- LIM
-
|- !f y0 x0.
(f -> y0) x0 =
(!e.
& 0 |<| e ==>
(?d.
& 0 |<| d /\
(!x.
& 0 |<| abs (x |-| x0) /\ abs (x |-| x0) |<| d ==>
abs (f x |-| y0) |<| e)))
- LIM_CONST
-
|- !k x. ((\x. k) -> k) x
- LIM_ADD
-
|- !f g l m x. (f -> l) x /\ (g -> m) x ==> ((\x. f x |+| g x) -> (l |+| m)) x
- LIM_MUL
-
|- !f g l m x. (f -> l) x /\ (g -> m) x ==> ((\x. f x |*| g x) -> (l |*| m)) x
- LIM_NEG
-
|- !f l x. (f -> l) x = ((\x. -- (f x)) -> -- l) x
- LIM_INV
-
|- !f l x. (f -> l) x /\ ~(l = & 0) ==> ((\x. inv (f x)) -> inv l) x
- LIM_SUB
-
|- !f g l m x. (f -> l) x /\ (g -> m) x ==> ((\x. f x |-| g x) -> (l |-| m)) x
- LIM_DIV
-
|- !f g l m x.
(f -> l) x /\ (g -> m) x /\ ~(m = & 0) ==> ((\x. f x / g x) -> (l / m)) x
- LIM_NULL
-
|- !f l x. (f -> l) x = ((\x. f x |-| l) -> & 0) x
- LIM_X
-
|- !x0. ((\x. x) -> x0) x0
- LIM_UNIQ
-
|- !f l m x. (f -> l) x /\ (f -> m) x ==> (l = m)
- LIM_EQUAL
-
|- !f g l x0. (!x. ~(x = x0) ==> (f x = g x)) ==> ((f -> l) x0 = (g -> l) x0)
- LIM_TRANSFORM
-
|- !f g x0 l. ((\x. f x |-| g x) -> & 0) x0 /\ (g -> l) x0 ==> (f -> l) x0
- DIFF_UNIQ
-
|- !f l m x. (f diffl l) x /\ (f diffl m) x ==> (l = m)
- DIFF_CONT
-
|- !f l x. (f diffl l) x ==> f contl x
- CONTL_LIM
-
|- !f x. f contl x = (f -> f x) x
- DIFF_CARAT
-
|- !f l x.
(f diffl l) x =
(?g. (!z. f z |-| f x = g z |*| (z |-| x)) /\ g contl x /\ (g x = l))
- CONT_CONST
-
|- !k x. (\x. k) contl x
- CONT_ADD
-
|- !f g x. f contl x /\ g contl x ==> (\x. f x |+| g x) contl x
- CONT_MUL
-
|- !f g x. f contl x /\ g contl x ==> (\x. f x |*| g x) contl x
- CONT_NEG
-
|- !f x. f contl x ==> (\x. -- (f x)) contl x
- CONT_INV
-
|- !f x. f contl x /\ ~(f x = & 0) ==> (\x. inv (f x)) contl x
- CONT_SUB
-
|- !f g x. f contl x /\ g contl x ==> (\x. f x |-| g x) contl x
- CONT_DIV
-
|- !f g x. f contl x /\ g contl x /\ ~(g x = & 0) ==> (\x. f x / g x) contl x
- CONT_COMPOSE
-
|- !f g x. f contl x /\ g contl f x ==> (\x. g (f x)) contl x
- IVT
-
|- !f a b y.
a |<=| b /\
(f a |<=| y /\ y |<=| f b) /\
(!x. a |<=| x /\ x |<=| b ==> f contl x) ==>
(?x. a |<=| x /\ x |<=| b /\ (f x = y))
- IVT2
-
|- !f a b y.
a |<=| b /\
(f b |<=| y /\ y |<=| f a) /\
(!x. a |<=| x /\ x |<=| b ==> f contl x) ==>
(?x. a |<=| x /\ x |<=| b /\ (f x = y))
- DIFF_CONST
-
|- !k x. ((\x. k) diffl & 0) x
- DIFF_ADD
-
|- !f g l m x.
(f diffl l) x /\ (g diffl m) x ==> ((\x. f x |+| g x) diffl (l |+| m)) x
- DIFF_MUL
-
|- !f g l m x.
(f diffl l) x /\ (g diffl m) x ==>
((\x. f x |*| g x) diffl (l |*| g x |+| m |*| f x)) x
- DIFF_CMUL
-
|- !f c l x. (f diffl l) x ==> ((\x. c |*| f x) diffl (c |*| l)) x
- DIFF_NEG
-
|- !f l x. (f diffl l) x ==> ((\x. -- (f x)) diffl -- l) x
- DIFF_SUB
-
|- !f g l m x.
(f diffl l) x /\ (g diffl m) x ==> ((\x. f x |-| g x) diffl (l |-| m)) x
- DIFF_CHAIN
-
|- !f g l m x.
(f diffl l) (g x) /\ (g diffl m) x ==> ((\x. f (g x)) diffl (l |*| m)) x
- DIFF_X
-
|- !x. ((\x. x) diffl & 1) x
- DIFF_POW
-
|- !n x. ((\x. x pow n) diffl (& n |*| x pow (n - 1))) x
- DIFF_XM1
-
|- !x. ~(x = & 0) ==> ((\x. inv x) diffl -- (inv x pow 2)) x
- DIFF_INV
-
|- !f l x.
(f diffl l) x /\ ~(f x = & 0) ==>
((\x. inv (f x)) diffl -- (l / f x pow 2)) x
- DIFF_DIV
-
|- !f g l m x.
(f diffl l) x /\ (g diffl m) x /\ ~(g x = & 0) ==>
((\x. f x / g x) diffl ((l |*| g x |-| m |*| f x) / g x pow 2)) x
- DIFF_SUM
-
|- !f f' m n x.
(!r. m <= r /\ r < m + n ==> ((\x. f r x) diffl f' r x) x) ==>
((\x. sum (m,n) (\n. f n x)) diffl sum (m,n) (\r. f' r x)) x
- CONT_BOUNDED
-
|- !f a b.
a |<=| b /\ (!x. a |<=| x /\ x |<=| b ==> f contl x) ==>
(?M. !x. a |<=| x /\ x |<=| b ==> f x |<=| M)
- CONT_HASSUP
-
|- !f a b.
a |<=| b /\ (!x. a |<=| x /\ x |<=| b ==> f contl x) ==>
(?M.
(!x. a |<=| x /\ x |<=| b ==> f x |<=| M) /\
(!N. N |<| M ==> (?x. a |<=| x /\ x |<=| b /\ N |<| f x)))
- CONT_ATTAINS
-
|- !f a b.
a |<=| b /\ (!x. a |<=| x /\ x |<=| b ==> f contl x) ==>
(?M.
(!x. a |<=| x /\ x |<=| b ==> f x |<=| M) /\
(?x. a |<=| x /\ x |<=| b /\ (f x = M)))
- CONT_ATTAINS2
-
|- !f a b.
a |<=| b /\ (!x. a |<=| x /\ x |<=| b ==> f contl x) ==>
(?M.
(!x. a |<=| x /\ x |<=| b ==> M |<=| f x) /\
(?x. a |<=| x /\ x |<=| b /\ (f x = M)))
- CONT_ATTAINS_ALL
-
|- !f a b.
a |<=| b /\ (!x. a |<=| x /\ x |<=| b ==> f contl x) ==>
(?L M.
L |<=| M /\
(!y.
L |<=| y /\ y |<=| M ==> (?x. a |<=| x /\ x |<=| b /\ (f x = y))) /\
(!x. a |<=| x /\ x |<=| b ==> L |<=| f x /\ f x |<=| M))
- DIFF_LINC
-
|- !f x l.
(f diffl l) x /\ & 0 |<| l ==>
(?d. & 0 |<| d /\ (!h. & 0 |<| h /\ h |<| d ==> f x |<| f (x |+| h)))
- DIFF_LDEC
-
|- !f x l.
(f diffl l) x /\ l |<| & 0 ==>
(?d. & 0 |<| d /\ (!h. & 0 |<| h /\ h |<| d ==> f x |<| f (x |-| h)))
- DIFF_LMAX
-
|- !f x l.
(f diffl l) x /\
(?d. & 0 |<| d /\ (!y. abs (x |-| y) |<| d ==> f y |<=| f x)) ==>
(l = & 0)
- DIFF_LMIN
-
|- !f x l.
(f diffl l) x /\
(?d. & 0 |<| d /\ (!y. abs (x |-| y) |<| d ==> f x |<=| f y)) ==>
(l = & 0)
- DIFF_LCONST
-
|- !f x l.
(f diffl l) x /\
(?d. & 0 |<| d /\ (!y. abs (x |-| y) |<| d ==> (f y = f x))) ==>
(l = & 0)
- INTERVAL_LEMMA
-
|- !a b x.
a |<| x /\ x |<| b ==>
(?d. & 0 |<| d /\ (!y. abs (x |-| y) |<| d ==> a |<=| y /\ y |<=| b))
- ROLLE
-
|- !f a b.
a |<| b /\
(f a = f b) /\
(!x. a |<=| x /\ x |<=| b ==> f contl x) /\
(!x. a |<| x /\ x |<| b ==> f differentiable x) ==>
(?z. a |<| z /\ z |<| b /\ (f diffl & 0) z)
- MVT_LEMMA
-
|- !f a b.
(\x. f x |-| ((f b |-| f a) / (b |-| a)) |*| x) a =
(\x. f x |-| ((f b |-| f a) / (b |-| a)) |*| x) b
- MVT
-
|- !f a b.
a |<| b /\
(!x. a |<=| x /\ x |<=| b ==> f contl x) /\
(!x. a |<| x /\ x |<| b ==> f differentiable x) ==>
(?l z.
a |<| z /\ z |<| b /\ (f diffl l) z /\ (f b |-| f a = (b |-| a) |*| l))
- DIFF_ISCONST_END
-
|- !f a b.
a |<| b /\
(!x. a |<=| x /\ x |<=| b ==> f contl x) /\
(!x. a |<| x /\ x |<| b ==> (f diffl & 0) x) ==>
(f b = f a)
- DIFF_ISCONST
-
|- !f a b.
a |<| b /\
(!x. a |<=| x /\ x |<=| b ==> f contl x) /\
(!x. a |<| x /\ x |<| b ==> (f diffl & 0) x) ==>
(!x. a |<=| x /\ x |<=| b ==> (f x = f a))
- DIFF_ISCONST_ALL
-
|- !f. (!x. (f diffl & 0) x) ==> (!x y. f x = f y)
- INTERVAL_ABS
-
|- !x z d. x |-| d |<=| z /\ z |<=| x |+| d = abs (z |-| x) |<=| d
- CONT_INJ_LEMMA
-
|- !f g x d.
& 0 |<| d /\
(!z. abs (z |-| x) |<=| d ==> (g (f z) = z)) /\
(!z. abs (z |-| x) |<=| d ==> f contl z) ==>
~(!z. abs (z |-| x) |<=| d ==> f z |<=| f x)
- CONT_INJ_LEMMA2
-
|- !f g x d.
& 0 |<| d /\
(!z. abs (z |-| x) |<=| d ==> (g (f z) = z)) /\
(!z. abs (z |-| x) |<=| d ==> f contl z) ==>
~(!z. abs (z |-| x) |<=| d ==> f x |<=| f z)
- CONT_INJ_RANGE
-
|- !f g x d.
& 0 |<| d /\
(!z. abs (z |-| x) |<=| d ==> (g (f z) = z)) /\
(!z. abs (z |-| x) |<=| d ==> f contl z) ==>
(?e.
& 0 |<| e /\
(!y.
abs (y |-| f x) |<=| e ==> (?z. abs (z |-| x) |<=| d /\ (f z = y))))
- CONT_INVERSE
-
|- !f g x d.
& 0 |<| d /\
(!z. abs (z |-| x) |<=| d ==> (g (f z) = z)) /\
(!z. abs (z |-| x) |<=| d ==> f contl z) ==>
g contl f x
- DIFF_INVERSE
-
|- !f g l x d.
& 0 |<| d /\
(!z. abs (z |-| x) |<=| d ==> (g (f z) = z)) /\
(!z. abs (z |-| x) |<=| d ==> f contl z) /\
(f diffl l) x /\
~(l = & 0) ==>
(g diffl inv l) (f x)
- INTERVAL_CLEMMA
-
|- !a b x.
a |<| x /\ x |<| b ==>
(?d. & 0 |<| d /\ (!y. abs (y |-| x) |<=| d ==> a |<| y /\ y |<| b))
- DIFF_INVERSE_OPEN
-
|- !f g l a x b.
a |<| x /\
x |<| b /\
(!z. a |<| z /\ z |<| b ==> (g (f z) = z) /\ f contl z) /\
(f diffl l) x /\
~(l = & 0) ==>
(g diffl inv l) (f x)