(pp_patterns patterns)
| Ast.Cast (t1, t2) -> sprintf "(%s: %s)" (pp_term ~pp_parens:true t1) (pp_term ~pp_parens:true t2)
| Ast.LetIn ((var,t2), t1, t3) ->
- let t2 = match t2 with None -> Ast.Implicit | Some t -> t in
- sprintf "let %s : %s \\def %s in %s" (pp_term var)
- (pp_term ~pp_parens:true t2)
+(* let t2 = match t2 with None -> Ast.Implicit | Some t -> t in *)
+ sprintf "let %s \\def %s in %s" (pp_term var)
+(* (pp_term ~pp_parens:true t2) *)
(pp_term ~pp_parens:true t1)
(pp_term ~pp_parens:true t3)
| Ast.LetRec (kind, definitions, term) ->
include "Base-2/preamble.ma".
-inline procedural "Base-1/blt/defs.ma".
-
include "Base-2/preamble.ma".
-inline procedural "Base-1/plist/defs.ma".
-
include "Base-2/preamble.ma".
-inline procedural "Base-1/types/defs.ma".
-
b) v) (CHead d2 (Bind b0) w))))))).(\lambda (H1: (getl (S h0) (CHead c1 (Bind
b) v) (CHead d1 (Bind b0) w))).(let H_x \def (H b0 d1 w (r (Bind b) h0)
(getl_gen_S (Bind b) c1 (CHead d1 (Bind b0) w) v h0 H1)) in (let H2 \def H_x
-in (ex_ind C (\lambda (d2: C).(getl (r (Bind b) h0) c2 (CHead d2 (Bind b0)
-w))) (ex C (\lambda (d2: C).(getl (S h0) (CHead c2 (Bind b) v) (CHead d2
-(Bind b0) w)))) (\lambda (x: C).(\lambda (H3: (getl (r (Bind b) h0) c2 (CHead
-x (Bind b0) w))).(ex_intro C (\lambda (d2: C).(getl (S h0) (CHead c2 (Bind b)
-v) (CHead d2 (Bind b0) w))) x (getl_head (Bind b) h0 c2 (CHead x (Bind b0) w)
-H3 v)))) H2)))))) h H0)))))))))).
+in (ex_ind C (\lambda (d2: C).(getl h0 c2 (CHead d2 (Bind b0) w))) (ex C
+(\lambda (d2: C).(getl (S h0) (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w))))
+(\lambda (x: C).(\lambda (H3: (getl h0 c2 (CHead x (Bind b0) w))).(ex_intro C
+(\lambda (d2: C).(getl (S h0) (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w)))
+x (getl_head (Bind b) h0 c2 (CHead x (Bind b0) w) H3 v)))) H2)))))) h
+H0)))))))))).
theorem cimp_getl_conf:
\forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall
d1 (CHead d0 (Bind b0) w0))).(let H_y \def (getl_trans (S h) c1 (CHead d1
(Bind b) w) i H0) in (let H_x0 \def (H b0 d0 w0 (plus (S h) i) (H_y (CHead d0
(Bind b0) w0) (getl_head (Bind b) h d1 (CHead d0 (Bind b0) w0) H3 w))) in
-(let H4 \def H_x0 in (ex_ind C (\lambda (d2: C).(getl (plus (S h) i) c2
+(let H4 \def H_x0 in (ex_ind C (\lambda (d2: C).(getl (S (plus h i)) c2
(CHead d2 (Bind b0) w0))) (ex C (\lambda (d2: C).(getl h x (CHead d2 (Bind
-b0) w0)))) (\lambda (x0: C).(\lambda (H5: (getl (plus (S h) i) c2 (CHead x0
-(Bind b0) w0))).(let H_y0 \def (getl_conf_le (plus (S h) i) (CHead x0 (Bind
-b0) w0) c2 H5 (CHead x (Bind b) w) i H2) in (let H6 \def (eq_ind nat (minus
-(plus (S h) i) i) (\lambda (n: nat).(getl n (CHead x (Bind b) w) (CHead x0
-(Bind b0) w0))) (H_y0 (le_plus_r (S h) i)) (S h) (minus_plus_r (S h) i)) in
-(ex_intro C (\lambda (d2: C).(getl h x (CHead d2 (Bind b0) w0))) x0
-(getl_gen_S (Bind b) x (CHead x0 (Bind b0) w0) w h H6)))))) H4))))))))) H2)))
-H1)))))))))).
+b0) w0)))) (\lambda (x0: C).(\lambda (H5: (getl (S (plus h i)) c2 (CHead x0
+(Bind b0) w0))).(let H_y0 \def (getl_conf_le (S (plus h i)) (CHead x0 (Bind
+b0) w0) c2 H5 (CHead x (Bind b) w) i H2) in (let H6 \def (refl_equal nat
+(plus (S h) i)) in (let H7 \def (eq_ind nat (S (plus h i)) (\lambda (n:
+nat).(getl (minus n i) (CHead x (Bind b) w) (CHead x0 (Bind b0) w0))) (H_y0
+(le_S i (plus h i) (le_plus_r h i))) (plus (S h) i) H6) in (let H8 \def
+(eq_ind nat (minus (plus (S h) i) i) (\lambda (n: nat).(getl n (CHead x (Bind
+b) w) (CHead x0 (Bind b0) w0))) H7 (S h) (minus_plus_r (S h) i)) in (ex_intro
+C (\lambda (d2: C).(getl h x (CHead d2 (Bind b0) w0))) x0 (getl_gen_S (Bind
+b) x (CHead x0 (Bind b0) w0) w h H8)))))))) H4))))))))) H2))) H1)))))))))).
\to (\forall (c3: C).((csubc g (CHead c k0 t) c3) \to (ex2 C (\lambda (e2:
C).(drop n O c3 e2)) (\lambda (e2: C).(csubc g e1 e2))))))) H0 (Bind Abst)
H5) in (let H_x0 \def (H e1 (r (Bind Abst) n) H10 x0 H7) in (let H12 \def
-H_x0 in (ex2_ind C (\lambda (e2: C).(drop (r (Bind Abst) n) O x0 e2))
-(\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: C).(drop (S n) O
-(CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda
-(x: C).(\lambda (H13: (drop (r (Bind Abst) n) O x0 x)).(\lambda (H14: (csubc
-g e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x0 (Bind Abbr)
-x1) e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind Abbr) n x0 x
-H13 x1) H14)))) H12))))) c2 H6))))))))) H4)) H3)))))))) h))))))) c1)).
+H_x0 in (ex2_ind C (\lambda (e2: C).(drop n O x0 e2)) (\lambda (e2: C).(csubc
+g e1 e2)) (ex2 C (\lambda (e2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1)
+e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: C).(\lambda (H13: (drop
+n O x0 x)).(\lambda (H14: (csubc g e1 x)).(ex_intro2 C (\lambda (e2: C).(drop
+(S n) O (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g e1 e2)) x
+(drop_drop (Bind Abbr) n x0 x H13 x1) H14)))) H12))))) c2 H6))))))))) H4))
+H3)))))))) h))))))) c1)).
theorem drop_csubc_trans:
\forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall
K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3)))
(\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1)))) (let H_x0
\def (H x0 (r (Bind Abst) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind
-C (\lambda (c1: C).(drop h (r (Bind Abst) n) c1 x2)) (\lambda (c1: C).(csubc
-g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3)))
-(\lambda (c1: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1))
-c1))) (\lambda (x: C).(\lambda (H19: (drop h (r (Bind Abst) n) x
-x2)).(\lambda (H20: (csubc g c x)).(ex_intro2 C (\lambda (c1: C).(drop h (S
-n) c1 (CHead x2 (Bind Abbr) x3))) (\lambda (c1: C).(csubc g (CHead c (Bind
-Abst) (lift h (r (Bind Abst) n) x1)) c1)) (CHead x (Bind Abbr) (lift h n x3))
-(drop_skip_bind h n x x2 H19 Abbr x3) (csubc_abst g c x H20 (lift h (r (Bind
-Abst) n) x1) x4 (sc3_lift g (asucc g x4) x0 x1 H14 c h (r (Bind Abst) n) H17)
-(lift h n x3) (sc3_lift g x4 x2 x3 H15 x h n H19)))))) H18))) k H11))) e1
-H12))))))))) H10)) H9))) t H4))))))))) (drop_gen_skip_l c e2 t h n k
-H1)))))))) d))))))) c2)).
+C (\lambda (c1: C).(drop h n c1 x2)) (\lambda (c1: C).(csubc g c c1)) (ex2 C
+(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3))) (\lambda (c1:
+C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c1)))
+(\lambda (x: C).(\lambda (H19: (drop h n x x2)).(\lambda (H20: (csubc g c
+x)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr)
+x3))) (\lambda (c1: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst)
+n) x1)) c1)) (CHead x (Bind Abbr) (lift h n x3)) (drop_skip_bind h n x x2 H19
+Abbr x3) (csubc_abst g c x H20 (lift h (r (Bind Abst) n) x1) x4 (sc3_lift g
+(asucc g x4) x0 x1 H14 c h (r (Bind Abst) n) H17) (lift h n x3) (sc3_lift g
+x4 x2 x3 H15 x h n H19)))))) H18))) k H11))) e1 H12))))))))) H10)) H9))) t
+H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
theorem csubc_drop_conf_rev:
\forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall
(c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc
g c1 (CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind
Abbr) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind C (\lambda (c1:
-C).(drop h (r (Bind Abbr) n) c1 x2)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C
+C).(drop h n c1 x2)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1:
+C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc g c1
+(CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1))))) (\lambda (x:
+C).(\lambda (H19: (drop h n x x2)).(\lambda (H20: (csubc g x c)).(ex_intro2 C
(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1:
-C).(csubc g c1 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1)))))
-(\lambda (x: C).(\lambda (H19: (drop h (r (Bind Abbr) n) x x2)).(\lambda
-(H20: (csubc g x c)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2
-(Bind Abst) x3))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind Abbr) (lift h
-(r (Bind Abbr) n) x1)))) (CHead x (Bind Abst) (lift h n x3)) (drop_skip_bind
-h n x x2 H19 Abst x3) (csubc_abst g x c H20 (lift h n x3) x4 (sc3_lift g
-(asucc g x4) x2 x3 H14 x h n H19) (lift h (r (Bind Abbr) n) x1) (sc3_lift g
-x4 x0 x1 H15 c h (r (Bind Abbr) n) H17)))))) H18))) k H11))) e1 H12)))))))))
-H10)) H9))) t H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d)))))))
-c2)).
+C).(csubc g c1 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1)))) (CHead x
+(Bind Abst) (lift h n x3)) (drop_skip_bind h n x x2 H19 Abst x3) (csubc_abst
+g x c H20 (lift h n x3) x4 (sc3_lift g (asucc g x4) x2 x3 H14 x h n H19)
+(lift h (r (Bind Abbr) n) x1) (sc3_lift g x4 x0 x1 H15 c h (r (Bind Abbr) n)
+H17)))))) H18))) k H11))) e1 H12))))))))) H10)) H9))) t H4)))))))))
+(drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
include "LambdaDelta-1/next_plus/defs.ma".
-include "LambdaDelta-1/tau1/defs.ma".
+include "LambdaDelta-1/sty1/defs.ma".
include "LambdaDelta-1/llt/defs.ma".
nat).(\lambda (_: ((\forall (n0: nat).((le n (S k0)) \to (eq A (asucc gz
(aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O \Rightarrow (S
k0) | (S l) \Rightarrow (minus k0 l)]) n0))))))).(\lambda (n0: nat).(\lambda
-(H0: (le (S n) (S k0))).(ex2_ind nat (\lambda (n1: nat).(eq nat (S k0) (S
-n1))) (\lambda (n1: nat).(le n n1)) (eq A (asucc gz (aplus gz (ASort (S n)
-n0) k0)) (ASort O (plus (minus k0 n) n0))) (\lambda (x: nat).(\lambda (H1:
-(eq nat (S k0) (S x))).(\lambda (H2: (le n x)).(let H3 \def (f_equal nat nat
-(\lambda (e: nat).(match e in nat return (\lambda (_: nat).nat) with [O
-\Rightarrow k0 | (S n1) \Rightarrow n1])) (S k0) (S x) H1) in (let H4 \def
-(eq_ind_r nat x (\lambda (n1: nat).(le n n1)) H2 k0 H3) in (eq_ind A (aplus
-gz (ASort n n0) k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n)
-n0) k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n) n0)) k0) (\lambda (a:
+(H0: (le (S n) (S k0))).(let H_y \def (le_S_n n k0 H0) in (eq_ind A (aplus gz
+(ASort n n0) k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n) n0)
+k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n) n0)) k0) (\lambda (a:
A).(eq A a (aplus gz (ASort n n0) k0))) (refl_equal A (aplus gz (ASort n n0)
k0)) (asucc gz (aplus gz (ASort (S n) n0) k0)) (aplus_asucc gz k0 (ASort (S
-n) n0))) (ASort O (plus (minus k0 n) n0)) (IH n n0 H4))))))) (le_gen_S n (S
-k0) H0)))))) h)))) k).
+n) n0))) (ASort O (plus (minus k0 n) n0)) (IH n n0 H_y))))))) h)))) k).
theorem aplus_gz_ge:
\forall (n: nat).(\forall (k: nat).(\forall (h: nat).((le k h) \to (eq A
I (S x) H0) in (False_ind (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O
n)) H2))))) (le_gen_S k0 O H))) (\lambda (n0: nat).(\lambda (_: (((le (S k0)
n0) \to (eq A (asucc gz (aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0))
-n))))).(\lambda (H0: (le (S k0) (S n0))).(ex2_ind nat (\lambda (n1: nat).(eq
-nat (S n0) (S n1))) (\lambda (n1: nat).(le k0 n1)) (eq A (asucc gz (aplus gz
-(ASort (S n0) n) k0)) (ASort (minus n0 k0) n)) (\lambda (x: nat).(\lambda
-(H1: (eq nat (S n0) (S x))).(\lambda (H2: (le k0 x)).(let H3 \def (f_equal
-nat nat (\lambda (e: nat).(match e in nat return (\lambda (_: nat).nat) with
-[O \Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x) H1) in (let H4 \def
-(eq_ind_r nat x (\lambda (n1: nat).(le k0 n1)) H2 n0 H3) in (eq_ind A (aplus
-gz (ASort n0 n) k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n0)
-n) k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n0) n)) k0) (\lambda (a:
-A).(eq A a (aplus gz (ASort n0 n) k0))) (refl_equal A (aplus gz (ASort n0 n)
-k0)) (asucc gz (aplus gz (ASort (S n0) n) k0)) (aplus_asucc gz k0 (ASort (S
-n0) n))) (ASort (minus n0 k0) n) (IH n0 H4))))))) (le_gen_S k0 (S n0) H0)))))
-h)))) k)).
+n))))).(\lambda (H0: (le (S k0) (S n0))).(let H_y \def (le_S_n k0 n0 H0) in
+(eq_ind A (aplus gz (ASort n0 n) k0) (\lambda (a: A).(eq A (asucc gz (aplus
+gz (ASort (S n0) n) k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n0) n))
+k0) (\lambda (a: A).(eq A a (aplus gz (ASort n0 n) k0))) (refl_equal A (aplus
+gz (ASort n0 n) k0)) (asucc gz (aplus gz (ASort (S n0) n) k0)) (aplus_asucc
+gz k0 (ASort (S n0) n))) (ASort (minus n0 k0) n) (IH n0 H_y)))))) h)))) k)).
theorem next_plus_gz:
\forall (n: nat).(\forall (h: nat).(eq nat (next_plus gz n h) (plus h n)))
x))).(let H_x \def (lift_gen_head (Bind b) u t x h d H) in (let H0 \def H_x
in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y
z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t (lift h (s (Bind b) d) z)))) (ex3_2 T T (\lambda
-(y: T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h (S d) z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda
-(H1: (eq T x (THead (Bind b) x0 x1))).(\lambda (H2: (eq T u (lift h d
-x0))).(\lambda (H3: (eq T t (lift h (s (Bind b) d) x1))).(eq_ind_r T (THead
-(Bind b) x0 x1) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z:
-T).(eq T t0 (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u
-(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))))
-(eq_ind_r T (lift h (s (Bind b) d) x1) (\lambda (t0: T).(ex3_2 T T (\lambda
-(y: T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind b) y z))))
-(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t0 (lift h (S d) z)))))) (eq_ind_r T (lift h d x0)
-(\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead
+T).(\lambda (z: T).(eq T t (lift h (S d) z)))) (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda
+(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift
+h (S d) z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead
+(Bind b) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t
+(lift h (S d) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0:
+T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Bind b) y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) (eq_ind_r T (lift h (S d)
+x1) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead
(Bind b) x0 x1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T
-t0 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s (Bind b)
-d) x1) (lift h (S d) z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z:
-T).(eq T (THead (Bind b) x0 x1) (THead (Bind b) y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T (lift h d x0) (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T (lift h (s (Bind b) d) x1) (lift h (S d) z)))) x0 x1
-(refl_equal T (THead (Bind b) x0 x1)) (refl_equal T (lift h d x0))
-(refl_equal T (lift h (S d) x1))) u H2) t H3) x H1)))))) H0))))))))).
+u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h (S d)
+z)))))) (eq_ind_r T (lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind b) y z))))
+(\lambda (y: T).(\lambda (_: T).(eq T t0 (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) z)))))) (ex3_2_intro
+T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind
+b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d x0) (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d)
+z)))) x0 x1 (refl_equal T (THead (Bind b) x0 x1)) (refl_equal T (lift h d
+x0)) (refl_equal T (lift h (S d) x1))) u H2) t H3) x H1)))))) H0))))))))).
theorem lift_gen_flat:
\forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
x))).(let H_x \def (lift_gen_head (Flat f) u t x h d H) in (let H0 \def H_x
in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y
z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t (lift h (s (Flat f) d) z)))) (ex3_2 T T (\lambda
-(y: T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h d z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1:
-(eq T x (THead (Flat f) x0 x1))).(\lambda (H2: (eq T u (lift h d
-x0))).(\lambda (H3: (eq T t (lift h (s (Flat f) d) x1))).(eq_ind_r T (THead
-(Flat f) x0 x1) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z:
-T).(eq T t0 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u
-(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))))
-(eq_ind_r T (lift h (s (Flat f) d) x1) (\lambda (t0: T).(ex3_2 T T (\lambda
-(y: T).(\lambda (z: T).(eq T (THead (Flat f) x0 x1) (THead (Flat f) y z))))
-(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t0 (lift h d z)))))) (eq_ind_r T (lift h d x0)
-(\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead
-(Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T
-t0 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s (Flat f)
-d) x1) (lift h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq
+T).(\lambda (z: T).(eq T t (lift h d z)))) (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda
+(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift
+h d z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead
+(Flat f) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t
+(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t0: T).(ex3_2 T
+T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) y z)))) (\lambda
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h d z)))))) (eq_ind_r T (lift h d x1) (\lambda (t0:
+T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) x0 x1)
+(THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h d z)))))) (eq_ind_r T
+(lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq
T (THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_:
+T).(eq T t0 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d
+x1) (lift h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T
+(THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_:
T).(eq T (lift h d x0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T
-(lift h (s (Flat f) d) x1) (lift h d z)))) x0 x1 (refl_equal T (THead (Flat
-f) x0 x1)) (refl_equal T (lift h d x0)) (refl_equal T (lift h d x1))) u H2) t
-H3) x H1)))))) H0))))))))).
+(lift h d x1) (lift h d z)))) x0 x1 (refl_equal T (THead (Flat f) x0 x1))
+(refl_equal T (lift h d x0)) (refl_equal T (lift h d x1))) u H2) t H3) x
+H1)))))) H0))))))))).
\def
\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (nf2 c (THead
(Flat Cast) u t))).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) u t (H t
-(pr2_free c (THead (Flat Cast) u t) t (pr0_epsilon t t (pr0_refl t) u)))
-P))))).
+(pr2_free c (THead (Flat Cast) u t) t (pr0_tau t t (pr0_refl t) u))) P))))).
theorem nf2_gen_beta:
\forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((nf2 c
(Flat Cast) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat
Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
(Flat Cast) t t0) t2)) t0 (\lambda (H1: (eq T (THead (Flat Cast) t t0)
-t0)).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) t t0 H1 P))) (pr0_epsilon t0
-t0 (pr0_refl t0) t))) f)) k)))))) t1).
+t0)).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) t t0 H1 P))) (pr0_tau t0 t0
+(pr0_refl t0) t))) f)) k)))))) t1).
| pr0_zeta: \forall (b: B).((not (eq B b Abst)) \to (\forall (t1: T).(\forall
(t2: T).((pr0 t1 t2) \to (\forall (u: T).(pr0 (THead (Bind b) u (lift (S O) O
t1)) t2))))))
-| pr0_epsilon: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (u:
+| pr0_tau: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (u:
T).(pr0 (THead (Flat Cast) u t1) t2)))).
(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t (lift h x1 t4))) (\lambda
(t4: T).(pr0 (THead (Flat Cast) x2 x3) t4)))) (ex_intro2 T (\lambda (t4:
T).(eq T (lift h x1 x4) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat
-Cast) x2 x3) t4)) x4 (refl_equal T (lift h x1 x4)) (pr0_epsilon x3 x4 H7 x2))
-t3 H_x)))) (H2 x3 x1 H6)) x0 H4)))))) (lift_gen_flat Cast u t2 x0 h x1
+Cast) x2 x3) t4)) x4 (refl_equal T (lift h x1 x4)) (pr0_tau x3 x4 H7 x2)) t3
+H_x)))) (H2 x3 x1 H6)) x0 H4)))))) (lift_gen_flat Cast u t2 x0 h x1
H3)))))))))) y x H0))))) H))))).
x2 x O H10 x1 H7))))) H8)) (pr0_subst0 t3 x0 H3 u2 w O H x H1))))) H5))
(pr0_subst0 t5 x0 H4 u3 w0 O H0 x H2))))))))))))))).
-theorem pr0_confluence__pr0_delta_epsilon:
+theorem pr0_confluence__pr0_delta_tau:
\forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to
(\forall (t4: T).((pr0 (lift (S O) O t4) t3) \to (\forall (t2: T).(ex2 T
(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2
T).(pr0 t8 t9)))))))))) H (THead (Bind b) u (lift (S O) O t4)) H7) in
(ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind b) u (lift (S O) O t4)) t6))
(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_zeta b H10 t4 t2 H11 u) (pr0_refl
-t2)))) t1 H12)))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H5 H6))) | (pr0_epsilon t4
-t5 H5 u) \Rightarrow (\lambda (H6: (eq T (THead (Flat Cast) u t4)
-t)).(\lambda (H7: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u t4) (\lambda
-(_: T).((eq T t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1
-t7)) (\lambda (t7: T).(pr0 t2 t7)))))) (\lambda (H8: (eq T t5 t2)).(eq_ind T
-t2 (\lambda (t6: T).((pr0 t4 t6) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7))
-(\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H9: (pr0 t4 t2)).(let H10 \def
-(eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead (Flat Cast) u t4) H6)
-in (eq_ind T (THead (Flat Cast) u t4) (\lambda (t6: T).(ex2 T (\lambda (t7:
-T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H11 \def (eq_ind_r T t
-(\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Cast) u t4) H6) in (let H12
-\def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall
-(t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9:
+t2)))) t1 H12)))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H5 H6))) | (pr0_tau t4 t5
+H5 u) \Rightarrow (\lambda (H6: (eq T (THead (Flat Cast) u t4) t)).(\lambda
+(H7: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u t4) (\lambda (_: T).((eq T
+t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda
+(t7: T).(pr0 t2 t7)))))) (\lambda (H8: (eq T t5 t2)).(eq_ind T t2 (\lambda
+(t6: T).((pr0 t4 t6) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7:
+T).(pr0 t2 t7))))) (\lambda (H9: (pr0 t4 t2)).(let H10 \def (eq_ind_r T t
+(\lambda (t6: T).(eq T t6 t1)) H4 (THead (Flat Cast) u t4) H6) in (eq_ind T
+(THead (Flat Cast) u t4) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6
+t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H11 \def (eq_ind_r T t (\lambda
+(t6: T).(eq T t3 t6)) H2 (THead (Flat Cast) u t4) H6) in (let H12 \def
+(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7:
+T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9:
T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u
t4) H6) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Cast) u t4) t6))
-(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_epsilon t4 t2 H9 u) (pr0_refl t2)))) t1
+(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_tau t4 t2 H9 u) (pr0_refl t2)))) t1
H10))) t5 (sym_eq T t5 t2 H8))) t H6 H7 H5)))]) in (H5 (refl_equal T t)
(refl_equal T t2))) t (sym_eq T t t1 H4))) t3 (sym_eq T t3 t H2) H3))) |
(pr0_comp u1 u2 H2 t3 t4 H3 k) \Rightarrow (\lambda (H4: (eq T (THead k u1
(t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead
(Bind Abbr) v2 t6) t9)))) with []) in H37))) t8 (sym_eq T t8 t4 H34))) t5
H33)) u0 (sym_eq T u0 u H32))) b (sym_eq B b Abst H31))) H30)) H29)) H28 H25
-H26))) | (pr0_epsilon t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead
-(Flat Cast) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H27: (eq T t8
-t4)).((let H28 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in
-K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind Abst) u t5) H26) in (False_ind ((eq T t8
-t4) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2
-t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) H28)) H27
-H25)))]) in (H25 (refl_equal T (THead (Bind Abst) u t5)) (refl_equal T
-t4))))) k H21))))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon
-b H9 v1 v2 H10 u0 u3 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead
-(Flat Appl) v1 (THead (Bind b) u0 t5)) t)).(\lambda (H14: (eq T (THead (Bind
-b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat
-Appl) v1 (THead (Bind b) u0 t5)) (\lambda (_: T).((eq T (THead (Bind b) u3
-(THead (Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to
-((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
-T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda
-(H15: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6))
-t2)).(eq_ind T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6))
-(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda
-(t8: T).(pr0 t7 t8)))))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17:
-(pr0 v1 v2)).(\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 t6)).(let H20
-\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead
-(Flat Appl) v1 (THead (Bind b) u0 t5)) H13) in (let H21 \def (f_equal T K
-(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
+H26))) | (pr0_tau t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Flat
+Cast) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H27: (eq T t8 t4)).((let
+H28 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: T).(match e in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Abst) u t5) H26) in (False_ind ((eq T t8 t4) \to
+((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) H28)) H27 H25)))]) in
+(H25 (refl_equal T (THead (Bind Abst) u t5)) (refl_equal T t4))))) k H21)))))
+H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b H9 v1 v2 H10 u0
+u3 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1
+(THead (Bind b) u0 t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u3 (THead
+(Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1
+(THead (Bind b) u0 t5)) (\lambda (_: T).((eq T (THead (Bind b) u3 (THead
+(Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1
+v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
+k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T
+(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T
+(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7:
+T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6)
+\to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0
+t7 t8)))))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 v1
+v2)).(\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 t6)).(let H20 \def
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat
+Appl) v1 (THead (Bind b) u0 t5)) H13) in (let H21 \def (f_equal T K (\lambda
+(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k
+| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3)
+(THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let H22 \def (f_equal
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7]))
(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let
-H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _)
+H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7)
\Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0
-t5)) H20) in ((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3
-| (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1
-(THead (Bind b) u0 t5)) H20) in (\lambda (H24: (eq T u1 v1)).(\lambda (H25:
-(eq K k (Flat Appl))).(let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall
-(v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9:
-T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10:
-T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H13)
-in (eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0
-(THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 (THead (Flat
-Appl) (lift (S O) O v2) t6)) t7)))) (let H27 \def (eq_ind T u1 (\lambda (t7:
+t5)) H20) in (\lambda (H24: (eq T u1 v1)).(\lambda (H25: (eq K k (Flat
+Appl))).(let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v
+t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to
+(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
+t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H13) in
+(eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead
+k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl)
+(lift (S O) O v2) t6)) t7)))) (let H27 \def (eq_ind T u1 (\lambda (t7:
T).(pr0 t7 u2)) H7 v1 H24) in (let H28 \def (eq_ind T t3 (\lambda (t7:
T).(pr0 t7 t4)) H8 (THead (Bind b) u0 t5) H23) in (let H29 \def (match H28 in
pr0 return (\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T
(THead (Bind b) u0 (lift (S O) O t7)))) x H45 t4 H40)) t6 H44))))
(pr0_gen_lift t7 t6 (S O) O H43))))))) t8 (sym_eq T t8 t4 H38))) t5 H37)) u
(sym_eq T u u0 H36))) b0 (sym_eq B b0 b H35))) H34)) H33)) H32 H29 H30))) |
-(pr0_epsilon t7 t8 H29 u) \Rightarrow (\lambda (H30: (eq T (THead (Flat Cast)
-u t7) (THead (Bind b) u0 t5))).(\lambda (H31: (eq T t8 t4)).((let H32 \def
+(pr0_tau t7 t8 H29 u) \Rightarrow (\lambda (H30: (eq T (THead (Flat Cast) u
+t7) (THead (Bind b) u0 t5))).(\lambda (H31: (eq T t8 t4)).((let H32 \def
(eq_ind T (THead (Flat Cast) u t7) (\lambda (e: T).(match e in T return
(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
(Bind b) u2 (lift (S O) O x)) t7)) (\lambda (t7: T).(pr0 t2 t7)) x0 (pr0_zeta
b H14 x x0 H27 u2) H28)))) (H22 t5 (lift_tlt_dx (Bind b) u t5 (S O) O) x H26
t2 H15)) t4 H25)))) (pr0_gen_lift t5 t4 (S O) O H24)))) k H21))))) H18))
-H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_epsilon t5 t6
-H9 u) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u t5) t)).(\lambda
-(H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T
-t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8))
+H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6 H9 u)
+\Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H11:
+(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6
+t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8))
(\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 t2)).(eq_ind T t2
(\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2
t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H13: (pr0 t5 t2)).(let
t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t4) t7)) (\lambda
(t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H23: (pr0 t4 x)).(\lambda
(H24: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2
-t4) t7)) (\lambda (t7: T).(pr0 t2 t7)) x (pr0_epsilon t4 x H23 u2) H24))))
-(H20 t5 (tlt_head_dx (Flat Cast) u t5) t4 H22 t2 H13)))) k H19))))) H16))
-H15)))) t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t)
+t4) t7)) (\lambda (t7: T).(pr0 t2 t7)) x (pr0_tau t4 x H23 u2) H24)))) (H20
+t5 (tlt_head_dx (Flat Cast) u t5) t4 H22 t2 H13)))) k H19))))) H16)) H15))))
+t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t)
(refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_beta u v1 v2 H2 t3 t4
H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind Abst)
u t3)) t)).(\lambda (H5: (eq T (THead (Bind Abbr) v2 t4) t1)).(eq_ind T
(t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead
(Flat Appl) u2 t6) t9)))) with []) in H36))) t8 (sym_eq T t8 t6 H33))) t3
H32)) u0 (sym_eq T u0 u H31))) b (sym_eq B b Abst H30))) H29)) H28)) H27 H24
-H25))) | (pr0_epsilon t7 t8 H24 u0) \Rightarrow (\lambda (H25: (eq T (THead
-(Flat Cast) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H26: (eq T t8
-t6)).((let H27 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in
-K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind Abst) u t3) H25) in (False_ind ((eq T t8
-t6) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2
-t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) H27)) H26
-H24)))]) in (H24 (refl_equal T (THead (Bind Abst) u t3)) (refl_equal T
-t6))))) k H21)))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u0
-v0 v3 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v0
-(THead (Bind Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v3 t6)
-t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) (\lambda (_:
-T).((eq T (THead (Bind Abbr) v3 t6) t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to
+H25))) | (pr0_tau t7 t8 H24 u0) \Rightarrow (\lambda (H25: (eq T (THead (Flat
+Cast) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H26: (eq T t8 t6)).((let
+H27 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: T).(match e in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Abst) u t3) H25) in (False_ind ((eq T t8 t6) \to
+((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9))
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) H27)) H26 H24)))]) in
+(H24 (refl_equal T (THead (Bind Abst) u t3)) (refl_equal T t6))))) k H21))))
+H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u0 v0 v3 H9 t5 t6
+H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v0 (THead (Bind
+Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind
+T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) (\lambda (_: T).((eq T
+(THead (Bind Abbr) v3 t6) t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T
+(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2
+t8))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T
+(THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to ((pr0 t5 t6) \to
(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8:
-T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v3 t6)
-t2)).(eq_ind T (THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to
+T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 v0 v3)).(\lambda (H15: (pr0 t5
+t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1
+(THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind Abst) u0
+t5)) H11) in (let H17 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1
+| (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u
+t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in ((let H18 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7)
+\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) \Rightarrow t8])]))
+(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead
+(Bind Abst) u0 t5)) H16) in ((let H19 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 |
+(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3
+| (THead _ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst)
+u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in (\lambda (_:
+(eq T u u0)).(\lambda (H21: (eq T v1 v0)).(let H22 \def (eq_ind_r T t
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8)
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind
+Abst) u0 t5)) H11) in (let H23 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7
+v2)) H7 v0 H21) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4))
+H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0
+t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda
+(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) (\lambda (x: T).(\lambda (H25:
+(pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 v2
+t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind
+Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))
+(\lambda (x0: T).(\lambda (H27: (pr0 v2 x0)).(\lambda (H28: (pr0 v3
+x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7))
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)) (THead (Bind Abbr) x0 x)
+(pr0_comp v2 x0 H27 t4 x H25 (Bind Abbr)) (pr0_comp v3 x0 H28 t6 x H26 (Bind
+Abbr)))))) (H22 v0 (tlt_head_sx (Flat Appl) v0 (THead (Bind Abst) u0 t5)) v2
+H23 v3 H14))))) (H22 t5 (tlt_trans (THead (Bind Abst) u0 t5) t5 (THead (Flat
+Appl) v0 (THead (Bind Abst) u0 t5)) (tlt_head_dx (Bind Abst) u0 t5)
+(tlt_head_dx (Flat Appl) v0 (THead (Bind Abst) u0 t5))) t4 H24 t6 H15))))))))
+H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b H9 v0 v3 H10 u1
+u2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v0
+(THead (Bind b) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Flat Appl) v0
+(THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v0
+v3) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
+(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15:
+(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6))
+t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6))
+(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to
((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8))
-(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 v0 v3)).(\lambda (H15:
-(pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat
-Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind
-Abst) u0 t5)) H11) in (let H17 \def (f_equal T T (\lambda (e: T).(match e in
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _)
-\Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1
-(THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5))
-H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
-(THead _ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _)
-\Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead
-(Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in ((let H19 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match
-t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
-\Rightarrow t3 | (THead _ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1
-(THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5))
-H16) in (\lambda (_: (eq T u u0)).(\lambda (H21: (eq T v1 v0)).(let H22 \def
-(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8:
-T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10:
-T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl)
-v0 (THead (Bind Abst) u0 t5)) H11) in (let H23 \def (eq_ind T v1 (\lambda
-(t7: T).(pr0 t7 v2)) H7 v0 H21) in (let H24 \def (eq_ind T t3 (\lambda (t7:
-T).(pr0 t7 t4)) H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7))
-(\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr)
-v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) (\lambda
-(x: T).(\lambda (H25: (pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T
-(\lambda (t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda
-(t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead
-(Bind Abbr) v3 t6) t7))) (\lambda (x0: T).(\lambda (H27: (pr0 v2
-x0)).(\lambda (H28: (pr0 v3 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead
-(Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))
-(THead (Bind Abbr) x0 x) (pr0_comp v2 x0 H27 t4 x H25 (Bind Abbr)) (pr0_comp
-v3 x0 H28 t6 x H26 (Bind Abbr)))))) (H22 v0 (tlt_head_sx (Flat Appl) v0
-(THead (Bind Abst) u0 t5)) v2 H23 v3 H14))))) (H22 t5 (tlt_trans (THead (Bind
-Abst) u0 t5) t5 (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) (tlt_head_dx
-(Bind Abst) u0 t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind Abst) u0 t5))) t4
-H24 t6 H15)))))))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon
-b H9 v0 v3 H10 u1 u2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead
-(Flat Appl) v0 (THead (Bind b) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind
-b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Flat
-Appl) v0 (THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b Abst)) \to
-((pr0 v0 v3) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
-T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))))
-(\lambda (H15: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3)
-t6)) t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3)
-t6)) (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2)
-\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4)
-t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (H16: (not (eq B b
+(\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (H16: (not (eq B b
Abst))).(\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0
t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl)
v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind b) u1
(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H16) in (False_ind (ex2 T
(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2
-t7))) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_epsilon
-t5 t6 H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0 t5)
+t7))) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6
+H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0 t5)
t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda
(_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq
u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7))) v2 H46
u3 H18))) t4 H44)))) (pr0_gen_lift t7 t4 (S O) O H43)))))))) t8 (sym_eq T t8
t6 H37))) t3 H36)) u (sym_eq T u u1 H35))) b0 (sym_eq B b0 b H34))) H33))
-H32)) H31 H28 H29))) | (pr0_epsilon t7 t8 H28 u) \Rightarrow (\lambda (H29:
-(eq T (THead (Flat Cast) u t7) (THead (Bind b) u1 t3))).(\lambda (H30: (eq T
-t8 t6)).((let H31 \def (eq_ind T (THead (Flat Cast) u t7) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in
-K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
+H32)) H31 H28 H29))) | (pr0_tau t7 t8 H28 u) \Rightarrow (\lambda (H29: (eq T
+(THead (Flat Cast) u t7) (THead (Bind b) u1 t3))).(\lambda (H30: (eq T t8
+t6)).((let H31 \def (eq_ind T (THead (Flat Cast) u t7) (\lambda (e: T).(match
+e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
\Rightarrow True])])) I (THead (Bind b) u1 t3) H29) in (False_ind ((eq T t8
t6) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2
(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead
(Flat _) \Rightarrow True])])) I (THead (Bind b0) u (lift (S O) O t5)) H20)
in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat
Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) H21))))) t6
-(sym_eq T t6 t2 H17))) t H15 H16 H13 H14))) | (pr0_epsilon t5 t6 H13 u)
+(sym_eq T t6 t2 H17))) t H15 H16 H13 H14))) | (pr0_tau t5 t6 H13 u)
\Rightarrow (\lambda (H14: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H15:
(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6
t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2
T).(pr0 x t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0
(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0:
T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t2
-x0)).(pr0_confluence__pr0_delta_epsilon u2 (lift (S O) O x) w H31 x (pr0_refl
+x0)).(pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H31 x (pr0_refl
(lift (S O) O x)) t2)))) (H26 t5 (lift_tlt_dx (Bind Abbr) u t5 (S O) O) x H30
t2 H17)))))) (pr0_gen_lift t5 t4 (S O) O H28)))))))))) H20)) H19))))) t6
-(sym_eq T t6 t2 H15))) t H13 H14 H11 H12))) | (pr0_epsilon t5 t6 H11 u)
+(sym_eq T t6 t2 H15))) t H13 H14 H11 H12))) | (pr0_tau t5 t6 H11 u)
\Rightarrow (\lambda (H12: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H13:
(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6
t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2
(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w)
t7))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t1
x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u2 w)) (pr0 t1)
-(pr0_confluence__pr0_delta_epsilon u2 (lift (S O) O x) w H29 x (pr0_refl
-(lift (S O) O x)) t1))))) (H28 t3 (lift_tlt_dx (Bind Abbr) u1 t3 (S O) O) x
-H26 t1 H8))))))))) (pr0_gen_lift t3 t6 (S O) O H24)))))) H20)) H19)))))) t2
-H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b0 H9 t5 t6 H10 u0) \Rightarrow
-(\lambda (H11: (eq T (THead (Bind b0) u0 (lift (S O) O t5)) t)).(\lambda
-(H12: (eq T t6 t2)).(eq_ind T (THead (Bind b0) u0 (lift (S O) O t5)) (\lambda
-(_: T).((eq T t6 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T
-(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda
-(H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b0 Abst)) \to
-((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0
-t2 t8)))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (H15: (pr0 t5
-t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u
-(lift (S O) O t3)) t7)) H4 (THead (Bind b0) u0 (lift (S O) O t5)) H11) in
-(let H17 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda (_:
-T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1)
-\Rightarrow b1 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O
-t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in ((let H18 \def (f_equal T
-T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) \Rightarrow t7]))
-(THead (Bind b) u (lift (S O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5))
-H16) in ((let H19 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
-\to nat))) (d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u1 t8)
-\Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t8))]) in
-lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match
-t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k u1 t8) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d)
-t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t7)
-\Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind b0) u0
-(lift (S O) O t5)) H16) in (\lambda (_: (eq T u u0)).(\lambda (H21: (eq B b
-b0)).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7)
-\to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T
-(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H
-(THead (Bind b0) u0 (lift (S O) O t5)) H11) in (let H23 \def (eq_ind T t3
-(\lambda (t7: T).(pr0 t7 t1)) H8 t5 (lift_inj t3 t5 (S O) O H19)) in (let H24
-\def (eq_ind B b (\lambda (b1: B).(not (eq B b1 Abst))) H7 b0 H21) in
-(ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x:
-T).(\lambda (H25: (pr0 t1 x)).(\lambda (H26: (pr0 t2 x)).(ex_intro2 T
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) x H25 H26))))
-(H22 t5 (lift_tlt_dx (Bind b0) u0 t5 (S O) O) t1 H23 t2 H15)))))))) H18))
-H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_epsilon t5 t6
-H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0 t5)
-t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda
-(_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1
-t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 t2)).(eq_ind T
-t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
-(\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 t2)).(let H14 \def
+(pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H29 x (pr0_refl (lift (S
+O) O x)) t1))))) (H28 t3 (lift_tlt_dx (Bind Abbr) u1 t3 (S O) O) x H26 t1
+H8))))))))) (pr0_gen_lift t3 t6 (S O) O H24)))))) H20)) H19)))))) t2 H14)) t
+H12 H13 H9 H10 H11))) | (pr0_zeta b0 H9 t5 t6 H10 u0) \Rightarrow (\lambda
+(H11: (eq T (THead (Bind b0) u0 (lift (S O) O t5)) t)).(\lambda (H12: (eq T
+t6 t2)).(eq_ind T (THead (Bind b0) u0 (lift (S O) O t5)) (\lambda (_: T).((eq
+T t6 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
+T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6
+t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 t5 t7) \to
+(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))
+(\lambda (_: (not (eq B b0 Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def
(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3))
-t7)) H4 (THead (Flat Cast) u0 t5) H10) in (let H15 \def (eq_ind T (THead
-(Bind b) u (lift (S O) O t3)) (\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) u0 t5) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda
-(t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in
-(H9 (refl_equal T t) (refl_equal T t2))))) t4 (sym_eq T t4 t1 H6))) t H4 H5
-H2 H3))) | (pr0_epsilon t3 t4 H2 u) \Rightarrow (\lambda (H3: (eq T (THead
-(Flat Cast) u t3) t)).(\lambda (H4: (eq T t4 t1)).(eq_ind T (THead (Flat
-Cast) u t3) (\lambda (_: T).((eq T t4 t1) \to ((pr0 t3 t4) \to (ex2 T
-(\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H5:
-(eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((pr0 t3 t5) \to (ex2 T (\lambda
-(t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))) (\lambda (H6: (pr0 t3
-t1)).(let H7 \def (match H1 in pr0 return (\lambda (t5: T).(\lambda (t6:
-T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with
-[(pr0_refl t5) \Rightarrow (\lambda (H7: (eq T t5 t)).(\lambda (H8: (eq T t5
-t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7:
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H9: (eq T t
-t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7))
-(\lambda (t7: T).(pr0 t2 t7)))) (let H10 \def (eq_ind_r T t (\lambda (t6:
-T).(eq T t6 t2)) H9 (THead (Flat Cast) u t3) H3) in (eq_ind T (THead (Flat
-Cast) u t3) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda
-(t7: T).(pr0 t6 t7)))) (let H11 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5
-t6)) H7 (THead (Flat Cast) u t3) H3) in (let H12 \def (eq_ind_r T t (\lambda
-(t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to
-(\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9))
-(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u t3) H3) in
-(ex_intro2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 (THead (Flat
-Cast) u t3) t6)) t1 (pr0_refl t1) (pr0_epsilon t3 t1 H6 u)))) t2 H10)) t
-(sym_eq T t t2 H9))) t5 (sym_eq T t5 t H7) H8))) | (pr0_comp u1 u2 H7 t5 t6
-H8 k) \Rightarrow (\lambda (H9: (eq T (THead k u1 t5) t)).(\lambda (H10: (eq
-T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T
-(THead k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda
-(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T
-(THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2)
-\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8:
-T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13: (pr0 t5
-t6)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u
-t3) t7)) H3 (THead k u1 t5) H9) in (let H15 \def (f_equal T K (\lambda (e:
-T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Flat
-Cast) | (TLRef _) \Rightarrow (Flat Cast) | (THead k0 _ _) \Rightarrow k0]))
-(THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H16 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) \Rightarrow t7]))
-(THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H17 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7]))
-(THead (Flat Cast) u t3) (THead k u1 t5) H14) in (\lambda (_: (eq T u
-u1)).(\lambda (H19: (eq K (Flat Cast) k)).(eq_ind K (Flat Cast) (\lambda (k0:
-K).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead k0 u2
-t6) t7)))) (let H20 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u1
-t5) t)) H9 (Flat Cast) H19) in (let H21 \def (eq_ind_r T t (\lambda (t7:
+t7)) H4 (THead (Bind b0) u0 (lift (S O) O t5)) H11) in (let H17 \def (f_equal
+T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _)
+\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k
+in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _)
+\Rightarrow b])])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind b0) u0
+(lift (S O) O t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _)
+\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S
+O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in ((let H19 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7:
+T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i)
+\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false
+\Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f d u1)
+(lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O
+t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat)
+(t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) |
+(TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i |
+false \Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f
+d u1) (lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S
+O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O
+t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in (\lambda (_: (eq T u
+u0)).(\lambda (H21: (eq B b b0)).(let H22 \def (eq_ind_r T t (\lambda (t7:
T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall
(t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10:
-T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u1 t5) H20) in (let H22 \def
-(eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 H17) in (ex2_ind T (\lambda
-(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7:
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6) t7)))
-(\lambda (x: T).(\lambda (H23: (pr0 t1 x)).(\lambda (H24: (pr0 t6
-x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead
-(Flat Cast) u2 t6) t7)) x H23 (pr0_epsilon t6 x H24 u2))))) (H21 t5
-(tlt_head_dx (Flat Cast) u1 t5) t1 H22 t6 H13))))) k H19)))) H16)) H15)))))
-t2 H11)) t H9 H10 H7 H8))) | (pr0_beta u0 v1 v2 H7 t5 t6 H8) \Rightarrow
-(\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5))
-t)).(\lambda (H10: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead
-(Flat Appl) v1 (THead (Bind Abst) u0 t5)) (\lambda (_: T).((eq T (THead (Bind
-Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T
-(THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda
-(t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1
-t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_:
+T).(pr0 t9 t10)))))))))) H (THead (Bind b0) u0 (lift (S O) O t5)) H11) in
+(let H23 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H8 t5 (lift_inj t3
+t5 (S O) O H19)) in (let H24 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1
+Abst))) H7 b0 H21) in (ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7:
+T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2
+t7))) (\lambda (x: T).(\lambda (H25: (pr0 t1 x)).(\lambda (H26: (pr0 t2
+x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))
+x H25 H26)))) (H22 t5 (lift_tlt_dx (Bind b0) u0 t5 (S O) O) t1 H23 t2
+H15)))))))) H18)) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) |
+(pr0_tau t5 t6 H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0
+t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5)
+(\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
+T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6
+t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8:
+T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5
+t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u
+(lift (S O) O t3)) t7)) H4 (THead (Flat Cast) u0 t5) H10) in (let H15 \def
+(eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\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) u0 t5) H14) in (False_ind (ex2 T (\lambda
+(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T t6
+t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t4
+(sym_eq T t4 t1 H6))) t H4 H5 H2 H3))) | (pr0_tau t3 t4 H2 u) \Rightarrow
+(\lambda (H3: (eq T (THead (Flat Cast) u t3) t)).(\lambda (H4: (eq T t4
+t1)).(eq_ind T (THead (Flat Cast) u t3) (\lambda (_: T).((eq T t4 t1) \to
+((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0
+t2 t6)))))) (\lambda (H5: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((pr0
+t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2
+t6))))) (\lambda (H6: (pr0 t3 t1)).(let H7 \def (match H1 in pr0 return
+(\lambda (t5: T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to
+((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0
+t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H7: (eq T t5
+t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2)
+\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))
+(\lambda (H9: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7:
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H10 \def (eq_ind_r T t
+(\lambda (t6: T).(eq T t6 t2)) H9 (THead (Flat Cast) u t3) H3) in (eq_ind T
+(THead (Flat Cast) u t3) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t1
+t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H11 \def (eq_ind_r T t (\lambda
+(t6: T).(eq T t5 t6)) H7 (THead (Flat Cast) u t3) H3) in (let H12 \def
+(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7:
+T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9:
+T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u
+t3) H3) in (ex_intro2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0
+(THead (Flat Cast) u t3) t6)) t1 (pr0_refl t1) (pr0_tau t3 t1 H6 u)))) t2
+H10)) t (sym_eq T t t2 H9))) t5 (sym_eq T t5 t H7) H8))) | (pr0_comp u1 u2 H7
+t5 t6 H8 k) \Rightarrow (\lambda (H9: (eq T (THead k u1 t5) t)).(\lambda
+(H10: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_:
+T).((eq T (THead k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T
+(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda
+(H11: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7:
+T).((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
+(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13:
(pr0 t5 t6)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat
-Cast) u t3) t7)) H3 (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) H9) in
-(let H15 \def (eq_ind T (THead (Flat Cast) u t3) (\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) v1 (THead (Bind Abst) u0
-t5)) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7:
-T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H15))))) t2 H11)) t H9 H10 H7 H8))) |
-(pr0_upsilon b H7 v1 v2 H8 u1 u2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq
-T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) t)).(\lambda (H12: (eq T
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T
-(THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b
-Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda
-(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H13: (eq
-T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7:
-T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6)
-\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))))
-(\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0
-u1 u2)).(\lambda (_: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7:
+Cast) u t3) t7)) H3 (THead k u1 t5) H9) in (let H15 \def (f_equal T K
+(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow (Flat Cast) | (TLRef _) \Rightarrow (Flat Cast) | (THead k0 _ _)
+\Rightarrow k0])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H16
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _)
+\Rightarrow t7])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H17
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7)
+\Rightarrow t7])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in (\lambda
+(_: (eq T u u1)).(\lambda (H19: (eq K (Flat Cast) k)).(eq_ind K (Flat Cast)
+(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0
+(THead k0 u2 t6) t7)))) (let H20 \def (eq_ind_r K k (\lambda (k0: K).(eq T
+(THead k0 u1 t5) t)) H9 (Flat Cast) H19) in (let H21 \def (eq_ind_r T t
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8)
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u1 t5) H20) in
+(let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 H17) in
+(ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6)
+t7))) (\lambda (x: T).(\lambda (H23: (pr0 t1 x)).(\lambda (H24: (pr0 t6
+x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead
+(Flat Cast) u2 t6) t7)) x H23 (pr0_tau t6 x H24 u2))))) (H21 t5 (tlt_head_dx
+(Flat Cast) u1 t5) t1 H22 t6 H13))))) k H19)))) H16)) H15))))) t2 H11)) t H9
+H10 H7 H8))) | (pr0_beta u0 v1 v2 H7 t5 t6 H8) \Rightarrow (\lambda (H9: (eq
+T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) t)).(\lambda (H10: (eq T
+(THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind
+Abst) u0 t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0
+v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda
+(t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2 t6)
+t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0
+t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H14 \def
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead
+(Flat Appl) v1 (THead (Bind Abst) u0 t5)) H9) in (let H15 \def (eq_ind T
+(THead (Flat Cast) u t3) (\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) v1 (THead (Bind Abst) u0 t5)) H14) in
+(False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead
+(Bind Abbr) v2 t6) t7))) H15))))) t2 H11)) t H9 H10 H7 H8))) | (pr0_upsilon b
+H7 v1 v2 H8 u1 u2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat
+Appl) v1 (THead (Bind b) u1 t5)) t)).(\lambda (H12: (eq T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl)
+v1 (THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1
+v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1
+t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H13: (eq T (THead (Bind
+b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind
+b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B
+b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T
+(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda
+(_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1
+u2)).(\lambda (_: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7:
T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Appl) v1 (THead (Bind
b) u1 t5)) H11) in (let H19 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda
(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
\Rightarrow True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H14) in
(False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2
-t7))) H15))))) t6 (sym_eq T t6 t2 H11))) t H9 H10 H7 H8))) | (pr0_epsilon t5
-t6 H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Flat Cast) u0 t5)
-t)).(\lambda (H9: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda
-(_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1
-t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H10: (eq T t6 t2)).(eq_ind T
-t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
-(\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H11: (pr0 t5 t2)).(let H12 \def
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead
-(Flat Cast) u0 t5) H8) in (let H13 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _)
-\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t3)
-(THead (Flat Cast) u0 t5) H12) in ((let H14 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 |
-(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Flat
-Cast) u t3) (THead (Flat Cast) u0 t5) H12) in (\lambda (_: (eq T u u0)).(let
-H16 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to
-(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T
-(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H
-(THead (Flat Cast) u0 t5) H8) in (let H17 \def (eq_ind T t3 (\lambda (t7:
-T).(pr0 t7 t1)) H6 t5 H14) in (ex2_ind T (\lambda (t7: T).(pr0 t1 t7))
-(\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda
-(t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H18: (pr0 t1 x)).(\lambda
-(H19: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7:
-T).(pr0 t2 t7)) x H18 H19)))) (H16 t5 (tlt_head_dx (Flat Cast) u0 t5) t1 H17
-t2 H11)))))) H13)))) t6 (sym_eq T t6 t2 H10))) t H8 H9 H7)))]) in (H7
-(refl_equal T t) (refl_equal T t2)))) t4 (sym_eq T t4 t1 H5))) t H3 H4
-H2)))]) in (H2 (refl_equal T t) (refl_equal T t1))))))))) t0).
+t7))) H15))))) t6 (sym_eq T t6 t2 H11))) t H9 H10 H7 H8))) | (pr0_tau t5 t6
+H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Flat Cast) u0 t5) t)).(\lambda
+(H9: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda (_: T).((eq T
+t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda
+(t8: T).(pr0 t2 t8)))))) (\lambda (H10: (eq T t6 t2)).(eq_ind T t2 (\lambda
+(t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8:
+T).(pr0 t2 t8))))) (\lambda (H11: (pr0 t5 t2)).(let H12 \def (eq_ind_r T t
+(\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Cast) u0
+t5) H8) in (let H13 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
+(THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t3) (THead (Flat Cast)
+u0 t5) H12) in ((let H14 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Cast) u t3)
+(THead (Flat Cast) u0 t5) H12) in (\lambda (_: (eq T u u0)).(let H16 \def
+(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8:
+T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10:
+T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast)
+u0 t5) H8) in (let H17 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5
+H14) in (ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2
+t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))
+(\lambda (x: T).(\lambda (H18: (pr0 t1 x)).(\lambda (H19: (pr0 t2
+x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))
+x H18 H19)))) (H16 t5 (tlt_head_dx (Flat Cast) u0 t5) t1 H17 t2 H11))))))
+H13)))) t6 (sym_eq T t6 t2 H10))) t H8 H9 H7)))]) in (H7 (refl_equal T t)
+(refl_equal T t2)))) t4 (sym_eq T t4 t1 H5))) t H3 H4 H2)))]) in (H2
+(refl_equal T t) (refl_equal T t1))))))))) t0).
(pr0 t3 t4)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h
d t3) (lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d:
nat).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d)
-t3)) (\lambda (t: T).(pr0 t (lift h d t4))) (pr0_epsilon (lift h (s (Flat
-Cast) d) t3) (lift h d t4) (H1 h d) (lift h d u)) (lift h d (THead (Flat
-Cast) u t3)) (lift_head (Flat Cast) u t3 h d))))))))) t1 t2 H))).
+t3)) (\lambda (t: T).(pr0 t (lift h d t4))) (pr0_tau (lift h (s (Flat Cast)
+d) t3) (lift h d t4) (H1 h d) (lift h d u)) (lift h d (THead (Flat Cast) u
+t3)) (lift_head (Flat Cast) u t3 h d))))))))) t1 t2 H))).
theorem pr0_subst0_back:
\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0
T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2:
T).(subst0 i v2 t4 w2))))) (or_introl (pr0 (THead (Flat Cast) x t3) t4) (ex2
T (\lambda (w2: T).(pr0 (THead (Flat Cast) x t3) w2)) (\lambda (w2:
-T).(subst0 i v2 t4 w2))) (pr0_epsilon t3 t4 H0 x)) w1 H5)))) H4)) (\lambda
-(H4: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda
-(t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5)))).(ex2_ind T (\lambda (t5:
-T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 (s (Flat
-Cast) i) v1 t3 t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2))
-(\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H5: (eq T
-w1 (THead (Flat Cast) u x))).(\lambda (H6: (subst0 (s (Flat Cast) i) v1 t3
-x)).(eq_ind_r T (THead (Flat Cast) u x) (\lambda (t: T).(or (pr0 t t4) (ex2 T
-(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (or_ind
-(pr0 x t4) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s
-(Flat Cast) i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T
+T).(subst0 i v2 t4 w2))) (pr0_tau t3 t4 H0 x)) w1 H5)))) H4)) (\lambda (H4:
+(ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5:
+T).(subst0 (s (Flat Cast) i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T
+w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 (s (Flat Cast) i) v1
+t3 t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat
+Cast) u x))).(\lambda (H6: (subst0 (s (Flat Cast) i) v1 t3 x)).(eq_ind_r T
+(THead (Flat Cast) u x) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2:
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (or_ind (pr0 x t4)
+(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast)
+i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2:
+T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))
+(\lambda (H7: (pr0 x t4)).(or_introl (pr0 (THead (Flat Cast) u x) t4) (ex2 T
(\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i
-v2 t4 w2)))) (\lambda (H7: (pr0 x t4)).(or_introl (pr0 (THead (Flat Cast) u
-x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda
-(w2: T).(subst0 i v2 t4 w2))) (pr0_epsilon x t4 H7 u))) (\lambda (H7: (ex2 T
+v2 t4 w2))) (pr0_tau x t4 H7 u))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0
+x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T
(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s
-(Flat Cast) i) v2 t4 w2)) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T
-(\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i
-v2 t4 w2)))) (\lambda (x0: T).(\lambda (H8: (pr0 x x0)).(\lambda (H9: (subst0
-(s (Flat Cast) i) v2 t4 x0)).(or_intror (pr0 (THead (Flat Cast) u x) t4) (ex2
-T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0
-i v2 t4 w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2))
-(\lambda (w2: T).(subst0 i v2 t4 w2)) x0 (pr0_epsilon x x0 H8 u) H9))))) H7))
-(H1 v1 x (s (Flat Cast) i) H6 v2 H3)) w1 H5)))) H4)) (\lambda (H4: (ex3_2 T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_:
-T).(\lambda (t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_:
-T).(\lambda (t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (or (pr0 w1 t4)
-(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T w1 (THead (Flat Cast)
-x0 x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H7: (subst0 (s (Flat
-Cast) i) v1 t3 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or
-(pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2
-t4 w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda
+w2)) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead
+(Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0:
+T).(\lambda (H8: (pr0 x x0)).(\lambda (H9: (subst0 (s (Flat Cast) i) v2 t4
+x0)).(or_intror (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0
+(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))
+(ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2)) x0 (pr0_tau x x0 H8 u) H9))))) H7)) (H1 v1 x (s (Flat
+Cast) i) H6 v2 H3)) w1 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5:
+T).(subst0 (s (Flat Cast) i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5:
+T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (or (pr0 w1 t4) (ex2 T (\lambda (w2:
+T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x0
+x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H7: (subst0 (s (Flat Cast)
+i) v1 t3 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (pr0
+t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4
+w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda
(w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) x0
x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda
(w2: T).(subst0 i v2 t4 w2)))) (\lambda (H8: (pr0 x1 t4)).(or_introl (pr0
(THead (Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast)
-x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_epsilon x1 t4 H8
-x0))) (\lambda (H8: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2:
-T).(subst0 (s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1
-w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)) (or (pr0 (THead
+x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_tau x1 t4 H8 x0)))
+(\lambda (H8: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0
+(s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2))
+(\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)) (or (pr0 (THead (Flat
+Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2))
+(\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (pr0 x1
+x)).(\lambda (H10: (subst0 (s (Flat Cast) i) v2 t4 x)).(or_intror (pr0 (THead
(Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1)
-w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H9:
-(pr0 x1 x)).(\lambda (H10: (subst0 (s (Flat Cast) i) v2 t4 x)).(or_intror
-(pr0 (THead (Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat
-Cast) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T
-(\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda (w2: T).(subst0
-i v2 t4 w2)) x (pr0_epsilon x1 x H9 x0) H10))))) H8)) (H1 v1 x1 (s (Flat
-Cast) i) H7 v2 H3)) w1 H5)))))) H4)) (subst0_gen_head (Flat Cast) v1 u t3 w1
-i H2))))))))))))) t1 t2 H))).
+w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda (w2:
+T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))
+x (pr0_tau x1 x H9 x0) H10))))) H8)) (H1 v1 x1 (s (Flat Cast) i) H7 v2 H3))
+w1 H5)))))) H4)) (subst0_gen_head (Flat Cast) v1 u t3 w1 i H2))))))))))))) t1
+t2 H))).
\lambda (g: G).(\lambda (a: A).(\lambda (c: C).(\lambda (t: T).(\lambda (H:
(sc3 g a c t)).(let H_x \def (sc3_props__sc3_sn3_abst g a) in (let H0 \def
H_x in (land_ind (\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3
-c0 t0)))) (\forall (vs: TList).(\forall (i: nat).(let t0 \def (THeads (Flat
-Appl) vs (TLRef i)) in (\forall (c0: C).((arity g c0 t0 a) \to ((nf2 c0
-(TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a c0 t0)))))))) (sn3 c t) (\lambda
-(H1: ((\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 c0
-t0)))))).(\lambda (_: ((\forall (vs: TList).(\forall (i: nat).(let t0 \def
-(THeads (Flat Appl) vs (TLRef i)) in (\forall (c0: C).((arity g c0 t0 a) \to
-((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a c0 t0)))))))))).(H1 c t
-H))) H0))))))).
+c0 t0)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g
+c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0
+vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t)
+(\lambda (H1: ((\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 c0
+t0)))))).(\lambda (_: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0:
+C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i))
+\to ((sns3 c0 vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef
+i))))))))))).(H1 c t H))) H0))))))).
theorem sc3_abst:
\forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall
a)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(let H_x \def
(sc3_props__sc3_sn3_abst g a) in (let H2 \def H_x in (land_ind (\forall (c0:
C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0 t)))) (\forall (vs0:
-TList).(\forall (i0: nat).(let t \def (THeads (Flat Appl) vs0 (TLRef i0)) in
-(\forall (c0: C).((arity g c0 t a) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0
-vs0) \to (sc3 g a c0 t)))))))) (sc3 g a c (THeads (Flat Appl) vs (TLRef i)))
-(\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0
-t)))))).(\lambda (H4: ((\forall (vs0: TList).(\forall (i0: nat).(let t \def
-(THeads (Flat Appl) vs0 (TLRef i0)) in (\forall (c0: C).((arity g c0 t a) \to
-((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a c0 t)))))))))).(H4 vs i
-c H H0 H1))) H2)))))))))).
+TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl)
+vs0 (TLRef i0)) a) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a
+c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a c (THeads (Flat Appl)
+vs (TLRef i))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a c0 t)
+\to (sn3 c0 t)))))).(\lambda (H4: ((\forall (vs0: TList).(\forall (i0:
+nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a) \to
+((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a c0 (THeads (Flat Appl)
+vs0 (TLRef i0))))))))))).(H4 vs i c H H0 H1))) H2)))))))))).
theorem sc3_bind:
\forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1:
(Flat Appl) (TCons t1 t2) u)))).(\lambda (t3: T).(\lambda (H2: (sn3 c (THead
(Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) t3)))).(let H_x \def
(sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) t3) H2) in (let H3
-\def H_x in (land_ind (sn3 c t) (sn3 c (THeads (Flat Appl) (TCons t1 t2) t3))
-(sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) (THead (Flat
-Cast) u t3)))) (\lambda (_: (sn3 c t)).(\lambda (H5: (sn3 c (THeads (Flat
-Appl) (TCons t1 t2) t3))).(let H6 \def H5 in (let H_x0 \def (sn3_gen_flat
-Appl c t (THeads (Flat Appl) (TCons t1 t2) u) H1) in (let H7 \def H_x0 in
-(land_ind (sn3 c t) (sn3 c (THeads (Flat Appl) (TCons t1 t2) u)) (sn3 c
-(THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u
-t3)))) (\lambda (H8: (sn3 c t)).(\lambda (H9: (sn3 c (THeads (Flat Appl)
-(TCons t1 t2) u))).(let H10 \def H9 in (sn3_appl_appls t1 (THead (Flat Cast)
-u t3) t2 c (H0 u H10 t3 H6) t H8 (\lambda (u2: T).(\lambda (H11: (pr3 c
-(THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u t3)) u2)).(\lambda
-(H12: (((iso (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u t3)) u2)
-\to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t (THeads
-(Flat Appl) (TCons t1 t2) t3)) H2 (THead (Flat Appl) t u2) (pr3_thin_dx c
-(THeads (Flat Appl) (TCons t1 t2) t3) u2 (pr3_iso_appls_cast c u t3 (TCons t1
-t2) u2 H11 H12) t Appl))))))))) H7)))))) H3))))))))))) t0))) vs)).
+\def H_x in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads (Flat
+Appl) t2 t3))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2)
+(THead (Flat Cast) u t3)))) (\lambda (_: (sn3 c t)).(\lambda (H5: (sn3 c
+(THead (Flat Appl) t1 (THeads (Flat Appl) t2 t3)))).(let H6 \def H5 in (let
+H_x0 \def (sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) u) H1) in
+(let H7 \def H_x0 in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads
+(Flat Appl) t2 u))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1
+t2) (THead (Flat Cast) u t3)))) (\lambda (H8: (sn3 c t)).(\lambda (H9: (sn3 c
+(THead (Flat Appl) t1 (THeads (Flat Appl) t2 u)))).(let H10 \def H9 in
+(sn3_appl_appls t1 (THead (Flat Cast) u t3) t2 c (H0 u H10 t3 H6) t H8
+(\lambda (u2: T).(\lambda (H11: (pr3 c (THeads (Flat Appl) (TCons t1 t2)
+(THead (Flat Cast) u t3)) u2)).(\lambda (H12: (((iso (THeads (Flat Appl)
+(TCons t1 t2) (THead (Flat Cast) u t3)) u2) \to (\forall (P:
+Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t (THeads (Flat Appl) (TCons
+t1 t2) t3)) H2 (THead (Flat Appl) t u2) (pr3_thin_dx c (THeads (Flat Appl)
+(TCons t1 t2) t3) u2 (pr3_iso_appls_cast c u t3 (TCons t1 t2) u2 H11 H12) t
+Appl))))))))) H7)))))) H3))))))))))) t0))) vs)).
theorem sn3_appls_bind:
\forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u:
v) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))).(let H_x \def
(sn3_gen_flat Appl (CHead c (Bind b) u) (lift (S O) O v) (THeads (Flat Appl)
(lifts (S O) O (TCons t t0)) t1) H3) in (let H4 \def H_x in (land_ind (sn3
-(CHead c (Bind b) u) (lift (S O) O v)) (sn3 (CHead c (Bind b) u) (THeads
-(Flat Appl) (lifts (S O) O (TCons t t0)) t1)) (sn3 c (THead (Flat Appl) v
-(THeads (Flat Appl) (TCons t t0) (THead (Bind b) u t1)))) (\lambda (H5: (sn3
-(CHead c (Bind b) u) (lift (S O) O v))).(\lambda (H6: (sn3 (CHead c (Bind b)
-u) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1))).(let H_y \def
-(sn3_gen_lift (CHead c (Bind b) u) v (S O) O H5 c) in (sn3_appl_appls t
-(THead (Bind b) u t1) t0 c (H2 t1 H6) v (H_y (drop_drop (Bind b) O c c
-(drop_refl c) u)) (\lambda (u2: T).(\lambda (H7: (pr3 c (THeads (Flat Appl)
-(TCons t t0) (THead (Bind b) u t1)) u2)).(\lambda (H8: (((iso (THeads (Flat
-Appl) (TCons t t0) (THead (Bind b) u t1)) u2) \to (\forall (P:
-Prop).P)))).(let H9 \def (pr3_iso_appls_bind b H (TCons t t0) u t1 c u2 H7
-H8) in (sn3_pr3_trans c (THead (Flat Appl) v (THead (Bind b) u (THeads (Flat
-Appl) (lifts (S O) O (TCons t t0)) t1))) (sn3_appl_bind b H c u H0 (THeads
-(Flat Appl) (lifts (S O) O (TCons t t0)) t1) v H3) (THead (Flat Appl) v u2)
-(pr3_flat c v v (pr3_refl c v) (THead (Bind b) u (THeads (Flat Appl) (lifts
-(S O) O (TCons t t0)) t1)) u2 H9 Appl)))))))))) H4))))))))) vs0))) vs)))))).
+(CHead c (Bind b) u) (lift (S O) O v)) (sn3 (CHead c (Bind b) u) (THead (Flat
+Appl) (lift (S O) O t) (THeads (Flat Appl) (lifts (S O) O t0) t1))) (sn3 c
+(THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u
+t1)))) (\lambda (H5: (sn3 (CHead c (Bind b) u) (lift (S O) O v))).(\lambda
+(H6: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O t) (THeads
+(Flat Appl) (lifts (S O) O t0) t1)))).(let H_y \def (sn3_gen_lift (CHead c
+(Bind b) u) v (S O) O H5 c) in (sn3_appl_appls t (THead (Bind b) u t1) t0 c
+(H2 t1 H6) v (H_y (drop_drop (Bind b) O c c (drop_refl c) u)) (\lambda (u2:
+T).(\lambda (H7: (pr3 c (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u
+t1)) u2)).(\lambda (H8: (((iso (THeads (Flat Appl) (TCons t t0) (THead (Bind
+b) u t1)) u2) \to (\forall (P: Prop).P)))).(let H9 \def (pr3_iso_appls_bind b
+H (TCons t t0) u t1 c u2 H7 H8) in (sn3_pr3_trans c (THead (Flat Appl) v
+(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))
+(sn3_appl_bind b H c u H0 (THeads (Flat Appl) (lifts (S O) O (TCons t t0))
+t1) v H3) (THead (Flat Appl) v u2) (pr3_flat c v v (pr3_refl c v) (THead
+(Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) u2 H9
+Appl)))))))))) H4))))))))) vs0))) vs)))))).
theorem sn3_appls_beta:
\forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (us: TList).((sn3 c
(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t))))).(\lambda (w:
T).(\lambda (H2: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THeads
(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t)) H1) in (let H3 \def H_x in
-(land_ind (sn3 c u) (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Bind
-Abbr) v t))) (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1)
-(THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H4: (sn3 c
-u)).(\lambda (H5: (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr)
-v t)))).(sn3_appl_appls t0 (THead (Flat Appl) v (THead (Bind Abst) w t)) t1 c
-(H0 H5 w H2) u H4 (\lambda (u2: T).(\lambda (H6: (pr3 c (THeads (Flat Appl)
-(TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w t))) u2)).(\lambda
-(H7: (((iso (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead
-(Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8 \def
-(pr3_iso_appls_beta (TCons t0 t1) v w t c u2 H6 H7) in (sn3_pr3_trans c
+(land_ind (sn3 c u) (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
+(THead (Bind Abbr) v t)))) (sn3 c (THead (Flat Appl) u (THeads (Flat Appl)
+(TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H4:
+(sn3 c u)).(\lambda (H5: (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
+(THead (Bind Abbr) v t))))).(sn3_appl_appls t0 (THead (Flat Appl) v (THead
+(Bind Abst) w t)) t1 c (H0 H5 w H2) u H4 (\lambda (u2: T).(\lambda (H6: (pr3
+c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w
+t))) u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t0 t1) (THead (Flat
+Appl) v (THead (Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8
+\def (pr3_iso_appls_beta (TCons t0 t1) v w t c u2 H6 H7) in (sn3_pr3_trans c
(THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v
t))) H1 (THead (Flat Appl) u u2) (pr3_thin_dx c (THeads (Flat Appl) (TCons t0
t1) (THead (Bind Abbr) v t)) u2 H8 u Appl))))))))) H3)))))))))) us0))) us)))).
(H2: (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (lift (S i)
O w))))).(let H_x \def (sn3_gen_flat Appl c v (THeads (Flat Appl) (TCons t
t0) (lift (S i) O w)) H2) in (let H3 \def H_x in (land_ind (sn3 c v) (sn3 c
-(THeads (Flat Appl) (TCons t t0) (lift (S i) O w))) (sn3 c (THead (Flat Appl)
-v (THeads (Flat Appl) (TCons t t0) (TLRef i)))) (\lambda (H4: (sn3 c
-v)).(\lambda (H5: (sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i) O
-w)))).(sn3_appl_appls t (TLRef i) t0 c (H1 H5) v H4 (\lambda (u2: T).(\lambda
-(H6: (pr3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2)).(\lambda (H7:
-(((iso (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2) \to (\forall (P:
-Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) v (THeads (Flat Appl) (TCons
-t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2) (pr3_thin_dx c (THeads
-(Flat Appl) (TCons t t0) (lift (S i) O w)) u2 (pr3_iso_appls_abbr c d w i H
-(TCons t t0) u2 H6 H7) v Appl)))))))) H3)))))))) vs0))) vs)))))).
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w)))) (sn3 c (THead
+(Flat Appl) v (THeads (Flat Appl) (TCons t t0) (TLRef i)))) (\lambda (H4:
+(sn3 c v)).(\lambda (H5: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t0
+(lift (S i) O w))))).(sn3_appl_appls t (TLRef i) t0 c (H1 H5) v H4 (\lambda
+(u2: T).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t t0) (TLRef i))
+u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2) \to
+(\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) v (THeads (Flat
+Appl) (TCons t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2)
+(pr3_thin_dx c (THeads (Flat Appl) (TCons t t0) (lift (S i) O w)) u2
+(pr3_iso_appls_abbr c d w i H (TCons t t0) u2 H6 H7) v Appl))))))))
+H3)))))))) vs0))) vs)))))).
theorem sns3_lifts:
\forall (c: C).(\forall (d: C).(\forall (h: nat).(\forall (i: nat).((drop h
H1) in (let H2 \def H_x in (ex_ind T (\lambda (u0: T).(eq T (TLRef j) (lift
(S i) O u0))) (lt i j) (\lambda (x: T).(\lambda (H3: (eq T (TLRef j) (lift (S
i) O x))).(let H_x0 \def (lift_gen_lref x O (S i) j H3) in (let H4 \def H_x0
-in (or_ind (land (lt j O) (eq T x (TLRef j))) (land (le (plus O (S i)) j) (eq
-T x (TLRef (minus j (S i))))) (lt i j) (\lambda (H5: (land (lt j O) (eq T x
+in (or_ind (land (lt j O) (eq T x (TLRef j))) (land (le (S i) j) (eq T x
+(TLRef (minus j (S i))))) (lt i j) (\lambda (H5: (land (lt j O) (eq T x
(TLRef j)))).(land_ind (lt j O) (eq T x (TLRef j)) (lt i j) (\lambda (H6: (lt
j O)).(\lambda (_: (eq T x (TLRef j))).(lt_x_O j H6 (lt i j)))) H5)) (\lambda
-(H5: (land (le (plus O (S i)) j) (eq T x (TLRef (minus j (S i)))))).(land_ind
-(le (plus O (S i)) j) (eq T x (TLRef (minus j (S i)))) (lt i j) (\lambda (H6:
-(le (plus O (S i)) j)).(\lambda (_: (eq T x (TLRef (minus j (S i))))).H6))
-H5)) H4))))) H2))))))))).
+(H5: (land (le (S i) j) (eq T x (TLRef (minus j (S i)))))).(land_ind (le (S
+i) j) (eq T x (TLRef (minus j (S i)))) (lt i j) (\lambda (H6: (le (S i)
+j)).(\lambda (_: (eq T x (TLRef (minus j (S i))))).H6)) H5)) H4)))))
+H2))))))))).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/G/defs.ma".
+
+include "LambdaDelta-1/getl/defs.ma".
+
+inductive sty0 (g: G): C \to (T \to (T \to Prop)) \def
+| sty0_sort: \forall (c: C).(\forall (n: nat).(sty0 g c (TSort n) (TSort
+(next g n))))
+| sty0_abbr: \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 (sty0 g c (TLRef i) (lift (S i) O w))))))))
+| sty0_abst: \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 (sty0 g c (TLRef i) (lift (S i) O v))))))))
+| sty0_bind: \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 (sty0 g c (THead
+(Bind b) v t1) (THead (Bind b) v t2)))))))
+| sty0_appl: \forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2:
+T).((sty0 g c t1 t2) \to (sty0 g c (THead (Flat Appl) v t1) (THead (Flat
+Appl) v t2))))))
+| sty0_cast: \forall (c: C).(\forall (v1: T).(\forall (v2: T).((sty0 g c v1
+v2) \to (\forall (t1: T).(\forall (t2: T).((sty0 g c t1 t2) \to (sty0 g c
+(THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t2)))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/sty0/defs.ma".
+
+theorem 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
+ \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda
+(H: (sty0 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(sty0 g c
+t x)) (\lambda (_: T).(eq T x (TSort (next g n)))) (\lambda (y: T).(\lambda
+(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
+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 _)
+\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))))).
+
+theorem 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:
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(t: T).(eq T x (lift (S n) O t)))))) (ex3_3 C T T (\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).(sty0 g e u t)))) (\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq T x (lift (S n) O u)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda
+(H: (sty0 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(sty0 g c
+t x)) (\lambda (_: T).(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: T).(\lambda (t0: T).(sty0 g e u t0)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(eq T x (lift (S n) O t0)))))) (ex3_3 C T
+T (\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).(sty0 g e u
+t0)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T x (lift (S n) O
+u)))))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g (\lambda
+(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (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 (t1: T).(sty0 g e u
+t1)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(eq T t0 (lift (S n)
+O t1)))))) (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
+(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:
+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 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
+(\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:
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq T (lift (S n0) O v) (lift (S n) O u)))))))) (or_intror (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 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:
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq T (lift (S n) O v) (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
+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 v)
+(lift (S n) O u))))) d v w H6 H2 (refl_equal T (lift (S n) O v)))) i
+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 (TLRef n)) \to (or (ex3_3 C T T (\lambda (e:
+C).(\lambda (u: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) v) (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 (CHead c0 (Bind b) v) (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 (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 _ _ _)
+\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 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))))).
+
+theorem 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
+(Bind b) u t2))))))))))
+\def
+ \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1:
+T).(\lambda (x: T).(\lambda (H: (sty0 g c (THead (Bind b) u t1)
+x)).(insert_eq T (THead (Bind b) u t1) (\lambda (t: T).(sty0 g c t x))
+(\lambda (_: T).(ex2 T (\lambda (t2: T).(sty0 g (CHead c (Bind b) u) t1 t2))
+(\lambda (t2: T).(eq T x (THead (Bind b) u t2))))) (\lambda (y: T).(\lambda
+(H0: (sty0 g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda
+(t0: T).((eq T t (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 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 _)
+\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
+w) (THead (Bind b) 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 (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 (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))))))).
+
+theorem 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)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (x:
+T).(\lambda (H: (sty0 g c (THead (Flat Appl) u t1) x)).(insert_eq T (THead
+(Flat Appl) u t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_: T).(ex2 T
+(\lambda (t2: T).(sty0 g c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat
+Appl) u t2))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g
+(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl)
+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
+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 _)
+\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)))))).
+
+theorem 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
+g c t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) v2
+t2))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (t1: T).(\lambda
+(x: T).(\lambda (H: (sty0 g c (THead (Flat Cast) v1 t1) x)).(insert_eq T
+(THead (Flat Cast) v1 t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_:
+T).(ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda
+(_: T).(\lambda (t2: T).(sty0 g c t1 t2))) (\lambda (v2: T).(\lambda (t2:
+T).(eq T x (THead (Flat Cast) v2 t2)))))) (\lambda (y: T).(\lambda (H0: (sty0
+g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq
+T t (THead (Flat Cast) v1 t1)) \to (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 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)))
+(\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
+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 (_:
+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)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/sty0/defs.ma".
+
+include "LambdaDelta-1/getl/drop.ma".
+
+theorem sty0_lift:
+ \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty0 g e
+t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c
+e) \to (sty0 g c (lift h d t1) (lift h d t2))))))))))
+\def
+ \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (sty0 g e t1 t2)).(sty0_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
+(t0: T).(\forall (c0: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 c)
+\to (sty0 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda
+(n: nat).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_:
+(drop h d c0 c)).(eq_ind_r T (TSort n) (\lambda (t: T).(sty0 g c0 t (lift h d
+(TSort (next g n))))) (eq_ind_r T (TSort (next g n)) (\lambda (t: T).(sty0 g
+c0 (TSort n) t)) (sty0_sort g c0 n) (lift h d (TSort (next g n))) (lift_sort
+(next g n) h d)) (lift h d (TSort n)) (lift_sort n h d)))))))) (\lambda (c:
+C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c
+(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v
+w)).(\lambda (H2: ((\forall (c0: C).(\forall (h: nat).(\forall (d0:
+nat).((drop h d0 c0 d) \to (sty0 g c0 (lift h d0 v) (lift h d0
+w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3:
+(drop h d0 c0 c)).(lt_le_e i d0 (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0
+(lift (S i) O w))) (\lambda (H4: (lt i d0)).(let H5 \def (drop_getl_trans_le
+i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) v) H0)
+in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i O c0 e0)))
+(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) (\lambda (_:
+C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) v)))) (sty0 g c0 (lift h
+d0 (TLRef i)) (lift h d0 (lift (S i) O w))) (\lambda (x0: C).(\lambda (x1:
+C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h (minus d0 i) x0
+x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) v))).(let H9 \def (eq_ind
+nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i)))
+(minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i))
+H9 Abbr d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind
+Abbr) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h (minus d0 (S
+i)) c1 d)) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O w)))
+(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus
+d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x d)).(eq_ind_r T
+(TLRef i) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind
+nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(sty0 g c0 (TLRef i)
+(lift h n (lift (S i) O w)))) (eq_ind_r T (lift (S i) O (lift h (minus d0 (S
+i)) w)) (\lambda (t: T).(sty0 g c0 (TLRef i) t)) (eq_ind nat d0 (\lambda (_:
+nat).(sty0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i)) w))))
+(sty0_abbr g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead x
+(Bind Abbr) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S i))
+w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i)))
+(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S
+i) O w)) (lift_d w h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0
+(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0
+H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i
+h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind nat
+(S i) (\lambda (_: nat).(sty0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i)
+O w)))) (eq_ind_r T (lift (plus h (S i)) O w) (\lambda (t: T).(sty0 g c0
+(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(sty0 g
+c0 (TLRef (plus i h)) (lift n O w))) (sty0_abbr g c0 d v (plus i h)
+(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abbr) v) H0 H4) w H1) (plus
+h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O w)) (lift_free w (S i)
+h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O)
+i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus
+i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0
+H4)))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda
+(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) v))).(\lambda (w:
+T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: ((\forall (c0: C).(\forall (h:
+nat).(\forall (d0: nat).((drop h d0 c0 d) \to (sty0 g c0 (lift h d0 v) (lift
+h d0 w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda
+(H3: (drop h d0 c0 c)).(lt_le_e i d0 (sty0 g c0 (lift h d0 (TLRef i)) (lift h
+d0 (lift (S i) O v))) (\lambda (H4: (lt i d0)).(let H5 \def
+(drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d
+(Bind Abst) v) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i
+O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1)))
+(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) v)))) (sty0 g
+c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O v))) (\lambda (x0:
+C).(\lambda (x1: C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h
+(minus d0 i) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) v))).(let
+H9 \def (eq_ind nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S
+(minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0
+h (minus d0 (S i)) H9 Abst d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0
+(CHead c1 (Bind Abst) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h
+(minus d0 (S i)) c1 d)) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S
+i) O v))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift
+h (minus d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x
+d)).(eq_ind_r T (TLRef i) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i)
+O v)))) (eq_ind nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(sty0 g
+c0 (TLRef i) (lift h n (lift (S i) O v)))) (eq_ind_r T (lift (S i) O (lift h
+(minus d0 (S i)) v)) (\lambda (t: T).(sty0 g c0 (TLRef i) t)) (eq_ind nat d0
+(\lambda (_: nat).(sty0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i))
+v)))) (sty0_abst g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead
+x (Bind Abst) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S
+i)) w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i)))
+(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S
+i) O v)) (lift_d v h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0
+(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0
+H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i
+h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O v)))) (eq_ind nat
+(S i) (\lambda (_: nat).(sty0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i)
+O v)))) (eq_ind_r T (lift (plus h (S i)) O v) (\lambda (t: T).(sty0 g c0
+(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(sty0 g
+c0 (TLRef (plus i h)) (lift n O v))) (sty0_abst g c0 d v (plus i h)
+(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abst) v) H0 H4) w H1) (plus
+h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O v)) (lift_free v (S i)
+h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O)
+i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus
+i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0
+H4)))))))))))))))) (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda
+(t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g (CHead c (Bind b) v) t3
+t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c0 (CHead c (Bind b) v)) \to (sty0 g c0 (lift h d t3) (lift h
+d t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H2: (drop h d c0 c)).(eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s
+(Bind b) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Bind b) v
+t4)))) (eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s (Bind b) d) t4))
+(\lambda (t: T).(sty0 g c0 (THead (Bind b) (lift h d v) (lift h (s (Bind b)
+d) t3)) t)) (sty0_bind g b c0 (lift h d v) (lift h (S d) t3) (lift h (S d)
+t4) (H1 (CHead c0 (Bind b) (lift h d v)) h (S d) (drop_skip_bind h d c0 c H2
+b v))) (lift h d (THead (Bind b) v t4)) (lift_head (Bind b) v t4 h d)) (lift
+h d (THead (Bind b) v t3)) (lift_head (Bind b) v t3 h d))))))))))))) (\lambda
+(c: C).(\lambda (v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g
+c t3 t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d
+t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2:
+(drop h d c0 c)).(eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat
+Appl) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Appl) v
+t4)))) (eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat Appl) d)
+t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Appl) (lift h d v) (lift h (s
+(Flat Appl) d) t3)) t)) (sty0_appl g c0 (lift h d v) (lift h (s (Flat Appl)
+d) t3) (lift h (s (Flat Appl) d) t4) (H1 c0 h (s (Flat Appl) d) H2)) (lift h
+d (THead (Flat Appl) v t4)) (lift_head (Flat Appl) v t4 h d)) (lift h d
+(THead (Flat Appl) v t3)) (lift_head (Flat Appl) v t3 h d))))))))))))
+(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c v1
+v2)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c0 c) \to (sty0 g c0 (lift h d v1) (lift h d
+v2)))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g c t3
+t4)).(\lambda (H3: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d
+t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4:
+(drop h d c0 c)).(eq_ind_r T (THead (Flat Cast) (lift h d v1) (lift h (s
+(Flat Cast) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Cast)
+v2 t4)))) (eq_ind_r T (THead (Flat Cast) (lift h d v2) (lift h (s (Flat Cast)
+d) t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Cast) (lift h d v1) (lift h
+(s (Flat Cast) d) t3)) t)) (sty0_cast g c0 (lift h d v1) (lift h d v2) (H1 c0
+h d H4) (lift h (s (Flat Cast) d) t3) (lift h (s (Flat Cast) d) t4) (H3 c0 h
+(s (Flat Cast) d) H4)) (lift h d (THead (Flat Cast) v2 t4)) (lift_head (Flat
+Cast) v2 t4 h d)) (lift h d (THead (Flat Cast) v1 t3)) (lift_head (Flat Cast)
+v1 t3 h d))))))))))))))) e t1 t2 H))))).
+
+theorem sty0_correct:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c
+t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
+(sty0 g c t1 t)).(sty0_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (t2:
+T).(ex T (\lambda (t3: T).(sty0 g c0 t2 t3)))))) (\lambda (c0: C).(\lambda
+(n: nat).(ex_intro T (\lambda (t2: T).(sty0 g c0 (TSort (next g n)) t2))
+(TSort (next g (next g n))) (sty0_sort g c0 (next g n))))) (\lambda (c0:
+C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0
+(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v
+w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g d w t2)))).(let H3 \def H2
+in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex T (\lambda (t2: T).(sty0 g
+c0 (lift (S i) O w) t2))) (\lambda (x: T).(\lambda (H4: (sty0 g d w
+x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O w) t2)) (lift (S i)
+O x) (sty0_lift g d w x H4 c0 (S i) O (getl_drop Abbr c0 d v i H0)))))
+H3)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w:
+T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g
+d w t2)))).(let H3 \def H2 in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex
+T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v) t2))) (\lambda (x: T).(\lambda
+(_: (sty0 g d w x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v)
+t2)) (lift (S i) O w) (sty0_lift g d v w H1 c0 (S i) O (getl_drop Abst c0 d v
+i H0))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v:
+T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g (CHead c0 (Bind b)
+v) t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g (CHead c0 (Bind b) v)
+t3 t4)))).(let H2 \def H1 in (ex_ind T (\lambda (t4: T).(sty0 g (CHead c0
+(Bind b) v) t3 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3)
+t4))) (\lambda (x: T).(\lambda (H3: (sty0 g (CHead c0 (Bind b) v) t3
+x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3) t4)) (THead
+(Bind b) v x) (sty0_bind g b c0 v t3 x H3)))) H2))))))))) (\lambda (c0:
+C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g c0
+t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H2
+\def H1 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4:
+T).(sty0 g c0 (THead (Flat Appl) v t3) t4))) (\lambda (x: T).(\lambda (H3:
+(sty0 g c0 t3 x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Appl)
+v t3) t4)) (THead (Flat Appl) v x) (sty0_appl g c0 v t3 x H3)))) H2))))))))
+(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1
+v2)).(\lambda (H1: (ex T (\lambda (t2: T).(sty0 g c0 v2 t2)))).(\lambda (t2:
+T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 t2 t3)).(\lambda (H3: (ex T
+(\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H4 \def H1 in (ex_ind T (\lambda
+(t4: T).(sty0 g c0 v2 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Flat
+Cast) v2 t3) t4))) (\lambda (x: T).(\lambda (H5: (sty0 g c0 v2 x)).(let H6
+\def H3 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4:
+T).(sty0 g c0 (THead (Flat Cast) v2 t3) t4))) (\lambda (x0: T).(\lambda (H7:
+(sty0 g c0 t3 x0)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Cast)
+v2 t3) t4)) (THead (Flat Cast) x x0) (sty0_cast g c0 v2 x H5 t3 x0 H7))))
+H6)))) H4))))))))))) c t1 t H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/sty1/props.ma".
+
+include "LambdaDelta-1/cnt/props.ma".
+
+theorem sty1_cnt:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c
+t1 t) \to (ex2 T (\lambda (t2: T).(sty1 g c t1 t2)) (\lambda (t2: T).(cnt
+t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
+(sty0 g c t1 t)).(sty0_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
+T).(ex2 T (\lambda (t3: T).(sty1 g c0 t0 t3)) (\lambda (t3: T).(cnt t3))))))
+(\lambda (c0: C).(\lambda (n: nat).(ex_intro2 T (\lambda (t2: T).(sty1 g c0
+(TSort n) t2)) (\lambda (t2: T).(cnt t2)) (TSort (next g n)) (sty1_sty0 g c0
+(TSort n) (TSort (next g n)) (sty0_sort g c0 n)) (cnt_sort (next g n)))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0
+g d v w)).(\lambda (H2: (ex2 T (\lambda (t2: T).(sty1 g d v t2)) (\lambda
+(t2: T).(cnt t2)))).(let H3 \def H2 in (ex2_ind T (\lambda (t2: T).(sty1 g d
+v t2)) (\lambda (t2: T).(cnt t2)) (ex2 T (\lambda (t2: T).(sty1 g c0 (TLRef
+i) t2)) (\lambda (t2: T).(cnt t2))) (\lambda (x: T).(\lambda (H4: (sty1 g d v
+x)).(\lambda (H5: (cnt x)).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 (TLRef i)
+t2)) (\lambda (t2: T).(cnt t2)) (lift (S i) O x) (sty1_abbr g c0 d v i H0 x
+H4) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
+Abst) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: (ex2 T
+(\lambda (t2: T).(sty1 g d v t2)) (\lambda (t2: T).(cnt t2)))).(let H3 \def
+H2 in (ex2_ind T (\lambda (t2: T).(sty1 g d v t2)) (\lambda (t2: T).(cnt t2))
+(ex2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) t2)) (\lambda (t2: T).(cnt t2)))
+(\lambda (x: T).(\lambda (H4: (sty1 g d v x)).(\lambda (H5: (cnt
+x)).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) t2)) (\lambda (t2:
+T).(cnt t2)) (lift (S i) O x) (sty1_trans g c0 (TLRef i) (lift (S i) O v)
+(sty1_sty0 g c0 (TLRef i) (lift (S i) O v) (sty0_abst g c0 d v i H0 w H1))
+(lift (S i) O x) (sty1_lift g d v x H4 c0 (S i) O (getl_drop Abst c0 d v i
+H0))) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (b: B).(\lambda (c0:
+C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g
+(CHead c0 (Bind b) v) t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4: T).(sty1 g
+(CHead c0 (Bind b) v) t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in
+(ex2_ind T (\lambda (t4: T).(sty1 g (CHead c0 (Bind b) v) t2 t4)) (\lambda
+(t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Bind b) v t2)
+t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (sty1 g (CHead
+c0 (Bind b) v) t2 x)).(\lambda (H4: (cnt x)).(ex_intro2 T (\lambda (t4:
+T).(sty1 g c0 (THead (Bind b) v t2) t4)) (\lambda (t4: T).(cnt t4)) (THead
+(Bind b) v x) (sty1_bind g b c0 v t2 x H3) (cnt_head x H4 (Bind b) v)))))
+H2))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3:
+T).(\lambda (_: (sty0 g c0 t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4:
+T).(sty1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in
+(ex2_ind T (\lambda (t4: T).(sty1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4))
+(ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Appl) v t2) t4)) (\lambda
+(t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (sty1 g c0 t2 x)).(\lambda
+(H4: (cnt x)).(ex_intro2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Appl) v
+t2) t4)) (\lambda (t4: T).(cnt t4)) (THead (Flat Appl) v x) (sty1_appl g c0 v
+t2 x H3) (cnt_head x H4 (Flat Appl) v))))) H2)))))))) (\lambda (c0:
+C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (sty0 g c0 v1
+v2)).(\lambda (_: (ex2 T (\lambda (t2: T).(sty1 g c0 v1 t2)) (\lambda (t2:
+T).(cnt t2)))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 t2
+t3)).(\lambda (H3: (ex2 T (\lambda (t4: T).(sty1 g c0 t2 t4)) (\lambda (t4:
+T).(cnt t4)))).(let H4 \def H3 in (ex2_ind T (\lambda (t4: T).(sty1 g c0 t2
+t4)) (\lambda (t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead
+(Flat Cast) v1 t2) t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda
+(H5: (sty1 g c0 t2 x)).(\lambda (H6: (cnt x)).(let H_x \def (sty1_cast2 g c0
+t2 x H5 v1 v2 H0) in (let H7 \def H_x in (ex2_ind T (\lambda (v3: T).(sty1 g
+c0 v1 v3)) (\lambda (v3: T).(sty1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat
+Cast) v3 x))) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Cast) v1 t2)
+t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x0: T).(\lambda (_: (sty1 g c0 v1
+x0)).(\lambda (H9: (sty1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat Cast) x0
+x))).(ex_intro2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Cast) v1 t2) t4))
+(\lambda (t4: T).(cnt t4)) (THead (Flat Cast) x0 x) H9 (cnt_head x H6 (Flat
+Cast) x0))))) H7)))))) H4))))))))))) c t1 t H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/sty0/defs.ma".
+
+inductive sty1 (g: G) (c: C) (t1: T): T \to Prop \def
+| sty1_sty0: \forall (t2: T).((sty0 g c t1 t2) \to (sty1 g c t1 t2))
+| sty1_sing: \forall (t: T).((sty1 g c t1 t) \to (\forall (t2: T).((sty0 g c
+t t2) \to (sty1 g c t1 t2)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/sty1/defs.ma".
+
+include "LambdaDelta-1/sty0/props.ma".
+
+theorem sty1_trans:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty1 g c
+t1 t) \to (\forall (t2: T).((sty1 g c t t2) \to (sty1 g c t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
+(sty1 g c t1 t)).(\lambda (t2: T).(\lambda (H0: (sty1 g c t t2)).(sty1_ind g
+c t (\lambda (t0: T).(sty1 g c t1 t0)) (\lambda (t3: T).(\lambda (H1: (sty0 g
+c t t3)).(sty1_sing g c t1 t H t3 H1))) (\lambda (t0: T).(\lambda (_: (sty1 g
+c t t0)).(\lambda (H2: (sty1 g c t1 t0)).(\lambda (t3: T).(\lambda (H3: (sty0
+g c t0 t3)).(sty1_sing g c t1 t0 H2 t3 H3)))))) t2 H0))))))).
+
+theorem sty1_bind:
+ \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1:
+T).(\forall (t2: T).((sty1 g (CHead c (Bind b) v) t1 t2) \to (sty1 g c (THead
+(Bind b) v t1) (THead (Bind b) v t2))))))))
+\def
+ \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H: (sty1 g (CHead c (Bind b) v) t1
+t2)).(sty1_ind g (CHead c (Bind b) v) t1 (\lambda (t: T).(sty1 g c (THead
+(Bind b) v t1) (THead (Bind b) v t))) (\lambda (t3: T).(\lambda (H0: (sty0 g
+(CHead c (Bind b) v) t1 t3)).(sty1_sty0 g c (THead (Bind b) v t1) (THead
+(Bind b) v t3) (sty0_bind g b c v t1 t3 H0)))) (\lambda (t: T).(\lambda (_:
+(sty1 g (CHead c (Bind b) v) t1 t)).(\lambda (H1: (sty1 g c (THead (Bind b) v
+t1) (THead (Bind b) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g (CHead c
+(Bind b) v) t t3)).(sty1_sing g c (THead (Bind b) v t1) (THead (Bind b) v t)
+H1 (THead (Bind b) v t3) (sty0_bind g b c v t t3 H2))))))) t2 H))))))).
+
+theorem sty1_appl:
+ \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall
+(t2: T).((sty1 g c t1 t2) \to (sty1 g c (THead (Flat Appl) v t1) (THead (Flat
+Appl) v t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (H: (sty1 g c t1 t2)).(sty1_ind g c t1 (\lambda (t: T).(sty1
+g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t))) (\lambda (t3:
+T).(\lambda (H0: (sty0 g c t1 t3)).(sty1_sty0 g c (THead (Flat Appl) v t1)
+(THead (Flat Appl) v t3) (sty0_appl g c v t1 t3 H0)))) (\lambda (t:
+T).(\lambda (_: (sty1 g c t1 t)).(\lambda (H1: (sty1 g c (THead (Flat Appl) v
+t1) (THead (Flat Appl) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g c t
+t3)).(sty1_sing g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t) H1
+(THead (Flat Appl) v t3) (sty0_appl g c v t t3 H2))))))) t2 H)))))).
+
+theorem sty1_lift:
+ \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty1 g e
+t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c
+e) \to (sty1 g c (lift h d t1) (lift h d t2))))))))))
+\def
+ \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (sty1 g e t1 t2)).(sty1_ind g e t1 (\lambda (t: T).(\forall (c:
+C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (sty1 g c (lift h
+d t1) (lift h d t))))))) (\lambda (t3: T).(\lambda (H0: (sty0 g e t1
+t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop
+h d c e)).(sty1_sty0 g c (lift h d t1) (lift h d t3) (sty0_lift g e t1 t3 H0
+c h d H1)))))))) (\lambda (t: T).(\lambda (_: (sty1 g e t1 t)).(\lambda (H1:
+((\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to
+(sty1 g c (lift h d t1) (lift h d t)))))))).(\lambda (t3: T).(\lambda (H2:
+(sty0 g e t t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H3: (drop h d c e)).(sty1_sing g c (lift h d t1) (lift h d t) (H1 c h d H3)
+(lift h d t3) (sty0_lift g e t t3 H2 c h d H3))))))))))) t2 H))))).
+
+theorem sty1_correct:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty1 g c
+t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
+(sty1 g c t1 t)).(sty1_ind g c t1 (\lambda (t0: T).(ex T (\lambda (t2:
+T).(sty0 g c t0 t2)))) (\lambda (t2: T).(\lambda (H0: (sty0 g c t1
+t2)).(sty0_correct g c t1 t2 H0))) (\lambda (t0: T).(\lambda (_: (sty1 g c t1
+t0)).(\lambda (_: (ex T (\lambda (t2: T).(sty0 g c t0 t2)))).(\lambda (t2:
+T).(\lambda (H2: (sty0 g c t0 t2)).(sty0_correct g c t0 t2 H2)))))) t H))))).
+
+theorem sty1_abbr:
+ \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i:
+nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty1 g d v w)
+\to (sty1 g c (TLRef i) (lift (S i) O w)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (w:
+T).(\lambda (H0: (sty1 g d v w)).(sty1_ind g d v (\lambda (t: T).(sty1 g c
+(TLRef i) (lift (S i) O t))) (\lambda (t2: T).(\lambda (H1: (sty0 g d v
+t2)).(sty1_sty0 g c (TLRef i) (lift (S i) O t2) (sty0_abbr g c d v i H t2
+H1)))) (\lambda (t: T).(\lambda (_: (sty1 g d v t)).(\lambda (H2: (sty1 g c
+(TLRef i) (lift (S i) O t))).(\lambda (t2: T).(\lambda (H3: (sty0 g d t
+t2)).(sty1_sing g c (TLRef i) (lift (S i) O t) H2 (lift (S i) O t2)
+(sty0_lift g d t t2 H3 c (S i) O (getl_drop Abbr c d v i H)))))))) w
+H0)))))))).
+
+theorem sty1_cast2:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((sty1 g c
+t1 t2) \to (\forall (v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to (ex2 T
+(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat
+Cast) v1 t1) (THead (Flat Cast) v3 t2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (sty1 g c t1 t2)).(sty1_ind g c t1 (\lambda (t: T).(\forall (v1:
+T).(\forall (v2: T).((sty0 g c v1 v2) \to (ex2 T (\lambda (v3: T).(sty1 g c
+v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat
+Cast) v3 t)))))))) (\lambda (t3: T).(\lambda (H0: (sty0 g c t1 t3)).(\lambda
+(v1: T).(\lambda (v2: T).(\lambda (H1: (sty0 g c v1 v2)).(ex_intro2 T
+(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat
+Cast) v1 t1) (THead (Flat Cast) v3 t3))) v2 (sty1_sty0 g c v1 v2 H1)
+(sty1_sty0 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t3) (sty0_cast
+g c v1 v2 H1 t1 t3 H0)))))))) (\lambda (t: T).(\lambda (_: (sty1 g c t1
+t)).(\lambda (H1: ((\forall (v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to
+(ex2 T (\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead
+(Flat Cast) v1 t1) (THead (Flat Cast) v3 t))))))))).(\lambda (t3: T).(\lambda
+(H2: (sty0 g c t t3)).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H3: (sty0 g
+c v1 v2)).(let H_x \def (H1 v1 v2 H3) in (let H4 \def H_x in (ex2_ind T
+(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat
+Cast) v1 t1) (THead (Flat Cast) v3 t))) (ex2 T (\lambda (v3: T).(sty1 g c v1
+v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast)
+v3 t3)))) (\lambda (x: T).(\lambda (H5: (sty1 g c v1 x)).(\lambda (H6: (sty1
+g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) x t))).(let H_x0 \def
+(sty1_correct g c v1 x H5) in (let H7 \def H_x0 in (ex_ind T (\lambda (t4:
+T).(sty0 g c x t4)) (ex2 T (\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3:
+T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v3 t3)))) (\lambda
+(x0: T).(\lambda (H8: (sty0 g c x x0)).(ex_intro2 T (\lambda (v3: T).(sty1 g
+c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat
+Cast) v3 t3))) x0 (sty1_sing g c v1 x H5 x0 H8) (sty1_sing g c (THead (Flat
+Cast) v1 t1) (THead (Flat Cast) x t) H6 (THead (Flat Cast) x0 t3) (sty0_cast
+g c x x0 H8 t t3 H2))))) H7)))))) H4))))))))))) t2 H))))).
+
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/G/defs.ma".
-
-include "LambdaDelta-1/getl/defs.ma".
-
-inductive tau0 (g: G): C \to (T \to (T \to Prop)) \def
-| tau0_sort: \forall (c: C).(\forall (n: nat).(tau0 g c (TSort n) (TSort
-(next g n))))
-| tau0_abbr: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i:
-nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((tau0 g d v w)
-\to (tau0 g c (TLRef i) (lift (S i) O w))))))))
-| tau0_abst: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i:
-nat).((getl i c (CHead d (Bind Abst) v)) \to (\forall (w: T).((tau0 g d v w)
-\to (tau0 g c (TLRef i) (lift (S i) O v))))))))
-| tau0_bind: \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1:
-T).(\forall (t2: T).((tau0 g (CHead c (Bind b) v) t1 t2) \to (tau0 g c (THead
-(Bind b) v t1) (THead (Bind b) v t2)))))))
-| tau0_appl: \forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2:
-T).((tau0 g c t1 t2) \to (tau0 g c (THead (Flat Appl) v t1) (THead (Flat
-Appl) v t2))))))
-| tau0_cast: \forall (c: C).(\forall (v1: T).(\forall (v2: T).((tau0 g c v1
-v2) \to (\forall (t1: T).(\forall (t2: T).((tau0 g c t1 t2) \to (tau0 g c
-(THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t2)))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/tau0/defs.ma".
-
-theorem tau0_gen_sort:
- \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((tau0 g c
-(TSort n) x) \to (eq T x (TSort (next g n)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda
-(H: (tau0 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(tau0 g c
-t x)) (\lambda (_: T).(eq T x (TSort (next g n)))) (\lambda (y: T).(\lambda
-(H0: (tau0 g c y x)).(tau0_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 (_: (tau0 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 (_: (tau0 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 (_: (tau0 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 _)
-\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
-(_: (tau0 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 (_: (tau0 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 (_: (tau0 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))))).
-
-theorem tau0_gen_lref:
- \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((tau0 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:
-T).(\lambda (t: T).(tau0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(t: T).(eq T x (lift (S n) O t)))))) (ex3_3 C T T (\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).(tau0 g e u t)))) (\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq T x (lift (S n) O u)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda
-(H: (tau0 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(tau0 g c
-t x)) (\lambda (_: T).(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: T).(\lambda (t0: T).(tau0 g e u t0)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (t0: T).(eq T x (lift (S n) O t0)))))) (ex3_3 C T
-T (\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).(tau0 g e u
-t0)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T x (lift (S n) O
-u)))))))) (\lambda (y: T).(\lambda (H0: (tau0 g c y x)).(tau0_ind g (\lambda
-(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (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 (t1: T).(tau0 g e u
-t1)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(eq T t0 (lift (S n)
-O t1)))))) (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
-(t1: T).(tau0 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).(tau0 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).(tau0 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: (tau0 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).(tau0 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).(tau0 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:
-T).(\lambda (t: T).(tau0 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).(tau0 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).(tau0 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).(tau0 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).(tau0 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: (tau0 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).(tau0 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).(tau0 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
-(\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).(tau0 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:
-T).(\lambda (t: T).(tau0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(eq T (lift (S n0) O v) (lift (S n) O u)))))))) (or_intror (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).(tau0 g e u
-t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n) 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:
-T).(\lambda (t: T).(tau0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(eq T (lift (S n) O v) (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
-Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(tau0 g e u
-t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n) O v)
-(lift (S n) O u))))) d v w H6 H2 (refl_equal T (lift (S n) O v)))) i
-H5)))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (_: (tau0 g (CHead c0 (Bind b) v) 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 (CHead c0 (Bind b) v) (CHead e
-(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(tau0 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 (CHead c0 (Bind b) v) (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda
-(u: T).(\lambda (t: T).(tau0 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 _ _ _)
-\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).(tau0 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).(tau0 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
-(_: (tau0 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).(tau0 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).(tau0 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).(tau0 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).(tau0 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 (_: (tau0 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).(tau0 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).(tau0 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 (_: (tau0 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).(tau0 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).(tau0 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).(tau0 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).(tau0 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 tau0_gen_bind:
- \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1:
-T).(\forall (x: T).((tau0 g c (THead (Bind b) u t1) x) \to (ex2 T (\lambda
-(t2: T).(tau0 g (CHead c (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T x (THead
-(Bind b) u t2))))))))))
-\def
- \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1:
-T).(\lambda (x: T).(\lambda (H: (tau0 g c (THead (Bind b) u t1)
-x)).(insert_eq T (THead (Bind b) u t1) (\lambda (t: T).(tau0 g c t x))
-(\lambda (_: T).(ex2 T (\lambda (t2: T).(tau0 g (CHead c (Bind b) u) t1 t2))
-(\lambda (t2: T).(eq T x (THead (Bind b) u t2))))) (\lambda (y: T).(\lambda
-(H0: (tau0 g c y x)).(tau0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda
-(t0: T).((eq T t (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(tau0 g
-(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).(tau0 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 (_: (tau0 g d v w)).(\lambda (_: (((eq T v
-(THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(tau0 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).(tau0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O
-w) (THead (Bind b) 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 (_: (tau0 g d v w)).(\lambda (_: (((eq T
-v (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(tau0 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).(tau0 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: (tau0 g
-(CHead c0 (Bind b0) v) t0 t2)).(\lambda (H2: (((eq T t0 (THead (Bind b) u
-t1)) \to (ex2 T (\lambda (t3: T).(tau0 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).(tau0 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).(tau0 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).(tau0 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).(tau0 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).(tau0 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).(tau0 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).(tau0 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).(tau0 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).(tau0 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 (_: (tau0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u
-t1)) \to (ex2 T (\lambda (t3: T).(tau0 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).(tau0 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 (_: (tau0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u
-t1)) \to (ex2 T (\lambda (t2: T).(tau0 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 (_: (tau0 g c0 t0 t2)).(\lambda (_: (((eq T t0
-(THead (Bind b) u t1)) \to (ex2 T (\lambda (t3: T).(tau0 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).(tau0 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 tau0_gen_appl:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (x:
-T).((tau0 g c (THead (Flat Appl) u t1) x) \to (ex2 T (\lambda (t2: T).(tau0 g
-c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat Appl) u t2)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (x:
-T).(\lambda (H: (tau0 g c (THead (Flat Appl) u t1) x)).(insert_eq T (THead
-(Flat Appl) u t1) (\lambda (t: T).(tau0 g c t x)) (\lambda (_: T).(ex2 T
-(\lambda (t2: T).(tau0 g c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat
-Appl) u t2))))) (\lambda (y: T).(\lambda (H0: (tau0 g c y x)).(tau0_ind g
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl)
-u t1)) \to (ex2 T (\lambda (t2: T).(tau0 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).(tau0 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 (_: (tau0 g d v
-w)).(\lambda (_: (((eq T v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2:
-T).(tau0 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).(tau0 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 (_: (tau0 g d v
-w)).(\lambda (_: (((eq T v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2:
-T).(tau0 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).(tau0 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 (_: (tau0 g (CHead c0 (Bind b) v) t0 t2)).(\lambda (_: (((eq T t0
-(THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(tau0 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).(tau0 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: (tau0 g c0 t0 t2)).(\lambda (H2: (((eq
-T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(tau0 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 _)
-\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).(tau0 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).(tau0 g c0 t t2)) H1 t1 H5) in (eq_ind_r T
-u (\lambda (t: T).(ex2 T (\lambda (t3: T).(tau0 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).(tau0 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 (_: (tau0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Flat Appl)
-u t1)) \to (ex2 T (\lambda (t2: T).(tau0 g c0 t1 t2)) (\lambda (t2: T).(eq T
-v2 (THead (Flat Appl) u t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda
-(_: (tau0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t1)) \to
-(ex2 T (\lambda (t3: T).(tau0 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).(tau0 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 tau0_gen_cast:
- \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (t1: T).(\forall
-(x: T).((tau0 g c (THead (Flat Cast) v1 t1) x) \to (ex3_2 T T (\lambda (v2:
-T).(\lambda (_: T).(tau0 g c v1 v2))) (\lambda (_: T).(\lambda (t2: T).(tau0
-g c t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) v2
-t2))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (t1: T).(\lambda
-(x: T).(\lambda (H: (tau0 g c (THead (Flat Cast) v1 t1) x)).(insert_eq T
-(THead (Flat Cast) v1 t1) (\lambda (t: T).(tau0 g c t x)) (\lambda (_:
-T).(ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(tau0 g c v1 v2))) (\lambda
-(_: T).(\lambda (t2: T).(tau0 g c t1 t2))) (\lambda (v2: T).(\lambda (t2:
-T).(eq T x (THead (Flat Cast) v2 t2)))))) (\lambda (y: T).(\lambda (H0: (tau0
-g c y x)).(tau0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq
-T t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_:
-T).(tau0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(tau0 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).(tau0 g
-c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(tau0 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 (_: (tau0 g d v w)).(\lambda (_: (((eq T v (THead (Flat Cast) v1
-t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(tau0 g d v1 v2)))
-(\lambda (_: T).(\lambda (t2: T).(tau0 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).(tau0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(tau0
-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 (_: (tau0 g d v w)).(\lambda (_: (((eq T
-v (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_:
-T).(tau0 g d v1 v2))) (\lambda (_: T).(\lambda (t2: T).(tau0 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).(tau0 g c0 v1 v2)))
-(\lambda (_: T).(\lambda (t2: T).(tau0 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 (_: (tau0 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).(tau0 g (CHead c0 (Bind b) v) v1 v2)))
-(\lambda (_: T).(\lambda (t3: T).(tau0 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).(tau0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t3:
-T).(tau0 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 (_: (tau0 g c0 t0
-t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T T
-(\lambda (v2: T).(\lambda (_: T).(tau0 g c0 v1 v2))) (\lambda (_: T).(\lambda
-(t3: T).(tau0 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).(tau0 g c0 v1
-v2))) (\lambda (_: T).(\lambda (t3: T).(tau0 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: (tau0 g c0 v0 v2)).(\lambda (H2: (((eq T v0 (THead (Flat
-Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: T).(tau0 g c0 v1
-v3))) (\lambda (_: T).(\lambda (t2: T).(tau0 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: (tau0 g c0 t0 t2)).(\lambda (H4: (((eq T t0
-(THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_:
-T).(tau0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(tau0 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).(tau0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(tau0
-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).(tau0 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).(tau0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(tau0 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).(tau0 g c0 t v2)) H1 v1
-H8) in (ex3_2_intro T T (\lambda (v3: T).(\lambda (_: T).(tau0 g c0 v1 v3)))
-(\lambda (_: T).(\lambda (t3: T).(tau0 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)))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/tau0/defs.ma".
-
-include "LambdaDelta-1/getl/drop.ma".
-
-theorem tau0_lift:
- \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((tau0 g e
-t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c
-e) \to (tau0 g c (lift h d t1) (lift h d t2))))))))))
-\def
- \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (tau0 g e t1 t2)).(tau0_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
-(t0: T).(\forall (c0: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 c)
-\to (tau0 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda
-(n: nat).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_:
-(drop h d c0 c)).(eq_ind_r T (TSort n) (\lambda (t: T).(tau0 g c0 t (lift h d
-(TSort (next g n))))) (eq_ind_r T (TSort (next g n)) (\lambda (t: T).(tau0 g
-c0 (TSort n) t)) (tau0_sort g c0 n) (lift h d (TSort (next g n))) (lift_sort
-(next g n) h d)) (lift h d (TSort n)) (lift_sort n h d)))))))) (\lambda (c:
-C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c
-(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (H1: (tau0 g d v
-w)).(\lambda (H2: ((\forall (c0: C).(\forall (h: nat).(\forall (d0:
-nat).((drop h d0 c0 d) \to (tau0 g c0 (lift h d0 v) (lift h d0
-w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3:
-(drop h d0 c0 c)).(lt_le_e i d0 (tau0 g c0 (lift h d0 (TLRef i)) (lift h d0
-(lift (S i) O w))) (\lambda (H4: (lt i d0)).(let H5 \def (drop_getl_trans_le
-i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) v) H0)
-in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i O c0 e0)))
-(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) (\lambda (_:
-C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) v)))) (tau0 g c0 (lift h
-d0 (TLRef i)) (lift h d0 (lift (S i) O w))) (\lambda (x0: C).(\lambda (x1:
-C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h (minus d0 i) x0
-x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) v))).(let H9 \def (eq_ind
-nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i)))
-(minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i))
-H9 Abbr d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind
-Abbr) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h (minus d0 (S
-i)) c1 d)) (tau0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O w)))
-(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus
-d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x d)).(eq_ind_r T
-(TLRef i) (\lambda (t: T).(tau0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind
-nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(tau0 g c0 (TLRef i)
-(lift h n (lift (S i) O w)))) (eq_ind_r T (lift (S i) O (lift h (minus d0 (S
-i)) w)) (\lambda (t: T).(tau0 g c0 (TLRef i) t)) (eq_ind nat d0 (\lambda (_:
-nat).(tau0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i)) w))))
-(tau0_abbr g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead x
-(Bind Abbr) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S i))
-w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i)))
-(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S
-i) O w)) (lift_d w h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0
-(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0
-H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i
-h)) (\lambda (t: T).(tau0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind nat
-(S i) (\lambda (_: nat).(tau0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i)
-O w)))) (eq_ind_r T (lift (plus h (S i)) O w) (\lambda (t: T).(tau0 g c0
-(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(tau0 g
-c0 (TLRef (plus i h)) (lift n O w))) (tau0_abbr g c0 d v (plus i h)
-(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abbr) v) H0 H4) w H1) (plus
-h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O w)) (lift_free w (S i)
-h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O)
-i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus
-i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0
-H4)))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda
-(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) v))).(\lambda (w:
-T).(\lambda (H1: (tau0 g d v w)).(\lambda (H2: ((\forall (c0: C).(\forall (h:
-nat).(\forall (d0: nat).((drop h d0 c0 d) \to (tau0 g c0 (lift h d0 v) (lift
-h d0 w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda
-(H3: (drop h d0 c0 c)).(lt_le_e i d0 (tau0 g c0 (lift h d0 (TLRef i)) (lift h
-d0 (lift (S i) O v))) (\lambda (H4: (lt i d0)).(let H5 \def
-(drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d
-(Bind Abst) v) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i
-O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1)))
-(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) v)))) (tau0 g
-c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O v))) (\lambda (x0:
-C).(\lambda (x1: C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h
-(minus d0 i) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) v))).(let
-H9 \def (eq_ind nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S
-(minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0
-h (minus d0 (S i)) H9 Abst d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0
-(CHead c1 (Bind Abst) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h
-(minus d0 (S i)) c1 d)) (tau0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S
-i) O v))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift
-h (minus d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x
-d)).(eq_ind_r T (TLRef i) (\lambda (t: T).(tau0 g c0 t (lift h d0 (lift (S i)
-O v)))) (eq_ind nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(tau0 g
-c0 (TLRef i) (lift h n (lift (S i) O v)))) (eq_ind_r T (lift (S i) O (lift h
-(minus d0 (S i)) v)) (\lambda (t: T).(tau0 g c0 (TLRef i) t)) (eq_ind nat d0
-(\lambda (_: nat).(tau0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i))
-v)))) (tau0_abst g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead
-x (Bind Abst) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S
-i)) w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i)))
-(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S
-i) O v)) (lift_d v h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0
-(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0
-H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i
-h)) (\lambda (t: T).(tau0 g c0 t (lift h d0 (lift (S i) O v)))) (eq_ind nat
-(S i) (\lambda (_: nat).(tau0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i)
-O v)))) (eq_ind_r T (lift (plus h (S i)) O v) (\lambda (t: T).(tau0 g c0
-(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(tau0 g
-c0 (TLRef (plus i h)) (lift n O v))) (tau0_abst g c0 d v (plus i h)
-(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abst) v) H0 H4) w H1) (plus
-h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O v)) (lift_free v (S i)
-h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O)
-i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus
-i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0
-H4)))))))))))))))) (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda
-(t3: T).(\lambda (t4: T).(\lambda (_: (tau0 g (CHead c (Bind b) v) t3
-t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
-nat).((drop h d c0 (CHead c (Bind b) v)) \to (tau0 g c0 (lift h d t3) (lift h
-d t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
-(H2: (drop h d c0 c)).(eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s
-(Bind b) d) t3)) (\lambda (t: T).(tau0 g c0 t (lift h d (THead (Bind b) v
-t4)))) (eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s (Bind b) d) t4))
-(\lambda (t: T).(tau0 g c0 (THead (Bind b) (lift h d v) (lift h (s (Bind b)
-d) t3)) t)) (tau0_bind g b c0 (lift h d v) (lift h (S d) t3) (lift h (S d)
-t4) (H1 (CHead c0 (Bind b) (lift h d v)) h (S d) (drop_skip_bind h d c0 c H2
-b v))) (lift h d (THead (Bind b) v t4)) (lift_head (Bind b) v t4 h d)) (lift
-h d (THead (Bind b) v t3)) (lift_head (Bind b) v t3 h d))))))))))))) (\lambda
-(c: C).(\lambda (v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (tau0 g
-c t3 t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
-nat).((drop h d c0 c) \to (tau0 g c0 (lift h d t3) (lift h d
-t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2:
-(drop h d c0 c)).(eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat
-Appl) d) t3)) (\lambda (t: T).(tau0 g c0 t (lift h d (THead (Flat Appl) v
-t4)))) (eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat Appl) d)
-t4)) (\lambda (t: T).(tau0 g c0 (THead (Flat Appl) (lift h d v) (lift h (s
-(Flat Appl) d) t3)) t)) (tau0_appl g c0 (lift h d v) (lift h (s (Flat Appl)
-d) t3) (lift h (s (Flat Appl) d) t4) (H1 c0 h (s (Flat Appl) d) H2)) (lift h
-d (THead (Flat Appl) v t4)) (lift_head (Flat Appl) v t4 h d)) (lift h d
-(THead (Flat Appl) v t3)) (lift_head (Flat Appl) v t3 h d))))))))))))
-(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (tau0 g c v1
-v2)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
-nat).((drop h d c0 c) \to (tau0 g c0 (lift h d v1) (lift h d
-v2)))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (tau0 g c t3
-t4)).(\lambda (H3: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
-nat).((drop h d c0 c) \to (tau0 g c0 (lift h d t3) (lift h d
-t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4:
-(drop h d c0 c)).(eq_ind_r T (THead (Flat Cast) (lift h d v1) (lift h (s
-(Flat Cast) d) t3)) (\lambda (t: T).(tau0 g c0 t (lift h d (THead (Flat Cast)
-v2 t4)))) (eq_ind_r T (THead (Flat Cast) (lift h d v2) (lift h (s (Flat Cast)
-d) t4)) (\lambda (t: T).(tau0 g c0 (THead (Flat Cast) (lift h d v1) (lift h
-(s (Flat Cast) d) t3)) t)) (tau0_cast g c0 (lift h d v1) (lift h d v2) (H1 c0
-h d H4) (lift h (s (Flat Cast) d) t3) (lift h (s (Flat Cast) d) t4) (H3 c0 h
-(s (Flat Cast) d) H4)) (lift h d (THead (Flat Cast) v2 t4)) (lift_head (Flat
-Cast) v2 t4 h d)) (lift h d (THead (Flat Cast) v1 t3)) (lift_head (Flat Cast)
-v1 t3 h d))))))))))))))) e t1 t2 H))))).
-
-theorem tau0_correct:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau0 g c
-t1 t) \to (ex T (\lambda (t2: T).(tau0 g c t t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
-(tau0 g c t1 t)).(tau0_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (t2:
-T).(ex T (\lambda (t3: T).(tau0 g c0 t2 t3)))))) (\lambda (c0: C).(\lambda
-(n: nat).(ex_intro T (\lambda (t2: T).(tau0 g c0 (TSort (next g n)) t2))
-(TSort (next g (next g n))) (tau0_sort g c0 (next g n))))) (\lambda (c0:
-C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0
-(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (tau0 g d v
-w)).(\lambda (H2: (ex T (\lambda (t2: T).(tau0 g d w t2)))).(let H3 \def H2
-in (ex_ind T (\lambda (t2: T).(tau0 g d w t2)) (ex T (\lambda (t2: T).(tau0 g
-c0 (lift (S i) O w) t2))) (\lambda (x: T).(\lambda (H4: (tau0 g d w
-x)).(ex_intro T (\lambda (t2: T).(tau0 g c0 (lift (S i) O w) t2)) (lift (S i)
-O x) (tau0_lift g d w x H4 c0 (S i) O (getl_drop Abbr c0 d v i H0)))))
-H3)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
-nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w:
-T).(\lambda (H1: (tau0 g d v w)).(\lambda (H2: (ex T (\lambda (t2: T).(tau0 g
-d w t2)))).(let H3 \def H2 in (ex_ind T (\lambda (t2: T).(tau0 g d w t2)) (ex
-T (\lambda (t2: T).(tau0 g c0 (lift (S i) O v) t2))) (\lambda (x: T).(\lambda
-(_: (tau0 g d w x)).(ex_intro T (\lambda (t2: T).(tau0 g c0 (lift (S i) O v)
-t2)) (lift (S i) O w) (tau0_lift g d v w H1 c0 (S i) O (getl_drop Abst c0 d v
-i H0))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v:
-T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (tau0 g (CHead c0 (Bind b)
-v) t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(tau0 g (CHead c0 (Bind b) v)
-t3 t4)))).(let H2 \def H1 in (ex_ind T (\lambda (t4: T).(tau0 g (CHead c0
-(Bind b) v) t3 t4)) (ex T (\lambda (t4: T).(tau0 g c0 (THead (Bind b) v t3)
-t4))) (\lambda (x: T).(\lambda (H3: (tau0 g (CHead c0 (Bind b) v) t3
-x)).(ex_intro T (\lambda (t4: T).(tau0 g c0 (THead (Bind b) v t3) t4)) (THead
-(Bind b) v x) (tau0_bind g b c0 v t3 x H3)))) H2))))))))) (\lambda (c0:
-C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (tau0 g c0
-t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(tau0 g c0 t3 t4)))).(let H2
-\def H1 in (ex_ind T (\lambda (t4: T).(tau0 g c0 t3 t4)) (ex T (\lambda (t4:
-T).(tau0 g c0 (THead (Flat Appl) v t3) t4))) (\lambda (x: T).(\lambda (H3:
-(tau0 g c0 t3 x)).(ex_intro T (\lambda (t4: T).(tau0 g c0 (THead (Flat Appl)
-v t3) t4)) (THead (Flat Appl) v x) (tau0_appl g c0 v t3 x H3)))) H2))))))))
-(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (tau0 g c0 v1
-v2)).(\lambda (H1: (ex T (\lambda (t2: T).(tau0 g c0 v2 t2)))).(\lambda (t2:
-T).(\lambda (t3: T).(\lambda (_: (tau0 g c0 t2 t3)).(\lambda (H3: (ex T
-(\lambda (t4: T).(tau0 g c0 t3 t4)))).(let H4 \def H1 in (ex_ind T (\lambda
-(t4: T).(tau0 g c0 v2 t4)) (ex T (\lambda (t4: T).(tau0 g c0 (THead (Flat
-Cast) v2 t3) t4))) (\lambda (x: T).(\lambda (H5: (tau0 g c0 v2 x)).(let H6
-\def H3 in (ex_ind T (\lambda (t4: T).(tau0 g c0 t3 t4)) (ex T (\lambda (t4:
-T).(tau0 g c0 (THead (Flat Cast) v2 t3) t4))) (\lambda (x0: T).(\lambda (H7:
-(tau0 g c0 t3 x0)).(ex_intro T (\lambda (t4: T).(tau0 g c0 (THead (Flat Cast)
-v2 t3) t4)) (THead (Flat Cast) x x0) (tau0_cast g c0 v2 x H5 t3 x0 H7))))
-H6)))) H4))))))))))) c t1 t H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/tau1/props.ma".
-
-include "LambdaDelta-1/cnt/props.ma".
-
-theorem tau1_cnt:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau0 g c
-t1 t) \to (ex2 T (\lambda (t2: T).(tau1 g c t1 t2)) (\lambda (t2: T).(cnt
-t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
-(tau0 g c t1 t)).(tau0_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
-T).(ex2 T (\lambda (t3: T).(tau1 g c0 t0 t3)) (\lambda (t3: T).(cnt t3))))))
-(\lambda (c0: C).(\lambda (n: nat).(ex_intro2 T (\lambda (t2: T).(tau1 g c0
-(TSort n) t2)) (\lambda (t2: T).(cnt t2)) (TSort (next g n)) (tau1_tau0 g c0
-(TSort n) (TSort (next g n)) (tau0_sort g c0 n)) (cnt_sort (next g n)))))
-(\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
-(H0: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (tau0
-g d v w)).(\lambda (H2: (ex2 T (\lambda (t2: T).(tau1 g d v t2)) (\lambda
-(t2: T).(cnt t2)))).(let H3 \def H2 in (ex2_ind T (\lambda (t2: T).(tau1 g d
-v t2)) (\lambda (t2: T).(cnt t2)) (ex2 T (\lambda (t2: T).(tau1 g c0 (TLRef
-i) t2)) (\lambda (t2: T).(cnt t2))) (\lambda (x: T).(\lambda (H4: (tau1 g d v
-x)).(\lambda (H5: (cnt x)).(ex_intro2 T (\lambda (t2: T).(tau1 g c0 (TLRef i)
-t2)) (\lambda (t2: T).(cnt t2)) (lift (S i) O x) (tau1_abbr g c0 d v i H0 x
-H4) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
-Abst) v))).(\lambda (w: T).(\lambda (H1: (tau0 g d v w)).(\lambda (H2: (ex2 T
-(\lambda (t2: T).(tau1 g d v t2)) (\lambda (t2: T).(cnt t2)))).(let H3 \def
-H2 in (ex2_ind T (\lambda (t2: T).(tau1 g d v t2)) (\lambda (t2: T).(cnt t2))
-(ex2 T (\lambda (t2: T).(tau1 g c0 (TLRef i) t2)) (\lambda (t2: T).(cnt t2)))
-(\lambda (x: T).(\lambda (H4: (tau1 g d v x)).(\lambda (H5: (cnt
-x)).(ex_intro2 T (\lambda (t2: T).(tau1 g c0 (TLRef i) t2)) (\lambda (t2:
-T).(cnt t2)) (lift (S i) O x) (tau1_trans g c0 (TLRef i) (lift (S i) O v)
-(tau1_tau0 g c0 (TLRef i) (lift (S i) O v) (tau0_abst g c0 d v i H0 w H1))
-(lift (S i) O x) (tau1_lift g d v x H4 c0 (S i) O (getl_drop Abst c0 d v i
-H0))) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (b: B).(\lambda (c0:
-C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (tau0 g
-(CHead c0 (Bind b) v) t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4: T).(tau1 g
-(CHead c0 (Bind b) v) t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in
-(ex2_ind T (\lambda (t4: T).(tau1 g (CHead c0 (Bind b) v) t2 t4)) (\lambda
-(t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(tau1 g c0 (THead (Bind b) v t2)
-t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (tau1 g (CHead
-c0 (Bind b) v) t2 x)).(\lambda (H4: (cnt x)).(ex_intro2 T (\lambda (t4:
-T).(tau1 g c0 (THead (Bind b) v t2) t4)) (\lambda (t4: T).(cnt t4)) (THead
-(Bind b) v x) (tau1_bind g b c0 v t2 x H3) (cnt_head x H4 (Bind b) v)))))
-H2))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3:
-T).(\lambda (_: (tau0 g c0 t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4:
-T).(tau1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in
-(ex2_ind T (\lambda (t4: T).(tau1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4))
-(ex2 T (\lambda (t4: T).(tau1 g c0 (THead (Flat Appl) v t2) t4)) (\lambda
-(t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (tau1 g c0 t2 x)).(\lambda
-(H4: (cnt x)).(ex_intro2 T (\lambda (t4: T).(tau1 g c0 (THead (Flat Appl) v
-t2) t4)) (\lambda (t4: T).(cnt t4)) (THead (Flat Appl) v x) (tau1_appl g c0 v
-t2 x H3) (cnt_head x H4 (Flat Appl) v))))) H2)))))))) (\lambda (c0:
-C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (tau0 g c0 v1
-v2)).(\lambda (_: (ex2 T (\lambda (t2: T).(tau1 g c0 v1 t2)) (\lambda (t2:
-T).(cnt t2)))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (tau0 g c0 t2
-t3)).(\lambda (H3: (ex2 T (\lambda (t4: T).(tau1 g c0 t2 t4)) (\lambda (t4:
-T).(cnt t4)))).(let H4 \def H3 in (ex2_ind T (\lambda (t4: T).(tau1 g c0 t2
-t4)) (\lambda (t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(tau1 g c0 (THead
-(Flat Cast) v1 t2) t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda
-(H5: (tau1 g c0 t2 x)).(\lambda (H6: (cnt x)).(let H_x \def (tau1_cast2 g c0
-t2 x H5 v1 v2 H0) in (let H7 \def H_x in (ex2_ind T (\lambda (v3: T).(tau1 g
-c0 v1 v3)) (\lambda (v3: T).(tau1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat
-Cast) v3 x))) (ex2 T (\lambda (t4: T).(tau1 g c0 (THead (Flat Cast) v1 t2)
-t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x0: T).(\lambda (_: (tau1 g c0 v1
-x0)).(\lambda (H9: (tau1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat Cast) x0
-x))).(ex_intro2 T (\lambda (t4: T).(tau1 g c0 (THead (Flat Cast) v1 t2) t4))
-(\lambda (t4: T).(cnt t4)) (THead (Flat Cast) x0 x) H9 (cnt_head x H6 (Flat
-Cast) x0))))) H7)))))) H4))))))))))) c t1 t H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/tau0/defs.ma".
-
-inductive tau1 (g: G) (c: C) (t1: T): T \to Prop \def
-| tau1_tau0: \forall (t2: T).((tau0 g c t1 t2) \to (tau1 g c t1 t2))
-| tau1_sing: \forall (t: T).((tau1 g c t1 t) \to (\forall (t2: T).((tau0 g c
-t t2) \to (tau1 g c t1 t2)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/tau1/defs.ma".
-
-include "LambdaDelta-1/tau0/props.ma".
-
-theorem tau1_trans:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau1 g c
-t1 t) \to (\forall (t2: T).((tau1 g c t t2) \to (tau1 g c t1 t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
-(tau1 g c t1 t)).(\lambda (t2: T).(\lambda (H0: (tau1 g c t t2)).(tau1_ind g
-c t (\lambda (t0: T).(tau1 g c t1 t0)) (\lambda (t3: T).(\lambda (H1: (tau0 g
-c t t3)).(tau1_sing g c t1 t H t3 H1))) (\lambda (t0: T).(\lambda (_: (tau1 g
-c t t0)).(\lambda (H2: (tau1 g c t1 t0)).(\lambda (t3: T).(\lambda (H3: (tau0
-g c t0 t3)).(tau1_sing g c t1 t0 H2 t3 H3)))))) t2 H0))))))).
-
-theorem tau1_bind:
- \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1:
-T).(\forall (t2: T).((tau1 g (CHead c (Bind b) v) t1 t2) \to (tau1 g c (THead
-(Bind b) v t1) (THead (Bind b) v t2))))))))
-\def
- \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (H: (tau1 g (CHead c (Bind b) v) t1
-t2)).(tau1_ind g (CHead c (Bind b) v) t1 (\lambda (t: T).(tau1 g c (THead
-(Bind b) v t1) (THead (Bind b) v t))) (\lambda (t3: T).(\lambda (H0: (tau0 g
-(CHead c (Bind b) v) t1 t3)).(tau1_tau0 g c (THead (Bind b) v t1) (THead
-(Bind b) v t3) (tau0_bind g b c v t1 t3 H0)))) (\lambda (t: T).(\lambda (_:
-(tau1 g (CHead c (Bind b) v) t1 t)).(\lambda (H1: (tau1 g c (THead (Bind b) v
-t1) (THead (Bind b) v t))).(\lambda (t3: T).(\lambda (H2: (tau0 g (CHead c
-(Bind b) v) t t3)).(tau1_sing g c (THead (Bind b) v t1) (THead (Bind b) v t)
-H1 (THead (Bind b) v t3) (tau0_bind g b c v t t3 H2))))))) t2 H))))))).
-
-theorem tau1_appl:
- \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall
-(t2: T).((tau1 g c t1 t2) \to (tau1 g c (THead (Flat Appl) v t1) (THead (Flat
-Appl) v t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
-(t2: T).(\lambda (H: (tau1 g c t1 t2)).(tau1_ind g c t1 (\lambda (t: T).(tau1
-g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t))) (\lambda (t3:
-T).(\lambda (H0: (tau0 g c t1 t3)).(tau1_tau0 g c (THead (Flat Appl) v t1)
-(THead (Flat Appl) v t3) (tau0_appl g c v t1 t3 H0)))) (\lambda (t:
-T).(\lambda (_: (tau1 g c t1 t)).(\lambda (H1: (tau1 g c (THead (Flat Appl) v
-t1) (THead (Flat Appl) v t))).(\lambda (t3: T).(\lambda (H2: (tau0 g c t
-t3)).(tau1_sing g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t) H1
-(THead (Flat Appl) v t3) (tau0_appl g c v t t3 H2))))))) t2 H)))))).
-
-theorem tau1_lift:
- \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((tau1 g e
-t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c
-e) \to (tau1 g c (lift h d t1) (lift h d t2))))))))))
-\def
- \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (tau1 g e t1 t2)).(tau1_ind g e t1 (\lambda (t: T).(\forall (c:
-C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (tau1 g c (lift h
-d t1) (lift h d t))))))) (\lambda (t3: T).(\lambda (H0: (tau0 g e t1
-t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop
-h d c e)).(tau1_tau0 g c (lift h d t1) (lift h d t3) (tau0_lift g e t1 t3 H0
-c h d H1)))))))) (\lambda (t: T).(\lambda (_: (tau1 g e t1 t)).(\lambda (H1:
-((\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to
-(tau1 g c (lift h d t1) (lift h d t)))))))).(\lambda (t3: T).(\lambda (H2:
-(tau0 g e t t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
-(H3: (drop h d c e)).(tau1_sing g c (lift h d t1) (lift h d t) (H1 c h d H3)
-(lift h d t3) (tau0_lift g e t t3 H2 c h d H3))))))))))) t2 H))))).
-
-theorem tau1_correct:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau1 g c
-t1 t) \to (ex T (\lambda (t2: T).(tau0 g c t t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
-(tau1 g c t1 t)).(tau1_ind g c t1 (\lambda (t0: T).(ex T (\lambda (t2:
-T).(tau0 g c t0 t2)))) (\lambda (t2: T).(\lambda (H0: (tau0 g c t1
-t2)).(tau0_correct g c t1 t2 H0))) (\lambda (t0: T).(\lambda (_: (tau1 g c t1
-t0)).(\lambda (_: (ex T (\lambda (t2: T).(tau0 g c t0 t2)))).(\lambda (t2:
-T).(\lambda (H2: (tau0 g c t0 t2)).(tau0_correct g c t0 t2 H2)))))) t H))))).
-
-theorem tau1_abbr:
- \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i:
-nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((tau1 g d v w)
-\to (tau1 g c (TLRef i) (lift (S i) O w)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
-nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (w:
-T).(\lambda (H0: (tau1 g d v w)).(tau1_ind g d v (\lambda (t: T).(tau1 g c
-(TLRef i) (lift (S i) O t))) (\lambda (t2: T).(\lambda (H1: (tau0 g d v
-t2)).(tau1_tau0 g c (TLRef i) (lift (S i) O t2) (tau0_abbr g c d v i H t2
-H1)))) (\lambda (t: T).(\lambda (_: (tau1 g d v t)).(\lambda (H2: (tau1 g c
-(TLRef i) (lift (S i) O t))).(\lambda (t2: T).(\lambda (H3: (tau0 g d t
-t2)).(tau1_sing g c (TLRef i) (lift (S i) O t) H2 (lift (S i) O t2)
-(tau0_lift g d t t2 H3 c (S i) O (getl_drop Abbr c d v i H)))))))) w
-H0)))))))).
-
-theorem tau1_cast2:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((tau1 g c
-t1 t2) \to (\forall (v1: T).(\forall (v2: T).((tau0 g c v1 v2) \to (ex2 T
-(\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat
-Cast) v1 t1) (THead (Flat Cast) v3 t2)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (tau1 g c t1 t2)).(tau1_ind g c t1 (\lambda (t: T).(\forall (v1:
-T).(\forall (v2: T).((tau0 g c v1 v2) \to (ex2 T (\lambda (v3: T).(tau1 g c
-v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat Cast) v1 t1) (THead (Flat
-Cast) v3 t)))))))) (\lambda (t3: T).(\lambda (H0: (tau0 g c t1 t3)).(\lambda
-(v1: T).(\lambda (v2: T).(\lambda (H1: (tau0 g c v1 v2)).(ex_intro2 T
-(\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat
-Cast) v1 t1) (THead (Flat Cast) v3 t3))) v2 (tau1_tau0 g c v1 v2 H1)
-(tau1_tau0 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t3) (tau0_cast
-g c v1 v2 H1 t1 t3 H0)))))))) (\lambda (t: T).(\lambda (_: (tau1 g c t1
-t)).(\lambda (H1: ((\forall (v1: T).(\forall (v2: T).((tau0 g c v1 v2) \to
-(ex2 T (\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3: T).(tau1 g c (THead
-(Flat Cast) v1 t1) (THead (Flat Cast) v3 t))))))))).(\lambda (t3: T).(\lambda
-(H2: (tau0 g c t t3)).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H3: (tau0 g
-c v1 v2)).(let H_x \def (H1 v1 v2 H3) in (let H4 \def H_x in (ex2_ind T
-(\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat
-Cast) v1 t1) (THead (Flat Cast) v3 t))) (ex2 T (\lambda (v3: T).(tau1 g c v1
-v3)) (\lambda (v3: T).(tau1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast)
-v3 t3)))) (\lambda (x: T).(\lambda (H5: (tau1 g c v1 x)).(\lambda (H6: (tau1
-g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) x t))).(let H_x0 \def
-(tau1_correct g c v1 x H5) in (let H7 \def H_x0 in (ex_ind T (\lambda (t4:
-T).(tau0 g c x t4)) (ex2 T (\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3:
-T).(tau1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v3 t3)))) (\lambda
-(x0: T).(\lambda (H8: (tau0 g c x x0)).(ex_intro2 T (\lambda (v3: T).(tau1 g
-c v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat Cast) v1 t1) (THead (Flat
-Cast) v3 t3))) x0 (tau1_sing g c v1 x H5 x0 H8) (tau1_sing g c (THead (Flat
-Cast) v1 t1) (THead (Flat Cast) x t) H6 (THead (Flat Cast) x0 t3) (tau0_cast
-g c x x0 H8 t t3 H2))))) H7)))))) H4))))))))))) t2 H))))).
-
include "LambdaDelta-1/subst/props.ma".
-include "LambdaDelta-1/tau1/cnt.ma".
+include "LambdaDelta-1/sty1/cnt.ma".
include "LambdaDelta-1/ex0/props.ma".
include "LambdaDelta-1/ex1/props.ma".
-include "LambdaDelta-1/ty3/tau0.ma".
+include "LambdaDelta-1/ty3/sty0.ma".
include "LambdaDelta-1/ty3/nf2.ma".
(H18 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b))
(lift (S O) O t4) (pr0_lift t5 t4 H17 (S O) O)))))))) t6 (sym_eq T t6 t4
H15))) t2 H14)) u0 (sym_eq T u0 u H13))) b0 (sym_eq B b0 b H12))) H11)) H10))
-H9 H6 H7))) | (pr0_epsilon t5 t6 H6 u0) \Rightarrow (\lambda (H7: (eq T
-(THead (Flat Cast) u0 t5) (THead (Bind b) u t2))).(\lambda (H8: (eq T t6
-t4)).((let H9 \def (eq_ind T (THead (Flat Cast) u0 t5) (\lambda (e: T).(match
-e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+H9 H6 H7))) | (pr0_tau t5 t6 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead
+(Flat Cast) u0 t5) (THead (Bind b) u t2))).(\lambda (H8: (eq T t6 t4)).((let
+H9 \def (eq_ind T (THead (Flat Cast) u0 t5) (\lambda (e: T).(match e in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
True])])) I (THead (Bind b) u t2) H7) in (False_ind ((eq T t6 t4) \to ((pr0
t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))) H9)) H8 H6)))]) in (H6
(refl_equal T (THead (Bind b) u t2)) (refl_equal T t4)))))))))))))))))
(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
False])])) I (THead (Flat Appl) w v) H8) in (False_ind ((eq T t4 t2) \to
((not (eq B b Abst)) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w
-(THead (Bind Abst) u t0)))))) H10)) H9 H6 H7))) | (pr0_epsilon t3 t4 H6 u0)
+(THead (Bind Abst) u t0)))))) H10)) H9 H6 H7))) | (pr0_tau t3 t4 H6 u0)
\Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t3) (THead (Flat Appl)
w v))).(\lambda (H8: (eq T t4 t2)).((let H9 \def (eq_ind T (THead (Flat Cast)
u0 t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
Cast) t3 t2) H8) in (False_ind ((eq T t6 t4) \to ((not (eq B b Abst)) \to
((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7)))
-| (pr0_epsilon t5 t6 H6 u) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast)
-u t5) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T t6 t4)).((let H9 \def
+| (pr0_tau t5 t6 H6 u) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u
+t5) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T t6 t4)).((let H9 \def
(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
[(TSort _) \Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t7)
\Rightarrow t7])) (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) in
t0 x)).(ty3_conv g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0)
(ty3_cast g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) x H14) t4 t3 (H1 c2 H4 t4
H13) (pc3_pr2_x c2 t3 (THead (Flat Cast) t0 t3) (pr2_free c2 (THead (Flat
-Cast) t0 t3) t3 (pr0_epsilon t3 t3 (pr0_refl t3) t0)))))) (ty3_correct g c2
-t3 t0 (H3 c2 H4 t3 (pr0_refl t3))))) t6 (sym_eq T t6 t4 H12))) t5 (sym_eq T
-t5 t2 H11))) u (sym_eq T u t3 H10))) H9)) H8 H6)))]) in (H6 (refl_equal T
-(THead (Flat Cast) t3 t2)) (refl_equal T t4))))))))))))))) c1 t1 t H))))).
+Cast) t0 t3) t3 (pr0_tau t3 t3 (pr0_refl t3) t0)))))) (ty3_correct g c2 t3 t0
+(H3 c2 H4 t3 (pr0_refl t3))))) t6 (sym_eq T t6 t4 H12))) t5 (sym_eq T t5 t2
+H11))) u (sym_eq T u t3 H10))) H9)) H8 H6)))]) in (H6 (refl_equal T (THead
+(Flat Cast) t3 t2)) (refl_equal T t4))))))))))))))) c1 t1 t H))))).
theorem ty3_sred_pr0:
\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (g: G).(\forall
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/ty3/pr3_props.ma".
+
+include "LambdaDelta-1/sty0/fwd.ma".
+
+theorem ty3_sty0:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u
+t1) \to (\forall (t2: T).((sty0 g c u t2) \to (ty3 g c u t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H:
+(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (_:
+T).(\forall (t2: T).((sty0 g c0 t t2) \to (ty3 g c0 t t2)))))) (\lambda (c0:
+C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda
+(_: ((\forall (t3: T).((sty0 g c0 t2 t3) \to (ty3 g c0 t2 t3))))).(\lambda
+(u0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c0 u0 t3)).(\lambda (H3:
+((\forall (t4: T).((sty0 g c0 u0 t4) \to (ty3 g c0 u0 t4))))).(\lambda (_:
+(pc3 c0 t3 t2)).(\lambda (t0: T).(\lambda (H5: (sty0 g c0 u0 t0)).(H3 t0
+H5))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (t2: T).(\lambda
+(H0: (sty0 g c0 (TSort m) t2)).(let H_y \def (sty0_gen_sort g c0 t2 m H0) in
+(let H1 \def (f_equal T T (\lambda (e: T).e) t2 (TSort (next g m)) H_y) in
+(eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 (TSort m) t))
+(ty3_sort g c0 m) t2 H1))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda
+(d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr)
+u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall
+(t2: T).((sty0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: T).(\lambda
+(H3: (sty0 g c0 (TLRef n) t2)).(let H_x \def (sty0_gen_lref g c0 t2 n H3) in
+(let H4 \def H_x in (or_ind (ex3_3 C T T (\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e:
+C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0)))))) (ex3_3 C
+T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g
+e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift
+(S n) O u1)))))) (ty3 g c0 (TLRef n) t2) (\lambda (H5: (ex3_3 C T T (\lambda
+(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
+u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O
+t0))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1:
+T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))) (ty3 g c0 (TLRef n) t2)
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0
+(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (sty0 g x0 x1 x2)).(\lambda (H8:
+(eq T t2 (lift (S n) O x2))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2
+(lift (S n) O x2) H8) in (eq_ind_r T (lift (S n) O x2) (\lambda (t0: T).(ty3
+g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda
+(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d
+(Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H11 \def (f_equal
+C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0)
+(CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead
+x0 (Bind Abbr) x1) H6)) in ((let H12 \def (f_equal C T (\lambda (e: C).(match
+e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _
+t0) \Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1)
+(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in
+(\lambda (H13: (eq C d x0)).(let H14 \def (eq_ind_r T x1 (\lambda (t0:
+T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H10 u0 H12) in (let H15 \def
+(eq_ind_r T x1 (\lambda (t0: T).(sty0 g x0 t0 x2)) H7 u0 H12) in (let H16
+\def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0)))
+H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1: C).(sty0 g c1 u0
+x2)) H15 d H13) in (ty3_abbr g n c0 d u0 H16 x2 (H2 x2 H17)))))))) H11))) t2
+H9)))))))) H5)) (\lambda (H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e:
+C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O
+u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1:
+T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0 (TLRef n) t2)
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0
+(CHead x0 (Bind Abst) x1))).(\lambda (_: (sty0 g x0 x1 x2)).(\lambda (H8: (eq
+T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2
+(lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1) (\lambda (t0: T).(ty3
+g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda
+(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d
+(Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H11 \def (eq_ind
+C (CHead d (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda
+(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match
+b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst
+\Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow
+False])])) I (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0)
+n H0 (CHead x0 (Bind Abst) x1) H6)) in (False_ind (ty3 g c0 (TLRef n) (lift
+(S n) O x1)) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\lambda (n:
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n
+c0 (CHead d (Bind Abst) u0))).(\lambda (t: T).(\lambda (H1: (ty3 g d u0
+t)).(\lambda (_: ((\forall (t2: T).((sty0 g d u0 t2) \to (ty3 g d u0
+t2))))).(\lambda (t2: T).(\lambda (H3: (sty0 g c0 (TLRef n) t2)).(let H_x
+\def (sty0_gen_lref g c0 t2 n H3) in (let H4 \def H_x in (or_ind (ex3_3 C T T
+(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1
+t0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n)
+O t0)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1:
+T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(eq T t2 (lift (S n) O u1)))))) (ty3 g c0 (TLRef n) t2)
+(\lambda (H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1:
+T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))))).(ex3_3_ind C T T
+(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1
+t0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n)
+O t0))))) (ty3 g c0 (TLRef n) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda
+(x2: T).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (_:
+(sty0 g x0 x1 x2)).(\lambda (H8: (eq T t2 (lift (S n) O x2))).(let H9 \def
+(f_equal T T (\lambda (e: T).e) t2 (lift (S n) O x2) H8) in (eq_ind_r T (lift
+(S n) O x2) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (let H10 \def (eq_ind C
+(CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind
+Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr)
+x1) H6)) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee:
+C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
+False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop)
+with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with
+[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) |
+(Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0
+(CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind
+(ty3 g c0 (TLRef n) (lift (S n) O x2)) H11))) t2 H9)))))))) H5)) (\lambda
+(H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0
+(CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0:
+T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T
+t2 (lift (S n) O u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e:
+C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0
+(TLRef n) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda
+(H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7: (sty0 g x0 x1
+x2)).(\lambda (H8: (eq T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T
+(\lambda (e: T).e) t2 (lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1)
+(\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d
+(Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1)
+(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in
+(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_:
+C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead
+d (Bind Abst) u0) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind
+Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in ((let H12 \def (f_equal C T
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abst) u0)
+(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead
+x0 (Bind Abst) x1) H6)) in (\lambda (H13: (eq C d x0)).(let H14 \def
+(eq_ind_r T x1 (\lambda (t0: T).(getl n c0 (CHead x0 (Bind Abst) t0))) H10 u0
+H12) in (let H15 \def (eq_ind_r T x1 (\lambda (t0: T).(sty0 g x0 t0 x2)) H7
+u0 H12) in (eq_ind T u0 (\lambda (t0: T).(ty3 g c0 (TLRef n) (lift (S n) O
+t0))) (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1
+(Bind Abst) u0))) H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1:
+C).(sty0 g c1 u0 x2)) H15 d H13) in (ty3_abst g n c0 d u0 H16 t H1))) x1
+H12))))) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\lambda (c0: C).(\lambda
+(u0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u0 t)).(\lambda (_: ((\forall
+(t2: T).((sty0 g c0 u0 t2) \to (ty3 g c0 u0 t2))))).(\lambda (b: B).(\lambda
+(t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t2
+t3)).(\lambda (H3: ((\forall (t4: T).((sty0 g (CHead c0 (Bind b) u0) t2 t4)
+\to (ty3 g (CHead c0 (Bind b) u0) t2 t4))))).(\lambda (t0: T).(\lambda (H4:
+(sty0 g c0 (THead (Bind b) u0 t2) t0)).(let H_x \def (sty0_gen_bind g b c0 u0
+t2 t0 H4) in (let H5 \def H_x in (ex2_ind T (\lambda (t4: T).(sty0 g (CHead
+c0 (Bind b) u0) t2 t4)) (\lambda (t4: T).(eq T t0 (THead (Bind b) u0 t4)))
+(ty3 g c0 (THead (Bind b) u0 t2) t0) (\lambda (x: T).(\lambda (H6: (sty0 g
+(CHead c0 (Bind b) u0) t2 x)).(\lambda (H7: (eq T t0 (THead (Bind b) u0
+x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u0 x)
+H7) in (eq_ind_r T (THead (Bind b) u0 x) (\lambda (t4: T).(ty3 g c0 (THead
+(Bind b) u0 t2) t4)) (ty3_bind g c0 u0 t H0 b t2 x (H3 x H6)) t0 H8)))))
+H5))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda
+(H0: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: T).((sty0 g c0 w t2) \to
+(ty3 g c0 w t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v
+(THead (Bind Abst) u0 t))).(\lambda (H3: ((\forall (t2: T).((sty0 g c0 v t2)
+\to (ty3 g c0 v t2))))).(\lambda (t2: T).(\lambda (H4: (sty0 g c0 (THead
+(Flat Appl) w v) t2)).(let H_x \def (sty0_gen_appl g c0 w v t2 H4) in (let H5
+\def H_x in (ex2_ind T (\lambda (t3: T).(sty0 g c0 v t3)) (\lambda (t3:
+T).(eq T t2 (THead (Flat Appl) w t3))) (ty3 g c0 (THead (Flat Appl) w v) t2)
+(\lambda (x: T).(\lambda (H6: (sty0 g c0 v x)).(\lambda (H7: (eq T t2 (THead
+(Flat Appl) w x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t2 (THead
+(Flat Appl) w x) H7) in (eq_ind_r T (THead (Flat Appl) w x) (\lambda (t0:
+T).(ty3 g c0 (THead (Flat Appl) w v) t0)) (let H_y \def (H3 x H6) in (let H9
+\def (ty3_unique g c0 v x H_y (THead (Bind Abst) u0 t) H2) in (ex_ind T
+(\lambda (t0: T).(ty3 g c0 x t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead
+(Flat Appl) w x)) (\lambda (x0: T).(\lambda (H10: (ty3 g c0 x x0)).(ex_ind T
+(\lambda (t0: T).(ty3 g c0 u0 t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead
+(Flat Appl) w x)) (\lambda (x1: T).(\lambda (_: (ty3 g c0 u0 x1)).(ex_ind T
+(\lambda (t0: T).(ty3 g c0 (THead (Bind Abst) u0 t) t0)) (ty3 g c0 (THead
+(Flat Appl) w v) (THead (Flat Appl) w x)) (\lambda (x2: T).(\lambda (H12:
+(ty3 g c0 (THead (Bind Abst) u0 t) x2)).(ex3_2_ind T T (\lambda (t3:
+T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u0 t3) x2))) (\lambda (_:
+T).(\lambda (t0: T).(ty3 g c0 u0 t0))) (\lambda (t3: T).(\lambda (_: T).(ty3
+g (CHead c0 (Bind Abst) u0) t t3))) (ty3 g c0 (THead (Flat Appl) w v) (THead
+(Flat Appl) w x)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (_: (pc3 c0
+(THead (Bind Abst) u0 x3) x2)).(\lambda (H14: (ty3 g c0 u0 x4)).(\lambda
+(H15: (ty3 g (CHead c0 (Bind Abst) u0) t x3)).(ty3_conv g c0 (THead (Flat
+Appl) w x) (THead (Flat Appl) w (THead (Bind Abst) u0 x3)) (ty3_appl g c0 w
+u0 H0 x x3 (ty3_sconv g c0 x x0 H10 (THead (Bind Abst) u0 t) (THead (Bind
+Abst) u0 x3) (ty3_bind g c0 u0 x4 H14 Abst t x3 H15) H9)) (THead (Flat Appl)
+w v) (THead (Flat Appl) w (THead (Bind Abst) u0 t)) (ty3_appl g c0 w u0 H0 v
+t H2) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) x (ty3_unique g c0 v (THead
+(Bind Abst) u0 t) H2 x H_y) w Appl))))))) (ty3_gen_bind g Abst c0 u0 t x2
+H12)))) (ty3_correct g c0 v (THead (Bind Abst) u0 t) H2)))) (ty3_correct g c0
+w u0 H0)))) (ty3_correct g c0 v x H_y)))) t2 H8))))) H5))))))))))))))
+(\lambda (c0: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (ty3 g c0 t2
+t3)).(\lambda (H1: ((\forall (t4: T).((sty0 g c0 t2 t4) \to (ty3 g c0 t2
+t4))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t3 t0)).(\lambda (H3:
+((\forall (t4: T).((sty0 g c0 t3 t4) \to (ty3 g c0 t3 t4))))).(\lambda (t4:
+T).(\lambda (H4: (sty0 g c0 (THead (Flat Cast) t3 t2) t4)).(let H_x \def
+(sty0_gen_cast g c0 t3 t2 t4 H4) in (let H5 \def H_x in (ex3_2_ind T T
+(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 t3 v2))) (\lambda (_: T).(\lambda
+(t5: T).(sty0 g c0 t2 t5))) (\lambda (v2: T).(\lambda (t5: T).(eq T t4 (THead
+(Flat Cast) v2 t5)))) (ty3 g c0 (THead (Flat Cast) t3 t2) t4) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H6: (sty0 g c0 t3 x0)).(\lambda (H7: (sty0 g c0
+t2 x1)).(\lambda (H8: (eq T t4 (THead (Flat Cast) x0 x1))).(let H9 \def
+(f_equal T T (\lambda (e: T).e) t4 (THead (Flat Cast) x0 x1) H8) in (eq_ind_r
+T (THead (Flat Cast) x0 x1) (\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t3
+t2) t)) (let H_y \def (H1 x1 H7) in (let H_y0 \def (H3 x0 H6) in (let H10
+\def (ty3_unique g c0 t2 x1 H_y t3 H0) in (ex_ind T (\lambda (t: T).(ty3 g c0
+x0 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0 x1))
+(\lambda (x: T).(\lambda (H11: (ty3 g c0 x0 x)).(ex_ind T (\lambda (t:
+T).(ty3 g c0 x1 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0
+x1)) (\lambda (x2: T).(\lambda (H12: (ty3 g c0 x1 x2)).(ty3_conv g c0 (THead
+(Flat Cast) x0 x1) (THead (Flat Cast) x x0) (ty3_cast g c0 x1 x0 (ty3_sconv g
+c0 x1 x2 H12 t3 x0 H_y0 H10) x H11) (THead (Flat Cast) t3 t2) (THead (Flat
+Cast) x0 t3) (ty3_cast g c0 t2 t3 H0 x0 H_y0) (pc3_thin_dx c0 t3 x1
+(ty3_unique g c0 t2 t3 H0 x1 H_y) x0 Cast)))) (ty3_correct g c0 t2 x1 H_y))))
+(ty3_correct g c0 t3 x0 H_y0))))) t4 H9))))))) H5))))))))))))) c u t1 H))))).
+
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/ty3/pr3_props.ma".
-
-include "LambdaDelta-1/tau0/fwd.ma".
-
-theorem ty3_tau0:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u
-t1) \to (\forall (t2: T).((tau0 g c u t2) \to (ty3 g c u t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H:
-(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (_:
-T).(\forall (t2: T).((tau0 g c0 t t2) \to (ty3 g c0 t t2)))))) (\lambda (c0:
-C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda
-(_: ((\forall (t3: T).((tau0 g c0 t2 t3) \to (ty3 g c0 t2 t3))))).(\lambda
-(u0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c0 u0 t3)).(\lambda (H3:
-((\forall (t4: T).((tau0 g c0 u0 t4) \to (ty3 g c0 u0 t4))))).(\lambda (_:
-(pc3 c0 t3 t2)).(\lambda (t0: T).(\lambda (H5: (tau0 g c0 u0 t0)).(H3 t0
-H5))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (t2: T).(\lambda
-(H0: (tau0 g c0 (TSort m) t2)).(let H_y \def (tau0_gen_sort g c0 t2 m H0) in
-(let H1 \def (f_equal T T (\lambda (e: T).e) t2 (TSort (next g m)) H_y) in
-(eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 (TSort m) t))
-(ty3_sort g c0 m) t2 H1))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda
-(d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr)
-u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall
-(t2: T).((tau0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: T).(\lambda
-(H3: (tau0 g c0 (TLRef n) t2)).(let H_x \def (tau0_gen_lref g c0 t2 n H3) in
-(let H4 \def H_x in (or_ind (ex3_3 C T T (\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e:
-C).(\lambda (u1: T).(\lambda (t0: T).(tau0 g e u1 t0)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0)))))) (ex3_3 C
-T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e
-(Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(tau0 g
-e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift
-(S n) O u1)))))) (ty3 g c0 (TLRef n) t2) (\lambda (H5: (ex3_3 C T T (\lambda
-(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
-u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(tau0 g e u1 t0))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O
-t0))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1:
-T).(\lambda (t0: T).(tau0 g e u1 t0)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))) (ty3 g c0 (TLRef n) t2)
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0
-(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (tau0 g x0 x1 x2)).(\lambda (H8:
-(eq T t2 (lift (S n) O x2))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2
-(lift (S n) O x2) H8) in (eq_ind_r T (lift (S n) O x2) (\lambda (t0: T).(ty3
-g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda
-(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d
-(Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H11 \def (f_equal
-C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0)
-(CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead
-x0 (Bind Abbr) x1) H6)) in ((let H12 \def (f_equal C T (\lambda (e: C).(match
-e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _
-t0) \Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1)
-(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in
-(\lambda (H13: (eq C d x0)).(let H14 \def (eq_ind_r T x1 (\lambda (t0:
-T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H10 u0 H12) in (let H15 \def
-(eq_ind_r T x1 (\lambda (t0: T).(tau0 g x0 t0 x2)) H7 u0 H12) in (let H16
-\def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0)))
-H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1: C).(tau0 g c1 u0
-x2)) H15 d H13) in (ty3_abbr g n c0 d u0 H16 x2 (H2 x2 H17)))))))) H11))) t2
-H9)))))))) H5)) (\lambda (H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e:
-C).(\lambda (u1: T).(\lambda (t0: T).(tau0 g e u1 t0)))) (\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O
-u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1:
-T).(\lambda (t0: T).(tau0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0 (TLRef n) t2)
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0
-(CHead x0 (Bind Abst) x1))).(\lambda (_: (tau0 g x0 x1 x2)).(\lambda (H8: (eq
-T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2
-(lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1) (\lambda (t0: T).(ty3
-g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda
-(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d
-(Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H11 \def (eq_ind
-C (CHead d (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda
-(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match
-b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst
-\Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow
-False])])) I (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0)
-n H0 (CHead x0 (Bind Abst) x1) H6)) in (False_ind (ty3 g c0 (TLRef n) (lift
-(S n) O x1)) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\lambda (n:
-nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n
-c0 (CHead d (Bind Abst) u0))).(\lambda (t: T).(\lambda (H1: (ty3 g d u0
-t)).(\lambda (_: ((\forall (t2: T).((tau0 g d u0 t2) \to (ty3 g d u0
-t2))))).(\lambda (t2: T).(\lambda (H3: (tau0 g c0 (TLRef n) t2)).(let H_x
-\def (tau0_gen_lref g c0 t2 n H3) in (let H4 \def H_x in (or_ind (ex3_3 C T T
-(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(tau0 g e u1
-t0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n)
-O t0)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1:
-T).(\lambda (t0: T).(tau0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(eq T t2 (lift (S n) O u1)))))) (ty3 g c0 (TLRef n) t2)
-(\lambda (H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1:
-T).(\lambda (t0: T).(tau0 g e u1 t0)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))))).(ex3_3_ind C T T
-(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(tau0 g e u1
-t0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n)
-O t0))))) (ty3 g c0 (TLRef n) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda
-(x2: T).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (_:
-(tau0 g x0 x1 x2)).(\lambda (H8: (eq T t2 (lift (S n) O x2))).(let H9 \def
-(f_equal T T (\lambda (e: T).e) t2 (lift (S n) O x2) H8) in (eq_ind_r T (lift
-(S n) O x2) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (let H10 \def (eq_ind C
-(CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind
-Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr)
-x1) H6)) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee:
-C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
-False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop)
-with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with
-[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) |
-(Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0
-(CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind
-(ty3 g c0 (TLRef n) (lift (S n) O x2)) H11))) t2 H9)))))))) H5)) (\lambda
-(H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0
-(CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0:
-T).(tau0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T
-t2 (lift (S n) O u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e:
-C).(\lambda (u1: T).(\lambda (t0: T).(tau0 g e u1 t0)))) (\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0
-(TLRef n) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda
-(H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7: (tau0 g x0 x1
-x2)).(\lambda (H8: (eq T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T
-(\lambda (e: T).e) t2 (lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1)
-(\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d
-(Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1)
-(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in
-(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_:
-C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead
-d (Bind Abst) u0) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind
-Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in ((let H12 \def (f_equal C T
-(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abst) u0)
-(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead
-x0 (Bind Abst) x1) H6)) in (\lambda (H13: (eq C d x0)).(let H14 \def
-(eq_ind_r T x1 (\lambda (t0: T).(getl n c0 (CHead x0 (Bind Abst) t0))) H10 u0
-H12) in (let H15 \def (eq_ind_r T x1 (\lambda (t0: T).(tau0 g x0 t0 x2)) H7
-u0 H12) in (eq_ind T u0 (\lambda (t0: T).(ty3 g c0 (TLRef n) (lift (S n) O
-t0))) (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1
-(Bind Abst) u0))) H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1:
-C).(tau0 g c1 u0 x2)) H15 d H13) in (ty3_abst g n c0 d u0 H16 t H1))) x1
-H12))))) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\lambda (c0: C).(\lambda
-(u0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u0 t)).(\lambda (_: ((\forall
-(t2: T).((tau0 g c0 u0 t2) \to (ty3 g c0 u0 t2))))).(\lambda (b: B).(\lambda
-(t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t2
-t3)).(\lambda (H3: ((\forall (t4: T).((tau0 g (CHead c0 (Bind b) u0) t2 t4)
-\to (ty3 g (CHead c0 (Bind b) u0) t2 t4))))).(\lambda (t0: T).(\lambda (H4:
-(tau0 g c0 (THead (Bind b) u0 t2) t0)).(let H_x \def (tau0_gen_bind g b c0 u0
-t2 t0 H4) in (let H5 \def H_x in (ex2_ind T (\lambda (t4: T).(tau0 g (CHead
-c0 (Bind b) u0) t2 t4)) (\lambda (t4: T).(eq T t0 (THead (Bind b) u0 t4)))
-(ty3 g c0 (THead (Bind b) u0 t2) t0) (\lambda (x: T).(\lambda (H6: (tau0 g
-(CHead c0 (Bind b) u0) t2 x)).(\lambda (H7: (eq T t0 (THead (Bind b) u0
-x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u0 x)
-H7) in (eq_ind_r T (THead (Bind b) u0 x) (\lambda (t4: T).(ty3 g c0 (THead
-(Bind b) u0 t2) t4)) (ty3_bind g c0 u0 t H0 b t2 x (H3 x H6)) t0 H8)))))
-H5))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda
-(H0: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: T).((tau0 g c0 w t2) \to
-(ty3 g c0 w t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v
-(THead (Bind Abst) u0 t))).(\lambda (H3: ((\forall (t2: T).((tau0 g c0 v t2)
-\to (ty3 g c0 v t2))))).(\lambda (t2: T).(\lambda (H4: (tau0 g c0 (THead
-(Flat Appl) w v) t2)).(let H_x \def (tau0_gen_appl g c0 w v t2 H4) in (let H5
-\def H_x in (ex2_ind T (\lambda (t3: T).(tau0 g c0 v t3)) (\lambda (t3:
-T).(eq T t2 (THead (Flat Appl) w t3))) (ty3 g c0 (THead (Flat Appl) w v) t2)
-(\lambda (x: T).(\lambda (H6: (tau0 g c0 v x)).(\lambda (H7: (eq T t2 (THead
-(Flat Appl) w x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t2 (THead
-(Flat Appl) w x) H7) in (eq_ind_r T (THead (Flat Appl) w x) (\lambda (t0:
-T).(ty3 g c0 (THead (Flat Appl) w v) t0)) (let H_y \def (H3 x H6) in (let H9
-\def (ty3_unique g c0 v x H_y (THead (Bind Abst) u0 t) H2) in (ex_ind T
-(\lambda (t0: T).(ty3 g c0 x t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead
-(Flat Appl) w x)) (\lambda (x0: T).(\lambda (H10: (ty3 g c0 x x0)).(ex_ind T
-(\lambda (t0: T).(ty3 g c0 u0 t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead
-(Flat Appl) w x)) (\lambda (x1: T).(\lambda (_: (ty3 g c0 u0 x1)).(ex_ind T
-(\lambda (t0: T).(ty3 g c0 (THead (Bind Abst) u0 t) t0)) (ty3 g c0 (THead
-(Flat Appl) w v) (THead (Flat Appl) w x)) (\lambda (x2: T).(\lambda (H12:
-(ty3 g c0 (THead (Bind Abst) u0 t) x2)).(ex3_2_ind T T (\lambda (t3:
-T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u0 t3) x2))) (\lambda (_:
-T).(\lambda (t0: T).(ty3 g c0 u0 t0))) (\lambda (t3: T).(\lambda (_: T).(ty3
-g (CHead c0 (Bind Abst) u0) t t3))) (ty3 g c0 (THead (Flat Appl) w v) (THead
-(Flat Appl) w x)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (_: (pc3 c0
-(THead (Bind Abst) u0 x3) x2)).(\lambda (H14: (ty3 g c0 u0 x4)).(\lambda
-(H15: (ty3 g (CHead c0 (Bind Abst) u0) t x3)).(ty3_conv g c0 (THead (Flat
-Appl) w x) (THead (Flat Appl) w (THead (Bind Abst) u0 x3)) (ty3_appl g c0 w
-u0 H0 x x3 (ty3_sconv g c0 x x0 H10 (THead (Bind Abst) u0 t) (THead (Bind
-Abst) u0 x3) (ty3_bind g c0 u0 x4 H14 Abst t x3 H15) H9)) (THead (Flat Appl)
-w v) (THead (Flat Appl) w (THead (Bind Abst) u0 t)) (ty3_appl g c0 w u0 H0 v
-t H2) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) x (ty3_unique g c0 v (THead
-(Bind Abst) u0 t) H2 x H_y) w Appl))))))) (ty3_gen_bind g Abst c0 u0 t x2
-H12)))) (ty3_correct g c0 v (THead (Bind Abst) u0 t) H2)))) (ty3_correct g c0
-w u0 H0)))) (ty3_correct g c0 v x H_y)))) t2 H8))))) H5))))))))))))))
-(\lambda (c0: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (ty3 g c0 t2
-t3)).(\lambda (H1: ((\forall (t4: T).((tau0 g c0 t2 t4) \to (ty3 g c0 t2
-t4))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t3 t0)).(\lambda (H3:
-((\forall (t4: T).((tau0 g c0 t3 t4) \to (ty3 g c0 t3 t4))))).(\lambda (t4:
-T).(\lambda (H4: (tau0 g c0 (THead (Flat Cast) t3 t2) t4)).(let H_x \def
-(tau0_gen_cast g c0 t3 t2 t4 H4) in (let H5 \def H_x in (ex3_2_ind T T
-(\lambda (v2: T).(\lambda (_: T).(tau0 g c0 t3 v2))) (\lambda (_: T).(\lambda
-(t5: T).(tau0 g c0 t2 t5))) (\lambda (v2: T).(\lambda (t5: T).(eq T t4 (THead
-(Flat Cast) v2 t5)))) (ty3 g c0 (THead (Flat Cast) t3 t2) t4) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H6: (tau0 g c0 t3 x0)).(\lambda (H7: (tau0 g c0
-t2 x1)).(\lambda (H8: (eq T t4 (THead (Flat Cast) x0 x1))).(let H9 \def
-(f_equal T T (\lambda (e: T).e) t4 (THead (Flat Cast) x0 x1) H8) in (eq_ind_r
-T (THead (Flat Cast) x0 x1) (\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t3
-t2) t)) (let H_y \def (H1 x1 H7) in (let H_y0 \def (H3 x0 H6) in (let H10
-\def (ty3_unique g c0 t2 x1 H_y t3 H0) in (ex_ind T (\lambda (t: T).(ty3 g c0
-x0 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0 x1))
-(\lambda (x: T).(\lambda (H11: (ty3 g c0 x0 x)).(ex_ind T (\lambda (t:
-T).(ty3 g c0 x1 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0
-x1)) (\lambda (x2: T).(\lambda (H12: (ty3 g c0 x1 x2)).(ty3_conv g c0 (THead
-(Flat Cast) x0 x1) (THead (Flat Cast) x x0) (ty3_cast g c0 x1 x0 (ty3_sconv g
-c0 x1 x2 H12 t3 x0 H_y0 H10) x H11) (THead (Flat Cast) t3 t2) (THead (Flat
-Cast) x0 t3) (ty3_cast g c0 t2 t3 H0 x0 H_y0) (pc3_thin_dx c0 t3 x1
-(ty3_unique g c0 t2 t3 H0 x1 H_y) x0 Cast)))) (ty3_correct g c0 t2 x1 H_y))))
-(ty3_correct g c0 t3 x0 H_y0))))) t4 H9))))))) H5))))))))))))) c u t1 H))))).
-
include "Legacy-2/preamble.ma".
-inline procedural "Legacy-1/coq/defs.ma".
-
default "absurd"
cic:/matita/LAMBDA-TYPES/Legacy-1/coq/props/absurd.con.
-interpretation "Coq's natural plus" 'plus x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/plus.con x y).
-interpretation "Coq's natural minus" 'minus x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/minus.con x y).
-interpretation "Coq's logical and" 'and x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/land.ind#xpointer(1/1) x y).
-interpretation "Coq's logical or" 'or x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/or.ind#xpointer(1/1) x y).
-interpretation "Coq's logical not" 'not x = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/not.con x).
-interpretation "Coq's exists" 'exists \eta.x = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/ex.ind#xpointer(1/1) _ x).
-interpretation "Coq's natural 'less or equal to'" 'leq x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/le.ind#xpointer(1/1) x y).
-interpretation "Coq's natural 'less than'" 'lt x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/lt.con x y).
-interpretation "Coq's leibnitz's equality" 'eq x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/eq.ind#xpointer(1/1) _ x y).
+interpretation "Coq 7.3.1 natural plus" 'plus x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/plus.con x y).
+interpretation "Coq 7.3.1 natural minus" 'minus x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/minus.con x y).
+interpretation "Coq 7.3.1 logical and" 'and x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/land.ind#xpointer(1/1) x y).
+interpretation "Coq 7.3.1 logical or" 'or x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/or.ind#xpointer(1/1) x y).
+interpretation "Coq 7.3.1 logical not" 'not x = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/not.con x).
+interpretation "Coq 7.3.1 exists" 'exists \eta.x = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/ex.ind#xpointer(1/1) _ x).
+interpretation "Coq 7.3.1 natural 'less or equal to'" 'leq x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/le.ind#xpointer(1/1) x y).
+interpretation "Coq 7.3.1 natural 'less than'" 'lt x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/lt.con x y).
+interpretation "Coq 7.3.1 leibnitz's equality" 'eq x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/eq.ind#xpointer(1/1) _ x y).
+
+alias symbol "plus" = "Coq 7.3.1 natural plus".
+alias symbol "minus" = "Coq 7.3.1 natural minus".
+alias symbol "and" = "Coq 7.3.1 logical and".
+alias symbol "or" = "Coq 7.3.1 logical or".
+alias symbol "not" = "Coq 7.3.1 logical not".
+alias symbol "exists" = "Coq 7.3.1 exists".
+alias symbol "leq" = "Coq 7.3.1 natural 'less or equal to'".
+alias symbol "lt" = "Coq 7.3.1 natural 'less than'".
+alias symbol "eq" = "Coq 7.3.1 leibnitz's equality".
theorem f_equal_sym: \forall A,B:Set.\forall f:A\to B.\forall x,y.
x = y \to (f y) = (f x).
DIR=$(shell basename $$PWD)
-MMAS = $(shell find Legacy-2 -name "*.mma") # Base-2
+MMAS = $(shell find Legacy-2 -name "*.mma")
+# Base-2
MAS = $(MMAS:%.mma=%.ma)
-XMAS = Legacy-2/theory.ma # Base-2/theory.ma LambdaDelta-2/theory.ma
+XMAS = Legacy-2/theory.ma Base-2/theory.ma LambdaDelta-2/theory.ma
$(DIR) all: depends
$(H)$(MAKE) H=$(H) --no-print-directory build
depend:
@echo matitadep
$(H)$(BIN)matitadep $(foreach FILE,$(XMAS),-exclude $(FILE))
- $(H)cat Legacy-2/depends >> depends # Base-2/depends
+ $(H)cat Legacy-2/depends >> depends
+# Base-2/depends
depend.opt:
@echo matitadep.opt
$(H)$(BIN)matitadep.opt $(foreach FILE,$(XMAS),-exclude $(FILE))
- $(H)cat Legacy-2/depends >> depends # Base-2/depends
+ $(H)cat Legacy-2/depends >> depends
+# Base-2/depends
depends: depend.opt
projects / helm.git / commitdiff