--- /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))))).
+
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