in (\lambda (H6: (eq A a3 x0)).(let H7 \def (eq_ind_r A x1 (\lambda (a:
A).(leq g a2 a)) H2 a4 H5) in (let H8 \def (eq_ind_r A x0 (\lambda (a:
A).(leq g a1 a)) H1 a3 H6) in H7)))) H4))))))) H0)))))))).
in (\lambda (H6: (eq A a3 x0)).(let H7 \def (eq_ind_r A x1 (\lambda (a:
A).(leq g a2 a)) H2 a4 H5) in (let H8 \def (eq_ind_r A x0 (\lambda (a:
A).(leq g a1 a)) H1 a3 H6) in H7)))) H4))))))) H0)))))))).
(aplus g (ASort n n0) O))))) (\lambda (a0: A).(\lambda (H: (leq g a0
a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1
H0))))) a)).
(aplus g (ASort n n0) O))))) (\lambda (a0: A).(\lambda (H: (leq g a0
a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1
H0))))) a)).
\def
\lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1
a2)).(eq_ind A a1 (\lambda (a: A).(leq g a1 a)) (leq_refl g a1) a2 H)))).
\def
\lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1
a2)).(eq_ind A a1 (\lambda (a: A).(leq g a1 a)) (leq_refl g a1) a2 H)))).
(leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: A).(\lambda (a6:
A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3
H1 a6 a5 H3))))))))) a1 a2 H)))).
(leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: A).(\lambda (a6:
A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3
H1 a6 a5 H3))))))))) a1 a2 H)))).
(AHead x0 x1) H8) in (eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead
a3 a5) a)) (leq_head g a3 x0 (H1 x0 H6) a5 x1 (H3 x1 H7)) a0 H9)))))))
H5))))))))))))) a1 a2 H)))).
(AHead x0 x1) H8) in (eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead
a3 a5) a)) (leq_head g a3 x0 (H1 x0 H6) a5 x1 (H3 x1 H7)) a0 H9)))))))
H5))))))))))))) a1 a2 H)))).
theorem leq_ahead_false_1:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1)
theorem leq_ahead_false_1:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1)
(eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) H4 a0 H7) in (let H10 \def
(eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a a0) a3)) H3 a H8) in (H a0
H10 P))))) H6))))))) H2)))))))))) a1)).
(eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) H4 a0 H7) in (let H10 \def
(eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a a0) a3)) H3 a H8) in (H a0
H10 P))))) H6))))))) H2)))))))))) a1)).
theorem leq_ahead_false_2:
\forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2)
theorem leq_ahead_false_2:
\forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2)
(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)).
(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)).