/* Comparison operators. They call the internal comparison routines when * both arguments are numbers. The value Infinity is also understood. */ 10 # n_IsNumber < m_IsNumber <-- LessThan(n,m); 20 # (n1_IsNumber/n2_IsNumber) < (m1_IsNumber/m2_IsNumber) <-- n1*m2 < m1*n2; 20 # (n1_IsNumber ) < (m1_IsNumber/m2_IsNumber) <-- n1*m2 < m1; 20 # (n1_IsNumber/n2_IsNumber) < (m1_IsNumber ) <-- n1 < m1*n2; 30 # n_IsNumber < Infinity <-- True; 30 # Infinity < n_IsNumber <-- False; 30 # n_IsNumber < -Infinity <-- False; 30 # -Infinity < n_IsNumber <-- True; 10 # n_IsNumber > m_IsNumber <-- GreaterThan(n,m); 20 # (n1_IsNumber/n2_IsNumber) > (m1_IsNumber/m2_IsNumber) <-- n1*m2 > m1*n2; 20 # (n1_IsNumber ) > (m1_IsNumber/m2_IsNumber) <-- n1*m2 > m1; 20 # (n1_IsNumber/n2_IsNumber) > (m1_IsNumber ) <-- n1 > m1*n2; 30 # n_IsNumber > Infinity <-- False; 30 # Infinity > n_IsNumber <-- True; 30 # n_IsNumber > -Infinity <-- True; 30 # -Infinity > n_IsNumber <-- False; 10 # n_IsNumber <= m_IsNumber <-- MathOr(Equals(n,m), LessThan(n,m)); 20 # (n1_IsNumber/n2_IsNumber) <= (m1_IsNumber/m2_IsNumber) <-- n1*m2 <= m1*n2; 20 # (n1_IsNumber ) <= (m1_IsNumber/m2_IsNumber) <-- n1*m2 <= m1; 20 # (n1_IsNumber/n2_IsNumber) <= (m1_IsNumber ) <-- n1 <= m1*n2; 30 # n_IsNumber <= Infinity <-- True; 30 # Infinity <= n_IsNumber <-- False; 30 # n_IsNumber <= -Infinity <-- False; 30 # -Infinity <= n_IsNumber <-- True; 10 # n_IsNumber >= m_IsNumber <-- MathOr(Equals(n,m), GreaterThan(n,m)); 20 # (n1_IsNumber/n2_IsNumber) >= (m1_IsNumber/m2_IsNumber) <-- n1*m2 <= m1*n2; 20 # (n1_IsNumber ) >= (m1_IsNumber/m2_IsNumber) <-- n1*m2 >= m1; 20 # (n1_IsNumber/n2_IsNumber) >= (m1_IsNumber ) <-- n1 >= m1*n2; 30 # n_IsNumber >= Infinity <-- False; 30 # Infinity >= n_IsNumber <-- True; 30 # n_IsNumber >= -Infinity <-- True; 30 # -Infinity >= n_IsNumber <-- False; Function("!=",{aLeft,aRight}) Not(aLeft=aRight); /* Shifting operators */ n_IsInteger << m_IsInteger <-- ShiftLeft(n,m); n_IsInteger >> m_IsInteger <-- ShiftRight(n,m); 0 # Sqrt(0) <-- 0; 1 # Sqrt(x_IsPositiveInteger)_(IsInteger(MathSqrt(x))) <-- MathSqrt(x); 2 # Sqrt(x_IsPositiveNumber)_(Numeric) <-- MathSqrt(x); 2 # Sqrt(x_IsNegativeNumber) <-- Complex(0,Sqrt(-x)); /* 3 # Sqrt(x_IsNumber/y_IsNumber) <-- Sqrt(x)/Sqrt(y); */ 3 # Sqrt(x_IsComplex)_(Numeric) <-- x^(1/2); /* Threading */ Sqrt(xlist_IsList) <-- MapSingle("Sqrt",xlist); 400 # x_IsRationalOrNumber * Sqrt(y_IsRationalOrNumber) <-- Sign(x)*Sqrt(x^2*y); 400 # Sqrt(y_IsRationalOrNumber) * x_IsRationalOrNumber <-- Sign(x)*Sqrt(x^2*y); 400 # x_IsRationalOrNumber / Sqrt(y_IsRationalOrNumber) <-- Sign(x)*Sqrt(x^2/y); 400 # Sqrt(y_IsRationalOrNumber) / x_IsRationalOrNumber <-- Sign(x)*Sqrt(y/(x^2)); 400 # Sqrt(y_IsRationalOrNumber) / Sqrt(x_IsRationalOrNumber) <-- Sqrt(y/x); 400 # Sqrt(y_IsRationalOrNumber) * Sqrt(x_IsRationalOrNumber) <-- Sqrt(y*x); 400 # Sqrt(x_IsInteger)_IsInteger(MathSqrt(x)) <-- MathSqrt(x); 400 # Sqrt(x_IsInteger/y_IsInteger)_(IsInteger(MathSqrt(x)) And IsInteger(MathSqrt(y))) <-- MathSqrt(x)/MathSqrt(y); /* Integer divisions */ 0 # Div(n_IsInteger,m_IsInteger) <-- MathDiv(n,m); 1 # Div(0 ,_m) <-- 0; 2 # Div(n_IsRationalOrNumber,m_IsRationalOrNumber) <-- [ Local(n1,n2,m1,m2); n1:=Numer(n); n2:=Denom(n); m1:=Numer(m); m2:=Denom(m); Div(n1*m2,m1*n2); ]; 30 # Div(n_CanBeUni,m_CanBeUni) <-- [ Local(vars,nl,ml); vars:=VarList(n+m); nl := MakeUni(n,vars); ml := MakeUni(m,vars); NormalForm(Div(nl,ml)); ]; 0 # Mod(n_IsNegativeInteger,m_IsInteger) <-- m+MathMod(n,m); 0 # Mod(n_IsPositiveInteger,m_IsInteger) <-- MathMod(n,m); 1 # Mod(0,_m) <-- 0; 2 # Mod(n_IsRationalOrNumber,m_IsRationalOrNumber) <-- [ Local(n1,n2,m1,m2); n1:=Numer(n); n2:=Denom(n); m1:=Numer(m); m2:=Denom(m); Mod(n1*m2,m1*n2)/(n2*m2); ]; 30 # Mod(n_CanBeUni,m_CanBeUni) <-- [ Local(vars); vars:=VarList(n+m); NormalForm(Mod(MakeUni(n,vars),MakeUni(m,vars))); ]; 0 # Rem(n_IsNegativeInteger,m_IsInteger) <-- MathMod(n,m); 0 # Rem(n_IsPositiveInteger,m_IsInteger) <-- MathMod(n,m); 1 # Rem(0,_m) <-- 0; 30 # Rem(n_CanBeUni,m_CanBeUni) <-- Mod(n,m); 0 # Gcd(0,_m) <-- m; 0 # Gcd(_n,0) <-- n; 1 # Gcd(n_IsInteger,m_IsInteger) <-- MathGcd(n,m); 5 # Gcd(n_IsRationalOrNumber,m_IsRationalOrNumber) <-- [ Gcd(Numer(n),Numer(m))/Gcd(Denom(n),Denom(m)); ]; 10 # Gcd(list_IsList)_(Length(list)>2) <-- [ Local(first); first:=Gcd(list[1],list[2]); Gcd(first:Tail(Tail(list))); ]; 15 # Gcd({_head}) <-- head; 20 # Gcd(list_IsList)_(Length(list)=2) <-- Gcd(list[1],list[2]); 30 # Gcd(n_CanBeUni,m_CanBeUni) <-- [ Local(vars); vars:=VarList(n+m); NormalForm(Gcd(MakeUni(n,vars),MakeUni(m,vars))); ]; /* Least common multiple */ Lcm(a,b):=Div(a*b,Gcd(a,b)); /* Expand expands polynomials. */ 10 # Expand(expr_CanBeUni) <-- NormalForm(MakeUni(expr)); 20 # Expand(_expr) <-- expr; 10 # Expand(_expr,_var)_CanBeUni(expr,var) <-- NormalForm(MakeUni(expr,var)); 20 # Expand(_expr,_var) <-- expr; RuleBase("Object",{pred,x}); Rule("Object",2,0,Apply(pred,{x})=True) x; /*TODO move to math */ Rule("Gcd",2,100,True) [ NormalForm(Gcd(MakeUni(n),MakeUni(m))); ]; 10 # Abs(n_IsNumber) <-- MathAbs(n); 20 # Abs(n_IsList) <-- MapSingle("Abs",n); 100 # Abs(_a)^n_IsEven <-- a^n; 100 # Abs(_a)^n_IsOdd <-- Sign(a)*a^n; 10 # Sign(n_IsPositiveNumber) <-- 1; 10 # Sign(n_IsZero) <-- 1; 20 # Sign(n_IsNumber) <-- -1; 15 # Sign(n_IsNumber/m_IsNumber) <-- Sign(n)*Sign(m); 20 # Sign(n_IsList) <-- MapSingle("Sign",n); 100 # Sign(_a)^n_IsEven <-- 1; 100 # Sign(_a)^n_IsOdd <-- Sign(a); 10 # Floor(x_IsNumber) <-- MathFloor(x); 10 # Ceil (x_IsNumber) <-- MathCeil (x); 10 # Round(x_IsNumber) <-- MathFloor(x+0.5);