(* This file was automatically generated: do not edit *********************)
-set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubt/fwd".
-
-include "csubt/defs.ma".
-
-theorem csubt_inv_coq:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).(\forall (P: ((G \to (C \to
-(C \to Prop))))).((((csubt g c1 c2) \to (\forall (n: nat).((eq C (CSort n)
-c1) \to ((eq C (CSort n) c2) \to (P g c1 c2)))))) \to ((((csubt g c1 c2) \to
-(\forall (c0: C).(\forall (c3: C).(\forall (k: K).(\forall (u: T).((eq C
-(CHead c0 k u) c1) \to ((eq C (CHead c3 k u) c2) \to ((csubt g c0 c3) \to (P
-g c1 c2)))))))))) \to ((((csubt g c1 c2) \to (\forall (c0: C).(\forall (c3:
-C).(\forall (b: B).(\forall (u1: T).(\forall (u2: T).((eq C (CHead c0 (Bind
-Void) u1) c1) \to ((eq C (CHead c3 (Bind b) u2) c2) \to ((csubt g c0 c3) \to
-((not (eq B b Void)) \to (P g c1 c2)))))))))))) \to ((((csubt g c1 c2) \to
-(\forall (c0: C).(\forall (c3: C).(\forall (u: T).(\forall (t: T).((eq C
-(CHead c0 (Bind Abst) t) c1) \to ((eq C (CHead c3 (Bind Abbr) u) c2) \to
-((csubt g c0 c3) \to ((ty3 g c3 u t) \to (P g c1 c2))))))))))) \to ((csubt g
-c1 c2) \to (P g c1 c2)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (P: ((G \to (C \to
-(C \to Prop))))).(\lambda (H: (((csubt g c1 c2) \to (\forall (n: nat).((eq C
-(CSort n) c1) \to ((eq C (CSort n) c2) \to (P g c1 c2))))))).(\lambda (H0:
-(((csubt g c1 c2) \to (\forall (c0: C).(\forall (c3: C).(\forall (k:
-K).(\forall (u: T).((eq C (CHead c0 k u) c1) \to ((eq C (CHead c3 k u) c2)
-\to ((csubt g c0 c3) \to (P g c1 c2))))))))))).(\lambda (H1: (((csubt g c1
-c2) \to (\forall (c0: C).(\forall (c3: C).(\forall (b: B).(\forall (u1:
-T).(\forall (u2: T).((eq C (CHead c0 (Bind Void) u1) c1) \to ((eq C (CHead c3
-(Bind b) u2) c2) \to ((csubt g c0 c3) \to ((not (eq B b Void)) \to (P g c1
-c2))))))))))))).(\lambda (H2: (((csubt g c1 c2) \to (\forall (c0: C).(\forall
-(c3: C).(\forall (u: T).(\forall (t: T).((eq C (CHead c0 (Bind Abst) t) c1)
-\to ((eq C (CHead c3 (Bind Abbr) u) c2) \to ((csubt g c0 c3) \to ((ty3 g c3 u
-t) \to (P g c1 c2)))))))))))).(\lambda (H3: (csubt g c1 c2)).(let H4 \def
-(match H3 in csubt return (\lambda (c: C).(\lambda (c0: C).(\lambda (_:
-(csubt ? c c0)).((eq C c c1) \to ((eq C c0 c2) \to (P g c1 c2)))))) with
-[(csubt_sort n) \Rightarrow (\lambda (H4: (eq C (CSort n) c1)).(\lambda (H5:
-(eq C (CSort n) c2)).(H H3 n H4 H5))) | (csubt_head c0 c3 H4 k u) \Rightarrow
-(\lambda (H5: (eq C (CHead c0 k u) c1)).(\lambda (H6: (eq C (CHead c3 k u)
-c2)).(H0 H3 c0 c3 k u H5 H6 H4))) | (csubt_void c0 c3 H4 b H5 u1 u2)
-\Rightarrow (\lambda (H6: (eq C (CHead c0 (Bind Void) u1) c1)).(\lambda (H7:
-(eq C (CHead c3 (Bind b) u2) c2)).(H1 H3 c0 c3 b u1 u2 H6 H7 H4 H5))) |
-(csubt_abst c0 c3 H4 u t H5) \Rightarrow (\lambda (H6: (eq C (CHead c0 (Bind
-Abst) t) c1)).(\lambda (H7: (eq C (CHead c3 (Bind Abbr) u) c2)).(H2 H3 c0 c3
-u t H6 H7 H4 H5)))]) in (H4 (refl_equal C c1) (refl_equal C c2))))))))))).
+include "LambdaDelta-1/csubt/defs.ma".
theorem csubt_gen_abbr:
\forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csubt g
(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))))
\def
\lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
-(H: (csubt g (CHead e1 (Bind Abbr) v) c2)).(csubt_inv_coq g (CHead e1 (Bind
-Abbr) v) c2 (\lambda (g0: G).(\lambda (_: C).(\lambda (c0: C).(ex2 C (\lambda
-(e2: C).(eq C c0 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g0 e1
-e2)))))) (\lambda (H0: (csubt g (CHead e1 (Bind Abbr) v) c2)).(\lambda (n:
-nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind Abbr) v))).(\lambda (H2:
-(eq C (CSort n) c2)).(let H3 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g
-(CHead e1 (Bind Abbr) v) c)) H0 (CSort n) H2) in (let H4 \def (eq_ind_r C c2
-(\lambda (c: C).(csubt g (CHead e1 (Bind Abbr) v) c)) H (CSort n) H2) in
-(eq_ind C (CSort n) (\lambda (c: C).(ex2 C (\lambda (e2: C).(eq C c (CHead e2
-(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (let H5 \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 Abbr) v) H1) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CSort n)
-(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H5)) c2
-H2))))))) (\lambda (H0: (csubt g (CHead e1 (Bind Abbr) v) c2)).(\lambda (c0:
-C).(\lambda (c3: C).(\lambda (k: K).(\lambda (u: T).(\lambda (H1: (eq C
-(CHead c0 k u) (CHead e1 (Bind Abbr) v))).(\lambda (H2: (eq C (CHead c3 k u)
-c2)).(\lambda (H3: (csubt g c0 c3)).(let H4 \def (eq_ind_r C c2 (\lambda (c:
-C).(csubt g (CHead e1 (Bind Abbr) v) c)) H0 (CHead c3 k u) H2) in (let H5
-\def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind Abbr) v) c)) H
-(CHead c3 k u) H2) in (eq_ind C (CHead c3 k u) (\lambda (c: C).(ex2 C
-(\lambda (e2: C).(eq C c (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g
-e1 e2)))) (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 Abbr) v) H1) in ((let H7 \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 Abbr) v) H1) in ((let H8 \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) (CHead e1 (Bind Abbr) v) H1) in (\lambda (H9:
-(eq K k (Bind Abbr))).(\lambda (H10: (eq C c0 e1)).(let H11 \def (eq_ind T u
-(\lambda (t: T).(csubt g (CHead e1 (Bind Abbr) v) (CHead c3 k t))) H5 v H8)
-in (let H12 \def (eq_ind T u (\lambda (t: T).(csubt g (CHead e1 (Bind Abbr)
-v) (CHead c3 k t))) H4 v H8) in (eq_ind_r T v (\lambda (t: T).(ex2 C (\lambda
-(e2: C).(eq C (CHead c3 k t) (CHead e2 (Bind Abbr) v))) (\lambda (e2:
-C).(csubt g e1 e2)))) (let H13 \def (eq_ind K k (\lambda (k0: K).(csubt g
-(CHead e1 (Bind Abbr) v) (CHead c3 k0 v))) H11 (Bind Abbr) H9) in (let H14
-\def (eq_ind K k (\lambda (k0: K).(csubt g (CHead e1 (Bind Abbr) v) (CHead c3
-k0 v))) H12 (Bind Abbr) H9) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2
-C (\lambda (e2: C).(eq C (CHead c3 k0 v) (CHead e2 (Bind Abbr) v))) (\lambda
-(e2: C).(csubt g e1 e2)))) (let H15 \def (eq_ind C c0 (\lambda (c: C).(csubt
-g c c3)) H3 e1 H10) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind
-Abbr) v) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)) c3
-(refl_equal C (CHead c3 (Bind Abbr) v)) H15)) k H9))) u H8)))))) H7)) H6)) c2
-H2))))))))))) (\lambda (H0: (csubt g (CHead e1 (Bind Abbr) v) c2)).(\lambda
-(c0: C).(\lambda (c3: C).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2:
-T).(\lambda (H2: (eq C (CHead c0 (Bind Void) u1) (CHead e1 (Bind Abbr)
-v))).(\lambda (H3: (eq C (CHead c3 (Bind b) u2) c2)).(\lambda (_: (csubt g c0
-c3)).(\lambda (_: (not (eq B b Void))).(let H5 \def (eq_ind_r C c2 (\lambda
-(c: C).(csubt g (CHead e1 (Bind Abbr) v) c)) H0 (CHead c3 (Bind b) u2) H3) in
-(let H6 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind Abbr) v)
-c)) H (CHead c3 (Bind b) u2) H3) in (eq_ind C (CHead c3 (Bind b) u2) (\lambda
-(c: C).(ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abbr) v))) (\lambda
-(e2: C).(csubt g e1 e2)))) (let H7 \def (eq_ind C (CHead c0 (Bind Void) u1)
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+(H: (csubt g (CHead e1 (Bind Abbr) v) c2)).(insert_eq C (CHead e1 (Bind Abbr)
+v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C (\lambda (e2:
+C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))))
+(\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda (c:
+C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda
+(e2: C).(eq C c0 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1
+e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind
+Abbr) v))).(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 Abbr) v) H1) in (False_ind (ex2 C
+(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abbr) v))) (\lambda (e2:
+C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1:
+(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2
+C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2:
+C).(csubt g e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C
+(CHead c1 k u) (CHead e1 (Bind Abbr) v))).(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 Abbr) v) 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 Abbr) v) 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
+Abbr) v) H3) in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1
+e1)).(eq_ind_r T v (\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k
+t) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K
+(Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v)
+(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def
+(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C
+(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt
+g e1 e2))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g
+c c3)) H1 e1 H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr)
+v) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)) c3
+(refl_equal C (CHead c3 (Bind Abbr) v)) H10))) k H7) u H6)))) H5))
+H4))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1
+c3)).(\lambda (_: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda
+(e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt 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 Abbr) v))).(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 e1 (Bind Abbr)
-v) H2) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2)
-(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H7)) c2
-H3))))))))))))) (\lambda (H0: (csubt g (CHead e1 (Bind Abbr) v) c2)).(\lambda
-(c0: C).(\lambda (c3: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (eq C
-(CHead c0 (Bind Abst) t) (CHead e1 (Bind Abbr) v))).(\lambda (H3: (eq C
-(CHead c3 (Bind Abbr) u) c2)).(\lambda (_: (csubt g c0 c3)).(\lambda (_: (ty3
-g c3 u t)).(let H5 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1
-(Bind Abbr) v) c)) H0 (CHead c3 (Bind Abbr) u) H3) in (let H6 \def (eq_ind_r
-C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind Abbr) v) c)) H (CHead c3 (Bind
-Abbr) u) H3) in (eq_ind C (CHead c3 (Bind Abbr) u) (\lambda (c: C).(ex2 C
-(\lambda (e2: C).(eq C c (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g
-e1 e2)))) (let H7 \def (eq_ind C (CHead c0 (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 e1 (Bind Abbr) v) H2) in (False_ind
-(ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr)
-v))) (\lambda (e2: C).(csubt g e1 e2))) H7)) c2 H3)))))))))))) H))))).
+v) H4) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2)
+(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H5)))))))))))
+(\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_:
+(((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3
+(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (u:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (_: (ty3 g c3 u
+t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abbr)
+v))).(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 e1 (Bind Abbr) v) H5) in (False_ind (ex2 C
+(\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v)))
+(\lambda (e2: C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H))))).
theorem csubt_gen_abst:
\forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csubt g
(CHead e1 (Bind Abst) v1) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead
-e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda
+e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda
(e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
-C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g
-e2 v2 v1)))))))))
+C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g
+e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))))
\def
\lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda
-(H: (csubt g (CHead e1 (Bind Abst) v1) c2)).(csubt_inv_coq g (CHead e1 (Bind
-Abst) v1) c2 (\lambda (g0: G).(\lambda (_: C).(\lambda (c0: C).(or (ex2 C
-(\lambda (e2: C).(eq C c0 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt
-g0 e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c0 (CHead e2
-(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g0 e1 e2)))
-(\lambda (e2: C).(\lambda (v2: T).(ty3 g0 e2 v2 v1)))))))) (\lambda (H0:
-(csubt g (CHead e1 (Bind Abst) v1) c2)).(\lambda (n: nat).(\lambda (H1: (eq C
-(CSort n) (CHead e1 (Bind Abst) v1))).(\lambda (H2: (eq C (CSort n) c2)).(let
-H3 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind Abst) v1) c))
-H0 (CSort n) H2) in (let H4 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g
-(CHead e1 (Bind Abst) v1) c)) H (CSort n) H2) in (eq_ind C (CSort n) (\lambda
-(c: C).(or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abst) v1)))
-(\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2:
-T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
-T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))
-(let H5 \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 Abst) v1) H1) in (False_ind (or (ex2 C
-(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abst) v1))) (\lambda (e2:
-C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C
-(CSort n) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
-T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))
-H5)) c2 H2))))))) (\lambda (H0: (csubt g (CHead e1 (Bind Abst) v1)
-c2)).(\lambda (c0: C).(\lambda (c3: C).(\lambda (k: K).(\lambda (u:
-T).(\lambda (H1: (eq C (CHead c0 k u) (CHead e1 (Bind Abst) v1))).(\lambda
-(H2: (eq C (CHead c3 k u) c2)).(\lambda (H3: (csubt g c0 c3)).(let H4 \def
-(eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind Abst) v1) c)) H0
-(CHead c3 k u) H2) in (let H5 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g
-(CHead e1 (Bind Abst) v1) c)) H (CHead c3 k u) H2) in (eq_ind C (CHead c3 k
-u) (\lambda (c: C).(or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abst)
-v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda
-(v2: T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
-T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))
-(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 Abst) v1) H1) in ((let H7 \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 Abst) v1)
-H1) in ((let H8 \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) (CHead e1 (Bind Abst) v1) H1) in (\lambda (H9: (eq K k
-(Bind Abst))).(\lambda (H10: (eq C c0 e1)).(let H11 \def (eq_ind T u (\lambda
-(t: T).(csubt g (CHead e1 (Bind Abst) v1) (CHead c3 k t))) H5 v1 H8) in (let
-H12 \def (eq_ind T u (\lambda (t: T).(csubt g (CHead e1 (Bind Abst) v1)
-(CHead c3 k t))) H4 v1 H8) in (eq_ind_r T v1 (\lambda (t: T).(or (ex2 C
-(\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 (Bind Abst) v1))) (\lambda
-(e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C
-(CHead c3 k t) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
-T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))
-(let H13 \def (eq_ind K k (\lambda (k0: K).(csubt g (CHead e1 (Bind Abst) v1)
-(CHead c3 k0 v1))) H11 (Bind Abst) H9) in (let H14 \def (eq_ind K k (\lambda
-(k0: K).(csubt g (CHead e1 (Bind Abst) v1) (CHead c3 k0 v1))) H12 (Bind Abst)
-H9) in (eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (e2:
-C).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt
-g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0
-v1) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1
-e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) (let H15 \def
-(eq_ind C c0 (\lambda (c: C).(csubt g c c3)) H3 e1 H10) in (or_introl (ex2 C
-(\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst) v1)))
-(\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2:
-T).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
-C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g
-e2 v2 v1)))) (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1)
-(CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal
-C (CHead c3 (Bind Abst) v1)) H15))) k H9))) u H8)))))) H7)) H6)) c2
-H2))))))))))) (\lambda (H0: (csubt g (CHead e1 (Bind Abst) v1) c2)).(\lambda
-(c0: C).(\lambda (c3: C).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2:
-T).(\lambda (H2: (eq C (CHead c0 (Bind Void) u1) (CHead e1 (Bind Abst)
-v1))).(\lambda (H3: (eq C (CHead c3 (Bind b) u2) c2)).(\lambda (_: (csubt g
-c0 c3)).(\lambda (_: (not (eq B b Void))).(let H5 \def (eq_ind_r C c2
-(\lambda (c: C).(csubt g (CHead e1 (Bind Abst) v1) c)) H0 (CHead c3 (Bind b)
-u2) H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1
-(Bind Abst) v1) c)) H (CHead c3 (Bind b) u2) H3) in (eq_ind C (CHead c3 (Bind
-b) u2) (\lambda (c: C).(or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind
-Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2:
-C).(\lambda (v2: T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
-C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g
-e2 v2 v1)))))) (let H7 \def (eq_ind C (CHead c0 (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 e1 (Bind Abst) v1) H2) in (False_ind
-(or (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Bind
-Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2:
-C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Bind Abbr) v2))))
-(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda
-(v2: T).(ty3 g e2 v2 v1))))) H7)) c2 H3))))))))))))) (\lambda (H0: (csubt g
-(CHead e1 (Bind Abst) v1) c2)).(\lambda (c0: C).(\lambda (c3: C).(\lambda (u:
-T).(\lambda (t: T).(\lambda (H2: (eq C (CHead c0 (Bind Abst) t) (CHead e1
-(Bind Abst) v1))).(\lambda (H3: (eq C (CHead c3 (Bind Abbr) u) c2)).(\lambda
-(H1: (csubt g c0 c3)).(\lambda (H4: (ty3 g c3 u t)).(let H5 \def (eq_ind_r C
-c2 (\lambda (c: C).(csubt g (CHead e1 (Bind Abst) v1) c)) H0 (CHead c3 (Bind
-Abbr) u) H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead
-e1 (Bind Abst) v1) c)) H (CHead c3 (Bind Abbr) u) H3) in (eq_ind C (CHead c3
-(Bind Abbr) u) (\lambda (c: C).(or (ex2 C (\lambda (e2: C).(eq C c (CHead e2
-(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2:
-C).(\lambda (v2: T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
-C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g
-e2 v2 v1)))))) (let H7 \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 Abst) t) (CHead e1 (Bind Abst) v1) H2) in ((let H8 \def
+(H: (csubt g (CHead e1 (Bind Abst) v1) c2)).(insert_eq C (CHead e1 (Bind
+Abst) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(or (ex2 C (\lambda
+(e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1
+e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
+Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_:
+C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3
+g e2 v2 v1)))))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g
+(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or
+(ex2 C (\lambda (e2: C).(eq C c0 (CHead e2 (Bind Abst) v1))) (\lambda (e2:
+C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c0
+(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1
+e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2:
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (n: nat).(\lambda (H1:
+(eq C (CSort n) (CHead e1 (Bind Abst) 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 Abst)
+v1) H1) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CSort n) (CHead e2
+(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2:
+C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 (Bind Abbr) v2)))) (\lambda
+(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2:
+T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))
+H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1
+c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C
+(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt
+g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2
+(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))
+(\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda
+(v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3:
+(eq C (CHead c1 k u) (CHead e1 (Bind Abst) 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 Abst) 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 Abst) 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 Abst) v1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda
+(H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(or (ex2 C (\lambda (e2:
+C).(eq C (CHead c3 k t) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g
+e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t)
+(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1
+e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2:
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) (eq_ind_r K (Bind Abst) (\lambda
+(k0: K).(or (ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v1) (CHead e2 (Bind
+Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2:
+C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abbr) v2))))
+(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda
+(v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2
+v1)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind
+Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst)
+v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda
+(v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
+T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1)))
+(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let
+H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_introl
+(ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst)
+v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda
+(v2: T).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda
+(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2:
+T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))
+(ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind
+Abst) v1))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3
+(Bind Abst) v1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1
+(CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2
+(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2:
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
+C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g
+e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2
+v1)))))))).(\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 Abst) v1))).(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 e1 (Bind Abst)
+v1) H4) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b)
+u2) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T
+(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) u2) (CHead e2
+(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))
+(\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda
+(v2: T).(ty3 g e2 v2 v1))))) H5))))))))))) (\lambda (c1: C).(\lambda (c3:
+C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind
+Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst)
+v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda
+(v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
+T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1)))
+(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (u:
+T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (H4: (ty3 g c3 u
+t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst)
+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 Abst) v1) 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 c0 (Bind
-Abst) t) (CHead e1 (Bind Abst) v1) H2) in (\lambda (H9: (eq C c0 e1)).(let
-H10 \def (eq_ind T t (\lambda (t0: T).(ty3 g c3 u t0)) H4 v1 H8) in (let H11
-\def (eq_ind C c0 (\lambda (c: C).(csubt g c c3)) H1 e1 H9) in (or_intror
-(ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abst)
-v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda
-(v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda
-(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2:
-T).(ty3 g e2 v2 v1)))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq
-C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
-C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g
-e2 v2 v1))) c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H11 H10)))))) H7))
-c2 H3)))))))))))) H))))).
+[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind
+Abst) t) (CHead e1 (Bind Abst) v1) H5) in (\lambda (H8: (eq C c1 e1)).(let H9
+\def (eq_ind T t (\lambda (t0: T).(ty3 g c3 u t0)) H4 v1 H7) in (let H10 \def
+(eq_ind T t (\lambda (t0: T).(ty3 g c1 u t0)) H3 v1 H7) in (let H11 \def
+(eq_ind C c1 (\lambda (c: C).(ty3 g c u v1)) H10 e1 H8) in (let H12 \def
+(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or (ex2
+C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2:
+C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3
+(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1
+e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2:
+C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let H13 \def (eq_ind
+C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_intror (ex2 C (\lambda
+(e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda
+(e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C
+(CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
+C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g
+e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))
+(ex4_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind
+Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt
+g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2:
+C).(\lambda (v2: T).(ty3 g e2 v2 v1))) c3 u (refl_equal C (CHead c3 (Bind
+Abbr) u)) H13 H11 H9))))))))) H6))))))))))) y c2 H0))) H))))).
+
+theorem csubt_gen_flat:
+ \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).(\forall
+(f: F).((csubt g (CHead e1 (Flat f) v) c2) \to (ex2 C (\lambda (e2: C).(eq C
+c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))))))))
+\def
+ \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
+(f: F).(\lambda (H: (csubt g (CHead e1 (Flat f) v) c2)).(insert_eq C (CHead
+e1 (Flat f) v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C
+(\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g
+e1 e2)))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda
+(c: C).(\lambda (c0: C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda
+(e2: C).(eq C c0 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1
+e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Flat f)
+v))).(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 (Flat f) v) H1) in (False_ind (ex2 C
+(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Flat f) v))) (\lambda (e2:
+C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1:
+(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Flat f) v)) \to (ex2 C
+(\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g
+e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k
+u) (CHead e1 (Flat f) v))).(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 (Flat f) v) 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 (Flat f) v) 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 (Flat f) v) H3) in
+(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v
+(\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 (Flat
+f) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K (Flat f) (\lambda
+(k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v) (CHead e2 (Flat f) v)))
+(\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def (eq_ind C c1 (\lambda (c:
+C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda (e2: C).(eq C c3
+(CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))))) H2 e1 H8) in
+(let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in
+(ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Flat f) v) (CHead e2 (Flat f)
+v))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 (Flat f)
+v)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c3:
+C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Flat f)
+v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda
+(e2: C).(csubt 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 (Flat f) v))).(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 _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (CHead e1 (Flat f) v) H4) in (False_ind (ex2 C (\lambda (e2:
+C).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Flat f) v))) (\lambda (e2:
+C).(csubt g e1 e2))) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda
+(_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Flat f) v)) \to (ex2
+C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g
+e1 e2)))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u
+t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t)
+(CHead e1 (Flat f) v))).(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 e1 (Flat f) v) H5) in (False_ind (ex2 C (\lambda (e2:
+C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Flat f) v))) (\lambda (e2:
+C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H)))))).
theorem csubt_gen_bind:
\forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall
(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))))))
\def
\lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda
-(v1: T).(\lambda (H: (csubt g (CHead e1 (Bind b1) v1) c2)).(csubt_inv_coq g
-(CHead e1 (Bind b1) v1) c2 (\lambda (g0: G).(\lambda (_: C).(\lambda (c0:
-C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c0
+(v1: T).(\lambda (H: (csubt g (CHead e1 (Bind b1) v1) c2)).(insert_eq C
+(CHead e1 (Bind b1) v1) (\lambda (c: C).(csubt 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).(csubt g0 e1 e2)))))))) (\lambda (H0: (csubt g (CHead e1 (Bind b1) v1)
-c2)).(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind b1)
-v1))).(\lambda (H2: (eq C (CSort n) c2)).(let H3 \def (eq_ind_r C c2 (\lambda
-(c: C).(csubt g (CHead e1 (Bind b1) v1) c)) H0 (CSort n) H2) in (let H4 \def
-(eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind b1) v1) c)) H (CSort
-n) H2) in (eq_ind C (CSort n) (\lambda (c: C).(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).(csubt g e1 e2)))))) (let H5
-\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).(csubt g e1 e2)))))
-H5)) c2 H2))))))) (\lambda (H0: (csubt g (CHead e1 (Bind b1) v1)
-c2)).(\lambda (c0: C).(\lambda (c3: C).(\lambda (k: K).(\lambda (u:
-T).(\lambda (H1: (eq C (CHead c0 k u) (CHead e1 (Bind b1) v1))).(\lambda (H2:
-(eq C (CHead c3 k u) c2)).(\lambda (H3: (csubt g c0 c3)).(let H4 \def
-(eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind b1) v1) c)) H0 (CHead
-c3 k u) H2) in (let H5 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1
-(Bind b1) v1) c)) H (CHead c3 k u) H2) in (eq_ind C (CHead c3 k u) (\lambda
-(c: C).(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).(csubt g e1 e2)))))) (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) H1) in ((let H7 \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) H1) in ((let H8 \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) (CHead e1 (Bind b1) v1) H1) in
-(\lambda (H9: (eq K k (Bind b1))).(\lambda (H10: (eq C c0 e1)).(let H11 \def
-(eq_ind T u (\lambda (t: T).(csubt g (CHead e1 (Bind b1) v1) (CHead c3 k t)))
-H5 v1 H8) in (let H12 \def (eq_ind T u (\lambda (t: T).(csubt g (CHead e1
-(Bind b1) v1) (CHead c3 k t))) H4 v1 H8) in (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).(csubt g e1 e2)))))) (let H13 \def (eq_ind K k (\lambda
-(k0: K).(csubt g (CHead e1 (Bind b1) v1) (CHead c3 k0 v1))) H11 (Bind b1) H9)
-in (let H14 \def (eq_ind K k (\lambda (k0: K).(csubt g (CHead e1 (Bind b1)
-v1) (CHead c3 k0 v1))) H12 (Bind b1) H9) in (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).(csubt g e1 e2)))))) (let H15 \def (eq_ind C c0 (\lambda
-(c: C).(csubt g c c3)) H3 e1 H10) in (ex2_3_intro B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b1) v1) (CHead e2
-(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g
-e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3 (Bind b1) v1)) H15)) k H9))) u
-H8)))))) H7)) H6)) c2 H2))))))))))) (\lambda (H0: (csubt g (CHead e1 (Bind
-b1) v1) c2)).(\lambda (c0: C).(\lambda (c3: C).(\lambda (b: B).(\lambda (u1:
-T).(\lambda (u2: T).(\lambda (H2: (eq C (CHead c0 (Bind Void) u1) (CHead e1
-(Bind b1) v1))).(\lambda (H3: (eq C (CHead c3 (Bind b) u2) c2)).(\lambda (H1:
-(csubt g c0 c3)).(\lambda (_: (not (eq B b Void))).(let H5 \def (eq_ind_r C
-c2 (\lambda (c: C).(csubt g (CHead e1 (Bind b1) v1) c)) H0 (CHead c3 (Bind b)
-u2) H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1
-(Bind b1) v1) c)) H (CHead c3 (Bind b) u2) H3) in (eq_ind C (CHead c3 (Bind
-b) u2) (\lambda (c: C).(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).(csubt g e1 e2)))))) (let H7 \def
+T).(csubt g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csubt g y
+c2)).(csubt_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).(csubt 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).(csubt g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda
+(c3: C).(\lambda (H1: (csubt 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).(csubt 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 c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Bind
-Void) u1) (CHead e1 (Bind b1) v1) H2) in ((let H8 \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 c0 (Bind
-Void) u1) (CHead e1 (Bind b1) v1) H2) in ((let H9 \def (f_equal C T (\lambda
-(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u1
-| (CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind Void) u1) (CHead e1 (Bind
-b1) v1) H2) in (\lambda (H10: (eq B Void b1)).(\lambda (H11: (eq C c0
-e1)).(let H12 \def (eq_ind C c0 (\lambda (c: C).(csubt g c c3)) H1 e1 H11) in
-(let H13 \def (eq_ind_r B b1 (\lambda (b0: B).(csubt g (CHead e1 (Bind b0)
-v1) (CHead c3 (Bind b) u2))) H6 Void H10) in (let H14 \def (eq_ind_r B b1
-(\lambda (b0: B).(csubt g (CHead e1 (Bind b0) v1) (CHead c3 (Bind b) u2))) H5
-Void H10) 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).(csubt g e1 e2)))) b c3 u2 (refl_equal C
-(CHead c3 (Bind b) u2)) H12))))))) H8)) H7)) c2 H3))))))))))))) (\lambda (H0:
-(csubt g (CHead e1 (Bind b1) v1) c2)).(\lambda (c0: C).(\lambda (c3:
-C).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (eq C (CHead c0 (Bind Abst)
-t) (CHead e1 (Bind b1) v1))).(\lambda (H3: (eq C (CHead c3 (Bind Abbr) u)
-c2)).(\lambda (H1: (csubt g c0 c3)).(\lambda (H4: (ty3 g c3 u t)).(let H5
-\def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind b1) v1) c)) H0
-(CHead c3 (Bind Abbr) u) H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c:
-C).(csubt g (CHead e1 (Bind b1) v1) c)) H (CHead c3 (Bind Abbr) u) H3) in
-(eq_ind C (CHead c3 (Bind Abbr) u) (\lambda (c: C).(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).(csubt g e1 e2)))))) (let H7
-\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 Abst) t) (CHead e1 (Bind b1) v1) H2) in ((let H8 \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 c0 (Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in ((let H9 \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 c0 (Bind
-Abst) t) (CHead e1 (Bind b1) v1) H2) in (\lambda (H10: (eq B Abst
-b1)).(\lambda (H11: (eq C c0 e1)).(let H12 \def (eq_ind T t (\lambda (t0:
-T).(ty3 g c3 u t0)) H4 v1 H9) in (let H13 \def (eq_ind C c0 (\lambda (c:
-C).(csubt g c c3)) H1 e1 H11) in (let H14 \def (eq_ind_r B b1 (\lambda (b:
-B).(csubt g (CHead e1 (Bind b) v1) (CHead c3 (Bind Abbr) u))) H6 Abst H10) in
-(let H15 \def (eq_ind_r B b1 (\lambda (b: B).(csubt g (CHead e1 (Bind b) v1)
-(CHead c3 (Bind Abbr) u))) H5 Abst H10) in (ex2_3_intro B C T (\lambda (b2:
+[(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).(csubt 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).(csubt 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).(csubt g e1 e2))))))) H2 e1 H8) in (let
+H10 \def (eq_ind C c1 (\lambda (c: C).(csubt 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 c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
+(e2: C).(\lambda (_: T).(csubt 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: (csubt 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).(csubt 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 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).(csubt g e1 e2))))))) H2 e1
+H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csubt 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).(csubt 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).(csubt 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:
+(csubt 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).(csubt g
+e1 e2)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u
+t)).(\lambda (H4: (ty3 g c3 u t)).(\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) 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).(ty3 g c3 u t0)) H4 v1 H8) in (let H12 \def (eq_ind T t (\lambda (t0:
+T).(ty3 g c1 u t0)) H3 v1 H8) in (let H13 \def (eq_ind C c1 (\lambda (c:
+C).(ty3 g c u v1)) H12 e1 H10) in (let H14 \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).(csubt g e1 e2))))))) H2 e1
+H10) in (let H15 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H10)
+in (let H16 \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).(csubt g e1 e2))))))) H14 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).(csubt g
-e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H13)))))))) H8))
-H7)) c2 H3)))))))))))) H)))))).
+e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H15))))))))))
+H7)) H6))))))))))) y c2 H0))) H)))))).