Theory: POWSER

Parents


Type constants


Term constants


Axioms


Definitions

diffs
|- !c. diffs c = (\n. & (SUC n) |*| c (SUC n))

Theorems

POWDIFF_LEMMA
|- !n x y.
     sum (0,SUC n) (\p. x pow p |*| y pow (SUC n - p)) =
     y |*| sum (0,SUC n) (\p. x pow p |*| y pow (n - p))
POWDIFF
|- !n x y.
     x pow SUC n |-| y pow SUC n =
     (x |-| y) |*| sum (0,SUC n) (\p. x pow p |*| y pow (n - p))
POWREV
|- !n x y.
     sum (0,SUC n) (\p. x pow p |*| y pow (n - p)) =
     sum (0,SUC n) (\p. x pow (n - p) |*| y pow p)
POWSER_INSIDEA
|- !f x z.
     summable (\n. f n |*| x pow n) /\ abs z |<| abs x ==>
     summable (\n. abs (f n) |*| z pow n)
POWSER_INSIDE
|- !f x z.
     summable (\n. f n |*| x pow n) /\ abs z |<| abs x ==>
     summable (\n. f n |*| z pow n)
DIFFS_NEG
|- !c. diffs (\n. -- (c n)) = (\n. -- (diffs c n))
DIFFS_LEMMA
|- !n c x.
     sum (0,n) (\n. diffs c n |*| x pow n) =
     sum (0,n) (\n. & n |*| c n |*| x pow (n - 1)) |+|
     & n |*| c n |*| x pow (n - 1)
DIFFS_LEMMA2
|- !n c x.
     sum (0,n) (\n. & n |*| c n |*| x pow (n - 1)) =
     sum (0,n) (\n. diffs c n |*| x pow n) |-| & n |*| c n |*| x pow (n - 1)
DIFFS_EQUIV
|- !c x.
     summable (\n. diffs c n |*| x pow n) ==>
     (\n. & n |*| c n |*| x pow (n - 1)) sums
     suminf (\n. diffs c n |*| x pow n)
TERMDIFF_LEMMA1
|- !m z h.
     sum (0,m) (\p. (z |+| h) pow (m - p) |*| z pow p |-| z pow m) =
     sum (0,m) (\p. z pow p |*| ((z |+| h) pow (m - p) |-| z pow (m - p)))
TERMDIFF_LEMMA2
|- !z h n.
     ~(h = & 0) ==>
     (((z |+| h) pow n |-| z pow n) / h |-| & n |*| z pow (n - 1) =
      h |*|
      sum (0,n - 1)
        (\p.
          z pow p |*|
          sum (0,(n - 1) - p)
            (\q. (z |+| h) pow q |*| z pow (((n - 2) - p) - q))))
TERMDIFF_LEMMA3
|- !z h n k'.
     ~(h = & 0) /\ abs z |<=| k' /\ abs (z |+| h) |<=| k' ==>
     abs (((z |+| h) pow n |-| z pow n) / h |-| & n |*| z pow (n - 1)) |<=|
     & n |*| & (n - 1) |*| k' pow (n - 2) |*| abs h
TERMDIFF_LEMMA4
|- !f k' k.
     & 0 |<| k /\
     (!h. & 0 |<| abs h /\ abs h |<| k ==> abs (f h) |<=| k' |*| abs h) ==>
     (f -> & 0) (& 0)
TERMDIFF_LEMMA5
|- !f g k.
     & 0 |<| k /\
     summable f /\
     (!h.
       & 0 |<| abs h /\ abs h |<| k ==>
       (!n. abs (g h n) |<=| f n |*| abs h)) ==>
     ((\h. suminf (g h)) -> & 0) (& 0)
TERMDIFF
|- !c k' x.
     summable (\n. c n |*| k' pow n) /\
     summable (\n. diffs c n |*| k' pow n) /\
     summable (\n. diffs (diffs c) n |*| k' pow n) /\
     abs x |<| abs k' ==>
     ((\x. suminf (\n. c n |*| x pow n)) diffl
      suminf (\n. diffs c n |*| x pow n))
       x