Theory "nets"

Signature

Definitions

dorder

|- ∀g. dorder g ⇔ ∀x y. g x x ∧ g y y ⇒ ∃z. g z z ∧ ∀w. g w z ⇒ g w x ∧ g w y

tends

|- ∀s l top g.
     (s tends l) (top,g) ⇔
     ∀N. neigh top (N,l) ⇒ ∃n. g n n ∧ ∀m. g m n ⇒ N (s m)

bounded

|- ∀m g f. bounded (m,g) f ⇔ ∃k x N. g N N ∧ ∀n. g n N ⇒ dist m (f n,x) < k

tendsto

|- ∀m x y z.
     tendsto (m,x) y z ⇔ 0 < dist m (x,y) ∧ dist m (x,y) ≤ dist m (x,z)

Theorems

DORDER_LEMMA

|- ∀g.
     dorder g ⇒
     ∀P Q.
       (∃n. g n n ∧ ∀m. g m n ⇒ P m) ∧ (∃n. g n n ∧ ∀m. g m n ⇒ Q m) ⇒
       ∃n. g n n ∧ ∀m. g m n ⇒ P m ∧ Q m

DORDER_NGE

|- dorder $>=

DORDER_TENDSTO

|- ∀m x. dorder (tendsto (m,x))

MTOP_TENDS

|- ∀d g x x0.
     (x tends x0) (mtop d,g) ⇔
     ∀e. 0 < e ⇒ ∃n. g n n ∧ ∀m. g m n ⇒ dist d (x m,x0) < e

MTOP_TENDS_UNIQ

|- ∀g d.
     dorder g ⇒ (x tends x0) (mtop d,g) ∧ (x tends x1) (mtop d,g) ⇒ (x0 = x1)

SEQ_TENDS

|- ∀d x x0.
     (x tends x0) (mtop d,$>=) ⇔
     ∀e. 0 < e ⇒ ∃N. ∀n. n ≥ N ⇒ dist d (x n,x0) < e

LIM_TENDS

|- ∀m1 m2 f x0 y0.
     limpt (mtop m1) x0 𝕌(:α) ⇒
     ((f tends y0) (mtop m2,tendsto (m1,x0)) ⇔
      ∀e.
        0 < e ⇒
        ∃d.
          0 < d ∧
          ∀x. 0 < dist m1 (x,x0) ∧ dist m1 (x,x0) ≤ d ⇒ dist m2 (f x,y0) < e)

LIM_TENDS2

|- ∀m1 m2 f x0 y0.
     limpt (mtop m1) x0 𝕌(:α) ⇒
     ((f tends y0) (mtop m2,tendsto (m1,x0)) ⇔
      ∀e.
        0 < e ⇒
        ∃d.
          0 < d ∧
          ∀x. 0 < dist m1 (x,x0) ∧ dist m1 (x,x0) < d ⇒ dist m2 (f x,y0) < e)

MR1_BOUNDED

|- ∀g f. bounded (mr1,g) f ⇔ ∃k N. g N N ∧ ∀n. g n N ⇒ abs (f n) < k

NET_NULL

|- ∀g x x0. (x tends x0) (mtop mr1,g) ⇔ ((λn. x n − x0) tends 0) (mtop mr1,g)

NET_CONV_BOUNDED

|- ∀g x x0. (x tends x0) (mtop mr1,g) ⇒ bounded (mr1,g) x

NET_CONV_NZ

|- ∀g x x0.
     (x tends x0) (mtop mr1,g) ∧ x0 ≠ 0 ⇒ ∃N. g N N ∧ ∀n. g n N ⇒ x n ≠ 0

NET_CONV_IBOUNDED

|- ∀g x x0.
     (x tends x0) (mtop mr1,g) ∧ x0 ≠ 0 ⇒ bounded (mr1,g) (λn. inv (x n))

NET_NULL_ADD

|- ∀g.
     dorder g ⇒
     ∀x y.
       (x tends 0) (mtop mr1,g) ∧ (y tends 0) (mtop mr1,g) ⇒
       ((λn. x n + y n) tends 0) (mtop mr1,g)

NET_NULL_MUL

|- ∀g.
     dorder g ⇒
     ∀x y.
       bounded (mr1,g) x ∧ (y tends 0) (mtop mr1,g) ⇒
       ((λn. x n * y n) tends 0) (mtop mr1,g)

NET_NULL_CMUL

|- ∀g k x. (x tends 0) (mtop mr1,g) ⇒ ((λn. k * x n) tends 0) (mtop mr1,g)

NET_ADD

|- ∀g.
     dorder g ⇒
     ∀x x0 y y0.
       (x tends x0) (mtop mr1,g) ∧ (y tends y0) (mtop mr1,g) ⇒
       ((λn. x n + y n) tends (x0 + y0)) (mtop mr1,g)

NET_NEG

|- ∀g.
     dorder g ⇒
     ∀x x0. (x tends x0) (mtop mr1,g) ⇔ ((λn. -x n) tends -x0) (mtop mr1,g)

NET_SUB

|- ∀g.
     dorder g ⇒
     ∀x x0 y y0.
       (x tends x0) (mtop mr1,g) ∧ (y tends y0) (mtop mr1,g) ⇒
       ((λn. x n − y n) tends (x0 − y0)) (mtop mr1,g)

NET_MUL

|- ∀g.
     dorder g ⇒
     ∀x y x0 y0.
       (x tends x0) (mtop mr1,g) ∧ (y tends y0) (mtop mr1,g) ⇒
       ((λn. x n * y n) tends (x0 * y0)) (mtop mr1,g)

NET_INV

|- ∀g.
     dorder g ⇒
     ∀x x0.
       (x tends x0) (mtop mr1,g) ∧ x0 ≠ 0 ⇒
       ((λn. inv (x n)) tends inv x0) (mtop mr1,g)

NET_DIV

|- ∀g.
     dorder g ⇒
     ∀x x0 y y0.
       (x tends x0) (mtop mr1,g) ∧ (y tends y0) (mtop mr1,g) ∧ y0 ≠ 0 ⇒
       ((λn. x n / y n) tends (x0 / y0)) (mtop mr1,g)

NET_ABS

|- ∀g x x0.
     (x tends x0) (mtop mr1,g) ⇒ ((λn. abs (x n)) tends abs x0) (mtop mr1,g)

NET_LE

|- ∀g.
     dorder g ⇒
     ∀x x0 y y0.
       (x tends x0) (mtop mr1,g) ∧ (y tends y0) (mtop mr1,g) ∧
       (∃N. g N N ∧ ∀n. g n N ⇒ x n ≤ y n) ⇒
       x0 ≤ y0