YES 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()) EDG 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()) CDG 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#(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#(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(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#(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(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(x,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#(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#(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(x,b()),b()) f#(f(f(a(),x),b()),b()) -> f#(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(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#(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(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#(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#(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(),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(),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#(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(),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(),x) -> f#(a(),f(a(),f(a(),f(x,b())))) -> f#(a(),x) SCC Processor: #sccs: 2 #rules: 5 #arcs: 24/64 DPs: 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()) Arctic Interpretation Processor: dimension: 1 usable rules: 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()) interpretation: [f#](x0, x1) = x0 + x1 + 0, [f](x0, x1) = x0 + 1x1 + 0, [b] = 2, [a] = 4 orientation: f#(a(),f(a(),f(a(),f(x,b())))) = 2x + 5 >= 1x + 4 = f#(a(),f(a(),x)) f#(a(),f(a(),f(a(),f(x,b())))) = 2x + 5 >= x + 4 = f#(a(),x) f(a(),f(a(),f(a(),f(x,b())))) = 3x + 6 >= 3x + 6 = f(f(a(),f(a(),f(a(),x))),b()) f(f(f(a(),x),b()),b()) = 1x + 4 >= 1x + 4 = f(f(a(),f(f(x,b()),b())),b()) problem: DPs: 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()) Qed 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#(x,b()) f#(f(f(a(),x),b()),b()) -> f#(f(x,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()) Bounds Processor: bound: 0 enrichment: match-dp automaton: final states: {1} transitions: f120() -> 3* f{#,0}(7,2) -> 1* f0(4,2) -> 5* f0(3,2) -> 4* f0(6,5) -> 7* f0(7,2) -> 5* f0(2,2) -> 4* a0() -> 6* b0() -> 2* 5 -> 4* problem: DPs: f#(f(f(a(),x),b()),b()) -> f#(x,b()) f#(f(f(a(),x),b()),b()) -> f#(f(x,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()) Arctic Interpretation Processor: dimension: 1 usable rules: 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()) interpretation: [f#](x0, x1) = -1x0 + 3x1 + 0, [f](x0, x1) = x0 + 1x1 + 4, [b] = 0, [a] = 5 orientation: f#(f(f(a(),x),b()),b()) = x + 4 >= -1x + 3 = f#(x,b()) f#(f(f(a(),x),b()),b()) = x + 4 >= -1x + 3 = f#(f(x,b()),b()) f(a(),f(a(),f(a(),f(x,b())))) = 3x + 7 >= 3x + 7 = f(f(a(),f(a(),f(a(),x))),b()) f(f(f(a(),x),b()),b()) = 1x + 5 >= 1x + 5 = f(f(a(),f(f(x,b()),b())),b()) problem: DPs: 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()) Qed