# Library Coq.Sets.Relations_2_facts

Require Export Relations_1.
Require Export Relations_1_facts.
Require Export Relations_2.
Theorem Rstar_reflexive :
forall (U:Type) (R:Relation U), Reflexive U (Rstar U R).
Theorem Rplus_contains_R :
forall (U:Type) (R:Relation U), contains U (Rplus U R) R.
Theorem Rstar_contains_R :
forall (U:Type) (R:Relation U), contains U (Rstar U R) R.
Theorem Rstar_contains_Rplus :
forall (U:Type) (R:Relation U), contains U (Rstar U R) (Rplus U R).
Theorem Rstar_transitive :
forall (U:Type) (R:Relation U), Transitive U (Rstar U R).
Theorem Rstar_cases :
forall (U:Type) (R:Relation U) (x y:U),
Rstar U R x y -> x = y \/ ( exists u : _ , R x u /\ Rstar U R u y ).
Theorem Rstar_equiv_Rstar1 :
forall (U:Type) (R:Relation U), same_relation U (Rstar U R) (Rstar1 U R).
Theorem Rsym_imp_Rstarsym :
forall (U:Type) (R:Relation U), Symmetric U R -> Symmetric U (Rstar U R).
Theorem Sstar_contains_Rstar :
forall (U:Type) (R S:Relation U),
contains U (Rstar U S) R -> contains U (Rstar U S) (Rstar U R).
Theorem star_monotone :
forall (U:Type) (R S:Relation U),
contains U S R -> contains U (Rstar U S) (Rstar U R).
Theorem RstarRplus_RRstar :
forall (U:Type) (R:Relation U) (x y z:U),
Rstar U R x y -> Rplus U R y z -> exists u : _ , R x u /\ Rstar U R u z.
Theorem Lemma1 :
forall (U:Type) (R:Relation U),
Strongly_confluent U R ->
forall x b:U,
Rstar U R x b ->
forall a:U, R x a -> exists z : _ , Rstar U R a z /\ R b z.