(* plus *)
theorem monotonic_le_plus_r:
-\forall n:nat.monotonic nat le (\lambda m.plus n m).
+\forall n:nat.monotonic nat le (\lambda m.n+m).
simplify.intros.elim n.
simplify.assumption.
simplify.apply le_S_S.assumption.
qed.
-theorem le_plus_r: \forall p,n,m:nat. le n m \to le (plus p n) (plus p m)
+theorem le_plus_r: \forall p,n,m:nat. n \leq m \to p+n \leq p+m
\def monotonic_le_plus_r.
theorem monotonic_le_plus_l:
-\forall m:nat.monotonic nat le (\lambda n.plus n m).
+\forall m:nat.monotonic nat le (\lambda n.n+m).
simplify.intros.
rewrite < sym_plus.rewrite < sym_plus m.
apply le_plus_r.assumption.
qed.
-theorem le_plus_l: \forall p,n,m:nat. le n m \to le (plus n p) (plus m p)
+theorem le_plus_l: \forall p,n,m:nat. n \leq m \to n+p \leq m+p
\def monotonic_le_plus_l.
-theorem le_plus: \forall n1,n2,m1,m2:nat. le n1 n2 \to le m1 m2
-\to le (plus n1 m1) (plus n2 m2).
+theorem le_plus: \forall n1,n2,m1,m2:nat. n1 \leq n2 \to m1 \leq m2
+\to n1+m1 \leq n2+m2.
intros.
-apply trans_le ? (plus n2 m1).
+apply trans_le ? (n2+m1).
apply le_plus_l.assumption.
apply le_plus_r.assumption.
qed.
-theorem le_plus_n :\forall n,m:nat. le m (plus n m).
-intros.change with le (plus O m) (plus n m).
+theorem le_plus_n :\forall n,m:nat. m \leq n+m.
+intros.change with O+m \leq n+m.
apply le_plus_l.apply le_O_n.
qed.
-theorem eq_plus_to_le: \forall n,m,p:nat.eq nat n (plus m p)
-\to le m n.
+theorem eq_plus_to_le: \forall n,m,p:nat.n=m+p \to m \leq n.
intros.rewrite > H.
rewrite < sym_plus.
apply le_plus_n.
(* times *)
theorem monotonic_le_times_r:
-\forall n:nat.monotonic nat le (\lambda m.times n m).
+\forall n:nat.monotonic nat le (\lambda m.n*m).
simplify.intros.elim n.
simplify.apply le_O_n.
simplify.apply le_plus.
assumption.
qed.
-theorem le_times_r: \forall p,n,m:nat. le n m \to le (times p n) (times p m)
+theorem le_times_r: \forall p,n,m:nat. n \leq m \to p*n \leq p*m
\def monotonic_le_times_r.
theorem monotonic_le_times_l:
-\forall m:nat.monotonic nat le (\lambda n.times n m).
+\forall m:nat.monotonic nat le (\lambda n.n*m).
simplify.intros.
rewrite < sym_times.rewrite < sym_times m.
apply le_times_r.assumption.
qed.
-theorem le_times_l: \forall p,n,m:nat. le n m \to le (times n p) (times m p)
+theorem le_times_l: \forall p,n,m:nat. n \leq m \to n*p \leq m*p
\def monotonic_le_times_l.
-theorem le_times: \forall n1,n2,m1,m2:nat. le n1 n2 \to le m1 m2
-\to le (times n1 m1) (times n2 m2).
+theorem le_times: \forall n1,n2,m1,m2:nat. n1 \leq n2 \to m1 \leq m2
+\to n1*m1 \leq n2*m2.
intros.
-apply trans_le ? (times n2 m1).
+apply trans_le ? (n2*m1).
apply le_times_l.assumption.
apply le_times_r.assumption.
qed.
-theorem le_times_n: \forall n,m:nat.le (S O) n \to le m (times n m).
+theorem le_times_n: \forall n,m:nat.S O \leq n \to m \leq n*m.
intros.elim H.simplify.
elim (plus_n_O ?).apply le_n.
simplify.rewrite < sym_plus.apply le_plus_n.