YES Problem: cond(true(),x,y) -> cond(gr(x,y),p(x),y) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x Proof: DP Processor: DPs: cond#(true(),x,y) -> p#(x) cond#(true(),x,y) -> gr#(x,y) cond#(true(),x,y) -> cond#(gr(x,y),p(x),y) gr#(s(x),s(y)) -> gr#(x,y) TRS: cond(true(),x,y) -> cond(gr(x,y),p(x),y) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x TDG Processor: DPs: cond#(true(),x,y) -> p#(x) cond#(true(),x,y) -> gr#(x,y) cond#(true(),x,y) -> cond#(gr(x,y),p(x),y) gr#(s(x),s(y)) -> gr#(x,y) TRS: cond(true(),x,y) -> cond(gr(x,y),p(x),y) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x graph: gr#(s(x),s(y)) -> gr#(x,y) -> gr#(s(x),s(y)) -> gr#(x,y) cond#(true(),x,y) -> gr#(x,y) -> gr#(s(x),s(y)) -> gr#(x,y) cond#(true(),x,y) -> cond#(gr(x,y),p(x),y) -> cond#(true(),x,y) -> cond#(gr(x,y),p(x),y) cond#(true(),x,y) -> cond#(gr(x,y),p(x),y) -> cond#(true(),x,y) -> gr#(x,y) cond#(true(),x,y) -> cond#(gr(x,y),p(x),y) -> cond#(true(),x,y) -> p#(x) SCC Processor: #sccs: 2 #rules: 2 #arcs: 5/16 DPs: cond#(true(),x,y) -> cond#(gr(x,y),p(x),y) TRS: cond(true(),x,y) -> cond(gr(x,y),p(x),y) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x Arctic Interpretation Processor: dimension: 1 interpretation: [cond#](x0, x1, x2) = 1x0 + -1x1 + 2, [s](x0) = 10x0 + 9, [false] = 0, [0] = 5, [p](x0) = -6x0 + 5, [gr](x0, x1) = -5x0 + 0, [cond](x0, x1, x2) = 2x0 + 1x1 + 4x2 + 8, [true] = 4 orientation: cond#(true(),x,y) = -1x + 5 >= -4x + 4 = cond#(gr(x,y),p(x),y) cond(true(),x,y) = 1x + 4y + 8 >= -3x + 4y + 8 = cond(gr(x,y),p(x),y) gr(0(),x) = 0 >= 0 = false() gr(s(x),0()) = 5x + 4 >= 4 = true() gr(s(x),s(y)) = 5x + 4 >= -5x + 0 = gr(x,y) p(0()) = 5 >= 5 = 0() p(s(x)) = 4x + 5 >= x = x problem: DPs: TRS: cond(true(),x,y) -> cond(gr(x,y),p(x),y) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x Qed DPs: gr#(s(x),s(y)) -> gr#(x,y) TRS: cond(true(),x,y) -> cond(gr(x,y),p(x),y) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x Subterm Criterion Processor: simple projection: pi(gr#) = 1 problem: DPs: TRS: cond(true(),x,y) -> cond(gr(x,y),p(x),y) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x Qed