Theory: NETS

Parents


Type constants


Term constants


Axioms


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 re_universe ==>
     ((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 re_universe ==>
     ((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) = ((\n. 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) (\n. inv (x n))
NET_NULL_ADD
|- !g.
     dorder g ==>
     (!x y.
       (x tends & 0) (mtop mr1,g) /\ (y tends & 0) (mtop mr1,g) ==>
       ((\n. 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) ==>
       ((\n. x n |*| y n) tends & 0) (mtop mr1,g))
NET_NULL_CMUL
|- !g k x.
     (x tends & 0) (mtop mr1,g) ==> ((\n. 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) ==>
       ((\n. x n |+| y n) tends (x0 |+| y0)) (mtop mr1,g))
NET_NEG
|- !g.
     dorder g ==>
     (!x x0.
       (x tends x0) (mtop mr1,g) = ((\n. -- (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) ==>
       ((\n. 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) ==>
       ((\n. 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) ==>
       ((\n. 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) ==>
       ((\n. x n / y n) tends (x0 / y0)) (mtop mr1,g))
NET_ABS
|- !g x x0.
     (x tends x0) (mtop mr1,g) ==> ((\n. 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)