Theory: TC

Parents


Types


Constants


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)