Theory: LIM

Parents


Type constants


Term constants


Axioms


Definitions

tends_real_real
|- !f l x0. (f -> l) x0 = (f tends l) (mtop mr1,tendsto (mr1,x0))
diffl
|- !f l x. (f diffl l) x = ((\h. (f (x |+| h) |-| f x) / h) -> l) (& 0)
contl
|- !f x. f contl x = ((\h. f (x |+| h)) -> f x) (& 0)
differentiable
|- !f x. f differentiable x = (?l. (f diffl l) x)

Theorems

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)