Prop).P)))
\def
\lambda (x: C).(\lambda (n: nat).(\lambda (H: (clear (CSort n) x)).(\lambda
-(P: Prop).(let H0 \def (match H in clear return (\lambda (c: C).(\lambda (c0:
-C).(\lambda (_: (clear c c0)).((eq C c (CSort n)) \to ((eq C c0 x) \to P)))))
-with [(clear_bind b e u) \Rightarrow (\lambda (H0: (eq C (CHead e (Bind b) u)
-(CSort n))).(\lambda (H1: (eq C (CHead e (Bind b) u) x)).((let H2 \def
-(eq_ind C (CHead e (Bind b) u) (\lambda (e0: C).(match e0 in C return
+(P: Prop).(insert_eq C (CSort n) (\lambda (c: C).(clear c x)) (\lambda (_:
+C).P) (\lambda (y: C).(\lambda (H0: (clear y x)).(clear_ind (\lambda (c:
+C).(\lambda (_: C).((eq C c (CSort n)) \to P))) (\lambda (b: B).(\lambda (e:
+C).(\lambda (u: T).(\lambda (H1: (eq C (CHead e (Bind b) u) (CSort n))).(let
+H2 \def (eq_ind C (CHead e (Bind b) u) (\lambda (ee: C).(match ee in C return
(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _)
-\Rightarrow True])) I (CSort n) H0) in (False_ind ((eq C (CHead e (Bind b) u)
-x) \to P) H2)) H1))) | (clear_flat e c H0 f u) \Rightarrow (\lambda (H1: (eq
-C (CHead e (Flat f) u) (CSort n))).(\lambda (H2: (eq C c x)).((let H3 \def
-(eq_ind C (CHead e (Flat f) u) (\lambda (e0: C).(match e0 in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _)
-\Rightarrow True])) I (CSort n) H1) in (False_ind ((eq C c x) \to ((clear e
-c) \to P)) H3)) H2 H0)))]) in (H0 (refl_equal C (CSort n)) (refl_equal C
-x)))))).
+\Rightarrow True])) I (CSort n) H1) in (False_ind P H2)))))) (\lambda (e:
+C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (_: (((eq C e (CSort
+n)) \to P))).(\lambda (f: F).(\lambda (u: T).(\lambda (H3: (eq C (CHead e
+(Flat f) u) (CSort n))).(let H4 \def (eq_ind C (CHead e (Flat f) u) (\lambda
+(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in
+(False_ind P H4))))))))) y x H0))) H)))).
theorem clear_gen_bind:
\forall (b: B).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear
(CHead e (Bind b) u) x) \to (eq C x (CHead e (Bind b) u))))))
\def
\lambda (b: B).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H:
-(clear (CHead e (Bind b) u) x)).(let H0 \def (match H in clear return
-(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (clear c c0)).((eq C c (CHead e
-(Bind b) u)) \to ((eq C c0 x) \to (eq C x (CHead e (Bind b) u))))))) with
-[(clear_bind b0 e0 u0) \Rightarrow (\lambda (H0: (eq C (CHead e0 (Bind b0)
-u0) (CHead e (Bind b) u))).(\lambda (H1: (eq C (CHead e0 (Bind b0) u0)
-x)).((let H2 \def (f_equal C T (\lambda (e1: C).(match e1 in C return
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow
-t])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H0) in ((let H3 \def
+(clear (CHead e (Bind b) u) x)).(insert_eq C (CHead e (Bind b) u) (\lambda
+(c: C).(clear c x)) (\lambda (c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0:
+(clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e
+(Bind b) u)) \to (eq C c0 c)))) (\lambda (b0: B).(\lambda (e0: C).(\lambda
+(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b0) u0) (CHead e (Bind b)
+u))).(let H2 \def (f_equal C C (\lambda (e1: C).(match e1 in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow e0 | (CHead c _ _) \Rightarrow
+c])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in ((let H3 \def
(f_equal C B (\lambda (e1: C).(match e1 in C return (\lambda (_: C).B) with
[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return
(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
-b0])])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H0) in ((let H4 \def
-(f_equal C C (\lambda (e1: C).(match e1 in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow e0 | (CHead c _ _) \Rightarrow c])) (CHead e0 (Bind
-b0) u0) (CHead e (Bind b) u) H0) in (eq_ind C e (\lambda (c: C).((eq B b0 b)
-\to ((eq T u0 u) \to ((eq C (CHead c (Bind b0) u0) x) \to (eq C x (CHead e
-(Bind b) u)))))) (\lambda (H5: (eq B b0 b)).(eq_ind B b (\lambda (b1: B).((eq
-T u0 u) \to ((eq C (CHead e (Bind b1) u0) x) \to (eq C x (CHead e (Bind b)
-u))))) (\lambda (H6: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead e
-(Bind b) t) x) \to (eq C x (CHead e (Bind b) u)))) (\lambda (H7: (eq C (CHead
-e (Bind b) u) x)).(eq_ind C (CHead e (Bind b) u) (\lambda (c: C).(eq C c
-(CHead e (Bind b) u))) (refl_equal C (CHead e (Bind b) u)) x H7)) u0 (sym_eq
-T u0 u H6))) b0 (sym_eq B b0 b H5))) e0 (sym_eq C e0 e H4))) H3)) H2)) H1)))
-| (clear_flat e0 c H0 f u0) \Rightarrow (\lambda (H1: (eq C (CHead e0 (Flat
-f) u0) (CHead e (Bind b) u))).(\lambda (H2: (eq C c x)).((let H3 \def (eq_ind
-C (CHead e0 (Flat f) u0) (\lambda (e1: C).(match e1 in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
-k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat
-_) \Rightarrow True])])) I (CHead e (Bind b) u) H1) in (False_ind ((eq C c x)
-\to ((clear e0 c) \to (eq C x (CHead e (Bind b) u)))) H3)) H2 H0)))]) in (H0
-(refl_equal C (CHead e (Bind b) u)) (refl_equal C x))))))).
+b0])])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in ((let H4 \def
+(f_equal C T (\lambda (e1: C).(match e1 in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e0 (Bind
+b0) u0) (CHead e (Bind b) u) H1) in (\lambda (H5: (eq B b0 b)).(\lambda (H6:
+(eq C e0 e)).(eq_ind_r T u (\lambda (t: T).(eq C (CHead e0 (Bind b0) t)
+(CHead e0 (Bind b0) t))) (eq_ind_r C e (\lambda (c: C).(eq C (CHead c (Bind
+b0) u) (CHead c (Bind b0) u))) (eq_ind_r B b (\lambda (b1: B).(eq C (CHead e
+(Bind b1) u) (CHead e (Bind b1) u))) (refl_equal C (CHead e (Bind b) u)) b0
+H5) e0 H6) u0 H4)))) H3)) H2)))))) (\lambda (e0: C).(\lambda (c: C).(\lambda
+(_: (clear e0 c)).(\lambda (_: (((eq C e0 (CHead e (Bind b) u)) \to (eq C c
+e0)))).(\lambda (f: F).(\lambda (u0: T).(\lambda (H3: (eq C (CHead e0 (Flat
+f) u0) (CHead e (Bind b) u))).(let H4 \def (eq_ind C (CHead e0 (Flat f) 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 False | (Flat _) \Rightarrow
+True])])) I (CHead e (Bind b) u) H3) in (False_ind (eq C c (CHead e0 (Flat f)
+u0)) H4))))))))) y x H0))) H))))).
theorem clear_gen_flat:
\forall (f: F).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear
(CHead e (Flat f) u) x) \to (clear e x)))))
\def
\lambda (f: F).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H:
-(clear (CHead e (Flat f) u) x)).(let H0 \def (match H in clear return
-(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (clear c c0)).((eq C c (CHead e
-(Flat f) u)) \to ((eq C c0 x) \to (clear e x)))))) with [(clear_bind b e0 u0)
-\Rightarrow (\lambda (H0: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f)
-u))).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) x)).((let H2 \def (eq_ind C
-(CHead e0 (Bind b) u0) (\lambda (e1: C).(match e1 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 e (Flat f) u) H0) in (False_ind ((eq C
-(CHead e0 (Bind b) u0) x) \to (clear e x)) H2)) H1))) | (clear_flat e0 c H0
-f0 u0) \Rightarrow (\lambda (H1: (eq C (CHead e0 (Flat f0) u0) (CHead e (Flat
-f) u))).(\lambda (H2: (eq C c x)).((let H3 \def (f_equal C T (\lambda (e1:
+(clear (CHead e (Flat f) u) x)).(insert_eq C (CHead e (Flat f) u) (\lambda
+(c: C).(clear c x)) (\lambda (_: C).(clear e x)) (\lambda (y: C).(\lambda
+(H0: (clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead
+e (Flat f) u)) \to (clear e c0)))) (\lambda (b: B).(\lambda (e0: C).(\lambda
+(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f)
+u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) 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 e (Flat f) u) H1)
+in (False_ind (clear e (CHead e0 (Bind b) u0)) H2)))))) (\lambda (e0:
+C).(\lambda (c: C).(\lambda (H1: (clear e0 c)).(\lambda (H2: (((eq C e0
+(CHead e (Flat f) u)) \to (clear e c)))).(\lambda (f0: F).(\lambda (u0:
+T).(\lambda (H3: (eq C (CHead e0 (Flat f0) u0) (CHead e (Flat f) u))).(let H4
+\def (f_equal C C (\lambda (e1: C).(match e1 in C return (\lambda (_: C).C)
+with [(CSort _) \Rightarrow e0 | (CHead c0 _ _) \Rightarrow c0])) (CHead e0
+(Flat f0) u0) (CHead e (Flat f) u) H3) in ((let H5 \def (f_equal C F (\lambda
+(e1: C).(match e1 in C return (\lambda (_: C).F) with [(CSort _) \Rightarrow
+f0 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).F) with
+[(Bind _) \Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead e0 (Flat f0)
+u0) (CHead e (Flat f) u) H3) in ((let H6 \def (f_equal C T (\lambda (e1:
C).(match e1 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
(CHead _ _ t) \Rightarrow t])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u)
-H1) in ((let H4 \def (f_equal C F (\lambda (e1: C).(match e1 in C return
-(\lambda (_: C).F) with [(CSort _) \Rightarrow f0 | (CHead _ k _) \Rightarrow
-(match k in K return (\lambda (_: K).F) with [(Bind _) \Rightarrow f0 | (Flat
-f1) \Rightarrow f1])])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u) H1) in
-((let H5 \def (f_equal C C (\lambda (e1: C).(match e1 in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow e0 | (CHead c0 _ _) \Rightarrow c0]))
-(CHead e0 (Flat f0) u0) (CHead e (Flat f) u) H1) in (eq_ind C e (\lambda (c0:
-C).((eq F f0 f) \to ((eq T u0 u) \to ((eq C c x) \to ((clear c0 c) \to (clear
-e x)))))) (\lambda (H6: (eq F f0 f)).(eq_ind F f (\lambda (_: F).((eq T u0 u)
-\to ((eq C c x) \to ((clear e c) \to (clear e x))))) (\lambda (H7: (eq T u0
-u)).(eq_ind T u (\lambda (_: T).((eq C c x) \to ((clear e c) \to (clear e
-x)))) (\lambda (H8: (eq C c x)).(eq_ind C x (\lambda (c0: C).((clear e c0)
-\to (clear e x))) (\lambda (H9: (clear e x)).H9) c (sym_eq C c x H8))) u0
-(sym_eq T u0 u H7))) f0 (sym_eq F f0 f H6))) e0 (sym_eq C e0 e H5))) H4))
-H3)) H2 H0)))]) in (H0 (refl_equal C (CHead e (Flat f) u)) (refl_equal C
-x))))))).
+H3) in (\lambda (_: (eq F f0 f)).(\lambda (H8: (eq C e0 e)).(let H9 \def
+(eq_ind C e0 (\lambda (c0: C).((eq C c0 (CHead e (Flat f) u)) \to (clear e
+c))) H2 e H8) in (let H10 \def (eq_ind C e0 (\lambda (c0: C).(clear c0 c)) H1
+e H8) in H10))))) H5)) H4))))))))) y x H0))) H))))).
theorem clear_gen_flat_r:
\forall (f: F).(\forall (x: C).(\forall (e: C).(\forall (u: T).((clear x
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