YES Problem: op(i(x),op(y,z)) -> op(f(x,i(x)),op(i(i(y)),z)) op(i(x),op(y,op(z,w))) -> op(f(x,i(x)),op(z,op(y,w))) i(x) -> x f(x,y) -> x Proof: DP Processor: DPs: op#(i(x),op(y,z)) -> i#(y) op#(i(x),op(y,z)) -> i#(i(y)) op#(i(x),op(y,z)) -> op#(i(i(y)),z) op#(i(x),op(y,z)) -> f#(x,i(x)) op#(i(x),op(y,z)) -> op#(f(x,i(x)),op(i(i(y)),z)) op#(i(x),op(y,op(z,w))) -> op#(y,w) op#(i(x),op(y,op(z,w))) -> op#(z,op(y,w)) op#(i(x),op(y,op(z,w))) -> f#(x,i(x)) op#(i(x),op(y,op(z,w))) -> op#(f(x,i(x)),op(z,op(y,w))) TRS: op(i(x),op(y,z)) -> op(f(x,i(x)),op(i(i(y)),z)) op(i(x),op(y,op(z,w))) -> op(f(x,i(x)),op(z,op(y,w))) i(x) -> x f(x,y) -> x TDG Processor: DPs: op#(i(x),op(y,z)) -> i#(y) op#(i(x),op(y,z)) -> i#(i(y)) op#(i(x),op(y,z)) -> op#(i(i(y)),z) op#(i(x),op(y,z)) -> f#(x,i(x)) op#(i(x),op(y,z)) -> op#(f(x,i(x)),op(i(i(y)),z)) op#(i(x),op(y,op(z,w))) -> op#(y,w) op#(i(x),op(y,op(z,w))) -> op#(z,op(y,w)) op#(i(x),op(y,op(z,w))) -> f#(x,i(x)) op#(i(x),op(y,op(z,w))) -> op#(f(x,i(x)),op(z,op(y,w))) TRS: op(i(x),op(y,z)) -> op(f(x,i(x)),op(i(i(y)),z)) op(i(x),op(y,op(z,w))) -> op(f(x,i(x)),op(z,op(y,w))) i(x) -> x f(x,y) -> x graph: op#(i(x),op(y,op(z,w))) -> op#(f(x,i(x)),op(z,op(y,w))) -> op#(i(x),op(y,op(z,w))) -> op#(f(x,i(x)),op(z,op(y,w))) op#(i(x),op(y,op(z,w))) -> op#(f(x,i(x)),op(z,op(y,w))) -> op#(i(x),op(y,op(z,w))) -> f#(x,i(x)) op#(i(x),op(y,op(z,w))) -> op#(f(x,i(x)),op(z,op(y,w))) -> op#(i(x),op(y,op(z,w))) -> op#(z,op(y,w)) op#(i(x),op(y,op(z,w))) -> op#(f(x,i(x)),op(z,op(y,w))) -> op#(i(x),op(y,op(z,w))) -> op#(y,w) op#(i(x),op(y,op(z,w))) -> op#(f(x,i(x)),op(z,op(y,w))) -> op#(i(x),op(y,z)) -> op#(f(x,i(x)),op(i(i(y)),z)) op#(i(x),op(y,op(z,w))) -> op#(f(x,i(x)),op(z,op(y,w))) -> op#(i(x),op(y,z)) -> f#(x,i(x)) op#(i(x),op(y,op(z,w))) -> op#(f(x,i(x)),op(z,op(y,w))) -> op#(i(x),op(y,z)) -> op#(i(i(y)),z) op#(i(x),op(y,op(z,w))) -> op#(f(x,i(x)),op(z,op(y,w))) -> op#(i(x),op(y,z)) -> i#(i(y)) op#(i(x),op(y,op(z,w))) -> op#(f(x,i(x)),op(z,op(y,w))) -> op#(i(x),op(y,z)) -> i#(y) op#(i(x),op(y,op(z,w))) -> op#(z,op(y,w)) -> op#(i(x),op(y,op(z,w))) -> op#(f(x,i(x)),op(z,op(y,w))) op#(i(x),op(y,op(z,w))) -> op#(z,op(y,w)) -> op#(i(x),op(y,op(z,w))) -> f#(x,i(x)) op#(i(x),op(y,op(z,w))) -> op#(z,op(y,w)) -> op#(i(x),op(y,op(z,w))) -> op#(z,op(y,w)) op#(i(x),op(y,op(z,w))) -> op#(z,op(y,w)) -> op#(i(x),op(y,op(z,w))) -> op#(y,w) op#(i(x),op(y,op(z,w))) -> op#(z,op(y,w)) -> op#(i(x),op(y,z)) -> op#(f(x,i(x)),op(i(i(y)),z)) op#(i(x),op(y,op(z,w))) -> op#(z,op(y,w)) -> op#(i(x),op(y,z)) -> f#(x,i(x)) op#(i(x),op(y,op(z,w))) -> op#(z,op(y,w)) -> op#(i(x),op(y,z)) -> op#(i(i(y)),z) op#(i(x),op(y,op(z,w))) -> op#(z,op(y,w)) -> op#(i(x),op(y,z)) -> i#(i(y)) op#(i(x),op(y,op(z,w))) -> op#(z,op(y,w)) -> op#(i(x),op(y,z)) -> i#(y) op#(i(x),op(y,op(z,w))) -> op#(y,w) -> op#(i(x),op(y,op(z,w))) -> op#(f(x,i(x)),op(z,op(y,w))) op#(i(x),op(y,op(z,w))) -> op#(y,w) -> op#(i(x),op(y,op(z,w))) -> f#(x,i(x)) op#(i(x),op(y,op(z,w))) -> op#(y,w) -> op#(i(x),op(y,op(z,w))) -> op#(z,op(y,w)) op#(i(x),op(y,op(z,w))) -> op#(y,w) -> op#(i(x),op(y,op(z,w))) -> op#(y,w) op#(i(x),op(y,op(z,w))) -> op#(y,w) -> op#(i(x),op(y,z)) -> op#(f(x,i(x)),op(i(i(y)),z)) op#(i(x),op(y,op(z,w))) -> op#(y,w) -> op#(i(x),op(y,z)) -> f#(x,i(x)) op#(i(x),op(y,op(z,w))) -> op#(y,w) -> op#(i(x),op(y,z)) -> op#(i(i(y)),z) op#(i(x),op(y,op(z,w))) -> op#(y,w) -> op#(i(x),op(y,z)) -> i#(i(y)) op#(i(x),op(y,op(z,w))) -> op#(y,w) -> op#(i(x),op(y,z)) -> i#(y) op#(i(x),op(y,z)) -> op#(f(x,i(x)),op(i(i(y)),z)) -> op#(i(x),op(y,op(z,w))) -> op#(f(x,i(x)),op(z,op(y,w))) op#(i(x),op(y,z)) -> op#(f(x,i(x)),op(i(i(y)),z)) -> op#(i(x),op(y,op(z,w))) -> f#(x,i(x)) op#(i(x),op(y,z)) -> op#(f(x,i(x)),op(i(i(y)),z)) -> op#(i(x),op(y,op(z,w))) -> op#(z,op(y,w)) op#(i(x),op(y,z)) -> op#(f(x,i(x)),op(i(i(y)),z)) -> op#(i(x),op(y,op(z,w))) -> op#(y,w) op#(i(x),op(y,z)) -> op#(f(x,i(x)),op(i(i(y)),z)) -> op#(i(x),op(y,z)) -> op#(f(x,i(x)),op(i(i(y)),z)) op#(i(x),op(y,z)) -> op#(f(x,i(x)),op(i(i(y)),z)) -> op#(i(x),op(y,z)) -> f#(x,i(x)) op#(i(x),op(y,z)) -> op#(f(x,i(x)),op(i(i(y)),z)) -> op#(i(x),op(y,z)) -> op#(i(i(y)),z) op#(i(x),op(y,z)) -> op#(f(x,i(x)),op(i(i(y)),z)) -> op#(i(x),op(y,z)) -> i#(i(y)) op#(i(x),op(y,z)) -> op#(f(x,i(x)),op(i(i(y)),z)) -> op#(i(x),op(y,z)) -> i#(y) op#(i(x),op(y,z)) -> op#(i(i(y)),z) -> op#(i(x),op(y,op(z,w))) -> op#(f(x,i(x)),op(z,op(y,w))) op#(i(x),op(y,z)) -> op#(i(i(y)),z) -> op#(i(x),op(y,op(z,w))) -> f#(x,i(x)) op#(i(x),op(y,z)) -> op#(i(i(y)),z) -> op#(i(x),op(y,op(z,w))) -> op#(z,op(y,w)) op#(i(x),op(y,z)) -> op#(i(i(y)),z) -> op#(i(x),op(y,op(z,w))) -> op#(y,w) op#(i(x),op(y,z)) -> op#(i(i(y)),z) -> op#(i(x),op(y,z)) -> op#(f(x,i(x)),op(i(i(y)),z)) op#(i(x),op(y,z)) -> op#(i(i(y)),z) -> op#(i(x),op(y,z)) -> f#(x,i(x)) op#(i(x),op(y,z)) -> op#(i(i(y)),z) -> op#(i(x),op(y,z)) -> op#(i(i(y)),z) op#(i(x),op(y,z)) -> op#(i(i(y)),z) -> op#(i(x),op(y,z)) -> i#(i(y)) op#(i(x),op(y,z)) -> op#(i(i(y)),z) -> op#(i(x),op(y,z)) -> i#(y) SCC Processor: #sccs: 1 #rules: 5 #arcs: 45/81 DPs: op#(i(x),op(y,op(z,w))) -> op#(f(x,i(x)),op(z,op(y,w))) op#(i(x),op(y,z)) -> op#(i(i(y)),z) op#(i(x),op(y,z)) -> op#(f(x,i(x)),op(i(i(y)),z)) op#(i(x),op(y,op(z,w))) -> op#(y,w) op#(i(x),op(y,op(z,w))) -> op#(z,op(y,w)) TRS: op(i(x),op(y,z)) -> op(f(x,i(x)),op(i(i(y)),z)) op(i(x),op(y,op(z,w))) -> op(f(x,i(x)),op(z,op(y,w))) i(x) -> x f(x,y) -> x Arctic Interpretation Processor: dimension: 1 interpretation: [op#](x0, x1) = 1x1, [f](x0, x1) = x0 + x1 + 0, [op](x0, x1) = 1x1 + 0, [i](x0) = x0 + 0 orientation: op#(i(x),op(y,op(z,w))) = 3w + 2 >= 3w + 2 = op#(f(x,i(x)),op(z,op(y,w))) op#(i(x),op(y,z)) = 2z + 1 >= 1z = op#(i(i(y)),z) op#(i(x),op(y,z)) = 2z + 1 >= 2z + 1 = op#(f(x,i(x)),op(i(i(y)),z)) op#(i(x),op(y,op(z,w))) = 3w + 2 >= 1w = op#(y,w) op#(i(x),op(y,op(z,w))) = 3w + 2 >= 2w + 1 = op#(z,op(y,w)) op(i(x),op(y,z)) = 2z + 1 >= 2z + 1 = op(f(x,i(x)),op(i(i(y)),z)) op(i(x),op(y,op(z,w))) = 3w + 2 >= 3w + 2 = op(f(x,i(x)),op(z,op(y,w))) i(x) = x + 0 >= x = x f(x,y) = x + y + 0 >= x = x problem: DPs: op#(i(x),op(y,op(z,w))) -> op#(f(x,i(x)),op(z,op(y,w))) op#(i(x),op(y,z)) -> op#(f(x,i(x)),op(i(i(y)),z)) op#(i(x),op(y,op(z,w))) -> op#(z,op(y,w)) TRS: op(i(x),op(y,z)) -> op(f(x,i(x)),op(i(i(y)),z)) op(i(x),op(y,op(z,w))) -> op(f(x,i(x)),op(z,op(y,w))) i(x) -> x f(x,y) -> x KBO Processor: argument filtering: pi(i) = [0] pi(op) = [1] pi(f) = 0 pi(op#) = 1 weight function: w0 = 1 w(op#) = w(f) = w(op) = w(i) = 1 precedence: op# ~ f ~ op ~ i problem: DPs: op#(i(x),op(y,op(z,w))) -> op#(f(x,i(x)),op(z,op(y,w))) op#(i(x),op(y,z)) -> op#(f(x,i(x)),op(i(i(y)),z)) TRS: op(i(x),op(y,z)) -> op(f(x,i(x)),op(i(i(y)),z)) op(i(x),op(y,op(z,w))) -> op(f(x,i(x)),op(z,op(y,w))) i(x) -> x f(x,y) -> x KBO Processor: argument filtering: pi(i) = [0] pi(op) = [] pi(f) = [0] pi(op#) = 0 weight function: w0 = 1 w(op#) = w(f) = w(op) = w(i) = 1 precedence: op ~ i > op# ~ f problem: DPs: TRS: op(i(x),op(y,z)) -> op(f(x,i(x)),op(i(i(y)),z)) op(i(x),op(y,op(z,w))) -> op(f(x,i(x)),op(z,op(y,w))) i(x) -> x f(x,y) -> x Qed