include "ground_1/blt/defs.ma".
-theorem lt_blt:
+lemma lt_blt:
\forall (x: nat).(\forall (y: nat).((lt y x) \to (eq bool (blt y x) true)))
\def
\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to
with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true)))).(\lambda
(H1: (lt (S n0) (S n))).(H n0 (le_S_n (S n0) n H1))))) y)))) x).
-theorem le_bge:
+lemma le_bge:
\forall (x: nat).(\forall (y: nat).((le x y) \to (eq bool (blt y x) false)))
\def
\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to
(eq bool (blt n0 (S n)) false)))).(\lambda (H1: (le (S n) (S n0))).(H n0
(le_S_n n n0 H1))))) y)))) x).
-theorem blt_lt:
+lemma blt_lt:
\forall (x: nat).(\forall (y: nat).((eq bool (blt y x) true) \to (lt y x)))
\def
\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt
\Rightarrow (blt m n)]) true) \to (lt n0 (S n))))).(\lambda (H1: (eq bool
(blt n0 n) true)).(lt_n_S n0 n (H n0 H1))))) y)))) x).
-theorem bge_le:
+lemma bge_le:
\forall (x: nat).(\forall (y: nat).((eq bool (blt y x) false) \to (le x y)))
\def
\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt