proposition: Theorem correct: forall A i j, bounded A -> belong i A -> belong j A -> (equiv A i j = None <-> eval (change_initial A i) == eval (change_initial A j)).
proof: Proof. intros A i j HA Hi Hj. split. apply valid; assumption. case_eq (equiv A i j); auto. intros c Hc H. exfalso. apply (completeness HA Hi Hj). rewrite Hc. intro. discriminate. assumption. Qed.
imports: ATBR(Common) ATBR(Classes) ATBR(Graph) ATBR(Monoid) ATBR(SemiLattice) ATBR(SemiRing) ATBR(KleeneAlgebra) ATBR(MxGraph) ATBR(MxSemiLattice) ATBR(MxSemiRing) ATBR(MxKleeneAlgebra) ATBR(DKA_Definitions) ATBR(DKA_StateSetSets) ATBR(Utils_WF) Coq(Relations) Coq(List)