YES Time: 1.462 Problem: Equations: fAC(fAC(x2,x3),x4) -> fAC(x2,fAC(x3,x4)) fAC(x2,x3) -> fAC(x3,x2) fAC(x2,fAC(x3,x4)) -> fAC(fAC(x2,x3),x4) fAC(x3,x2) -> fAC(x2,x3) 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(fAC(a(),a()),a())) -> fAC(g(fAC(h(a()),a())),a()) fAC(h(a()),a()) -> fAC(h(a()),b()) fAC(i(x,y),fAC(a(),y)) -> fAC(g(i(x,y)),y) Proof: DP Processor: Equations#: f{AC,#}(fAC(x2,x3),x4) -> f{AC,#}(x2,fAC(x3,x4)) f{AC,#}(x2,x3) -> f{AC,#}(x3,x2) f{AC,#}(x2,fAC(x3,x4)) -> f{AC,#}(fAC(x2,x3),x4) f{AC,#}(x3,x2) -> f{AC,#}(x2,x3) 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(fAC(a(),a()),a())) -> f{AC,#}(h(a()),a()) f{AC,#}(g(h(a())),fAC(fAC(a(),a()),a())) -> f{AC,#}(g(fAC(h(a()),a())),a()) f{AC,#}(h(a()),a()) -> f{AC,#}(h(a()),b()) f{AC,#}(i(x,y),fAC(a(),y)) -> f{AC,#}(g(i(x,y)),y) f{AC,#}(x5,fAC(g(fAC(h(a()),a())),a())) -> f{AC,#}(a(),a()) f{AC,#}(x5,fAC(g(fAC(h(a()),a())),a())) -> f{AC,#}(h(a()),fAC(a(),a())) f{AC,#}(x5,fAC(g(fAC(h(a()),a())),a())) -> f{AC,#}(x5,fAC(h(a()),fAC(a(),a()))) f{AC,#}(x6,fAC(h(a()),g(a()))) -> f{AC,#}(g(h(a())),a()) f{AC,#}(x6,fAC(h(a()),g(a()))) -> f{AC,#}(x6,fAC(g(h(a())),a())) f{AC,#}(x7,fAC(g(h(a())),fAC(fAC(a(),a()),a()))) -> f{AC,#}(h(a()),a()) f{AC,#}(x7,fAC(g(h(a())),fAC(fAC(a(),a()),a()))) -> f{AC,#}(g(fAC(h(a()),a())),a()) f{AC,#}(x7,fAC(g(h(a())),fAC(fAC(a(),a()),a()))) -> f{AC,#}(x7,fAC(g(fAC(h(a()),a())),a())) f{AC,#}(x8,fAC(h(a()),a())) -> f{AC,#}(h(a()),b()) f{AC,#}(x8,fAC(h(a()),a())) -> f{AC,#}(x8,fAC(h(a()),b())) f{AC,#}(x9,fAC(i(x,y),fAC(a(),y))) -> f{AC,#}(g(i(x,y)),y) f{AC,#}(x9,fAC(i(x,y),fAC(a(),y))) -> f{AC,#}(x9,fAC(g(i(x,y)),y)) Equations: fAC(fAC(x2,x3),x4) -> fAC(x2,fAC(x3,x4)) fAC(x2,x3) -> fAC(x3,x2) fAC(x2,fAC(x3,x4)) -> fAC(fAC(x2,x3),x4) fAC(x3,x2) -> fAC(x2,x3) 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(fAC(a(),a()),a())) -> fAC(g(fAC(h(a()),a())),a()) fAC(h(a()),a()) -> fAC(h(a()),b()) fAC(i(x,y),fAC(a(),y)) -> fAC(g(i(x,y)),y) S: f{AC,#}(fAC(x10,x11),x12) -> f{AC,#}(x10,x11) f{AC,#}(x10,fAC(x11,x12)) -> f{AC,#}(x11,x12) AC-EDG Processor: Equations#: f{AC,#}(fAC(x2,x3),x4) -> f{AC,#}(x2,fAC(x3,x4)) f{AC,#}(x2,x3) -> f{AC,#}(x3,x2) f{AC,#}(x2,fAC(x3,x4)) -> f{AC,#}(fAC(x2,x3),x4) f{AC,#}(x3,x2) -> f{AC,#}(x2,x3) 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(fAC(a(),a()),a())) -> f{AC,#}(h(a()),a()) f{AC,#}(g(h(a())),fAC(fAC(a(),a()),a())) -> f{AC,#}(g(fAC(h(a()),a())),a()) f{AC,#}(h(a()),a()) -> f{AC,#}(h(a()),b()) f{AC,#}(i(x,y),fAC(a(),y)) -> f{AC,#}(g(i(x,y)),y) f{AC,#}(x5,fAC(g(fAC(h(a()),a())),a())) -> f{AC,#}(a(),a()) f{AC,#}(x5,fAC(g(fAC(h(a()),a())),a())) -> f{AC,#}(h(a()),fAC(a(),a())) f{AC,#}(x5,fAC(g(fAC(h(a()),a())),a())) -> f{AC,#}(x5,fAC(h(a()),fAC(a(),a()))) f{AC,#}(x6,fAC(h(a()),g(a()))) -> f{AC,#}(g(h(a())),a()) f{AC,#}(x6,fAC(h(a()),g(a()))) -> f{AC,#}(x6,fAC(g(h(a())),a())) f{AC,#}(x7,fAC(g(h(a())),fAC(fAC(a(),a()),a()))) -> f{AC,#}(h(a()),a()) f{AC,#}(x7,fAC(g(h(a())),fAC(fAC(a(),a()),a()))) -> f{AC,#}(g(fAC(h(a()),a())),a()) f{AC,#}(x7,fAC(g(h(a())),fAC(fAC(a(),a()),a()))) -> f{AC,#}(x7,fAC(g(fAC(h(a()),a())),a())) f{AC,#}(x8,fAC(h(a()),a())) -> f{AC,#}(h(a()),b()) f{AC,#}(x8,fAC(h(a()),a())) -> f{AC,#}(x8,fAC(h(a()),b())) f{AC,#}(x9,fAC(i(x,y),fAC(a(),y))) -> f{AC,#}(g(i(x,y)),y) f{AC,#}(x9,fAC(i(x,y),fAC(a(),y))) -> f{AC,#}(x9,fAC(g(i(x,y)),y)) Equations: fAC(fAC(x2,x3),x4) -> fAC(x2,fAC(x3,x4)) fAC(x2,x3) -> fAC(x3,x2) fAC(x2,fAC(x3,x4)) -> fAC(fAC(x2,x3),x4) fAC(x3,x2) -> fAC(x2,x3) 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(fAC(a(),a()),a())) -> fAC(g(fAC(h(a()),a())),a()) fAC(h(a()),a()) -> fAC(h(a()),b()) fAC(i(x,y),fAC(a(),y)) -> fAC(g(i(x,y)),y) S: f{AC,#}(fAC(x10,x11),x12) -> f{AC,#}(x10,x11) f{AC,#}(x10,fAC(x11,x12)) -> f{AC,#}(x11,x12) SCC Processor: #sccs: 1 #rules: 11 #arcs: 177/361 Equations#: f{AC,#}(fAC(x2,x3),x4) -> f{AC,#}(x2,fAC(x3,x4)) f{AC,#}(x2,x3) -> f{AC,#}(x3,x2) f{AC,#}(x2,fAC(x3,x4)) -> f{AC,#}(fAC(x2,x3),x4) f{AC,#}(x3,x2) -> f{AC,#}(x2,x3) DPs: f{AC,#}(i(x,y),fAC(a(),y)) -> f{AC,#}(g(i(x,y)),y) f{AC,#}(x9,fAC(i(x,y),fAC(a(),y))) -> f{AC,#}(x9,fAC(g(i(x,y)),y)) f{AC,#}(x9,fAC(i(x,y),fAC(a(),y))) -> f{AC,#}(g(i(x,y)),y) f{AC,#}(x8,fAC(h(a()),a())) -> f{AC,#}(x8,fAC(h(a()),b())) f{AC,#}(x7,fAC(g(h(a())),fAC(fAC(a(),a()),a()))) -> f{AC,#}(x7,fAC(g(fAC(h(a()),a())),a())) f{AC,#}(x7,fAC(g(h(a())),fAC(fAC(a(),a()),a()))) -> 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()),g(a()))) -> f{AC,#}(x6,fAC(g(h(a())),a())) f{AC,#}(x5,fAC(g(fAC(h(a()),a())),a())) -> f{AC,#}(x5,fAC(h(a()),fAC(a(),a()))) f{AC,#}(x5,fAC(g(fAC(h(a()),a())),a())) -> f{AC,#}(h(a()),fAC(a(),a())) f{AC,#}(g(h(a())),fAC(fAC(a(),a()),a())) -> f{AC,#}(g(fAC(h(a()),a())),a()) Equations: fAC(fAC(x2,x3),x4) -> fAC(x2,fAC(x3,x4)) fAC(x2,x3) -> fAC(x3,x2) fAC(x2,fAC(x3,x4)) -> fAC(fAC(x2,x3),x4) fAC(x3,x2) -> fAC(x2,x3) 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(fAC(a(),a()),a())) -> fAC(g(fAC(h(a()),a())),a()) fAC(h(a()),a()) -> fAC(h(a()),b()) fAC(i(x,y),fAC(a(),y)) -> fAC(g(i(x,y)),y) S: f{AC,#}(fAC(x10,x11),x12) -> f{AC,#}(x10,x11) f{AC,#}(x10,fAC(x11,x12)) -> f{AC,#}(x11,x12) AC-DP unlabeling: Equations#: fAC(fAC(x2,x3),x4) -> fAC(x2,fAC(x3,x4)) fAC(x2,x3) -> fAC(x3,x2) fAC(x2,fAC(x3,x4)) -> fAC(fAC(x2,x3),x4) fAC(x3,x2) -> fAC(x2,x3) DPs: fAC(i(x,y),fAC(a(),y)) -> fAC(g(i(x,y)),y) fAC(x9,fAC(i(x,y),fAC(a(),y))) -> fAC(x9,fAC(g(i(x,y)),y)) fAC(x9,fAC(i(x,y),fAC(a(),y))) -> fAC(g(i(x,y)),y) fAC(x8,fAC(h(a()),a())) -> fAC(x8,fAC(h(a()),b())) fAC(x7,fAC(g(h(a())),fAC(fAC(a(),a()),a()))) -> fAC(x7,fAC(g(fAC(h(a()),a())),a())) fAC(x7,fAC(g(h(a())),fAC(fAC(a(),a()),a()))) -> 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()),g(a()))) -> fAC(x6,fAC(g(h(a())),a())) fAC(x5,fAC(g(fAC(h(a()),a())),a())) -> fAC(x5,fAC(h(a()),fAC(a(),a()))) fAC(x5,fAC(g(fAC(h(a()),a())),a())) -> fAC(h(a()),fAC(a(),a())) fAC(g(h(a())),fAC(fAC(a(),a()),a())) -> fAC(g(fAC(h(a()),a())),a()) Equations: fAC(fAC(x2,x3),x4) -> fAC(x2,fAC(x3,x4)) fAC(x2,x3) -> fAC(x3,x2) fAC(x2,fAC(x3,x4)) -> fAC(fAC(x2,x3),x4) fAC(x3,x2) -> fAC(x2,x3) 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(fAC(a(),a()),a())) -> fAC(g(fAC(h(a()),a())),a()) fAC(h(a()),a()) -> fAC(h(a()),b()) fAC(i(x,y),fAC(a(),y)) -> fAC(g(i(x,y)),y) S: fAC(fAC(x10,x11),x12) -> fAC(x10,x11) fAC(x10,fAC(x11,x12)) -> fAC(x11,x12) AC-KBO Processor: argument filtering: pi(fAC) = [0,1] pi(a) = [] pi(h) = [0] pi(g) = [0] pi(b) = [] pi(i) = 1 precedence: b > i > h ~ fAC > g ~ a weight function: [i](x0, x1) = x1, [b] = 1, [g](x0) = x0 + 1, [h](x0) = 6x0 + 4, [a] = 2, [fAC](x0, x1) = x0 + x1 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(fAC(a(),a()),a())) -> fAC(g(fAC(h(a()),a())),a()) fAC(h(a()),a()) -> fAC(h(a()),b()) fAC(i(x,y),fAC(a(),y)) -> fAC(g(i(x,y)),y) problem: Equations#: fAC(fAC(x2,x3),x4) -> fAC(x2,fAC(x3,x4)) fAC(x2,x3) -> fAC(x3,x2) fAC(x2,fAC(x3,x4)) -> fAC(fAC(x2,x3),x4) fAC(x3,x2) -> fAC(x2,x3) DPs: Equations: fAC(fAC(x2,x3),x4) -> fAC(x2,fAC(x3,x4)) fAC(x2,x3) -> fAC(x3,x2) fAC(x2,fAC(x3,x4)) -> fAC(fAC(x2,x3),x4) fAC(x3,x2) -> fAC(x2,x3) 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(fAC(a(),a()),a())) -> fAC(g(fAC(h(a()),a())),a()) fAC(h(a()),a()) -> fAC(h(a()),b()) fAC(i(x,y),fAC(a(),y)) -> fAC(g(i(x,y)),y) S: fAC(fAC(x10,x11),x12) -> fAC(x10,x11) fAC(x10,fAC(x11,x12)) -> fAC(x11,x12) Qed