0.01/1.52 YES 0.01/1.52 0.01/1.52 Problem 1: 0.01/1.52 0.01/1.52 (VAR v_NonEmpty:S X:S) 0.01/1.52 (RULES 0.01/1.52 active(f(f(a))) -> mark(f(g(f(a)))) 0.01/1.52 f(active(X:S)) -> f(X:S) 0.01/1.52 f(mark(X:S)) -> f(X:S) 0.01/1.52 g(active(X:S)) -> g(X:S) 0.01/1.52 g(mark(X:S)) -> g(X:S) 0.01/1.52 mark(f(X:S)) -> active(f(mark(X:S))) 0.01/1.52 mark(g(X:S)) -> active(g(X:S)) 0.01/1.52 mark(a) -> active(a) 0.01/1.52 ) 0.01/1.52 (STRATEGY INNERMOST) 0.01/1.52 0.01/1.52 Problem 1: 0.01/1.52 0.01/1.52 Dependency Pairs Processor: 0.01/1.52 -> Pairs: 0.01/1.52 ACTIVE(f(f(a))) -> MARK(f(g(f(a)))) 0.01/1.52 F(active(X:S)) -> F(X:S) 0.01/1.52 F(mark(X:S)) -> F(X:S) 0.01/1.52 G(active(X:S)) -> G(X:S) 0.01/1.52 G(mark(X:S)) -> G(X:S) 0.01/1.52 MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) 0.01/1.52 MARK(f(X:S)) -> F(mark(X:S)) 0.01/1.52 MARK(f(X:S)) -> MARK(X:S) 0.01/1.52 MARK(g(X:S)) -> ACTIVE(g(X:S)) 0.01/1.52 -> Rules: 0.01/1.52 active(f(f(a))) -> mark(f(g(f(a)))) 0.01/1.52 f(active(X:S)) -> f(X:S) 0.01/1.52 f(mark(X:S)) -> f(X:S) 0.01/1.52 g(active(X:S)) -> g(X:S) 0.01/1.52 g(mark(X:S)) -> g(X:S) 0.01/1.52 mark(f(X:S)) -> active(f(mark(X:S))) 0.01/1.52 mark(g(X:S)) -> active(g(X:S)) 0.01/1.52 mark(a) -> active(a) 0.01/1.52 0.01/1.52 Problem 1: 0.01/1.52 0.01/1.52 SCC Processor: 0.01/1.52 -> Pairs: 0.01/1.52 ACTIVE(f(f(a))) -> MARK(f(g(f(a)))) 0.01/1.52 F(active(X:S)) -> F(X:S) 0.01/1.52 F(mark(X:S)) -> F(X:S) 0.01/1.52 G(active(X:S)) -> G(X:S) 0.01/1.52 G(mark(X:S)) -> G(X:S) 0.01/1.52 MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) 0.01/1.52 MARK(f(X:S)) -> F(mark(X:S)) 0.01/1.52 MARK(f(X:S)) -> MARK(X:S) 0.01/1.52 MARK(g(X:S)) -> ACTIVE(g(X:S)) 0.01/1.52 -> Rules: 0.01/1.52 active(f(f(a))) -> mark(f(g(f(a)))) 0.01/1.52 f(active(X:S)) -> f(X:S) 0.01/1.52 f(mark(X:S)) -> f(X:S) 0.01/1.52 g(active(X:S)) -> g(X:S) 0.01/1.52 g(mark(X:S)) -> g(X:S) 0.01/1.52 mark(f(X:S)) -> active(f(mark(X:S))) 0.01/1.52 mark(g(X:S)) -> active(g(X:S)) 0.01/1.52 mark(a) -> active(a) 0.01/1.52 ->Strongly Connected Components: 0.01/1.52 ->->Cycle: 0.01/1.52 ->->-> Pairs: 0.01/1.52 G(active(X:S)) -> G(X:S) 0.01/1.52 G(mark(X:S)) -> G(X:S) 0.01/1.52 ->->-> Rules: 0.01/1.52 active(f(f(a))) -> mark(f(g(f(a)))) 0.01/1.52 f(active(X:S)) -> f(X:S) 0.01/1.52 f(mark(X:S)) -> f(X:S) 0.01/1.52 g(active(X:S)) -> g(X:S) 0.01/1.52 g(mark(X:S)) -> g(X:S) 0.01/1.52 mark(f(X:S)) -> active(f(mark(X:S))) 0.01/1.52 mark(g(X:S)) -> active(g(X:S)) 0.01/1.52 mark(a) -> active(a) 0.01/1.52 ->->Cycle: 0.01/1.52 ->->-> Pairs: 0.01/1.52 F(active(X:S)) -> F(X:S) 0.01/1.52 F(mark(X:S)) -> F(X:S) 0.01/1.52 ->->-> Rules: 0.01/1.52 active(f(f(a))) -> mark(f(g(f(a)))) 0.01/1.52 f(active(X:S)) -> f(X:S) 0.01/1.52 f(mark(X:S)) -> f(X:S) 0.01/1.52 g(active(X:S)) -> g(X:S) 0.01/1.52 g(mark(X:S)) -> g(X:S) 0.01/1.52 mark(f(X:S)) -> active(f(mark(X:S))) 0.01/1.52 mark(g(X:S)) -> active(g(X:S)) 0.01/1.52 mark(a) -> active(a) 0.01/1.52 ->->Cycle: 0.01/1.52 ->->-> Pairs: 0.01/1.52 ACTIVE(f(f(a))) -> MARK(f(g(f(a)))) 0.01/1.52 MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) 0.01/1.52 MARK(f(X:S)) -> MARK(X:S) 0.01/1.52 ->->-> Rules: 0.01/1.52 active(f(f(a))) -> mark(f(g(f(a)))) 0.01/1.52 f(active(X:S)) -> f(X:S) 0.01/1.52 f(mark(X:S)) -> f(X:S) 0.01/1.52 g(active(X:S)) -> g(X:S) 0.01/1.52 g(mark(X:S)) -> g(X:S) 0.01/1.52 mark(f(X:S)) -> active(f(mark(X:S))) 0.01/1.52 mark(g(X:S)) -> active(g(X:S)) 0.01/1.52 mark(a) -> active(a) 0.01/1.52 0.01/1.52 0.01/1.52 The problem is decomposed in 3 subproblems. 0.01/1.52 0.01/1.52 Problem 1.1: 0.01/1.52 0.01/1.52 Subterm Processor: 0.01/1.52 -> Pairs: 0.01/1.52 G(active(X:S)) -> G(X:S) 0.01/1.52 G(mark(X:S)) -> G(X:S) 0.01/1.52 -> Rules: 0.01/1.52 active(f(f(a))) -> mark(f(g(f(a)))) 0.01/1.52 f(active(X:S)) -> f(X:S) 0.01/1.52 f(mark(X:S)) -> f(X:S) 0.01/1.52 g(active(X:S)) -> g(X:S) 0.01/1.52 g(mark(X:S)) -> g(X:S) 0.01/1.52 mark(f(X:S)) -> active(f(mark(X:S))) 0.01/1.52 mark(g(X:S)) -> active(g(X:S)) 0.01/1.52 mark(a) -> active(a) 0.01/1.52 ->Projection: 0.01/1.52 pi(G) = 1 0.01/1.52 0.01/1.52 Problem 1.1: 0.01/1.52 0.01/1.52 SCC Processor: 0.01/1.52 -> Pairs: 0.01/1.52 Empty 0.01/1.52 -> Rules: 0.01/1.52 active(f(f(a))) -> mark(f(g(f(a)))) 0.01/1.52 f(active(X:S)) -> f(X:S) 0.01/1.52 f(mark(X:S)) -> f(X:S) 0.01/1.52 g(active(X:S)) -> g(X:S) 0.01/1.52 g(mark(X:S)) -> g(X:S) 0.01/1.52 mark(f(X:S)) -> active(f(mark(X:S))) 0.01/1.52 mark(g(X:S)) -> active(g(X:S)) 0.01/1.52 mark(a) -> active(a) 0.01/1.52 ->Strongly Connected Components: 0.01/1.52 There is no strongly connected component 0.01/1.52 0.01/1.52 The problem is finite. 0.01/1.52 0.01/1.52 Problem 1.2: 0.01/1.52 0.01/1.52 Subterm Processor: 0.01/1.52 -> Pairs: 0.01/1.52 F(active(X:S)) -> F(X:S) 0.01/1.52 F(mark(X:S)) -> F(X:S) 0.01/1.52 -> Rules: 0.01/1.52 active(f(f(a))) -> mark(f(g(f(a)))) 0.01/1.52 f(active(X:S)) -> f(X:S) 0.01/1.52 f(mark(X:S)) -> f(X:S) 0.01/1.52 g(active(X:S)) -> g(X:S) 0.01/1.52 g(mark(X:S)) -> g(X:S) 0.01/1.52 mark(f(X:S)) -> active(f(mark(X:S))) 0.01/1.52 mark(g(X:S)) -> active(g(X:S)) 0.01/1.52 mark(a) -> active(a) 0.01/1.52 ->Projection: 0.01/1.52 pi(F) = 1 0.01/1.52 0.01/1.52 Problem 1.2: 0.01/1.52 0.01/1.52 SCC Processor: 0.01/1.52 -> Pairs: 0.01/1.52 Empty 0.01/1.52 -> Rules: 0.01/1.52 active(f(f(a))) -> mark(f(g(f(a)))) 0.01/1.52 f(active(X:S)) -> f(X:S) 0.01/1.52 f(mark(X:S)) -> f(X:S) 0.01/1.52 g(active(X:S)) -> g(X:S) 0.01/1.52 g(mark(X:S)) -> g(X:S) 0.01/1.52 mark(f(X:S)) -> active(f(mark(X:S))) 0.01/1.52 mark(g(X:S)) -> active(g(X:S)) 0.01/1.52 mark(a) -> active(a) 0.01/1.52 ->Strongly Connected Components: 0.01/1.52 There is no strongly connected component 0.01/1.52 0.01/1.52 The problem is finite. 0.01/1.52 0.01/1.52 Problem 1.3: 0.01/1.52 0.01/1.52 Reduction Pairs Processor: 0.01/1.52 -> Pairs: 0.01/1.52 ACTIVE(f(f(a))) -> MARK(f(g(f(a)))) 0.01/1.52 MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) 0.01/1.52 MARK(f(X:S)) -> MARK(X:S) 0.01/1.52 -> Rules: 0.01/1.52 active(f(f(a))) -> mark(f(g(f(a)))) 0.01/1.52 f(active(X:S)) -> f(X:S) 0.01/1.52 f(mark(X:S)) -> f(X:S) 0.01/1.52 g(active(X:S)) -> g(X:S) 0.01/1.52 g(mark(X:S)) -> g(X:S) 0.01/1.52 mark(f(X:S)) -> active(f(mark(X:S))) 0.01/1.52 mark(g(X:S)) -> active(g(X:S)) 0.01/1.52 mark(a) -> active(a) 0.01/1.52 -> Usable rules: 0.01/1.52 active(f(f(a))) -> mark(f(g(f(a)))) 0.01/1.52 f(active(X:S)) -> f(X:S) 0.01/1.52 f(mark(X:S)) -> f(X:S) 0.01/1.52 g(active(X:S)) -> g(X:S) 0.01/1.52 g(mark(X:S)) -> g(X:S) 0.01/1.52 mark(f(X:S)) -> active(f(mark(X:S))) 0.01/1.52 mark(g(X:S)) -> active(g(X:S)) 0.01/1.52 mark(a) -> active(a) 0.01/1.52 ->Interpretation type: 0.01/1.52 Linear 0.01/1.52 ->Coefficients: 0.01/1.52 Natural Numbers 0.01/1.52 ->Dimension: 0.01/1.52 1 0.01/1.52 ->Bound: 0.01/1.52 2 0.01/1.52 ->Interpretation: 0.01/1.52 0.01/1.52 [active](X) = X 0.01/1.52 [f](X) = X + 2 0.01/1.52 [g](X) = 0 0.01/1.52 [mark](X) = X + 1 0.01/1.52 [a] = 2 0.01/1.52 [fSNonEmpty] = 0 0.01/1.52 [ACTIVE](X) = X 0.01/1.52 [F](X) = 0 0.01/1.52 [G](X) = 0 0.01/1.52 [MARK](X) = 2.X 0.01/1.52 0.01/1.52 Problem 1.3: 0.01/1.52 0.01/1.52 SCC Processor: 0.01/1.52 -> Pairs: 0.01/1.52 MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) 0.01/1.52 MARK(f(X:S)) -> MARK(X:S) 0.01/1.52 -> Rules: 0.01/1.52 active(f(f(a))) -> mark(f(g(f(a)))) 0.01/1.52 f(active(X:S)) -> f(X:S) 0.01/1.52 f(mark(X:S)) -> f(X:S) 0.01/1.52 g(active(X:S)) -> g(X:S) 0.01/1.52 g(mark(X:S)) -> g(X:S) 0.01/1.52 mark(f(X:S)) -> active(f(mark(X:S))) 0.01/1.52 mark(g(X:S)) -> active(g(X:S)) 0.01/1.52 mark(a) -> active(a) 0.01/1.52 ->Strongly Connected Components: 0.01/1.52 ->->Cycle: 0.01/1.52 ->->-> Pairs: 0.01/1.52 MARK(f(X:S)) -> MARK(X:S) 0.01/1.52 ->->-> Rules: 0.01/1.52 active(f(f(a))) -> mark(f(g(f(a)))) 0.01/1.52 f(active(X:S)) -> f(X:S) 0.01/1.52 f(mark(X:S)) -> f(X:S) 0.01/1.52 g(active(X:S)) -> g(X:S) 0.01/1.52 g(mark(X:S)) -> g(X:S) 0.01/1.52 mark(f(X:S)) -> active(f(mark(X:S))) 0.01/1.52 mark(g(X:S)) -> active(g(X:S)) 0.01/1.52 mark(a) -> active(a) 0.01/1.52 0.01/1.52 Problem 1.3: 0.01/1.52 0.01/1.52 Subterm Processor: 0.01/1.52 -> Pairs: 0.01/1.52 MARK(f(X:S)) -> MARK(X:S) 0.01/1.52 -> Rules: 0.01/1.52 active(f(f(a))) -> mark(f(g(f(a)))) 0.01/1.52 f(active(X:S)) -> f(X:S) 0.01/1.52 f(mark(X:S)) -> f(X:S) 0.01/1.52 g(active(X:S)) -> g(X:S) 0.01/1.52 g(mark(X:S)) -> g(X:S) 0.01/1.52 mark(f(X:S)) -> active(f(mark(X:S))) 0.01/1.52 mark(g(X:S)) -> active(g(X:S)) 0.01/1.52 mark(a) -> active(a) 0.01/1.52 ->Projection: 0.01/1.52 pi(MARK) = 1 0.01/1.52 0.01/1.52 Problem 1.3: 0.01/1.52 0.01/1.52 SCC Processor: 0.01/1.52 -> Pairs: 0.01/1.52 Empty 0.01/1.52 -> Rules: 0.01/1.52 active(f(f(a))) -> mark(f(g(f(a)))) 0.01/1.52 f(active(X:S)) -> f(X:S) 0.01/1.52 f(mark(X:S)) -> f(X:S) 0.01/1.52 g(active(X:S)) -> g(X:S) 0.01/1.52 g(mark(X:S)) -> g(X:S) 0.01/1.52 mark(f(X:S)) -> active(f(mark(X:S))) 0.01/1.52 mark(g(X:S)) -> active(g(X:S)) 0.01/1.52 mark(a) -> active(a) 0.01/1.52 ->Strongly Connected Components: 0.01/1.52 There is no strongly connected component 0.01/1.52 0.01/1.52 The problem is finite. 0.04/1.52 EOF