YES Time: 1.456 Problem: Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: pAC(x,zero()) -> x pAC(i(x),x) -> zero() m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) gAC(x,n()) -> x gAC(x,inv(x)) -> n() sm(x,sm(y,z)) -> sm(m(x,y),z) sm(one(),z) -> z gAC(sm(x,z),sm(y,z)) -> sm(pAC(x,y),z) sm(x,gAC(y,z)) -> gAC(sm(x,y),sm(x,z)) Proof: DP Processor: Equations#: p{AC,#}(pAC(x3,x4),x5) -> p{AC,#}(x3,pAC(x4,x5)) p{AC,#}(x3,x4) -> p{AC,#}(x4,x3) g{AC,#}(gAC(x3,x4),x5) -> g{AC,#}(x3,gAC(x4,x5)) g{AC,#}(x3,x4) -> g{AC,#}(x4,x3) p{AC,#}(x3,pAC(x4,x5)) -> p{AC,#}(pAC(x3,x4),x5) p{AC,#}(x4,x3) -> p{AC,#}(x3,x4) g{AC,#}(x3,gAC(x4,x5)) -> g{AC,#}(gAC(x3,x4),x5) g{AC,#}(x4,x3) -> g{AC,#}(x3,x4) DPs: m#(x,m(y,z)) -> m#(x,y) m#(x,m(y,z)) -> m#(m(x,y),z) m#(z,pAC(x,y)) -> m#(z,y) m#(z,pAC(x,y)) -> m#(z,x) m#(z,pAC(x,y)) -> p{AC,#}(m(z,x),m(z,y)) m#(pAC(x,y),z) -> m#(y,z) m#(pAC(x,y),z) -> m#(x,z) m#(pAC(x,y),z) -> p{AC,#}(m(x,z),m(y,z)) sm#(x,sm(y,z)) -> m#(x,y) sm#(x,sm(y,z)) -> sm#(m(x,y),z) g{AC,#}(sm(x,z),sm(y,z)) -> p{AC,#}(x,y) g{AC,#}(sm(x,z),sm(y,z)) -> sm#(pAC(x,y),z) sm#(x,gAC(y,z)) -> sm#(x,z) sm#(x,gAC(y,z)) -> sm#(x,y) sm#(x,gAC(y,z)) -> g{AC,#}(sm(x,y),sm(x,z)) p{AC,#}(x6,pAC(x,zero())) -> p{AC,#}(x6,x) p{AC,#}(x7,pAC(i(x),x)) -> p{AC,#}(x7,zero()) g{AC,#}(x8,gAC(x,n())) -> g{AC,#}(x8,x) g{AC,#}(x9,gAC(x,inv(x))) -> g{AC,#}(x9,n()) g{AC,#}(x10,gAC(sm(x,z),sm(y,z))) -> p{AC,#}(x,y) g{AC,#}(x10,gAC(sm(x,z),sm(y,z))) -> sm#(pAC(x,y),z) g{AC,#}(x10,gAC(sm(x,z),sm(y,z))) -> g{AC,#}(x10,sm(pAC(x,y),z)) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: pAC(x,zero()) -> x pAC(i(x),x) -> zero() m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) gAC(x,n()) -> x gAC(x,inv(x)) -> n() sm(x,sm(y,z)) -> sm(m(x,y),z) sm(one(),z) -> z gAC(sm(x,z),sm(y,z)) -> sm(pAC(x,y),z) sm(x,gAC(y,z)) -> gAC(sm(x,y),sm(x,z)) S: p{AC,#}(pAC(x11,x12),x13) -> p{AC,#}(x11,x12) p{AC,#}(x11,pAC(x12,x13)) -> p{AC,#}(x12,x13) g{AC,#}(gAC(x11,x12),x13) -> g{AC,#}(x11,x12) g{AC,#}(x11,gAC(x12,x13)) -> g{AC,#}(x12,x13) AC-EDG Processor: Equations#: p{AC,#}(pAC(x3,x4),x5) -> p{AC,#}(x3,pAC(x4,x5)) p{AC,#}(x3,x4) -> p{AC,#}(x4,x3) g{AC,#}(gAC(x3,x4),x5) -> g{AC,#}(x3,gAC(x4,x5)) g{AC,#}(x3,x4) -> g{AC,#}(x4,x3) p{AC,#}(x3,pAC(x4,x5)) -> p{AC,#}(pAC(x3,x4),x5) p{AC,#}(x4,x3) -> p{AC,#}(x3,x4) g{AC,#}(x3,gAC(x4,x5)) -> g{AC,#}(gAC(x3,x4),x5) g{AC,#}(x4,x3) -> g{AC,#}(x3,x4) DPs: m#(x,m(y,z)) -> m#(x,y) m#(x,m(y,z)) -> m#(m(x,y),z) m#(z,pAC(x,y)) -> m#(z,y) m#(z,pAC(x,y)) -> m#(z,x) m#(z,pAC(x,y)) -> p{AC,#}(m(z,x),m(z,y)) m#(pAC(x,y),z) -> m#(y,z) m#(pAC(x,y),z) -> m#(x,z) m#(pAC(x,y),z) -> p{AC,#}(m(x,z),m(y,z)) sm#(x,sm(y,z)) -> m#(x,y) sm#(x,sm(y,z)) -> sm#(m(x,y),z) g{AC,#}(sm(x,z),sm(y,z)) -> p{AC,#}(x,y) g{AC,#}(sm(x,z),sm(y,z)) -> sm#(pAC(x,y),z) sm#(x,gAC(y,z)) -> sm#(x,z) sm#(x,gAC(y,z)) -> sm#(x,y) sm#(x,gAC(y,z)) -> g{AC,#}(sm(x,y),sm(x,z)) p{AC,#}(x6,pAC(x,zero())) -> p{AC,#}(x6,x) p{AC,#}(x7,pAC(i(x),x)) -> p{AC,#}(x7,zero()) g{AC,#}(x8,gAC(x,n())) -> g{AC,#}(x8,x) g{AC,#}(x9,gAC(x,inv(x))) -> g{AC,#}(x9,n()) g{AC,#}(x10,gAC(sm(x,z),sm(y,z))) -> p{AC,#}(x,y) g{AC,#}(x10,gAC(sm(x,z),sm(y,z))) -> sm#(pAC(x,y),z) g{AC,#}(x10,gAC(sm(x,z),sm(y,z))) -> g{AC,#}(x10,sm(pAC(x,y),z)) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: pAC(x,zero()) -> x pAC(i(x),x) -> zero() m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) gAC(x,n()) -> x gAC(x,inv(x)) -> n() sm(x,sm(y,z)) -> sm(m(x,y),z) sm(one(),z) -> z gAC(sm(x,z),sm(y,z)) -> sm(pAC(x,y),z) sm(x,gAC(y,z)) -> gAC(sm(x,y),sm(x,z)) S: p{AC,#}(pAC(x11,x12),x13) -> p{AC,#}(x11,x12) p{AC,#}(x11,pAC(x12,x13)) -> p{AC,#}(x12,x13) g{AC,#}(gAC(x11,x12),x13) -> g{AC,#}(x11,x12) g{AC,#}(x11,gAC(x12,x13)) -> g{AC,#}(x12,x13) SCC Processor: #sccs: 3 #rules: 17 #arcs: 121/484 Equations#: p{AC,#}(pAC(x3,x4),x5) -> p{AC,#}(x3,pAC(x4,x5)) p{AC,#}(x3,x4) -> p{AC,#}(x4,x3) g{AC,#}(gAC(x3,x4),x5) -> g{AC,#}(x3,gAC(x4,x5)) g{AC,#}(x3,x4) -> g{AC,#}(x4,x3) p{AC,#}(x3,pAC(x4,x5)) -> p{AC,#}(pAC(x3,x4),x5) p{AC,#}(x4,x3) -> p{AC,#}(x3,x4) g{AC,#}(x3,gAC(x4,x5)) -> g{AC,#}(gAC(x3,x4),x5) g{AC,#}(x4,x3) -> g{AC,#}(x3,x4) DPs: sm#(x,sm(y,z)) -> sm#(m(x,y),z) sm#(x,gAC(y,z)) -> g{AC,#}(sm(x,y),sm(x,z)) g{AC,#}(x10,gAC(sm(x,z),sm(y,z))) -> g{AC,#}(x10,sm(pAC(x,y),z)) g{AC,#}(x10,gAC(sm(x,z),sm(y,z))) -> sm#(pAC(x,y),z) sm#(x,gAC(y,z)) -> sm#(x,y) sm#(x,gAC(y,z)) -> sm#(x,z) g{AC,#}(x9,gAC(x,inv(x))) -> g{AC,#}(x9,n()) g{AC,#}(x8,gAC(x,n())) -> g{AC,#}(x8,x) g{AC,#}(sm(x,z),sm(y,z)) -> sm#(pAC(x,y),z) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: pAC(x,zero()) -> x pAC(i(x),x) -> zero() m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) gAC(x,n()) -> x gAC(x,inv(x)) -> n() sm(x,sm(y,z)) -> sm(m(x,y),z) sm(one(),z) -> z gAC(sm(x,z),sm(y,z)) -> sm(pAC(x,y),z) sm(x,gAC(y,z)) -> gAC(sm(x,y),sm(x,z)) S: p{AC,#}(pAC(x11,x12),x13) -> p{AC,#}(x11,x12) p{AC,#}(x11,pAC(x12,x13)) -> p{AC,#}(x12,x13) g{AC,#}(gAC(x11,x12),x13) -> g{AC,#}(x11,x12) g{AC,#}(x11,gAC(x12,x13)) -> g{AC,#}(x12,x13) AC-DP unlabeling: Equations#: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) DPs: sm#(x,sm(y,z)) -> sm#(m(x,y),z) sm#(x,gAC(y,z)) -> gAC(sm(x,y),sm(x,z)) gAC(x10,gAC(sm(x,z),sm(y,z))) -> gAC(x10,sm(pAC(x,y),z)) gAC(x10,gAC(sm(x,z),sm(y,z))) -> sm#(pAC(x,y),z) sm#(x,gAC(y,z)) -> sm#(x,y) sm#(x,gAC(y,z)) -> sm#(x,z) gAC(x9,gAC(x,inv(x))) -> gAC(x9,n()) gAC(x8,gAC(x,n())) -> gAC(x8,x) gAC(sm(x,z),sm(y,z)) -> sm#(pAC(x,y),z) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: pAC(x,zero()) -> x pAC(i(x),x) -> zero() m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) gAC(x,n()) -> x gAC(x,inv(x)) -> n() sm(x,sm(y,z)) -> sm(m(x,y),z) sm(one(),z) -> z gAC(sm(x,z),sm(y,z)) -> sm(pAC(x,y),z) sm(x,gAC(y,z)) -> gAC(sm(x,y),sm(x,z)) S: pAC(pAC(x11,x12),x13) -> pAC(x11,x12) pAC(x11,pAC(x12,x13)) -> pAC(x12,x13) gAC(gAC(x11,x12),x13) -> gAC(x11,x12) gAC(x11,gAC(x12,x13)) -> gAC(x12,x13) AC-RPO Processor: argument filtering: pi(pAC) = [] pi(gAC) = [0,1] pi(zero) = [] pi(i) = [] pi(one) = [] pi(m) = 0 pi(n) = [] pi(inv) = [] pi(sm) = 1 pi(sm#) = 1 precedence: sm > i > sm# > gAC > n > inv > one > zero > pAC > m status: sm#:mul sm:mul m:mul problem: Equations#: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) DPs: sm#(x,sm(y,z)) -> sm#(m(x,y),z) sm#(x,gAC(y,z)) -> gAC(sm(x,y),sm(x,z)) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: pAC(x,zero()) -> x pAC(i(x),x) -> zero() m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) gAC(x,n()) -> x gAC(x,inv(x)) -> n() sm(x,sm(y,z)) -> sm(m(x,y),z) sm(one(),z) -> z gAC(sm(x,z),sm(y,z)) -> sm(pAC(x,y),z) sm(x,gAC(y,z)) -> gAC(sm(x,y),sm(x,z)) S: pAC(pAC(x11,x12),x13) -> pAC(x11,x12) pAC(x11,pAC(x12,x13)) -> pAC(x12,x13) gAC(gAC(x11,x12),x13) -> gAC(x11,x12) gAC(x11,gAC(x12,x13)) -> gAC(x12,x13) Restore Modifier: Equations#: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) DPs: sm#(x,sm(y,z)) -> sm#(m(x,y),z) sm#(x,gAC(y,z)) -> gAC(sm(x,y),sm(x,z)) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: pAC(x,zero()) -> x pAC(i(x),x) -> zero() m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) gAC(x,n()) -> x gAC(x,inv(x)) -> n() sm(x,sm(y,z)) -> sm(m(x,y),z) sm(one(),z) -> z gAC(sm(x,z),sm(y,z)) -> sm(pAC(x,y),z) sm(x,gAC(y,z)) -> gAC(sm(x,y),sm(x,z)) S: pAC(pAC(x11,x12),x13) -> pAC(x11,x12) pAC(x11,pAC(x12,x13)) -> pAC(x12,x13) gAC(gAC(x11,x12),x13) -> gAC(x11,x12) gAC(x11,gAC(x12,x13)) -> gAC(x12,x13) AC-DP unlabeling: Equations#: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) DPs: sm#(x,sm(y,z)) -> sm#(m(x,y),z) sm#(x,gAC(y,z)) -> gAC(sm(x,y),sm(x,z)) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: pAC(x,zero()) -> x pAC(i(x),x) -> zero() m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) gAC(x,n()) -> x gAC(x,inv(x)) -> n() sm(x,sm(y,z)) -> sm(m(x,y),z) sm(one(),z) -> z gAC(sm(x,z),sm(y,z)) -> sm(pAC(x,y),z) sm(x,gAC(y,z)) -> gAC(sm(x,y),sm(x,z)) S: pAC(pAC(x11,x12),x13) -> pAC(x11,x12) pAC(x11,pAC(x12,x13)) -> pAC(x12,x13) gAC(gAC(x11,x12),x13) -> gAC(x11,x12) gAC(x11,gAC(x12,x13)) -> gAC(x12,x13) AC-RPO Processor: argument filtering: pi(pAC) = [] pi(gAC) = [0,1] pi(zero) = [] pi(i) = 0 pi(one) = [] pi(m) = 0 pi(n) = [] pi(inv) = [] pi(sm) = 1 pi(sm#) = [1] precedence: m > inv > sm# > n > one > zero > i > sm > gAC > pAC status: sm#:mul sm:mul m:mul problem: Equations#: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) DPs: sm#(x,sm(y,z)) -> sm#(m(x,y),z) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: pAC(x,zero()) -> x pAC(i(x),x) -> zero() m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) gAC(x,n()) -> x gAC(x,inv(x)) -> n() sm(x,sm(y,z)) -> sm(m(x,y),z) sm(one(),z) -> z gAC(sm(x,z),sm(y,z)) -> sm(pAC(x,y),z) sm(x,gAC(y,z)) -> gAC(sm(x,y),sm(x,z)) S: pAC(pAC(x11,x12),x13) -> pAC(x11,x12) pAC(x11,pAC(x12,x13)) -> pAC(x12,x13) gAC(gAC(x11,x12),x13) -> gAC(x11,x12) gAC(x11,gAC(x12,x13)) -> gAC(x12,x13) Restore Modifier: Equations#: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) DPs: sm#(x,sm(y,z)) -> sm#(m(x,y),z) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: pAC(x,zero()) -> x pAC(i(x),x) -> zero() m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) gAC(x,n()) -> x gAC(x,inv(x)) -> n() sm(x,sm(y,z)) -> sm(m(x,y),z) sm(one(),z) -> z gAC(sm(x,z),sm(y,z)) -> sm(pAC(x,y),z) sm(x,gAC(y,z)) -> gAC(sm(x,y),sm(x,z)) S: pAC(pAC(x11,x12),x13) -> pAC(x11,x12) pAC(x11,pAC(x12,x13)) -> pAC(x12,x13) gAC(gAC(x11,x12),x13) -> gAC(x11,x12) gAC(x11,gAC(x12,x13)) -> gAC(x12,x13) AC-DP unlabeling: Equations#: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) DPs: sm#(x,sm(y,z)) -> sm#(m(x,y),z) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: pAC(x,zero()) -> x pAC(i(x),x) -> zero() m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) gAC(x,n()) -> x gAC(x,inv(x)) -> n() sm(x,sm(y,z)) -> sm(m(x,y),z) sm(one(),z) -> z gAC(sm(x,z),sm(y,z)) -> sm(pAC(x,y),z) sm(x,gAC(y,z)) -> gAC(sm(x,y),sm(x,z)) S: pAC(pAC(x11,x12),x13) -> pAC(x11,x12) pAC(x11,pAC(x12,x13)) -> pAC(x12,x13) gAC(gAC(x11,x12),x13) -> gAC(x11,x12) gAC(x11,gAC(x12,x13)) -> gAC(x12,x13) Usable Rule Processor: Equations#: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) DPs: sm#(x,sm(y,z)) -> sm#(m(x,y),z) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) pAC(x,zero()) -> x pAC(i(x),x) -> zero() S: pAC(pAC(x11,x12),x13) -> pAC(x11,x12) pAC(x11,pAC(x12,x13)) -> pAC(x12,x13) gAC(gAC(x11,x12),x13) -> gAC(x11,x12) gAC(x11,gAC(x12,x13)) -> gAC(x12,x13) AC-RPO Processor: argument filtering: pi(pAC) = [] pi(gAC) = [] pi(zero) = [] pi(i) = 0 pi(one) = [] pi(m) = [0,1] pi(sm) = [1] pi(sm#) = [1] precedence: m > sm > sm# > one > zero > gAC > i > pAC status: sm#:lex sm:mul m:lex problem: Equations#: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) DPs: Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) pAC(x,zero()) -> x pAC(i(x),x) -> zero() S: pAC(pAC(x11,x12),x13) -> pAC(x11,x12) pAC(x11,pAC(x12,x13)) -> pAC(x12,x13) gAC(gAC(x11,x12),x13) -> gAC(x11,x12) gAC(x11,gAC(x12,x13)) -> gAC(x12,x13) Qed Equations#: p{AC,#}(pAC(x3,x4),x5) -> p{AC,#}(x3,pAC(x4,x5)) p{AC,#}(x3,x4) -> p{AC,#}(x4,x3) g{AC,#}(gAC(x3,x4),x5) -> g{AC,#}(x3,gAC(x4,x5)) g{AC,#}(x3,x4) -> g{AC,#}(x4,x3) p{AC,#}(x3,pAC(x4,x5)) -> p{AC,#}(pAC(x3,x4),x5) p{AC,#}(x4,x3) -> p{AC,#}(x3,x4) g{AC,#}(x3,gAC(x4,x5)) -> g{AC,#}(gAC(x3,x4),x5) g{AC,#}(x4,x3) -> g{AC,#}(x3,x4) DPs: m#(pAC(x,y),z) -> m#(x,z) m#(pAC(x,y),z) -> m#(y,z) m#(z,pAC(x,y)) -> m#(z,x) m#(z,pAC(x,y)) -> m#(z,y) m#(x,m(y,z)) -> m#(m(x,y),z) m#(x,m(y,z)) -> m#(x,y) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: pAC(x,zero()) -> x pAC(i(x),x) -> zero() m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) gAC(x,n()) -> x gAC(x,inv(x)) -> n() sm(x,sm(y,z)) -> sm(m(x,y),z) sm(one(),z) -> z gAC(sm(x,z),sm(y,z)) -> sm(pAC(x,y),z) sm(x,gAC(y,z)) -> gAC(sm(x,y),sm(x,z)) S: p{AC,#}(pAC(x11,x12),x13) -> p{AC,#}(x11,x12) p{AC,#}(x11,pAC(x12,x13)) -> p{AC,#}(x12,x13) g{AC,#}(gAC(x11,x12),x13) -> g{AC,#}(x11,x12) g{AC,#}(x11,gAC(x12,x13)) -> g{AC,#}(x12,x13) AC-DP unlabeling: Equations#: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) DPs: m#(pAC(x,y),z) -> m#(x,z) m#(pAC(x,y),z) -> m#(y,z) m#(z,pAC(x,y)) -> m#(z,x) m#(z,pAC(x,y)) -> m#(z,y) m#(x,m(y,z)) -> m#(m(x,y),z) m#(x,m(y,z)) -> m#(x,y) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: pAC(x,zero()) -> x pAC(i(x),x) -> zero() m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) gAC(x,n()) -> x gAC(x,inv(x)) -> n() sm(x,sm(y,z)) -> sm(m(x,y),z) sm(one(),z) -> z gAC(sm(x,z),sm(y,z)) -> sm(pAC(x,y),z) sm(x,gAC(y,z)) -> gAC(sm(x,y),sm(x,z)) S: pAC(pAC(x11,x12),x13) -> pAC(x11,x12) pAC(x11,pAC(x12,x13)) -> pAC(x12,x13) gAC(gAC(x11,x12),x13) -> gAC(x11,x12) gAC(x11,gAC(x12,x13)) -> gAC(x12,x13) Usable Rule Processor: Equations#: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) DPs: m#(pAC(x,y),z) -> m#(x,z) m#(pAC(x,y),z) -> m#(y,z) m#(z,pAC(x,y)) -> m#(z,x) m#(z,pAC(x,y)) -> m#(z,y) m#(x,m(y,z)) -> m#(m(x,y),z) m#(x,m(y,z)) -> m#(x,y) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) pAC(x,zero()) -> x pAC(i(x),x) -> zero() S: pAC(pAC(x11,x12),x13) -> pAC(x11,x12) pAC(x11,pAC(x12,x13)) -> pAC(x12,x13) gAC(gAC(x11,x12),x13) -> gAC(x11,x12) gAC(x11,gAC(x12,x13)) -> gAC(x12,x13) AC-RPO Processor: argument filtering: pi(pAC) = [0,1] pi(gAC) = [0,1] pi(zero) = [] pi(i) = [] pi(one) = [] pi(m) = [0,1] pi(m#) = 1 precedence: m# > one > gAC > m > pAC > i > zero status: m#:mul m:lex problem: Equations#: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) DPs: m#(pAC(x,y),z) -> m#(x,z) m#(pAC(x,y),z) -> m#(y,z) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) pAC(x,zero()) -> x pAC(i(x),x) -> zero() S: pAC(pAC(x11,x12),x13) -> pAC(x11,x12) pAC(x11,pAC(x12,x13)) -> pAC(x12,x13) gAC(gAC(x11,x12),x13) -> gAC(x11,x12) gAC(x11,gAC(x12,x13)) -> gAC(x12,x13) Restore Modifier: Equations#: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) DPs: m#(pAC(x,y),z) -> m#(x,z) m#(pAC(x,y),z) -> m#(y,z) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) pAC(x,zero()) -> x pAC(i(x),x) -> zero() S: pAC(pAC(x11,x12),x13) -> pAC(x11,x12) pAC(x11,pAC(x12,x13)) -> pAC(x12,x13) gAC(gAC(x11,x12),x13) -> gAC(x11,x12) gAC(x11,gAC(x12,x13)) -> gAC(x12,x13) AC-DP unlabeling: Equations#: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) DPs: m#(pAC(x,y),z) -> m#(x,z) m#(pAC(x,y),z) -> m#(y,z) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) pAC(x,zero()) -> x pAC(i(x),x) -> zero() S: pAC(pAC(x11,x12),x13) -> pAC(x11,x12) pAC(x11,pAC(x12,x13)) -> pAC(x12,x13) gAC(gAC(x11,x12),x13) -> gAC(x11,x12) gAC(x11,gAC(x12,x13)) -> gAC(x12,x13) Usable Rule Processor: Equations#: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) DPs: m#(pAC(x,y),z) -> m#(x,z) m#(pAC(x,y),z) -> m#(y,z) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: S: pAC(pAC(x11,x12),x13) -> pAC(x11,x12) pAC(x11,pAC(x12,x13)) -> pAC(x12,x13) gAC(gAC(x11,x12),x13) -> gAC(x11,x12) gAC(x11,gAC(x12,x13)) -> gAC(x12,x13) AC-RPO Processor: argument filtering: pi(pAC) = [0,1] pi(gAC) = [0,1] pi(m#) = 0 precedence: pAC > m# > gAC status: m#:mul problem: Equations#: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) DPs: Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: S: pAC(pAC(x11,x12),x13) -> pAC(x11,x12) pAC(x11,pAC(x12,x13)) -> pAC(x12,x13) gAC(gAC(x11,x12),x13) -> gAC(x11,x12) gAC(x11,gAC(x12,x13)) -> gAC(x12,x13) Qed Equations#: p{AC,#}(pAC(x3,x4),x5) -> p{AC,#}(x3,pAC(x4,x5)) p{AC,#}(x3,x4) -> p{AC,#}(x4,x3) g{AC,#}(gAC(x3,x4),x5) -> g{AC,#}(x3,gAC(x4,x5)) g{AC,#}(x3,x4) -> g{AC,#}(x4,x3) p{AC,#}(x3,pAC(x4,x5)) -> p{AC,#}(pAC(x3,x4),x5) p{AC,#}(x4,x3) -> p{AC,#}(x3,x4) g{AC,#}(x3,gAC(x4,x5)) -> g{AC,#}(gAC(x3,x4),x5) g{AC,#}(x4,x3) -> g{AC,#}(x3,x4) DPs: p{AC,#}(x7,pAC(i(x),x)) -> p{AC,#}(x7,zero()) p{AC,#}(x6,pAC(x,zero())) -> p{AC,#}(x6,x) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: pAC(x,zero()) -> x pAC(i(x),x) -> zero() m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) gAC(x,n()) -> x gAC(x,inv(x)) -> n() sm(x,sm(y,z)) -> sm(m(x,y),z) sm(one(),z) -> z gAC(sm(x,z),sm(y,z)) -> sm(pAC(x,y),z) sm(x,gAC(y,z)) -> gAC(sm(x,y),sm(x,z)) S: p{AC,#}(pAC(x11,x12),x13) -> p{AC,#}(x11,x12) p{AC,#}(x11,pAC(x12,x13)) -> p{AC,#}(x12,x13) g{AC,#}(gAC(x11,x12),x13) -> g{AC,#}(x11,x12) g{AC,#}(x11,gAC(x12,x13)) -> g{AC,#}(x12,x13) AC-DP unlabeling: Equations#: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) DPs: pAC(x7,pAC(i(x),x)) -> pAC(x7,zero()) pAC(x6,pAC(x,zero())) -> pAC(x6,x) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: pAC(x,zero()) -> x pAC(i(x),x) -> zero() m(one(),x) -> x m(x,one()) -> x m(x,m(y,z)) -> m(m(x,y),z) m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y)) m(pAC(x,y),z) -> pAC(m(x,z),m(y,z)) gAC(x,n()) -> x gAC(x,inv(x)) -> n() sm(x,sm(y,z)) -> sm(m(x,y),z) sm(one(),z) -> z gAC(sm(x,z),sm(y,z)) -> sm(pAC(x,y),z) sm(x,gAC(y,z)) -> gAC(sm(x,y),sm(x,z)) S: pAC(pAC(x11,x12),x13) -> pAC(x11,x12) pAC(x11,pAC(x12,x13)) -> pAC(x12,x13) gAC(gAC(x11,x12),x13) -> gAC(x11,x12) gAC(x11,gAC(x12,x13)) -> gAC(x12,x13) Usable Rule Processor: Equations#: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) DPs: pAC(x7,pAC(i(x),x)) -> pAC(x7,zero()) pAC(x6,pAC(x,zero())) -> pAC(x6,x) Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: pAC(x,zero()) -> x pAC(i(x),x) -> zero() S: pAC(pAC(x11,x12),x13) -> pAC(x11,x12) pAC(x11,pAC(x12,x13)) -> pAC(x12,x13) gAC(gAC(x11,x12),x13) -> gAC(x11,x12) gAC(x11,gAC(x12,x13)) -> gAC(x12,x13) AC-RPO Processor: argument filtering: pi(pAC) = [0,1] pi(gAC) = [0,1] pi(zero) = [] pi(i) = [] precedence: i > zero > pAC > gAC status: problem: Equations#: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) DPs: Equations: pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5)) pAC(x3,x4) -> pAC(x4,x3) gAC(gAC(x3,x4),x5) -> gAC(x3,gAC(x4,x5)) gAC(x3,x4) -> gAC(x4,x3) pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5) pAC(x4,x3) -> pAC(x3,x4) gAC(x3,gAC(x4,x5)) -> gAC(gAC(x3,x4),x5) gAC(x4,x3) -> gAC(x3,x4) TRS: pAC(x,zero()) -> x pAC(i(x),x) -> zero() S: pAC(pAC(x11,x12),x13) -> pAC(x11,x12) pAC(x11,pAC(x12,x13)) -> pAC(x12,x13) gAC(gAC(x11,x12),x13) -> gAC(x11,x12) gAC(x11,gAC(x12,x13)) -> gAC(x12,x13) Qed