41.80/10.90 YES 41.83/10.92 41.83/10.92 Problem: 41.83/10.92 b(a(x1)) -> b(b(x1)) 41.83/10.92 b(a(b(x1))) -> b(a(a(x1))) 41.83/10.92 a(a(a(x1))) -> a(b(b(x1))) 41.83/10.92 41.83/10.92 Proof: 41.83/10.92 DP Processor: 41.83/10.92 DPs: 41.83/10.92 b#(a(x1)) -> b#(x1) 41.83/10.92 b#(a(x1)) -> b#(b(x1)) 41.83/10.92 b#(a(b(x1))) -> a#(x1) 41.83/10.92 b#(a(b(x1))) -> a#(a(x1)) 41.83/10.92 b#(a(b(x1))) -> b#(a(a(x1))) 41.83/10.92 a#(a(a(x1))) -> b#(x1) 41.83/10.92 a#(a(a(x1))) -> b#(b(x1)) 41.83/10.92 a#(a(a(x1))) -> a#(b(b(x1))) 41.83/10.92 TRS: 41.83/10.92 b(a(x1)) -> b(b(x1)) 41.83/10.92 b(a(b(x1))) -> b(a(a(x1))) 41.83/10.92 a(a(a(x1))) -> a(b(b(x1))) 41.83/10.92 TDG Processor: 41.83/10.92 DPs: 41.83/10.92 b#(a(x1)) -> b#(x1) 41.83/10.92 b#(a(x1)) -> b#(b(x1)) 41.83/10.92 b#(a(b(x1))) -> a#(x1) 41.83/10.92 b#(a(b(x1))) -> a#(a(x1)) 41.83/10.92 b#(a(b(x1))) -> b#(a(a(x1))) 41.83/10.92 a#(a(a(x1))) -> b#(x1) 41.83/10.92 a#(a(a(x1))) -> b#(b(x1)) 41.83/10.92 a#(a(a(x1))) -> a#(b(b(x1))) 41.83/10.92 TRS: 41.83/10.92 b(a(x1)) -> b(b(x1)) 41.83/10.92 b(a(b(x1))) -> b(a(a(x1))) 41.83/10.92 a(a(a(x1))) -> a(b(b(x1))) 41.83/10.92 graph: 41.83/10.92 a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) 41.83/10.92 a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(b(x1)) 41.83/10.92 a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(x1) 41.83/10.92 a#(a(a(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> b#(a(a(x1))) 41.83/10.92 a#(a(a(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> a#(a(x1)) 41.83/10.92 a#(a(a(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> a#(x1) 41.83/10.92 a#(a(a(x1))) -> b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) 41.83/10.92 a#(a(a(x1))) -> b#(b(x1)) -> b#(a(x1)) -> b#(x1) 41.83/10.92 a#(a(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> b#(a(a(x1))) 41.83/10.92 a#(a(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> a#(a(x1)) 41.83/10.92 a#(a(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> a#(x1) 41.83/10.92 a#(a(a(x1))) -> b#(x1) -> b#(a(x1)) -> b#(b(x1)) 41.83/10.92 a#(a(a(x1))) -> b#(x1) -> b#(a(x1)) -> b#(x1) 41.83/10.92 b#(a(b(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) 41.83/10.92 b#(a(b(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(b(x1)) 41.83/10.92 b#(a(b(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(x1) 41.83/10.92 b#(a(b(x1))) -> a#(x1) -> a#(a(a(x1))) -> a#(b(b(x1))) 41.83/10.92 b#(a(b(x1))) -> a#(x1) -> a#(a(a(x1))) -> b#(b(x1)) 41.83/10.92 b#(a(b(x1))) -> a#(x1) -> a#(a(a(x1))) -> b#(x1) 41.83/10.92 b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(b(x1))) -> b#(a(a(x1))) 41.83/10.92 b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(b(x1))) -> a#(a(x1)) 41.83/10.92 b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(b(x1))) -> a#(x1) 41.83/10.92 b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(x1)) -> b#(b(x1)) 41.83/10.92 b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(x1)) -> b#(x1) 41.83/10.92 b#(a(x1)) -> b#(b(x1)) -> b#(a(b(x1))) -> b#(a(a(x1))) 41.83/10.92 b#(a(x1)) -> b#(b(x1)) -> b#(a(b(x1))) -> a#(a(x1)) 41.83/10.92 b#(a(x1)) -> b#(b(x1)) -> b#(a(b(x1))) -> a#(x1) 41.83/10.92 b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) 41.83/10.92 b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(x1) 41.83/10.92 b#(a(x1)) -> b#(x1) -> b#(a(b(x1))) -> b#(a(a(x1))) 41.83/10.92 b#(a(x1)) -> b#(x1) -> b#(a(b(x1))) -> a#(a(x1)) 41.83/10.92 b#(a(x1)) -> b#(x1) -> b#(a(b(x1))) -> a#(x1) 41.83/10.92 b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(b(x1)) 41.83/10.92 b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1) 41.83/10.92 Polynomial Interpretation Processor: 41.83/10.92 dimension: 1 41.83/10.92 interpretation: 41.83/10.92 [a#](x0) = x0, 41.83/10.92 41.83/10.92 [b#](x0) = x0, 41.83/10.92 41.83/10.92 [b](x0) = x0 + 1, 41.83/10.92 41.83/10.92 [a](x0) = x0 + 1 41.83/10.92 orientation: 41.83/10.92 b#(a(x1)) = x1 + 1 >= x1 = b#(x1) 41.83/10.92 41.83/10.92 b#(a(x1)) = x1 + 1 >= x1 + 1 = b#(b(x1)) 41.83/10.92 41.83/10.92 b#(a(b(x1))) = x1 + 2 >= x1 = a#(x1) 41.83/10.92 41.83/10.92 b#(a(b(x1))) = x1 + 2 >= x1 + 1 = a#(a(x1)) 41.83/10.92 41.83/10.92 b#(a(b(x1))) = x1 + 2 >= x1 + 2 = b#(a(a(x1))) 41.83/10.92 41.83/10.92 a#(a(a(x1))) = x1 + 2 >= x1 = b#(x1) 41.83/10.92 41.83/10.92 a#(a(a(x1))) = x1 + 2 >= x1 + 1 = b#(b(x1)) 41.83/10.92 41.83/10.92 a#(a(a(x1))) = x1 + 2 >= x1 + 2 = a#(b(b(x1))) 41.83/10.92 41.83/10.92 b(a(x1)) = x1 + 2 >= x1 + 2 = b(b(x1)) 41.83/10.92 41.83/10.92 b(a(b(x1))) = x1 + 3 >= x1 + 3 = b(a(a(x1))) 41.83/10.92 41.83/10.92 a(a(a(x1))) = x1 + 3 >= x1 + 3 = a(b(b(x1))) 41.83/10.92 problem: 41.83/10.92 DPs: 41.83/10.92 b#(a(x1)) -> b#(b(x1)) 41.83/10.92 b#(a(b(x1))) -> b#(a(a(x1))) 41.83/10.92 a#(a(a(x1))) -> a#(b(b(x1))) 41.83/10.92 TRS: 41.83/10.92 b(a(x1)) -> b(b(x1)) 41.83/10.92 b(a(b(x1))) -> b(a(a(x1))) 41.83/10.92 a(a(a(x1))) -> a(b(b(x1))) 41.83/10.92 SCC Processor: 41.83/10.92 #sccs: 2 41.83/10.92 #rules: 3 41.83/10.92 #arcs: 34/9 41.83/10.92 DPs: 41.83/10.92 b#(a(b(x1))) -> b#(a(a(x1))) 41.83/10.92 b#(a(x1)) -> b#(b(x1)) 41.83/10.92 TRS: 41.83/10.92 b(a(x1)) -> b(b(x1)) 41.83/10.92 b(a(b(x1))) -> b(a(a(x1))) 41.83/10.92 a(a(a(x1))) -> a(b(b(x1))) 41.83/10.92 Polynomial Interpretation Processor: 41.83/10.92 dimension: 1 41.83/10.92 usable rules: 41.83/10.92 b(a(x1)) -> b(b(x1)) 41.83/10.92 b(a(b(x1))) -> b(a(a(x1))) 41.83/10.92 a(a(a(x1))) -> a(b(b(x1))) 41.83/10.94 interpretation: 41.83/10.94 [b#](x0) = x0, 41.83/10.94 41.83/10.94 [b](x0) = 0, 41.83/10.94 41.83/10.94 [a](x0) = 1 41.83/10.94 orientation: 41.83/10.94 b#(a(b(x1))) = 1 >= 1 = b#(a(a(x1))) 41.83/10.94 41.83/10.94 b#(a(x1)) = 1 >= 0 = b#(b(x1)) 41.83/10.94 41.83/10.94 b(a(x1)) = 0 >= 0 = b(b(x1)) 41.83/10.94 41.83/10.94 b(a(b(x1))) = 0 >= 0 = b(a(a(x1))) 41.83/10.94 41.83/10.94 a(a(a(x1))) = 1 >= 1 = a(b(b(x1))) 41.83/10.94 problem: 41.83/10.94 DPs: 41.83/10.94 b#(a(b(x1))) -> b#(a(a(x1))) 41.83/10.94 TRS: 41.83/10.94 b(a(x1)) -> b(b(x1)) 41.83/10.94 b(a(b(x1))) -> b(a(a(x1))) 41.83/10.94 a(a(a(x1))) -> a(b(b(x1))) 41.83/10.94 Root-Labeling Processor: 41.83/10.94 DPs: 41.83/10.94 b{#,(f4)}(f4(a)(a(b)(b(f4)(x1)))) -> b{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) 41.83/10.94 b{#,(f4)}(f4(a)(a(b)(b(b)(x1)))) -> b{#,(f4)}(f4(a)(a(a)(a(b)(x1)))) 41.83/10.94 b{#,(f4)}(f4(a)(a(b)(b(a)(x1)))) -> b{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) 41.83/10.94 TRS: 41.83/10.94 b(a)(a(f4)(x1)) -> b(b)(b(f4)(x1)) 41.83/10.94 b(a)(a(b)(x1)) -> b(b)(b(b)(x1)) 41.83/10.94 b(a)(a(a)(x1)) -> b(b)(b(a)(x1)) 41.83/10.94 b(a)(a(b)(b(f4)(x1))) -> b(a)(a(a)(a(f4)(x1))) 41.83/10.94 b(a)(a(b)(b(b)(x1))) -> b(a)(a(a)(a(b)(x1))) 41.83/10.94 b(a)(a(b)(b(a)(x1))) -> b(a)(a(a)(a(a)(x1))) 41.83/10.94 a(a)(a(a)(a(f4)(x1))) -> a(b)(b(b)(b(f4)(x1))) 41.83/10.94 a(a)(a(a)(a(b)(x1))) -> a(b)(b(b)(b(b)(x1))) 41.83/10.94 a(a)(a(a)(a(a)(x1))) -> a(b)(b(b)(b(a)(x1))) 41.83/10.94 Polynomial Interpretation Processor: 41.83/10.94 dimension: 1 41.83/10.94 usable rules: 41.83/10.94 a(a)(a(a)(a(f4)(x1))) -> a(b)(b(b)(b(f4)(x1))) 41.83/10.94 a(a)(a(a)(a(b)(x1))) -> a(b)(b(b)(b(b)(x1))) 41.83/10.94 a(a)(a(a)(a(a)(x1))) -> a(b)(b(b)(b(a)(x1))) 41.83/10.94 interpretation: 41.83/10.94 [b(a)](x0) = 1, 41.83/10.94 41.83/10.94 [b(b)](x0) = 0, 41.83/10.94 41.83/10.94 [a(f4)](x0) = 0, 41.83/10.94 41.83/10.94 [a(a)](x0) = 1, 41.83/10.94 41.83/10.94 [b(f4)](x0) = 0, 41.83/10.94 41.83/10.94 [a(b)](x0) = x0 + 1, 41.83/10.94 41.83/10.94 [f4(a)](x0) = x0, 41.83/10.94 41.83/10.94 [b{#,(f4)}](x0) = x0 + 1 41.83/10.94 orientation: 41.83/10.94 b{#,(f4)}(f4(a)(a(b)(b(f4)(x1)))) = 2 >= 2 = b{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) 41.83/10.94 41.83/10.94 b{#,(f4)}(f4(a)(a(b)(b(b)(x1)))) = 2 >= 2 = b{#,(f4)}(f4(a)(a(a)(a(b)(x1)))) 41.83/10.94 41.83/10.94 b{#,(f4)}(f4(a)(a(b)(b(a)(x1)))) = 3 >= 2 = b{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) 41.83/10.94 41.83/10.94 b(a)(a(f4)(x1)) = 1 >= 0 = b(b)(b(f4)(x1)) 41.83/10.94 41.83/10.94 b(a)(a(b)(x1)) = 1 >= 0 = b(b)(b(b)(x1)) 41.83/10.94 41.83/10.94 b(a)(a(a)(x1)) = 1 >= 0 = b(b)(b(a)(x1)) 41.83/10.94 41.83/10.94 b(a)(a(b)(b(f4)(x1))) = 1 >= 1 = b(a)(a(a)(a(f4)(x1))) 41.83/10.94 41.83/10.94 b(a)(a(b)(b(b)(x1))) = 1 >= 1 = b(a)(a(a)(a(b)(x1))) 41.83/10.94 41.83/10.94 b(a)(a(b)(b(a)(x1))) = 1 >= 1 = b(a)(a(a)(a(a)(x1))) 41.83/10.94 41.83/10.94 a(a)(a(a)(a(f4)(x1))) = 1 >= 1 = a(b)(b(b)(b(f4)(x1))) 41.83/10.94 41.83/10.94 a(a)(a(a)(a(b)(x1))) = 1 >= 1 = a(b)(b(b)(b(b)(x1))) 41.83/10.94 41.83/10.94 a(a)(a(a)(a(a)(x1))) = 1 >= 1 = a(b)(b(b)(b(a)(x1))) 41.83/10.94 problem: 41.83/10.94 DPs: 41.83/10.94 b{#,(f4)}(f4(a)(a(b)(b(f4)(x1)))) -> b{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) 41.83/10.94 b{#,(f4)}(f4(a)(a(b)(b(b)(x1)))) -> b{#,(f4)}(f4(a)(a(a)(a(b)(x1)))) 41.83/10.94 TRS: 41.83/10.94 b(a)(a(f4)(x1)) -> b(b)(b(f4)(x1)) 41.83/10.94 b(a)(a(b)(x1)) -> b(b)(b(b)(x1)) 41.83/10.94 b(a)(a(a)(x1)) -> b(b)(b(a)(x1)) 41.83/10.94 b(a)(a(b)(b(f4)(x1))) -> b(a)(a(a)(a(f4)(x1))) 41.83/10.94 b(a)(a(b)(b(b)(x1))) -> b(a)(a(a)(a(b)(x1))) 41.83/10.94 b(a)(a(b)(b(a)(x1))) -> b(a)(a(a)(a(a)(x1))) 41.83/10.94 a(a)(a(a)(a(f4)(x1))) -> a(b)(b(b)(b(f4)(x1))) 41.83/10.94 a(a)(a(a)(a(b)(x1))) -> a(b)(b(b)(b(b)(x1))) 41.83/10.94 a(a)(a(a)(a(a)(x1))) -> a(b)(b(b)(b(a)(x1))) 41.83/10.94 Usable Rule Processor: 41.83/10.94 DPs: 41.83/10.94 b{#,(f4)}(f4(a)(a(b)(b(f4)(x1)))) -> b{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) 41.83/10.94 b{#,(f4)}(f4(a)(a(b)(b(b)(x1)))) -> b{#,(f4)}(f4(a)(a(a)(a(b)(x1)))) 41.83/10.94 TRS: 41.83/10.94 41.83/10.94 Polynomial Interpretation Processor: 41.83/10.94 dimension: 1 41.83/10.94 interpretation: 41.83/10.94 [b(b)](x0) = x0 + 1, 41.83/10.94 41.83/10.94 [a(f4)](x0) = x0 + 1, 41.83/10.94 41.83/10.94 [a(a)](x0) = x0, 41.83/10.94 41.83/10.94 [b(f4)](x0) = x0 + 1, 41.83/10.94 41.83/10.94 [a(b)](x0) = x0, 41.83/10.94 41.83/10.94 [f4(a)](x0) = x0, 41.83/10.94 41.83/10.94 [b{#,(f4)}](x0) = x0 41.83/10.94 orientation: 41.83/10.94 b{#,(f4)}(f4(a)(a(b)(b(f4)(x1)))) = x1 + 1 >= x1 + 1 = b{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) 41.83/10.95 41.83/10.95 b{#,(f4)}(f4(a)(a(b)(b(b)(x1)))) = x1 + 1 >= x1 = b{#,(f4)}(f4(a)(a(a)(a(b)(x1)))) 41.83/10.95 problem: 41.83/10.95 DPs: 41.83/10.95 b{#,(f4)}(f4(a)(a(b)(b(f4)(x1)))) -> b{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) 41.83/10.95 TRS: 41.83/10.95 41.83/10.95 Polynomial Interpretation Processor: 41.83/10.95 dimension: 1 41.83/10.95 interpretation: 41.83/10.95 [a(f4)](x0) = x0, 41.83/10.95 41.83/10.95 [a(a)](x0) = x0, 41.83/10.95 41.83/10.95 [b(f4)](x0) = x0, 41.83/10.95 41.83/10.95 [a(b)](x0) = x0 + 1, 41.83/10.95 41.83/10.95 [f4(a)](x0) = x0, 41.83/10.95 41.83/10.95 [b{#,(f4)}](x0) = x0 41.83/10.95 orientation: 41.83/10.95 b{#,(f4)}(f4(a)(a(b)(b(f4)(x1)))) = x1 + 1 >= x1 = b{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) 41.83/10.95 problem: 41.83/10.95 DPs: 41.83/10.95 41.83/10.95 TRS: 41.83/10.95 41.83/10.95 Qed 41.83/10.95 41.83/10.95 DPs: 41.83/10.95 a#(a(a(x1))) -> a#(b(b(x1))) 41.83/10.95 TRS: 41.83/10.95 b(a(x1)) -> b(b(x1)) 41.83/10.95 b(a(b(x1))) -> b(a(a(x1))) 41.83/10.95 a(a(a(x1))) -> a(b(b(x1))) 41.83/10.95 Polynomial Interpretation Processor: 41.83/10.95 dimension: 1 41.83/10.95 usable rules: 41.83/10.95 b(a(x1)) -> b(b(x1)) 41.83/10.95 b(a(b(x1))) -> b(a(a(x1))) 41.83/10.95 interpretation: 41.83/10.95 [a#](x0) = x0, 41.83/10.95 41.83/10.95 [b](x0) = 0, 41.83/10.95 41.83/10.95 [a](x0) = 1 41.83/10.95 orientation: 41.83/10.95 a#(a(a(x1))) = 1 >= 0 = a#(b(b(x1))) 41.83/10.95 41.83/10.95 b(a(x1)) = 0 >= 0 = b(b(x1)) 41.83/10.95 41.83/10.95 b(a(b(x1))) = 0 >= 0 = b(a(a(x1))) 41.83/10.95 41.83/10.95 a(a(a(x1))) = 1 >= 1 = a(b(b(x1))) 41.83/10.95 problem: 41.83/10.95 DPs: 41.83/10.95 41.83/10.95 TRS: 41.83/10.95 b(a(x1)) -> b(b(x1)) 41.83/10.95 b(a(b(x1))) -> b(a(a(x1))) 41.83/10.95 a(a(a(x1))) -> a(b(b(x1))) 41.83/10.95 Qed 41.83/10.95 EOF