% Simple natural number proofs in the Oyster system
% Edinburgh proof development system
% Reference: The Oyster-CLaM System, Bundy et al., CADE-10 (1990)

% Verify that zero plus n equals n
prove all n:nat. plus(0, n) = n
by eval.

% Verify simple arithmetic: 2 + 1 = 3
prove plus(s(s(0)), s(0)) = s(s(s(0)))
by eval.

% Prove identity: every natural number equals itself
prove all n:nat. n = n
by intro_forall n; refl.

% Prove the successor of a number is not zero
prove all n:nat. not(s(n) = 0)
by intro_forall n; discriminate.