+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* Problematic objects for disambiguation/typechecking ********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/problems".
+
+include "LambdaDelta/theory.ma".
+
+theorem asucc_inj:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc
+g a2)) \to (leq g a1 a2))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
+A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g
+(asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda
+(n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0))
+(asucc g (ASort n1 n2)))).((match n in nat return (\lambda (n3: nat).((leq g
+(asucc g (ASort n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0)
+(ASort n1 n2)))) with [O \Rightarrow (\lambda (H0: (leq g (asucc g (ASort O
+n0)) (asucc g (ASort n1 n2)))).((match n1 in nat return (\lambda (n3:
+nat).((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2))) \to (leq g
+(ASort O n0) (ASort n3 n2)))) with [O \Rightarrow (\lambda (H1: (leq g (asucc
+g (ASort O n0)) (asucc g (ASort O n2)))).(let H2 \def (match H1 in leq return
+(\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort O
+(next g n0))) \to ((eq A a0 (ASort O (next g n2))) \to (leq g (ASort O n0)
+(ASort O n2))))))) with [(leq_sort h1 h2 n1 n3 k H0) \Rightarrow (\lambda
+(H1: (eq A (ASort h1 n1) (ASort O (next g n0)))).(\lambda (H2: (eq A (ASort
+h2 n3) (ASort O (next g n2)))).((let H3 \def (f_equal A nat (\lambda (e:
+A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n
+| (AHead _ _) \Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H1) in
+((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda
+(_: A).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1]))
+(ASort h1 n1) (ASort O (next g n0)) H1) in (eq_ind nat O (\lambda (n:
+nat).((eq nat n1 (next g n0)) \to ((eq A (ASort h2 n3) (ASort O (next g n2)))
+\to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n3) k)) \to (leq g
+(ASort O n0) (ASort O n2)))))) (\lambda (H5: (eq nat n1 (next g n0))).(eq_ind
+nat (next g n0) (\lambda (n: nat).((eq A (ASort h2 n3) (ASort O (next g n2)))
+\to ((eq A (aplus g (ASort O n) k) (aplus g (ASort h2 n3) k)) \to (leq g
+(ASort O n0) (ASort O n2))))) (\lambda (H6: (eq A (ASort h2 n3) (ASort O
+(next g n2)))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e in A
+return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
+\Rightarrow n3])) (ASort h2 n3) (ASort O (next g n2)) H6) in ((let H8 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n3)
+(ASort O (next g n2)) H6) in (eq_ind nat O (\lambda (n: nat).((eq nat n3
+(next g n2)) \to ((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n
+n3) k)) \to (leq g (ASort O n0) (ASort O n2))))) (\lambda (H9: (eq nat n3
+(next g n2))).(eq_ind nat (next g n2) (\lambda (n: nat).((eq A (aplus g
+(ASort O (next g n0)) k) (aplus g (ASort O n) k)) \to (leq g (ASort O n0)
+(ASort O n2)))) (\lambda (H10: (eq A (aplus g (ASort O (next g n0)) k) (aplus
+g (ASort O (next g n2)) k))).(let H \def (eq_ind_r A (aplus g (ASort O (next
+g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n2)) k))) H10
+(aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0 k)) in (let H11 \def
+(eq_ind_r A (aplus g (ASort O (next g n2)) k) (\lambda (a: A).(eq A (aplus g
+(ASort O n0) (S k)) a)) H (aplus g (ASort O n2) (S k)) (aplus_sort_O_S_simpl
+g n2 k)) in (leq_sort g O O n0 n2 (S k) H11)))) n3 (sym_eq nat n3 (next g n2)
+H9))) h2 (sym_eq nat h2 O H8))) H7))) n1 (sym_eq nat n1 (next g n0) H5))) h1
+(sym_eq nat h1 O H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1)
+\Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort O (next g
+n0)))).(\lambda (H3: (eq A (AHead a2 a4) (ASort O (next g n2)))).((let H4
+\def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e in A return (\lambda
+(_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
+True])) I (ASort O (next g n0)) H2) in (False_ind ((eq A (AHead a2 a4) (ASort
+O (next g n2))) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort O n0)
+(ASort O n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort O (next g
+n0))) (refl_equal A (ASort O (next g n2)))))) | (S n3) \Rightarrow (\lambda
+(H1: (leq g (asucc g (ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H2 \def
+(match H1 in leq return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ?
+a a0)).((eq A a (ASort O (next g n0))) \to ((eq A a0 (ASort n3 n2)) \to (leq
+g (ASort O n0) (ASort (S n3) n2))))))) with [(leq_sort h1 h2 n1 n3 k H0)
+\Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (ASort O (next g
+n0)))).(\lambda (H2: (eq A (ASort h2 n3) (ASort n3 n2))).((let H3 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) (ASort h1 n1)
+(ASort O (next g n0)) H1) in ((let H4 \def (f_equal A nat (\lambda (e:
+A).(match e in A return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n
+| (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort O (next g n0)) H1) in
+(eq_ind nat O (\lambda (n: nat).((eq nat n1 (next g n0)) \to ((eq A (ASort h2
+n3) (ASort n3 n2)) \to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n3)
+k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))))) (\lambda (H5: (eq nat n1
+(next g n0))).(eq_ind nat (next g n0) (\lambda (n: nat).((eq A (ASort h2 n3)
+(ASort n3 n2)) \to ((eq A (aplus g (ASort O n) k) (aplus g (ASort h2 n3) k))
+\to (leq g (ASort O n0) (ASort (S n3) n2))))) (\lambda (H6: (eq A (ASort h2
+n3) (ASort n3 n2))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e in A
+return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
+\Rightarrow n3])) (ASort h2 n3) (ASort n3 n2) H6) in ((let H8 \def (f_equal A
+nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort n
+_) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n3) (ASort n3 n2)
+H6) in (eq_ind nat n3 (\lambda (n: nat).((eq nat n3 n2) \to ((eq A (aplus g
+(ASort O (next g n0)) k) (aplus g (ASort n n3) k)) \to (leq g (ASort O n0)
+(ASort (S n3) n2))))) (\lambda (H9: (eq nat n3 n2)).(eq_ind nat n2 (\lambda
+(n: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n3 n) k))
+\to (leq g (ASort O n0) (ASort (S n3) n2)))) (\lambda (H10: (eq A (aplus g
+(ASort O (next g n0)) k) (aplus g (ASort n3 n2) k))).(let H \def (eq_ind_r A
+(aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort n3
+n2) k))) H10 (aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0 k)) in
+(let H11 \def (eq_ind_r A (aplus g (ASort n3 n2) k) (\lambda (a: A).(eq A
+(aplus g (ASort O n0) (S k)) a)) H (aplus g (ASort (S n3) n2) (S k))
+(aplus_sort_S_S_simpl g n2 n3 k)) in (leq_sort g O (S n3) n0 n2 (S k) H11))))
+n3 (sym_eq nat n3 n2 H9))) h2 (sym_eq nat h2 n3 H8))) H7))) n1 (sym_eq nat n1
+(next g n0) H5))) h1 (sym_eq nat h1 O H4))) H3)) H2 H0))) | (leq_head a1 a2
+H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort O (next g
+n0)))).(\lambda (H3: (eq A (AHead a2 a4) (ASort n3 n2))).((let H4 \def
+(eq_ind A (AHead a1 a3) (\lambda (e: A).(match e in A return (\lambda (_:
+A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
+True])) I (ASort O (next g n0)) H2) in (False_ind ((eq A (AHead a2 a4) (ASort
+n3 n2)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort O n0) (ASort
+(S n3) n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort O (next g n0)))
+(refl_equal A (ASort n3 n2)))))]) H0)) | (S n3) \Rightarrow (\lambda (H0:
+(leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).((match n1 in
+nat return (\lambda (n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g
+(ASort n4 n2))) \to (leq g (ASort (S n3) n0) (ASort n4 n2)))) with [O
+\Rightarrow (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort
+O n2)))).(let H2 \def (match H1 in leq return (\lambda (a: A).(\lambda (a0:
+A).(\lambda (_: (leq ? a a0)).((eq A a (ASort n3 n0)) \to ((eq A a0 (ASort O
+(next g n2))) \to (leq g (ASort (S n3) n0) (ASort O n2))))))) with [(leq_sort
+h1 h2 n1 n3 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (ASort n3
+n0))).(\lambda (H2: (eq A (ASort h2 n3) (ASort O (next g n2)))).((let H3 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) (ASort h1 n1)
+(ASort n3 n0) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in
+A return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n | (AHead _ _)
+\Rightarrow h1])) (ASort h1 n1) (ASort n3 n0) H1) in (eq_ind nat n3 (\lambda
+(n: nat).((eq nat n1 n0) \to ((eq A (ASort h2 n3) (ASort O (next g n2))) \to
+((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort
+(S n3) n0) (ASort O n2)))))) (\lambda (H5: (eq nat n1 n0)).(eq_ind nat n0
+(\lambda (n: nat).((eq A (ASort h2 n3) (ASort O (next g n2))) \to ((eq A
+(aplus g (ASort n3 n) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort (S n3)
+n0) (ASort O n2))))) (\lambda (H6: (eq A (ASort h2 n3) (ASort O (next g
+n2)))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e in A return
+(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
+\Rightarrow n3])) (ASort h2 n3) (ASort O (next g n2)) H6) in ((let H8 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n3)
+(ASort O (next g n2)) H6) in (eq_ind nat O (\lambda (n: nat).((eq nat n3
+(next g n2)) \to ((eq A (aplus g (ASort n3 n0) k) (aplus g (ASort n n3) k))
+\to (leq g (ASort (S n3) n0) (ASort O n2))))) (\lambda (H9: (eq nat n3 (next
+g n2))).(eq_ind nat (next g n2) (\lambda (n: nat).((eq A (aplus g (ASort n3
+n0) k) (aplus g (ASort O n) k)) \to (leq g (ASort (S n3) n0) (ASort O n2))))
+(\lambda (H10: (eq A (aplus g (ASort n3 n0) k) (aplus g (ASort O (next g n2))
+k))).(let H \def (eq_ind_r A (aplus g (ASort n3 n0) k) (\lambda (a: A).(eq A
+a (aplus g (ASort O (next g n2)) k))) H10 (aplus g (ASort (S n3) n0) (S k))
+(aplus_sort_S_S_simpl g n0 n3 k)) in (let H11 \def (eq_ind_r A (aplus g
+(ASort O (next g n2)) k) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S
+k)) a)) H (aplus g (ASort O n2) (S k)) (aplus_sort_O_S_simpl g n2 k)) in
+(leq_sort g (S n3) O n0 n2 (S k) H11)))) n3 (sym_eq nat n3 (next g n2) H9)))
+h2 (sym_eq nat h2 O H8))) H7))) n1 (sym_eq nat n1 n0 H5))) h1 (sym_eq nat h1
+n3 H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda
+(H2: (eq A (AHead a1 a3) (ASort n3 n0))).(\lambda (H3: (eq A (AHead a2 a4)
+(ASort O (next g n2)))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e:
+A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow
+False | (AHead _ _) \Rightarrow True])) I (ASort n3 n0) H2) in (False_ind
+((eq A (AHead a2 a4) (ASort O (next g n2))) \to ((leq g a1 a2) \to ((leq g a3
+a4) \to (leq g (ASort (S n3) n0) (ASort O n2))))) H4)) H3 H0 H1)))]) in (H2
+(refl_equal A (ASort n3 n0)) (refl_equal A (ASort O (next g n2)))))) | (S n4)
+\Rightarrow (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort
+(S n4) n2)))).(let H2 \def (match H1 in leq return (\lambda (a: A).(\lambda
+(a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort n3 n0)) \to ((eq A a0
+(ASort n4 n2)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))))) with
+[(leq_sort h1 h2 n3 n4 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n3)
+(ASort n3 n0))).(\lambda (H2: (eq A (ASort h2 n4) (ASort n4 n2))).((let H3
+\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat)
+with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n3])) (ASort h1 n3)
+(ASort n3 n0) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in
+A return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n | (AHead _ _)
+\Rightarrow h1])) (ASort h1 n3) (ASort n3 n0) H1) in (eq_ind nat n3 (\lambda
+(n: nat).((eq nat n3 n0) \to ((eq A (ASort h2 n4) (ASort n4 n2)) \to ((eq A
+(aplus g (ASort n n3) k) (aplus g (ASort h2 n4) k)) \to (leq g (ASort (S n3)
+n0) (ASort (S n4) n2)))))) (\lambda (H5: (eq nat n3 n0)).(eq_ind nat n0
+(\lambda (n: nat).((eq A (ASort h2 n4) (ASort n4 n2)) \to ((eq A (aplus g
+(ASort n3 n) k) (aplus g (ASort h2 n4) k)) \to (leq g (ASort (S n3) n0)
+(ASort (S n4) n2))))) (\lambda (H6: (eq A (ASort h2 n4) (ASort n4 n2))).(let
+H7 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_:
+A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n4]))
+(ASort h2 n4) (ASort n4 n2) H6) in ((let H8 \def (f_equal A nat (\lambda (e:
+A).(match e in A return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n
+| (AHead _ _) \Rightarrow h2])) (ASort h2 n4) (ASort n4 n2) H6) in (eq_ind
+nat n4 (\lambda (n: nat).((eq nat n4 n2) \to ((eq A (aplus g (ASort n3 n0) k)
+(aplus g (ASort n n4) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2)))))
+(\lambda (H9: (eq nat n4 n2)).(eq_ind nat n2 (\lambda (n: nat).((eq A (aplus
+g (ASort n3 n0) k) (aplus g (ASort n4 n) k)) \to (leq g (ASort (S n3) n0)
+(ASort (S n4) n2)))) (\lambda (H10: (eq A (aplus g (ASort n3 n0) k) (aplus g
+(ASort n4 n2) k))).(let H \def (eq_ind_r A (aplus g (ASort n3 n0) k) (\lambda
+(a: A).(eq A a (aplus g (ASort n4 n2) k))) H10 (aplus g (ASort (S n3) n0) (S
+k)) (aplus_sort_S_S_simpl g n0 n3 k)) in (let H11 \def (eq_ind_r A (aplus g
+(ASort n4 n2) k) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S k)) a))
+H (aplus g (ASort (S n4) n2) (S k)) (aplus_sort_S_S_simpl g n2 n4 k)) in
+(leq_sort g (S n3) (S n4) n0 n2 (S k) H11)))) n4 (sym_eq nat n4 n2 H9))) h2
+(sym_eq nat h2 n4 H8))) H7))) n3 (sym_eq nat n3 n0 H5))) h1 (sym_eq nat h1 n3
+H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2:
+(eq A (AHead a1 a3) (ASort n3 n0))).(\lambda (H3: (eq A (AHead a2 a4) (ASort
+n4 n2))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e in A
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
+_) \Rightarrow True])) I (ASort n3 n0) H2) in (False_ind ((eq A (AHead a2 a4)
+(ASort n4 n2)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort (S n3)
+n0) (ASort (S n4) n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort n3
+n0)) (refl_equal A (ASort n4 n2)))))]) H0))]) H)))) (\lambda (a: A).(\lambda
+(H: (((leq g (asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0)
+a)))).(\lambda (a0: A).(\lambda (H0: (((leq g (asucc g (ASort n n0)) (asucc g
+a0)) \to (leq g (ASort n n0) a0)))).(\lambda (H1: (leq g (asucc g (ASort n
+n0)) (asucc g (AHead a a0)))).((match n in nat return (\lambda (n1:
+nat).((((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)))))) with [O \Rightarrow (\lambda (_:
+(((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O n0)
+a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq g
+(ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g
+(AHead a a0)))).(let H5 \def (match H4 in leq return (\lambda (a1:
+A).(\lambda (a2: A).(\lambda (_: (leq ? a1 a2)).((eq A a1 (ASort O (next g
+n0))) \to ((eq A a2 (AHead a (asucc g a0))) \to (leq g (ASort O n0) (AHead a
+a0))))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A
+(ASort h1 n1) (ASort O (next g n0)))).(\lambda (H4: (eq A (ASort h2 n2)
+(AHead a (asucc g a0)))).((let H5 \def (f_equal A nat (\lambda (e: A).(match
+e in A return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _
+_) \Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H3) in ((let H6 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1)
+(ASort O (next g n0)) H3) in (eq_ind nat O (\lambda (n: nat).((eq nat n1
+(next g n0)) \to ((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A
+(aplus g (ASort n n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0)
+(AHead a a0)))))) (\lambda (H7: (eq nat n1 (next g n0))).(eq_ind nat (next g
+n0) (\lambda (n: nat).((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A
+(aplus g (ASort O n) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0)
+(AHead a a0))))) (\lambda (H8: (eq A (ASort h2 n2) (AHead a (asucc g
+a0)))).(let H9 \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)) H8) 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))) H9))) n1 (sym_eq nat n1 (next g n0) H7))) h1 (sym_eq nat h1 O
+H6))) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4:
+(eq A (AHead a1 a3) (ASort O (next g n0)))).(\lambda (H5: (eq A (AHead a2 a4)
+(AHead a (asucc g a0)))).((let H6 \def (eq_ind A (AHead a1 a3) (\lambda (e:
+A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow
+False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H4) in
+(False_ind ((eq A (AHead a2 a4) (AHead a (asucc g a0))) \to ((leq g a1 a2)
+\to ((leq g a3 a4) \to (leq g (ASort O n0) (AHead a a0))))) H6)) H5 H2
+H3)))]) in (H5 (refl_equal A (ASort O (next g n0))) (refl_equal A (AHead a
+(asucc g a0)))))))) | (S n1) \Rightarrow (\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 (a1:
+A).(\lambda (a2: A).(\lambda (_: (leq ? a1 a2)).((eq A a1 (ASort n1 n0)) \to
+((eq A a2 (AHead a (asucc g a0))) \to (leq g (ASort (S n1) n0) (AHead a
+a0))))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A
+(ASort h1 n1) (ASort n1 n0))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a
+(asucc g a0)))).((let H5 \def (f_equal A nat (\lambda (e: A).(match e in A
+return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
+\Rightarrow n1])) (ASort h1 n1) (ASort n1 n0) H3) in ((let H6 \def (f_equal A
+nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort n
+_) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort n1 n0)
+H3) in (eq_ind nat n1 (\lambda (n: nat).((eq nat n1 n0) \to ((eq A (ASort h2
+n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n n1) k) (aplus g
+(ASort h2 n2) k)) \to (leq g (ASort (S n1) n0) (AHead a a0)))))) (\lambda
+(H7: (eq nat n1 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (ASort h2 n2)
+(AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n1 n) k) (aplus g (ASort
+h2 n2) k)) \to (leq g (ASort (S n1) n0) (AHead a a0))))) (\lambda (H8: (eq A
+(ASort h2 n2) (AHead a (asucc g a0)))).(let H9 \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))
+H8) in (False_ind ((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) k))
+\to (leq g (ASort (S n1) n0) (AHead a a0))) H9))) n1 (sym_eq nat n1 n0 H7)))
+h1 (sym_eq nat h1 n1 H6))) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3)
+\Rightarrow (\lambda (H4: (eq A (AHead a1 a3) (ASort n1 n0))).(\lambda (H5:
+(eq A (AHead a2 a4) (AHead a (asucc g a0)))).((let H6 \def (eq_ind A (AHead
+a1 a3) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
+[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1
+n0) H4) in (False_ind ((eq A (AHead a2 a4) (AHead a (asucc g a0))) \to ((leq
+g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort (S n1) n0) (AHead a a0)))))
+H6)) H5 H2 H3)))]) in (H5 (refl_equal A (ASort n1 n0)) (refl_equal A (AHead a
+(asucc g a0))))))))]) 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)))).((match n in nat return (\lambda (n1:
+nat).((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g
+(AHead a a0) (ASort n1 n0)))) with [O \Rightarrow (\lambda (H2: (leq g (asucc
+g (AHead a a0)) (asucc g (ASort O n0)))).(let H3 \def (match H2 in leq return
+(\lambda (a1: A).(\lambda (a2: A).(\lambda (_: (leq ? a1 a2)).((eq A a1
+(AHead a (asucc g a0))) \to ((eq A a2 (ASort O (next g n0))) \to (leq g
+(AHead a a0) (ASort O n0))))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow
+(\lambda (H3: (eq A (ASort h1 n1) (AHead a (asucc g a0)))).(\lambda (H4: (eq
+A (ASort h2 n2) (ASort O (next g n0)))).((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 a (asucc g a0))
+H3) 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)))) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow
+(\lambda (H4: (eq A (AHead a1 a3) (AHead a (asucc g a0)))).(\lambda (H5: (eq
+A (AHead a2 a4) (ASort O (next g n0)))).((let H6 \def (f_equal A A (\lambda
+(e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow
+a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead a (asucc g a0)) H4) in
+((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_:
+A).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead
+a1 a3) (AHead a (asucc g a0)) H4) in (eq_ind A a (\lambda (a5: A).((eq A a3
+(asucc g a0)) \to ((eq A (AHead a2 a4) (ASort O (next g n0))) \to ((leq g a5
+a2) \to ((leq g a3 a4) \to (leq g (AHead a a0) (ASort O n0))))))) (\lambda
+(H8: (eq A a3 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a5: A).((eq A
+(AHead a2 a4) (ASort O (next g n0))) \to ((leq g a a2) \to ((leq g a5 a4) \to
+(leq g (AHead a a0) (ASort O n0)))))) (\lambda (H9: (eq A (AHead a2 a4)
+(ASort O (next g n0)))).(let H10 \def (eq_ind A (AHead a2 a4) (\lambda (e:
+A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow
+False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H9) in
+(False_ind ((leq g a a2) \to ((leq g (asucc g a0) a4) \to (leq g (AHead a a0)
+(ASort O n0)))) H10))) a3 (sym_eq A a3 (asucc g a0) H8))) a1 (sym_eq A a1 a
+H7))) H6)) H5 H2 H3)))]) in (H3 (refl_equal A (AHead a (asucc g a0)))
+(refl_equal A (ASort O (next g n0)))))) | (S n1) \Rightarrow (\lambda (H2:
+(leq g (asucc g (AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H3 \def
+(match H2 in leq return (\lambda (a1: A).(\lambda (a2: A).(\lambda (_: (leq ?
+a1 a2)).((eq A a1 (AHead a (asucc g a0))) \to ((eq A a2 (ASort n1 n0)) \to
+(leq g (AHead a a0) (ASort (S n1) n0))))))) with [(leq_sort h1 h2 n1 n2 k H2)
+\Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead a (asucc g
+a0)))).(\lambda (H4: (eq A (ASort h2 n2) (ASort n1 n0))).((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 a (asucc g a0)) H3) in (False_ind ((eq A (ASort h2 n2)
+(ASort n1 n0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
+k)) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H5)) H4 H2))) | (leq_head a1
+a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a3) (AHead a (asucc
+g a0)))).(\lambda (H5: (eq A (AHead a2 a4) (ASort n1 n0))).((let H6 \def
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
+[(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3)
+(AHead a (asucc g a0)) H4) in ((let H7 \def (f_equal A A (\lambda (e:
+A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 |
+(AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead a (asucc g a0)) H4) in
+(eq_ind A a (\lambda (a5: A).((eq A a3 (asucc g a0)) \to ((eq A (AHead a2 a4)
+(ASort n1 n0)) \to ((leq g a5 a2) \to ((leq g a3 a4) \to (leq g (AHead a a0)
+(ASort (S n1) n0))))))) (\lambda (H8: (eq A a3 (asucc g a0))).(eq_ind A
+(asucc g a0) (\lambda (a5: A).((eq A (AHead a2 a4) (ASort n1 n0)) \to ((leq g
+a a2) \to ((leq g a5 a4) \to (leq g (AHead a a0) (ASort (S n1) n0))))))
+(\lambda (H9: (eq A (AHead a2 a4) (ASort n1 n0))).(let H10 \def (eq_ind A
+(AHead a2 a4) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
+[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1
+n0) H9) in (False_ind ((leq g a a2) \to ((leq g (asucc g a0) a4) \to (leq g
+(AHead a a0) (ASort (S n1) n0)))) H10))) a3 (sym_eq A a3 (asucc g a0) H8)))
+a1 (sym_eq A a1 a H7))) H6)) H5 H2 H3)))]) in (H3 (refl_equal A (AHead a
+(asucc g a0))) (refl_equal A (ASort n1 n0)))))]) 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 (a1: A).(\lambda (a2: A).(\lambda (_: (leq ? a1 a2)).((eq A
+a1 (AHead a (asucc g a0))) \to ((eq A a2 (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
+a3 a4 H4 a5 a6 H5) \Rightarrow (\lambda (H6: (eq A (AHead a3 a5) (AHead a
+(asucc g a0)))).(\lambda (H7: (eq A (AHead a4 a6) (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 a5 | (AHead _ a) \Rightarrow
+a])) (AHead a3 a5) (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 a3 | (AHead a _) \Rightarrow a])) (AHead a3 a5) (AHead a (asucc g
+a0)) H6) in (eq_ind A a (\lambda (a1: A).((eq A a5 (asucc g a0)) \to ((eq A
+(AHead a4 a6) (AHead a3 (asucc g a4))) \to ((leq g a1 a4) \to ((leq g a5 a6)
+\to (leq g (AHead a a0) (AHead a3 a4))))))) (\lambda (H10: (eq A a5 (asucc g
+a0))).(eq_ind A (asucc g a0) (\lambda (a1: A).((eq A (AHead a4 a6) (AHead a3
+(asucc g a4))) \to ((leq g a a4) \to ((leq g a1 a6) \to (leq g (AHead a a0)
+(AHead a3 a4)))))) (\lambda (H11: (eq A (AHead a4 a6) (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 a6 | (AHead _ a) \Rightarrow
+a])) (AHead a4 a6) (AHead a3 (asucc g a4)) H11) in ((let H13 \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 a4 a6) (AHead a3 (asucc
+g a4)) H11) in (eq_ind A a3 (\lambda (a1: A).((eq A a6 (asucc g a4)) \to
+((leq g a a1) \to ((leq g (asucc g a0) a6) \to (leq g (AHead a a0) (AHead a3
+a4)))))) (\lambda (H14: (eq A a6 (asucc g a4))).(eq_ind A (asucc g a4)
+(\lambda (a1: A).((leq g a a3) \to ((leq g (asucc g a0) a1) \to (leq g (AHead
+a a0) (AHead a3 a4))))) (\lambda (H15: (leq g a a3)).(\lambda (H16: (leq g
+(asucc g a0) (asucc g a4))).(leq_head g a a3 H15 a0 a4 (H0 a4 H16)))) a6
+(sym_eq A a6 (asucc g a4) H14))) a4 (sym_eq A a4 a3 H13))) H12))) a5 (sym_eq
+A a5 (asucc g a0) H10))) a3 (sym_eq A a3 a H9))) H8)) H7 H4 H5)))]) in (H4
+(refl_equal A (AHead a (asucc g a0))) (refl_equal A (AHead a3 (asucc g
+a4)))))))))) a2)))))) a1)).
+