Theory: SEQ

Parents


Type constants


Term constants


Axioms


Definitions

tends_num_real
|- !x x0. x --> x0 = (x tends x0) (mtop mr1,$>=)
convergent
|- !f. convergent f = (?l. f --> l)
cauchy
|- !f.
     cauchy f =
     (!e.
       & 0 |<| e ==> (?N. !m n. m >= N /\ n >= N ==> abs (f m |-| f n) |<| e))
lim
|- !f. lim f = (@l. f --> l)
subseq
|- !f. subseq f = (!m n. m < n ==> f m < f n)
mono
|- !f.
     mono f =
     (!m n. m <= n ==> f m |<=| f n) \/ (!m n. m <= n ==> f m |>=| f n)
sums
|- !f s. f sums s = (\n. sum (0,n) f) --> s
summable
|- !f. summable f = (?s. f sums s)
suminf
|- !f. suminf f = (@s. f sums s)

Theorems

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