YES Time: 1.101 Problem: Equations: fAC(fAC(x0,x1),x2) -> fAC(x0,fAC(x1,x2)) fAC(x0,x1) -> fAC(x1,x0) fAC(x0,fAC(x1,x2)) -> fAC(fAC(x0,x1),x2) fAC(x1,x0) -> fAC(x0,x1) TRS: fAC(g(fAC(h(a()),a())),a()) -> fAC(h(a()),fAC(a(),a())) fAC(h(a()),g(a())) -> fAC(g(h(a())),a()) fAC(g(h(a())),fAC(b(),fAC(b(),b()))) -> fAC(g(fAC(h(a()),a())),a()) fAC(h(a()),a()) -> fAC(h(a()),b()) Proof: DP Processor: Equations#: f{AC,#}(fAC(x0,x1),x2) -> f{AC,#}(x0,fAC(x1,x2)) f{AC,#}(x0,x1) -> f{AC,#}(x1,x0) f{AC,#}(x0,fAC(x1,x2)) -> f{AC,#}(fAC(x0,x1),x2) f{AC,#}(x1,x0) -> f{AC,#}(x0,x1) DPs: f{AC,#}(g(fAC(h(a()),a())),a()) -> f{AC,#}(a(),a()) f{AC,#}(g(fAC(h(a()),a())),a()) -> f{AC,#}(h(a()),fAC(a(),a())) f{AC,#}(h(a()),g(a())) -> f{AC,#}(g(h(a())),a()) f{AC,#}(g(h(a())),fAC(b(),fAC(b(),b()))) -> f{AC,#}(h(a()),a()) f{AC,#}(g(h(a())),fAC(b(),fAC(b(),b()))) -> f{AC,#}(g(fAC(h(a()),a())),a()) f{AC,#}(h(a()),a()) -> f{AC,#}(h(a()),b()) f{AC,#}(x3,fAC(g(fAC(h(a()),a())),a())) -> f{AC,#}(a(),a()) f{AC,#}(x3,fAC(g(fAC(h(a()),a())),a())) -> f{AC,#}(h(a()),fAC(a(),a())) f{AC,#}(x3,fAC(g(fAC(h(a()),a())),a())) -> f{AC,#}(x3,fAC(h(a()),fAC(a(),a()))) f{AC,#}(x4,fAC(h(a()),g(a()))) -> f{AC,#}(g(h(a())),a()) f{AC,#}(x4,fAC(h(a()),g(a()))) -> f{AC,#}(x4,fAC(g(h(a())),a())) f{AC,#}(x5,fAC(g(h(a())),fAC(b(),fAC(b(),b())))) -> f{AC,#}(h(a()),a()) f{AC,#}(x5,fAC(g(h(a())),fAC(b(),fAC(b(),b())))) -> f{AC,#}(g(fAC(h(a()),a())),a()) f{AC,#}(x5,fAC(g(h(a())),fAC(b(),fAC(b(),b())))) -> f{AC,#}(x5,fAC(g(fAC(h(a()),a())),a())) f{AC,#}(x6,fAC(h(a()),a())) -> f{AC,#}(h(a()),b()) f{AC,#}(x6,fAC(h(a()),a())) -> f{AC,#}(x6,fAC(h(a()),b())) Equations: fAC(fAC(x0,x1),x2) -> fAC(x0,fAC(x1,x2)) fAC(x0,x1) -> fAC(x1,x0) fAC(x0,fAC(x1,x2)) -> fAC(fAC(x0,x1),x2) fAC(x1,x0) -> fAC(x0,x1) TRS: fAC(g(fAC(h(a()),a())),a()) -> fAC(h(a()),fAC(a(),a())) fAC(h(a()),g(a())) -> fAC(g(h(a())),a()) fAC(g(h(a())),fAC(b(),fAC(b(),b()))) -> fAC(g(fAC(h(a()),a())),a()) fAC(h(a()),a()) -> fAC(h(a()),b()) S: f{AC,#}(fAC(x7,x8),x9) -> f{AC,#}(x7,x8) f{AC,#}(x7,fAC(x8,x9)) -> f{AC,#}(x8,x9) AC-EDG Processor: Equations#: f{AC,#}(fAC(x0,x1),x2) -> f{AC,#}(x0,fAC(x1,x2)) f{AC,#}(x0,x1) -> f{AC,#}(x1,x0) f{AC,#}(x0,fAC(x1,x2)) -> f{AC,#}(fAC(x0,x1),x2) f{AC,#}(x1,x0) -> f{AC,#}(x0,x1) DPs: f{AC,#}(g(fAC(h(a()),a())),a()) -> f{AC,#}(a(),a()) f{AC,#}(g(fAC(h(a()),a())),a()) -> f{AC,#}(h(a()),fAC(a(),a())) f{AC,#}(h(a()),g(a())) -> f{AC,#}(g(h(a())),a()) f{AC,#}(g(h(a())),fAC(b(),fAC(b(),b()))) -> f{AC,#}(h(a()),a()) f{AC,#}(g(h(a())),fAC(b(),fAC(b(),b()))) -> f{AC,#}(g(fAC(h(a()),a())),a()) f{AC,#}(h(a()),a()) -> f{AC,#}(h(a()),b()) f{AC,#}(x3,fAC(g(fAC(h(a()),a())),a())) -> f{AC,#}(a(),a()) f{AC,#}(x3,fAC(g(fAC(h(a()),a())),a())) -> f{AC,#}(h(a()),fAC(a(),a())) f{AC,#}(x3,fAC(g(fAC(h(a()),a())),a())) -> f{AC,#}(x3,fAC(h(a()),fAC(a(),a()))) f{AC,#}(x4,fAC(h(a()),g(a()))) -> f{AC,#}(g(h(a())),a()) f{AC,#}(x4,fAC(h(a()),g(a()))) -> f{AC,#}(x4,fAC(g(h(a())),a())) f{AC,#}(x5,fAC(g(h(a())),fAC(b(),fAC(b(),b())))) -> f{AC,#}(h(a()),a()) f{AC,#}(x5,fAC(g(h(a())),fAC(b(),fAC(b(),b())))) -> f{AC,#}(g(fAC(h(a()),a())),a()) f{AC,#}(x5,fAC(g(h(a())),fAC(b(),fAC(b(),b())))) -> f{AC,#}(x5,fAC(g(fAC(h(a()),a())),a())) f{AC,#}(x6,fAC(h(a()),a())) -> f{AC,#}(h(a()),b()) f{AC,#}(x6,fAC(h(a()),a())) -> f{AC,#}(x6,fAC(h(a()),b())) Equations: fAC(fAC(x0,x1),x2) -> fAC(x0,fAC(x1,x2)) fAC(x0,x1) -> fAC(x1,x0) fAC(x0,fAC(x1,x2)) -> fAC(fAC(x0,x1),x2) fAC(x1,x0) -> fAC(x0,x1) TRS: fAC(g(fAC(h(a()),a())),a()) -> fAC(h(a()),fAC(a(),a())) fAC(h(a()),g(a())) -> fAC(g(h(a())),a()) fAC(g(h(a())),fAC(b(),fAC(b(),b()))) -> fAC(g(fAC(h(a()),a())),a()) fAC(h(a()),a()) -> fAC(h(a()),b()) S: f{AC,#}(fAC(x7,x8),x9) -> f{AC,#}(x7,x8) f{AC,#}(x7,fAC(x8,x9)) -> f{AC,#}(x8,x9) SCC Processor: #sccs: 1 #rules: 8 #arcs: 102/256 Equations#: f{AC,#}(fAC(x0,x1),x2) -> f{AC,#}(x0,fAC(x1,x2)) f{AC,#}(x0,x1) -> f{AC,#}(x1,x0) f{AC,#}(x0,fAC(x1,x2)) -> f{AC,#}(fAC(x0,x1),x2) f{AC,#}(x1,x0) -> f{AC,#}(x0,x1) DPs: f{AC,#}(g(h(a())),fAC(b(),fAC(b(),b()))) -> f{AC,#}(g(fAC(h(a()),a())),a()) f{AC,#}(g(fAC(h(a()),a())),a()) -> f{AC,#}(h(a()),fAC(a(),a())) f{AC,#}(x6,fAC(h(a()),a())) -> f{AC,#}(x6,fAC(h(a()),b())) f{AC,#}(x5,fAC(g(h(a())),fAC(b(),fAC(b(),b())))) -> f{AC,#}(x5,fAC(g(fAC(h(a()),a())),a())) f{AC,#}(x5,fAC(g(h(a())),fAC(b(),fAC(b(),b())))) -> f{AC,#}(g(fAC(h(a()),a())),a()) f{AC,#}(x4,fAC(h(a()),g(a()))) -> f{AC,#}(x4,fAC(g(h(a())),a())) f{AC,#}(x3,fAC(g(fAC(h(a()),a())),a())) -> f{AC,#}(x3,fAC(h(a()),fAC(a(),a()))) f{AC,#}(x3,fAC(g(fAC(h(a()),a())),a())) -> f{AC,#}(h(a()),fAC(a(),a())) Equations: fAC(fAC(x0,x1),x2) -> fAC(x0,fAC(x1,x2)) fAC(x0,x1) -> fAC(x1,x0) fAC(x0,fAC(x1,x2)) -> fAC(fAC(x0,x1),x2) fAC(x1,x0) -> fAC(x0,x1) TRS: fAC(g(fAC(h(a()),a())),a()) -> fAC(h(a()),fAC(a(),a())) fAC(h(a()),g(a())) -> fAC(g(h(a())),a()) fAC(g(h(a())),fAC(b(),fAC(b(),b()))) -> fAC(g(fAC(h(a()),a())),a()) fAC(h(a()),a()) -> fAC(h(a()),b()) S: f{AC,#}(fAC(x7,x8),x9) -> f{AC,#}(x7,x8) f{AC,#}(x7,fAC(x8,x9)) -> f{AC,#}(x8,x9) AC-DP unlabeling: Equations#: fAC(fAC(x0,x1),x2) -> fAC(x0,fAC(x1,x2)) fAC(x0,x1) -> fAC(x1,x0) fAC(x0,fAC(x1,x2)) -> fAC(fAC(x0,x1),x2) fAC(x1,x0) -> fAC(x0,x1) DPs: fAC(g(h(a())),fAC(b(),fAC(b(),b()))) -> fAC(g(fAC(h(a()),a())),a()) fAC(g(fAC(h(a()),a())),a()) -> fAC(h(a()),fAC(a(),a())) fAC(x6,fAC(h(a()),a())) -> fAC(x6,fAC(h(a()),b())) fAC(x5,fAC(g(h(a())),fAC(b(),fAC(b(),b())))) -> fAC(x5,fAC(g(fAC(h(a()),a())),a())) fAC(x5,fAC(g(h(a())),fAC(b(),fAC(b(),b())))) -> fAC(g(fAC(h(a()),a())),a()) fAC(x4,fAC(h(a()),g(a()))) -> fAC(x4,fAC(g(h(a())),a())) fAC(x3,fAC(g(fAC(h(a()),a())),a())) -> fAC(x3,fAC(h(a()),fAC(a(),a()))) fAC(x3,fAC(g(fAC(h(a()),a())),a())) -> fAC(h(a()),fAC(a(),a())) Equations: fAC(fAC(x0,x1),x2) -> fAC(x0,fAC(x1,x2)) fAC(x0,x1) -> fAC(x1,x0) fAC(x0,fAC(x1,x2)) -> fAC(fAC(x0,x1),x2) fAC(x1,x0) -> fAC(x0,x1) TRS: fAC(g(fAC(h(a()),a())),a()) -> fAC(h(a()),fAC(a(),a())) fAC(h(a()),g(a())) -> fAC(g(h(a())),a()) fAC(g(h(a())),fAC(b(),fAC(b(),b()))) -> fAC(g(fAC(h(a()),a())),a()) fAC(h(a()),a()) -> fAC(h(a()),b()) S: fAC(fAC(x7,x8),x9) -> fAC(x7,x8) fAC(x7,fAC(x8,x9)) -> fAC(x8,x9) AC-KBO Processor: argument filtering: pi(fAC) = [0,1] pi(a) = [] pi(h) = [] pi(g) = [] pi(b) = [] precedence: h ~ fAC > g > b ~ a weight function: w0 = 1 w(g) = 6 w(h) = w(a) = 2 w(b) = 1 w(fAC) = 0 usable rules: fAC(g(fAC(h(a()),a())),a()) -> fAC(h(a()),fAC(a(),a())) fAC(h(a()),g(a())) -> fAC(g(h(a())),a()) fAC(g(h(a())),fAC(b(),fAC(b(),b()))) -> fAC(g(fAC(h(a()),a())),a()) fAC(h(a()),a()) -> fAC(h(a()),b()) problem: Equations#: fAC(fAC(x0,x1),x2) -> fAC(x0,fAC(x1,x2)) fAC(x0,x1) -> fAC(x1,x0) fAC(x0,fAC(x1,x2)) -> fAC(fAC(x0,x1),x2) fAC(x1,x0) -> fAC(x0,x1) DPs: Equations: fAC(fAC(x0,x1),x2) -> fAC(x0,fAC(x1,x2)) fAC(x0,x1) -> fAC(x1,x0) fAC(x0,fAC(x1,x2)) -> fAC(fAC(x0,x1),x2) fAC(x1,x0) -> fAC(x0,x1) TRS: fAC(g(fAC(h(a()),a())),a()) -> fAC(h(a()),fAC(a(),a())) fAC(h(a()),g(a())) -> fAC(g(h(a())),a()) fAC(g(h(a())),fAC(b(),fAC(b(),b()))) -> fAC(g(fAC(h(a()),a())),a()) fAC(h(a()),a()) -> fAC(h(a()),b()) S: fAC(fAC(x7,x8),x9) -> fAC(x7,x8) fAC(x7,fAC(x8,x9)) -> fAC(x8,x9) Qed