-(AHead a a0)))).(let H5 \def (match H4 in leq return (\lambda (a3:
-A).(\lambda (a4: A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (ASort O (next g
-n0))) \to ((eq A a4 (AHead a (asucc g a0))) \to (leq g (ASort O n0) (AHead a
-a0))))))) with [(leq_sort h1 h2 n1 n2 k H5) \Rightarrow (\lambda (H6: (eq A
-(ASort h1 n1) (ASort O (next g n0)))).(\lambda (H7: (eq A (ASort h2 n2)
-(AHead a (asucc g a0)))).((let H8 \def (f_equal A nat (\lambda (e: A).(match
-e in A return (\lambda (_: A).nat) with [(ASort _ n3) \Rightarrow n3 | (AHead
-_ _) \Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H6) in ((let H9
-\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat)
-with [(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow h1])) (ASort h1
-n1) (ASort O (next g n0)) H6) in (eq_ind nat O (\lambda (n3: nat).((eq nat n1
-(next g n0)) \to ((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A
-(aplus g (ASort n3 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0)
-(AHead a a0)))))) (\lambda (H10: (eq nat n1 (next g n0))).(eq_ind nat (next g
-n0) (\lambda (n3: nat).((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq
-A (aplus g (ASort O n3) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0)
-(AHead a a0))))) (\lambda (H11: (eq A (ASort h2 n2) (AHead a (asucc g
-a0)))).(let H12 \def (eq_ind A (ASort h2 n2) (\lambda (e: A).(match e in A
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
-\Rightarrow False])) I (AHead a (asucc g a0)) H11) in (False_ind ((eq A
-(aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n2) k)) \to (leq g
-(ASort O n0) (AHead a a0))) H12))) n1 (sym_eq nat n1 (next g n0) H10))) h1
-(sym_eq nat h1 O H9))) H8)) H7 H5))) | (leq_head a3 a4 H5 a5 a6 H6)
-\Rightarrow (\lambda (H7: (eq A (AHead a3 a5) (ASort O (next g
-n0)))).(\lambda (H8: (eq A (AHead a4 a6) (AHead a (asucc g a0)))).((let H9
-\def (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda
-(_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
-True])) I (ASort O (next g n0)) H7) in (False_ind ((eq A (AHead a4 a6) (AHead
-a (asucc g a0))) \to ((leq g a3 a4) \to ((leq g a5 a6) \to (leq g (ASort O
-n0) (AHead a a0))))) H9)) H8 H5 H6)))]) in (H5 (refl_equal A (ASort O (next g
-n0))) (refl_equal A (AHead a (asucc g a0)))))))) (\lambda (n1: nat).(\lambda
-(_: (((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0)
-a))) \to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1
-n0) a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to
-(leq g (ASort n1 n0) (AHead a a0))))))).(\lambda (_: (((leq g (asucc g (ASort
-(S n1) n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda (_:
-(((leq g (asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1)
-n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort (S n1) n0)) (asucc g (AHead a
-a0)))).(let H5 \def (match H4 in leq return (\lambda (a3: A).(\lambda (a4:
-A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (ASort n1 n0)) \to ((eq A a4 (AHead
-a (asucc g a0))) \to (leq g (ASort (S n1) n0) (AHead a a0))))))) with
-[(leq_sort h1 h2 n2 n3 k H5) \Rightarrow (\lambda (H6: (eq A (ASort h1 n2)
-(ASort n1 n0))).(\lambda (H7: (eq A (ASort h2 n3) (AHead a (asucc g
-a0)))).((let H8 \def (f_equal A nat (\lambda (e: A).(match e in A return
-(\lambda (_: A).nat) with [(ASort _ n4) \Rightarrow n4 | (AHead _ _)
-\Rightarrow n2])) (ASort h1 n2) (ASort n1 n0) H6) in ((let H9 \def (f_equal A
-nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort n4
-_) \Rightarrow n4 | (AHead _ _) \Rightarrow h1])) (ASort h1 n2) (ASort n1 n0)
-H6) in (eq_ind nat n1 (\lambda (n4: nat).((eq nat n2 n0) \to ((eq A (ASort h2
-n3) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n4 n2) k) (aplus g
-(ASort h2 n3) k)) \to (leq g (ASort (S n1) n0) (AHead a a0)))))) (\lambda
-(H10: (eq nat n2 n0)).(eq_ind nat n0 (\lambda (n4: nat).((eq A (ASort h2 n3)
-(AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n1 n4) k) (aplus g (ASort
-h2 n3) k)) \to (leq g (ASort (S n1) n0) (AHead a a0))))) (\lambda (H11: (eq A
-(ASort h2 n3) (AHead a (asucc g a0)))).(let H12 \def (eq_ind A (ASort h2 n3)
-(\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
-\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0))
-H11) in (False_ind ((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n3)
-k)) \to (leq g (ASort (S n1) n0) (AHead a a0))) H12))) n2 (sym_eq nat n2 n0
-H10))) h1 (sym_eq nat h1 n1 H9))) H8)) H7 H5))) | (leq_head a3 a4 H5 a5 a6
-H6) \Rightarrow (\lambda (H7: (eq A (AHead a3 a5) (ASort n1 n0))).(\lambda
-(H8: (eq A (AHead a4 a6) (AHead a (asucc g a0)))).((let H9 \def (eq_ind A
-(AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
-[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1
-n0) H7) in (False_ind ((eq A (AHead a4 a6) (AHead a (asucc g a0))) \to ((leq
-g a3 a4) \to ((leq g a5 a6) \to (leq g (ASort (S n1) n0) (AHead a a0)))))
-H9)) H8 H5 H6)))]) in (H5 (refl_equal A (ASort n1 n0)) (refl_equal A (AHead a
-(asucc g a0)))))))))) n H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_:
-((\forall (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq g a
-a2))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0)
-(asucc g a2)) \to (leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3:
-A).((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))
-(\lambda (n: nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a
-a0)) (asucc g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g
-(AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1
-n0)))) (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O
-n0)))).(let H3 \def (match H2 in leq return (\lambda (a3: A).(\lambda (a4:
-A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (AHead a (asucc g a0))) \to ((eq A
-a4 (ASort O (next g n0))) \to (leq g (AHead a a0) (ASort O n0))))))) with
-[(leq_sort h1 h2 n1 n2 k H3) \Rightarrow (\lambda (H4: (eq A (ASort h1 n1)
-(AHead a (asucc g a0)))).(\lambda (H5: (eq A (ASort h2 n2) (ASort O (next g
-n0)))).((let H6 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
-\Rightarrow False])) I (AHead a (asucc g a0)) H4) in (False_ind ((eq A (ASort
-h2 n2) (ASort O (next g n0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
-(ASort h2 n2) k)) \to (leq g (AHead a a0) (ASort O n0)))) H6)) H5 H3))) |
-(leq_head a3 a4 H3 a5 a6 H4) \Rightarrow (\lambda (H5: (eq A (AHead a3 a5)
-(AHead a (asucc g a0)))).(\lambda (H6: (eq A (AHead a4 a6) (ASort O (next g
-n0)))).((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return
-(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a7)
-\Rightarrow a7])) (AHead a3 a5) (AHead a (asucc g a0)) H5) in ((let H8 \def
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
-[(ASort _ _) \Rightarrow a3 | (AHead a7 _) \Rightarrow a7])) (AHead a3 a5)
-(AHead a (asucc g a0)) H5) in (eq_ind A a (\lambda (a7: A).((eq A a5 (asucc g
-a0)) \to ((eq A (AHead a4 a6) (ASort O (next g n0))) \to ((leq g a7 a4) \to
-((leq g a5 a6) \to (leq g (AHead a a0) (ASort O n0))))))) (\lambda (H9: (eq A
-a5 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a7: A).((eq A (AHead a4
-a6) (ASort O (next g n0))) \to ((leq g a a4) \to ((leq g a7 a6) \to (leq g
-(AHead a a0) (ASort O n0)))))) (\lambda (H10: (eq A (AHead a4 a6) (ASort O
-(next g n0)))).(let H11 \def (eq_ind A (AHead a4 a6) (\lambda (e: A).(match e
-in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
-(AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H10) in (False_ind
-((leq g a a4) \to ((leq g (asucc g a0) a6) \to (leq g (AHead a a0) (ASort O
-n0)))) H11))) a5 (sym_eq A a5 (asucc g a0) H9))) a3 (sym_eq A a3 a H8))) H7))
-H6 H3 H4)))]) in (H3 (refl_equal A (AHead a (asucc g a0))) (refl_equal A
-(ASort O (next g n0)))))) (\lambda (n1: nat).(\lambda (_: (((leq g (asucc g
-(AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1
-n0))))).(\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort (S n1)
-n0)))).(let H3 \def (match H2 in leq return (\lambda (a3: A).(\lambda (a4:
-A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (AHead a (asucc g a0))) \to ((eq A
-a4 (ASort n1 n0)) \to (leq g (AHead a a0) (ASort (S n1) n0))))))) with
-[(leq_sort h1 h2 n2 n3 k H3) \Rightarrow (\lambda (H4: (eq A (ASort h1 n2)
-(AHead a (asucc g a0)))).(\lambda (H5: (eq A (ASort h2 n3) (ASort n1
-n0))).((let H6 \def (eq_ind A (ASort h1 n2) (\lambda (e: A).(match e in A
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
-\Rightarrow False])) I (AHead a (asucc g a0)) H4) in (False_ind ((eq A (ASort
-h2 n3) (ASort n1 n0)) \to ((eq A (aplus g (ASort h1 n2) k) (aplus g (ASort h2
-n3) k)) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H6)) H5 H3))) |
-(leq_head a3 a4 H3 a5 a6 H4) \Rightarrow (\lambda (H5: (eq A (AHead a3 a5)
-(AHead a (asucc g a0)))).(\lambda (H6: (eq A (AHead a4 a6) (ASort n1
-n0))).((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return
-(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a7)
-\Rightarrow a7])) (AHead a3 a5) (AHead a (asucc g a0)) H5) in ((let H8 \def
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
-[(ASort _ _) \Rightarrow a3 | (AHead a7 _) \Rightarrow a7])) (AHead a3 a5)
-(AHead a (asucc g a0)) H5) in (eq_ind A a (\lambda (a7: A).((eq A a5 (asucc g
-a0)) \to ((eq A (AHead a4 a6) (ASort n1 n0)) \to ((leq g a7 a4) \to ((leq g
-a5 a6) \to (leq g (AHead a a0) (ASort (S n1) n0))))))) (\lambda (H9: (eq A a5
-(asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a7: A).((eq A (AHead a4 a6)
-(ASort n1 n0)) \to ((leq g a a4) \to ((leq g a7 a6) \to (leq g (AHead a a0)
-(ASort (S n1) n0)))))) (\lambda (H10: (eq A (AHead a4 a6) (ASort n1
-n0))).(let H11 \def (eq_ind A (AHead a4 a6) (\lambda (e: A).(match e in A
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
-_) \Rightarrow True])) I (ASort n1 n0) H10) in (False_ind ((leq g a a4) \to
-((leq g (asucc g a0) a6) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H11)))
-a5 (sym_eq A a5 (asucc g a0) H9))) a3 (sym_eq A a3 a H8))) H7)) H6 H3 H4)))])
-in (H3 (refl_equal A (AHead a (asucc g a0))) (refl_equal A (ASort n1
-n0))))))) n H1)))) (\lambda (a3: A).(\lambda (_: (((leq g (asucc g (AHead a
-a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda (a4: A).(\lambda
-(_: (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g (AHead a a0)
-a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g (AHead a3
-a4)))).(let H4 \def (match H3 in leq return (\lambda (a5: A).(\lambda (a6:
-A).(\lambda (_: (leq ? a5 a6)).((eq A a5 (AHead a (asucc g a0))) \to ((eq A
-a6 (AHead a3 (asucc g a4))) \to (leq g (AHead a a0) (AHead a3 a4))))))) with
-[(leq_sort h1 h2 n1 n2 k H4) \Rightarrow (\lambda (H5: (eq A (ASort h1 n1)
-(AHead a (asucc g a0)))).(\lambda (H6: (eq A (ASort h2 n2) (AHead a3 (asucc g
-a4)))).((let H7 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
-\Rightarrow False])) I (AHead a (asucc g a0)) H5) in (False_ind ((eq A (ASort
-h2 n2) (AHead a3 (asucc g a4))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
-(ASort h2 n2) k)) \to (leq g (AHead a a0) (AHead a3 a4)))) H7)) H6 H4))) |
-(leq_head a5 a6 H4 a7 a8 H5) \Rightarrow (\lambda (H6: (eq A (AHead a5 a7)
-(AHead a (asucc g a0)))).(\lambda (H7: (eq A (AHead a6 a8) (AHead a3 (asucc g
-a4)))).((let H8 \def (f_equal A A (\lambda (e: A).(match e in A return
-(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 | (AHead _ a9)
-\Rightarrow a9])) (AHead a5 a7) (AHead a (asucc g a0)) H6) in ((let H9 \def
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
-[(ASort _ _) \Rightarrow a5 | (AHead a9 _) \Rightarrow a9])) (AHead a5 a7)
-(AHead a (asucc g a0)) H6) in (eq_ind A a (\lambda (a9: A).((eq A a7 (asucc g
-a0)) \to ((eq A (AHead a6 a8) (AHead a3 (asucc g a4))) \to ((leq g a9 a6) \to
-((leq g a7 a8) \to (leq g (AHead a a0) (AHead a3 a4))))))) (\lambda (H10: (eq
-A a7 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a9: A).((eq A (AHead a6
-a8) (AHead a3 (asucc g a4))) \to ((leq g a a6) \to ((leq g a9 a8) \to (leq g
-(AHead a a0) (AHead a3 a4)))))) (\lambda (H11: (eq A (AHead a6 a8) (AHead a3
-(asucc g a4)))).(let H12 \def (f_equal A A (\lambda (e: A).(match e in A
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a8 | (AHead _ a9)
-\Rightarrow a9])) (AHead a6 a8) (AHead a3 (asucc g a4)) H11) in ((let H13
+(AHead a a0)))).(let H_x \def (leq_gen_sort1 g O (next g n0) (AHead a (asucc
+g a0)) H4) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2:
+nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort O (next g
+n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2:
+nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0)) (ASort h2 n2))))) (leq g
+(ASort O n0) (AHead a a0)) (\lambda (x0: nat).(\lambda (x1: nat).(\lambda
+(x2: nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g
+(ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1
+x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda (ee: A).(match
+ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
+(AHead _ _) \Rightarrow True])) I (ASort x1 x0) H7) in (False_ind (leq g
+(ASort O n0) (AHead a a0)) H8))))))) H5)))))) (\lambda (n1: nat).(\lambda (_:
+(((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0) a)))
+\to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1 n0)
+a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to (leq g
+(ASort n1 n0) (AHead a a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1)
+n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g
+(asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0)
+a0)))).(\lambda (H4: (leq g (asucc g (ASort (S n1) n0)) (asucc g (AHead a
+a0)))).(let H_x \def (leq_gen_sort1 g n1 n0 (AHead a (asucc g a0)) H4) in
+(let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2:
+nat).(\lambda (k: nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2)
+k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a
+(asucc g a0)) (ASort h2 n2))))) (leq g (ASort (S n1) n0) (AHead a a0))
+(\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A
+(aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A
+(AHead a (asucc g a0)) (ASort x1 x0))).(let H8 \def (eq_ind A (AHead a (asucc
+g a0)) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with
+[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort x1
+x0) H7) in (False_ind (leq g (ASort (S n1) n0) (AHead a a0)) H8)))))))
+H5)))))))) n H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_: ((\forall (a2:
+A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2))))).(\lambda (a0:
+A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0) (asucc g a2)) \to
+(leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3: A).((leq g (asucc g
+(AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3))) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a a0)) (asucc g
+(ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g (AHead a a0))
+(asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 n0)))) (\lambda
+(H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O n0)))).(let H_x \def
+(leq_gen_head1 g a (asucc g a0) (ASort O (next g n0)) H2) in (let H3 \def H_x
+in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a a3))) (\lambda
+(_: A).(\lambda (a4: A).(leq g (asucc g a0) a4))) (\lambda (a3: A).(\lambda
+(a4: A).(eq A (ASort O (next g n0)) (AHead a3 a4)))) (leq g (AHead a a0)
+(ASort O n0)) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a
+x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda (H6: (eq A (ASort O (next
+g n0)) (AHead x0 x1))).(let H7 \def (eq_ind A (ASort O (next g n0)) (\lambda
+(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _)
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in
+(False_ind (leq g (AHead a a0) (ASort O n0)) H7))))))) H3)))) (\lambda (n1:
+nat).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0)))
+\to (leq g (AHead a a0) (ASort n1 n0))))).(\lambda (H2: (leq g (asucc g
+(AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H_x \def (leq_gen_head1 g a
+(asucc g a0) (ASort n1 n0) H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda
+(a3: A).(\lambda (_: A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq
+g (asucc g a0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0)
+(AHead a3 a4)))) (leq g (AHead a a0) (ASort (S n1) n0)) (\lambda (x0:
+A).(\lambda (x1: A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g
+a0) x1)).(\lambda (H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let H7 \def
+(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_:
+A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
+False])) I (AHead x0 x1) H6) in (False_ind (leq g (AHead a a0) (ASort (S n1)
+n0)) H7))))))) H3)))))) n H1)))) (\lambda (a3: A).(\lambda (_: (((leq g
+(asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda
+(a4: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g
+(AHead a a0) a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g
+(AHead a3 a4)))).(let H_x \def (leq_gen_head1 g a (asucc g a0) (AHead a3
+(asucc g a4)) H3) in (let H4 \def H_x in (ex3_2_ind A A (\lambda (a5:
+A).(\lambda (_: A).(leq g a a5))) (\lambda (_: A).(\lambda (a6: A).(leq g
+(asucc g a0) a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 (asucc g
+a4)) (AHead a5 a6)))) (leq g (AHead a a0) (AHead a3 a4)) (\lambda (x0:
+A).(\lambda (x1: A).(\lambda (H5: (leq g a x0)).(\lambda (H6: (leq g (asucc g
+a0) x1)).(\lambda (H7: (eq A (AHead a3 (asucc g a4)) (AHead x0 x1))).(let H8