MAYBE Problem: times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0()))),times(x,s(z))) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) Proof: DP Processor: DPs: times#(x,plus(y,s(z))) -> times#(x,s(z)) times#(x,plus(y,s(z))) -> times#(s(z),0()) times#(x,plus(y,s(z))) -> plus#(y,times(s(z),0())) times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) times#(x,plus(y,s(z))) -> plus#(times(x,plus(y,times(s(z),0()))),times(x,s(z))) times#(x,s(y)) -> times#(x,y) times#(x,s(y)) -> plus#(times(x,y),x) plus#(x,s(y)) -> plus#(x,y) TRS: times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0()))),times(x,s(z))) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) TDG Processor: DPs: times#(x,plus(y,s(z))) -> times#(x,s(z)) times#(x,plus(y,s(z))) -> times#(s(z),0()) times#(x,plus(y,s(z))) -> plus#(y,times(s(z),0())) times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) times#(x,plus(y,s(z))) -> plus#(times(x,plus(y,times(s(z),0()))),times(x,s(z))) times#(x,s(y)) -> times#(x,y) times#(x,s(y)) -> plus#(times(x,y),x) plus#(x,s(y)) -> plus#(x,y) TRS: times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0()))),times(x,s(z))) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) graph: plus#(x,s(y)) -> plus#(x,y) -> plus#(x,s(y)) -> plus#(x,y) times#(x,plus(y,s(z))) -> plus#(times(x,plus(y,times(s(z),0()))),times(x,s(z))) -> plus#(x,s(y)) -> plus#(x,y) times#(x,plus(y,s(z))) -> plus#(y,times(s(z),0())) -> plus#(x,s(y)) -> plus#(x,y) times#(x,plus(y,s(z))) -> times#(s(z),0()) -> times#(x,s(y)) -> plus#(times(x,y),x) times#(x,plus(y,s(z))) -> times#(s(z),0()) -> times#(x,s(y)) -> times#(x,y) times#(x,plus(y,s(z))) -> times#(s(z),0()) -> times#(x,plus(y,s(z))) -> plus#(times(x,plus(y,times(s(z),0()))),times(x,s(z))) times#(x,plus(y,s(z))) -> times#(s(z),0()) -> times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) times#(x,plus(y,s(z))) -> times#(s(z),0()) -> times#(x,plus(y,s(z))) -> plus#(y,times(s(z),0())) times#(x,plus(y,s(z))) -> times#(s(z),0()) -> times#(x,plus(y,s(z))) -> times#(s(z),0()) times#(x,plus(y,s(z))) -> times#(s(z),0()) -> times#(x,plus(y,s(z))) -> times#(x,s(z)) times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) -> times#(x,s(y)) -> plus#(times(x,y),x) times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) -> times#(x,s(y)) -> times#(x,y) times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) -> times#(x,plus(y,s(z))) -> plus#(times(x,plus(y,times(s(z),0()))),times(x,s(z))) times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) -> times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) -> times#(x,plus(y,s(z))) -> plus#(y,times(s(z),0())) times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) -> times#(x,plus(y,s(z))) -> times#(s(z),0()) times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) -> times#(x,plus(y,s(z))) -> times#(x,s(z)) times#(x,plus(y,s(z))) -> times#(x,s(z)) -> times#(x,s(y)) -> plus#(times(x,y),x) times#(x,plus(y,s(z))) -> times#(x,s(z)) -> times#(x,s(y)) -> times#(x,y) times#(x,plus(y,s(z))) -> times#(x,s(z)) -> times#(x,plus(y,s(z))) -> plus#(times(x,plus(y,times(s(z),0()))),times(x,s(z))) times#(x,plus(y,s(z))) -> times#(x,s(z)) -> times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) times#(x,plus(y,s(z))) -> times#(x,s(z)) -> times#(x,plus(y,s(z))) -> plus#(y,times(s(z),0())) times#(x,plus(y,s(z))) -> times#(x,s(z)) -> times#(x,plus(y,s(z))) -> times#(s(z),0()) times#(x,plus(y,s(z))) -> times#(x,s(z)) -> times#(x,plus(y,s(z))) -> times#(x,s(z)) times#(x,s(y)) -> plus#(times(x,y),x) -> plus#(x,s(y)) -> plus#(x,y) times#(x,s(y)) -> times#(x,y) -> times#(x,s(y)) -> plus#(times(x,y),x) times#(x,s(y)) -> times#(x,y) -> times#(x,s(y)) -> times#(x,y) times#(x,s(y)) -> times#(x,y) -> times#(x,plus(y,s(z))) -> plus#(times(x,plus(y,times(s(z),0()))),times(x,s(z))) times#(x,s(y)) -> times#(x,y) -> times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) times#(x,s(y)) -> times#(x,y) -> times#(x,plus(y,s(z))) -> plus#(y,times(s(z),0())) times#(x,s(y)) -> times#(x,y) -> times#(x,plus(y,s(z))) -> times#(s(z),0()) times#(x,s(y)) -> times#(x,y) -> times#(x,plus(y,s(z))) -> times#(x,s(z)) SCC Processor: #sccs: 2 #rules: 5 #arcs: 32/64 DPs: times#(x,plus(y,s(z))) -> times#(s(z),0()) times#(x,plus(y,s(z))) -> times#(x,s(z)) times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) times#(x,s(y)) -> times#(x,y) TRS: times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0()))),times(x,s(z))) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) EDG Processor: DPs: times#(x,plus(y,s(z))) -> times#(s(z),0()) times#(x,plus(y,s(z))) -> times#(x,s(z)) times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) times#(x,s(y)) -> times#(x,y) TRS: times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0()))),times(x,s(z))) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) graph: times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) -> times#(x,plus(y,s(z))) -> times#(x,s(z)) times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) -> times#(x,plus(y,s(z))) -> times#(s(z),0()) times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) -> times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) -> times#(x,s(y)) -> times#(x,y) times#(x,plus(y,s(z))) -> times#(x,s(z)) -> times#(x,s(y)) -> times#(x,y) times#(x,s(y)) -> times#(x,y) -> times#(x,plus(y,s(z))) -> times#(x,s(z)) times#(x,s(y)) -> times#(x,y) -> times#(x,plus(y,s(z))) -> times#(s(z),0()) times#(x,s(y)) -> times#(x,y) -> times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) times#(x,s(y)) -> times#(x,y) -> times#(x,s(y)) -> times#(x,y) SCC Processor: #sccs: 1 #rules: 3 #arcs: 9/16 DPs: times#(x,plus(y,s(z))) -> times#(x,plus(y,times(s(z),0()))) times#(x,s(y)) -> times#(x,y) times#(x,plus(y,s(z))) -> times#(x,s(z)) TRS: times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0()))),times(x,s(z))) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) Open DPs: plus#(x,s(y)) -> plus#(x,y) TRS: times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0()))),times(x,s(z))) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) Subterm Criterion Processor: simple projection: pi(plus#) = 1 problem: DPs: TRS: times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0()))),times(x,s(z))) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) Qed