- transcript: patched to generate CoRN_notation.ma instead of CoRN.ma

[helm.git] / matita / contribs / LAMBDA-TYPES / Level-1 / LambdaDelta / ty3 / dec.ma

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/dec.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/dec.ma
index efbd85e19ff246a3de0f74c36e4dffa028bada04..e46c2bacf8bc985d12bff16d9857c188d6978566 100644 (file)
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/dec.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/dec.ma
@@ -32,28 +32,28 @@ Prop).P))))))
  \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: 
+(t2: T).(T_ind (\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)))))) (\lambda (n: nat).(\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))))) (\lambda (n: nat).(\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: 
@@ -63,68 +63,67 @@ T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to
 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 
+x)).(B_ind (\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)))))) (\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)))) (\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)))) (\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)))) 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))) 
+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))))))) 
+x1 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: 
@@ -226,48 +225,56 @@ t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
 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 
+P))))))) H3)) (ty3_gen_lref g c2 t3 n H2))))))) H0))))) (\lambda (k: 
+K).(\lambda (t: T).(\lambda (_: ((((\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))))))).(\lambda (t0: T).(\lambda (_: ((((\forall (c1: C).(\forall 
+(t3: T).((flt c1 t3 c2 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 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) 
+\to (\forall (P: Prop).P))))))).(\lambda (H1: ((\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))))))))).(K_ind (\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)))))) (\lambda (b: 
+B).(\lambda (H2: ((\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 H3 \def (H2 
+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 (H4: (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 (H5: (ty3 g c2 t x)).(let 
+H6 \def (H2 (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 (H7: (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 
+t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x0: T).(\lambda (H8: (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) 
+(P: Prop).P)))) (\lambda (x1: T).(\lambda (H9: (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: 
+(THead (Bind b) t x0) (ty3_bind g c2 t x H5 b t0 x0 H8 x1 H9))))) 
+(ty3_correct g (CHead c2 (Bind b) t) t0 x0 H8)))) H7)) (\lambda (H7: 
 ((\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) 
+(P: Prop).P))) (\lambda (t3: T).(\lambda (H8: (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: 
@@ -275,182 +282,181 @@ 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 
+(H11: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(\lambda (_: (ty3 g (CHead c2 
+(Bind b) t) x0 x2)).(H7 x0 H11 P)))))))) (ty3_gen_bind g b c2 t t0 t3 
+H8))))))) H6)))) H4)) (\lambda (H4: ((\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 (H5: (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 (H7: (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 
+(CHead c2 (Bind b) t) x0 x2)).(H4 x1 H7 P)))))))) (ty3_gen_bind g b c2 t t0 
+t3 H5))))))) H3)))) (\lambda (f: F).(\lambda (H2: ((\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))))))))).(F_ind (\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)))))) (\lambda (H3: ((\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 H4 \def (H3 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 (H5: (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 (x0: 
-T).(\lambda (H7: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0 
+(THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x: 
+T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 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 (H8: (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 (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 
+Prop).P)))) (\lambda (x0: T).(\lambda (H9: (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 (H16: (ex4_2 T T (\lambda (u: 
+t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x1: T).(\lambda (H10: (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 (H11: (ty3 g c2 x x2)).(let H12 \def (ty3_sn3 g c2 x x2 H11) in 
+(let H_x \def (nf2_sn3 c2 x H12) in (let H13 \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 (H14: (pr3 c2 x x3)).(\lambda (H15: (nf2 c2 x3)).(let H16 \def 
+(ty3_sred_pr3 c2 x x3 H14 g x2 H11) in (let H_x0 \def (pc3_abst_dec g c2 x0 
+x1 H10 x3 x2 H16) in (let H17 \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 (H18: (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 
+T).(\lambda (H19: (pc3 c2 x0 (THead (Bind Abst) x3 x4))).(\lambda (H20: (ty3 
+g c2 (THead (Bind Abst) x5 x4) x1)).(\lambda (H21: (pr3 c2 x3 x5)).(\lambda 
+(_: (nf2 c2 x5)).(let H_y \def (nf2_pr3_unfold c2 x3 x5 H21 H15) in (let H23 
+\def (eq_ind_r T x5 (\lambda (t3: T).(pr3 c2 x3 t3)) H21 x3 H_y) in (let H24 
 \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) 
+H20 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: 
+c2 t x3 (ty3_tred g c2 t x H6 x3 H14) t0 x4 (ty3_conv g c2 (THead (Bind Abst) 
+x3 x4) x1 H24 t0 x0 H9 H19))))))))))))) H18)) (\lambda (H18: ((\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 
+(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H19: (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 
+Abst) x4 x5)) t3)).(\lambda (H21: (ty3 g c2 t0 (THead (Bind Abst) x4 
+x5))).(\lambda (H22: (ty3 g c2 t x4)).(let H_y \def (ty3_unique g c2 t x4 H22 
+x H6) in (let H_y0 \def (ty3_unique g c2 t0 (THead (Bind Abst) x4 x5) H21 x0 
+H9) in (H18 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: 
+(pc3_t x c2 x4 H_y x3 (pc3_pr3_r c2 x x3 H14)) (Bind Abst) x5)) P)))))))) 
+(ty3_gen_appl g c2 t t0 t3 H19))))))) H17))))))) H13)))))) (ty3_correct g c2 
+t x H6)))) (ty3_correct g c2 t0 x0 H9)))) H8)) (\lambda (H8: ((\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: 
+T).(\lambda (H9: (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: 
+Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (H11: (ty3 g c2 t0 (THead 
+(Bind Abst) x0 x1))).(\lambda (_: (ty3 g c2 t x0)).(H8 (THead (Bind Abst) x0 
+x1) H11 P)))))) (ty3_gen_appl g c2 t t0 t3 H9))))))) H7)))) H5)) (\lambda 
+(H5: ((\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 (H6: (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)).(\lambda (_: (ty3 g c2 t0 (THead (Bind Abst) x0 x1))).(\lambda (H9: (ty3 
+g c2 t x0)).(H5 x0 H9 P)))))) (ty3_gen_appl g c2 t t0 t3 H6))))))) H4))) 
+(\lambda (H3: ((\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 H4 \def (H3 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 (H5: (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 (H6: (ty3 g c2 t 
+x)).(let H7 \def (H3 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 (H8: (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 (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 
+(THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x0: 
+T).(\lambda (H9: (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 (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 
+Prop).P)))) (\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(let H_x \def 
+(pc3_dec g c2 x0 x1 H10 t x H6) in (let H11 \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 (H12: (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 H6 t0 x0 H9 H12) x H6)))) 
+(\lambda (H12: (((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 (H10: (pc3 c2 x0 t)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 
+T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P))) 
+(\lambda (t3: T).(\lambda (H13: (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 (H15: (ty3 g c2 t0 t)).(let H_y \def (ty3_unique g c2 
+t0 t H15 x0 H9) in (H12 (pc3_s c2 x0 t H_y) P)))) (ty3_gen_cast g c2 t0 t t3 
+H13))))))) H11))))) (ty3_correct g c2 t0 x0 H9)))) H8)) (\lambda (H8: 
+((\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 (H9: (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 (H11: (ty3 g c2 t0 t)).(H8 t H11 P))) (ty3_gen_cast g 
+c2 t0 t t3 H9))))))) H7)))) H5)) (\lambda (H5: ((\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))) (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))).
+t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H6: (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 (H8: (ty3 g c2 t0 
+t)).(ex_ind T (\lambda (t4: T).(ty3 g c2 t t4)) P (\lambda (x: T).(\lambda 
+(H9: (ty3 g c2 t x)).(H5 x H9 P))) (ty3_correct g c2 t0 t H8)))) 
+(ty3_gen_cast g c2 t0 t t3 H6))))))) H4))) f H2))) k H1))))))) t2))) c t1))).