
+t2 t (pc3_refl c0 (THead (Bind b) u t2)) H18 H16) u0 H9))))) b0 H10))))))))
+H7)) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0:
+T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (_: (((eq T w (THead (Bind b) u
+t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b)
+u t2) u0))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (v:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0
+t))).(\lambda (_: (((eq T v (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
+(t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (THead (Bind Abst) u0
+t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (H5:
+(eq T (THead (Flat Appl) w v) (THead (Bind b) u t1))).(let H6 \def (eq_ind T
+(THead (Flat Appl) w v) (\lambda (ee: T).(match ee 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) u t1) H5) in (False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3
+c0 (THead (Bind b) u t2) (THead (Flat Appl) w (THead (Bind Abst) u0 t)))))
+(\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda
+(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) H6)))))))))))) (\lambda (c0:
+C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda
+(_: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3:
+T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) t2))) (\lambda (_:
+T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t3: T).(\lambda (_: T).(ty3 g
+(CHead c0 (Bind b) u) t1 t3))))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2
+t3)).(\lambda (_: (((eq T t2 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
+(t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t3))) (\lambda (_:
+T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t4: T).(\lambda (_: T).(ty3 g
+(CHead c0 (Bind b) u) t1 t4))))))).(\lambda (H5: (eq T (THead (Flat Cast) t2
+t0) (THead (Bind b) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t0)
+(\lambda (ee: T).(match ee 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) u t1) H5) in (False_ind
+(ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4)
+(THead (Flat Cast) t3 t2)))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t)))
+(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4))))
+H6))))))))))) c y x H0))) H))))))).
theorem ty3_gen_appl:
\forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x:
\def
\lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x:
T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(insert_eq T (THead
-(Flat Appl) w v) (\lambda (t: T).(ty3 g c t x)) (ex3_2 T T (\lambda (u:
-T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
-(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
-(\lambda (u: T).(\lambda (_: T).(ty3 g c w u)))) (\lambda (y: T).(\lambda
-(H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
-T).((eq T t (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda
-(t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t1)) t0))) (\lambda
-(u: T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u t1)))) (\lambda (u:
-T).(\lambda (_: T).(ty3 g c0 w u)))))))) (\lambda (c0: C).(\lambda (t2:
-T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2
-(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t0: T).(pc3
-c0 (THead (Flat Appl) w (THead (Bind Abst) u t0)) t))) (\lambda (u:
-T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u t0)))) (\lambda (u:
-T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (u: T).(\lambda (t1:
-T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (THead (Flat Appl)
-w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat
-Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0:
+(Flat Appl) w v) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2 T T
+(\lambda (u: T).(\lambda (t0: T).(pc3 c (THead (Flat Appl) w (THead (Bind
+Abst) u t0)) x))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c v (THead (Bind
+Abst) u t0)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))))) (\lambda (y:
+T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).((eq T t (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda
+(u: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u
+t1)) t0))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u
+t1)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))))))) (\lambda (c0:
+C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda
+(_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u:
+T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t0))
+t))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u t0))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (u: T).(\lambda
+(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (THead (Flat
+Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead
+(Flat Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0:
T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_:
T).(ty3 g c0 w u0))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: (eq T u
(THead (Flat Appl) w v))).(let H7 \def (f_equal T T (\lambda (e: T).e) u
(u0: T).(\lambda (t0: T).(pc3 (CHead c0 (Bind b) u) (THead (Flat Appl) w
(THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g
(CHead c0 (Bind b) u) v (THead (Bind Abst) u0 t0)))) (\lambda (u0:
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) w u0))))))).(\lambda (t0:
-T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t2 t0)).(\lambda (_: (((eq T t2
-(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t3:
-T).(pc3 (CHead c0 (Bind b) u) (THead (Flat Appl) w (THead (Bind Abst) u0 t3))
-t0))) (\lambda (u0: T).(\lambda (t3: T).(ty3 g (CHead c0 (Bind b) u) v (THead
-(Bind Abst) u0 t3)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind
-b) u) w u0))))))).(\lambda (H7: (eq T (THead (Bind b) u t1) (THead (Flat
-Appl) w v))).(let H8 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee:
-T).(match ee 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) H7) in (False_ind (ex3_2 T T
-(\lambda (u0: T).(\lambda (t3: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind
-Abst) u0 t3)) (THead (Bind b) u t2)))) (\lambda (u0: T).(\lambda (t3: T).(ty3
-g c0 v (THead (Bind Abst) u0 t3)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g
-c0 w u0)))) H8)))))))))))))))) (\lambda (c0: C).(\lambda (w0: T).(\lambda (u:
-T).(\lambda (H1: (ty3 g c0 w0 u)).(\lambda (H2: (((eq T w0 (THead (Flat Appl)
-w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t: T).(pc3 c0 (THead (Flat
-Appl) w (THead (Bind Abst) u0 t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3
-g c0 v (THead (Bind Abst) u0 t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0
-w u0))))))).(\lambda (v0: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v0
-(THead (Bind Abst) u t))).(\lambda (H4: (((eq T v0 (THead (Flat Appl) w v))
-\to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w
-(THead (Bind Abst) u0 t0)) (THead (Bind Abst) u t)))) (\lambda (u0:
-T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0:
-T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (H5: (eq T (THead (Flat
-Appl) w0 v0) (THead (Flat Appl) w v))).(let H6 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow w0 |
-(TLRef _) \Rightarrow w0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat
-Appl) w0 v0) (THead (Flat Appl) w v) H5) in ((let H7 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ _ t0) \Rightarrow t0]))
-(THead (Flat Appl) w0 v0) (THead (Flat Appl) w v) H5) in (\lambda (H8: (eq T
-w0 w)).(let H9 \def (eq_ind T v0 (\lambda (t0: T).((eq T t0 (THead (Flat
-Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead
-(Flat Appl) w (THead (Bind Abst) u0 t1)) (THead (Bind Abst) u t)))) (\lambda
-(u0: T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda
-(u0: T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 v H7) in (let H10 \def
-(eq_ind T v0 (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 v H7)
-in (let H11 \def (eq_ind T w0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w
-v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat
-Appl) w (THead (Bind Abst) u0 t1)) u))) (\lambda (u0: T).(\lambda (t1:
-T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda (_:
-T).(ty3 g c0 w u0)))))) H2 w H8) in (let H12 \def (eq_ind T w0 (\lambda (t0:
-T).(ty3 g c0 t0 u)) H1 w H8) in (eq_ind_r T w (\lambda (t0: T).(ex3_2 T T
-(\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind
-Abst) u0 t1)) (THead (Flat Appl) t0 (THead (Bind Abst) u t))))) (\lambda (u0:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) w u0))))))).(\lambda (H5: (eq
+T (THead (Bind b) u t1) (THead (Flat Appl) w v))).(let H6 \def (eq_ind T
+(THead (Bind b) u t1) (\lambda (ee: T).(match ee 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) H5) in (False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0:
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (THead (Bind b) u
+t2)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0
+t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) H6)))))))))))))
+(\lambda (c0: C).(\lambda (w0: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w0
+u)).(\lambda (H2: (((eq T w0 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda
+(u0: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0
+t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u0
+t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (v0:
+T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v0 (THead (Bind Abst) u
+t))).(\lambda (H4: (((eq T v0 (THead (Flat Appl) w v)) \to (ex3_2 T T
+(\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind
+Abst) u0 t0)) (THead (Bind Abst) u t)))) (\lambda (u0: T).(\lambda (t0:
+T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_:
+T).(ty3 g c0 w u0))))))).(\lambda (H5: (eq T (THead (Flat Appl) w0 v0) (THead
+(Flat Appl) w v))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow w0 | (TLRef _)
+\Rightarrow w0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) w0 v0)
+(THead (Flat Appl) w v) H5) in ((let H7 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v0 |
+(TLRef _) \Rightarrow v0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
+Appl) w0 v0) (THead (Flat Appl) w v) H5) in (\lambda (H8: (eq T w0 w)).(let
+H9 \def (eq_ind T v0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w v)) \to
+(ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w
+(THead (Bind Abst) u0 t1)) (THead (Bind Abst) u t)))) (\lambda (u0:
T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0:
-T).(\lambda (_: T).(ty3 g c0 w u0))))) (ex3_2_intro T T (\lambda (u0:
-T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0))
-(THead (Flat Appl) w (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda
-(t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda
-(_: T).(ty3 g c0 w u0))) u t (pc3_refl c0 (THead (Flat Appl) w (THead (Bind
-Abst) u t))) H10 H12) w0 H8))))))) H6)))))))))))) (\lambda (c0: C).(\lambda
-(t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T
-t1 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t:
-T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) t2))) (\lambda (u:
-T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u:
-T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (t0: T).(\lambda (_: (ty3 g
-c0 t2 t0)).(\lambda (_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T
-(\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind
-Abst) u t)) t0))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind
-Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda
-(H5: (eq T (THead (Flat Cast) t2 t1) (THead (Flat Appl) w v))).(let H6 \def
-(eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee 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) H5) in (False_ind (ex3_2 T
-T (\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind
-Abst) u t)) (THead (Flat Cast) t0 t2)))) (\lambda (u: T).(\lambda (t: T).(ty3
-g c0 v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w
-u)))) H6))))))))))) c y x H0))) H)))))).
+T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 v H7) in (let H10 \def (eq_ind T
+v0 (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 v H7) in (let
+H11 \def (eq_ind T w0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w v)) \to
+(ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w
+(THead (Bind Abst) u0 t1)) u))) (\lambda (u0: T).(\lambda (t1: T).(ty3 g c0 v
+(THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w
+u0)))))) H2 w H8) in (let H12 \def (eq_ind T w0 (\lambda (t0: T).(ty3 g c0 t0
+u)) H1 w H8) in (eq_ind_r T w (\lambda (t0: T).(ex3_2 T T (\lambda (u0:
+T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t1))
+(THead (Flat Appl) t0 (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda
+(t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda
+(_: T).(ty3 g c0 w u0))))) (ex3_2_intro T T (\lambda (u0: T).(\lambda (t0:
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (THead (Flat Appl)
+w (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v
+(THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w
+u0))) u t (pc3_refl c0 (THead (Flat Appl) w (THead (Bind Abst) u t))) H10
+H12) w0 H8))))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (THead (Flat
+Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead
+(Flat Appl) w (THead (Bind Abst) u t)) t2))) (\lambda (u: T).(\lambda (t:
+T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_:
+T).(ty3 g c0 w u))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2
+t0)).(\lambda (_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda
+(u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t))
+t0))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (H5: (eq T
+(THead (Flat Cast) t2 t1) (THead (Flat Appl) w v))).(let H6 \def (eq_ind T
+(THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee 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) H5) in (False_ind (ex3_2 T T (\lambda
+(u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t))
+(THead (Flat Cast) t0 t2)))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v
+(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))
+H6))))))))))) c y x H0))) H)))))).
theorem ty3_gen_cast:
\forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall
\def
\lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
(x: T).(\lambda (H: (ty3 g c (THead (Flat Cast) t2 t1) x)).(insert_eq T
-(THead (Flat Cast) t2 t1) (\lambda (t: T).(ty3 g c t x)) (ex3 T (\lambda (t0:
-T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 g c t1 t2))
-(\lambda (t0: T).(ty3 g c t2 t0))) (\lambda (y: T).(\lambda (H0: (ty3 g c y
-x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t
-(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0 (THead (Flat
-Cast) t3 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t3: T).(ty3 g
-c0 t2 t3))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t: T).(\lambda
-(_: (ty3 g c0 t0 t)).(\lambda (_: (((eq T t0 (THead (Flat Cast) t2 t1)) \to
-(ex3 T (\lambda (t3: T).(pc3 c0 (THead (Flat Cast) t3 t2) t)) (\lambda (_:
-T).(ty3 g c0 t1 t2)) (\lambda (t3: T).(ty3 g c0 t2 t3)))))).(\lambda (u:
-T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u t3)).(\lambda (H4: (((eq T u
-(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat
-Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g
-c0 t2 t4)))))).(\lambda (H5: (pc3 c0 t3 t0)).(\lambda (H6: (eq T u (THead
-(Flat Cast) t2 t1))).(let H7 \def (f_equal T T (\lambda (e: T).e) u (THead
-(Flat Cast) t2 t1) H6) in (let H8 \def (eq_ind T u (\lambda (t4: T).((eq T t4
-(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat
-Cast) t5 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g
-c0 t2 t5))))) H4 (THead (Flat Cast) t2 t1) H7) in (let H9 \def (eq_ind T u
-(\lambda (t4: T).(ty3 g c0 t4 t3)) H3 (THead (Flat Cast) t2 t1) H7) in (let
-H10 \def (H8 (refl_equal T (THead (Flat Cast) t2 t1))) in (ex3_ind T (\lambda
+(THead (Flat Cast) t2 t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3
+T (\lambda (t0: T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3
+g c t1 t2)) (\lambda (t0: T).(ty3 g c t2 t0)))) (\lambda (y: T).(\lambda (H0:
+(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).((eq T t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0
+(THead (Flat Cast) t3 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda
+(t3: T).(ty3 g c0 t2 t3))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 t0 t)).(\lambda (_: (((eq T t0 (THead (Flat Cast)
+t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0 (THead (Flat Cast) t3 t2) t))
+(\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t3: T).(ty3 g c0 t2
+t3)))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u
+t3)).(\lambda (H4: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda
(t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1
-t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)) (ex3 T (\lambda (t4: T).(pc3 c0
-(THead (Flat Cast) t4 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda
-(t4: T).(ty3 g c0 t2 t4))) (\lambda (x0: T).(\lambda (H11: (pc3 c0 (THead
-(Flat Cast) x0 t2) t3)).(\lambda (H12: (ty3 g c0 t1 t2)).(\lambda (H13: (ty3
-g c0 t2 x0)).(ex3_intro T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 t2)
-t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)) x0
-(pc3_t t3 c0 (THead (Flat Cast) x0 t2) H11 t0 H5) H12 H13)))))
-H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T
-(TSort m) (THead (Flat Cast) t2 t1))).(let H2 \def (eq_ind T (TSort m)
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-False])) I (THead (Flat Cast) t2 t1) H1) in (False_ind (ex3 T (\lambda (t0:
-T).(pc3 c0 (THead (Flat Cast) t0 t2) (TSort (next g m)))) (\lambda (_:
-T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) H2))))) (\lambda (n:
+t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)))))).(\lambda (H5: (pc3 c0 t3
+t0)).(\lambda (H6: (eq T u (THead (Flat Cast) t2 t1))).(let H7 \def (f_equal
+T T (\lambda (e: T).e) u (THead (Flat Cast) t2 t1) H6) in (let H8 \def
+(eq_ind T u (\lambda (t4: T).((eq T t4 (THead (Flat Cast) t2 t1)) \to (ex3 T
+(\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t3)) (\lambda (_: T).(ty3
+g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H4 (THead (Flat Cast) t2
+t1) H7) in (let H9 \def (eq_ind T u (\lambda (t4: T).(ty3 g c0 t4 t3)) H3
+(THead (Flat Cast) t2 t1) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat
+Cast) t2 t1))) in (ex3_ind T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4
+t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4))
+(ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_:
+T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4))) (\lambda (x0:
+T).(\lambda (H11: (pc3 c0 (THead (Flat Cast) x0 t2) t3)).(\lambda (H12: (ty3
+g c0 t1 t2)).(\lambda (H13: (ty3 g c0 t2 x0)).(ex3_intro T (\lambda (t4:
+T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2))
+(\lambda (t4: T).(ty3 g c0 t2 t4)) x0 (pc3_t t3 c0 (THead (Flat Cast) x0 t2)
+H11 t0 H5) H12 H13))))) H10)))))))))))))))) (\lambda (c0: C).(\lambda (m:
+nat).(\lambda (H1: (eq T (TSort m) (THead (Flat Cast) t2 t1))).(let H2 \def
+(eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t2 t1) H1) in
+(False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (TSort
+(next g m)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2
+t0))) H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda
+(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3
+T (\lambda (t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3
+g d t1 t2)) (\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef
+n) (THead (Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
+(THead (Flat Cast) t2 t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0
+(THead (Flat Cast) t0 t2) (lift (S n) O t))) (\lambda (_: T).(ty3 g c0 t1
+t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (n:
nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0
-(CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u
+(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u
t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda
(t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 g d t1 t2))
(\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef n) (THead
ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead
(Flat Cast) t2 t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead
-(Flat Cast) t0 t2) (lift (S n) O t))) (\lambda (_: T).(ty3 g c0 t1 t2))
-(\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (n: nat).(\lambda
-(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 (CHead d
-(Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_:
-(((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 d
-(THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 g d t1 t2)) (\lambda (t0:
-T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Cast) t2
-t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t2
-t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0
-t2) (lift (S n) O u))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0:
-T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda
-(t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (THead (Flat Cast)
-t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) t))
-(\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2
-t0)))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3
-g (CHead c0 (Bind b) u) t0 t3)).(\lambda (_: (((eq T t0 (THead (Flat Cast) t2
-t1)) \to (ex3 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (THead (Flat
-Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))
-(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t2 t4)))))).(\lambda (t4:
-T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (_: (((eq T t3
-(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 (CHead c0 (Bind
-b) u) (THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g (CHead c0 (Bind
-b) u) t1 t2)) (\lambda (t5: T).(ty3 g (CHead c0 (Bind b) u) t2
-t5)))))).(\lambda (H7: (eq T (THead (Bind b) u t0) (THead (Flat Cast) t2
-t1))).(let H8 \def (eq_ind T (THead (Bind b) u t0) (\lambda (ee: T).(match ee
-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) t2 t1) H7) in (False_ind (ex3 T (\lambda (t5:
-T).(pc3 c0 (THead (Flat Cast) t5 t2) (THead (Bind b) u t3))) (\lambda (_:
-T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))) H8))))))))))))))))
-(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w
-u)).(\lambda (_: (((eq T w (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda
-(t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) u)) (\lambda (_: T).(ty3 g c0 t1
-t2)) (\lambda (t0: T).(ty3 g c0 t2 t0)))))).(\lambda (v: T).(\lambda (t:
-T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u t))).(\lambda (_: (((eq T v
+(Flat Cast) t0 t2) (lift (S n) O u))) (\lambda (_: T).(ty3 g c0 t1 t2))
+(\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (c0: C).(\lambda
+(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u
(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat
-Cast) t0 t2) (THead (Bind Abst) u t))) (\lambda (_: T).(ty3 g c0 t1 t2))
-(\lambda (t0: T).(ty3 g c0 t2 t0)))))).(\lambda (H5: (eq T (THead (Flat Appl)
-w v) (THead (Flat Cast) t2 t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w
-v) (\lambda (ee: T).(match ee 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) t2 t1) H5) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead
-(Flat Cast) t0 t2) (THead (Flat Appl) w (THead (Bind Abst) u t)))) (\lambda
-(_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) H6))))))))))))
-(\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t0
-t3)).(\lambda (H2: (((eq T t0 (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda
-(t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1
-t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)))))).(\lambda (t4: T).(\lambda (H3:
-(ty3 g c0 t3 t4)).(\lambda (H4: (((eq T t3 (THead (Flat Cast) t2 t1)) \to
+Cast) t0 t2) t)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0
+t2 t0)))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_:
+(ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (_: (((eq T t0 (THead (Flat
+Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (THead
+(Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))
+(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t2 t4)))))).(\lambda (H5: (eq T
+(THead (Bind b) u t0) (THead (Flat Cast) t2 t1))).(let H6 \def (eq_ind T
+(THead (Bind b) u t0) (\lambda (ee: T).(match ee 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) t2 t1) H5) in (False_ind (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat
+Cast) t4 t2) (THead (Bind b) u t3))) (\lambda (_: T).(ty3 g c0 t1 t2))
+(\lambda (t4: T).(ty3 g c0 t2 t4))) H6))))))))))))) (\lambda (c0: C).(\lambda
+(w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w
+(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat
+Cast) t0 t2) u)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0
+t2 t0)))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead
+(Bind Abst) u t))).(\lambda (_: (((eq T v (THead (Flat Cast) t2 t1)) \to (ex3
+T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Bind Abst) u
+t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2
+t0)))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (THead (Flat Cast) t2
+t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match
+ee 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) t2 t1) H5) in (False_ind (ex3 T
+(\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Flat Appl) w
+(THead (Bind Abst) u t)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0:
+T).(ty3 g c0 t2 t0))) H6)))))))))))) (\lambda (c0: C).(\lambda (t0:
+T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t0 t3)).(\lambda (H2: (((eq T t0
+(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat
+Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g
+c0 t2 t4)))))).(\lambda (t4: T).(\lambda (H3: (ty3 g c0 t3 t4)).(\lambda (H4:
+(((eq T t3 (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0
+(THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda
+(t5: T).(ty3 g c0 t2 t5)))))).(\lambda (H5: (eq T (THead (Flat Cast) t3 t0)
+(THead (Flat Cast) t2 t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
+\Rightarrow t3 | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) t3 t0)
+(THead (Flat Cast) t2 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
+(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast)
+t3 t0) (THead (Flat Cast) t2 t1) H5) in (\lambda (H8: (eq T t3 t2)).(let H9
+\def (eq_ind T t3 (\lambda (t: T).((eq T t (THead (Flat Cast) t2 t1)) \to
(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4)) (\lambda (_:
-T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)))))).(\lambda (H5: (eq
-T (THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1))).(let H6 \def (f_equal
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ t _) \Rightarrow t]))
-(THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1) H5) in ((let H7 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
-\Rightarrow t])) (THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1) H5) in
-(\lambda (H8: (eq T t3 t2)).(let H9 \def (eq_ind T t3 (\lambda (t: T).((eq T
+T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H4 t2 H8) in (let
+H10 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t4)) H3 t2 H8) in (let H11
+\def (eq_ind T t3 (\lambda (t: T).((eq T t0 (THead (Flat Cast) t2 t1)) \to
+(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t)) (\lambda (_:
+T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H2 t2 H8) in (let
+H12 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t0 t)) H1 t2 H8) in (eq_ind_r
+T t2 (\lambda (t: T).(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5
+t2) (THead (Flat Cast) t4 t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda
+(t5: T).(ty3 g c0 t2 t5)))) (let H13 \def (eq_ind T t0 (\lambda (t: T).((eq T
t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat
-Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g
-c0 t2 t5))))) H4 t2 H8) in (let H10 \def (eq_ind T t3 (\lambda (t: T).(ty3 g
-c0 t t4)) H3 t2 H8) in (let H11 \def (eq_ind T t3 (\lambda (t: T).((eq T t0
-(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat
-Cast) t5 t2) t)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0
-t2 t5))))) H2 t2 H8) in (let H12 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0
-t0 t)) H1 t2 H8) in (eq_ind_r T t2 (\lambda (t: T).(ex3 T (\lambda (t5:
-T).(pc3 c0 (THead (Flat Cast) t5 t2) (THead (Flat Cast) t4 t))) (\lambda (_:
-T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)))) (let H13 \def
-(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Cast) t2 t1)) \to (ex3 T
-(\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t2)) (\lambda (_: T).(ty3
-g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H11 t1 H7) in (let H14
-\def (eq_ind T t0 (\lambda (t: T).(ty3 g c0 t t2)) H12 t1 H7) in (ex3_intro T
-(\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) (THead (Flat Cast) t4
-t2))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)) t4
-(pc3_refl c0 (THead (Flat Cast) t4 t2)) H14 H10))) t3 H8))))))) H6)))))))))))
-c y x H0))) H)))))).
+Cast) t5 t2) t2)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g
+c0 t2 t5))))) H11 t1 H7) in (let H14 \def (eq_ind T t0 (\lambda (t: T).(ty3 g
+c0 t t2)) H12 t1 H7) in (ex3_intro T (\lambda (t5: T).(pc3 c0 (THead (Flat
+Cast) t5 t2) (THead (Flat Cast) t4 t2))) (\lambda (_: T).(ty3 g c0 t1 t2))
+(\lambda (t5: T).(ty3 g c0 t2 t5)) t4 (pc3_refl c0 (THead (Flat Cast) t4 t2))
+H14 H10))) t3 H8))))))) H6))))))))))) c y x H0))) H)))))).