YES Problem: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) +(0(),y) -> y +(s(x),y) -> s(+(x,y)) +(x,+(y,z)) -> +(+(x,y),z) f(g(f(x))) -> f(h(s(0()),x)) f(g(h(x,y))) -> f(h(s(x),y)) f(h(x,h(y,z))) -> f(h(+(x,y),z)) Proof: DP Processor: DPs: +#(x,s(y)) -> +#(x,y) +#(s(x),y) -> +#(x,y) +#(x,+(y,z)) -> +#(x,y) +#(x,+(y,z)) -> +#(+(x,y),z) f#(g(f(x))) -> f#(h(s(0()),x)) f#(g(h(x,y))) -> f#(h(s(x),y)) f#(h(x,h(y,z))) -> +#(x,y) f#(h(x,h(y,z))) -> f#(h(+(x,y),z)) TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) +(0(),y) -> y +(s(x),y) -> s(+(x,y)) +(x,+(y,z)) -> +(+(x,y),z) f(g(f(x))) -> f(h(s(0()),x)) f(g(h(x,y))) -> f(h(s(x),y)) f(h(x,h(y,z))) -> f(h(+(x,y),z)) TDG Processor: DPs: +#(x,s(y)) -> +#(x,y) +#(s(x),y) -> +#(x,y) +#(x,+(y,z)) -> +#(x,y) +#(x,+(y,z)) -> +#(+(x,y),z) f#(g(f(x))) -> f#(h(s(0()),x)) f#(g(h(x,y))) -> f#(h(s(x),y)) f#(h(x,h(y,z))) -> +#(x,y) f#(h(x,h(y,z))) -> f#(h(+(x,y),z)) TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) +(0(),y) -> y +(s(x),y) -> s(+(x,y)) +(x,+(y,z)) -> +(+(x,y),z) f(g(f(x))) -> f(h(s(0()),x)) f(g(h(x,y))) -> f(h(s(x),y)) f(h(x,h(y,z))) -> f(h(+(x,y),z)) graph: f#(h(x,h(y,z))) -> f#(h(+(x,y),z)) -> f#(h(x,h(y,z))) -> f#(h(+(x,y),z)) f#(h(x,h(y,z))) -> f#(h(+(x,y),z)) -> f#(h(x,h(y,z))) -> +#(x,y) f#(h(x,h(y,z))) -> f#(h(+(x,y),z)) -> f#(g(h(x,y))) -> f#(h(s(x),y)) f#(h(x,h(y,z))) -> f#(h(+(x,y),z)) -> f#(g(f(x))) -> f#(h(s(0()),x)) f#(h(x,h(y,z))) -> +#(x,y) -> +#(x,+(y,z)) -> +#(+(x,y),z) f#(h(x,h(y,z))) -> +#(x,y) -> +#(x,+(y,z)) -> +#(x,y) f#(h(x,h(y,z))) -> +#(x,y) -> +#(s(x),y) -> +#(x,y) f#(h(x,h(y,z))) -> +#(x,y) -> +#(x,s(y)) -> +#(x,y) f#(g(h(x,y))) -> f#(h(s(x),y)) -> f#(h(x,h(y,z))) -> f#(h(+(x,y),z)) f#(g(h(x,y))) -> f#(h(s(x),y)) -> f#(h(x,h(y,z))) -> +#(x,y) f#(g(h(x,y))) -> f#(h(s(x),y)) -> f#(g(h(x,y))) -> f#(h(s(x),y)) f#(g(h(x,y))) -> f#(h(s(x),y)) -> f#(g(f(x))) -> f#(h(s(0()),x)) f#(g(f(x))) -> f#(h(s(0()),x)) -> f#(h(x,h(y,z))) -> f#(h(+(x,y),z)) f#(g(f(x))) -> f#(h(s(0()),x)) -> f#(h(x,h(y,z))) -> +#(x,y) f#(g(f(x))) -> f#(h(s(0()),x)) -> f#(g(h(x,y))) -> f#(h(s(x),y)) f#(g(f(x))) -> f#(h(s(0()),x)) -> f#(g(f(x))) -> f#(h(s(0()),x)) +#(s(x),y) -> +#(x,y) -> +#(x,+(y,z)) -> +#(+(x,y),z) +#(s(x),y) -> +#(x,y) -> +#(x,+(y,z)) -> +#(x,y) +#(s(x),y) -> +#(x,y) -> +#(s(x),y) -> +#(x,y) +#(s(x),y) -> +#(x,y) -> +#(x,s(y)) -> +#(x,y) +#(x,s(y)) -> +#(x,y) -> +#(x,+(y,z)) -> +#(+(x,y),z) +#(x,s(y)) -> +#(x,y) -> +#(x,+(y,z)) -> +#(x,y) +#(x,s(y)) -> +#(x,y) -> +#(s(x),y) -> +#(x,y) +#(x,s(y)) -> +#(x,y) -> +#(x,s(y)) -> +#(x,y) +#(x,+(y,z)) -> +#(+(x,y),z) -> +#(x,+(y,z)) -> +#(+(x,y),z) +#(x,+(y,z)) -> +#(+(x,y),z) -> +#(x,+(y,z)) -> +#(x,y) +#(x,+(y,z)) -> +#(+(x,y),z) -> +#(s(x),y) -> +#(x,y) +#(x,+(y,z)) -> +#(+(x,y),z) -> +#(x,s(y)) -> +#(x,y) +#(x,+(y,z)) -> +#(x,y) -> +#(x,+(y,z)) -> +#(+(x,y),z) +#(x,+(y,z)) -> +#(x,y) -> +#(x,+(y,z)) -> +#(x,y) +#(x,+(y,z)) -> +#(x,y) -> +#(s(x),y) -> +#(x,y) +#(x,+(y,z)) -> +#(x,y) -> +#(x,s(y)) -> +#(x,y) EDG Processor: DPs: +#(x,s(y)) -> +#(x,y) +#(s(x),y) -> +#(x,y) +#(x,+(y,z)) -> +#(x,y) +#(x,+(y,z)) -> +#(+(x,y),z) f#(g(f(x))) -> f#(h(s(0()),x)) f#(g(h(x,y))) -> f#(h(s(x),y)) f#(h(x,h(y,z))) -> +#(x,y) f#(h(x,h(y,z))) -> f#(h(+(x,y),z)) TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) +(0(),y) -> y +(s(x),y) -> s(+(x,y)) +(x,+(y,z)) -> +(+(x,y),z) f(g(f(x))) -> f(h(s(0()),x)) f(g(h(x,y))) -> f(h(s(x),y)) f(h(x,h(y,z))) -> f(h(+(x,y),z)) graph: f#(h(x,h(y,z))) -> f#(h(+(x,y),z)) -> f#(h(x,h(y,z))) -> +#(x,y) f#(h(x,h(y,z))) -> f#(h(+(x,y),z)) -> f#(h(x,h(y,z))) -> f#(h(+(x,y),z)) f#(h(x,h(y,z))) -> +#(x,y) -> +#(x,s(y)) -> +#(x,y) f#(h(x,h(y,z))) -> +#(x,y) -> +#(s(x),y) -> +#(x,y) f#(h(x,h(y,z))) -> +#(x,y) -> +#(x,+(y,z)) -> +#(x,y) f#(h(x,h(y,z))) -> +#(x,y) -> +#(x,+(y,z)) -> +#(+(x,y),z) f#(g(h(x,y))) -> f#(h(s(x),y)) -> f#(h(x,h(y,z))) -> +#(x,y) f#(g(h(x,y))) -> f#(h(s(x),y)) -> f#(h(x,h(y,z))) -> f#(h(+(x,y),z)) f#(g(f(x))) -> f#(h(s(0()),x)) -> f#(h(x,h(y,z))) -> +#(x,y) f#(g(f(x))) -> f#(h(s(0()),x)) -> f#(h(x,h(y,z))) -> f#(h(+(x,y),z)) +#(s(x),y) -> +#(x,y) -> +#(x,s(y)) -> +#(x,y) +#(s(x),y) -> +#(x,y) -> +#(s(x),y) -> +#(x,y) +#(s(x),y) -> +#(x,y) -> +#(x,+(y,z)) -> +#(x,y) +#(s(x),y) -> +#(x,y) -> +#(x,+(y,z)) -> +#(+(x,y),z) +#(x,s(y)) -> +#(x,y) -> +#(x,s(y)) -> +#(x,y) +#(x,s(y)) -> +#(x,y) -> +#(s(x),y) -> +#(x,y) +#(x,s(y)) -> +#(x,y) -> +#(x,+(y,z)) -> +#(x,y) +#(x,s(y)) -> +#(x,y) -> +#(x,+(y,z)) -> +#(+(x,y),z) +#(x,+(y,z)) -> +#(+(x,y),z) -> +#(x,s(y)) -> +#(x,y) +#(x,+(y,z)) -> +#(+(x,y),z) -> +#(s(x),y) -> +#(x,y) +#(x,+(y,z)) -> +#(+(x,y),z) -> +#(x,+(y,z)) -> +#(x,y) +#(x,+(y,z)) -> +#(+(x,y),z) -> +#(x,+(y,z)) -> +#(+(x,y),z) +#(x,+(y,z)) -> +#(x,y) -> +#(x,s(y)) -> +#(x,y) +#(x,+(y,z)) -> +#(x,y) -> +#(s(x),y) -> +#(x,y) +#(x,+(y,z)) -> +#(x,y) -> +#(x,+(y,z)) -> +#(x,y) +#(x,+(y,z)) -> +#(x,y) -> +#(x,+(y,z)) -> +#(+(x,y),z) CDG Processor: DPs: +#(x,s(y)) -> +#(x,y) +#(s(x),y) -> +#(x,y) +#(x,+(y,z)) -> +#(x,y) +#(x,+(y,z)) -> +#(+(x,y),z) f#(g(f(x))) -> f#(h(s(0()),x)) f#(g(h(x,y))) -> f#(h(s(x),y)) f#(h(x,h(y,z))) -> +#(x,y) f#(h(x,h(y,z))) -> f#(h(+(x,y),z)) TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) +(0(),y) -> y +(s(x),y) -> s(+(x,y)) +(x,+(y,z)) -> +(+(x,y),z) f(g(f(x))) -> f(h(s(0()),x)) f(g(h(x,y))) -> f(h(s(x),y)) f(h(x,h(y,z))) -> f(h(+(x,y),z)) graph: f#(h(x,h(y,z))) -> +#(x,y) -> +#(s(x),y) -> +#(x,y) +#(s(x),y) -> +#(x,y) -> +#(s(x),y) -> +#(x,y) +#(x,s(y)) -> +#(x,y) -> +#(s(x),y) -> +#(x,y) +#(x,+(y,z)) -> +#(+(x,y),z) -> +#(s(x),y) -> +#(x,y) +#(x,+(y,z)) -> +#(x,y) -> +#(s(x),y) -> +#(x,y) SCC Processor: #sccs: 1 #rules: 1 #arcs: 5/64 DPs: +#(s(x),y) -> +#(x,y) TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) +(0(),y) -> y +(s(x),y) -> s(+(x,y)) +(x,+(y,z)) -> +(+(x,y),z) f(g(f(x))) -> f(h(s(0()),x)) f(g(h(x,y))) -> f(h(s(x),y)) f(h(x,h(y,z))) -> f(h(+(x,y),z)) Matrix Interpretation Processor: dim=1 interpretation: [+#](x0, x1) = x0, [h](x0, x1) = 0, [g](x0) = 4, [f](x0) = 2, [s](x0) = x0 + 1, [+](x0, x1) = x0 + 2x1, [0] = 0 orientation: +#(s(x),y) = x + 1 >= x = +#(x,y) +(x,0()) = x >= x = x +(x,s(y)) = x + 2y + 2 >= x + 2y + 1 = s(+(x,y)) +(0(),y) = 2y >= y = y +(s(x),y) = x + 2y + 1 >= x + 2y + 1 = s(+(x,y)) +(x,+(y,z)) = x + 2y + 4z >= x + 2y + 2z = +(+(x,y),z) f(g(f(x))) = 2 >= 2 = f(h(s(0()),x)) f(g(h(x,y))) = 2 >= 2 = f(h(s(x),y)) f(h(x,h(y,z))) = 2 >= 2 = f(h(+(x,y),z)) problem: DPs: TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) +(0(),y) -> y +(s(x),y) -> s(+(x,y)) +(x,+(y,z)) -> +(+(x,y),z) f(g(f(x))) -> f(h(s(0()),x)) f(g(h(x,y))) -> f(h(s(x),y)) f(h(x,h(y,z))) -> f(h(+(x,y),z)) Qed