Theory "ieee"

Signature

Definitions

expwidth

|- ∀ew fw. expwidth (ew,fw) = ew

fracwidth

|- ∀ew fw. fracwidth (ew,fw) = fw

wordlength

|- ∀X. wordlength X = expwidth X + fracwidth X + 1

emax

|- ∀X. emax X = 2 ** expwidth X − 1

bias

|- ∀X. bias X = 2 ** (expwidth X − 1) − 1

is_single

|- ∀X. is_single X ⇔ (expwidth X = 8) ∧ (wordlength X = 32)

is_double

|- ∀X. is_double X ⇔ (expwidth X = 11) ∧ (wordlength X = 64)

is_single_extended

|- ∀X. is_single_extended X ⇔ expwidth X ≥ 11 ∧ wordlength X ≥ 43

is_double_extended

|- ∀X. is_double_extended X ⇔ expwidth X ≥ 15 ∧ wordlength X ≥ 79

sign

|- ∀s e f. sign (s,e,f) = s

exponent

|- ∀s e f. exponent (s,e,f) = e

fraction

|- ∀s e f. fraction (s,e,f) = f

is_nan

|- ∀X a. is_nan X a ⇔ (exponent a = emax X) ∧ fraction a ≠ 0

is_infinity

|- ∀X a. is_infinity X a ⇔ (exponent a = emax X) ∧ (fraction a = 0)

is_normal

|- ∀X a. is_normal X a ⇔ 0 < exponent a ∧ exponent a < emax X

is_denormal

|- ∀X a. is_denormal X a ⇔ (exponent a = 0) ∧ fraction a ≠ 0

is_zero

|- ∀X a. is_zero X a ⇔ (exponent a = 0) ∧ (fraction a = 0)

is_valid

|- ∀X s e f.
     is_valid X (s,e,f) ⇔
     s < SUC (SUC 0) ∧ e < 2 ** expwidth X ∧ f < 2 ** fracwidth X

is_finite

|- ∀X a.
     is_finite X a ⇔
     is_valid X a ∧ (is_normal X a ∨ is_denormal X a ∨ is_zero X a)

plus_infinity

|- ∀X. plus_infinity X = (0,emax X,0)

minus_infinity

|- ∀X. minus_infinity X = (1,emax X,0)

plus_zero

|- ∀X. plus_zero X = (0,0,0)

minus_zero

|- ∀X. minus_zero X = (1,0,0)

topfloat

|- ∀X. topfloat X = (0,emax X − 1,2 ** fracwidth X − 1)

bottomfloat

|- ∀X. bottomfloat X = (1,emax X − 1,2 ** fracwidth X − 1)

minus

|- ∀X a. minus X a = (1 − sign a,exponent a,fraction a)

encoding

|- ∀X s e f.
     encoding X (s,e,f) =
     s * 2 ** (wordlength X − 1) + e * 2 ** fracwidth X + f

valof

|- ∀X s e f.
     valof X (s,e,f) =
     if e = 0 then -1 pow s * (2 / 2 pow bias X) * (&f / 2 pow fracwidth X)
     else -1 pow s * (2 pow e / 2 pow bias X) * (1 + &f / 2 pow fracwidth X)

largest

|- ∀X.
     largest X =
     2 pow (emax X − 1) / 2 pow bias X * (2 − inv (2 pow fracwidth X))

threshold

|- ∀X.
     threshold X =
     2 pow (emax X − 1) / 2 pow bias X * (2 − inv (2 pow SUC (fracwidth X)))

ulp

|- ∀X a. ulp X a = valof X (0,exponent a,1) − valof X (0,exponent a,0)

roundmode_TY_DEF

|- ∃rep. TYPE_DEFINITION (λn. n < 4) rep

roundmode_BIJ

|- (∀a. num2roundmode (roundmode2num a) = a) ∧
   ∀r. (λn. n < 4) r ⇔ (roundmode2num (num2roundmode r) = r)

roundmode_size_def

|- ∀x. roundmode_size x = 0

roundmode_CASE

|- ∀x v0 v1 v2 v3.
     (case x of
        To_nearest => v0
      | float_To_zero => v1
      | To_pinfinity => v2
      | To_ninfinity => v3) =
     (λm.
        if m < 1 then v0 else if m < 2 then v1 else if m = 2 then v2 else v3)
       (roundmode2num x)

is_closest

|- ∀v s x a.
     is_closest v s x a ⇔ a ∈ s ∧ ∀b. b ∈ s ⇒ abs (v a − x) ≤ abs (v b − x)

closest

|- ∀v p s x.
     closest v p s x =
     @a. is_closest v s x a ∧ ((∃b. is_closest v s x b ∧ p b) ⇒ p a)

round_def

|- (∀X x.
      round X To_nearest x =
      if x ≤ -threshold X then minus_infinity X
      else if x ≥ threshold X then plus_infinity X
      else closest (valof X) (λa. EVEN (fraction a)) {a | is_finite X a} x) ∧
   (∀X x.
      round X float_To_zero x =
      if x < -largest X then bottomfloat X
      else if x > largest X then topfloat X
      else
        closest (valof X) (λx. T)
          {a | is_finite X a ∧ abs (valof X a) ≤ abs x} x) ∧
   (∀X x.
      round X To_pinfinity x =
      if x < -largest X then bottomfloat X
      else if x > largest X then plus_infinity X
      else closest (valof X) (λx. T) {a | is_finite X a ∧ valof X a ≥ x} x) ∧
   ∀X x.
     round X To_ninfinity x =
     if x < -largest X then minus_infinity X
     else if x > largest X then topfloat X
     else closest (valof X) (λx. T) {a | is_finite X a ∧ valof X a ≤ x} x

is_integral

|- ∀X a. is_integral X a ⇔ is_finite X a ∧ ∃n. abs (valof X a) = &n

intround_def

|- (∀X x.
      intround X To_nearest x =
      if x ≤ -threshold X then minus_infinity X
      else if x ≥ threshold X then plus_infinity X
      else
        closest (valof X) (λa. ∃n. EVEN n ∧ (abs (valof X a) = &n))
          {a | is_integral X a} x) ∧
   (∀X x.
      intround X float_To_zero x =
      if x < -largest X then bottomfloat X
      else if x > largest X then topfloat X
      else
        closest (valof X) (λx. T)
          {a | is_integral X a ∧ abs (valof X a) ≤ abs x} x) ∧
   (∀X x.
      intround X To_pinfinity x =
      if x < -largest X then bottomfloat X
      else if x > largest X then plus_infinity X
      else
        closest (valof X) (λx. T) {a | is_integral X a ∧ valof X a ≥ x} x) ∧
   ∀X x.
     intround X To_ninfinity x =
     if x < -largest X then minus_infinity X
     else if x > largest X then topfloat X
     else closest (valof X) (λx. T) {a | is_integral X a ∧ valof X a ≤ x} x

some_nan

|- ∀X. some_nan X = @a. is_nan X a

zerosign

|- ∀X s a.
     zerosign X s a =
     if is_zero X a then if s = 0 then plus_zero X else minus_zero X else a

rem

|- ∀x y.
     x rem y =
     (let n =
            closest I (λx. ∃n. EVEN n ∧ (abs x = &n)) {x | ∃n. abs x = &n}
              (x / y)
      in
        x − n * y)

fintrnd

|- ∀X m a.
     fintrnd X m a =
     if is_nan X a then some_nan X
     else if is_infinity X a then a
     else zerosign X (sign a) (intround X m (valof X a))

fadd

|- ∀X m a b.
     fadd X m a b =
     if
       is_nan X a ∨ is_nan X b ∨
       is_infinity X a ∧ is_infinity X b ∧ sign a ≠ sign b
     then
       some_nan X
     else if is_infinity X a then a
     else if is_infinity X b then b
     else
       zerosign X
         (if is_zero X a ∧ is_zero X b ∧ (sign a = sign b) then sign a
          else if m = To_ninfinity then 1
          else 0) (round X m (valof X a + valof X b))

fsub

|- ∀X m a b.
     fsub X m a b =
     if
       is_nan X a ∨ is_nan X b ∨
       is_infinity X a ∧ is_infinity X b ∧ (sign a = sign b)
     then
       some_nan X
     else if is_infinity X a then a
     else if is_infinity X b then minus X b
     else
       zerosign X
         (if is_zero X a ∧ is_zero X b ∧ sign a ≠ sign b then sign a
          else if m = To_ninfinity then 1
          else 0) (round X m (valof X a − valof X b))

fmul

|- ∀X m a b.
     fmul X m a b =
     if
       is_nan X a ∨ is_nan X b ∨ is_zero X a ∧ is_infinity X b ∨
       is_infinity X a ∧ is_zero X b
     then
       some_nan X
     else if is_infinity X a ∨ is_infinity X b then
       if sign a = sign b then plus_infinity X else minus_infinity X
     else
       zerosign X (if sign a = sign b then 0 else 1)
         (round X m (valof X a * valof X b))

fdiv

|- ∀X m a b.
     fdiv X m a b =
     if
       is_nan X a ∨ is_nan X b ∨ is_zero X a ∧ is_zero X b ∨
       is_infinity X a ∧ is_infinity X b
     then
       some_nan X
     else if is_infinity X a ∨ is_zero X b then
       if sign a = sign b then plus_infinity X else minus_infinity X
     else if is_infinity X b then
       if sign a = sign b then plus_zero X else minus_zero X
     else
       zerosign X (if sign a = sign b then 0 else 1)
         (round X m (valof X a / valof X b))

fsqrt

|- ∀X m a.
     fsqrt X m a =
     if is_nan X a then some_nan X
     else if is_zero X a ∨ is_infinity X a ∧ (sign a = 0) then a
     else if sign a = 1 then some_nan X
     else zerosign X (sign a) (round X m (sqrt (valof X a)))

frem

|- ∀X m a b.
     frem X m a b =
     if is_nan X a ∨ is_nan X b ∨ is_infinity X a ∨ is_zero X b then
       some_nan X
     else if is_infinity X b then a
     else zerosign X (sign a) (round X m (valof X a rem valof X b))

fneg

|- ∀X m a. fneg X m a = (1 − sign a,exponent a,fraction a)

ccode_TY_DEF

|- ∃rep. TYPE_DEFINITION (λn. n < 4) rep

ccode_BIJ

|- (∀a. num2ccode (ccode2num a) = a) ∧
   ∀r. (λn. n < 4) r ⇔ (ccode2num (num2ccode r) = r)

ccode_size_def

|- ∀x. ccode_size x = 0

ccode_CASE

|- ∀x v0 v1 v2 v3.
     (case x of Gt => v0 | Lt => v1 | Eq => v2 | Un => v3) =
     (λm.
        if m < 1 then v0 else if m < 2 then v1 else if m = 2 then v2 else v3)
       (ccode2num x)

fcompare

|- ∀X a b.
     fcompare X a b =
     if is_nan X a ∨ is_nan X b then Un
     else if is_infinity X a ∧ (sign a = 1) then
       if is_infinity X b ∧ (sign b = 1) then Eq else Lt
     else if is_infinity X a ∧ (sign a = 0) then
       if is_infinity X b ∧ (sign b = 0) then Eq else Gt
     else if is_infinity X b ∧ (sign b = 1) then Gt
     else if is_infinity X b ∧ (sign b = 0) then Lt
     else if valof X a < valof X b then Lt
     else if valof X a = valof X b then Eq
     else Gt

flt

|- ∀X a b. flt X a b ⇔ (fcompare X a b = Lt)

fle

|- ∀X a b. fle X a b ⇔ (fcompare X a b = Lt) ∨ (fcompare X a b = Eq)

fgt

|- ∀X a b. fgt X a b ⇔ (fcompare X a b = Gt)

fge

|- ∀X a b. fge X a b ⇔ (fcompare X a b = Gt) ∨ (fcompare X a b = Eq)

feq

|- ∀X a b. feq X a b ⇔ (fcompare X a b = Eq)

float_format

|- float_format = (8,23)

float_TY_DEF

|- ∃rep. TYPE_DEFINITION (is_valid float_format) rep

float_tybij

|- (∀a. float (defloat a) = a) ∧
   ∀r. is_valid float_format r ⇔ (defloat (float r) = r)

Val

|- ∀a. Val a = valof float_format (defloat a)

Float

|- ∀x. Float x = float (round float_format To_nearest x)

Sign

|- ∀a. Sign a = sign (defloat a)

Exponent

|- ∀a. Exponent a = exponent (defloat a)

Fraction

|- ∀a. Fraction a = fraction (defloat a)

Ulp

|- ∀a. Ulp a = ulp float_format (defloat a)

Isnan

|- ∀a. Isnan a ⇔ is_nan float_format (defloat a)

Infinity

|- ∀a. Infinity a ⇔ is_infinity float_format (defloat a)

Isnormal

|- ∀a. Isnormal a ⇔ is_normal float_format (defloat a)

Isdenormal

|- ∀a. Isdenormal a ⇔ is_denormal float_format (defloat a)

Iszero

|- ∀a. Iszero a ⇔ is_zero float_format (defloat a)

Finite

|- ∀a. Finite a ⇔ Isnormal a ∨ Isdenormal a ∨ Iszero a

Isintegral

|- ∀a. Isintegral a ⇔ is_integral float_format (defloat a)

Plus_zero

|- Plus_zero = float (plus_zero float_format)

Minus_zero

|- Minus_zero = float (minus_zero float_format)

Minus_infinity

|- Minus_infinity = float (minus_infinity float_format)

Plus_infinity

|- Plus_infinity = float (plus_infinity float_format)

float_add

|- ∀a b. a + b = float (fadd float_format To_nearest (defloat a) (defloat b))

float_sub

|- ∀a b. a − b = float (fsub float_format To_nearest (defloat a) (defloat b))

float_mul

|- ∀a b. a * b = float (fmul float_format To_nearest (defloat a) (defloat b))

float_div

|- ∀a b. a / b = float (fdiv float_format To_nearest (defloat a) (defloat b))

float_rem

|- ∀a b.
     a float_rem b =
     float (frem float_format To_nearest (defloat a) (defloat b))

float_sqrt

|- ∀a. float_sqrt a = float (fsqrt float_format To_nearest (defloat a))

ROUNDFLOAT

|- ∀a. ROUNDFLOAT a = float (fintrnd float_format To_nearest (defloat a))

float_lt

|- ∀a b. a < b ⇔ flt float_format (defloat a) (defloat b)

float_le

|- ∀a b. a ≤ b ⇔ fle float_format (defloat a) (defloat b)

float_gt

|- ∀a b. a > b ⇔ fgt float_format (defloat a) (defloat b)

float_ge

|- ∀a b. a ≥ b ⇔ fge float_format (defloat a) (defloat b)

float_eq

|- ∀a b. a == b ⇔ feq float_format (defloat a) (defloat b)

float_neg

|- ∀a. ¬a = float (fneg float_format To_nearest (defloat a))

float_abs

|- ∀a. float_abs a = if a ≥ Plus_zero then a else ¬a

Theorems

num2roundmode_roundmode2num

|- ∀a. num2roundmode (roundmode2num a) = a

roundmode2num_num2roundmode

|- ∀r. r < 4 ⇔ (roundmode2num (num2roundmode r) = r)

num2roundmode_11

|- ∀r r'. r < 4 ⇒ r' < 4 ⇒ ((num2roundmode r = num2roundmode r') ⇔ (r = r'))

roundmode2num_11

|- ∀a a'. (roundmode2num a = roundmode2num a') ⇔ (a = a')

num2roundmode_ONTO

|- ∀a. ∃r. (a = num2roundmode r) ∧ r < 4

roundmode2num_ONTO

|- ∀r. r < 4 ⇔ ∃a. r = roundmode2num a

num2roundmode_thm

|- (num2roundmode 0 = To_nearest) ∧ (num2roundmode 1 = float_To_zero) ∧
   (num2roundmode 2 = To_pinfinity) ∧ (num2roundmode 3 = To_ninfinity)

roundmode2num_thm

|- (roundmode2num To_nearest = 0) ∧ (roundmode2num float_To_zero = 1) ∧
   (roundmode2num To_pinfinity = 2) ∧ (roundmode2num To_ninfinity = 3)

roundmode_EQ_roundmode

|- ∀a a'. (a = a') ⇔ (roundmode2num a = roundmode2num a')

roundmode_case_def

|- (∀v0 v1 v2 v3.
      (case To_nearest of
         To_nearest => v0
       | float_To_zero => v1
       | To_pinfinity => v2
       | To_ninfinity => v3) =
      v0) ∧
   (∀v0 v1 v2 v3.
      (case float_To_zero of
         To_nearest => v0
       | float_To_zero => v1
       | To_pinfinity => v2
       | To_ninfinity => v3) =
      v1) ∧
   (∀v0 v1 v2 v3.
      (case To_pinfinity of
         To_nearest => v0
       | float_To_zero => v1
       | To_pinfinity => v2
       | To_ninfinity => v3) =
      v2) ∧
   ∀v0 v1 v2 v3.
     (case To_ninfinity of
        To_nearest => v0
      | float_To_zero => v1
      | To_pinfinity => v2
      | To_ninfinity => v3) =
     v3

datatype_roundmode

|- DATATYPE (roundmode To_nearest float_To_zero To_pinfinity To_ninfinity)

roundmode_distinct

|- To_nearest ≠ float_To_zero ∧ To_nearest ≠ To_pinfinity ∧
   To_nearest ≠ To_ninfinity ∧ float_To_zero ≠ To_pinfinity ∧
   float_To_zero ≠ To_ninfinity ∧ To_pinfinity ≠ To_ninfinity

roundmode_case_cong

|- ∀M M' v0 v1 v2 v3.
     (M = M') ∧ ((M' = To_nearest) ⇒ (v0 = v0')) ∧
     ((M' = float_To_zero) ⇒ (v1 = v1')) ∧
     ((M' = To_pinfinity) ⇒ (v2 = v2')) ∧ ((M' = To_ninfinity) ⇒ (v3 = v3')) ⇒
     ((case M of
         To_nearest => v0
       | float_To_zero => v1
       | To_pinfinity => v2
       | To_ninfinity => v3) =
      case M' of
        To_nearest => v0'
      | float_To_zero => v1'
      | To_pinfinity => v2'
      | To_ninfinity => v3')

roundmode_nchotomy

|- ∀a.
     (a = To_nearest) ∨ (a = float_To_zero) ∨ (a = To_pinfinity) ∨
     (a = To_ninfinity)

roundmode_Axiom

|- ∀x0 x1 x2 x3.
     ∃f.
       (f To_nearest = x0) ∧ (f float_To_zero = x1) ∧ (f To_pinfinity = x2) ∧
       (f To_ninfinity = x3)

roundmode_induction

|- ∀P.
     P To_nearest ∧ P To_ninfinity ∧ P To_pinfinity ∧ P float_To_zero ⇒
     ∀a. P a

num2ccode_ccode2num

|- ∀a. num2ccode (ccode2num a) = a

ccode2num_num2ccode

|- ∀r. r < 4 ⇔ (ccode2num (num2ccode r) = r)

num2ccode_11

|- ∀r r'. r < 4 ⇒ r' < 4 ⇒ ((num2ccode r = num2ccode r') ⇔ (r = r'))

ccode2num_11

|- ∀a a'. (ccode2num a = ccode2num a') ⇔ (a = a')

num2ccode_ONTO

|- ∀a. ∃r. (a = num2ccode r) ∧ r < 4

ccode2num_ONTO

|- ∀r. r < 4 ⇔ ∃a. r = ccode2num a

num2ccode_thm

|- (num2ccode 0 = Gt) ∧ (num2ccode 1 = Lt) ∧ (num2ccode 2 = Eq) ∧
   (num2ccode 3 = Un)

ccode2num_thm

|- (ccode2num Gt = 0) ∧ (ccode2num Lt = 1) ∧ (ccode2num Eq = 2) ∧
   (ccode2num Un = 3)

ccode_EQ_ccode

|- ∀a a'. (a = a') ⇔ (ccode2num a = ccode2num a')

ccode_case_def

|- (∀v0 v1 v2 v3.
      (case Gt of Gt => v0 | Lt => v1 | Eq => v2 | Un => v3) = v0) ∧
   (∀v0 v1 v2 v3.
      (case Lt of Gt => v0 | Lt => v1 | Eq => v2 | Un => v3) = v1) ∧
   (∀v0 v1 v2 v3.
      (case Eq of Gt => v0 | Lt => v1 | Eq => v2 | Un => v3) = v2) ∧
   ∀v0 v1 v2 v3. (case Un of Gt => v0 | Lt => v1 | Eq => v2 | Un => v3) = v3

datatype_ccode

|- DATATYPE (ccode Gt Lt Eq Un)

ccode_distinct

|- Gt ≠ Lt ∧ Gt ≠ Eq ∧ Gt ≠ Un ∧ Lt ≠ Eq ∧ Lt ≠ Un ∧ Eq ≠ Un

ccode_case_cong

|- ∀M M' v0 v1 v2 v3.
     (M = M') ∧ ((M' = Gt) ⇒ (v0 = v0')) ∧ ((M' = Lt) ⇒ (v1 = v1')) ∧
     ((M' = Eq) ⇒ (v2 = v2')) ∧ ((M' = Un) ⇒ (v3 = v3')) ⇒
     ((case M of Gt => v0 | Lt => v1 | Eq => v2 | Un => v3) =
      case M' of Gt => v0' | Lt => v1' | Eq => v2' | Un => v3')

ccode_nchotomy

|- ∀a. (a = Gt) ∨ (a = Lt) ∨ (a = Eq) ∨ (a = Un)

ccode_Axiom

|- ∀x0 x1 x2 x3. ∃f. (f Gt = x0) ∧ (f Lt = x1) ∧ (f Eq = x2) ∧ (f Un = x3)

ccode_induction

|- ∀P. P Eq ∧ P Gt ∧ P Lt ∧ P Un ⇒ ∀a. P a