0.03/1.05 YES 0.03/1.05 0.03/1.05 Problem 1: 0.03/1.05 0.03/1.05 (VAR v_NonEmpty:S X:S X1:S X2:S Y:S) 0.03/1.05 (RULES 0.03/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.03/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.03/1.05 active(g(X:S)) -> g(active(X:S)) 0.03/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.03/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.03/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.03/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.03/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.03/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.03/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.03/1.05 top(ok(X:S)) -> top(active(X:S)) 0.03/1.05 ) 0.03/1.05 (STRATEGY INNERMOST) 0.03/1.05 0.03/1.05 Problem 1: 0.03/1.05 0.03/1.05 Dependency Pairs Processor: 0.03/1.05 -> Pairs: 0.03/1.05 ACTIVE(f(g(X:S),Y:S)) -> F(X:S,f(g(X:S),Y:S)) 0.03/1.05 ACTIVE(f(X1:S,X2:S)) -> ACTIVE(X1:S) 0.03/1.05 ACTIVE(f(X1:S,X2:S)) -> F(active(X1:S),X2:S) 0.03/1.05 ACTIVE(g(X:S)) -> ACTIVE(X:S) 0.03/1.05 ACTIVE(g(X:S)) -> G(active(X:S)) 0.03/1.05 F(mark(X1:S),X2:S) -> F(X1:S,X2:S) 0.03/1.05 F(ok(X1:S),ok(X2:S)) -> F(X1:S,X2:S) 0.03/1.05 G(mark(X:S)) -> G(X:S) 0.03/1.05 G(ok(X:S)) -> G(X:S) 0.03/1.05 PROPER(f(X1:S,X2:S)) -> F(proper(X1:S),proper(X2:S)) 0.03/1.05 PROPER(f(X1:S,X2:S)) -> PROPER(X1:S) 0.03/1.05 PROPER(f(X1:S,X2:S)) -> PROPER(X2:S) 0.03/1.05 PROPER(g(X:S)) -> G(proper(X:S)) 0.03/1.05 PROPER(g(X:S)) -> PROPER(X:S) 0.03/1.05 TOP(mark(X:S)) -> PROPER(X:S) 0.03/1.05 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.03/1.05 TOP(ok(X:S)) -> ACTIVE(X:S) 0.03/1.05 TOP(ok(X:S)) -> TOP(active(X:S)) 0.03/1.05 -> Rules: 0.03/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.03/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.03/1.05 active(g(X:S)) -> g(active(X:S)) 0.03/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.03/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.03/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.03/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.03/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.03/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.03/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.03/1.05 top(ok(X:S)) -> top(active(X:S)) 0.03/1.05 0.03/1.05 Problem 1: 0.03/1.05 0.03/1.05 SCC Processor: 0.03/1.05 -> Pairs: 0.03/1.05 ACTIVE(f(g(X:S),Y:S)) -> F(X:S,f(g(X:S),Y:S)) 0.03/1.05 ACTIVE(f(X1:S,X2:S)) -> ACTIVE(X1:S) 0.03/1.05 ACTIVE(f(X1:S,X2:S)) -> F(active(X1:S),X2:S) 0.03/1.05 ACTIVE(g(X:S)) -> ACTIVE(X:S) 0.03/1.05 ACTIVE(g(X:S)) -> G(active(X:S)) 0.03/1.05 F(mark(X1:S),X2:S) -> F(X1:S,X2:S) 0.03/1.05 F(ok(X1:S),ok(X2:S)) -> F(X1:S,X2:S) 0.03/1.05 G(mark(X:S)) -> G(X:S) 0.03/1.05 G(ok(X:S)) -> G(X:S) 0.03/1.05 PROPER(f(X1:S,X2:S)) -> F(proper(X1:S),proper(X2:S)) 0.05/1.05 PROPER(f(X1:S,X2:S)) -> PROPER(X1:S) 0.05/1.05 PROPER(f(X1:S,X2:S)) -> PROPER(X2:S) 0.05/1.05 PROPER(g(X:S)) -> G(proper(X:S)) 0.05/1.05 PROPER(g(X:S)) -> PROPER(X:S) 0.05/1.05 TOP(mark(X:S)) -> PROPER(X:S) 0.05/1.05 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.05/1.05 TOP(ok(X:S)) -> ACTIVE(X:S) 0.05/1.05 TOP(ok(X:S)) -> TOP(active(X:S)) 0.05/1.05 -> Rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.05/1.05 top(ok(X:S)) -> top(active(X:S)) 0.05/1.05 ->Strongly Connected Components: 0.05/1.05 ->->Cycle: 0.05/1.05 ->->-> Pairs: 0.05/1.05 G(mark(X:S)) -> G(X:S) 0.05/1.05 G(ok(X:S)) -> G(X:S) 0.05/1.05 ->->-> Rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.05/1.05 top(ok(X:S)) -> top(active(X:S)) 0.05/1.05 ->->Cycle: 0.05/1.05 ->->-> Pairs: 0.05/1.05 F(mark(X1:S),X2:S) -> F(X1:S,X2:S) 0.05/1.05 F(ok(X1:S),ok(X2:S)) -> F(X1:S,X2:S) 0.05/1.05 ->->-> Rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.05/1.05 top(ok(X:S)) -> top(active(X:S)) 0.05/1.05 ->->Cycle: 0.05/1.05 ->->-> Pairs: 0.05/1.05 PROPER(f(X1:S,X2:S)) -> PROPER(X1:S) 0.05/1.05 PROPER(f(X1:S,X2:S)) -> PROPER(X2:S) 0.05/1.05 PROPER(g(X:S)) -> PROPER(X:S) 0.05/1.05 ->->-> Rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.05/1.05 top(ok(X:S)) -> top(active(X:S)) 0.05/1.05 ->->Cycle: 0.05/1.05 ->->-> Pairs: 0.05/1.05 ACTIVE(f(X1:S,X2:S)) -> ACTIVE(X1:S) 0.05/1.05 ACTIVE(g(X:S)) -> ACTIVE(X:S) 0.05/1.05 ->->-> Rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.05/1.05 top(ok(X:S)) -> top(active(X:S)) 0.05/1.05 ->->Cycle: 0.05/1.05 ->->-> Pairs: 0.05/1.05 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.05/1.05 TOP(ok(X:S)) -> TOP(active(X:S)) 0.05/1.05 ->->-> Rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.05/1.05 top(ok(X:S)) -> top(active(X:S)) 0.05/1.05 0.05/1.05 0.05/1.05 The problem is decomposed in 5 subproblems. 0.05/1.05 0.05/1.05 Problem 1.1: 0.05/1.05 0.05/1.05 Subterm Processor: 0.05/1.05 -> Pairs: 0.05/1.05 G(mark(X:S)) -> G(X:S) 0.05/1.05 G(ok(X:S)) -> G(X:S) 0.05/1.05 -> Rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.05/1.05 top(ok(X:S)) -> top(active(X:S)) 0.05/1.05 ->Projection: 0.05/1.05 pi(G) = 1 0.05/1.05 0.05/1.05 Problem 1.1: 0.05/1.05 0.05/1.05 SCC Processor: 0.05/1.05 -> Pairs: 0.05/1.05 Empty 0.05/1.05 -> Rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.05/1.05 top(ok(X:S)) -> top(active(X:S)) 0.05/1.05 ->Strongly Connected Components: 0.05/1.05 There is no strongly connected component 0.05/1.05 0.05/1.05 The problem is finite. 0.05/1.05 0.05/1.05 Problem 1.2: 0.05/1.05 0.05/1.05 Subterm Processor: 0.05/1.05 -> Pairs: 0.05/1.05 F(mark(X1:S),X2:S) -> F(X1:S,X2:S) 0.05/1.05 F(ok(X1:S),ok(X2:S)) -> F(X1:S,X2:S) 0.05/1.05 -> Rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.05/1.05 top(ok(X:S)) -> top(active(X:S)) 0.05/1.05 ->Projection: 0.05/1.05 pi(F) = 1 0.05/1.05 0.05/1.05 Problem 1.2: 0.05/1.05 0.05/1.05 SCC Processor: 0.05/1.05 -> Pairs: 0.05/1.05 Empty 0.05/1.05 -> Rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.05/1.05 top(ok(X:S)) -> top(active(X:S)) 0.05/1.05 ->Strongly Connected Components: 0.05/1.05 There is no strongly connected component 0.05/1.05 0.05/1.05 The problem is finite. 0.05/1.05 0.05/1.05 Problem 1.3: 0.05/1.05 0.05/1.05 Subterm Processor: 0.05/1.05 -> Pairs: 0.05/1.05 PROPER(f(X1:S,X2:S)) -> PROPER(X1:S) 0.05/1.05 PROPER(f(X1:S,X2:S)) -> PROPER(X2:S) 0.05/1.05 PROPER(g(X:S)) -> PROPER(X:S) 0.05/1.05 -> Rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.05/1.05 top(ok(X:S)) -> top(active(X:S)) 0.05/1.05 ->Projection: 0.05/1.05 pi(PROPER) = 1 0.05/1.05 0.05/1.05 Problem 1.3: 0.05/1.05 0.05/1.05 SCC Processor: 0.05/1.05 -> Pairs: 0.05/1.05 Empty 0.05/1.05 -> Rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.05/1.05 top(ok(X:S)) -> top(active(X:S)) 0.05/1.05 ->Strongly Connected Components: 0.05/1.05 There is no strongly connected component 0.05/1.05 0.05/1.05 The problem is finite. 0.05/1.05 0.05/1.05 Problem 1.4: 0.05/1.05 0.05/1.05 Subterm Processor: 0.05/1.05 -> Pairs: 0.05/1.05 ACTIVE(f(X1:S,X2:S)) -> ACTIVE(X1:S) 0.05/1.05 ACTIVE(g(X:S)) -> ACTIVE(X:S) 0.05/1.05 -> Rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.05/1.05 top(ok(X:S)) -> top(active(X:S)) 0.05/1.05 ->Projection: 0.05/1.05 pi(ACTIVE) = 1 0.05/1.05 0.05/1.05 Problem 1.4: 0.05/1.05 0.05/1.05 SCC Processor: 0.05/1.05 -> Pairs: 0.05/1.05 Empty 0.05/1.05 -> Rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.05/1.05 top(ok(X:S)) -> top(active(X:S)) 0.05/1.05 ->Strongly Connected Components: 0.05/1.05 There is no strongly connected component 0.05/1.05 0.05/1.05 The problem is finite. 0.05/1.05 0.05/1.05 Problem 1.5: 0.05/1.05 0.05/1.05 Reduction Pairs Processor: 0.05/1.05 -> Pairs: 0.05/1.05 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.05/1.05 TOP(ok(X:S)) -> TOP(active(X:S)) 0.05/1.05 -> Rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.05/1.05 top(ok(X:S)) -> top(active(X:S)) 0.05/1.05 -> Usable rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 ->Interpretation type: 0.05/1.05 Linear 0.05/1.05 ->Coefficients: 0.05/1.05 Natural Numbers 0.05/1.05 ->Dimension: 0.05/1.05 1 0.05/1.05 ->Bound: 0.05/1.05 2 0.05/1.05 ->Interpretation: 0.05/1.05 0.05/1.05 [active](X) = 2.X + 1 0.05/1.05 [f](X1,X2) = X1 + X2 0.05/1.05 [g](X) = X 0.05/1.05 [proper](X) = 0 0.05/1.05 [top](X) = 0 0.05/1.05 [fSNonEmpty] = 0 0.05/1.05 [mark](X) = 1 0.05/1.05 [ok](X) = 2.X + 2 0.05/1.05 [ACTIVE](X) = 0 0.05/1.05 [F](X1,X2) = 0 0.05/1.05 [G](X) = 0 0.05/1.05 [PROPER](X) = 0 0.05/1.05 [TOP](X) = 2.X 0.05/1.05 0.05/1.05 Problem 1.5: 0.05/1.05 0.05/1.05 SCC Processor: 0.05/1.05 -> Pairs: 0.05/1.05 TOP(ok(X:S)) -> TOP(active(X:S)) 0.05/1.05 -> Rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.05/1.05 top(ok(X:S)) -> top(active(X:S)) 0.05/1.05 ->Strongly Connected Components: 0.05/1.05 ->->Cycle: 0.05/1.05 ->->-> Pairs: 0.05/1.05 TOP(ok(X:S)) -> TOP(active(X:S)) 0.05/1.05 ->->-> Rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.05/1.05 top(ok(X:S)) -> top(active(X:S)) 0.05/1.05 0.05/1.05 Problem 1.5: 0.05/1.05 0.05/1.05 Reduction Pairs Processor: 0.05/1.05 -> Pairs: 0.05/1.05 TOP(ok(X:S)) -> TOP(active(X:S)) 0.05/1.05 -> Rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.05/1.05 top(ok(X:S)) -> top(active(X:S)) 0.05/1.05 -> Usable rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 ->Interpretation type: 0.05/1.05 Linear 0.05/1.05 ->Coefficients: 0.05/1.05 Natural Numbers 0.05/1.05 ->Dimension: 0.05/1.05 1 0.05/1.05 ->Bound: 0.05/1.05 2 0.05/1.05 ->Interpretation: 0.05/1.05 0.05/1.05 [active](X) = X 0.05/1.05 [f](X1,X2) = X2 + 2 0.05/1.05 [g](X) = 2.X 0.05/1.05 [proper](X) = 0 0.05/1.05 [top](X) = 0 0.05/1.05 [fSNonEmpty] = 0 0.05/1.05 [mark](X) = 1 0.05/1.05 [ok](X) = X + 2 0.05/1.05 [ACTIVE](X) = 0 0.05/1.05 [F](X1,X2) = 0 0.05/1.05 [G](X) = 0 0.05/1.05 [PROPER](X) = 0 0.05/1.05 [TOP](X) = 2.X 0.05/1.05 0.05/1.05 Problem 1.5: 0.05/1.05 0.05/1.05 SCC Processor: 0.05/1.05 -> Pairs: 0.05/1.05 Empty 0.05/1.05 -> Rules: 0.05/1.05 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S))) 0.05/1.05 active(f(X1:S,X2:S)) -> f(active(X1:S),X2:S) 0.05/1.05 active(g(X:S)) -> g(active(X:S)) 0.05/1.05 f(mark(X1:S),X2:S) -> mark(f(X1:S,X2:S)) 0.05/1.05 f(ok(X1:S),ok(X2:S)) -> ok(f(X1:S,X2:S)) 0.05/1.05 g(mark(X:S)) -> mark(g(X:S)) 0.05/1.05 g(ok(X:S)) -> ok(g(X:S)) 0.05/1.05 proper(f(X1:S,X2:S)) -> f(proper(X1:S),proper(X2:S)) 0.05/1.05 proper(g(X:S)) -> g(proper(X:S)) 0.05/1.05 top(mark(X:S)) -> top(proper(X:S)) 0.05/1.05 top(ok(X:S)) -> top(active(X:S)) 0.05/1.05 ->Strongly Connected Components: 0.05/1.05 There is no strongly connected component 0.05/1.05 0.05/1.05 The problem is finite. 0.05/1.05 EOF