- SEQ
-
|- !x x0.
x --> x0 =
(!e. & 0 |<| e ==> (?N. !n. n >= N ==> abs (x n |-| x0) |<| e))
- SEQ_CONST
-
|- !k. (\x. k) --> k
- SEQ_ADD
-
|- !x x0 y y0. x --> x0 /\ y --> y0 ==> (\n. x n |+| y n) --> (x0 |+| y0)
- SEQ_MUL
-
|- !x x0 y y0. x --> x0 /\ y --> y0 ==> (\n. x n |*| y n) --> (x0 |*| y0)
- SEQ_NEG
-
|- !x x0. x --> x0 = (\n. -- (x n)) --> -- x0
- SEQ_INV
-
|- !x x0. x --> x0 /\ ~(x0 = & 0) ==> (\n. inv (x n)) --> inv x0
- SEQ_SUB
-
|- !x x0 y y0. x --> x0 /\ y --> y0 ==> (\n. x n |-| y n) --> (x0 |-| y0)
- SEQ_DIV
-
|- !x x0 y y0.
x --> x0 /\ y --> y0 /\ ~(y0 = & 0) ==> (\n. x n / y n) --> (x0 / y0)
- SEQ_UNIQ
-
|- !x x1 x2. x --> x1 /\ x --> x2 ==> (x1 = x2)
- SEQ_LIM
-
|- !f. convergent f = f --> lim f
- SUBSEQ_SUC
-
|- !f. subseq f = (!n. f n < f (SUC n))
- MONO_SUC
-
|- !f. mono f = (!n. f (SUC n) |>=| f n) \/ (!n. f (SUC n) |<=| f n)
- MAX_LEMMA
-
|- !s N. ?k. !n. n < N ==> abs (s n) |<| k
- SEQ_BOUNDED
-
|- !s. bounded (mr1,$>=) s = (?k. !n. abs (s n) |<| k)
- SEQ_BOUNDED_2
-
|- !f k k'. (!n. k |<=| f n /\ f n |<=| k') ==> bounded (mr1,$>=) f
- SEQ_CBOUNDED
-
|- !f. cauchy f ==> bounded (mr1,$>=) f
- SEQ_ICONV
-
|- !f. bounded (mr1,$>=) f /\ (!m n. m >= n ==> f m |>=| f n) ==> convergent f
- SEQ_NEG_CONV
-
|- !f. convergent f = convergent (\n. -- (f n))
- SEQ_NEG_BOUNDED
-
|- !f. bounded (mr1,$>=) (\n. -- (f n)) = bounded (mr1,$>=) f
- SEQ_BCONV
-
|- !f. bounded (mr1,$>=) f /\ mono f ==> convergent f
- SEQ_MONOSUB
-
|- !s. ?f. subseq f /\ mono (\n. s (f n))
- SEQ_SBOUNDED
-
|- !s f. bounded (mr1,$>=) s ==> bounded (mr1,$>=) (\n. s (f n))
- SEQ_SUBLE
-
|- !f. subseq f ==> (!n. n <= f n)
- SEQ_DIRECT
-
|- !f. subseq f ==> (!N1 N2. ?n. n >= N1 /\ f n >= N2)
- SEQ_CAUCHY
-
|- !f. cauchy f = convergent f
- SEQ_LE
-
|- !f g l m.
f --> l /\ g --> m /\ (?N. !n. n >= N ==> f n |<=| g n) ==> l |<=| m
- SEQ_SUC
-
|- !f l. f --> l = (\n. f (SUC n)) --> l
- SEQ_ABS
-
|- !f. (\n. abs (f n)) --> & 0 = f --> & 0
- SEQ_ABS_IMP
-
|- !f l. f --> l ==> (\n. abs (f n)) --> abs l
- SEQ_INV0
-
|- !f. (!y. ?N. !n. n >= N ==> f n |>| y) ==> (\n. inv (f n)) --> & 0
- SEQ_POWER_ABS
-
|- !c. abs c |<| & 1 ==> (\n. abs c pow n) --> & 0
- SEQ_POWER
-
|- !c. abs c |<| & 1 ==> (\n. c pow n) --> & 0
- NEST_LEMMA
-
|- !f g.
(!n. f (SUC n) |>=| f n) /\
(!n. g (SUC n) |<=| g n) /\
(!n. f n |<=| g n) ==>
(?l m.
l |<=| m /\
((!n. f n |<=| l) /\ f --> l) /\
(!n. m |<=| g n) /\
g --> m)
- NEST_LEMMA_UNIQ
-
|- !f g.
(!n. f (SUC n) |>=| f n) /\
(!n. g (SUC n) |<=| g n) /\
(!n. f n |<=| g n) /\
(\n. f n |-| g n) --> & 0 ==>
(?l. ((!n. f n |<=| l) /\ f --> l) /\ (!n. l |<=| g n) /\ g --> l)
- BOLZANO_LEMMA
-
|- !P.
(!a b c. a |<=| b /\ b |<=| c /\ P (a,b) /\ P (b,c) ==> P (a,c)) /\
(!x.
?d.
& 0 |<| d /\
(!a b. a |<=| x /\ x |<=| b /\ b |-| a |<| d ==> P (a,b))) ==>
(!a b. a |<=| b ==> P (a,b))
- SUM_SUMMABLE
-
|- !f l. f sums l ==> summable f
- SUMMABLE_SUM
-
|- !f. summable f ==> f sums suminf f
- SUM_UNIQ
-
|- !f x. f sums x ==> (x = suminf f)
- SER_0
-
|- !f n. (!m. n <= m ==> (f m = & 0)) ==> f sums sum (0,n) f
- SER_POS_LE
-
|- !f n.
summable f /\ (!m. n <= m ==> & 0 |<=| f m) ==> sum (0,n) f |<=| suminf f
- SER_POS_LT
-
|- !f n.
summable f /\ (!m. n <= m ==> & 0 |<| f m) ==> sum (0,n) f |<| suminf f
- SER_GROUP
-
|- !f k. summable f /\ 0 < k ==> (\n. sum (n * k,k) f) sums suminf f
- SER_PAIR
-
|- !f. summable f ==> (\n. sum (2 * n,2) f) sums suminf f
- SER_OFFSET
-
|- !f. summable f ==> (!k. (\n. f (n + k)) sums (suminf f |-| sum (0,k) f))
- SER_POS_LT_PAIR
-
|- !f n.
summable f /\ (!d. & 0 |<| f (n + 2 * d) |+| f (n + 2 * d + 1)) ==>
sum (0,n) f |<| suminf f
- SER_ADD
-
|- !x x0 y y0. x sums x0 /\ y sums y0 ==> (\n. x n |+| y n) sums (x0 |+| y0)
- SER_CMUL
-
|- !x x0 c. x sums x0 ==> (\n. c |*| x n) sums (c |*| x0)
- SER_NEG
-
|- !x x0. x sums x0 ==> (\n. -- (x n)) sums -- x0
- SER_SUB
-
|- !x x0 y y0. x sums x0 /\ y sums y0 ==> (\n. x n |-| y n) sums (x0 |-| y0)
- SER_CDIV
-
|- !x x0 c. x sums x0 ==> (\n. x n / c) sums (x0 / c)
- SER_CAUCHY
-
|- !f.
summable f =
(!e. & 0 |<| e ==> (?N. !m n. m >= N ==> abs (sum (m,n) f) |<| e))
- SER_ZERO
-
|- !f. summable f ==> f --> & 0
- SER_COMPAR
-
|- !f g. (?N. !n. n >= N ==> abs (f n) |<=| g n) /\ summable g ==> summable f
- SER_COMPARA
-
|- !f g.
(?N. !n. n >= N ==> abs (f n) |<=| g n) /\ summable g ==>
summable (\k. abs (f k))
- SER_LE
-
|- !f g.
(!n. f n |<=| g n) /\ summable f /\ summable g ==> suminf f |<=| suminf g
- SER_LE2
-
|- !f g.
(!n. abs (f n) |<=| g n) /\ summable g ==>
summable f /\ suminf f |<=| suminf g
- SER_ACONV
-
|- !f. summable (\n. abs (f n)) ==> summable f
- SER_ABS
-
|- !f. summable (\n. abs (f n)) ==> abs (suminf f) |<=| suminf (\n. abs (f n))
- GP_FINITE
-
|- !x.
~(x = & 1) ==>
(!n. sum (0,n) (\n. x pow n) = (x pow n |-| & 1) / (x |-| & 1))
- GP
-
|- !x. abs x |<| & 1 ==> (\n. x pow n) sums inv (& 1 |-| x)
- ABS_NEG_LEMMA
-
|- !c. c |<=| & 0 ==> (!x y. abs x |<=| c |*| abs y ==> (x = & 0))
- SER_RATIO
-
|- !f c N.
c |<| & 1 /\ (!n. n >= N ==> abs (f (SUC n)) |<=| c |*| abs (f n)) ==>
summable f