MAYBE Problem: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) *(x,0()) -> 0() *(x,s(y)) -> +(x,*(x,y)) twice(0()) -> 0() twice(s(x)) -> s(s(twice(x))) -(x,0()) -> x -(s(x),s(y)) -> -(x,y) f(s(x)) -> f(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) Proof: DP Processor: DPs: +#(s(x),y) -> +#(x,y) *#(x,s(y)) -> *#(x,y) *#(x,s(y)) -> +#(x,*(x,y)) twice#(s(x)) -> twice#(x) -#(s(x),s(y)) -> -#(x,y) f#(s(x)) -> *#(s(x),s(s(x))) f#(s(x)) -> +#(*(s(x),s(s(x))),s(s(0()))) f#(s(x)) -> *#(s(s(x)),s(s(x))) f#(s(x)) -> -#(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0())))) f#(s(x)) -> f#(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) *(x,0()) -> 0() *(x,s(y)) -> +(x,*(x,y)) twice(0()) -> 0() twice(s(x)) -> s(s(twice(x))) -(x,0()) -> x -(s(x),s(y)) -> -(x,y) f(s(x)) -> f(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) TDG Processor: DPs: +#(s(x),y) -> +#(x,y) *#(x,s(y)) -> *#(x,y) *#(x,s(y)) -> +#(x,*(x,y)) twice#(s(x)) -> twice#(x) -#(s(x),s(y)) -> -#(x,y) f#(s(x)) -> *#(s(x),s(s(x))) f#(s(x)) -> +#(*(s(x),s(s(x))),s(s(0()))) f#(s(x)) -> *#(s(s(x)),s(s(x))) f#(s(x)) -> -#(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0())))) f#(s(x)) -> f#(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) *(x,0()) -> 0() *(x,s(y)) -> +(x,*(x,y)) twice(0()) -> 0() twice(s(x)) -> s(s(twice(x))) -(x,0()) -> x -(s(x),s(y)) -> -(x,y) f(s(x)) -> f(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) graph: f#(s(x)) -> f#(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) -> f#(s(x)) -> f#(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) f#(s(x)) -> f#(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) -> f#(s(x)) -> -#(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0())))) f#(s(x)) -> f#(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) -> f#(s(x)) -> *#(s(s(x)),s(s(x))) f#(s(x)) -> f#(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) -> f#(s(x)) -> +#(*(s(x),s(s(x))),s(s(0()))) f#(s(x)) -> f#(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) -> f#(s(x)) -> *#(s(x),s(s(x))) f#(s(x)) -> -#(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0())))) -> -#(s(x),s(y)) -> -#(x,y) f#(s(x)) -> *#(s(s(x)),s(s(x))) -> *#(x,s(y)) -> +#(x,*(x,y)) f#(s(x)) -> *#(s(s(x)),s(s(x))) -> *#(x,s(y)) -> *#(x,y) f#(s(x)) -> *#(s(x),s(s(x))) -> *#(x,s(y)) -> +#(x,*(x,y)) f#(s(x)) -> *#(s(x),s(s(x))) -> *#(x,s(y)) -> *#(x,y) f#(s(x)) -> +#(*(s(x),s(s(x))),s(s(0()))) -> +#(s(x),y) -> +#(x,y) -#(s(x),s(y)) -> -#(x,y) -> -#(s(x),s(y)) -> -#(x,y) twice#(s(x)) -> twice#(x) -> twice#(s(x)) -> twice#(x) *#(x,s(y)) -> *#(x,y) -> *#(x,s(y)) -> +#(x,*(x,y)) *#(x,s(y)) -> *#(x,y) -> *#(x,s(y)) -> *#(x,y) *#(x,s(y)) -> +#(x,*(x,y)) -> +#(s(x),y) -> +#(x,y) +#(s(x),y) -> +#(x,y) -> +#(s(x),y) -> +#(x,y) SCC Processor: #sccs: 5 #rules: 5 #arcs: 17/100 DPs: twice#(s(x)) -> twice#(x) TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) *(x,0()) -> 0() *(x,s(y)) -> +(x,*(x,y)) twice(0()) -> 0() twice(s(x)) -> s(s(twice(x))) -(x,0()) -> x -(s(x),s(y)) -> -(x,y) f(s(x)) -> f(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) Subterm Criterion Processor: simple projection: pi(twice#) = 0 problem: DPs: TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) *(x,0()) -> 0() *(x,s(y)) -> +(x,*(x,y)) twice(0()) -> 0() twice(s(x)) -> s(s(twice(x))) -(x,0()) -> x -(s(x),s(y)) -> -(x,y) f(s(x)) -> f(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) Qed DPs: f#(s(x)) -> f#(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) *(x,0()) -> 0() *(x,s(y)) -> +(x,*(x,y)) twice(0()) -> 0() twice(s(x)) -> s(s(twice(x))) -(x,0()) -> x -(s(x),s(y)) -> -(x,y) f(s(x)) -> f(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) Open DPs: -#(s(x),s(y)) -> -#(x,y) TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) *(x,0()) -> 0() *(x,s(y)) -> +(x,*(x,y)) twice(0()) -> 0() twice(s(x)) -> s(s(twice(x))) -(x,0()) -> x -(s(x),s(y)) -> -(x,y) f(s(x)) -> f(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) Subterm Criterion Processor: simple projection: pi(-#) = 1 problem: DPs: TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) *(x,0()) -> 0() *(x,s(y)) -> +(x,*(x,y)) twice(0()) -> 0() twice(s(x)) -> s(s(twice(x))) -(x,0()) -> x -(s(x),s(y)) -> -(x,y) f(s(x)) -> f(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) Qed DPs: *#(x,s(y)) -> *#(x,y) TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) *(x,0()) -> 0() *(x,s(y)) -> +(x,*(x,y)) twice(0()) -> 0() twice(s(x)) -> s(s(twice(x))) -(x,0()) -> x -(s(x),s(y)) -> -(x,y) f(s(x)) -> f(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) Subterm Criterion Processor: simple projection: pi(*#) = 1 problem: DPs: TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) *(x,0()) -> 0() *(x,s(y)) -> +(x,*(x,y)) twice(0()) -> 0() twice(s(x)) -> s(s(twice(x))) -(x,0()) -> x -(s(x),s(y)) -> -(x,y) f(s(x)) -> f(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) Qed DPs: +#(s(x),y) -> +#(x,y) TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) *(x,0()) -> 0() *(x,s(y)) -> +(x,*(x,y)) twice(0()) -> 0() twice(s(x)) -> s(s(twice(x))) -(x,0()) -> x -(s(x),s(y)) -> -(x,y) f(s(x)) -> f(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) Subterm Criterion Processor: simple projection: pi(+#) = 0 problem: DPs: TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) *(x,0()) -> 0() *(x,s(y)) -> +(x,*(x,y)) twice(0()) -> 0() twice(s(x)) -> s(s(twice(x))) -(x,0()) -> x -(s(x),s(y)) -> -(x,y) f(s(x)) -> f(-(*(s(s(x)),s(s(x))),+(*(s(x),s(s(x))),s(s(0()))))) Qed