- EXP_CONVERGES
-
|- !x. (\n. (\n. inv (& (FACT n))) n |*| x pow n) sums exp x
- SIN_CONVERGES
-
|- !x.
(\n.
(\n. (EVEN n) => (& 0) | (-- (& 1) pow ((n - 1) DIV 2) / & (FACT n)))
n |*|
x pow n) sums
sin x
- COS_CONVERGES
-
|- !x.
(\n.
(\n. (EVEN n) => (-- (& 1) pow (n DIV 2) / & (FACT n)) | (& 0)) n |*|
x pow n) sums
cos x
- EXP_FDIFF
-
|- diffs (\n. inv (& (FACT n))) = (\n. inv (& (FACT n)))
- SIN_FDIFF
-
|- diffs
(\n. (EVEN n) => (& 0) | (-- (& 1) pow ((n - 1) DIV 2) / & (FACT n))) =
(\n. (EVEN n) => (-- (& 1) pow (n DIV 2) / & (FACT n)) | (& 0))
- COS_FDIFF
-
|- diffs (\n. (EVEN n) => (-- (& 1) pow (n DIV 2) / & (FACT n)) | (& 0)) =
(\n.
--
((\n. (EVEN n) => (& 0) | (-- (& 1) pow ((n - 1) DIV 2) / & (FACT n)))
n))
- SIN_NEGLEMMA
-
|- !x.
-- (sin x) =
suminf
(\n.
--
((\n.
(EVEN n) => (& 0) | (-- (& 1) pow ((n - 1) DIV 2) / & (FACT n)))
n |*|
x pow n))
- DIFF_EXP
-
|- !x. (exp diffl exp x) x
- DIFF_SIN
-
|- !x. (sin diffl cos x) x
- DIFF_COS
-
|- !x. (cos diffl -- (sin x)) x
- EXP_0
-
|- exp (& 0) = & 1
- EXP_LE_X
-
|- !x. & 0 |<=| x ==> & 1 |+| x |<=| exp x
- EXP_LT_1
-
|- !x. & 0 |<| x ==> & 1 |<| exp x
- EXP_ADD_MUL
-
|- !x y. exp (x |+| y) |*| exp (-- x) = exp y
- EXP_NEG_MUL
-
|- !x. exp x |*| exp (-- x) = & 1
- EXP_NEG_MUL2
-
|- !x. exp (-- x) |*| exp x = & 1
- EXP_NEG
-
|- !x. exp (-- x) = inv (exp x)
- EXP_ADD
-
|- !x y. exp (x |+| y) = exp x |*| exp y
- EXP_POS_LE
-
|- !x. & 0 |<=| exp x
- EXP_NZ
-
|- !x. ~(exp x = & 0)
- EXP_POS_LT
-
|- !x. & 0 |<| exp x
- EXP_N
-
|- !n x. exp (& n |*| x) = exp x pow n
- EXP_SUB
-
|- !x y. exp (x |-| y) = exp x / exp y
- EXP_MONO_IMP
-
|- !x y. x |<| y ==> exp x |<| exp y
- EXP_MONO_LT
-
|- !x y. exp x |<| exp y = x |<| y
- EXP_MONO_LE
-
|- !x y. exp x |<=| exp y = x |<=| y
- EXP_INJ
-
|- !x y. (exp x = exp y) = x = y
- EXP_TOTAL_LEMMA
-
|- !y. & 1 |<=| y ==> (?x. & 0 |<=| x /\ x |<=| y |-| & 1 /\ (exp x = y))
- EXP_TOTAL
-
|- !y. & 0 |<| y ==> (?x. exp x = y)
- LN_EXP
-
|- !x. ln (exp x) = x
- EXP_LN
-
|- !x. (exp (ln x) = x) = & 0 |<| x
- LN_MUL
-
|- !x y. & 0 |<| x /\ & 0 |<| y ==> (ln (x |*| y) = ln x |+| ln y)
- LN_INJ
-
|- !x y. & 0 |<| x /\ & 0 |<| y ==> ((ln x = ln y) = x = y)
- LN_1
-
|- ln (& 1) = & 0
- LN_INV
-
|- !x. & 0 |<| x ==> (ln (inv x) = -- (ln x))
- LN_DIV
-
|- !x y. & 0 |<| x /\ & 0 |<| y ==> (ln (x / y) = ln x |-| ln y)
- LN_MONO_LT
-
|- !x y. & 0 |<| x /\ & 0 |<| y ==> (ln x |<| ln y = x |<| y)
- LN_MONO_LE
-
|- !x y. & 0 |<| x /\ & 0 |<| y ==> (ln x |<=| ln y = x |<=| y)
- LN_POW
-
|- !n x. & 0 |<| x ==> (ln (x pow n) = & n |*| ln x)
- LN_LT_X
-
|- !x. & 0 |<| x ==> ln x |<| x
- ROOT_LT_LEMMA
-
|- !n x. & 0 |<| x ==> (exp (ln x / & (SUC n)) pow SUC n = x)
- ROOT_LN
-
|- !n x. & 0 |<| x ==> (root (SUC n) x = exp (ln x / & (SUC n)))
- ROOT_0
-
|- !n. root (SUC n) (& 0) = & 0
- ROOT_POS_LT
-
|- !n x. & 0 |<| x ==> & 0 |<| root (SUC n) x
- ROOT_POS
-
|- !n x. & 0 |<=| x ==> & 0 |<=| root (SUC n) x
- ROOT_1
-
|- !n. root (SUC n) (& 1) = & 1
- ROOT_POW_POS
-
|- !n x. & 0 |<=| x ==> (root (SUC n) x pow SUC n = x)
- SQRT_0
-
|- sqrt (& 0) = & 0
- SQRT_1
-
|- sqrt (& 1) = & 1
- SQRT_POW2
-
|- !x. (sqrt x pow 2 = x) = & 0 |<=| x
- POW_ROOT_POS
-
|- !n x. & 0 |<=| x ==> (root (SUC n) (x pow SUC n) = x)
- SQRT_EQ
-
|- !x y. (x pow 2 = y) /\ & 0 |<=| x ==> (x = sqrt y)
- SIN_0
-
|- sin (& 0) = & 0
- COS_0
-
|- cos (& 0) = & 1
- SIN_CIRCLE
-
|- !x. sin x pow 2 |+| cos x pow 2 = & 1
- SIN_BOUND
-
|- !x. abs (sin x) |<=| & 1
- SIN_BOUNDS
-
|- !x. -- (& 1) |<=| sin x /\ sin x |<=| & 1
- COS_BOUND
-
|- !x. abs (cos x) |<=| & 1
- COS_BOUNDS
-
|- !x. -- (& 1) |<=| cos x /\ cos x |<=| & 1
- SIN_COS_ADD
-
|- !x y.
(sin (x |+| y) |-| (sin x |*| cos y |+| cos x |*| sin y)) pow 2 |+|
(cos (x |+| y) |-| cos x |*| cos y |-| sin x |*| sin y) pow 2 =
& 0
- SIN_COS_NEG
-
|- !x. (sin (-- x) |+| sin x) pow 2 |+| (cos (-- x) |-| cos x) pow 2 = & 0
- SIN_ADD
-
|- !x y. sin (x |+| y) = sin x |*| cos y |+| cos x |*| sin y
- COS_ADD
-
|- !x y. cos (x |+| y) = cos x |*| cos y |-| sin x |*| sin y
- SIN_NEG
-
|- !x. sin (-- x) = -- (sin x)
- COS_NEG
-
|- !x. cos (-- x) = cos x
- SIN_DOUBLE
-
|- !x. sin (& 2 |*| x) = & 2 |*| sin x |*| cos x
- COS_DOUBLE
-
|- !x. cos (& 2 |*| x) = cos x pow 2 |-| sin x pow 2
- SIN_PAIRED
-
|- !x.
(\n. (-- (& 1) pow n / & (FACT (2 * n + 1))) |*| x pow (2 * n + 1)) sums
sin x
- SIN_POS
-
|- !x. & 0 |<| x /\ x |<| & 2 ==> & 0 |<| sin x
- COS_PAIRED
-
|- !x. (\n. (-- (& 1) pow n / & (FACT (2 * n))) |*| x pow (2 * n)) sums cos x
- COS_2
-
|- cos (& 2) |<| & 0
- COS_ISZERO
-
|- ?!x. & 0 |<=| x /\ x |<=| & 2 /\ (cos x = & 0)
- PI2
-
|- pi / & 2 = (@x. & 0 |<=| x /\ x |<=| & 2 /\ (cos x = & 0))
- COS_PI2
-
|- cos (pi / & 2) = & 0
- PI2_BOUNDS
-
|- & 0 |<| pi / & 2 /\ pi / & 2 |<| & 2
- PI_POS
-
|- & 0 |<| pi
- SIN_PI2
-
|- sin (pi / & 2) = & 1
- COS_PI
-
|- cos pi = -- (& 1)
- SIN_PI
-
|- sin pi = & 0
- SIN_COS
-
|- !x. sin x = cos (pi / & 2 |-| x)
- COS_SIN
-
|- !x. cos x = sin (pi / & 2 |-| x)
- SIN_PERIODIC_PI
-
|- !x. sin (x |+| pi) = -- (sin x)
- COS_PERIODIC_PI
-
|- !x. cos (x |+| pi) = -- (cos x)
- SIN_PERIODIC
-
|- !x. sin (x |+| & 2 |*| pi) = sin x
- COS_PERIODIC
-
|- !x. cos (x |+| & 2 |*| pi) = cos x
- COS_NPI
-
|- !n. cos (& n |*| pi) = -- (& 1) pow n
- SIN_NPI
-
|- !n. sin (& n |*| pi) = & 0
- SIN_POS_PI2
-
|- !x. & 0 |<| x /\ x |<| pi / & 2 ==> & 0 |<| sin x
- COS_POS_PI2
-
|- !x. & 0 |<| x /\ x |<| pi / & 2 ==> & 0 |<| cos x
- COS_POS_PI
-
|- !x. -- (pi / & 2) |<| x /\ x |<| pi / & 2 ==> & 0 |<| cos x
- SIN_POS_PI
-
|- !x. & 0 |<| x /\ x |<| pi ==> & 0 |<| sin x
- COS_POS_PI2_LE
-
|- !x. & 0 |<=| x /\ x |<=| pi / & 2 ==> & 0 |<=| cos x
- COS_POS_PI_LE
-
|- !x. -- (pi / & 2) |<=| x /\ x |<=| pi / & 2 ==> & 0 |<=| cos x
- SIN_POS_PI2_LE
-
|- !x. & 0 |<=| x /\ x |<=| pi / & 2 ==> & 0 |<=| sin x
- SIN_POS_PI_LE
-
|- !x. & 0 |<=| x /\ x |<=| pi ==> & 0 |<=| sin x
- COS_TOTAL
-
|- !y.
-- (& 1) |<=| y /\ y |<=| & 1 ==>
(?!x. & 0 |<=| x /\ x |<=| pi /\ (cos x = y))
- SIN_TOTAL
-
|- !y.
-- (& 1) |<=| y /\ y |<=| & 1 ==>
(?!x. -- (pi / & 2) |<=| x /\ x |<=| pi / & 2 /\ (sin x = y))
- COS_ZERO_LEMMA
-
|- !x.
& 0 |<=| x /\ (cos x = & 0) ==>
(?n. ~(EVEN n) /\ (x = & n |*| (pi / & 2)))
- SIN_ZERO_LEMMA
-
|- !x.
& 0 |<=| x /\ (sin x = & 0) ==> (?n. EVEN n /\ (x = & n |*| (pi / & 2)))
- COS_ZERO
-
|- !x.
(cos x = & 0) =
(?n. ~(EVEN n) /\ (x = & n |*| (pi / & 2))) \/
(?n. ~(EVEN n) /\ (x = -- (& n |*| (pi / & 2))))
- SIN_ZERO
-
|- !x.
(sin x = & 0) =
(?n. EVEN n /\ (x = & n |*| (pi / & 2))) \/
(?n. EVEN n /\ (x = -- (& n |*| (pi / & 2))))
- TAN_0
-
|- tan (& 0) = & 0
- TAN_PI
-
|- tan pi = & 0
- TAN_NPI
-
|- !n. tan (& n |*| pi) = & 0
- TAN_NEG
-
|- !x. tan (-- x) = -- (tan x)
- TAN_PERIODIC
-
|- !x. tan (x |+| & 2 |*| pi) = tan x
- TAN_ADD
-
|- !x y.
~(cos x = & 0) /\ ~(cos y = & 0) /\ ~(cos (x |+| y) = & 0) ==>
(tan (x |+| y) = (tan x |+| tan y) / (& 1 |-| tan x |*| tan y))
- TAN_DOUBLE
-
|- !x.
~(cos x = & 0) /\ ~(cos (& 2 |*| x) = & 0) ==>
(tan (& 2 |*| x) = (& 2 |*| tan x) / (& 1 |-| tan x pow 2))
- TAN_POS_PI2
-
|- !x. & 0 |<| x /\ x |<| pi / & 2 ==> & 0 |<| tan x
- DIFF_TAN
-
|- !x. ~(cos x = & 0) ==> (tan diffl inv (cos x pow 2)) x
- TAN_TOTAL_LEMMA
-
|- !y. & 0 |<| y ==> (?x. & 0 |<| x /\ x |<| pi / & 2 /\ y |<| tan x)
- TAN_TOTAL_POS
-
|- !y. & 0 |<=| y ==> (?x. & 0 |<=| x /\ x |<| pi / & 2 /\ (tan x = y))
- TAN_TOTAL
-
|- !y. ?!x. -- (pi / & 2) |<| x /\ x |<| pi / & 2 /\ (tan x = y)
- ASN
-
|- !y.
-- (& 1) |<=| y /\ y |<=| & 1 ==>
-- (pi / & 2) |<=| asn y /\ asn y |<=| pi / & 2 /\ (sin (asn y) = y)
- ASN_SIN
-
|- !y. -- (& 1) |<=| y /\ y |<=| & 1 ==> (sin (asn y) = y)
- ASN_BOUNDS
-
|- !y.
-- (& 1) |<=| y /\ y |<=| & 1 ==>
-- (pi / & 2) |<=| asn y /\ asn y |<=| pi / & 2
- ASN_BOUNDS_LT
-
|- !y.
-- (& 1) |<| y /\ y |<| & 1 ==>
-- (pi / & 2) |<| asn y /\ asn y |<| pi / & 2
- SIN_ASN
-
|- !x. -- (pi / & 2) |<=| x /\ x |<=| pi / & 2 ==> (asn (sin x) = x)
- ACS
-
|- !y.
-- (& 1) |<=| y /\ y |<=| & 1 ==>
& 0 |<=| acs y /\ acs y |<=| pi /\ (cos (acs y) = y)
- ACS_COS
-
|- !y. -- (& 1) |<=| y /\ y |<=| & 1 ==> (cos (acs y) = y)
- ACS_BOUNDS
-
|- !y. -- (& 1) |<=| y /\ y |<=| & 1 ==> & 0 |<=| acs y /\ acs y |<=| pi
- ACS_BOUNDS_LT
-
|- !y. -- (& 1) |<| y /\ y |<| & 1 ==> & 0 |<| acs y /\ acs y |<| pi
- COS_ACS
-
|- !x. & 0 |<=| x /\ x |<=| pi ==> (acs (cos x) = x)
- ATN
-
|- !y. -- (pi / & 2) |<| atn y /\ atn y |<| pi / & 2 /\ (tan (atn y) = y)
- ATN_TAN
-
|- !y. tan (atn y) = y
- ATN_BOUNDS
-
|- !y. -- (pi / & 2) |<| atn y /\ atn y |<| pi / & 2
- TAN_ATN
-
|- !x. -- (pi / & 2) |<| x /\ x |<| pi / & 2 ==> (atn (tan x) = x)
- TAN_SEC
-
|- !x. ~(cos x = & 0) ==> (& 1 |+| tan x pow 2 = inv (cos x) pow 2)
- SIN_COS_SQ
-
|- !x. & 0 |<=| x /\ x |<=| pi ==> (sin x = sqrt (& 1 |-| cos x pow 2))
- COS_SIN_SQ
-
|- !x.
-- (pi / & 2) |<=| x /\ x |<=| pi / & 2 ==>
(cos x = sqrt (& 1 |-| sin x pow 2))
- COS_ATN_NZ
-
|- !x. ~(cos (atn x) = & 0)
- COS_ASN_NZ
-
|- !x. -- (& 1) |<| x /\ x |<| & 1 ==> ~(cos (asn x) = & 0)
- SIN_ACS_NZ
-
|- !x. -- (& 1) |<| x /\ x |<| & 1 ==> ~(sin (acs x) = & 0)
- DIFF_LN
-
|- !x. & 0 |<| x ==> (ln diffl inv x) x
- DIFF_ASN_LEMMA
-
|- !x. -- (& 1) |<| x /\ x |<| & 1 ==> (asn diffl inv (cos (asn x))) x
- DIFF_ASN
-
|- !x.
-- (& 1) |<| x /\ x |<| & 1 ==>
(asn diffl inv (sqrt (& 1 |-| x pow 2))) x
- DIFF_ACS_LEMMA
-
|- !x. -- (& 1) |<| x /\ x |<| & 1 ==> (acs diffl inv (-- (sin (acs x)))) x
- DIFF_ACS
-
|- !x.
-- (& 1) |<| x /\ x |<| & 1 ==>
(acs diffl -- (inv (sqrt (& 1 |-| x pow 2)))) x
- DIFF_ATN
-
|- !x. (atn diffl inv (& 1 |+| x pow 2)) x