Scaling termination proofs by a characterisation of cycles in CHR

outputs relate to inputs by a match transition relation, given by a dependency analysis. Definition 4 (Match transition relation). A match transition of a CHR program P is an ordered pair M(i, j,k,l) = (out(i, j), in(k,l)), expressing a dependency between an output out(i, j) = (C(i, j),Gi) of OutP and an input in(k,l) = (C(k,l),Gk) of I nP, for which ∃θ : CT |= ((C(i, j) = C(k,l))∧Gi∧Gk)θ holds. The match transition relation M is the set of all match transitions M(i, j,k,l) in P. ¤ Notice that a match transition exists if the output and input it connects represent sets of constraints with a non-empty intersection: ℘(out(i, j))∩℘(in(k,l)) 6= / 0. Example 4 (GCD cont.). We obtain for the GCD program the following match transition relation: M = {M(2,1,2,1),M(2,1,2,2),M(2,2,1,1),M(2,2,2,1),M(2,2,2,2)}. Note that the first output of the second rule does not match with the input of the first rule as their intersection ℘(out(2,1))∩℘(in(1,1)) = / 0. ¤ For a program P, we call NP = 〈I nP,OutP,TP,MP〉 its CHR net.