MAYBE Problem: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) pred(s(x)) -> x minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(true(),x,y) -> gcd(minus(x,y),y) if_gcd(false(),x,y) -> gcd(minus(y,x),x) Proof: DP Processor: DPs: le#(s(x),s(y)) -> le#(x,y) minus#(x,s(y)) -> minus#(x,y) minus#(x,s(y)) -> pred#(minus(x,y)) gcd#(s(x),s(y)) -> le#(y,x) gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) if_gcd#(true(),x,y) -> minus#(x,y) if_gcd#(true(),x,y) -> gcd#(minus(x,y),y) if_gcd#(false(),x,y) -> minus#(y,x) if_gcd#(false(),x,y) -> gcd#(minus(y,x),x) TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) pred(s(x)) -> x minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(true(),x,y) -> gcd(minus(x,y),y) if_gcd(false(),x,y) -> gcd(minus(y,x),x) TDG Processor: DPs: le#(s(x),s(y)) -> le#(x,y) minus#(x,s(y)) -> minus#(x,y) minus#(x,s(y)) -> pred#(minus(x,y)) gcd#(s(x),s(y)) -> le#(y,x) gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) if_gcd#(true(),x,y) -> minus#(x,y) if_gcd#(true(),x,y) -> gcd#(minus(x,y),y) if_gcd#(false(),x,y) -> minus#(y,x) if_gcd#(false(),x,y) -> gcd#(minus(y,x),x) TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) pred(s(x)) -> x minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(true(),x,y) -> gcd(minus(x,y),y) if_gcd(false(),x,y) -> gcd(minus(y,x),x) graph: if_gcd#(false(),x,y) -> gcd#(minus(y,x),x) -> gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) if_gcd#(false(),x,y) -> gcd#(minus(y,x),x) -> gcd#(s(x),s(y)) -> le#(y,x) if_gcd#(false(),x,y) -> minus#(y,x) -> minus#(x,s(y)) -> pred#(minus(x,y)) if_gcd#(false(),x,y) -> minus#(y,x) -> minus#(x,s(y)) -> minus#(x,y) if_gcd#(true(),x,y) -> gcd#(minus(x,y),y) -> gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) if_gcd#(true(),x,y) -> gcd#(minus(x,y),y) -> gcd#(s(x),s(y)) -> le#(y,x) if_gcd#(true(),x,y) -> minus#(x,y) -> minus#(x,s(y)) -> pred#(minus(x,y)) if_gcd#(true(),x,y) -> minus#(x,y) -> minus#(x,s(y)) -> minus#(x,y) gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) -> if_gcd#(false(),x,y) -> gcd#(minus(y,x),x) gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) -> if_gcd#(false(),x,y) -> minus#(y,x) gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) -> if_gcd#(true(),x,y) -> gcd#(minus(x,y),y) gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) -> if_gcd#(true(),x,y) -> minus#(x,y) gcd#(s(x),s(y)) -> le#(y,x) -> le#(s(x),s(y)) -> le#(x,y) minus#(x,s(y)) -> minus#(x,y) -> minus#(x,s(y)) -> pred#(minus(x,y)) minus#(x,s(y)) -> minus#(x,y) -> minus#(x,s(y)) -> minus#(x,y) le#(s(x),s(y)) -> le#(x,y) -> le#(s(x),s(y)) -> le#(x,y) SCC Processor: #sccs: 3 #rules: 5 #arcs: 16/81 DPs: if_gcd#(false(),x,y) -> gcd#(minus(y,x),x) gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) if_gcd#(true(),x,y) -> gcd#(minus(x,y),y) TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) pred(s(x)) -> x minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(true(),x,y) -> gcd(minus(x,y),y) if_gcd(false(),x,y) -> gcd(minus(y,x),x) Open DPs: minus#(x,s(y)) -> minus#(x,y) TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) pred(s(x)) -> x minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(true(),x,y) -> gcd(minus(x,y),y) if_gcd(false(),x,y) -> gcd(minus(y,x),x) Subterm Criterion Processor: simple projection: pi(minus#) = 1 problem: DPs: TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) pred(s(x)) -> x minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(true(),x,y) -> gcd(minus(x,y),y) if_gcd(false(),x,y) -> gcd(minus(y,x),x) Qed DPs: le#(s(x),s(y)) -> le#(x,y) TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) pred(s(x)) -> x minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(true(),x,y) -> gcd(minus(x,y),y) if_gcd(false(),x,y) -> gcd(minus(y,x),x) Subterm Criterion Processor: simple projection: pi(le#) = 1 problem: DPs: TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) pred(s(x)) -> x minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(true(),x,y) -> gcd(minus(x,y),y) if_gcd(false(),x,y) -> gcd(minus(y,x),x) Qed