X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLevel-1%2FLambdaDelta%2Fty3%2Fdec.ma;fp=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLevel-1%2FLambdaDelta%2Fty3%2Fdec.ma;h=efbd85e19ff246a3de0f74c36e4dffa028bada04;hb=745585f18b0f5232214e1199ca1d4985f0238836;hp=0000000000000000000000000000000000000000;hpb=cb80cf01946bbaa887fd609b3c808032f1f5e29f;p=helm.git

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/dec.ma
new file mode 100644
index 000000000..efbd85e19
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/dec.ma
@@ -0,0 +1,456 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/dec".
+
+include "ty3/pr3_props.ma".
+
+include "pc3/dec.ma".
+
+include "getl/flt.ma".
+
+include "getl/dec.ma".
+
+theorem ty3_inference:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(or (ex T (\lambda (t2: 
+T).(ty3 g c t1 t2))) (\forall (t2: T).((ty3 g c t1 t2) \to (\forall (P: 
+Prop).P))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(flt_wf_ind (\lambda (c0: 
+C).(\lambda (t: T).(or (ex T (\lambda (t2: T).(ty3 g c0 t t2))) (\forall (t2: 
+T).((ty3 g c0 t t2) \to (\forall (P: Prop).P)))))) (\lambda (c2: C).(\lambda 
+(t2: T).(match t2 in T return (\lambda (t: T).(((\forall (c1: C).(\forall 
+(t3: T).((flt c1 t3 c2 t) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) 
+(\forall (t4: T).((ty3 g c1 t3 t4) \to (\forall (P: Prop).P)))))))) \to (or 
+(ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) 
+\to (\forall (P: Prop).P)))))) with [(TSort n) \Rightarrow (\lambda (_: 
+((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (TSort n)) \to (or (ex T 
+(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to 
+(\forall (P: Prop).P))))))))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 
+(TSort n) t3))) (\forall (t3: T).((ty3 g c2 (TSort n) t3) \to (\forall (P: 
+Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (TSort n) t3)) (TSort (next 
+g n)) (ty3_sort g c2 n)))) | (TLRef n) \Rightarrow (\lambda (H: ((\forall 
+(c1: C).(\forall (t3: T).((flt c1 t3 c2 (TLRef n)) \to (or (ex T (\lambda 
+(t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to (\forall 
+(P: Prop).P))))))))).(let H_x \def (getl_dec c2 n) in (let H0 \def H_x in 
+(or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n 
+c2 (CHead e (Bind b) v)))))) (\forall (d: C).((getl n c2 d) \to (\forall (P: 
+Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: 
+T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P)))) (\lambda (H1: (ex_3 
+C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n c2 (CHead e 
+(Bind b) v))))))).(ex_3_ind C B T (\lambda (e: C).(\lambda (b: B).(\lambda 
+(v: T).(getl n c2 (CHead e (Bind b) v))))) (or (ex T (\lambda (t3: T).(ty3 g 
+c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: 
+Prop).P)))) (\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: 
+(getl n c2 (CHead x0 (Bind x1) x2))).(let H3 \def (H x0 x2 (getl_flt x1 c2 x0 
+x2 n H2)) in (or_ind (ex T (\lambda (t3: T).(ty3 g x0 x2 t3))) (\forall (t3: 
+T).((ty3 g x0 x2 t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: 
+T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to 
+(\forall (P: Prop).P)))) (\lambda (H4: (ex T (\lambda (t3: T).(ty3 g x0 x2 
+t3)))).(ex_ind T (\lambda (t3: T).(ty3 g x0 x2 t3)) (or (ex T (\lambda (t3: 
+T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to 
+(\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H5: (ty3 g x0 x2 
+x)).((match x1 in B return (\lambda (b: B).((getl n c2 (CHead x0 (Bind b) 
+x2)) \to (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: 
+T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P)))))) with [Abbr 
+\Rightarrow (\lambda (H6: (getl n c2 (CHead x0 (Bind Abbr) x2))).(or_introl 
+(ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 
+(TLRef n) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g 
+c2 (TLRef n) t3)) (lift (S n) O x) (ty3_abbr g n c2 x0 x2 H6 x H5)))) | Abst 
+\Rightarrow (\lambda (H6: (getl n c2 (CHead x0 (Bind Abst) x2))).(or_introl 
+(ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 
+(TLRef n) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g 
+c2 (TLRef n) t3)) (lift (S n) O x2) (ty3_abst g n c2 x0 x2 H6 x H5)))) | Void 
+\Rightarrow (\lambda (H6: (getl n c2 (CHead x0 (Bind Void) x2))).(or_intror 
+(ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 
+(TLRef n) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H7: (ty3 
+g c2 (TLRef n) t3)).(\lambda (P: Prop).(or_ind (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda 
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T 
+T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) 
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t))))) P (\lambda (H8: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
+(t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) 
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t)))) P (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 
+c2 (lift (S n) O x5) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind Abbr) 
+x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 (Bind 
+Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abbr) x4) 
+(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) 
+in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\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 False | Void \Rightarrow True]) | (Flat 
+_) \Rightarrow False])])) I (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 
+(Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) in (False_ind P 
+H13))))))))) H8)) (\lambda (H8: (ex3_3 C T T (\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda 
+(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T 
+(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) 
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t)))) P (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 
+c2 (lift (S n) O x4) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind Abst) 
+x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 (Bind 
+Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abst) x4) 
+(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) 
+in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\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 False | Void \Rightarrow True]) | (Flat 
+_) \Rightarrow False])])) I (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 
+(Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) in (False_ind P 
+H13))))))))) H8)) (ty3_gen_lref g c2 t3 n H7)))))))]) H2))) H4)) (\lambda 
+(H4: ((\forall (t3: T).((ty3 g x0 x2 t3) \to (\forall (P: 
+Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) 
+(\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P))) 
+(\lambda (t3: T).(\lambda (H5: (ty3 g c2 (TLRef n) t3)).(\lambda (P: 
+Prop).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: 
+T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda 
+(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) P (\lambda 
+(H6: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 
+(lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
+n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
+T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: 
+T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda 
+(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) P (\lambda (x3: 
+C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (lift (S n) O x5) 
+t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abbr) x4))).(\lambda (H9: (ty3 
+g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind x1) x2) (\lambda (c0: 
+C).(getl n c2 c0)) H2 (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind 
+x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in (let H11 \def (f_equal C C 
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
+\Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0])) (CHead x0 (Bind x1) x2) 
+(CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead 
+x3 (Bind Abbr) x4) H8)) in ((let H12 \def (f_equal C B (\lambda (e: C).(match 
+e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow x1 | (CHead _ k 
+_) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0 (Bind x1) x2) (CHead 
+x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind 
+Abbr) x4) H8)) in ((let H13 \def (f_equal C T (\lambda (e: C).(match e in C 
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow x2 | (CHead _ _ t) 
+\Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono 
+c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in (\lambda 
+(_: (eq B x1 Abbr)).(\lambda (H15: (eq C x0 x3)).(let H16 \def (eq_ind_r T x4 
+(\lambda (t: T).(getl n c2 (CHead x3 (Bind Abbr) t))) H10 x2 H13) in (let H17 
+\def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) H9 x2 H13) in (let H18 
+\def (eq_ind_r C x3 (\lambda (c0: C).(getl n c2 (CHead c0 (Bind Abbr) x2))) 
+H16 x0 H15) in (let H19 \def (eq_ind_r C x3 (\lambda (c0: C).(ty3 g c0 x2 
+x5)) H17 x0 H15) in (H4 x5 H19 P)))))))) H12)) H11))))))))) H6)) (\lambda 
+(H6: (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 
+(lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
+n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
+T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda 
+(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) P (\lambda (x3: 
+C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H7: (pc3 c2 (lift (S n) O x4) 
+t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abst) x4))).(\lambda (H9: (ty3 
+g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind x1) x2) (\lambda (c0: 
+C).(getl n c2 c0)) H2 (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind 
+x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in (let H11 \def (f_equal C C 
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
+\Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0])) (CHead x0 (Bind x1) x2) 
+(CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead 
+x3 (Bind Abst) x4) H8)) in ((let H12 \def (f_equal C B (\lambda (e: C).(match 
+e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow x1 | (CHead _ k 
+_) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0 (Bind x1) x2) (CHead 
+x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind 
+Abst) x4) H8)) in ((let H13 \def (f_equal C T (\lambda (e: C).(match e in C 
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow x2 | (CHead _ _ t) 
+\Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono 
+c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in (\lambda 
+(_: (eq B x1 Abst)).(\lambda (H15: (eq C x0 x3)).(let H16 \def (eq_ind_r T x4 
+(\lambda (t: T).(getl n c2 (CHead x3 (Bind Abst) t))) H10 x2 H13) in (let H17 
+\def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) H9 x2 H13) in (let H18 
+\def (eq_ind_r T x4 (\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)) H7 x2 H13) 
+in (let H19 \def (eq_ind_r C x3 (\lambda (c0: C).(getl n c2 (CHead c0 (Bind 
+Abst) x2))) H16 x0 H15) in (let H20 \def (eq_ind_r C x3 (\lambda (c0: C).(ty3 
+g c0 x2 x5)) H17 x0 H15) in (H4 x5 H20 P))))))))) H12)) H11))))))))) H6)) 
+(ty3_gen_lref g c2 t3 n H5))))))) H3)))))) H1)) (\lambda (H1: ((\forall (d: 
+C).((getl n c2 d) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda 
+(t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) 
+\to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H2: (ty3 g c2 (TLRef 
+n) t3)).(\lambda (P: Prop).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: 
+T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda 
+(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda 
+(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind 
+Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t))))) P (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
+(t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) 
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t)))) P (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 
+c2 (lift (S n) O x2) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abbr) 
+x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abbr) x1) H5 
+P))))))) H3)) (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda 
+(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T 
+(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) 
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t)))) P (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 
+c2 (lift (S n) O x1) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abst) 
+x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abst) x1) H5 
+P))))))) H3)) (ty3_gen_lref g c2 t3 n H2))))))) H0)))) | (THead k t t0) 
+\Rightarrow (\lambda (H: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 
+(THead k t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall 
+(t4: T).((ty3 g c1 t3 t4) \to (\forall (P: Prop).P))))))))).((match k in K 
+return (\lambda (k0: K).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 
+(THead k0 t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall 
+(t4: T).((ty3 g c1 t3 t4) \to (\forall (P: Prop).P)))))))) \to (or (ex T 
+(\lambda (t3: T).(ty3 g c2 (THead k0 t t0) t3))) (\forall (t3: T).((ty3 g c2 
+(THead k0 t t0) t3) \to (\forall (P: Prop).P)))))) with [(Bind b) \Rightarrow 
+(\lambda (H0: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Bind 
+b) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: 
+T).((ty3 g c1 t3 t4) \to (\forall (P: Prop).P))))))))).(let H1 \def (H0 c2 t 
+(flt_thead_sx (Bind b) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 
+t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: Prop).P))) (or (ex 
+T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: 
+T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda 
+(H2: (ex T (\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda (t3: 
+T).(ty3 g c2 t t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t 
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall 
+(P: Prop).P)))) (\lambda (x: T).(\lambda (H3: (ty3 g c2 t x)).(let H4 \def 
+(H0 (CHead c2 (Bind b) t) t0 (flt_shift (Bind b) c2 t t0)) in (or_ind (ex T 
+(\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3))) (\forall (t3: T).((ty3 
+g (CHead c2 (Bind b) t) t0 t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda 
+(t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 
+(THead (Bind b) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H5: (ex T 
+(\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3)))).(ex_ind T (\lambda 
+(t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3)) (or (ex T (\lambda (t3: T).(ty3 
+g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) 
+t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x0: T).(\lambda (H6: (ty3 g 
+(CHead c2 (Bind b) t) t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g (CHead c2 
+(Bind b) t) x0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t 
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall 
+(P: Prop).P)))) (\lambda (x1: T).(\lambda (H7: (ty3 g (CHead c2 (Bind b) t) 
+x0 x1)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) 
+t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: 
+Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3)) 
+(THead (Bind b) t x0) (ty3_bind g c2 t x H3 b t0 x0 H6 x1 H7))))) 
+(ty3_correct g (CHead c2 (Bind b) t) t0 x0 H6)))) H5)) (\lambda (H5: 
+((\forall (t3: T).((ty3 g (CHead c2 (Bind b) t) t0 t3) \to (\forall (P: 
+Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t 
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall 
+(P: Prop).P))) (\lambda (t3: T).(\lambda (H6: (ty3 g c2 (THead (Bind b) t t0) 
+t3)).(\lambda (P: Prop).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: 
+T).(\lambda (_: T).(pc3 c2 (THead (Bind b) t t4) t3)))) (\lambda (_: 
+T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c2 t t5)))) (\lambda (t4: 
+T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 t4)))) 
+(\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead c2 (Bind b) 
+t) t4 t6)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda 
+(_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda (_: (ty3 g c2 t x1)).(\lambda 
+(H9: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(\lambda (_: (ty3 g (CHead c2 (Bind 
+b) t) x0 x2)).(H5 x0 H9 P)))))))) (ty3_gen_bind g b c2 t t0 t3 H6))))))) 
+H4)))) H2)) (\lambda (H2: ((\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: 
+Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t 
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall 
+(P: Prop).P))) (\lambda (t3: T).(\lambda (H3: (ty3 g c2 (THead (Bind b) t t0) 
+t3)).(\lambda (P: Prop).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: 
+T).(\lambda (_: T).(pc3 c2 (THead (Bind b) t t4) t3)))) (\lambda (_: 
+T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c2 t t5)))) (\lambda (t4: 
+T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 t4)))) 
+(\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead c2 (Bind b) 
+t) t4 t6)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda 
+(_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda (H5: (ty3 g c2 t 
+x1)).(\lambda (_: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(\lambda (_: (ty3 g 
+(CHead c2 (Bind b) t) x0 x2)).(H2 x1 H5 P)))))))) (ty3_gen_bind g b c2 t t0 
+t3 H3))))))) H1))) | (Flat f) \Rightarrow (\lambda (H0: ((\forall (c1: 
+C).(\forall (t3: T).((flt c1 t3 c2 (THead (Flat f) t t0)) \to (or (ex T 
+(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to 
+(\forall (P: Prop).P))))))))).((match f in F return (\lambda (f0: 
+F).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Flat f0) t t0)) 
+\to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 
+t3 t4) \to (\forall (P: Prop).P)))))))) \to (or (ex T (\lambda (t3: T).(ty3 g 
+c2 (THead (Flat f0) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat f0) 
+t t0) t3) \to (\forall (P: Prop).P)))))) with [Appl \Rightarrow (\lambda (H1: 
+((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Flat Appl) t t0)) 
+\to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 
+t3 t4) \to (\forall (P: Prop).P))))))))).(let H2 \def (H1 c2 t (flt_thead_sx 
+(Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) 
+(\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: Prop).P))) (or (ex T 
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: 
+T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) 
+(\lambda (H3: (ex T (\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda 
+(t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat 
+Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) 
+\to (\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H4: (ty3 g c2 t 
+x)).(let H5 \def (H1 c2 t0 (flt_thead_dx (Flat Appl) c2 t t0)) in (or_ind (ex 
+T (\lambda (t3: T).(ty3 g c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to 
+(\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat 
+Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) 
+\to (\forall (P: Prop).P)))) (\lambda (H6: (ex T (\lambda (t3: T).(ty3 g c2 
+t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T (\lambda 
+(t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 
+(THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x0: 
+T).(\lambda (H7: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0 
+t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) 
+(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: 
+Prop).P)))) (\lambda (x1: T).(\lambda (H8: (ty3 g c2 x0 x1)).(ex_ind T 
+(\lambda (t3: T).(ty3 g c2 x t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead 
+(Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) 
+t3) \to (\forall (P: Prop).P)))) (\lambda (x2: T).(\lambda (H9: (ty3 g c2 x 
+x2)).(let H10 \def (ty3_sn3 g c2 x x2 H9) in (let H_x \def (nf2_sn3 c2 x H10) 
+in (let H11 \def H_x in (ex2_ind T (\lambda (u: T).(pr3 c2 x u)) (\lambda (u: 
+T).(nf2 c2 u)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) 
+t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall 
+(P: Prop).P)))) (\lambda (x3: T).(\lambda (H12: (pr3 c2 x x3)).(\lambda (H13: 
+(nf2 c2 x3)).(let H14 \def (ty3_sred_pr3 c2 x x3 H12 g x2 H9) in (let H_x0 
+\def (pc3_abst_dec g c2 x0 x1 H8 x3 x2 H14) in (let H15 \def H_x0 in (or_ind 
+(ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 
+u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) 
+x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: 
+T).(\lambda (v2: T).(nf2 c2 v2)))) (\forall (u: T).((pc3 c2 x0 (THead (Bind 
+Abst) x3 u)) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 
+(THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) 
+t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H16: (ex4_2 T T (\lambda (u: 
+T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: 
+T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) (\lambda (_: 
+T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 
+v2))))).(ex4_2_ind T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead 
+(Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind 
+Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda 
+(_: T).(\lambda (v2: T).(nf2 c2 v2))) (or (ex T (\lambda (t3: T).(ty3 g c2 
+(THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) 
+t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (H17: (pc3 c2 x0 (THead (Bind Abst) x3 x4))).(\lambda (H18: (ty3 
+g c2 (THead (Bind Abst) x5 x4) x1)).(\lambda (H19: (pr3 c2 x3 x5)).(\lambda 
+(_: (nf2 c2 x5)).(let H_y \def (nf2_pr3_unfold c2 x3 x5 H19 H13) in (let H21 
+\def (eq_ind_r T x5 (\lambda (t3: T).(pr3 c2 x3 t3)) H19 x3 H_y) in (let H22 
+\def (eq_ind_r T x5 (\lambda (t3: T).(ty3 g c2 (THead (Bind Abst) t3 x4) x1)) 
+H18 x3 H_y) in (or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) 
+t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to 
+(\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat 
+Appl) t t0) t3)) (THead (Flat Appl) t (THead (Bind Abst) x3 x4)) (ty3_appl g 
+c2 t x3 (ty3_tred g c2 t x H4 x3 H12) t0 x4 (ty3_conv g c2 (THead (Bind Abst) 
+x3 x4) x1 H22 t0 x0 H7 H17))))))))))))) H16)) (\lambda (H16: ((\forall (u: 
+T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to (\forall (P: 
+Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t 
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to 
+(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H17: (ty3 g c2 (THead 
+(Flat Appl) t t0) t3)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u: 
+T).(\lambda (t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) 
+t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u 
+t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 t u))) P (\lambda (x4: 
+T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind 
+Abst) x4 x5)) t3)).(\lambda (H19: (ty3 g c2 t0 (THead (Bind Abst) x4 
+x5))).(\lambda (H20: (ty3 g c2 t x4)).(let H_y \def (ty3_unique g c2 t x4 H20 
+x H4) in (let H_y0 \def (ty3_unique g c2 t0 (THead (Bind Abst) x4 x5) H19 x0 
+H7) in (H16 x5 (pc3_t (THead (Bind Abst) x4 x5) c2 x0 (pc3_s c2 x0 (THead 
+(Bind Abst) x4 x5) H_y0) (THead (Bind Abst) x3 x5) (pc3_head_1 c2 x4 x3 
+(pc3_t x c2 x4 H_y x3 (pc3_pr3_r c2 x x3 H12)) (Bind Abst) x5)) P)))))))) 
+(ty3_gen_appl g c2 t t0 t3 H17))))))) H15))))))) H11)))))) (ty3_correct g c2 
+t x H4)))) (ty3_correct g c2 t0 x0 H7)))) H6)) (\lambda (H6: ((\forall (t3: 
+T).((ty3 g c2 t0 t3) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda 
+(t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 
+(THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: 
+T).(\lambda (H7: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(\lambda (P: 
+Prop).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat 
+Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 
+g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 
+t u))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Flat 
+Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (H9: (ty3 g c2 t0 (THead 
+(Bind Abst) x0 x1))).(\lambda (_: (ty3 g c2 t x0)).(H6 (THead (Bind Abst) x0 
+x1) H9 P)))))) (ty3_gen_appl g c2 t t0 t3 H7))))))) H5)))) H3)) (\lambda (H3: 
+((\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: Prop).P))))).(or_intror 
+(ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: 
+T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P))) 
+(\lambda (t3: T).(\lambda (H4: (ty3 g c2 (THead (Flat Appl) t t0) 
+t3)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 
+c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: 
+T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: 
+T).(\lambda (_: T).(ty3 g c2 t u))) P (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) 
+t3)).(\lambda (_: (ty3 g c2 t0 (THead (Bind Abst) x0 x1))).(\lambda (H7: (ty3 
+g c2 t x0)).(H3 x0 H7 P)))))) (ty3_gen_appl g c2 t t0 t3 H4))))))) H2))) | 
+Cast \Rightarrow (\lambda (H1: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3 
+c2 (THead (Flat Cast) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 
+t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to (\forall (P: 
+Prop).P))))))))).(let H2 \def (H1 c2 t (flt_thead_sx (Flat Cast) c2 t t0)) in 
+(or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 
+t t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead 
+(Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) 
+t3) \to (\forall (P: Prop).P)))) (\lambda (H3: (ex T (\lambda (t3: T).(ty3 g 
+c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda 
+(t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 
+(THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x: 
+T).(\lambda (H4: (ty3 g c2 t x)).(let H5 \def (H1 c2 t0 (flt_thead_dx (Flat 
+Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 t0 t3))) (\forall 
+(t3: T).((ty3 g c2 t0 t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: 
+T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 
+(THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H6: (ex T 
+(\lambda (t3: T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 
+t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) 
+(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: 
+Prop).P)))) (\lambda (x0: T).(\lambda (H7: (ty3 g c2 t0 x0)).(ex_ind T 
+(\lambda (t3: T).(ty3 g c2 x0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 
+(THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) 
+t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x1: T).(\lambda (H8: (ty3 g 
+c2 x0 x1)).(let H_x \def (pc3_dec g c2 x0 x1 H8 t x H4) in (let H9 \def H_x 
+in (or_ind (pc3 c2 x0 t) ((pc3 c2 x0 t) \to (\forall (P: Prop).P)) (or (ex T 
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: 
+T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) 
+(\lambda (H10: (pc3 c2 x0 t)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 
+(THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) 
+t t0) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 
+(THead (Flat Cast) t t0) t3)) t (ty3_cast g c2 t0 t (ty3_conv g c2 t x H4 t0 
+x0 H7 H10) x H4)))) (\lambda (H10: (((pc3 c2 x0 t) \to (\forall (P: 
+Prop).P)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t 
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to 
+(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H11: (ty3 g c2 (THead 
+(Flat Cast) t t0) t3)).(\lambda (P: Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 
+t) P (\lambda (_: (pc3 c2 t t3)).(\lambda (H13: (ty3 g c2 t0 t)).(let H_y 
+\def (ty3_unique g c2 t0 t H13 x0 H7) in (H10 (pc3_s c2 x0 t H_y) P)))) 
+(ty3_gen_cast g c2 t0 t t3 H11))))))) H9))))) (ty3_correct g c2 t0 x0 H7)))) 
+H6)) (\lambda (H6: ((\forall (t3: T).((ty3 g c2 t0 t3) \to (\forall (P: 
+Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t 
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to 
+(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H7: (ty3 g c2 (THead (Flat 
+Cast) t t0) t3)).(\lambda (P: Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 t) P 
+(\lambda (_: (pc3 c2 t t3)).(\lambda (H9: (ty3 g c2 t0 t)).(H6 t H9 P))) 
+(ty3_gen_cast g c2 t0 t t3 H7))))))) H5)))) H3)) (\lambda (H3: ((\forall (t3: 
+T).((ty3 g c2 t t3) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda 
+(t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 
+(THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: 
+T).(\lambda (H4: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(\lambda (P: 
+Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 t) P (\lambda (_: (pc3 c2 t 
+t3)).(\lambda (H6: (ty3 g c2 t0 t)).(ex_ind T (\lambda (t4: T).(ty3 g c2 t 
+t4)) P (\lambda (x: T).(\lambda (H7: (ty3 g c2 t x)).(H3 x H7 P))) 
+(ty3_correct g c2 t0 t H6)))) (ty3_gen_cast g c2 t0 t t3 H4))))))) H2)))]) 
+H0))]) H))]))) c t1))).
+