Peano's Axioms in Prolog
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% Peano's Axioms | |
:- module(peano, [ | |
is_zero/1, | |
is_natural/1, | |
equal/2, | |
pred/2 | |
]). | |
/** 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). | |
pred(succ(X), X) :- is_natural(X). | |
equal(X, succ(_)) :- is_natural(X), \+ is_zero(X). | |
equal(X, X) :- is_natural(X). | |
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