YES Time: 0.791 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) = [] precedence: i ~ b ~ g ~ h ~ a ~ fAC weight function: w0 = 2 w(a) = 4 w(b) = 3 w(i) = w(g) = 2 w(h) = 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(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: fAC(x6,fAC(h(a()),g(a()))) -> fAC(x6,fAC(g(h(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) Restore Modifier: 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(x6,fAC(h(a()),g(a()))) -> fAC(x6,fAC(g(h(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-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(x6,fAC(h(a()),g(a()))) -> fAC(x6,fAC(g(h(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) Usable Rule Processor: 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(x6,fAC(h(a()),g(a()))) -> fAC(x6,fAC(g(h(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: 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) = [] precedence: g ~ h ~ a ~ fAC weight function: w0 = 2 w(g) = 7 w(h) = 5 w(a) = w(fAC) = 3 usable rules: 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: S: fAC(fAC(x10,x11),x12) -> fAC(x10,x11) fAC(x10,fAC(x11,x12)) -> fAC(x11,x12) Qed