Theory: koenig

Parents


Type constants


Term constants


Axioms


Definitions

Bounded
|- !b P. Bounded b P = (?f. !x. P x ==> (?n. n < b /\ (x = f n)))
Finite
|- !P. Finite P = (?b. Bounded b P)
Infinite_Path
|- !x X. Infinite_Path x X = (?s. (s 0 = x) /\ (!d. X (s d) (s (SUC d))))
Unbounded_Path
|- !x X.
     Unbounded_Path x X =
     (!n. ?s. (s 0 = x) /\ (!d. d < n ==> X (s d) (s (SUC d))))

Theorems

Finite_EQ
|- !P. Finite P = (?b f. !x. P x = (?n. n < b /\ (x = f n)))
Koenig_Digraph_Lemma
|- !x X. (!s. Finite (X s)) ==> Unbounded_Path x X ==> Infinite_Path x X
Koenig_Original_Lemma
|- !E R.
     (!n.
       Finite (E n) /\
       (?x. E n x) /\
       (!x'. E (SUC n) x' ==> (?x. E n x /\ R x x'))) ==>
     (?a. !n. E n (a n) /\ R (a n) (a (SUC n)))