MathAction FunWithFunctions

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(1) -> <spad>

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)abbrev package MYEXP MyExp

MyExp(F: Algebra(Fraction(Integer))): with
    myexp: F -> F
    myexp: (F, NonNegativeInteger) -> F
 == add
    Map2 ==> ListFunctions2(NonNegativeInteger, F)
    myexp(x, n) ==
        a(i : NonNegativeInteger) : F == (1/(factorial i)) * x^i
        reduce(_+@((F, F) -> F), map(a, [i for i in 0..n])$Map2)

    myexp(x: F): F == myexp(x, 32)</spad>

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Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/8403278387071313463-25px001.spad
      using old system compiler.
   MYEXP abbreviates package MyExp 
------------------------------------------------------------------------
   initializing NRLIB MYEXP for MyExp 
   compiling into NRLIB MYEXP 
   processing macro definition Map2 ==> ListFunctions2(NonNegativeInteger,F) 
   compiling exported myexp : (F,NonNegativeInteger) -> F
Time: 0.04 SEC.

   compiling exported myexp : F -> F
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |MyExp| REDEFINED

;;;     ***       |MyExp| REDEFINED
Time: 0 SEC.

   Cumulative Statistics for Constructor MyExp
      Time: 0.04 seconds

   finalizing NRLIB MYEXP 
   Processing MyExp for Browser database:
--->-->MyExp(constructor): Not documented!!!!
--->-->MyExp((myexp (F F))): Not documented!!!!
--->-->MyExp((myexp (F F (NonNegativeInteger)))): Not documented!!!!
--->-->MyExp(): Missing Description
; compiling file "/var/aw/var/LatexWiki/MYEXP.NRLIB/MYEXP.lsp" (written 28 MAR 2024 07:05:26 PM):

; wrote /var/aw/var/LatexWiki/MYEXP.NRLIB/MYEXP.fasl
; compilation finished in 0:00:00.008
------------------------------------------------------------------------
   MyExp is now explicitly exposed in frame initial 
   MyExp will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/MYEXP.NRLIB/MYEXP

That's generic code and has to be compiled. Look at the types of the result.

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myexp(1.0)

$\label{eq1}2.7182818284 \<u> 590452354$

(1)

Type: Float

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myexp(1.0,4)

$\label{eq2}2.7083333333 \<u> 333333333$

(2)

Type: Float

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myexp(1)

$\label{eq3}\frac{28610550901763172837819811744646057}{105252334773477412 06688720486400000}$

(3)

Type: Fraction(Integer)

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myexp(matrix[[1,0],[0,2]] :: SQMATRIX(2,FRAC(INT))) :: SQMATRIX(2,Float)

$\label{eq4}\left[ \begin{array}{cc} {2.7182818284 \<u> 590452354}&{0.0} \ {0.0}&{7.3890560989 \</u> 306502273}$

(4)

Type: SquareMatrix?(2,Float)

And now we create a domain of functions and let it behave like an algebra over a field of rational numbers where multiplication is concatenation of functions and addition is... well, see below.

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)abbrev domain MYFUN MyFun

MyFun(F: Field): Algebra(Fraction(Integer)) with
    apply: (%, F) -> F
    coerce: (F -> F) -> %
  == add
    Rep := F -> F
    X ==> x pretend (F->F)
    Y ==> y pretend (F->F)
    zr(f: F): F == 0
    id(f: F): F == f
    plus(x: %, y: %): % == (f: F): F +-> X(f) + Y(f)
    times(x: %, y: %): % == (f: F): F +-> X(Y f)
    ltimes(l: Fraction(Integer), y: %): % == (f: F): F +-> l*Y(f)

    0: % == zr
    1: % == id
    (x: %) + (y: %) == plus(x,y)
    (x: %) * (y: %) == times(x,y)
    (l: Fraction(Integer)) * (y: %) == ltimes(l,y)
    coerce(x: %): OutputForm == message("myfun-object")$OutputForm
    apply(x: %, f: F): F == X f
    coerce(fun: F -> F): % == fun

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   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4752705461069213214-25px003.spad
      using old system compiler.
   MYFUN abbreviates domain MyFun 
------------------------------------------------------------------------
   initializing NRLIB MYFUN for MyFun 
   compiling into NRLIB MYFUN 
   processing macro definition X ==> pretend(x,F -> F) 
   processing macro definition Y ==> pretend(y,F -> F) 
   compiling local zr : F -> F
Time: 0 SEC.

   compiling local id : F -> F
      MYFUN;id is replaced by f 
Time: 0 SEC.

   compiling local plus : (%,%) -> %
Time: 0 SEC.

   compiling local times : (%,%) -> %
Time: 0 SEC.

   compiling local ltimes : (Fraction Integer,%) -> %
Time: 0 SEC.

   compiling exported Zero : () -> %
Time: 0 SEC.

   compiling exported One : () -> %
Time: 0 SEC.

   compiling exported + : (%,%) -> %
Time: 0 SEC.

   compiling exported * : (%,%) -> %
Time: 0 SEC.

   compiling exported * : (Fraction Integer,%) -> %
Time: 0 SEC.

   compiling exported coerce : % -> OutputForm
Time: 0 SEC.

   compiling exported apply : (%,F) -> F
      MYFUN;apply;%2F;12 is replaced by SPADCALLfx 
Time: 0 SEC.

   compiling exported coerce : F -> F -> %
      MYFUN;coerce;M%;13 is replaced by fun 
Time: 0 SEC.

(time taken in buildFunctor:  792)

;;;     ***       |MyFun| REDEFINED

;;;     ***       |MyFun| REDEFINED
Time: 0 SEC.

   Warnings: 
      [1] plus:  elt has no value
      [2] times:  elt has no value
      [3] apply:  elt has no value

   Cumulative Statistics for Constructor MyFun
      Time: 0.01 seconds

   finalizing NRLIB MYFUN 
   Processing MyFun for Browser database:
--->-->MyFun(constructor): Not documented!!!!
--->-->MyFun((apply (F % F))): Not documented!!!!
--->-->MyFun((coerce (% (Mapping F F)))): Not documented!!!!
--->-->MyFun(): Missing Description
; compiling file "/var/aw/var/LatexWiki/MYFUN.NRLIB/MYFUN.lsp" (written 28 MAR 2024 07:05:26 PM):

; wrote /var/aw/var/LatexWiki/MYFUN.NRLIB/MYFUN.fasl
; compilation finished in 0:00:00.012
------------------------------------------------------------------------
   MyFun is now explicitly exposed in frame initial 
   MyFun will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/MYFUN.NRLIB/MYFUN

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F := Fraction Integer

$\label{eq5}\hbox{\axiomType{Fraction}\ } \left({\hbox{\axiomType{Integer}\ }}\right)$

(5)

Type: Type

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H := MyFun(F)

$\label{eq6}\hbox{\axiomType{MyFun}\ } \left({\hbox{\axiomType{Fraction}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}\right)$

(6)

Type: Type

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phi(f: F): F == f/2

   Function declaration phi : Fraction(Integer) -> Fraction(Integer) 
      has been added to workspace.

Type: Void

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hphi:H := phi::H

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Compiling function phi with type Fraction(Integer) -> Fraction(
      Integer)

$\label{eq7}myfun - object$

(7)

Type: MyFun?(Fraction(Integer))

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apply(hphi, 1)

$\label{eq8}\frac{1}{2}$

(8)

Type: Fraction(Integer)

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a==>apply

Type: Void

Computing the first 3 terms of $1+\varphi(1)+\frac{1}{2}\varphi(\varphi(1))$ .

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1+a(hphi,1)+1/2 * a(hphi,a(hphi,1))

$\label{eq9}\frac{13}{8}$

(9)

Type: Fraction(Integer)

And here we first compute $1+\varphi+\frac{1}{2}\varphi^2$ and then apply this function to 1.

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hexp:=myexp(hphi,2)

$\label{eq10}myfun - object$

(10)

Type: MyFun?(Fraction(Integer))

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apply(hexp,1)

$\label{eq11}\frac{13}{8}$

(11)

Type: Fraction(Integer)

It also works with finite fields.

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P := PrimeField 17

$\label{eq12}\hbox{\axiomType{PrimeField}\ } \left({17}\right)$

(12)

Type: Type

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Q := MyFun(P)

$\label{eq13}\hbox{\axiomType{MyFun}\ } \left({\hbox{\axiomType{PrimeField}\ } \left({17}\right)}\right)$

(13)

Type: Type

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psi(f: P): P == f+1

   Function declaration psi : PrimeField(17) -> PrimeField(17) has been
      added to workspace.

Type: Void

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qpsi:Q := psi::Q

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Compiling function psi with type PrimeField(17) -> PrimeField(17)

$\label{eq14}myfun - object$

(14)

Type: MyFun?(PrimeField?(17))

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apply(qpsi, 1)

$\label{eq15}2$

(15)

Type: PrimeField?(17)

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qexp:=myexp(qpsi,2)

$\label{eq16}myfun - object$

(16)

Type: MyFun?(PrimeField?(17))

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apply(qexp,1)

$\label{eq17}13$

(17)

Type: PrimeField?(17)

Subject: Be Bold !!

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