MAYBE Problem: f(a(),f(a(),f(a(),f(x,b())))) -> f(f(a(),f(a(),f(a(),x))),b()) f(f(f(a(),x),b()),b()) -> f(f(a(),f(f(x,b()),b())),b()) Proof: DP Processor: DPs: f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),x) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),x)) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),f(a(),x))) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(f(a(),f(a(),f(a(),x))),b()) f#(f(f(a(),x),b()),b()) -> f#(x,b()) f#(f(f(a(),x),b()),b()) -> f#(f(x,b()),b()) f#(f(f(a(),x),b()),b()) -> f#(a(),f(f(x,b()),b())) f#(f(f(a(),x),b()),b()) -> f#(f(a(),f(f(x,b()),b())),b()) TRS: f(a(),f(a(),f(a(),f(x,b())))) -> f(f(a(),f(a(),f(a(),x))),b()) f(f(f(a(),x),b()),b()) -> f(f(a(),f(f(x,b()),b())),b()) ADG Processor: DPs: f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),x) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),x)) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),f(a(),x))) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(f(a(),f(a(),f(a(),x))),b()) f#(f(f(a(),x),b()),b()) -> f#(x,b()) f#(f(f(a(),x),b()),b()) -> f#(f(x,b()),b()) f#(f(f(a(),x),b()),b()) -> f#(a(),f(f(x,b()),b())) f#(f(f(a(),x),b()),b()) -> f#(f(a(),f(f(x,b()),b())),b()) TRS: f(a(),f(a(),f(a(),f(x,b())))) -> f(f(a(),f(a(),f(a(),x))),b()) f(f(f(a(),x),b()),b()) -> f(f(a(),f(f(x,b()),b())),b()) graph: f#(f(f(a(),x),b()),b()) -> f#(f(a(),f(f(x,b()),b())),b()) -> f#(f(f(a(),x),b()),b()) -> f#(x,b()) f#(f(f(a(),x),b()),b()) -> f#(f(a(),f(f(x,b()),b())),b()) -> f#(f(f(a(),x),b()),b()) -> f#(f(x,b()),b()) f#(f(f(a(),x),b()),b()) -> f#(f(a(),f(f(x,b()),b())),b()) -> f#(f(f(a(),x),b()),b()) -> f#(a(),f(f(x,b()),b())) f#(f(f(a(),x),b()),b()) -> f#(f(a(),f(f(x,b()),b())),b()) -> f#(f(f(a(),x),b()),b()) -> f#(f(a(),f(f(x,b()),b())),b()) f#(f(f(a(),x),b()),b()) -> f#(f(x,b()),b()) -> f#(f(f(a(),x),b()),b()) -> f#(x,b()) f#(f(f(a(),x),b()),b()) -> f#(f(x,b()),b()) -> f#(f(f(a(),x),b()),b()) -> f#(f(x,b()),b()) f#(f(f(a(),x),b()),b()) -> f#(f(x,b()),b()) -> f#(f(f(a(),x),b()),b()) -> f#(a(),f(f(x,b()),b())) f#(f(f(a(),x),b()),b()) -> f#(f(x,b()),b()) -> f#(f(f(a(),x),b()),b()) -> f#(f(a(),f(f(x,b()),b())),b()) f#(f(f(a(),x),b()),b()) -> f#(x,b()) -> f#(f(f(a(),x),b()),b()) -> f#(x,b()) f#(f(f(a(),x),b()),b()) -> f#(x,b()) -> f#(f(f(a(),x),b()),b()) -> f#(f(x,b()),b()) f#(f(f(a(),x),b()),b()) -> f#(x,b()) -> f#(f(f(a(),x),b()),b()) -> f#(a(),f(f(x,b()),b())) f#(f(f(a(),x),b()),b()) -> f#(x,b()) -> f#(f(f(a(),x),b()),b()) -> f#(f(a(),f(f(x,b()),b())),b()) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(f(a(),f(a(),f(a(),x))),b()) -> f#(f(f(a(),x),b()),b()) -> f#(x,b()) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(f(a(),f(a(),f(a(),x))),b()) -> f#(f(f(a(),x),b()),b()) -> f#(f(x,b()),b()) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(f(a(),f(a(),f(a(),x))),b()) -> f#(f(f(a(),x),b()),b()) -> f#(a(),f(f(x,b()),b())) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(f(a(),f(a(),f(a(),x))),b()) -> f#(f(f(a(),x),b()),b()) -> f#(f(a(),f(f(x,b()),b())),b()) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),f(a(),x))) -> f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),x) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),f(a(),x))) -> f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),x)) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),f(a(),x))) -> f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),f(a(),x))) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),f(a(),x))) -> f#(a(),f(a(),f(a(),f(x,b())))) -> f#(f(a(),f(a(),f(a(),x))),b()) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),x)) -> f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),x) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),x)) -> f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),x)) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),x)) -> f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),f(a(),x))) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),x)) -> f#(a(),f(a(),f(a(),f(x,b())))) -> f#(f(a(),f(a(),f(a(),x))),b()) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),x) -> f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),x) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),x) -> f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),x)) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),x) -> f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),f(a(),x))) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),x) -> f#(a(),f(a(),f(a(),f(x,b())))) -> f#(f(a(),f(a(),f(a(),x))),b()) Restore Modifier: DPs: f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),x) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),x)) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),f(a(),x))) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(f(a(),f(a(),f(a(),x))),b()) f#(f(f(a(),x),b()),b()) -> f#(x,b()) f#(f(f(a(),x),b()),b()) -> f#(f(x,b()),b()) f#(f(f(a(),x),b()),b()) -> f#(a(),f(f(x,b()),b())) f#(f(f(a(),x),b()),b()) -> f#(f(a(),f(f(x,b()),b())),b()) TRS: f(a(),f(a(),f(a(),f(x,b())))) -> f(f(a(),f(a(),f(a(),x))),b()) f(f(f(a(),x),b()),b()) -> f(f(a(),f(f(x,b()),b())),b()) SCC Processor: #sccs: 2 #rules: 6 #arcs: 28/64 DPs: f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),f(a(),x))) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),x)) f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),x) TRS: f(a(),f(a(),f(a(),f(x,b())))) -> f(f(a(),f(a(),f(a(),x))),b()) f(f(f(a(),x),b()),b()) -> f(f(a(),f(f(x,b()),b())),b()) Matrix Interpretation Processor: dimension: 1 interpretation: [f#](x0, x1) = x1, [f](x0, x1) = x0 + x1, [b] = 0, [a] = 1 orientation: f#(a(),f(a(),f(a(),f(x,b())))) = x + 2 >= x + 2 = f#(a(),f(a(),f(a(),x))) f#(a(),f(a(),f(a(),f(x,b())))) = x + 2 >= x + 1 = f#(a(),f(a(),x)) f#(a(),f(a(),f(a(),f(x,b())))) = x + 2 >= x = f#(a(),x) f(a(),f(a(),f(a(),f(x,b())))) = x + 3 >= x + 3 = f(f(a(),f(a(),f(a(),x))),b()) f(f(f(a(),x),b()),b()) = x + 1 >= x + 1 = f(f(a(),f(f(x,b()),b())),b()) problem: DPs: f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),f(a(),f(a(),x))) TRS: f(a(),f(a(),f(a(),f(x,b())))) -> f(f(a(),f(a(),f(a(),x))),b()) f(f(f(a(),x),b()),b()) -> f(f(a(),f(f(x,b()),b())),b()) Open DPs: f#(f(f(a(),x),b()),b()) -> f#(f(a(),f(f(x,b()),b())),b()) f#(f(f(a(),x),b()),b()) -> f#(f(x,b()),b()) f#(f(f(a(),x),b()),b()) -> f#(x,b()) TRS: f(a(),f(a(),f(a(),f(x,b())))) -> f(f(a(),f(a(),f(a(),x))),b()) f(f(f(a(),x),b()),b()) -> f(f(a(),f(f(x,b()),b())),b()) Matrix Interpretation Processor: dimension: 1 interpretation: [f#](x0, x1) = x0, [f](x0, x1) = x0 + x1, [b] = 0, [a] = 1 orientation: f#(f(f(a(),x),b()),b()) = x + 1 >= x + 1 = f#(f(a(),f(f(x,b()),b())),b()) f#(f(f(a(),x),b()),b()) = x + 1 >= x = f#(f(x,b()),b()) f#(f(f(a(),x),b()),b()) = x + 1 >= x = f#(x,b()) f(a(),f(a(),f(a(),f(x,b())))) = x + 3 >= x + 3 = f(f(a(),f(a(),f(a(),x))),b()) f(f(f(a(),x),b()),b()) = x + 1 >= x + 1 = f(f(a(),f(f(x,b()),b())),b()) problem: DPs: f#(f(f(a(),x),b()),b()) -> f#(f(a(),f(f(x,b()),b())),b()) TRS: f(a(),f(a(),f(a(),f(x,b())))) -> f(f(a(),f(a(),f(a(),x))),b()) f(f(f(a(),x),b()),b()) -> f(f(a(),f(f(x,b()),b())),b()) Open