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 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(i) = 1 w(op) = 0 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