- 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