Theory: TRANSC

Parents


Type constants


Term constants


Axioms


Definitions

exp
|- !x. exp x = suminf (\n. (\n. inv (& (FACT n))) n |*| x pow n)
sin
|- !x.
     sin x =
     suminf
       (\n.
         (\n. (EVEN n) => (& 0) | (-- (& 1) pow ((n - 1) DIV 2) / & (FACT n)))
           n |*|
         x pow n)
cos
|- !x.
     cos x =
     suminf
       (\n.
         (\n. (EVEN n) => (-- (& 1) pow (n DIV 2) / & (FACT n)) | (& 0)) n |*|
         x pow n)
ln
|- !x. ln x = (@u. exp u = x)
root
|- !n x. root n x = (@u. (& 0 |<| x ==> & 0 |<| u) /\ (u pow n = x))
sqrt
|- !x. sqrt x = root 2 x
pi
|- pi = & 2 |*| (@x. & 0 |<=| x /\ x |<=| & 2 /\ (cos x = & 0))
tan
|- !x. tan x = sin x / cos x
asn
|- !y. asn y = (@x. -- (pi / & 2) |<=| x /\ x |<=| pi / & 2 /\ (sin x = y))
acs
|- !y. acs y = (@x. & 0 |<=| x /\ x |<=| pi /\ (cos x = y))
atn
|- !y. atn y = (@x. -- (pi / & 2) |<| x /\ x |<| pi / & 2 /\ (tan x = y))

Theorems

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