include "basic_1/aplus/props.ma".
-theorem leq_refl:
+lemma leq_refl:
\forall (g: G).(\forall (a: A).(leq g a a))
\def
\lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(leq g a0 a0))
a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1
H0))))) a)).
-theorem leq_eq:
+lemma leq_eq:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1
a2))))
\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)))).
-theorem leq_sym:
+lemma leq_sym:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
a2 a1))))
\def
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:
+lemma leq_ahead_false_1:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1)
\to (\forall (P: Prop).P))))
\def
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:
+lemma leq_ahead_false_2:
\forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2)
\to (\forall (P: Prop).P))))
\def