(* This file was automatically generated: do not edit *********************)
-set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csuba/fwd".
-
-include "csuba/defs.ma".
+include "LambdaDelta-1/csuba/defs.ma".
theorem csuba_gen_abbr:
\forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g
(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))
\def
\lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
-(csuba g (CHead d1 (Bind Abbr) u) c)).(let H0 \def (match H in csuba return
-(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0
-(CHead d1 (Bind Abbr) u)) \to ((eq C c1 c) \to (ex2 C (\lambda (d2: C).(eq C
-c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) with
-[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind
-Abbr) u))).(\lambda (H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort
-n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abbr)
-u) H0) in (False_ind ((eq C (CSort n) c) \to (ex2 C (\lambda (d2: C).(eq C c
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) H2)) H1))) |
-(csuba_head c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0)
-(CHead d1 (Bind Abbr) u))).(\lambda (H2: (eq C (CHead c2 k u0) c)).((let H3
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k
-u0) (CHead d1 (Bind Abbr) u) H1) in ((let H4 \def (f_equal C K (\lambda (e:
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
-(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1)
-in ((let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0]))
-(CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in (eq_ind C d1 (\lambda (c0:
-C).((eq K k (Bind Abbr)) \to ((eq T u0 u) \to ((eq C (CHead c2 k u0) c) \to
-((csuba g c0 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr)
-u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (H6: (eq K k (Bind
-Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead
-c2 k0 u0) c) \to ((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead
-d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) (\lambda (H7: (eq
-T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c2 (Bind Abbr) t) c) \to
-((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr)
-u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H8: (eq C (CHead c2
-(Bind Abbr) u) c)).(eq_ind C (CHead c2 (Bind Abbr) u) (\lambda (c0:
-C).((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (H9: (csuba g d1
-c2)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2
-(Bind Abbr) u)) H9)) c H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind Abbr)
-H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a
-H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1
-(Bind Abbr) u))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) c)).((let H5
-\def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e 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 d1 (Bind Abbr) u) H3) in (False_ind ((eq C
-(CHead c2 (Bind Abbr) u0) c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g
-a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) H5)) H4 H0 H1 H2)))])
-in (H0 (refl_equal C (CHead d1 (Bind Abbr) u)) (refl_equal C c))))))).
+(csuba g (CHead d1 (Bind Abbr) u) c)).(insert_eq C (CHead d1 (Bind Abbr) u)
+(\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq
+C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (\lambda
+(y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda
+(c1: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C
+c1 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda
+(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr) u))).(let H2
+\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
+False])) I (CHead d1 (Bind Abbr) u) H1) in (False_ind (ex2 C (\lambda (d2:
+C).(eq C (CSort n) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
+c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c1 k u0)
+(CHead d1 (Bind Abbr) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3) in ((let H5
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k
+u0) (CHead d1 (Bind Abbr) u) H3) in ((let H6 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
+(CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3)
+in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r
+T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k t) (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (eq_ind_r K (Bind Abbr)
+(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u) (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H9 \def (eq_ind C
+c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0
+c2)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr)
+u) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) c2
+(refl_equal C (CHead c2 (Bind Abbr) u)) H10))) k H7) u0 H6)))) H5))
+H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1
+c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead d1
+(Bind Abbr) u))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\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 b0) \Rightarrow (match b0 in B return (\lambda (_:
+B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void
+\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr)
+u) H4) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind b) u2)
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H5)))))))))))
+(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_:
+(((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))).(\lambda (t:
+T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u0:
+T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst)
+t) (CHead d1 (Bind Abbr) u))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t)
+(\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 d1 (Bind Abbr)
+u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u0)
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H6))))))))))))
+y c H0))) H))))).
theorem csuba_gen_void:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g
-(CHead d1 (Bind Void) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2
-(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))))
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g
+(CHead d1 (Bind Void) u1) c) \to (ex2_3 B C T (\lambda (b: B).(\lambda (d2:
+C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2))))) (\lambda (_:
+B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))))
\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
-(csuba g (CHead d1 (Bind Void) u) c)).(let H0 \def (match H in csuba return
-(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0
-(CHead d1 (Bind Void) u)) \to ((eq C c1 c) \to (ex2 C (\lambda (d2: C).(eq C
-c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) with
-[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind
-Void) u))).(\lambda (H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort
-n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Void)
-u) H0) in (False_ind ((eq C (CSort n) c) \to (ex2 C (\lambda (d2: C).(eq C c
-(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) H2)) H1))) |
-(csuba_head c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0)
-(CHead d1 (Bind Void) u))).(\lambda (H2: (eq C (CHead c2 k u0) c)).((let H3
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k
-u0) (CHead d1 (Bind Void) u) H1) in ((let H4 \def (f_equal C K (\lambda (e:
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
-(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u0) (CHead d1 (Bind Void) u) H1)
-in ((let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0]))
-(CHead c1 k u0) (CHead d1 (Bind Void) u) H1) in (eq_ind C d1 (\lambda (c0:
-C).((eq K k (Bind Void)) \to ((eq T u0 u) \to ((eq C (CHead c2 k u0) c) \to
-((csuba g c0 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void)
-u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (H6: (eq K k (Bind
-Void))).(eq_ind K (Bind Void) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead
-c2 k0 u0) c) \to ((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead
-d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))))))) (\lambda (H7: (eq
-T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c2 (Bind Void) t) c) \to
-((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void)
-u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H8: (eq C (CHead c2
-(Bind Void) u) c)).(eq_ind C (CHead c2 (Bind Void) u) (\lambda (c0:
-C).((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind
-Void) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (H9: (csuba g d1
-c2)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Void) u) (CHead d2
-(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2
-(Bind Void) u)) H9)) c H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind Void)
-H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a
-H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1
-(Bind Void) u))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) c)).((let H5
-\def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e 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 d1 (Bind Void) u) H3) in (False_ind ((eq C
-(CHead c2 (Bind Abbr) u0) c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g
-a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2
-(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))))))) H5)) H4 H0 H1 H2)))])
-in (H0 (refl_equal C (CHead d1 (Bind Void) u)) (refl_equal C c))))))).
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
+(H: (csuba g (CHead d1 (Bind Void) u1) c)).(insert_eq C (CHead d1 (Bind Void)
+u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_3 B C T (\lambda
+(b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2)))))
+(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))
+(\lambda (y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0:
+C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T
+(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind b)
+u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind
+Void) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C
+return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead d1 (Bind Void) u1) H1) in (False_ind (ex2_3 B C
+T (\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2
+(Bind b) u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2))))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
+c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T
+(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b)
+u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u)
+(CHead d1 (Bind Void) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match
+e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c1 k u) (CHead d1 (Bind Void) u1) H3) in ((let H5
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k
+u) (CHead d1 (Bind Void) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Bind Void) u1) H3)
+in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r
+T u1 (\lambda (t: T).(ex2_3 B C T (\lambda (b: B).(\lambda (d2: C).(\lambda
+(u2: T).(eq C (CHead c2 k t) (CHead d2 (Bind b) u2))))) (\lambda (_:
+B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) (eq_ind_r K (Bind
+Void) (\lambda (k0: K).(ex2_3 B C T (\lambda (b: B).(\lambda (d2: C).(\lambda
+(u2: T).(eq C (CHead c2 k0 u1) (CHead d2 (Bind b) u2))))) (\lambda (_:
+B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) (let H9 \def (eq_ind
+C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T
+(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b)
+u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2))))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g
+c0 c2)) H1 d1 H8) in (ex2_3_intro B C T (\lambda (b: B).(\lambda (d2:
+C).(\lambda (u2: T).(eq C (CHead c2 (Bind Void) u1) (CHead d2 (Bind b)
+u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))
+Void c2 u1 (refl_equal C (CHead c2 (Bind Void) u1)) H10))) k H7) u H6))))
+H5)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
+c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T
+(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b)
+u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0:
+T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1
+(Bind Void) u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _)
+\Rightarrow c0])) (CHead c1 (Bind Void) u0) (CHead d1 (Bind Void) u1) H4) in
+((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead
+c1 (Bind Void) u0) (CHead d1 (Bind Void) u1) H4) in (\lambda (H7: (eq C c1
+d1)).(let H8 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind
+Void) u1)) \to (ex2_3 B C T (\lambda (b0: B).(\lambda (d2: C).(\lambda (u3:
+T).(eq C c2 (CHead d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2:
+C).(\lambda (_: T).(csuba g d1 d2))))))) H2 d1 H7) in (let H9 \def (eq_ind C
+c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H7) in (ex2_3_intro B C T (\lambda
+(b0: B).(\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c2 (Bind b) u2) (CHead
+d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba
+g d1 d2)))) b c2 u2 (refl_equal C (CHead c2 (Bind b) u2)) H9)))))
+H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1
+c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T
+(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b)
+u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc
+g a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C
+(CHead c1 (Bind Abst) t) (CHead d1 (Bind Void) u1))).(let H6 \def (eq_ind C
+(CHead c1 (Bind Abst) t) (\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 d1
+(Bind Void) u1) H5) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (d2:
+C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind b) u2)))))
+(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))
+H6)))))))))))) y c H0))) H))))).
theorem csuba_gen_abst:
\forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g
a))))))))))
\def
\lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
-(H: (csuba g (CHead d1 (Bind Abst) u1) c)).(let H0 \def (match H in csuba
-return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C
-c0 (CHead d1 (Bind Abst) u1)) \to ((eq C c1 c) \to (or (ex2 C (\lambda (d2:
+(H: (csuba g (CHead d1 (Bind Abst) u1) c)).(insert_eq C (CHead d1 (Bind Abst)
+u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(or (ex2 C (\lambda (d2:
C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead
d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 a))))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0:
-(eq C (CSort n) (CHead d1 (Bind Abst) u1))).(\lambda (H1: (eq C (CSort n)
-c)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
-\Rightarrow False])) I (CHead d1 (Bind Abst) u1) H0) in (False_ind ((eq C
-(CSort n) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst)
+A).(arity g d2 u2 a))))))) (\lambda (y: C).(\lambda (H0: (csuba g y
+c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind
+Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst)
u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2)))))
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abbr) u2)))))
(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))
-H2)) H1))) | (csuba_head c1 c2 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead
-c1 k u) (CHead d1 (Bind Abst) u1))).(\lambda (H2: (eq C (CHead c2 k u)
-c)).((let H3 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
-(CHead c1 k u) (CHead d1 (Bind Abst) u1) H1) in ((let H4 \def (f_equal C K
-(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _)
-\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead d1
-(Bind Abst) u1) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e in
-C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _)
-\Rightarrow c0])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H1) in (eq_ind C
-d1 (\lambda (c0: C).((eq K k (Bind Abst)) \to ((eq T u u1) \to ((eq C (CHead
-c2 k u) c) \to ((csuba g c0 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
-g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a))))))))))) (\lambda (H6: (eq K k (Bind Abst))).(eq_ind K (Bind Abst)
-(\lambda (k0: K).((eq T u u1) \to ((eq C (CHead c2 k0 u) c) \to ((csuba g d1
-c2) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))
+(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst)
+u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead d1 (Bind Abst) u1) H1) in (False_ind (or (ex2 C
+(\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(eq C (CSort n) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))) (\lambda
-(H7: (eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 (Bind Abst)
-t) c) \to ((csuba g d1 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a))))))))) (\lambda (H8: (eq C (CHead c2 (Bind Abst) u1) c)).(eq_ind C (CHead
-c2 (Bind Abst) u1) (\lambda (c0: C).((csuba g d1 c2) \to (or (ex2 C (\lambda
-(d2: C).(eq C c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c0
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) H2))))
+(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda
+(H2: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2:
+C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2
(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 a)))))))) (\lambda (H9: (csuba g d1 c2)).(or_introl (ex2 C
-(\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abst) u1) (CHead d2 (Bind Abbr)
+A).(arity g d2 u2 a))))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3:
+(eq C (CHead c1 k u) (CHead d1 (Bind Abst) u1))).(let H4 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 k u) (CHead d1
+(Bind Abst) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in
+C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _)
+\Rightarrow k0])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H3) in ((let H6
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u)
+(CHead d1 (Bind Abst) u1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda
+(H8: (eq C c1 d1)).(eq_ind_r T u1 (\lambda (t: T).(or (ex2 C (\lambda (d2:
+C).(eq C (CHead c2 k t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g
+d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C
+(CHead c2 k t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (eq_ind_r K (Bind Abst)
+(\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u1) (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 k0 u1) (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a))))))) (let H9 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1
+(Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))) H2
+d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1
+H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1)
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abst)
+u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2
+(Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))
+c2 (refl_equal C (CHead c2 (Bind Abst) u1)) H10)))) k H7) u H6)))) H5))
+H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1
+c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba
+g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
+C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(a: A).(arity g d2 u2 a))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b
+Void))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind
+Void) u0) (CHead d1 (Bind Abst) u1))).(let H5 \def (eq_ind C (CHead c1 (Bind
+Void) 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 b0) \Rightarrow (match b0 in B return
+(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False |
+Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind
+Abst) u1) H4) in (False_ind (or (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind
+b) u2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3
+C T A (\lambda (d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c2 (Bind
+b) u2) (CHead d2 (Bind Abbr) u3))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u3: T).(\lambda (a: A).(arity g d2 u3 a)))))) H5))))))))))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1
+(CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr)
u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a))))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) (CHead
-d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C
-(CHead c2 (Bind Abst) u1)) H9))) c H8)) u (sym_eq T u u1 H7))) k (sym_eq K k
-(Bind Abst) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst
-c1 c2 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst)
-t) (CHead d1 (Bind Abst) u1))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u)
-c)).((let H5 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0]))
-(CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1) H3) in ((let H6 \def
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind
-Abst) t) (CHead d1 (Bind Abst) u1) H3) in (eq_ind C d1 (\lambda (c0: C).((eq
-T t u1) \to ((eq C (CHead c2 (Bind Abbr) u) c) \to ((csuba g c0 c2) \to
-((arity g c0 t (asucc g a)) \to ((arity g c2 u a) \to (or (ex2 C (\lambda
-(d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity
-g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
-A).(arity g d2 u2 a0)))))))))))) (\lambda (H7: (eq T t u1)).(eq_ind T u1
-(\lambda (t0: T).((eq C (CHead c2 (Bind Abbr) u) c) \to ((csuba g d1 c2) \to
-((arity g d1 t0 (asucc g a)) \to ((arity g c2 u a) \to (or (ex2 C (\lambda
-(d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity
-g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
-A).(arity g d2 u2 a0))))))))))) (\lambda (H8: (eq C (CHead c2 (Bind Abbr) u)
-c)).(eq_ind C (CHead c2 (Bind Abbr) u) (\lambda (c0: C).((csuba g d1 c2) \to
-((arity g d1 u1 (asucc g a)) \to ((arity g c2 u a) \to (or (ex2 C (\lambda
-(d2: C).(eq C c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c0
+a))))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc
+g a))).(\lambda (u: T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C
+(CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1))).(let H6 \def (f_equal C
+C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind Abst) t)
+(CHead d1 (Bind Abst) u1) H5) in ((let H7 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t |
+(CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead d1 (Bind
+Abst) u1) H5) in (\lambda (H8: (eq C c1 d1)).(let H9 \def (eq_ind T t
+(\lambda (t0: T).(arity g c1 t0 (asucc g a))) H3 u1 H7) in (let H10 \def
+(eq_ind C c1 (\lambda (c0: C).(arity g c0 u1 (asucc g a))) H9 d1 H8) in (let
+H11 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u1))
+\to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))))))) H2 d1 H8)
+in (let H12 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in
+(or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u)
(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity
g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
-A).(arity g d2 u2 a0)))))))))) (\lambda (H9: (csuba g d1 c2)).(\lambda (H10:
-(arity g d1 u1 (asucc g a))).(\lambda (H11: (arity g c2 u a)).(or_intror (ex2
-C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr)
+A).(arity g d2 u2 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr)
u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g
a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2
-a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))) c2 u a
-(refl_equal C (CHead c2 (Bind Abbr) u)) H9 H10 H11))))) c H8)) t (sym_eq T t
-u1 H7))) c1 (sym_eq C c1 d1 H6))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C
-(CHead d1 (Bind Abst) u1)) (refl_equal C c))))))).
+a0)))) c2 u a (refl_equal C (CHead c2 (Bind Abbr) u)) H12 H10 H4))))))))
+H6)))))))))))) y c H0))) H))))).
theorem csuba_gen_flat:
\forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall
C).(\lambda (_: T).(csuba g d1 d2)))))))))
\def
\lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
-(f: F).(\lambda (H: (csuba g (CHead d1 (Flat f) u1) c)).(let H0 \def (match H
-in csuba return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0
-c1)).((eq C c0 (CHead d1 (Flat f) u1)) \to ((eq C c1 c) \to (ex2_2 C T
+(f: F).(\lambda (H: (csuba g (CHead d1 (Flat f) u1) c)).(insert_eq C (CHead
+d1 (Flat f) u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_2 C T
(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d1 d2))))))))) with [(csuba_sort n)
-\Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Flat f) u1))).(\lambda
-(H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort n) (\lambda (e:
-C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
-True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Flat f) u1) H0) in
-(False_ind ((eq C (CSort n) c) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba
-g d1 d2))))) H2)) H1))) | (csuba_head c1 c2 H0 k u) \Rightarrow (\lambda (H1:
-(eq C (CHead c1 k u) (CHead d1 (Flat f) u1))).(\lambda (H2: (eq C (CHead c2 k
-u) c)).((let H3 \def (f_equal C T (\lambda (e: C).(match e in C return
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow
-t])) (CHead c1 k u) (CHead d1 (Flat f) u1) H1) in ((let H4 \def (f_equal C K
-(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _)
-\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead d1
-(Flat f) u1) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _)
-\Rightarrow c0])) (CHead c1 k u) (CHead d1 (Flat f) u1) H1) in (eq_ind C d1
-(\lambda (c0: C).((eq K k (Flat f)) \to ((eq T u u1) \to ((eq C (CHead c2 k
-u) c) \to ((csuba g c0 c2) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba
-g d1 d2))))))))) (\lambda (H6: (eq K k (Flat f))).(eq_ind K (Flat f) (\lambda
-(k0: K).((eq T u u1) \to ((eq C (CHead c2 k0 u) c) \to ((csuba g d1 c2) \to
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))) (\lambda (H7:
-(eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 (Flat f) t) c) \to
-((csuba g d1 c2) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c
-(CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
-d2))))))) (\lambda (H8: (eq C (CHead c2 (Flat f) u1) c)).(eq_ind C (CHead c2
-(Flat f) u1) (\lambda (c0: C).((csuba g d1 c2) \to (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(eq C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d1 d2)))))) (\lambda (H9: (csuba g d1
-c2)).(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 (Flat
-f) u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
-d2))) c2 u1 (refl_equal C (CHead c2 (Flat f) u1)) H9)) c H8)) u (sym_eq T u
-u1 H7))) k (sym_eq K k (Flat f) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2
-H0))) | (csuba_abst c1 c2 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C
-(CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u1))).(\lambda (H4: (eq C (CHead
-c2 (Bind Abbr) u) c)).((let H5 \def (eq_ind C (CHead c1 (Bind Abst) t)
-(\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
-False])])) I (CHead d1 (Flat f) u1) H3) in (False_ind ((eq C (CHead c2 (Bind
-Abbr) u) c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity
-g c2 u a) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2
-(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))) H5))
-H4 H0 H1 H2)))]) in (H0 (refl_equal C (CHead d1 (Flat f) u1)) (refl_equal C
-c)))))))).
+(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (\lambda (y: C).(\lambda (H0:
+(csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f)
+u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) in (False_ind (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) H2)))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u)
+(CHead d1 (Flat f) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c1 k u) (CHead d1 (Flat f) u1) H3) in ((let H5
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k
+u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Flat f) u1) H3) in
+(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r T u1
+(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2
+k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C (CHead c2 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (let H9 \def (eq_ind C c1
+(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda
+(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d1 d2)))))) H2 d1 H8) in (let H10 \def (eq_ind C
+c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in (ex2_2_intro C T (\lambda
+(d2: C).(\lambda (u2: T).(eq C (CHead c2 (Flat f) u1) (CHead d2 (Flat f)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))) c2 u1 (refl_equal C
+(CHead c2 (Flat f) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0:
+T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1
+(Flat f) u1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) 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 _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1
+(Flat f) u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u3:
+T).(eq C (CHead c2 (Bind b) u2) (CHead d2 (Flat f) u3)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d1 d2)))) H5))))))))))) (\lambda (c1: C).(\lambda
+(c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 (CHead d1 (Flat
+f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2
+(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g
+a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C
+(CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C
+(CHead c1 (Bind Abst) t) (\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 _) \Rightarrow True | (Flat
+_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind (ex2_2 C
+T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2
+(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))
+H6)))))))))))) y c H0))) H)))))).
theorem csuba_gen_bind:
\forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall
(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))))))
\def
\lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda
-(v1: T).(\lambda (H: (csuba g (CHead e1 (Bind b1) v1) c2)).(let H0 \def
-(match H in csuba return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csuba
-? c c0)).((eq C c (CHead e1 (Bind b1) v1)) \to ((eq C c0 c2) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1
-e2)))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n)
-(CHead e1 (Bind b1) v1))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def
-(eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead e1 (Bind b1) v1) H0) in (False_ind ((eq C (CSort n) c2) \to
-(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
+(v1: T).(\lambda (H: (csuba g (CHead e1 (Bind b1) v1) c2)).(insert_eq C
+(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c c2)) (\lambda (_:
+C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csuba g e1 e2)))))) H2)) H1))) | (csuba_head c1 c0 H0 k u) \Rightarrow
-(\lambda (H1: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(\lambda (H2:
-(eq C (CHead c0 k u) c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e
-in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
-\Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in ((let H4 \def
-(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
-[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u)
-(CHead e1 (Bind b1) v1) H1) in ((let H5 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 |
-(CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in
-(eq_ind C e1 (\lambda (c: C).((eq K k (Bind b1)) \to ((eq T u v1) \to ((eq C
-(CHead c0 k u) c2) \to ((csuba g c c0) \to (ex2_3 B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2)))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))))))
-(\lambda (H6: (eq K k (Bind b1))).(eq_ind K (Bind b1) (\lambda (k0: K).((eq T
-u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csuba g e1 c0) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1
-e2))))))))) (\lambda (H7: (eq T u v1)).(eq_ind T v1 (\lambda (t: T).((eq C
-(CHead c0 (Bind b1) t) c2) \to ((csuba g e1 c0) \to (ex2_3 B C T (\lambda
-(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2)
-v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1
-e2)))))))) (\lambda (H8: (eq C (CHead c0 (Bind b1) v1) c2)).(eq_ind C (CHead
-c0 (Bind b1) v1) (\lambda (c: C).((csuba g e1 c0) \to (ex2_3 B C T (\lambda
-(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2)))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))
-(\lambda (H9: (csuba g e1 c0)).(let H10 \def (eq_ind_r C c2 (\lambda (c:
-C).(csuba g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind b1) v1) H8) in
+T).(csuba g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csuba g y
+c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind
+b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq
+C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n)
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1)
+v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
+(e2: C).(\lambda (_: T).(csuba g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda
+(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1
+(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (k: K).(\lambda (u:
+T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \def
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u)
+(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e:
+C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
+(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3)
+in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
+(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind
+b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t)
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1)
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 H8) in (let
+H10 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H8) in
(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C
-(CHead c0 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
-(e2: C).(\lambda (_: T).(csuba g e1 e2)))) b1 c0 v1 (refl_equal C (CHead c0
-(Bind b1) v1)) H9))) c2 H8)) u (sym_eq T u v1 H7))) k (sym_eq K k (Bind b1)
-H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c0 H0 t a
-H1 u H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead e1
-(Bind b1) v1))).(\lambda (H4: (eq C (CHead c0 (Bind Abbr) u) c2)).((let H5
+(CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
+(e2: C).(\lambda (_: T).(csuba g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3
+(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (b:
+B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1)
+v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c]))
+(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
+Void])])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7
\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c1
-(Bind Abst) t) (CHead e1 (Bind b1) v1) H3) in ((let H6 \def (f_equal C B
-(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
-\Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])]))
-(CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H3) in ((let H7 \def
+with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1
+(Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B Void
+b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind C c1 (\lambda (c:
+C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2:
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2)))))
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1
+H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H9)
+in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 (CHead e1 (Bind
+b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e1 e2))))))) H10 Void H8) in (ex2_3_intro B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b)
+u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 (Bind b) u2))
+H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1:
+(csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3
+B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2
+(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g
+e1 e2)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t
+(asucc g a))).(\lambda (u: T).(\lambda (_: (arity g c3 u a)).(\lambda (H5:
+(eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(let H6 \def
(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind
-Abst) t) (CHead e1 (Bind b1) v1) H3) in (eq_ind C e1 (\lambda (c: C).((eq B
-Abst b1) \to ((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csuba
-g c c0) \to ((arity g c t (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1
-e2)))))))))))) (\lambda (H8: (eq B Abst b1)).(eq_ind B Abst (\lambda (_:
-B).((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csuba g e1 c0)
-\to ((arity g e1 t (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1
-e2))))))))))) (\lambda (H9: (eq T t v1)).(eq_ind T v1 (\lambda (t0: T).((eq C
-(CHead c0 (Bind Abbr) u) c2) \to ((csuba g e1 c0) \to ((arity g e1 t0 (asucc
-g a)) \to ((arity g c0 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
-C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_:
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))) (\lambda (H10:
-(eq C (CHead c0 (Bind Abbr) u) c2)).(eq_ind C (CHead c0 (Bind Abbr) u)
-(\lambda (c: C).((csuba g e1 c0) \to ((arity g e1 v1 (asucc g a)) \to ((arity
-g c0 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
-T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (_: T).(csuba g e1 e2))))))))) (\lambda (H11: (csuba g e1
-c0)).(\lambda (_: (arity g e1 v1 (asucc g a))).(\lambda (_: (arity g c0 u
-a)).(let H14 \def (eq_ind_r C c2 (\lambda (c: C).(csuba g (CHead e1 (Bind b1)
-v1) c)) H (CHead c0 (Bind Abbr) u) H10) in (let H15 \def (eq_ind_r B b1
-(\lambda (b: B).(csuba g (CHead e1 (Bind b) v1) (CHead c0 (Bind Abbr) u)))
-H14 Abst H8) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
-(v2: T).(eq C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind b2) v2))))) (\lambda
-(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))) Abbr c0 u
-(refl_equal C (CHead c0 (Bind Abbr) u)) H11)))))) c2 H10)) t (sym_eq T t v1
-H9))) b1 H8)) c1 (sym_eq C c1 e1 H7))) H6)) H5)) H4 H0 H1 H2)))]) in (H0
-(refl_equal C (CHead e1 (Bind b1) v1)) (refl_equal C c2)))))))).
+Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def (f_equal C B (\lambda
+(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
+Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
+[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead c1 (Bind
+Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T (\lambda
+(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t
+| (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind
+b1) v1) H5) in (\lambda (H9: (eq B Abst b1)).(\lambda (H10: (eq C c1
+e1)).(let H11 \def (eq_ind T t (\lambda (t0: T).(arity g c1 t0 (asucc g a)))
+H3 v1 H8) in (let H12 \def (eq_ind C c1 (\lambda (c: C).(arity g c v1 (asucc
+g a))) H11 e1 H10) in (let H13 \def (eq_ind C c1 (\lambda (c: C).((eq C c
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 H10) in (let
+H14 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H10) in (let H15
+\def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to
+(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e1 e2))))))) H13 Abst H9) in (ex2_3_intro B C T (\lambda (b2:
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2
+(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g
+e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H14))))))))) H7))
+H6)))))))))))) y c2 H0))) H)))))).
theorem csuba_gen_abst_rev:
\forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c
-(CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
+(CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))))))))
\def
\lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
-(csuba g c (CHead d1 (Bind Abst) u))).(let H0 \def (match H in csuba return
-(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 c)
-\to ((eq C c1 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq C c
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) with
-[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1:
-(eq C (CSort n) (CHead d1 (Bind Abst) u))).(eq_ind C (CSort n) (\lambda (c0:
-C).((eq C (CSort n) (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq
-C c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda
-(H2: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H3 \def (eq_ind C (CSort
-n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abst)
-u) H2) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) H3))) c H0 H1))) | (csuba_head
-c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) c)).(\lambda
-(H2: (eq C (CHead c2 k u0) (CHead d1 (Bind Abst) u))).(eq_ind C (CHead c1 k
-u0) (\lambda (c0: C).((eq C (CHead c2 k u0) (CHead d1 (Bind Abst) u)) \to
-((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda (H3: (eq C (CHead c2 k
-u0) (CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
-(CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3)
+(csuba g c (CHead d1 (Bind Abst) u))).(insert_eq C (CHead d1 (Bind Abst) u)
+(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or (ex2 C (\lambda (d2:
+C).(eq C c (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (\lambda (y:
+C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1:
+C).((eq C c1 (CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C
+c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (n:
+nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H2 \def
+(eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
+False])) I (CHead d1 (Bind Abst) u) H1) in (False_ind (or (ex2 C (\lambda
+(d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g
+d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))
+H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
+c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (or (ex2 C
+(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba
+g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k
+u0) (CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 |
+(CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3)
in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0]))
-(CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) in ((let H6 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1
-(Bind Abst) u) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Abst)) \to
-((eq T u0 u) \to ((csuba g c1 c0) \to (ex2 C (\lambda (d2: C).(eq C (CHead c1
-k u0) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
-(\lambda (H7: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda (k0:
-K).((eq T u0 u) \to ((csuba g c1 d1) \to (ex2 C (\lambda (d2: C).(eq C (CHead
-c1 k0 u0) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))))
-(\lambda (H8: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((csuba g c1 d1) \to
-(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda (H9: (csuba g c1
-d1)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) u) (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1
-(Bind Abst) u)) H9)) u0 (sym_eq T u0 u H8))) k (sym_eq K k (Bind Abst) H7)))
-c2 (sym_eq C c2 d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a
-H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t)
-c)).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst)
-u))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2
-(Bind Abbr) u0) (CHead d1 (Bind Abst) u)) \to ((csuba g c1 c2) \to ((arity g
-c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c0
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) (\lambda
-(H5: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) u))).(let H6 \def
-(eq_ind C (CHead c2 (Bind Abbr) u0) (\lambda (e: C).(match e 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 d1 (Bind Abst) u) H5) in (False_ind ((csuba g
-c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C
-(\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1)))))) H6))) c H3 H4 H0 H1 H2)))]) in (H0
-(refl_equal C c) (refl_equal C (CHead d1 (Bind Abst) u)))))))).
+(CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) in ((let H6 \def (f_equal C T
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1
+(Bind Abst) u) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C
+c2 d1)).(eq_ind_r T u (\lambda (t: T).(or (ex2 C (\lambda (d2: C).(eq C
+(CHead c1 k t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t) (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))
+(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C
+(CHead c1 k0 u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k0 u) (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let
+H9 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to
+(or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g
+c1 c0)) H1 d1 H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind
+Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) u) (CHead
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))
+(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) u) (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind
+Abst) u)) H10)))) k H7) u0 H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda
+(c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 (CHead d1
+(Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b: B).(\lambda (H3: (not
+(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead
+c2 (Bind b) u2) (CHead d1 (Bind Abst) u))).(let H5 \def (f_equal C C (\lambda
+(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2
+| (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind
+Abst) u) H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e in C
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1
+(Bind Abst) u) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in
+C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t)
+\Rightarrow t])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abst) u) H4) in
+(\lambda (H8: (eq B b Abst)).(\lambda (H9: (eq C c2 d1)).(let H10 \def
+(eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 Abst H8) in (let H11
+\def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to
+(or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C c1
+(CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))))) H2 d1 H9) in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g
+c1 c0)) H1 d1 H9) in (or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind
+Void) u1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c1 (Bind Void) u1)
+(CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c1
+(Bind Void) u1) (CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))) c1 u1 (refl_equal C (CHead c1 (Bind Void) u1))
+H12)))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_:
+(csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (or
+(ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc
+g a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C
+(CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) u))).(let H6 \def (eq_ind C
+(CHead c2 (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
+d1 (Bind Abst) u) H5) in (False_ind (or (ex2 C (\lambda (d2: C).(eq C (CHead
+c1 (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
+d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind
+Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba
+g d2 d1))))) H6)))))))))))) c y H0))) H))))).
theorem csuba_gen_void_rev:
\forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c
Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
\def
\lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
-(csuba g c (CHead d1 (Bind Void) u))).(let H0 \def (match H in csuba return
-(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 c)
-\to ((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c
-(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) with
-[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1:
-(eq C (CSort n) (CHead d1 (Bind Void) u))).(eq_ind C (CSort n) (\lambda (c0:
-C).((eq C (CSort n) (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq
-C c0 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda
-(H2: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H3 \def (eq_ind C (CSort
-n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Void)
-u) H2) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind
-Void) u))) (\lambda (d2: C).(csuba g d2 d1))) H3))) c H0 H1))) | (csuba_head
-c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) c)).(\lambda
-(H2: (eq C (CHead c2 k u0) (CHead d1 (Bind Void) u))).(eq_ind C (CHead c1 k
-u0) (\lambda (c0: C).((eq C (CHead c2 k u0) (CHead d1 (Bind Void) u)) \to
-((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Void)
-u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda (H3: (eq C (CHead c2 k
-u0) (CHead d1 (Bind Void) u))).(let H4 \def (f_equal C T (\lambda (e:
+(csuba g c (CHead d1 (Bind Void) u))).(insert_eq C (CHead d1 (Bind Void) u)
+(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq
+C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (\lambda
+(y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda
+(c1: C).((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C
+c0 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda
+(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H2
+\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
+False])) I (CHead d1 (Bind Void) u) H1) in (False_ind (ex2 C (\lambda (d2:
+C).(eq C (CSort n) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2
+d1))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
+c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2
+d1)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k u0)
+(CHead d1 (Bind Void) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3) in ((let H5
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k
+u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C T (\lambda (e:
C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
(CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3)
-in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
-(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0]))
-(CHead c2 k u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1
-(Bind Void) u) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Void)) \to
-((eq T u0 u) \to ((csuba g c1 c0) \to (ex2 C (\lambda (d2: C).(eq C (CHead c1
-k u0) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
-(\lambda (H7: (eq K k (Bind Void))).(eq_ind K (Bind Void) (\lambda (k0:
-K).((eq T u0 u) \to ((csuba g c1 d1) \to (ex2 C (\lambda (d2: C).(eq C (CHead
-c1 k0 u0) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))))
-(\lambda (H8: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((csuba g c1 d1) \to
-(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) t) (CHead d2 (Bind Void)
-u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda (H9: (csuba g c1
-d1)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u) (CHead d2
-(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1
-(Bind Void) u)) H9)) u0 (sym_eq T u0 u H8))) k (sym_eq K k (Bind Void) H7)))
-c2 (sym_eq C c2 d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a
-H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t)
-c)).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void)
-u))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2
-(Bind Abbr) u0) (CHead d1 (Bind Void) u)) \to ((csuba g c1 c2) \to ((arity g
-c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c0
-(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) (\lambda
-(H5: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void) u))).(let H6 \def
-(eq_ind C (CHead c2 (Bind Abbr) u0) (\lambda (e: C).(match e 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 d1 (Bind Void) u) H5) in (False_ind ((csuba g
-c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C
-(\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u)))
-(\lambda (d2: C).(csuba g d2 d1)))))) H6))) c H3 H4 H0 H1 H2)))]) in (H0
-(refl_equal C c) (refl_equal C (CHead d1 (Bind Void) u)))))))).
+in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r
+T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead d2
+(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (eq_ind_r K (Bind Void)
+(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k0 u) (CHead d2
+(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H9 \def (eq_ind C
+c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2
+d1))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1
+c0)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void)
+u) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)) c1
+(refl_equal C (CHead c1 (Bind Void) u)) H10))) k H7) u0 H6)))) H5))
+H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
+c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2
+d1)))))).(\lambda (b: B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1
+(Bind Void) u))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _)
+\Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in
+((let H6 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_:
+C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k in K
+return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
+\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in
+((let H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead
+c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in (\lambda (H8: (eq B b
+Void)).(\lambda (H9: (eq C c2 d1)).(let H10 \def (eq_ind B b (\lambda (b0:
+B).(not (eq B b0 Void))) H3 Void H8) in (let H11 \def (eq_ind C c2 (\lambda
+(c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C
+c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))) H2 d1 H9)
+in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in
+(let H13 \def (match (H10 (refl_equal B Void)) in False return (\lambda (_:
+False).(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u1) (CHead d2
+(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) with []) in H13)))))))
+H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1
+c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2
+d1)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g
+a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C
+(CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void) u))).(let H6 \def (eq_ind C
+(CHead c2 (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
+d1 (Bind Void) u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c1
+(Bind Abst) t) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))
+H6)))))))))))) c y H0))) H))))).
theorem csuba_gen_abbr_rev:
\forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g c
-(CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2
+(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c (CHead d2
(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abst)
u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-a))))))))))
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))
\def
\lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
-(H: (csuba g c (CHead d1 (Bind Abbr) u1))).(let H0 \def (match H in csuba
-return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C
-c0 c) \to ((eq C c1 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2:
-C).(eq C c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead
-d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+(H: (csuba g c (CHead d1 (Bind Abbr) u1))).(insert_eq C (CHead d1 (Bind Abbr)
+u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or3 (ex2 C (\lambda
+(d2: C).(eq C c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))))))))) with [(csuba_sort n) \Rightarrow (\lambda
-(H0: (eq C (CSort n) c)).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr)
-u1))).(eq_ind C (CSort n) (\lambda (c0: C).((eq C (CSort n) (CHead d1 (Bind
-Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))
-(\lambda (H2: (eq C (CSort n) (CHead d1 (Bind Abbr) u1))).(let H3 \def
-(eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead d1 (Bind Abbr) u1) H2) in (False_ind (or (ex2 C (\lambda
-(d2: C).(eq C (CSort n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g
-d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C
-(CSort n) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda
+(c0: C).(\lambda (c1: C).((eq C c1 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C
+(\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
+C c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
+d2 d1)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind
+Abbr) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C
+return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead d1 (Bind Abbr) u1) H1) in (False_ind (or3 (ex2
+C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(eq C (CSort n) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) H2)))) (\lambda
+(c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C
+c2 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u)
+(CHead d1 (Bind Abbr) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match
+e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in ((let H5
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k
+u) (CHead d1 (Bind Abbr) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3)
+in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r
+T u1 (\lambda (t: T).(or3 (ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 k t) (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t)
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))))) (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or3 (ex2 C (\lambda (d2:
+C).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
+C (CHead c1 k0 u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) H3))) c H0 H1))) | (csuba_head
-c1 c2 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) c)).(\lambda
-(H2: (eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1))).(eq_ind C (CHead c1 k
-u) (\lambda (c0: C).((eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1)) \to
-((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))))))
-(\lambda (H3: (eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1))).(let H4 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u)
-(CHead d1 (Bind Abbr) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e:
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
-(CHead _ k0 _) \Rightarrow k0])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3)
-in ((let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0]))
-(CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in (eq_ind C d1 (\lambda (c0:
-C).((eq K k (Bind Abbr)) \to ((eq T u u1) \to ((csuba g c1 c0) \to (or (ex2 C
-(\lambda (d2: C).(eq C (CHead c1 k u) (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(eq C (CHead c1 k u) (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))))
-(\lambda (H7: (eq K k (Bind Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0:
-K).((eq T u u1) \to ((csuba g c1 d1) \to (or (ex2 C (\lambda (d2: C).(eq C
-(CHead c1 k0 u) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H9 \def (eq_ind C
+c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C
+(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
+C c1 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
+d2 d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba
+g c1 c0)) H1 d1 H8) in (or3_intro0 (ex2 C (\lambda (d2: C).(eq C (CHead c1
+(Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C
-(CHead c1 k0 u) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) (\lambda (H8: (eq T u
-u1)).(eq_ind T u1 (\lambda (t: T).((csuba g c1 d1) \to (or (ex2 C (\lambda
-(d2: C).(eq C (CHead c1 (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(eq C (CHead c1 (Bind Abbr) t) (CHead d2 (Bind Abst)
+(CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))
+(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1
+(Bind Abbr) u1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2
+(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst)
u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-a)))))))) (\lambda (H9: (csuba g c1 d1)).(or_introl (ex2 C (\lambda (d2:
-C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C
-(\lambda (d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind Abbr) u1))
-H9))) u (sym_eq T u u1 H8))) k (sym_eq K k (Bind Abbr) H7))) c2 (sym_eq C c2
-d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2)
-\Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) c)).(\lambda (H4:
-(eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1))).(eq_ind C (CHead
-c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2 (Bind Abbr) u) (CHead d1
-(Bind Abbr) u1)) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to
-((arity g c2 u a) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g a0)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 a0)))))))))))
-(\lambda (H5: (eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1))).(let
-H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c2
-(Bind Abbr) u) (CHead d1 (Bind Abbr) u1) H5) in ((let H7 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind Abbr) u)
-(CHead d1 (Bind Abbr) u1) H5) in (eq_ind C d1 (\lambda (c0: C).((eq T u u1)
-\to ((csuba g c1 c0) \to ((arity g c1 t (asucc g a)) \to ((arity g c0 u a)
-\to (or (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b:
+B).(\lambda (H3: (not (eq B b Void))).(\lambda (u0: T).(\lambda (u2:
+T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr)
+u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0]))
+(CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in ((let H6 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
+b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in ((let H7 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b)
+u2) (CHead d1 (Bind Abbr) u1) H4) in (\lambda (H8: (eq B b Abbr)).(\lambda
+(H9: (eq C c2 d1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0
+Void))) H3 Abbr H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0
+(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst)
+u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u3: T).(\lambda (a: A).(arity g d2 u3 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C c1 (CHead d2 (Bind Void)
+u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 d1 H9) in
+(let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in
+(or3_intro2 (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u0) (CHead d2
(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
-A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a0: A).(arity g d1 u1 a0))))))))))) (\lambda (H8: (eq T u u1)).(eq_ind T u1
-(\lambda (t0: T).((csuba g c1 d1) \to ((arity g c1 t (asucc g a)) \to ((arity
-g d1 t0 a) \to (or (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t)
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst)
-t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
-A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a0: A).(arity g d1 u1 a0)))))))))) (\lambda (H9: (csuba g c1 d1)).(\lambda
-(H10: (arity g c1 t (asucc g a))).(\lambda (H11: (arity g d1 u1
-a)).(or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead
+(d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u0)
+(CHead d2 (Bind Abst) u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u3: T).(\lambda (a:
+A).(arity g d2 u3 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq
+C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Void) u3)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2:
+C).(\lambda (u3: T).(eq C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Void)
+u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u0 (refl_equal C
+(CHead c1 (Bind Void) u0)) H12)))))))) H6)) H5))))))))))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2
+(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead
d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (t:
+T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc g a))).(\lambda (u:
+T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr)
+u) (CHead d1 (Bind Abbr) u1))).(let H6 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 |
+(CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind Abbr) u) (CHead d1 (Bind
+Abbr) u1) H5) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0)
+\Rightarrow t0])) (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1) H5) in
+(\lambda (H8: (eq C c2 d1)).(let H9 \def (eq_ind T u (\lambda (t0: T).(arity
+g c2 t0 a)) H4 u1 H7) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(arity g
+c0 u1 a)) H9 d1 H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0
+(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g
+a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1
+a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 d1 H8)
+in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in
+(or3_intro1 (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t)
(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a0: A).(arity g d1 u1 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g
-a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1
-a0)))) c1 t a (refl_equal C (CHead c1 (Bind Abst) t)) H9 H10 H11))))) u
-(sym_eq T u u1 H8))) c2 (sym_eq C c2 d1 H7))) H6))) c H3 H4 H0 H1 H2)))]) in
-(H0 (refl_equal C c) (refl_equal C (CHead d1 (Bind Abbr) u1)))))))).
+(a0: A).(arity g d1 u1 a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2
+(asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1
+u1 a0)))) c1 t a (refl_equal C (CHead c1 (Bind Abst) t)) H12 H3 H10))))))))
+H6)))))))))))) c y H0))) H))))).
theorem csuba_gen_flat_rev:
\forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall
C).(\lambda (_: T).(csuba g d2 d1)))))))))
\def
\lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
-(f: F).(\lambda (H: (csuba g c (CHead d1 (Flat f) u1))).(let H0 \def (match H
-in csuba return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0
-c1)).((eq C c0 c) \to ((eq C c1 (CHead d1 (Flat f) u1)) \to (ex2_2 C T
+(f: F).(\lambda (H: (csuba g c (CHead d1 (Flat f) u1))).(insert_eq C (CHead
+d1 (Flat f) u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2_2 C T
(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) with [(csuba_sort n)
-\Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1: (eq C (CSort n)
-(CHead d1 (Flat f) u1))).(eq_ind C (CSort n) (\lambda (c0: C).((eq C (CSort
-n) (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(eq C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba
-g d2 d1)))))) (\lambda (H2: (eq C (CSort n) (CHead d1 (Flat f) u1))).(let H3
-\def (eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead d1 (Flat f) u1) H2) in (False_ind (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))) H3))) c H0 H1))) | (csuba_head c1 c2 H0
-k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) c)).(\lambda (H2: (eq C
-(CHead c2 k u) (CHead d1 (Flat f) u1))).(eq_ind C (CHead c1 k u) (\lambda
-(c0: C).((eq C (CHead c2 k u) (CHead d1 (Flat f) u1)) \to ((csuba g c1 c2)
-\to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Flat f)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (H3:
-(eq C (CHead c2 k u) (CHead d1 (Flat f) u1))).(let H4 \def (f_equal C T
-(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Flat
-f) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return
-(\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow
-k0])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u) (CHead d1
-(Flat f) u1) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Flat f)) \to ((eq
-T u u1) \to ((csuba g c1 c0) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(eq C (CHead c1 k u) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1)))))))) (\lambda (H7: (eq K k (Flat f))).(eq_ind K
-(Flat f) (\lambda (k0: K).((eq T u u1) \to ((csuba g c1 d1) \to (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k0 u) (CHead d2 (Flat f)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (H8:
-(eq T u u1)).(eq_ind T u1 (\lambda (t: T).((csuba g c1 d1) \to (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Flat f) t) (CHead d2 (Flat
-f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (\lambda (H9:
-(csuba g c1 d1)).(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(eq C
-(CHead c1 (Flat f) u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1))) c1 u1 (refl_equal C (CHead c1 (Flat f) u1)) H9)) u
-(sym_eq T u u1 H8))) k (sym_eq K k (Flat f) H7))) c2 (sym_eq C c2 d1 H6)))
-H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2) \Rightarrow
-(\lambda (H3: (eq C (CHead c1 (Bind Abst) t) c)).(\lambda (H4: (eq C (CHead
-c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(eq_ind C (CHead c1 (Bind Abst) t)
-(\lambda (c0: C).((eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1)) \to
-((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u a) \to
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Flat f)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) (\lambda (H5:
-(eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind
-C (CHead c2 (Bind Abbr) u) (\lambda (e: C).(match e in C return (\lambda (_:
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (y: C).(\lambda (H0:
+(csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c1
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f)
+u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) in (False_ind (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) H2)))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u)
+(CHead d1 (Flat f) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in ((let H5
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k
+u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in
+(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r T u1
+(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1
+k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (let H9 \def (eq_ind C c2
+(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda
+(d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))))) H2 d1 H8) in (let H10 \def (eq_ind C
+c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (ex2_2_intro C T (\lambda
+(d2: C).(\lambda (u2: T).(eq C (CHead c1 (Flat f) u1) (CHead d2 (Flat f)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u1 (refl_equal C
+(CHead c1 (Flat f) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0:
+T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1
+(Flat f) u1))).(let H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\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 _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1
+(Flat f) u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u3:
+T).(eq C (CHead c1 (Bind Void) u0) (CHead d2 (Flat f) u3)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) H5))))))))))) (\lambda (c1: C).(\lambda
+(c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Flat
+f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2
+(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g
+a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C
+(CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C
+(CHead c2 (Bind Abbr) u) (\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 _) \Rightarrow True | (Flat
-_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind ((csuba
-g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u a) \to (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) t) (CHead d2
-(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H6)))
-c H3 H4 H0 H1 H2)))]) in (H0 (refl_equal C c) (refl_equal C (CHead d1 (Flat
-f) u1))))))))).
+_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind (ex2_2 C
+T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) t) (CHead d2
+(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))
+H6)))))))))))) c y H0))) H)))))).
theorem csuba_gen_bind_rev:
\forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall
(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))))))
\def
\lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda
-(v1: T).(\lambda (H: (csuba g c2 (CHead e1 (Bind b1) v1))).(let H0 \def
-(match H in csuba return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csuba
-? c c0)).((eq C c c2) \to ((eq C c0 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2
-e1)))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n)
-c2)).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind b1) v1))).(eq_ind C (CSort
-n) (\lambda (c: C).((eq C (CSort n) (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2
-e1))))))) (\lambda (H2: (eq C (CSort n) (CHead e1 (Bind b1) v1))).(let H3
-\def (eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead e1 (Bind b1) v1) H2) in (False_ind (ex2_3 B C T (\lambda
-(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 (Bind b2)
-v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))
-H3))) c2 H0 H1))) | (csuba_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C
-(CHead c1 k u) c2)).(\lambda (H2: (eq C (CHead c0 k u) (CHead e1 (Bind b1)
-v1))).(eq_ind C (CHead c1 k u) (\lambda (c: C).((eq C (CHead c0 k u) (CHead
-e1 (Bind b1) v1)) \to ((csuba g c1 c0) \to (ex2_3 B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2)))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))))
-(\lambda (H3: (eq C (CHead c0 k u) (CHead e1 (Bind b1) v1))).(let H4 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u)
+(v1: T).(\lambda (H: (csuba g c2 (CHead e1 (Bind b1) v1))).(insert_eq C
+(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c2 c)) (\lambda (_:
+C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e2 e1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c2
+y)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c0 (CHead e1 (Bind
+b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e2 e1)))))))) (\lambda (n: nat).(\lambda (H1: (eq
+C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n)
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1)
+v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
+(e2: C).(\lambda (_: T).(csuba g e2 e1))))) H2)))) (\lambda (c1: C).(\lambda
+(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 (CHead e1
+(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (k: K).(\lambda (u:
+T).(\lambda (H3: (eq C (CHead c3 k u) (CHead e1 (Bind b1) v1))).(let H4 \def
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k u)
(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e:
C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
-(CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u) (CHead e1 (Bind b1) v1) H3)
-in ((let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c]))
-(CHead c0 k u) (CHead e1 (Bind b1) v1) H3) in (eq_ind C e1 (\lambda (c:
-C).((eq K k (Bind b1)) \to ((eq T u v1) \to ((csuba g c1 c) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k u)
+(CHead _ k0 _) \Rightarrow k0])) (CHead c3 k u) (CHead e1 (Bind b1) v1) H3)
+in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
+(CHead c3 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind
+b1))).(\lambda (H8: (eq C c3 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k t)
(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csuba g e2 e1))))))))) (\lambda (H7: (eq K k (Bind b1))).(eq_ind K (Bind
-b1) (\lambda (k0: K).((eq T u v1) \to ((csuba g c1 e1) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k0 u)
+T).(csuba g e2 e1)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k0 v1)
(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csuba g e2 e1)))))))) (\lambda (H8: (eq T u v1)).(eq_ind T v1 (\lambda
-(t: T).((csuba g c1 e1) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
-C).(\lambda (v2: T).(eq C (CHead c1 (Bind b1) t) (CHead e2 (Bind b2) v2)))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))
-(\lambda (H9: (csuba g c1 e1)).(let H10 \def (eq_ind T u (\lambda (t: T).(eq
-C (CHead c1 k t) c2)) H1 v1 H8) in (let H11 \def (eq_ind K k (\lambda (k0:
-K).(eq C (CHead c1 k0 v1) c2)) H10 (Bind b1) H7) in (let H12 \def (eq_ind_r C
-c2 (\lambda (c: C).(csuba g c (CHead e1 (Bind b1) v1))) H (CHead c1 (Bind b1)
-v1) H11) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
-(v2: T).(eq C (CHead c1 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda
-(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))) b1 c1 v1
-(refl_equal C (CHead c1 (Bind b1) v1)) H9))))) u (sym_eq T u v1 H8))) k
-(sym_eq K k (Bind b1) H7))) c0 (sym_eq C c0 e1 H6))) H5)) H4))) c2 H1 H2
-H0))) | (csuba_abst c1 c0 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C
-(CHead c1 (Bind Abst) t) c2)).(\lambda (H4: (eq C (CHead c0 (Bind Abbr) u)
-(CHead e1 (Bind b1) v1))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c:
-C).((eq C (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1)) \to ((csuba g c1
-c0) \to ((arity g c1 t (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2
-e1)))))))))) (\lambda (H5: (eq C (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1)
-v1))).(let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0]))
-(CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
-Abbr])])) (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H8
-\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C)
-with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0
-(Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in (eq_ind C e1 (\lambda (c:
-C).((eq B Abbr b1) \to ((eq T u v1) \to ((csuba g c1 c) \to ((arity g c1 t
-(asucc g a)) \to ((arity g c u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda
-(e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2)
-v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2
-e1))))))))))) (\lambda (H9: (eq B Abbr b1)).(eq_ind B Abbr (\lambda (_:
-B).((eq T u v1) \to ((csuba g c1 e1) \to ((arity g c1 t (asucc g a)) \to
-((arity g e1 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
-(v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) v2))))) (\lambda
-(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))))) (\lambda
-(H10: (eq T u v1)).(eq_ind T v1 (\lambda (t0: T).((csuba g c1 e1) \to ((arity
-g c1 t (asucc g a)) \to ((arity g e1 t0 a) \to (ex2_3 B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2
+T).(csuba g e2 e1)))))) (let H9 \def (eq_ind C c3 (\lambda (c: C).((eq C c
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H8) in (let
+H10 \def (eq_ind C c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H8) in
+(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C
+(CHead c1 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
+(e2: C).(\lambda (_: T).(csuba g e2 e1)))) b1 c1 v1 (refl_equal C (CHead c1
+(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (b:
+B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (H4: (eq C (CHead c3 (Bind b) u2) (CHead e1 (Bind b1) v1))).(let
+H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C)
+with [(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3
+(Bind b) u2) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def (f_equal C B
+(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
+\Rightarrow b | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_:
+K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c3
+(Bind b) u2) (CHead e1 (Bind b1) v1) H4) in ((let H7 \def (f_equal C T
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c3 (Bind b) u2) (CHead
+e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B b b1)).(\lambda (H9: (eq C c3
+e1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 b1
+H8) in (let H11 \def (eq_ind C c3 (\lambda (c: C).((eq C c (CHead e1 (Bind
+b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H9) in (let H12 \def (eq_ind C
+c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H9) in (ex2_3_intro B C T (\lambda
+(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Void) u1)
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e2 e1)))) Void c1 u1 (refl_equal C (CHead c1 (Bind Void) u1))
+H12))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1:
+(csuba g c1 c3)).(\lambda (H2: (((eq C c3 (CHead e1 (Bind b1) v1)) \to (ex2_3
+B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2
(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g
-e2 e1))))))))) (\lambda (H11: (csuba g c1 e1)).(\lambda (_: (arity g c1 t
-(asucc g a))).(\lambda (_: (arity g e1 v1 a)).(let H14 \def (eq_ind_r C c2
-(\lambda (c: C).(csuba g c (CHead e1 (Bind b1) v1))) H (CHead c1 (Bind Abst)
-t) H3) in (let H15 \def (eq_ind_r B b1 (\lambda (b: B).(csuba g (CHead c1
-(Bind Abst) t) (CHead e1 (Bind b) v1))) H14 Abbr H9) in (ex2_3_intro B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind
-Abst) t) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (_: T).(csuba g e2 e1)))) Abst c1 t (refl_equal C (CHead c1 (Bind
-Abst) t)) H11)))))) u (sym_eq T u v1 H10))) b1 H9)) c0 (sym_eq C c0 e1 H8)))
-H7)) H6))) c2 H3 H4 H0 H1 H2)))]) in (H0 (refl_equal C c2) (refl_equal C
-(CHead e1 (Bind b1) v1))))))))).
+e2 e1)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t
+(asucc g a))).(\lambda (u: T).(\lambda (H4: (arity g c3 u a)).(\lambda (H5:
+(eq C (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1))).(let H6 \def
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 (Bind
+Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def (f_equal C B (\lambda
+(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
+Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
+[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead c3 (Bind
+Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T (\lambda
+(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u
+| (CHead _ _ t0) \Rightarrow t0])) (CHead c3 (Bind Abbr) u) (CHead e1 (Bind
+b1) v1) H5) in (\lambda (H9: (eq B Abbr b1)).(\lambda (H10: (eq C c3
+e1)).(let H11 \def (eq_ind T u (\lambda (t0: T).(arity g c3 t0 a)) H4 v1 H8)
+in (let H12 \def (eq_ind C c3 (\lambda (c: C).(arity g c v1 a)) H11 e1 H10)
+in (let H13 \def (eq_ind C c3 (\lambda (c: C).((eq C c (CHead e1 (Bind b1)
+v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq
+C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda
+(_: T).(csuba g e2 e1))))))) H2 e1 H10) in (let H14 \def (eq_ind C c3
+(\lambda (c: C).(csuba g c1 c)) H1 e1 H10) in (let H15 \def (eq_ind_r B b1
+(\lambda (b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to (ex2_3 B C T (\lambda
+(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2)
+v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2
+e1))))))) H13 Abbr H9) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2)
+v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))
+Abst c1 t (refl_equal C (CHead c1 (Bind Abst) t)) H14))))))))) H7))
+H6)))))))))))) c2 y H0))) H)))))).
projects / helm.git / blobdiff