set "baseuri" "cic:/matita/nat/le_arith".
-include "higher_order_defs/functions.ma".
include "nat/times.ma".
include "nat/orders.ma".
-include "nat/compare.ma".
(* 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 \le m \to p + n \le 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.
+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 \le m \to n + p \le 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 \le n2 \to m1 \le m2
+\to n1 + m1 \le 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 \le n + m.
+intros.change with (O+m \le 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 \le 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 \le m \to p*n \le 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.
+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 \le m \to n*p \le 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 \le n2 \to m1 \le m2
+\to n1*m1 \le 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) \le n \to m \le n*m.
intros.elim H.simplify.
elim (plus_n_O ?).apply le_n.
simplify.rewrite < sym_plus.apply le_plus_n.
qed.
-