YES Problem: minus(minus(x)) -> x minus(h(x)) -> h(minus(x)) minus(f(x,y)) -> f(minus(y),minus(x)) Proof: DP Processor: DPs: minus#(h(x)) -> minus#(x) minus#(f(x,y)) -> minus#(x) minus#(f(x,y)) -> minus#(y) TRS: minus(minus(x)) -> x minus(h(x)) -> h(minus(x)) minus(f(x,y)) -> f(minus(y),minus(x)) Usable Rule Processor: DPs: minus#(h(x)) -> minus#(x) minus#(f(x,y)) -> minus#(x) minus#(f(x,y)) -> minus#(y) TRS: Restore Modifier: DPs: minus#(h(x)) -> minus#(x) minus#(f(x,y)) -> minus#(x) minus#(f(x,y)) -> minus#(y) TRS: minus(minus(x)) -> x minus(h(x)) -> h(minus(x)) minus(f(x,y)) -> f(minus(y),minus(x)) SCC Processor: #sccs: 1 #rules: 3 #arcs: 9/9 DPs: minus#(h(x)) -> minus#(x) minus#(f(x,y)) -> minus#(x) minus#(f(x,y)) -> minus#(y) TRS: minus(minus(x)) -> x minus(h(x)) -> h(minus(x)) minus(f(x,y)) -> f(minus(y),minus(x)) Matrix Interpretation Processor: dimension: 1 interpretation: [minus#](x0) = x0 + 1, [f](x0, x1) = x0 + x1, [h](x0) = x0 + 1, [minus](x0) = x0 orientation: minus#(h(x)) = x + 2 >= x + 1 = minus#(x) minus#(f(x,y)) = x + y + 1 >= x + 1 = minus#(x) minus#(f(x,y)) = x + y + 1 >= y + 1 = minus#(y) minus(minus(x)) = x >= x = x minus(h(x)) = x + 1 >= x + 1 = h(minus(x)) minus(f(x,y)) = x + y >= x + y = f(minus(y),minus(x)) problem: DPs: minus#(f(x,y)) -> minus#(x) minus#(f(x,y)) -> minus#(y) TRS: minus(minus(x)) -> x minus(h(x)) -> h(minus(x)) minus(f(x,y)) -> f(minus(y),minus(x)) Matrix Interpretation Processor: dimension: 1 interpretation: [minus#](x0) = x0 + 1, [f](x0, x1) = x0 + x1 + 1, [h](x0) = 0, [minus](x0) = x0 orientation: minus#(f(x,y)) = x + y + 2 >= x + 1 = minus#(x) minus#(f(x,y)) = x + y + 2 >= y + 1 = minus#(y) minus(minus(x)) = x >= x = x minus(h(x)) = 0 >= 0 = h(minus(x)) minus(f(x,y)) = x + y + 1 >= x + y + 1 = f(minus(y),minus(x)) problem: DPs: TRS: minus(minus(x)) -> x minus(h(x)) -> h(minus(x)) minus(f(x,y)) -> f(minus(y),minus(x)) Qed