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# coreyhaines/peano.pl

Last active April 19, 2020 20:31
Peano's Axioms in Prolog
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 % Peano's Axioms :- module(peano, [ is_zero/1, is_natural/1, equal/2, add/3, subtract/3, multiply/3, divide/3 ]). /** Peano's Axioms * * 1. 0 is a natural number * 2. For every number x, x = x (reflexive property) * 3. For all numbers x and y, if x = y, then y = x (symmetric property) * 4. For all numbers x, y and z, if x = y and y = z, then x = z (transitive property) * 5. For all a and b, if b is a natural number and a = b, then a is also a natural number (closed under equality) * 6. For every number n, Successor(n) is a number (closed under a successor function) * 7. For all numbers m and n, m = n if and only if Successor(m) = Successor(n) (Successor is an injection) * 8. For every number n, Successor(n) = 0 is false (0 is the starting point of the numbers) */ is_zero(zero). is_natural(zero). is_natural(succ(X)) :- is_natural(X). equal(X, succ(_)) :- is_natural(X), \+ is_zero(X). equal(X, X) :- is_natural(X). add(zero, Addend, Addend). add(Addend, zero, Addend). add(succ(Addend1), Addend2, Sum) :- add(Addend1, succ(Addend2), Sum). subtract(zero, zero, zero). subtract(Minuend, zero, Minuend). subtract(succ(NewMinuend), succ(NewSubtrahend), Difference) :- subtract(NewMinuend, NewSubtrahend, Difference). multiply(zero, _, zero). multiply(_, zero, zero). multiply(Multiplicand, succ(zero), Multiplicand). multiply(Multiplicand, succ(Multiplier), Product) :- add(Multiplicand, OldProduct, Product), multiply(Multiplicand, Multiplier, OldProduct). divide(X, Y, Z) :- multiply(Y, Z, X).
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