--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Basic-1/ty3/defs.ma".
+
+include "Basic-1/pc3/props.ma".
+
+theorem ty3_gen_sort:
+ \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c
+(TSort n) x) \to (pc3 c (TSort (next g n)) x)))))
+\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)) (\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) 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
+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 Cast) t0 t2))
+H6))))))))))) c y x H0))) H))))).
+(* COMMENTS
+Initial nodes: 1179
+END *)
+
+theorem ty3_gen_lref:
+ \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c
+(TLRef n) x) \to (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))))))))))
+\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)) (\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 (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
+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)
+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) t1)))) (\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: (pc3 c0 t1 t2)).(\lambda (H6:
+(eq T u (TLRef n))).(let H7 \def (f_equal T T (\lambda (e: T).e) u (TLRef n)
+H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef n)) \to (or
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t3: T).(pc3 c0 (lift
+(S n) O t3) t1)))) (\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) t1)))) (\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)))))))) H4 (TLRef n) H7)
+in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef n)
+H7) in (let H10 \def (H8 (refl_equal T (TLRef n))) in (or_ind (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) 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) t1)))) (\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))))) (or (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) 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) 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)))))) (\lambda (H11: (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)
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
+t0)))))).(ex3_3_ind 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) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t0: T).(ty3 g e u0 t0)))) (or (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) 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) 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)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2:
+T).(\lambda (H12: (pc3 c0 (lift (S n) O x2) t1)).(\lambda (H13: (getl n c0
+(CHead x0 (Bind Abbr) x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_introl
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift
+(S n) O t0) 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) 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))))) (ex3_3_intro C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0)
+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)))) x0 x1 x2 (pc3_t t1 c0 (lift (S n) O x2) H12 t2 H5) H13
+H14)))))))) H11)) (\lambda (H11: (ex3_3 C T T (\lambda (_: C).(\lambda (u0:
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t1)))) (\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)))))).(ex3_3_ind C T T
+(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0)
+t1)))) (\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)))) (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
+T).(pc3 c0 (lift (S n) O t0) 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) 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))))))
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H12: (pc3 c0
+(lift (S n) O x1) t1)).(\lambda (H13: (getl n c0 (CHead x0 (Bind Abst)
+x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_intror (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) 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) 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))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0:
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) 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)))) x0 x1 x2 (pc3_t t1 c0
+(lift (S n) O x1) H12 t2 H5) H13 H14)))))))) H11)) H10))))))))))))))))
+(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (TLRef
+n))).(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 (TLRef n) H1) in
+(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t:
+T).(pc3 c0 (lift (S n) O t) (TSort (next g m)))))) (\lambda (e: C).(\lambda
+(u: T).(\lambda (_: T).(getl n c0 (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 c0 (lift (S n) O u) (TSort (next
+g m)))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t)))))) H2))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda
+(u: T).(\lambda (H1: (getl n0 c0 (CHead d (Bind Abbr) u))).(\lambda (t:
+T).(\lambda (H2: (ty3 g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S
+n) O t0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d
+(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 d (lift (S n) O u0) t)))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (H4:
+(eq T (TLRef n0) (TLRef n))).(let H5 \def (f_equal T nat (\lambda (e:
+T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n0 |
+(TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n0])) (TLRef n0) (TLRef
+n) H4) in (let H6 \def (eq_ind nat n0 (\lambda (n1: nat).(getl n1 c0 (CHead d
+(Bind Abbr) u))) H1 n H5) in (eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C
+T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O
+t0) (lift (S n1) O 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) (lift (S n1) O 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))))))) (or_introl (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda
+(t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O 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) (lift (S n) O 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))))) (ex3_3_intro C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O
+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)))) d u t (pc3_refl c0 (lift (S n) O t)) H6 H2)) n0 H5))))))))))))
+(\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(H1: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H2: (ty3
+g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S n) O t0) t)))) (\lambda
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d (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 d (lift (S
+n) O u0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d
+(CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0:
+T).(ty3 g e u0 t0))))))))).(\lambda (H4: (eq T (TLRef n0) (TLRef n))).(let H5
+\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat)
+with [(TSort _) \Rightarrow n0 | (TLRef n1) \Rightarrow n1 | (THead _ _ _)
+\Rightarrow n0])) (TLRef n0) (TLRef n) H4) in (let H6 \def (eq_ind nat n0
+(\lambda (n1: nat).(getl n1 c0 (CHead d (Bind Abst) u))) H1 n H5) in
+(eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C T T (\lambda (_: C).(\lambda
+(_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n1) O u)))))
+(\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) (lift (S n1) O u))))) (\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))))))) (or_intror (ex3_3
+C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O
+t0) (lift (S n) O u))))) (\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) (lift (S n) O u)))))
+(\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))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
+T).(pc3 c0 (lift (S n) O u0) (lift (S n) O u))))) (\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)))) d u t (pc3_refl c0
+(lift (S n) O u)) H6 H2)) n0 H5)))))))))))) (\lambda (c0: C).(\lambda (u:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((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) 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) 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 (b: B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_:
+(ty3 g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to
+(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 (CHead
+c0 (Bind b) u) (lift (S n) O t0) t2)))) (\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 (t0: T).(ty3 g e u0 t0))))) (ex3_3
+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 (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 _)
+\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 (Flat Appl) w (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 (Flat Appl) w (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)))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S
+n) O t) t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0
+(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 c0 (lift (S n) O u) t2)))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (t0:
+T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef n)) \to (or
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S
+n) O t) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0
+(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 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 (t: T).(ty3 g e u t))))))))).(\lambda (H5: (eq T
+(THead (Flat Cast) t2 t1) (TLRef n))).(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 _ _
+_) \Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t)
+(THead (Flat Cast) t0 t2))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c0 (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 c0 (lift (S n) O u) (THead (Flat Cast) t0 t2)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t)))))) H6))))))))))) c y x H0))) H))))).
+(* COMMENTS
+Initial nodes: 5569
+END *)
+
+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 (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)) (\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 (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 (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 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 (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 _ _ 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))))))).
+(* COMMENTS
+Initial nodes: 3389
+END *)
+
+theorem ty3_gen_appl:
+ \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x:
+T).((ty3 g c (THead (Flat Appl) w v) x) \to (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)))))))))
+\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)) (\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
+(THead (Flat Appl) w v) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq
+T t0 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t3:
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t3)) t1))) (\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)))))) H4 (THead (Flat Appl) w v) H7)
+in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (THead
+(Flat Appl) w v) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat Appl) w
+v))) in (ex3_2_ind 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))) (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0
+(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) 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)))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H11: (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) x0 x1))
+t1)).(\lambda (H12: (ty3 g c0 v (THead (Bind Abst) x0 x1))).(\lambda (H13:
+(ty3 g c0 w x0)).(ex3_2_intro T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0
+(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) 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))) x0 x1 (pc3_t t1 c0 (THead (Flat Appl) w
+(THead (Bind Abst) x0 x1)) H11 t2 H5) H12 H13)))))) H10))))))))))))))))
+(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (THead (Flat
+Appl) w v))).(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 Appl) w
+v) H1) in (False_ind (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0
+(THead (Flat Appl) w (THead (Bind Abst) u t)) (TSort (next g m))))) (\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)))) 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 Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0:
+T).(pc3 d (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t))) (\lambda (u0:
+T).(\lambda (t0: T).(ty3 g d v (THead (Bind Abst) u0 t0)))) (\lambda (u0:
+T).(\lambda (_: T).(ty3 g d w u0))))))).(\lambda (H4: (eq T (TLRef n) (THead
+(Flat Appl) w v))).(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 Appl) w v) H4) in (False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0:
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (lift (S n) O
+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)))) 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 Appl) w v)) \to (ex3_2 T T
+(\lambda (u0: T).(\lambda (t0: T).(pc3 d (THead (Flat Appl) w (THead (Bind
+Abst) u0 t0)) t))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g d v (THead (Bind
+Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g d w
+u0))))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Appl) w v))).(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 Appl) w v) H4) in
+(False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat
+Appl) w (THead (Bind Abst) u0 t0)) (lift (S n) O u)))) (\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)))) H5))))))))))) (\lambda (c0: C).(\lambda
+(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((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)) 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 (b: B).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1
+t2)).(\lambda (_: (((eq T t1 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda
+(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 (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)))))) 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)))))).
+(* COMMENTS
+Initial nodes: 3171
+END *)
+
+theorem ty3_gen_cast:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall
+(x: T).((ty3 g c (THead (Flat Cast) t2 t1) x) \to (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))))))))
+\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)) (\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)))))).(\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 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 (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))))) 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)))))).
+(* COMMENTS
+Initial nodes: 2609
+END *)
+
+theorem tys3_gen_nil:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).((tys3 g c TNil u) \to (ex T
+(\lambda (u0: T).(ty3 g c u u0))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (tys3 g c TNil
+u)).(insert_eq TList TNil (\lambda (t: TList).(tys3 g c t u)) (\lambda (_:
+TList).(ex T (\lambda (u0: T).(ty3 g c u u0)))) (\lambda (y: TList).(\lambda
+(H0: (tys3 g c y u)).(tys3_ind g c (\lambda (t: TList).(\lambda (t0: T).((eq
+TList t TNil) \to (ex T (\lambda (u0: T).(ty3 g c t0 u0)))))) (\lambda (u0:
+T).(\lambda (u1: T).(\lambda (H1: (ty3 g c u0 u1)).(\lambda (_: (eq TList
+TNil TNil)).(ex_intro T (\lambda (u2: T).(ty3 g c u0 u2)) u1 H1))))) (\lambda
+(t: T).(\lambda (u0: T).(\lambda (_: (ty3 g c t u0)).(\lambda (ts:
+TList).(\lambda (_: (tys3 g c ts u0)).(\lambda (_: (((eq TList ts TNil) \to
+(ex T (\lambda (u1: T).(ty3 g c u0 u1)))))).(\lambda (H4: (eq TList (TCons t
+ts) TNil)).(let H5 \def (eq_ind TList (TCons t ts) (\lambda (ee:
+TList).(match ee in TList return (\lambda (_: TList).Prop) with [TNil
+\Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H4) in (False_ind
+(ex T (\lambda (u1: T).(ty3 g c u0 u1))) H5))))))))) y u H0))) H)))).
+(* COMMENTS
+Initial nodes: 255
+END *)
+
+theorem tys3_gen_cons:
+ \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (t: T).(\forall
+(u: T).((tys3 g c (TCons t ts) u) \to (land (ty3 g c t u) (tys3 g c ts
+u)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (t: T).(\lambda
+(u: T).(\lambda (H: (tys3 g c (TCons t ts) u)).(insert_eq TList (TCons t ts)
+(\lambda (t0: TList).(tys3 g c t0 u)) (\lambda (_: TList).(land (ty3 g c t u)
+(tys3 g c ts u))) (\lambda (y: TList).(\lambda (H0: (tys3 g c y u)).(tys3_ind
+g c (\lambda (t0: TList).(\lambda (t1: T).((eq TList t0 (TCons t ts)) \to
+(land (ty3 g c t t1) (tys3 g c ts t1))))) (\lambda (u0: T).(\lambda (u1:
+T).(\lambda (_: (ty3 g c u0 u1)).(\lambda (H2: (eq TList TNil (TCons t
+ts))).(let H3 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList
+return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _)
+\Rightarrow False])) I (TCons t ts) H2) in (False_ind (land (ty3 g c t u0)
+(tys3 g c ts u0)) H3)))))) (\lambda (t0: T).(\lambda (u0: T).(\lambda (H1:
+(ty3 g c t0 u0)).(\lambda (ts0: TList).(\lambda (H2: (tys3 g c ts0
+u0)).(\lambda (H3: (((eq TList ts0 (TCons t ts)) \to (land (ty3 g c t u0)
+(tys3 g c ts u0))))).(\lambda (H4: (eq TList (TCons t0 ts0) (TCons t
+ts))).(let H5 \def (f_equal TList T (\lambda (e: TList).(match e in TList
+return (\lambda (_: TList).T) with [TNil \Rightarrow t0 | (TCons t1 _)
+\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H4) in ((let H6 \def (f_equal
+TList TList (\lambda (e: TList).(match e in TList return (\lambda (_:
+TList).TList) with [TNil \Rightarrow ts0 | (TCons _ t1) \Rightarrow t1]))
+(TCons t0 ts0) (TCons t ts) H4) in (\lambda (H7: (eq T t0 t)).(let H8 \def
+(eq_ind TList ts0 (\lambda (t1: TList).((eq TList t1 (TCons t ts)) \to (land
+(ty3 g c t u0) (tys3 g c ts u0)))) H3 ts H6) in (let H9 \def (eq_ind TList
+ts0 (\lambda (t1: TList).(tys3 g c t1 u0)) H2 ts H6) in (let H10 \def (eq_ind
+T t0 (\lambda (t1: T).(ty3 g c t1 u0)) H1 t H7) in (conj (ty3 g c t u0) (tys3
+g c ts u0) H10 H9)))))) H5))))))))) y u H0))) H)))))).
+(* COMMENTS
+Initial nodes: 479
+END *)
+
projects / helm.git / blobdiff