Theory: TC

Parents

• HOL

Types

Constants

• TC:('a -> 'a -> bool) -> 'a -> 'a -> bool
• transitive:('a -> 'a -> bool) -> bool

Axioms

Definitions

TC_DEF

|- !R a b.
     TC R a b =
     (!P.
       (!x y. R x y ==> P x y) /\ (!x y z. P x y /\ P y z ==> P x z) ==>
       P a b)

transitive_def

|- !R. transitive R = (!x y z. R x y /\ R y z ==> R x z)

Theorems

TC_TRANSITIVE

|- !R. transitive (TC R)

TC_SUBSET

|- !R x y. R x y ==> TC R x y

TC_INDUCT

|- !R P.
     (!x y. R x y ==> P x y) /\ (!x y z. P x y /\ P y z ==> P x z) ==>
     (!u v. TC R u v ==> P u v)

TC_CASES1

|- !R x z. TC R x z ==> R x z \/ (?y. R x y /\ TC R y z)

TC_CASES2

|- !R x z. TC R x z ==> R x z \/ (?y. TC R x y /\ R y z)