-a1 (ASort O n0)) (ASort O n0))).(let H1 \def (match H0 in leq return (\lambda
-(a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (AHead a1 (ASort
-O n0))) \to ((eq A a0 (ASort O n0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k
-H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n1) (AHead a1 (ASort O
-n0)))).(\lambda (H3: (eq A (ASort h2 n2) (ASort O n0))).((let H4 \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 a1 (ASort O n0)) H2) in (False_ind ((eq A (ASort h2 n2) (ASort O n0))
-\to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H4))
-H3 H1))) | (leq_head a0 a3 H1 a4 a5 H2) \Rightarrow (\lambda (H3: (eq A
-(AHead a0 a4) (AHead a1 (ASort O n0)))).(\lambda (H4: (eq A (AHead a3 a5)
-(ASort O n0))).((let H5 \def (f_equal A A (\lambda (e: A).(match e in A
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a)
-\Rightarrow a])) (AHead a0 a4) (AHead a1 (ASort O n0)) H3) in ((let H6 \def
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
-[(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a4)
-(AHead a1 (ASort O n0)) H3) in (eq_ind A a1 (\lambda (a: A).((eq A a4 (ASort
-O n0)) \to ((eq A (AHead a3 a5) (ASort O n0)) \to ((leq g a a3) \to ((leq g
-a4 a5) \to P))))) (\lambda (H7: (eq A a4 (ASort O n0))).(eq_ind A (ASort O
-n0) (\lambda (a: A).((eq A (AHead a3 a5) (ASort O n0)) \to ((leq g a1 a3) \to
-((leq g a a5) \to P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort O n0))).(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 n0) H8) in (False_ind ((leq g a1 a3) \to ((leq g (ASort O
-n0) a5) \to P)) H9))) a4 (sym_eq A a4 (ASort O n0) H7))) a0 (sym_eq A a0 a1
-H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead a1 (ASort O n0)))
-(refl_equal A (ASort O n0))))) (\lambda (n1: nat).(\lambda (_: (((leq g
-(AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead
-a1 (ASort (S n1) n0)) (ASort (S n1) n0))).(let H1 \def (match H0 in leq
-return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
-(AHead a1 (ASort (S n1) n0))) \to ((eq A a0 (ASort (S n1) n0)) \to P)))))
-with [(leq_sort h1 h2 n2 n3 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1
-n2) (AHead a1 (ASort (S n1) n0)))).(\lambda (H3: (eq A (ASort h2 n3) (ASort
-(S n1) n0))).((let H4 \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 a1 (ASort (S n1) n0)) H2) in (False_ind
-((eq A (ASort h2 n3) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort h1 n2) k)
-(aplus g (ASort h2 n3) k)) \to P)) H4)) H3 H1))) | (leq_head a0 a3 H1 a4 a5
-H2) \Rightarrow (\lambda (H3: (eq A (AHead a0 a4) (AHead a1 (ASort (S n1)
-n0)))).(\lambda (H4: (eq A (AHead a3 a5) (ASort (S n1) n0))).((let H5 \def
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
-[(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4)
-(AHead a1 (ASort (S n1) n0)) H3) in ((let H6 \def (f_equal A A (\lambda (e:
-A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 |
-(AHead a _) \Rightarrow a])) (AHead a0 a4) (AHead a1 (ASort (S n1) n0)) H3)
-in (eq_ind A a1 (\lambda (a: A).((eq A a4 (ASort (S n1) n0)) \to ((eq A
-(AHead a3 a5) (ASort (S n1) n0)) \to ((leq g a a3) \to ((leq g a4 a5) \to
-P))))) (\lambda (H7: (eq A a4 (ASort (S n1) n0))).(eq_ind A (ASort (S n1) n0)
-(\lambda (a: A).((eq A (AHead a3 a5) (ASort (S n1) n0)) \to ((leq g a1 a3)
-\to ((leq g a a5) \to P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort (S n1)
-n0))).(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 (S n1) n0) H8) in (False_ind ((leq g a1 a3)
-\to ((leq g (ASort (S n1) n0) a5) \to P)) H9))) a4 (sym_eq A a4 (ASort (S n1)
-n0) H7))) a0 (sym_eq A a0 a1 H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A
-(AHead a1 (ASort (S n1) n0))) (refl_equal A (ASort (S n1) n0))))))) n H))))))
-(\lambda (a: A).(\lambda (_: ((\forall (a1: A).((leq g (AHead a1 a) a) \to
-(\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1:
-A).((leq g (AHead a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1:
-A).(\lambda (H1: (leq g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P:
-Prop).(let H2 \def (match H1 in leq return (\lambda (a3: A).(\lambda (a4:
-A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (AHead a1 (AHead a a0))) \to ((eq A
-a4 (AHead a a0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow
-(\lambda (H3: (eq A (ASort h1 n1) (AHead a1 (AHead a a0)))).(\lambda (H4: (eq
-A (ASort h2 n2) (AHead a a0))).((let H5 \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 a1 (AHead a a0))
-H3) in (False_ind ((eq A (ASort h2 n2) (AHead a a0)) \to ((eq A (aplus g
-(ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head
-a3 a4 H2 a5 a6 H3) \Rightarrow (\lambda (H4: (eq A (AHead a3 a5) (AHead a1
-(AHead a a0)))).(\lambda (H5: (eq A (AHead a4 a6) (AHead a a0))).((let H6
-\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 a1 (AHead a a0)) H4) in ((let H7 \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 a1 (AHead a a0)) H4) in
-(eq_ind A a1 (\lambda (a7: A).((eq A a5 (AHead a a0)) \to ((eq A (AHead a4
-a6) (AHead a a0)) \to ((leq g a7 a4) \to ((leq g a5 a6) \to P))))) (\lambda
-(H8: (eq A a5 (AHead a a0))).(eq_ind A (AHead a a0) (\lambda (a7: A).((eq A
-(AHead a4 a6) (AHead a a0)) \to ((leq g a1 a4) \to ((leq g a7 a6) \to P))))
-(\lambda (H9: (eq A (AHead a4 a6) (AHead a a0))).(let H10 \def (f_equal A A
-(\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
-\Rightarrow a6 | (AHead _ a7) \Rightarrow a7])) (AHead a4 a6) (AHead a a0)
-H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e in A return
-(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead a7 _)
-\Rightarrow a7])) (AHead a4 a6) (AHead a a0) H9) in (eq_ind A a (\lambda (a7:
-A).((eq A a6 a0) \to ((leq g a1 a7) \to ((leq g (AHead a a0) a6) \to P))))
-(\lambda (H12: (eq A a6 a0)).(eq_ind A a0 (\lambda (a7: A).((leq g a1 a) \to
-((leq g (AHead a a0) a7) \to P))) (\lambda (_: (leq g a1 a)).(\lambda (H14:
-(leq g (AHead a a0) a0)).(H0 a H14 P))) a6 (sym_eq A a6 a0 H12))) a4 (sym_eq
-A a4 a H11))) H10))) a5 (sym_eq A a5 (AHead a a0) H8))) a3 (sym_eq A a3 a1
-H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead a1 (AHead a a0)))
-(refl_equal A (AHead a a0))))))))))) a2)).
+a1 (ASort O n0)) (ASort O n0))).(let H_x \def (leq_gen_head1 g a1 (ASort O
+n0) (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3:
+A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g
+(ASort O n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0)
+(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1
+x0)).(\lambda (_: (leq g (ASort O n0) x1)).(\lambda (H4: (eq A (ASort O n0)
+(AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: A).(match
+ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True |
+(AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P
+H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead a1 (ASort n1
+n0)) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead a1 (ASort (S n1)
+n0)) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g a1 (ASort (S n1) n0)
+(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3:
+A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g
+(ASort (S n1) n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1)
+n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g
+a1 x0)).(\lambda (_: (leq g (ASort (S n1) n0) x1)).(\lambda (H4: (eq A (ASort
+(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda
+(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _)
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in
+(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (_:
+((\forall (a1: A).((leq g (AHead a1 a) a) \to (\forall (P:
+Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1: A).((leq g (AHead
+a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H1: (leq
+g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P: Prop).(let H_x \def
+(leq_gen_head1 g a1 (AHead a a0) (AHead a a0) H1) in (let H2 \def H_x in
+(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_:
+A).(\lambda (a4: A).(leq g (AHead a a0) a4))) (\lambda (a3: A).(\lambda (a4:
+A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1:
+A).(\lambda (H3: (leq g a1 x0)).(\lambda (H4: (leq g (AHead a a0)
+x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A
+A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5)
+in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda
+(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3]))
+(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def
+(eq_ind_r A x1 (\lambda (a3: A).(leq g (AHead a a0) a3)) H4 a0 H7) in (let
+H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g a1 a3)) H3 a H8) in (H0 a H9
+P))))) H6))))))) H2)))))))))) a2)).