ASSORTED PRIMITIVE RECURSIVE FUNCTIONS:

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Minus(x,y)

Returns x-y (or 0 whenever y>=x).

First define minus_r which takes the arguments in reverse order

Example: 4-1 minus_r(1,4) =3

Minus_r(0,x) = p1-1(x)

Minus_r (y+1,x) =dec(P₂³(y,minus(y,x),x))

Minus(x,y) = minus(P₂²(x,y),P₁²(x,y))

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Equal(x,y) = not(or(lev(minus(x,y)),lev(minus(P₂²(x,y)P₁²(x,y)))))

Not_equal(x,y) = not(equal(x,y))

The above two can be reversed getting rid of the extra not.

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Mult

Mult(0,y) = Zero(y)

Mult (x+1,y) =add(P₂³(x,mult(x,y),y),P₃³(x,mult(x,y),y))

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Guard: returns 0 if x equals 0 and returns y otherwise.

Guard (0,y) = zero(y)

Guard (x+1,y) =P₃³(x,Guard(x,y),y)

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Lev: returns 1 if x is anything but zero.

Lev(0) = zero(0)

Lev(x+1) = succ(Zero(P₁³(x,lev(x))))

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Lev’ : opposite of above.

Lev(0) = succ(0)

Lev(x+1) = zero(P₁²(x,lev(x)))

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Max

Max(x,y) = add(x,Minus(y,x))

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Fact

Fact((0) = succ(o)

Fact(x+1) = mult(succ(x),fact(x))

Exponent is called pow and pow(x,y) is y^x

Pow(0,y) = succ(zero(y))

Pow(x+1,y) = Mult(P₂²(x,y),pow(x,y))

Min(x,y) = Minus(P₁²(x,y),Minus(x,y))