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SandBoxDomainOfComputation
  
last edited 7 years ago by Bill Page

Edit detail for SandBoxDomainOfComputation revision 2 of 7
1 2 3 4 5 6 7
Editor: Bill Page
Time: 2017/04/13 16:38:17 GMT+0
Note: subtypes 
added:
And "assume" system based on equational reasoning.

added:
markConstants(x:Expression Integer):Expression Integer ==
  if (s:=isPlus(x)) case List Expression Integer then
    return reduce(+,[markConstants i for i in s::List Expression Integer])
  if (s:=isTimes(x)) case List Expression Integer then
    return reduce(*,[markConstants i for i in s::List Expression Integer])
  if (r:=retractIfCan(x)@Union(Integer,"failed")) case Integer then
    if r<0 then return assume(x,'integer)
    if r>0 then return assume(x,'positiveInteger)
    return assume(x, 'nonNegativeInteger)
  return x

changed:
-  assume(x,'positiveInteger)+(y|integer?(y) and integer(y)>=0) == assume(x+y,'positiveInteger)
  -- subtypes
  assume(assume(x,'integer),'integer) == assume(x,'integer)
  assume(assume(x,'nonNegativeInteger),'integer) == assume(x,'integer)
  assume(assume(x,'positiveInteger),'integer) == assume(x,'integer)
  assume(assume(x,'nonNegativeInteger),'nonNegativeInteger) == assume(x,'nonNegativeinteger)
  assume(assume(x,'positiveInteger),'nonNegativeInteger) == assume(x,'nonNegativeInteger)
  -- operations
  assume(x,'integer) + assume(y,'integer) == assume(x+y,'integer)
  assume(x,'integer) + assume(y,'nonNegativeInteger) == assume(x+y,'integer)
  assume(x,'integer) + assume(y,'positiveInteger) == assume(x+y,'integer)
  assume(x,'nonNegativeInteger) + assume(y,'nonNegativeInteger) == assume(x+y,'nonNegativeInteger)
  assume(x,'nonNegativeInteger) + assume(y,'positiveInteger) == assume(x+y,'nonNegativeInteger)

added:
  assume(x,'positiveInteger)+assume(y,'nonNegativeInteger) == assume(x+y,'positiveInteger)

changed:
-  k:=retractIfCan(prove x)@Union(Kernel Expression Integer,"failed")
-  if k case Kernel Expression Integer then is?(k,t) else false
  --k:=retractIfCan(prove markConstants x)@Union(Kernel Expression Integer,"failed")
  --if k case Kernel Expression Integer then is?(k,t) else false
  prove markConstants x = assume(x,t)

added:
prove %
assume(assume(x,'positiveInteger),'integer)

And "assume" system based on equational reasoning.

fricas
domains:=empty()$Table(Symbol,BasicOperator)
\label{eq1} \mbox{\rm table} \left({}\right)(1)
Type: Table(Symbol,BasicOperator?)
fricas
assume(x:Expression Integer,t:Symbol):Expression Integer ==
  if not key?(t,domains) then domains.t := operator t
  (domains.t) x

   Function declaration assume : (Expression(Integer),Symbol) -> 
      Expression(Integer) has been added to workspace.
Type: Void
fricas
markConstants(x:Expression Integer):Expression Integer ==
  if (s:=isPlus(x)) case List Expression Integer then
    return reduce(+,[markConstants i for i in s::List Expression Integer])
  if (s:=isTimes(x)) case List Expression Integer then
    return reduce(*,[markConstants i for i in s::List Expression Integer])
  if (r:=retractIfCan(x)@Union(Integer,"failed")) case Integer then
    if r<0 then return assume(x,'integer)
    if r>0 then return assume(x,'positiveInteger)
    return assume(x, 'nonNegativeInteger)
  return x

   Function declaration markConstants : Expression(Integer) -> 
      Expression(Integer) has been added to workspace.
Type: Void
fricas
prove := rule
  -- subtypes
  assume(assume(x,'integer),'integer) == assume(x,'integer)
  assume(assume(x,'nonNegativeInteger),'integer) == assume(x,'integer)
  assume(assume(x,'positiveInteger),'integer) == assume(x,'integer)
  assume(assume(x,'nonNegativeInteger),'nonNegativeInteger) == assume(x,'nonNegativeinteger)
  assume(assume(x,'positiveInteger),'nonNegativeInteger) == assume(x,'nonNegativeInteger)
  -- operations
  assume(x,'integer) + assume(y,'integer) == assume(x+y,'integer)
  assume(x,'integer) + assume(y,'nonNegativeInteger) == assume(x+y,'integer)
  assume(x,'integer) + assume(y,'positiveInteger) == assume(x+y,'integer)
  assume(x,'nonNegativeInteger) + assume(y,'nonNegativeInteger) == assume(x+y,'nonNegativeInteger)
  assume(x,'nonNegativeInteger) + assume(y,'positiveInteger) == assume(x+y,'nonNegativeInteger)
  assume(x,'positiveInteger)+assume(y,'positiveInteger) == assume(x+y,'positiveInteger)
  assume(x,'positiveInteger)+assume(y,'nonNegativeInteger) == assume(x+y,'positiveInteger)
fricas
Compiling function assume with type (Expression(Integer),Symbol) -> 
      Expression(Integer)
\label{eq2}\begin{array}{@{}l} \displaystyle \left\{{{integer \left({integer \left({x}\right)}\right)}\mbox{\rm = =}{{{\tt'}assume}\left({x , \: integer}\right)}}, \: \right. \ \ \displaystyle \left.{{integer \left({nonNegativeInteger \left({x}\right)}\right)}\mbox{\rm = =}{{{\tt'}assume}\left({x , \: integer}\right)}}, \right. \ \ \displaystyle \left.\:{{integer \left({positiveInteger \left({x}\right)}\right)}\mbox{\rm = =}{{{\tt'}assume}\left({x , \: integer}\right)}}, \: \right. \ \ \displaystyle \left.{{nonNegativeInteger \left({nonNegativeInteger \left({x}\right)}\right)}\mbox{\rm = =}{{{\tt'}assume}\left({x , \: nonNegativeinteger}\right)}}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{{nonNegativeInteger \left({positiveInteger \left({x}\right)}\right)}\mbox{\rm = =}{{{\tt'}assume}\left({x , \: nonNegativeInteger}\right)}}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{{{integer \left({y}\right)}+{integer \left({x}\right)}+ \%B}\mbox{\rm = =}{{{{\tt'}assume}\left({{y + x}, \: integer}\right)}+ \%B}}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{{{nonNegativeInteger \left({y}\right)}+{integer \left({x}\right)}+ \%C}\mbox{\rm = =}{{{{\tt'}assume}\left({{y + x}, \: integer}\right)}+ \%C}}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{{{positiveInteger \left({y}\right)}+{integer \left({x}\right)}+ \%D}\mbox{\rm = =}{{{{\tt'}assume}\left({{y + x}, \: integer}\right)}+ \%D}}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{{{nonNegativeInteger \left({y}\right)}+{nonNegativeInteger \left({x}\right)}+ \%E}\mbox{\rm = =}{{{{\tt'}assume}\left({{y + x}, \: nonNegativeInteger}\right)}+ \%E}}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{{{positiveInteger \left({y}\right)}+{nonNegativeInteger \left({x}\right)}+ \%F}\mbox{\rm = =}{{{{\tt'}assume}\left({{y + x}, \: nonNegativeInteger}\right)}+ \%F}}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{{{positiveInteger \left({y}\right)}+{positiveInteger \left({x}\right)}+ \%G}\mbox{\rm = =}{{{{\tt'}assume}\left({{y + x}, \: positiveInteger}\right)}+ \%G}}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{{{positiveInteger \left({x}\right)}+{nonNegativeInteger \left({y}\right)}+ \%H}\mbox{\rm = =}{{{{\tt'}assume}\left({{y + x}, \: positiveInteger}\right)}+ \%H}}\right\} (2)
Type: Ruleset(Integer,Integer,Expression(Integer))
fricas
check(x:Expression Integer,t:Symbol):Boolean ==
  --k:=retractIfCan(prove markConstants x)@Union(Kernel Expression Integer,"failed")
  --if k case Kernel Expression Integer then is?(k,t) else false
  prove markConstants x = assume(x,t)

   Function declaration check : (Expression(Integer),Symbol) -> Boolean
      has been added to workspace.
Type: Void

fricas
assume(x,'positiveInteger)
\label{eq3}positiveInteger \left({x}\right)(3)
Type: Expression(Integer)
fricas
prove %
\label{eq4}positiveInteger \left({x}\right)(4)
Type: Expression(Integer)
fricas
assume(assume(x,'positiveInteger),'integer)
\label{eq5}integer \left({positiveInteger \left({x}\right)}\right)(5)
Type: Expression(Integer)
fricas
prove %
\label{eq6}integer \left({x}\right)(6)
Type: Expression(Integer)
fricas
check(assume(x,'positiveInteger),'positiveInteger)
fricas
Compiling function markConstants with type Expression(Integer) -> 
      Expression(Integer)
fricas
Compiling function check with type (Expression(Integer),Symbol) -> 
      Boolean
\label{eq7} \mbox{\rm false} (7)
Type: Boolean
fricas
check(assume(x,'positiveInteger)+1,'positiveInteger)
\label{eq8} \mbox{\rm false} (8)
Type: Boolean
fricas
check(3,'positiveInteger)
\label{eq9} \mbox{\rm true} (9)
Type: Boolean
fricas
domains
\label{eq10} \mbox{\rm table} \left({{positiveInteger = positiveInteger}, \:{nonNegativeInteger = nonNegativeInteger}, \:{integer = integer}}\right)(10)
Type: Table(Symbol,BasicOperator?)