(* This file was automatically generated: do not edit *********************)
-include "Basic-1/sty0/defs.ma".
+include "basic_1/sty0/defs.ma".
-theorem sty0_gen_sort:
+implied rec lemma sty0_ind (g: G) (P: (C \to (T \to (T \to Prop)))) (f:
+(\forall (c: C).(\forall (n: nat).(P c (TSort n) (TSort (next g n)))))) (f0:
+(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty0 g d v w) \to ((P d v w)
+\to (P c (TLRef i) (lift (S i) O w))))))))))) (f1: (\forall (c: C).(\forall
+(d: C).(\forall (v: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) v))
+\to (\forall (w: T).((sty0 g d v w) \to ((P d v w) \to (P c (TLRef i) (lift
+(S i) O v))))))))))) (f2: (\forall (b: B).(\forall (c: C).(\forall (v:
+T).(\forall (t1: T).(\forall (t2: T).((sty0 g (CHead c (Bind b) v) t1 t2) \to
+((P (CHead c (Bind b) v) t1 t2) \to (P c (THead (Bind b) v t1) (THead (Bind
+b) v t2)))))))))) (f3: (\forall (c: C).(\forall (v: T).(\forall (t1:
+T).(\forall (t2: T).((sty0 g c t1 t2) \to ((P c t1 t2) \to (P c (THead (Flat
+Appl) v t1) (THead (Flat Appl) v t2))))))))) (f4: (\forall (c: C).(\forall
+(v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to ((P c v1 v2) \to (\forall (t1:
+T).(\forall (t2: T).((sty0 g c t1 t2) \to ((P c t1 t2) \to (P c (THead (Flat
+Cast) v1 t1) (THead (Flat Cast) v2 t2)))))))))))) (c: C) (t: T) (t0: T) (s0:
+sty0 g c t t0) on s0: P c t t0 \def match s0 with [(sty0_sort c0 n)
+\Rightarrow (f c0 n) | (sty0_abbr c0 d v i g0 w s1) \Rightarrow (f0 c0 d v i
+g0 w s1 ((sty0_ind g P f f0 f1 f2 f3 f4) d v w s1)) | (sty0_abst c0 d v i g0
+w s1) \Rightarrow (f1 c0 d v i g0 w s1 ((sty0_ind g P f f0 f1 f2 f3 f4) d v w
+s1)) | (sty0_bind b c0 v t1 t2 s1) \Rightarrow (f2 b c0 v t1 t2 s1 ((sty0_ind
+g P f f0 f1 f2 f3 f4) (CHead c0 (Bind b) v) t1 t2 s1)) | (sty0_appl c0 v t1
+t2 s1) \Rightarrow (f3 c0 v t1 t2 s1 ((sty0_ind g P f f0 f1 f2 f3 f4) c0 t1
+t2 s1)) | (sty0_cast c0 v1 v2 s1 t1 t2 s2) \Rightarrow (f4 c0 v1 v2 s1
+((sty0_ind g P f f0 f1 f2 f3 f4) c0 v1 v2 s1) t1 t2 s2 ((sty0_ind g P f f0 f1
+f2 f3 f4) c0 t1 t2 s2))].
+
+lemma sty0_gen_sort:
\forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c
(TSort n) x) \to (eq T x (TSort (next g n)))))))
\def
(H0: (sty0 g c y x)).(sty0_ind g (\lambda (_: C).(\lambda (t: T).(\lambda
(t0: T).((eq T t (TSort n)) \to (eq T t0 (TSort (next g n))))))) (\lambda (_:
C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def
-(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
-[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n0 | (THead _ _ _)
-\Rightarrow n0])) (TSort n0) (TSort n) H1) in (eq_ind_r nat n (\lambda (n1:
-nat).(eq T (TSort (next g n1)) (TSort (next g n)))) (refl_equal T (TSort
-(next g n))) n0 H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v:
-T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr)
-v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v
-(TSort n)) \to (eq T w (TSort (next g n)))))).(\lambda (H4: (eq T (TLRef i)
-(TSort n))).(let H5 \def (eq_ind T (TLRef i) (\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 (eq T (lift (S i) O w) (TSort (next g n))) H5))))))))))) (\lambda
-(c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl
-i c0 (CHead d (Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v
+(f_equal T nat (\lambda (e: T).(match e with [(TSort n1) \Rightarrow n1 |
+(TLRef _) \Rightarrow n0 | (THead _ _ _) \Rightarrow n0])) (TSort n0) (TSort
+n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq T (TSort (next g n1)) (TSort
+(next g n)))) (refl_equal T (TSort (next g n))) n0 H2))))) (\lambda (c0:
+C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0
+(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v
w)).(\lambda (_: (((eq T v (TSort n)) \to (eq T w (TSort (next g
n)))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T
-(TLRef i) (\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 (eq T (lift (S i) O v)
-(TSort (next g n))) H5))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda
-(v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind
-b) v) t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort (next g
-n)))))).(\lambda (H3: (eq T (THead (Bind b) v t1) (TSort n))).(let H4 \def
-(eq_ind T (THead (Bind b) v t1) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+(TLRef i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n)
+H4) in (False_ind (eq T (lift (S i) O w) (TSort (next g n))) H5)))))))))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
+(_: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g
+d v w)).(\lambda (_: (((eq T v (TSort n)) \to (eq T w (TSort (next g
+n)))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T
+(TLRef i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n)
+H4) in (False_ind (eq T (lift (S i) O v) (TSort (next g n))) H5)))))))))))
+(\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t1 t2)).(\lambda (_: (((eq
+T t1 (TSort n)) \to (eq T t2 (TSort (next g n)))))).(\lambda (H3: (eq T
+(THead (Bind b) v t1) (TSort n))).(let H4 \def (eq_ind T (THead (Bind b) v
+t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in
(False_ind (eq T (THead (Bind b) v t2) (TSort (next g n))) H4))))))))))
(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
(_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort
(next g n)))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TSort n))).(let
-H4 \def (eq_ind T (THead (Flat Appl) v 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) H3) in
-(False_ind (eq T (THead (Flat Appl) v t2) (TSort (next g n))) H4)))))))))
-(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1
-v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 (TSort (next g
-n)))))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1
-t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort (next g
-n)))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t1) (TSort n))).(let H6
-\def (eq_ind T (THead (Flat Cast) v1 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 (eq T (THead (Flat Cast) v2 t2) (TSort (next g n))) H6))))))))))))
-c y x H0))) H))))).
-(* COMMENTS
-Initial nodes: 869
-END *)
+H4 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match ee with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TSort n) H3) in (False_ind (eq T (THead (Flat Appl) v
+t2) (TSort (next g n))) H4))))))))) (\lambda (c0: C).(\lambda (v1:
+T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1
+(TSort n)) \to (eq T v2 (TSort (next g n)))))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq
+T t2 (TSort (next g n)))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t1)
+(TSort n))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (ee:
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in (False_ind (eq T
+(THead (Flat Cast) v2 t2) (TSort (next g n))) H6)))))))))))) c y x H0)))
+H))))).
-theorem sty0_gen_lref:
+lemma sty0_gen_lref:
\forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c
(TLRef n) x) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda
(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
(t1: T).(sty0 g e u t1)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_:
T).(eq T t0 (lift (S n) O u))))))))))) (\lambda (c0: C).(\lambda (n0:
nat).(\lambda (H1: (eq T (TSort n0) (TLRef n))).(let H2 \def (eq_ind T (TSort
-n0) (\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 (e:
-C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u)))))
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (TSort (next g n0)) (lift (S n)
-O t)))))) (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T
-(TSort (next g n0)) (lift (S n) O u))))))) H2))))) (\lambda (c0: C).(\lambda
-(d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d
-(Bind Abbr) v))).(\lambda (w: T).(\lambda (H2: (sty0 g d v w)).(\lambda (_:
-(((eq T v (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u:
-T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (t: T).(eq T w (lift (S n) O t)))))) (ex3_3 C T T (\lambda
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u)))))
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq T w (lift (S n) O
-u)))))))))).(\lambda (H4: (eq T (TLRef i) (TLRef n))).(let H5 \def (f_equal T
-nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _)
-\Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i]))
-(TLRef i) (TLRef n) H4) in (let H6 \def (eq_ind nat i (\lambda (n0:
-nat).(getl n0 c0 (CHead d (Bind Abbr) v))) H1 n H5) in (eq_ind_r nat n
-(\lambda (n0: nat).(or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda
-(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+n0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n) H1) in
+(False_ind (or (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(t: T).(eq T (lift (S n0) O w) (lift (S n) O t)))))) (ex3_3 C T T (\lambda
+(t: T).(eq T (TSort (next g n0)) (lift (S n) O t)))))) (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n0) O w) (lift (S n) O
-u)))))))) (or_introl (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(t: T).(eq T (lift (S n) O w) (lift (S n) O t)))))) (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n) O w) (lift (S n) O
-u)))))) (ex3_3_intro C T T (\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).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(t: T).(eq T (lift (S n) O w) (lift (S n) O t))))) d v w H6 H2 (refl_equal T
-(lift (S n) O w)))) i H5)))))))))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind
-Abst) v))).(\lambda (w: T).(\lambda (H2: (sty0 g d v w)).(\lambda (_: (((eq T
-v (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda
-(_: T).(getl n d (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(t: T).(eq T w (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda
-(u: T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq T w (lift (S n) O u)))))))))).(\lambda (H4: (eq T
-(TLRef i) (TLRef n))).(let H5 \def (f_equal T nat (\lambda (e: T).(match e in
-T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | (TLRef n0)
-\Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H4) in
-(let H6 \def (eq_ind nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind
-Abst) v))) H1 n H5) in (eq_ind_r nat n (\lambda (n0: nat).(or (ex3_3 C T T
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq T (TSort (next g n0)) (lift (S n)
+O u))))))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda
+(i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w:
+T).(\lambda (H2: (sty0 g d v w)).(\lambda (_: (((eq T v (TLRef n)) \to (or
+(ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n d (CHead
+e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g
+e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T w (lift (S n)
+O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl
+n d (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
+T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T w
+(lift (S n) O u)))))))))).(\lambda (H4: (eq T (TLRef i) (TLRef n))).(let H5
+\def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow i |
+(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef
+n) H4) in (let H6 \def (eq_ind nat i (\lambda (n0: nat).(getl n0 c0 (CHead d
+(Bind Abbr) v))) H1 n H5) in (eq_ind_r nat n (\lambda (n0: nat).(or (ex3_3 C
+T T (\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).(sty0 g e u
+t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n0) O w)
+(lift (S n) O t)))))) (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq T (lift (S n0) O w) (lift (S n) O u)))))))) (or_introl (ex3_3 C T
+T (\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).(sty0 g e u
+t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n) O w)
+(lift (S n) O t)))))) (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq T (lift (S n) O w) (lift (S n) O u)))))) (ex3_3_intro C T T
(\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).(sty0 g e u
+t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n) O w)
+(lift (S n) O t))))) d v w H6 H2 (refl_equal T (lift (S n) O w)))) i
+H5)))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w:
+T).(\lambda (H2: (sty0 g d v w)).(\lambda (_: (((eq T v (TLRef n)) \to (or
+(ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n d (CHead
+e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g
+e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T w (lift (S n)
+O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl
+n d (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
+T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T w
+(lift (S n) O u)))))))))).(\lambda (H4: (eq T (TLRef i) (TLRef n))).(let H5
+\def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow i |
+(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef
+n) H4) in (let H6 \def (eq_ind nat i (\lambda (n0: nat).(getl n0 c0 (CHead d
+(Bind Abst) v))) H1 n H5) in (eq_ind_r nat n (\lambda (n0: nat).(or (ex3_3 C
+T T (\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).(sty0 g e u
t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n0) O v)
(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda
(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u:
(u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u:
T).(\lambda (_: T).(eq T t2 (lift (S n) O u)))))))))).(\lambda (H3: (eq T
(THead (Bind b) v t1) (TLRef n))).(let H4 \def (eq_ind T (THead (Bind b) v
-t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H3) in
+(False_ind (or (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(t: T).(eq T (THead (Bind b) v t2) (lift (S n) O t)))))) (ex3_3 C T T
+(\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).(sty0 g e u
+t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (THead (Bind b) v
+t2) (lift (S n) O u))))))) H4)))))))))) (\lambda (c0: C).(\lambda (v:
+T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1 t2)).(\lambda
+(_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (t: T).(eq T t2 (lift (S n) O t)))))) (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq T t2 (lift (S n) O
+u)))))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TLRef n))).(let H4
+\def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match ee with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
\Rightarrow True])) I (TLRef n) H3) in (False_ind (or (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Bind b) v t2) (lift (S
-n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_:
+(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Flat Appl) v t2) (lift
+(S n) O t)))))) (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(eq T (THead (Bind b) v t2) (lift (S n) O u))))))) H4))))))))))
-(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T
-T (\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).(sty0 g e u
-t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n) O
-t)))))) (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T t2
-(lift (S n) O u)))))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TLRef
-n))).(let H4 \def (eq_ind T (THead (Flat Appl) v 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)
-H3) in (False_ind (or (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(t: T).(eq T (THead (Flat Appl) v t2) (lift (S n) O t)))))) (ex3_3 C T T
-(\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).(sty0 g e u
-t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (THead (Flat
-Appl) v t2) (lift (S n) O u))))))) H4))))))))) (\lambda (c0: C).(\lambda (v1:
-T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1
-(TLRef n)) \to (or (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(t: T).(eq T v2 (lift (S n) O t)))))) (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq T v2 (lift (S n) O u)))))))))).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1
-(TLRef n)) \to (or (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(t: T).(eq T t2 (lift (S n) O t)))))) (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq T t2 (lift (S n) O u)))))))))).(\lambda (H5: (eq T
-(THead (Flat Cast) v1 t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Flat
-Cast) v1 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 (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u
-t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Flat
-Cast) v2 t2) (lift (S n) O t)))))) (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq T (THead (Flat Cast) v2 t2) (lift (S n) O u)))))))
-H6)))))))))))) c y x H0))) H))))).
-(* COMMENTS
-Initial nodes: 3231
-END *)
+(_: T).(eq T (THead (Flat Appl) v t2) (lift (S n) O u))))))) H4)))))))))
+(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1
+v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (or (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (t: T).(eq T v2 (lift (S n) O t)))))) (ex3_3
+C T T (\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).(sty0 g e
+u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T v2 (lift (S n)
+O u)))))))))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1
+t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n) O t)))))) (ex3_3
+C T T (\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).(sty0 g e
+u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T t2 (lift (S n)
+O u)))))))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t1) (TLRef n))).(let
+H6 \def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (ee: T).(match ee with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Flat Cast) v2 t2) (lift
+(S n) O t)))))) (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq T (THead (Flat Cast) v2 t2) (lift (S n) O u))))))) H6))))))))))))
+c y x H0))) H))))).
-theorem sty0_gen_bind:
+lemma sty0_gen_bind:
\forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1:
T).(\forall (x: T).((sty0 g c (THead (Bind b) u t1) x) \to (ex2 T (\lambda
(t2: T).(sty0 g (CHead c (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T x (THead
(CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T t0 (THead (Bind b) u
t2)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n)
(THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort n) (\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 (ex2 T (\lambda (t2: T).(sty0 g
-(CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (TSort (next g n))
-(THead (Bind b) u t2)))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
-(v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr)
-v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v
-(THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead d (Bind b)
-u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind b) u t2))))))).(\lambda (H4:
-(eq T (TLRef i) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef i)
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H1) in (False_ind
+(ex2 T (\lambda (t2: T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2:
+T).(eq T (TSort (next g n)) (THead (Bind b) u t2)))) H2))))) (\lambda (c0:
+C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0
+(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v
+w)).(\lambda (_: (((eq T v (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2:
+T).(sty0 g (CHead d (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind
+b) u t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Bind b) u t1))).(let H5
+\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort _)
\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
False])) I (THead (Bind b) u t1) H4) in (False_ind (ex2 T (\lambda (t2:
T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O
v (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead d (Bind
b) u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind b) u t2))))))).(\lambda
(H4: (eq T (TLRef i) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef i)
-(\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 (ex2 T (\lambda (t2:
-T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O
-v) (THead (Bind b) u t2)))) H5))))))))))) (\lambda (b0: B).(\lambda (c0:
-C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (sty0 g
-(CHead c0 (Bind b0) v) t0 t2)).(\lambda (H2: (((eq T t0 (THead (Bind b) u
-t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind
-b) u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda
-(H3: (eq T (THead (Bind b0) v t0) (THead (Bind b) u t1))).(let H4 \def
-(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with
-[(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1)
-\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) v t0) (THead
-(Bind b) u t1) H3) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v | (TLRef _)
-\Rightarrow v | (THead _ t _) \Rightarrow t])) (THead (Bind b0) v t0) (THead
-(Bind b) u t1) H3) in ((let H6 \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 (Bind b0) v t0) (THead
-(Bind b) u t1) H3) in (\lambda (H7: (eq T v u)).(\lambda (H8: (eq B b0
-b)).(let H9 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind b) u t1))
-\to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind b) u)
-t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3)))))) H2 t1 H6) in
-(let H10 \def (eq_ind T t0 (\lambda (t: T).(sty0 g (CHead c0 (Bind b0) v) t
-t2)) H1 t1 H6) in (let H11 \def (eq_ind T v (\lambda (t: T).((eq T t1 (THead
-(Bind b) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind
-b0) t) (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u
-t3)))))) H9 u H7) in (let H12 \def (eq_ind T v (\lambda (t: T).(sty0 g (CHead
-c0 (Bind b0) t) t1 t2)) H10 u H7) in (eq_ind_r T u (\lambda (t: T).(ex2 T
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1)
+H4) in (False_ind (ex2 T (\lambda (t2: T).(sty0 g (CHead c0 (Bind b) u) t1
+t2)) (\lambda (t2: T).(eq T (lift (S i) O v) (THead (Bind b) u t2))))
+H5))))))))))) (\lambda (b0: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0:
+T).(\lambda (t2: T).(\lambda (H1: (sty0 g (CHead c0 (Bind b0) v) t0
+t2)).(\lambda (H2: (((eq T t0 (THead (Bind b) u t1)) \to (ex2 T (\lambda (t3:
+T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind b) u) t1 t3)) (\lambda (t3:
+T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda (H3: (eq T (THead (Bind b0)
+v t0) (THead (Bind b) u t1))).(let H4 \def (f_equal T B (\lambda (e:
+T).(match e with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 |
+(THead k _ _) \Rightarrow (match k with [(Bind b1) \Rightarrow b1 | (Flat _)
+\Rightarrow b0])])) (THead (Bind b0) v t0) (THead (Bind b) u t1) H3) in ((let
+H5 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v |
+(TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t])) (THead (Bind b0) v
+t0) (THead (Bind b) u t1) H3) in ((let H6 \def (f_equal T T (\lambda (e:
+T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 |
+(THead _ _ t) \Rightarrow t])) (THead (Bind b0) v t0) (THead (Bind b) u t1)
+H3) in (\lambda (H7: (eq T v u)).(\lambda (H8: (eq B b0 b)).(let H9 \def
+(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind b) u t1)) \to (ex2 T
+(\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind b) u) t1 t3))
+(\lambda (t3: T).(eq T t2 (THead (Bind b) u t3)))))) H2 t1 H6) in (let H10
+\def (eq_ind T t0 (\lambda (t: T).(sty0 g (CHead c0 (Bind b0) v) t t2)) H1 t1
+H6) in (let H11 \def (eq_ind T v (\lambda (t: T).((eq T t1 (THead (Bind b) u
+t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) t) (Bind
+b) u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3)))))) H9 u H7)
+in (let H12 \def (eq_ind T v (\lambda (t: T).(sty0 g (CHead c0 (Bind b0) t)
+t1 t2)) H10 u H7) in (eq_ind_r T u (\lambda (t: T).(ex2 T (\lambda (t3:
+T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T (THead (Bind
+b0) t t2) (THead (Bind b) u t3))))) (let H13 \def (eq_ind B b0 (\lambda (b1:
+B).((eq T t1 (THead (Bind b) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g
+(CHead (CHead c0 (Bind b1) u) (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T t2
+(THead (Bind b) u t3)))))) H11 b H8) in (let H14 \def (eq_ind B b0 (\lambda
+(b1: B).(sty0 g (CHead c0 (Bind b1) u) t1 t2)) H12 b H8) in (eq_ind_r B b
+(\lambda (b1: B).(ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1
+t3)) (\lambda (t3: T).(eq T (THead (Bind b1) u t2) (THead (Bind b) u t3)))))
+(ex_intro2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda
+(t3: T).(eq T (THead (Bind b) u t2) (THead (Bind b) u t3))) t2 H14
+(refl_equal T (THead (Bind b) u t2))) b0 H8))) v H7)))))))) H5)) H4))))))))))
+(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda
+(_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u t1)) \to
+(ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3:
+T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda (H3: (eq T (THead (Flat
+Appl) v t0) (THead (Bind b) u t1))).(let H4 \def (eq_ind T (THead (Flat Appl)
+v t0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef
+_) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t1)
+H3) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1
+t3)) (\lambda (t3: T).(eq T (THead (Flat Appl) v t2) (THead (Bind b) u t3))))
+H4))))))))) (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_:
+(sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u t1)) \to (ex2 T
+(\lambda (t2: T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T
+v2 (THead (Bind b) u t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_:
+(sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u t1)) \to (ex2 T
(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T
-(THead (Bind b0) t t2) (THead (Bind b) u t3))))) (let H13 \def (eq_ind B b0
-(\lambda (b1: B).((eq T t1 (THead (Bind b) u t1)) \to (ex2 T (\lambda (t3:
-T).(sty0 g (CHead (CHead c0 (Bind b1) u) (Bind b) u) t1 t3)) (\lambda (t3:
-T).(eq T t2 (THead (Bind b) u t3)))))) H11 b H8) in (let H14 \def (eq_ind B
-b0 (\lambda (b1: B).(sty0 g (CHead c0 (Bind b1) u) t1 t2)) H12 b H8) in
-(eq_ind_r B b (\lambda (b1: B).(ex2 T (\lambda (t3: T).(sty0 g (CHead c0
-(Bind b) u) t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b1) u t2) (THead
-(Bind b) u t3))))) (ex_intro2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b)
-u) t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b) u t2) (THead (Bind b) u
-t3))) t2 H14 (refl_equal T (THead (Bind b) u t2))) b0 H8))) v H7)))))))) H5))
-H4)))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2:
-T).(\lambda (_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u
-t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3))
-(\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda (H3: (eq T
-(THead (Flat Appl) v t0) (THead (Bind b) u t1))).(let H4 \def (eq_ind T
-(THead (Flat Appl) v 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) H3) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b)
-u) t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Appl) v t2) (THead (Bind b) u
-t3)))) H4))))))))) (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2:
-T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u
-t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead c0 (Bind b) u) t1 t2))
-(\lambda (t2: T).(eq T v2 (THead (Bind b) u t2))))))).(\lambda (t0:
-T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0
-(THead (Bind b) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b)
-u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda
-(H5: (eq T (THead (Flat Cast) v1 t0) (THead (Bind b) u t1))).(let H6 \def
-(eq_ind T (THead (Flat Cast) v1 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 (ex2 T (\lambda (t3:
-T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T (THead (Flat
-Cast) v2 t2) (THead (Bind b) u t3)))) H6)))))))))))) c y x H0))) H))))))).
-(* COMMENTS
-Initial nodes: 1975
-END *)
+t2 (THead (Bind b) u t3))))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t0)
+(THead (Bind b) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t0)
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t1)
+H5) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1
+t3)) (\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead (Bind b) u
+t3)))) H6)))))))))))) c y x H0))) H))))))).
-theorem sty0_gen_appl:
+lemma sty0_gen_appl:
\forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (x:
T).((sty0 g c (THead (Flat Appl) u t1) x) \to (ex2 T (\lambda (t2: T).(sty0 g
c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat Appl) u t2)))))))))
u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T
t0 (THead (Flat Appl) u t2)))))))) (\lambda (c0: C).(\lambda (n:
nat).(\lambda (H1: (eq T (TSort n) (THead (Flat Appl) u t1))).(let H2 \def
-(eq_ind T (TSort n) (\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) u t1) H1) in
-(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T
-(TSort (next g n)) (THead (Flat Appl) u t2)))) H2))))) (\lambda (c0:
-C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0
-(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v
-w)).(\lambda (_: (((eq T v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2:
-T).(sty0 g d t1 t2)) (\lambda (t2: T).(eq T w (THead (Flat Appl) u
-t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u t1))).(let H5
-\def (eq_ind T (TLRef i) (\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) u t1) H4) in
-(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T
-(lift (S i) O w) (THead (Flat Appl) u t2)))) H5))))))))))) (\lambda (c0:
-C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0
-(CHead d (Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v
-w)).(\lambda (_: (((eq T v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2:
-T).(sty0 g d t1 t2)) (\lambda (t2: T).(eq T w (THead (Flat Appl) u
-t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u t1))).(let H5
-\def (eq_ind T (TLRef i) (\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) u t1) H4) in
-(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T
-(lift (S i) O v) (THead (Flat Appl) u t2)))) H5))))))))))) (\lambda (b:
-B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2:
-T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t0 t2)).(\lambda (_: (((eq T t0
-(THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind
-b) v) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u
+(eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
+(THead (Flat Appl) u t1) H1) in (False_ind (ex2 T (\lambda (t2: T).(sty0 g c0
+t1 t2)) (\lambda (t2: T).(eq T (TSort (next g n)) (THead (Flat Appl) u t2))))
+H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w:
+T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v (THead (Flat Appl) u
+t1)) \to (ex2 T (\lambda (t2: T).(sty0 g d t1 t2)) (\lambda (t2: T).(eq T w
+(THead (Flat Appl) u t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat
+Appl) u t1))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _
+_) \Rightarrow False])) I (THead (Flat Appl) u t1) H4) in (False_ind (ex2 T
+(\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O w)
+(THead (Flat Appl) u t2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
+Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T
+v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g d t1 t2))
+(\lambda (t2: T).(eq T w (THead (Flat Appl) u t2))))))).(\lambda (H4: (eq T
+(TLRef i) (THead (Flat Appl) u t1))).(let H5 \def (eq_ind T (TLRef i)
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u
+t1) H4) in (False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda
+(t2: T).(eq T (lift (S i) O v) (THead (Flat Appl) u t2)))) H5)))))))))))
+(\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda
+(t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t0 t2)).(\lambda (_: (((eq
+T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0
+(Bind b) v) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u
t3))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Appl) u
t1))).(let H4 \def (eq_ind T (THead (Bind b) v 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 Appl) u t1) H3) in (False_ind (ex2 T (\lambda (t3:
-T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b) v t2) (THead
-(Flat Appl) u t3)))) H4)))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda
-(t0: T).(\lambda (t2: T).(\lambda (H1: (sty0 g c0 t0 t2)).(\lambda (H2: (((eq
-T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3))
-(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))))))).(\lambda (H3: (eq T
-(THead (Flat Appl) v t0) (THead (Flat Appl) u t1))).(let H4 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
+_) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _)
+\Rightarrow False])])) I (THead (Flat Appl) u t1) H3) in (False_ind (ex2 T
+(\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b) v
+t2) (THead (Flat Appl) u t3)))) H4)))))))))) (\lambda (c0: C).(\lambda (v:
+T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (sty0 g c0 t0
+t2)).(\lambda (H2: (((eq T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda
+(t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u
+t3))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t0) (THead (Flat Appl) u
+t1))).(let H4 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _)
\Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t]))
(THead (Flat Appl) v t0) (THead (Flat Appl) u t1) H3) in ((let H5 \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 Appl) v t0) (THead (Flat Appl) u t1) H3) in
-(\lambda (H6: (eq T v u)).(let H7 \def (eq_ind T t0 (\lambda (t: T).((eq T t
-(THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3))
-(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3)))))) H2 t1 H5) in (let H8
-\def (eq_ind T t0 (\lambda (t: T).(sty0 g c0 t t2)) H1 t1 H5) in (eq_ind_r T
-u (\lambda (t: T).(ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3:
-T).(eq T (THead (Flat Appl) t t2) (THead (Flat Appl) u t3))))) (ex_intro2 T
-(\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Appl)
-u t2) (THead (Flat Appl) u t3))) t2 H8 (refl_equal T (THead (Flat Appl) u
-t2))) v H6))))) H4))))))))) (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2:
-T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Flat Appl)
-u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T
-v2 (THead (Flat Appl) u t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda
-(_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t1)) \to
-(ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T t2 (THead
-(Flat Appl) u t3))))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t0) (THead
-(Flat Appl) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 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 f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl
-\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t1)
-H5) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3:
-T).(eq T (THead (Flat Cast) v2 t2) (THead (Flat Appl) u t3)))) H6))))))))))))
-c y x H0))) H)))))).
-(* COMMENTS
-Initial nodes: 1489
-END *)
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef
+_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v t0)
+(THead (Flat Appl) u t1) H3) in (\lambda (H6: (eq T v u)).(let H7 \def
+(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Appl) u t1)) \to (ex2 T
+(\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat
+Appl) u t3)))))) H2 t1 H5) in (let H8 \def (eq_ind T t0 (\lambda (t: T).(sty0
+g c0 t t2)) H1 t1 H5) in (eq_ind_r T u (\lambda (t: T).(ex2 T (\lambda (t3:
+T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Appl) t t2) (THead
+(Flat Appl) u t3))))) (ex_intro2 T (\lambda (t3: T).(sty0 g c0 t1 t3))
+(\lambda (t3: T).(eq T (THead (Flat Appl) u t2) (THead (Flat Appl) u t3))) t2
+H8 (refl_equal T (THead (Flat Appl) u t2))) v H6))))) H4))))))))) (\lambda
+(c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1
+v2)).(\lambda (_: (((eq T v1 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda
+(t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T v2 (THead (Flat Appl) u
+t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t0
+t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda
+(t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u
+t3))))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t0) (THead (Flat Appl) u
+t1))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t0) (\lambda (ee:
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat
+f) \Rightarrow (match f with [Appl \Rightarrow False | Cast \Rightarrow
+True])])])) I (THead (Flat Appl) u t1) H5) in (False_ind (ex2 T (\lambda (t3:
+T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead
+(Flat Appl) u t3)))) H6)))))))))))) c y x H0))) H)))))).
-theorem sty0_gen_cast:
+lemma sty0_gen_cast:
\forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (t1: T).(\forall
(x: T).((sty0 g c (THead (Flat Cast) v1 t1) x) \to (ex3_2 T T (\lambda (v2:
T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0
T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2)))
(\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) v2 t2)))))))))
(\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (THead (Flat
-Cast) v1 t1))).(let H2 \def (eq_ind T (TSort n) (\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)
-v1 t1) H1) in (False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g
-c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda
-(v2: T).(\lambda (t2: T).(eq T (TSort (next g n)) (THead (Flat Cast) v2
-t2))))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
-nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w:
-T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v (THead (Flat Cast) v1
-t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g d v1 v2)))
-(\lambda (_: T).(\lambda (t2: T).(sty0 g d t1 t2))) (\lambda (v2: T).(\lambda
-(t2: T).(eq T w (THead (Flat Cast) v2 t2)))))))).(\lambda (H4: (eq T (TLRef
-i) (THead (Flat Cast) v1 t1))).(let H5 \def (eq_ind T (TLRef i) (\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) v1 t1) H4) in (False_ind (ex3_2 T T (\lambda (v2:
-T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0
-g c0 t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T (lift (S i) O w) (THead
-(Flat Cast) v2 t2))))) H5))))))))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
-Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T
-v (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_:
-T).(sty0 g d v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g d t1 t2)))
+Cast) v1 t1))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee
+with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow False])) I (THead (Flat Cast) v1 t1) H1) in (False_ind (ex3_2
+T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_:
+T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq
+T (TSort (next g n)) (THead (Flat Cast) v2 t2))))) H2))))) (\lambda (c0:
+C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0
+(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v
+w)).(\lambda (_: (((eq T v (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda
+(v2: T).(\lambda (_: T).(sty0 g d v1 v2))) (\lambda (_: T).(\lambda (t2:
+T).(sty0 g d t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T w (THead (Flat
+Cast) v2 t2)))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) v1
+t1))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
+\Rightarrow False])) I (THead (Flat Cast) v1 t1) H4) in (False_ind (ex3_2 T T
+(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda
+(t2: T).(sty0 g c0 t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T (lift (S
+i) O w) (THead (Flat Cast) v2 t2))))) H5))))))))))) (\lambda (c0: C).(\lambda
+(d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d
+(Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_:
+(((eq T v (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda
+(_: T).(sty0 g d v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g d t1 t2)))
(\lambda (v2: T).(\lambda (t2: T).(eq T w (THead (Flat Cast) v2
t2)))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) v1 t1))).(let H5
-\def (eq_ind T (TLRef i) (\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) v1 t1) H4) in
-(False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2)))
-(\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda (v2:
-T).(\lambda (t2: T).(eq T (lift (S i) O v) (THead (Flat Cast) v2 t2)))))
-H5))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0:
-T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t0
-t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T T
-(\lambda (v2: T).(\lambda (_: T).(sty0 g (CHead c0 (Bind b) v) v1 v2)))
-(\lambda (_: T).(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) v) t1 t3)))
-(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v2
+\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Flat Cast) v1 t1) H4) in (False_ind (ex3_2 T T (\lambda
+(v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2:
+T).(sty0 g c0 t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T (lift (S i) O
+v) (THead (Flat Cast) v2 t2))))) H5))))))))))) (\lambda (b: B).(\lambda (c0:
+C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (sty0 g
+(CHead c0 (Bind b) v) t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1
+t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g (CHead c0 (Bind
+b) v) v1 v2))) (\lambda (_: T).(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) v)
+t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v2
t3)))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Cast) v1
t1))).(let H4 \def (eq_ind T (THead (Bind b) v 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) v1 t1) H3) in (False_ind (ex3_2 T T (\lambda
-(v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t3:
-T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq T (THead (Bind
-b) v t2) (THead (Flat Cast) v2 t3))))) H4)))))))))) (\lambda (c0: C).(\lambda
-(v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t0
-t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T T
-(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda
-(t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Cast) v2 t3)))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t0) (THead
-(Flat Cast) v1 t1))).(let H4 \def (eq_ind T (THead (Flat Appl) v 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 f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl
-\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) v1 t1)
-H3) in (False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1
-v2))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v2:
-T).(\lambda (t3: T).(eq T (THead (Flat Appl) v t2) (THead (Flat Cast) v2
-t3))))) H4))))))))) (\lambda (c0: C).(\lambda (v0: T).(\lambda (v2:
-T).(\lambda (H1: (sty0 g c0 v0 v2)).(\lambda (H2: (((eq T v0 (THead (Flat
-Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1
-v3))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda (v3:
-T).(\lambda (t2: T).(eq T v2 (THead (Flat Cast) v3 t2)))))))).(\lambda (t0:
-T).(\lambda (t2: T).(\lambda (H3: (sty0 g c0 t0 t2)).(\lambda (H4: (((eq T t0
-(THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_:
-T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3)))
-(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v3
-t3)))))))).(\lambda (H5: (eq T (THead (Flat Cast) v0 t0) (THead (Flat Cast)
-v1 t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0
-| (THead _ t _) \Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast)
-v1 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) v0 t0) (THead (Flat Cast)
-v1 t1) H5) in (\lambda (H8: (eq T v0 v1)).(let H9 \def (eq_ind T t0 (\lambda
-(t: T).((eq T t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3:
-T).(\lambda (_: T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0
-g c0 t1 t3))) (\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast)
-v3 t3))))))) H4 t1 H7) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(sty0 g
-c0 t t2)) H3 t1 H7) in (let H11 \def (eq_ind T v0 (\lambda (t: T).((eq T t
-(THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_:
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
+_) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _)
+\Rightarrow False])])) I (THead (Flat Cast) v1 t1) H3) in (False_ind (ex3_2 T
+T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_:
+T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq
+T (THead (Bind b) v t2) (THead (Flat Cast) v2 t3))))) H4)))))))))) (\lambda
+(c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (sty0
+g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T
+T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_:
+T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq
+T t2 (THead (Flat Cast) v2 t3)))))))).(\lambda (H3: (eq T (THead (Flat Appl)
+v t0) (THead (Flat Cast) v1 t1))).(let H4 \def (eq_ind T (THead (Flat Appl) v
+t0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _)
+\Rightarrow False | (Flat f) \Rightarrow (match f with [Appl \Rightarrow True
+| Cast \Rightarrow False])])])) I (THead (Flat Cast) v1 t1) H3) in (False_ind
+(ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_:
+T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq
+T (THead (Flat Appl) v t2) (THead (Flat Cast) v2 t3))))) H4))))))))) (\lambda
+(c0: C).(\lambda (v0: T).(\lambda (v2: T).(\lambda (H1: (sty0 g c0 v0
+v2)).(\lambda (H2: (((eq T v0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T T
+(\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda
+(t2: T).(sty0 g c0 t1 t2))) (\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead
+(Flat Cast) v3 t2)))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3:
+(sty0 g c0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Cast) v1 t1)) \to
+(ex3_2 T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3))) (\lambda (_:
+T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v3: T).(\lambda (t3: T).(eq
+T t2 (THead (Flat Cast) v3 t3)))))))).(\lambda (H5: (eq T (THead (Flat Cast)
+v0 t0) (THead (Flat Cast) v1 t1))).(let H6 \def (f_equal T T (\lambda (e:
+T).(match e with [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 |
+(THead _ t _) \Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast)
+v1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
+\Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast) v1 t1) H5) in
+(\lambda (H8: (eq T v0 v1)).(let H9 \def (eq_ind T t0 (\lambda (t: T).((eq T
+t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_:
T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3)))
-(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Flat Cast) v3 t3))))))) H2
-v1 H8) in (let H12 \def (eq_ind T v0 (\lambda (t: T).(sty0 g c0 t v2)) H1 v1
-H8) in (ex3_2_intro T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3)))
+(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v3 t3))))))) H4
+t1 H7) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(sty0 g c0 t t2)) H3 t1
+H7) in (let H11 \def (eq_ind T v0 (\lambda (t: T).((eq T t (THead (Flat Cast)
+v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3)))
+(\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v3:
+T).(\lambda (t3: T).(eq T v2 (THead (Flat Cast) v3 t3))))))) H2 v1 H8) in
+(let H12 \def (eq_ind T v0 (\lambda (t: T).(sty0 g c0 t v2)) H1 v1 H8) in
+(ex3_2_intro T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3)))
(\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v3:
T).(\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead (Flat Cast) v3
t3)))) v2 t2 H12 H10 (refl_equal T (THead (Flat Cast) v2 t2)))))))))
H6)))))))))))) c y x H0))) H)))))).
-(* COMMENTS
-Initial nodes: 1855
-END *)