0.00/0.06 YES 0.00/0.06 0.00/0.06 Problem 1: 0.00/0.06 0.00/0.06 (VAR v_NonEmpty:S X:S) 0.00/0.06 (RULES 0.00/0.06 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.06 active(c) -> mark(f(g(c))) 0.00/0.06 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.06 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.06 proper(f(X:S)) -> f(proper(X:S)) 0.00/0.06 proper(g(X:S)) -> g(proper(X:S)) 0.00/0.06 proper(c) -> ok(c) 0.00/0.06 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.06 top(ok(X:S)) -> top(active(X:S)) 0.00/0.06 ) 0.00/0.06 (STRATEGY INNERMOST) 0.00/0.06 0.00/0.06 Problem 1: 0.00/0.06 0.00/0.06 Dependency Pairs Processor: 0.00/0.06 -> Pairs: 0.00/0.06 F(ok(X:S)) -> F(X:S) 0.00/0.06 G(ok(X:S)) -> G(X:S) 0.00/0.06 PROPER(f(X:S)) -> F(proper(X:S)) 0.00/0.06 PROPER(f(X:S)) -> PROPER(X:S) 0.00/0.06 PROPER(g(X:S)) -> G(proper(X:S)) 0.00/0.06 PROPER(g(X:S)) -> PROPER(X:S) 0.00/0.06 TOP(mark(X:S)) -> PROPER(X:S) 0.00/0.06 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.00/0.06 TOP(ok(X:S)) -> ACTIVE(X:S) 0.00/0.06 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.06 -> Rules: 0.00/0.06 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.06 active(c) -> mark(f(g(c))) 0.00/0.06 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.06 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.06 proper(f(X:S)) -> f(proper(X:S)) 0.00/0.06 proper(g(X:S)) -> g(proper(X:S)) 0.00/0.06 proper(c) -> ok(c) 0.00/0.06 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.06 top(ok(X:S)) -> top(active(X:S)) 0.00/0.06 0.00/0.06 Problem 1: 0.00/0.06 0.00/0.06 SCC Processor: 0.00/0.06 -> Pairs: 0.00/0.06 F(ok(X:S)) -> F(X:S) 0.00/0.06 G(ok(X:S)) -> G(X:S) 0.00/0.06 PROPER(f(X:S)) -> F(proper(X:S)) 0.00/0.06 PROPER(f(X:S)) -> PROPER(X:S) 0.00/0.06 PROPER(g(X:S)) -> G(proper(X:S)) 0.00/0.06 PROPER(g(X:S)) -> PROPER(X:S) 0.00/0.06 TOP(mark(X:S)) -> PROPER(X:S) 0.00/0.06 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.00/0.06 TOP(ok(X:S)) -> ACTIVE(X:S) 0.00/0.06 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.06 -> Rules: 0.00/0.06 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.06 active(c) -> mark(f(g(c))) 0.00/0.06 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.06 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.06 proper(f(X:S)) -> f(proper(X:S)) 0.00/0.06 proper(g(X:S)) -> g(proper(X:S)) 0.00/0.06 proper(c) -> ok(c) 0.00/0.06 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.06 top(ok(X:S)) -> top(active(X:S)) 0.00/0.06 ->Strongly Connected Components: 0.00/0.06 ->->Cycle: 0.00/0.06 ->->-> Pairs: 0.00/0.06 G(ok(X:S)) -> G(X:S) 0.00/0.06 ->->-> Rules: 0.00/0.06 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.06 active(c) -> mark(f(g(c))) 0.00/0.06 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.06 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.06 proper(f(X:S)) -> f(proper(X:S)) 0.00/0.06 proper(g(X:S)) -> g(proper(X:S)) 0.00/0.06 proper(c) -> ok(c) 0.00/0.06 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.06 top(ok(X:S)) -> top(active(X:S)) 0.00/0.06 ->->Cycle: 0.00/0.06 ->->-> Pairs: 0.00/0.06 F(ok(X:S)) -> F(X:S) 0.00/0.06 ->->-> Rules: 0.00/0.06 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.06 active(c) -> mark(f(g(c))) 0.00/0.06 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.06 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.06 proper(f(X:S)) -> f(proper(X:S)) 0.00/0.06 proper(g(X:S)) -> g(proper(X:S)) 0.00/0.06 proper(c) -> ok(c) 0.00/0.06 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.06 top(ok(X:S)) -> top(active(X:S)) 0.00/0.06 ->->Cycle: 0.00/0.06 ->->-> Pairs: 0.00/0.06 PROPER(f(X:S)) -> PROPER(X:S) 0.00/0.06 PROPER(g(X:S)) -> PROPER(X:S) 0.00/0.06 ->->-> Rules: 0.00/0.06 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.06 active(c) -> mark(f(g(c))) 0.00/0.06 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.06 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.06 proper(f(X:S)) -> f(proper(X:S)) 0.00/0.06 proper(g(X:S)) -> g(proper(X:S)) 0.00/0.06 proper(c) -> ok(c) 0.00/0.06 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.06 top(ok(X:S)) -> top(active(X:S)) 0.00/0.06 ->->Cycle: 0.00/0.06 ->->-> Pairs: 0.00/0.06 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.00/0.06 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.06 ->->-> Rules: 0.00/0.06 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.06 active(c) -> mark(f(g(c))) 0.00/0.06 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.06 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.06 proper(f(X:S)) -> f(proper(X:S)) 0.00/0.06 proper(g(X:S)) -> g(proper(X:S)) 0.00/0.06 proper(c) -> ok(c) 0.00/0.06 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.06 top(ok(X:S)) -> top(active(X:S)) 0.00/0.06 0.00/0.06 0.00/0.06 The problem is decomposed in 4 subproblems. 0.00/0.06 0.00/0.06 Problem 1.1: 0.00/0.06 0.00/0.06 Subterm Processor: 0.00/0.06 -> Pairs: 0.00/0.06 G(ok(X:S)) -> G(X:S) 0.00/0.06 -> Rules: 0.00/0.06 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.06 active(c) -> mark(f(g(c))) 0.00/0.06 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.06 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.06 proper(f(X:S)) -> f(proper(X:S)) 0.00/0.06 proper(g(X:S)) -> g(proper(X:S)) 0.00/0.06 proper(c) -> ok(c) 0.00/0.06 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.06 top(ok(X:S)) -> top(active(X:S)) 0.00/0.06 ->Projection: 0.00/0.06 pi(G) = 1 0.00/0.06 0.00/0.06 Problem 1.1: 0.00/0.06 0.00/0.06 SCC Processor: 0.00/0.06 -> Pairs: 0.00/0.06 Empty 0.00/0.06 -> Rules: 0.00/0.06 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.06 active(c) -> mark(f(g(c))) 0.00/0.06 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.06 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.06 proper(f(X:S)) -> f(proper(X:S)) 0.00/0.06 proper(g(X:S)) -> g(proper(X:S)) 0.00/0.06 proper(c) -> ok(c) 0.00/0.06 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.06 top(ok(X:S)) -> top(active(X:S)) 0.00/0.06 ->Strongly Connected Components: 0.00/0.06 There is no strongly connected component 0.00/0.06 0.00/0.06 The problem is finite. 0.00/0.06 0.00/0.06 Problem 1.2: 0.00/0.06 0.00/0.06 Subterm Processor: 0.00/0.06 -> Pairs: 0.00/0.06 F(ok(X:S)) -> F(X:S) 0.00/0.06 -> Rules: 0.00/0.06 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.06 active(c) -> mark(f(g(c))) 0.00/0.06 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.06 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.06 proper(f(X:S)) -> f(proper(X:S)) 0.00/0.06 proper(g(X:S)) -> g(proper(X:S)) 0.00/0.06 proper(c) -> ok(c) 0.00/0.06 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.06 top(ok(X:S)) -> top(active(X:S)) 0.00/0.06 ->Projection: 0.00/0.06 pi(F) = 1 0.00/0.06 0.00/0.06 Problem 1.2: 0.00/0.06 0.00/0.06 SCC Processor: 0.00/0.06 -> Pairs: 0.00/0.06 Empty 0.00/0.06 -> Rules: 0.00/0.06 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.06 active(c) -> mark(f(g(c))) 0.00/0.06 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.06 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.06 proper(f(X:S)) -> f(proper(X:S)) 0.00/0.06 proper(g(X:S)) -> g(proper(X:S)) 0.00/0.06 proper(c) -> ok(c) 0.00/0.06 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.06 top(ok(X:S)) -> top(active(X:S)) 0.00/0.06 ->Strongly Connected Components: 0.00/0.06 There is no strongly connected component 0.00/0.06 0.00/0.06 The problem is finite. 0.00/0.06 0.00/0.06 Problem 1.3: 0.00/0.06 0.00/0.06 Subterm Processor: 0.00/0.06 -> Pairs: 0.00/0.06 PROPER(f(X:S)) -> PROPER(X:S) 0.00/0.06 PROPER(g(X:S)) -> PROPER(X:S) 0.00/0.06 -> Rules: 0.00/0.06 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.06 active(c) -> mark(f(g(c))) 0.00/0.06 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.06 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.06 proper(f(X:S)) -> f(proper(X:S)) 0.00/0.06 proper(g(X:S)) -> g(proper(X:S)) 0.00/0.06 proper(c) -> ok(c) 0.00/0.06 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.06 top(ok(X:S)) -> top(active(X:S)) 0.00/0.06 ->Projection: 0.00/0.06 pi(PROPER) = 1 0.00/0.06 0.00/0.06 Problem 1.3: 0.00/0.06 0.00/0.06 SCC Processor: 0.00/0.06 -> Pairs: 0.00/0.06 Empty 0.00/0.06 -> Rules: 0.00/0.06 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.06 active(c) -> mark(f(g(c))) 0.00/0.06 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.06 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.06 proper(f(X:S)) -> f(proper(X:S)) 0.00/0.06 proper(g(X:S)) -> g(proper(X:S)) 0.00/0.06 proper(c) -> ok(c) 0.00/0.06 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.06 top(ok(X:S)) -> top(active(X:S)) 0.00/0.06 ->Strongly Connected Components: 0.00/0.06 There is no strongly connected component 0.00/0.06 0.00/0.06 The problem is finite. 0.00/0.06 0.00/0.06 Problem 1.4: 0.00/0.06 0.00/0.06 Reduction Pairs Processor: 0.00/0.06 -> Pairs: 0.00/0.06 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.00/0.06 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.06 -> Rules: 0.00/0.06 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.06 active(c) -> mark(f(g(c))) 0.00/0.06 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.06 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.06 proper(f(X:S)) -> f(proper(X:S)) 0.00/0.06 proper(g(X:S)) -> g(proper(X:S)) 0.00/0.06 proper(c) -> ok(c) 0.00/0.06 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.06 top(ok(X:S)) -> top(active(X:S)) 0.00/0.06 -> Usable rules: 0.00/0.06 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.06 active(c) -> mark(f(g(c))) 0.00/0.06 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.06 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.06 proper(f(X:S)) -> f(proper(X:S)) 0.00/0.06 proper(g(X:S)) -> g(proper(X:S)) 0.00/0.06 proper(c) -> ok(c) 0.00/0.06 ->Interpretation type: 0.00/0.06 Linear 0.00/0.06 ->Coefficients: 0.00/0.06 Natural Numbers 0.00/0.06 ->Dimension: 0.00/0.06 1 0.00/0.06 ->Bound: 0.00/0.06 2 0.00/0.06 ->Interpretation: 0.00/0.06 0.00/0.06 [active](X) = X 0.00/0.06 [f](X) = 1 0.00/0.06 [g](X) = 0 0.00/0.06 [proper](X) = X 0.00/0.06 [top](X) = 0 0.00/0.06 [c] = 2 0.00/0.06 [fSNonEmpty] = 0 0.00/0.06 [mark](X) = X + 1 0.00/0.06 [ok](X) = X 0.00/0.06 [ACTIVE](X) = 0 0.00/0.06 [F](X) = 0 0.00/0.06 [G](X) = 0 0.00/0.06 [PROPER](X) = 0 0.00/0.06 [TOP](X) = 2.X 0.00/0.06 0.00/0.06 Problem 1.4: 0.00/0.06 0.00/0.06 SCC Processor: 0.00/0.06 -> Pairs: 0.00/0.06 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.06 -> Rules: 0.00/0.06 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.06 active(c) -> mark(f(g(c))) 0.00/0.06 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.06 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.06 proper(f(X:S)) -> f(proper(X:S)) 0.00/0.06 proper(g(X:S)) -> g(proper(X:S)) 0.00/0.06 proper(c) -> ok(c) 0.00/0.06 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.06 top(ok(X:S)) -> top(active(X:S)) 0.00/0.06 ->Strongly Connected Components: 0.00/0.06 ->->Cycle: 0.00/0.06 ->->-> Pairs: 0.00/0.06 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.06 ->->-> Rules: 0.00/0.06 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.06 active(c) -> mark(f(g(c))) 0.00/0.06 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.06 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.06 proper(f(X:S)) -> f(proper(X:S)) 0.00/0.06 proper(g(X:S)) -> g(proper(X:S)) 0.00/0.06 proper(c) -> ok(c) 0.00/0.06 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.06 top(ok(X:S)) -> top(active(X:S)) 0.00/0.06 0.00/0.06 Problem 1.4: 0.00/0.06 0.00/0.06 Reduction Pairs Processor: 0.00/0.06 -> Pairs: 0.00/0.06 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.06 -> Rules: 0.00/0.06 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.06 active(c) -> mark(f(g(c))) 0.00/0.06 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.06 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.06 proper(f(X:S)) -> f(proper(X:S)) 0.00/0.06 proper(g(X:S)) -> g(proper(X:S)) 0.00/0.06 proper(c) -> ok(c) 0.00/0.06 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.06 top(ok(X:S)) -> top(active(X:S)) 0.00/0.06 -> Usable rules: 0.00/0.06 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.06 active(c) -> mark(f(g(c))) 0.00/0.06 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.06 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.06 ->Interpretation type: 0.00/0.06 Linear 0.00/0.06 ->Coefficients: 0.00/0.06 Natural Numbers 0.00/0.06 ->Dimension: 0.00/0.06 1 0.00/0.06 ->Bound: 0.00/0.06 2 0.00/0.06 ->Interpretation: 0.00/0.06 0.00/0.06 [active](X) = X + 1 0.00/0.06 [f](X) = X + 2 0.00/0.06 [g](X) = X 0.00/0.06 [proper](X) = 0 0.00/0.06 [top](X) = 0 0.00/0.06 [c] = 1 0.00/0.06 [fSNonEmpty] = 0 0.00/0.06 [mark](X) = 2 0.00/0.06 [ok](X) = X + 2 0.00/0.06 [ACTIVE](X) = 0 0.00/0.06 [F](X) = 0 0.00/0.06 [G](X) = 0 0.00/0.06 [PROPER](X) = 0 0.00/0.06 [TOP](X) = X 0.00/0.06 0.00/0.06 Problem 1.4: 0.00/0.06 0.00/0.06 SCC Processor: 0.00/0.06 -> Pairs: 0.00/0.06 Empty 0.00/0.06 -> Rules: 0.00/0.08 active(f(g(X:S))) -> mark(g(X:S)) 0.00/0.08 active(c) -> mark(f(g(c))) 0.00/0.08 f(ok(X:S)) -> ok(f(X:S)) 0.00/0.08 g(ok(X:S)) -> ok(g(X:S)) 0.00/0.08 proper(f(X:S)) -> f(proper(X:S)) 0.00/0.08 proper(g(X:S)) -> g(proper(X:S)) 0.00/0.08 proper(c) -> ok(c) 0.00/0.08 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.08 top(ok(X:S)) -> top(active(X:S)) 0.00/0.08 ->Strongly Connected Components: 0.00/0.08 There is no strongly connected component 0.00/0.08 0.00/0.08 The problem is finite. 0.00/0.08 EOF