- 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)