YES Problem: exp(x,0()) -> s(0()) exp(x,s(y)) -> *(x,exp(x,y)) *(0(),y) -> 0() *(s(x),y) -> +(y,*(x,y)) -(0(),y) -> 0() -(x,0()) -> x -(s(x),s(y)) -> -(x,y) Proof: DP Processor: DPs: exp#(x,s(y)) -> exp#(x,y) exp#(x,s(y)) -> *#(x,exp(x,y)) *#(s(x),y) -> *#(x,y) -#(s(x),s(y)) -> -#(x,y) TRS: exp(x,0()) -> s(0()) exp(x,s(y)) -> *(x,exp(x,y)) *(0(),y) -> 0() *(s(x),y) -> +(y,*(x,y)) -(0(),y) -> 0() -(x,0()) -> x -(s(x),s(y)) -> -(x,y) Matrix Interpretation Processor: dim=1 interpretation: [-#](x0, x1) = 2x0 + x1 + 1/2, [*#](x0, x1) = x0, [exp#](x0, x1) = 2x0 + x1 + 1/2, [-](x0, x1) = x0 + 3/2, [+](x0, x1) = x1, [*](x0, x1) = 3x0, [s](x0) = 3/2x0 + 1/2, [exp](x0, x1) = 3x0 + 1/2, [0] = 0 orientation: exp#(x,s(y)) = 2x + 3/2y + 1 >= 2x + y + 1/2 = exp#(x,y) exp#(x,s(y)) = 2x + 3/2y + 1 >= x = *#(x,exp(x,y)) *#(s(x),y) = 3/2x + 1/2 >= x = *#(x,y) -#(s(x),s(y)) = 3x + 3/2y + 2 >= 2x + y + 1/2 = -#(x,y) exp(x,0()) = 3x + 1/2 >= 1/2 = s(0()) exp(x,s(y)) = 3x + 1/2 >= 3x = *(x,exp(x,y)) *(0(),y) = 0 >= 0 = 0() *(s(x),y) = 9/2x + 3/2 >= 3x = +(y,*(x,y)) -(0(),y) = 3/2 >= 0 = 0() -(x,0()) = x + 3/2 >= x = x -(s(x),s(y)) = 3/2x + 2 >= x + 3/2 = -(x,y) problem: DPs: TRS: exp(x,0()) -> s(0()) exp(x,s(y)) -> *(x,exp(x,y)) *(0(),y) -> 0() *(s(x),y) -> +(y,*(x,y)) -(0(),y) -> 0() -(x,0()) -> x -(s(x),s(y)) -> -(x,y) Qed