Specification monotone-set-fixpoint

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enrich
   setelems
with
   functions 
       UN : ((elem → bool) → bool) → elem → bool;
       IS : ((elem → bool) → bool) → elem → bool;
       compl : ((elem → bool) → elem → bool) → (elem → bool) → elem → bool;
      comment: compl(f)(S) = X \ f(X \ S);
   
   predicates 
       . ⊆ . : (elem → bool) × (elem → bool);
       monotone : ((elem → bool) → elem → bool);
   
   variables 
       x : elem;
       x0 : elem;
       x1 : elem;
       iset : elem → bool;
       iset0 : elem → bool;
       iset1 : elem → bool;
       f : (elem → bool) → elem → bool;
       isetset : (elem → bool) → bool;
   
   partial functions 
       μ : ((elem → bool) → elem → bool) → elem → bool with λ f. λ x. monotone(f);
       ν : ((elem → bool) → elem → bool) → elem → bool with λ f. λ x. monotone(f);
   
   
   axioms 
      union-def: UN(isetset)(x) ↔ (∃ iset. isetset(iset) ∧ iset(x));
      
      intersection-def: IS(isetset)(x) ↔ (∀ iset. isetset(iset) → iset(x));
      
      subseteq-def: iset ⊆ iset0 ↔ (∀ x. iset(x) → iset0(x));
      
      monotone-def: monotone(f) ↔ (∀ iset, iset0. iset ⊆ iset0 → f(iset) ⊆ f(iset0));
      
      least-prefixpoint-def: monotone(f) → (μ(f)(x) ↔ (∀ iset. f(iset) ⊆ iset → iset(x)));
      
      greatest-prefixpoint-def: monotone(f) → (ν(f)(x) ↔ ¬ μ(compl(f))(x));
      
      compl-def: compl(f)(iset)(x) ↔ ¬ f(λ x0. ¬ iset(x0))(x);
   
   
end enrich