Reactive Systems: Modelling, Specification and Verification - Cs.ioc.ee

4.2. TARSKI’S FIXED POINT THEOREM 91
Thus, for every x ∈ A, it holds that x ⊑ zmax. As f is monotonic, x ⊑ zmax
implies that f(x) ⊑ f(zmax). It follows that, for every x ∈ A,
x ⊑ f(x) ⊑ f(zmax) .
Thus f(zmax) is an upper bound for the set A. By definition, zmax is the
least upper bound of A. Thus zmax ⊑ f(zmax), and we have shown (4.1).
To prove that (4.2) holds, note that, from (4.1) and the monotonicity of f, we
have that f(zmax) ⊑ f(f(zmax)). This implies that f(zmax) ∈ A. Therefore
f(zmax) ⊑ zmax, as zmax is an upper bound for A.
From (4.1) and (4.2), we have that zmax ⊑ f(zmax) ⊑ zmax. By antisymmetry,
it follows that zmax = f(zmax), i.e., zmax is a fixed point of f.
2. We now show that zmax is the largest fixed point of f. Let d be any fixed
point of f. Then, in particular, we have that d ⊑ f(d). This implies that
d ∈ A and therefore that d ⊑ A = zmax.
We have thus shown that zmax is the largest fixed point of f.
To show that zmin is the least fixed point of f, we proceed in a similar fashion by
proving the following two statements:
1. zmin is a fixed point of f, i.e., zmin = f(zmin), and
2. zmin ⊑ d, for every d ∈ D that is a fixed point of f.
To prove that zmin is a fixed point of f, it is sufficient to show that:
f(zmin) ⊑ zmin and (4.3)
zmin ⊑ f(zmin) . (4.4)
Claim (4.3) can be shown following the proof for (4.1), and claim (4.4) can be
shown following the proof for (4.2). The details are left as an exercise for the
reader. Having shown that zmin is a fixed point of f, it is a simple matter to prove
that it is indeed the least fixed point of f. (Do this as an exercise). ✷
Consider, for example, a complete lattice of the form (2 S , ⊆), where S is a set,
and a monotonic function f : S → S. If we instantiate the statement of the above
theorem to this setting, the largest and least fixed points for f can be characterized
thus:
zmax = {X ⊆ S | X ⊆ f(X)} and
zmin = {X ⊆ S | f(X) ⊆ X} .