0.64/0.74 YES 0.64/0.74 0.64/0.74 Problem 1: 0.64/0.74 0.64/0.74 (VAR v_NonEmpty:S X:S X1:S X2:S X3:S Y:S) 0.64/0.74 (RULES 0.64/0.74 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.74 active(g(b)) -> mark(c) 0.64/0.74 active(b) -> mark(c) 0.64/0.74 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.74 g(active(X:S)) -> g(X:S) 0.64/0.74 g(mark(X:S)) -> g(X:S) 0.64/0.74 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.74 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.74 mark(b) -> active(b) 0.64/0.74 mark(c) -> active(c) 0.64/0.74 ) 0.64/0.74 (STRATEGY INNERMOST) 0.64/0.74 0.64/0.74 Problem 1: 0.64/0.74 0.64/0.74 Dependency Pairs Processor: 0.64/0.74 -> Pairs: 0.64/0.74 ACTIVE(f(X:S,g(X:S),Y:S)) -> F(Y:S,Y:S,Y:S) 0.64/0.74 ACTIVE(f(X:S,g(X:S),Y:S)) -> MARK(f(Y:S,Y:S,Y:S)) 0.64/0.74 ACTIVE(g(b)) -> MARK(c) 0.64/0.74 ACTIVE(b) -> MARK(c) 0.64/0.74 F(active(X1:S),X2:S,X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.74 F(mark(X1:S),X2:S,X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.74 F(X1:S,active(X2:S),X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.74 F(X1:S,mark(X2:S),X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.74 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.74 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.74 G(active(X:S)) -> G(X:S) 0.64/0.74 G(mark(X:S)) -> G(X:S) 0.64/0.74 MARK(f(X1:S,X2:S,X3:S)) -> ACTIVE(f(X1:S,X2:S,X3:S)) 0.64/0.74 MARK(g(X:S)) -> ACTIVE(g(mark(X:S))) 0.64/0.74 MARK(g(X:S)) -> G(mark(X:S)) 0.64/0.74 MARK(g(X:S)) -> MARK(X:S) 0.64/0.74 MARK(b) -> ACTIVE(b) 0.64/0.74 -> Rules: 0.64/0.74 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.74 active(g(b)) -> mark(c) 0.64/0.74 active(b) -> mark(c) 0.64/0.74 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.74 g(active(X:S)) -> g(X:S) 0.64/0.74 g(mark(X:S)) -> g(X:S) 0.64/0.74 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.74 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.74 mark(b) -> active(b) 0.64/0.74 mark(c) -> active(c) 0.64/0.74 0.64/0.74 Problem 1: 0.64/0.74 0.64/0.74 SCC Processor: 0.64/0.74 -> Pairs: 0.64/0.74 ACTIVE(f(X:S,g(X:S),Y:S)) -> F(Y:S,Y:S,Y:S) 0.64/0.74 ACTIVE(f(X:S,g(X:S),Y:S)) -> MARK(f(Y:S,Y:S,Y:S)) 0.64/0.74 ACTIVE(g(b)) -> MARK(c) 0.64/0.74 ACTIVE(b) -> MARK(c) 0.64/0.74 F(active(X1:S),X2:S,X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.74 F(mark(X1:S),X2:S,X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.74 F(X1:S,active(X2:S),X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.74 F(X1:S,mark(X2:S),X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.74 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.74 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.74 G(active(X:S)) -> G(X:S) 0.64/0.74 G(mark(X:S)) -> G(X:S) 0.64/0.74 MARK(f(X1:S,X2:S,X3:S)) -> ACTIVE(f(X1:S,X2:S,X3:S)) 0.64/0.74 MARK(g(X:S)) -> ACTIVE(g(mark(X:S))) 0.64/0.74 MARK(g(X:S)) -> G(mark(X:S)) 0.64/0.74 MARK(g(X:S)) -> MARK(X:S) 0.64/0.74 MARK(b) -> ACTIVE(b) 0.64/0.74 -> Rules: 0.64/0.74 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.74 active(g(b)) -> mark(c) 0.64/0.74 active(b) -> mark(c) 0.64/0.74 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.74 g(active(X:S)) -> g(X:S) 0.64/0.74 g(mark(X:S)) -> g(X:S) 0.64/0.74 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.74 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.74 mark(b) -> active(b) 0.64/0.74 mark(c) -> active(c) 0.64/0.74 ->Strongly Connected Components: 0.64/0.74 ->->Cycle: 0.64/0.74 ->->-> Pairs: 0.64/0.74 G(active(X:S)) -> G(X:S) 0.64/0.74 G(mark(X:S)) -> G(X:S) 0.64/0.74 ->->-> Rules: 0.64/0.74 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.74 active(g(b)) -> mark(c) 0.64/0.74 active(b) -> mark(c) 0.64/0.74 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.74 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.74 g(active(X:S)) -> g(X:S) 0.64/0.74 g(mark(X:S)) -> g(X:S) 0.64/0.74 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.74 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.74 mark(b) -> active(b) 0.64/0.74 mark(c) -> active(c) 0.64/0.74 ->->Cycle: 0.64/0.74 ->->-> Pairs: 0.64/0.74 F(active(X1:S),X2:S,X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.74 F(mark(X1:S),X2:S,X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.74 F(X1:S,active(X2:S),X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.74 F(X1:S,mark(X2:S),X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.74 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.75 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.75 ->->-> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 ->->Cycle: 0.64/0.75 ->->-> Pairs: 0.64/0.75 ACTIVE(f(X:S,g(X:S),Y:S)) -> MARK(f(Y:S,Y:S,Y:S)) 0.64/0.75 MARK(f(X1:S,X2:S,X3:S)) -> ACTIVE(f(X1:S,X2:S,X3:S)) 0.64/0.75 MARK(g(X:S)) -> ACTIVE(g(mark(X:S))) 0.64/0.75 MARK(g(X:S)) -> MARK(X:S) 0.64/0.75 ->->-> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 0.64/0.75 0.64/0.75 The problem is decomposed in 3 subproblems. 0.64/0.75 0.64/0.75 Problem 1.1: 0.64/0.75 0.64/0.75 Subterm Processor: 0.64/0.75 -> Pairs: 0.64/0.75 G(active(X:S)) -> G(X:S) 0.64/0.75 G(mark(X:S)) -> G(X:S) 0.64/0.75 -> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 ->Projection: 0.64/0.75 pi(G) = 1 0.64/0.75 0.64/0.75 Problem 1.1: 0.64/0.75 0.64/0.75 SCC Processor: 0.64/0.75 -> Pairs: 0.64/0.75 Empty 0.64/0.75 -> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 ->Strongly Connected Components: 0.64/0.75 There is no strongly connected component 0.64/0.75 0.64/0.75 The problem is finite. 0.64/0.75 0.64/0.75 Problem 1.2: 0.64/0.75 0.64/0.75 Subterm Processor: 0.64/0.75 -> Pairs: 0.64/0.75 F(active(X1:S),X2:S,X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.75 F(mark(X1:S),X2:S,X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.75 F(X1:S,active(X2:S),X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.75 F(X1:S,mark(X2:S),X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.75 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.75 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.75 -> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 ->Projection: 0.64/0.75 pi(F) = 1 0.64/0.75 0.64/0.75 Problem 1.2: 0.64/0.75 0.64/0.75 SCC Processor: 0.64/0.75 -> Pairs: 0.64/0.75 F(X1:S,active(X2:S),X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.75 F(X1:S,mark(X2:S),X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.75 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.75 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.75 -> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 ->Strongly Connected Components: 0.64/0.75 ->->Cycle: 0.64/0.75 ->->-> Pairs: 0.64/0.75 F(X1:S,active(X2:S),X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.75 F(X1:S,mark(X2:S),X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.75 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.75 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.75 ->->-> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 0.64/0.75 Problem 1.2: 0.64/0.75 0.64/0.75 Subterm Processor: 0.64/0.75 -> Pairs: 0.64/0.75 F(X1:S,active(X2:S),X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.75 F(X1:S,mark(X2:S),X3:S) -> F(X1:S,X2:S,X3:S) 0.64/0.75 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.75 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.75 -> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 ->Projection: 0.64/0.75 pi(F) = 2 0.64/0.75 0.64/0.75 Problem 1.2: 0.64/0.75 0.64/0.75 SCC Processor: 0.64/0.75 -> Pairs: 0.64/0.75 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.75 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.75 -> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 ->Strongly Connected Components: 0.64/0.75 ->->Cycle: 0.64/0.75 ->->-> Pairs: 0.64/0.75 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.75 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.75 ->->-> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 0.64/0.75 Problem 1.2: 0.64/0.75 0.64/0.75 Subterm Processor: 0.64/0.75 -> Pairs: 0.64/0.75 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.75 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S) 0.64/0.75 -> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 ->Projection: 0.64/0.75 pi(F) = 3 0.64/0.75 0.64/0.75 Problem 1.2: 0.64/0.75 0.64/0.75 SCC Processor: 0.64/0.75 -> Pairs: 0.64/0.75 Empty 0.64/0.75 -> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 ->Strongly Connected Components: 0.64/0.75 There is no strongly connected component 0.64/0.75 0.64/0.75 The problem is finite. 0.64/0.75 0.64/0.75 Problem 1.3: 0.64/0.75 0.64/0.75 Reduction Pairs Processor: 0.64/0.75 -> Pairs: 0.64/0.75 ACTIVE(f(X:S,g(X:S),Y:S)) -> MARK(f(Y:S,Y:S,Y:S)) 0.64/0.75 MARK(f(X1:S,X2:S,X3:S)) -> ACTIVE(f(X1:S,X2:S,X3:S)) 0.64/0.75 MARK(g(X:S)) -> ACTIVE(g(mark(X:S))) 0.64/0.75 MARK(g(X:S)) -> MARK(X:S) 0.64/0.75 -> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 -> Usable rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 ->Interpretation type: 0.64/0.75 Linear 0.64/0.75 ->Coefficients: 0.64/0.75 Natural Numbers 0.64/0.75 ->Dimension: 0.64/0.75 1 0.64/0.75 ->Bound: 0.64/0.75 2 0.64/0.75 ->Interpretation: 0.64/0.75 0.64/0.75 [active](X) = X + 1 0.64/0.75 [f](X1,X2,X3) = 0 0.64/0.75 [g](X) = 2.X + 2 0.64/0.75 [mark](X) = 2.X + 1 0.64/0.75 [b] = 0 0.64/0.75 [c] = 0 0.64/0.75 [fSNonEmpty] = 0 0.64/0.75 [ACTIVE](X) = 2 0.64/0.75 [F](X1,X2,X3) = 0 0.64/0.75 [G](X) = 0 0.64/0.75 [MARK](X) = X + 2 0.64/0.75 0.64/0.75 Problem 1.3: 0.64/0.75 0.64/0.75 SCC Processor: 0.64/0.75 -> Pairs: 0.64/0.75 ACTIVE(f(X:S,g(X:S),Y:S)) -> MARK(f(Y:S,Y:S,Y:S)) 0.64/0.75 MARK(f(X1:S,X2:S,X3:S)) -> ACTIVE(f(X1:S,X2:S,X3:S)) 0.64/0.75 MARK(g(X:S)) -> MARK(X:S) 0.64/0.75 -> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 ->Strongly Connected Components: 0.64/0.75 ->->Cycle: 0.64/0.75 ->->-> Pairs: 0.64/0.75 ACTIVE(f(X:S,g(X:S),Y:S)) -> MARK(f(Y:S,Y:S,Y:S)) 0.64/0.75 MARK(f(X1:S,X2:S,X3:S)) -> ACTIVE(f(X1:S,X2:S,X3:S)) 0.64/0.75 MARK(g(X:S)) -> MARK(X:S) 0.64/0.75 ->->-> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 0.64/0.75 Problem 1.3: 0.64/0.75 0.64/0.75 Reduction Pairs Processor: 0.64/0.75 -> Pairs: 0.64/0.75 ACTIVE(f(X:S,g(X:S),Y:S)) -> MARK(f(Y:S,Y:S,Y:S)) 0.64/0.75 MARK(f(X1:S,X2:S,X3:S)) -> ACTIVE(f(X1:S,X2:S,X3:S)) 0.64/0.75 MARK(g(X:S)) -> MARK(X:S) 0.64/0.75 -> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 -> Usable rules: 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 ->Interpretation type: 0.64/0.75 Linear 0.64/0.75 ->Coefficients: 0.64/0.75 Natural Numbers 0.64/0.75 ->Dimension: 0.64/0.75 1 0.64/0.75 ->Bound: 0.64/0.75 2 0.64/0.75 ->Interpretation: 0.64/0.75 0.64/0.75 [active](X) = 0 0.64/0.75 [f](X1,X2,X3) = 0 0.64/0.75 [g](X) = 2.X + 2 0.64/0.75 [mark](X) = 0 0.64/0.75 [b] = 0 0.64/0.75 [c] = 0 0.64/0.75 [fSNonEmpty] = 0 0.64/0.75 [ACTIVE](X) = 2 0.64/0.75 [F](X1,X2,X3) = 0 0.64/0.75 [G](X) = 0 0.64/0.75 [MARK](X) = 2.X + 2 0.64/0.75 0.64/0.75 Problem 1.3: 0.64/0.75 0.64/0.75 SCC Processor: 0.64/0.75 -> Pairs: 0.64/0.75 ACTIVE(f(X:S,g(X:S),Y:S)) -> MARK(f(Y:S,Y:S,Y:S)) 0.64/0.75 MARK(f(X1:S,X2:S,X3:S)) -> ACTIVE(f(X1:S,X2:S,X3:S)) 0.64/0.75 -> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 ->Strongly Connected Components: 0.64/0.75 ->->Cycle: 0.64/0.75 ->->-> Pairs: 0.64/0.75 ACTIVE(f(X:S,g(X:S),Y:S)) -> MARK(f(Y:S,Y:S,Y:S)) 0.64/0.75 MARK(f(X1:S,X2:S,X3:S)) -> ACTIVE(f(X1:S,X2:S,X3:S)) 0.64/0.75 ->->-> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 0.64/0.75 Problem 1.3: 0.64/0.75 0.64/0.75 Reduction Pair Processor: 0.64/0.75 -> Pairs: 0.64/0.75 ACTIVE(f(X:S,g(X:S),Y:S)) -> MARK(f(Y:S,Y:S,Y:S)) 0.64/0.75 MARK(f(X1:S,X2:S,X3:S)) -> ACTIVE(f(X1:S,X2:S,X3:S)) 0.64/0.75 -> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 -> Usable rules: 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 ->Mace4 Output: 0.64/0.75 ============================== Mace4 ================================= 0.64/0.75 Mace4 (64) version 2009-11A, November 2009. 0.64/0.75 Process 52554 was started by sandbox2 on n007.star.cs.uiowa.edu, 0.64/0.75 Thu Mar 28 22:31:43 2019 0.64/0.75 The command was "./mace4 -c -f /tmp/mace49398195822001100545.in". 0.64/0.75 ============================== end of head =========================== 0.64/0.75 0.64/0.75 ============================== INPUT ================================= 0.64/0.75 0.64/0.75 % Reading from file /tmp/mace49398195822001100545.in 0.64/0.75 0.64/0.75 assign(max_seconds,20). 0.64/0.75 0.64/0.75 formulas(assumptions). 0.64/0.75 gtrsim_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility). 0.64/0.75 succeq_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility). 0.64/0.75 gtrsim_s0(x,y) & succeq_s0(y,z) -> gtrsim_s0(x,z) # label(compatibility). 0.64/0.75 arrow_s0(x1,y) -> arrow_s0(f2(x1),f2(y)) # label(congruence). 0.64/0.75 arrow_s0(x1,y) -> arrow_s0(f3(x1,x2,x3),f3(y,x2,x3)) # label(congruence). 0.64/0.75 arrow_s0(x2,y) -> arrow_s0(f3(x1,x2,x3),f3(x1,y,x3)) # label(congruence). 0.64/0.75 arrow_s0(x3,y) -> arrow_s0(f3(x1,x2,x3),f3(x1,x2,y)) # label(congruence). 0.64/0.75 arrow_s0(x1,y) -> arrow_s0(f4(x1),f4(y)) # label(congruence). 0.64/0.75 arrow_s0(x1,y) -> arrow_s0(f5(x1),f5(y)) # label(congruence). 0.64/0.75 arrow_s0(x1,y) -> arrow_s0(f10(x1),f10(y)) # label(congruence). 0.64/0.75 arrow_s0(x1,y) -> arrow_s0(f11(x1,x2,x3),f11(y,x2,x3)) # label(congruence). 0.64/0.75 arrow_s0(x2,y) -> arrow_s0(f11(x1,x2,x3),f11(x1,y,x3)) # label(congruence). 0.64/0.75 arrow_s0(x3,y) -> arrow_s0(f11(x1,x2,x3),f11(x1,x2,y)) # label(congruence). 0.64/0.75 arrow_s0(x1,y) -> arrow_s0(f12(x1),f12(y)) # label(congruence). 0.64/0.75 arrow_s0(x1,y) -> arrow_s0(f13(x1),f13(y)) # label(congruence). 0.64/0.75 arrow_s0(f3(f2(x2),x3,x4),f3(x2,x3,x4)) # label(replacement). 0.64/0.75 arrow_s0(f3(f5(x2),x3,x4),f3(x2,x3,x4)) # label(replacement). 0.64/0.75 arrow_s0(f3(x2,f2(x3),x4),f3(x2,x3,x4)) # label(replacement). 0.64/0.75 arrow_s0(f3(x2,f5(x3),x4),f3(x2,x3,x4)) # label(replacement). 0.64/0.75 arrow_s0(f3(x2,x3,f2(x4)),f3(x2,x3,x4)) # label(replacement). 0.64/0.75 arrow_s0(f3(x2,x3,f5(x4)),f3(x2,x3,x4)) # label(replacement). 0.64/0.75 arrow_s0(x,y) -> gtrsim_s0(x,y) # label(inclusion). 0.64/0.75 sqsupset_s0(f10(f3(x1,f4(x1),x5)),f13(f3(x5,x5,x5))) # label(replacement). 0.64/0.75 succeq_s0(f13(f3(x2,x3,x4)),f10(f3(x2,x3,x4))) # label(replacement). 0.64/0.75 sqsupset_s0(x,y) -> sqsupsetStar_s0(x,y) # label(inclusion). 0.64/0.75 sqsupset_s0(x,y) & sqsupsetStar_s0(y,z) -> sqsupsetStar_s0(x,z) # label(compatibility). 0.64/0.75 end_of_list. 0.64/0.75 0.64/0.75 formulas(goals). 0.64/0.75 (exists x sqsupsetStar_s0(x,x)) # label(wellfoundedness). 0.64/0.75 end_of_list. 0.64/0.75 0.64/0.75 ============================== end of input ========================== 0.64/0.75 0.64/0.75 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.64/0.75 0.64/0.75 % Formulas that are not ordinary clauses: 0.64/0.75 1 gtrsim_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility) # label(non_clause). [assumption]. 0.64/0.75 2 succeq_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility) # label(non_clause). [assumption]. 0.64/0.75 3 gtrsim_s0(x,y) & succeq_s0(y,z) -> gtrsim_s0(x,z) # label(compatibility) # label(non_clause). [assumption]. 0.64/0.75 4 arrow_s0(x1,y) -> arrow_s0(f2(x1),f2(y)) # label(congruence) # label(non_clause). [assumption]. 0.64/0.75 5 arrow_s0(x1,y) -> arrow_s0(f3(x1,x2,x3),f3(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 0.64/0.75 6 arrow_s0(x2,y) -> arrow_s0(f3(x1,x2,x3),f3(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 0.64/0.75 7 arrow_s0(x3,y) -> arrow_s0(f3(x1,x2,x3),f3(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 0.64/0.75 8 arrow_s0(x1,y) -> arrow_s0(f4(x1),f4(y)) # label(congruence) # label(non_clause). [assumption]. 0.64/0.75 9 arrow_s0(x1,y) -> arrow_s0(f5(x1),f5(y)) # label(congruence) # label(non_clause). [assumption]. 0.64/0.75 10 arrow_s0(x1,y) -> arrow_s0(f10(x1),f10(y)) # label(congruence) # label(non_clause). [assumption]. 0.64/0.75 11 arrow_s0(x1,y) -> arrow_s0(f11(x1,x2,x3),f11(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 0.64/0.75 12 arrow_s0(x2,y) -> arrow_s0(f11(x1,x2,x3),f11(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 0.64/0.75 13 arrow_s0(x3,y) -> arrow_s0(f11(x1,x2,x3),f11(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 0.64/0.75 14 arrow_s0(x1,y) -> arrow_s0(f12(x1),f12(y)) # label(congruence) # label(non_clause). [assumption]. 0.64/0.75 15 arrow_s0(x1,y) -> arrow_s0(f13(x1),f13(y)) # label(congruence) # label(non_clause). [assumption]. 0.64/0.75 16 arrow_s0(x,y) -> gtrsim_s0(x,y) # label(inclusion) # label(non_clause). [assumption]. 0.64/0.75 17 sqsupset_s0(x,y) -> sqsupsetStar_s0(x,y) # label(inclusion) # label(non_clause). [assumption]. 0.64/0.75 18 sqsupset_s0(x,y) & sqsupsetStar_s0(y,z) -> sqsupsetStar_s0(x,z) # label(compatibility) # label(non_clause). [assumption]. 0.64/0.75 19 (exists x sqsupsetStar_s0(x,x)) # label(wellfoundedness) # label(non_clause) # label(goal). [goal]. 0.64/0.75 0.64/0.75 ============================== end of process non-clausal formulas === 0.64/0.75 0.64/0.75 ============================== CLAUSES FOR SEARCH ==================== 0.64/0.75 0.64/0.75 formulas(mace4_clauses). 0.64/0.75 -gtrsim_s0(x,y) | -sqsupset_s0(y,z) | sqsupset_s0(x,z) # label(compatibility). 0.64/0.75 -succeq_s0(x,y) | -sqsupset_s0(y,z) | sqsupset_s0(x,z) # label(compatibility). 0.64/0.75 -gtrsim_s0(x,y) | -succeq_s0(y,z) | gtrsim_s0(x,z) # label(compatibility). 0.64/0.75 -arrow_s0(x,y) | arrow_s0(f2(x),f2(y)) # label(congruence). 0.64/0.75 -arrow_s0(x,y) | arrow_s0(f3(x,z,u),f3(y,z,u)) # label(congruence). 0.64/0.75 -arrow_s0(x,y) | arrow_s0(f3(z,x,u),f3(z,y,u)) # label(congruence). 0.64/0.75 -arrow_s0(x,y) | arrow_s0(f3(z,u,x),f3(z,u,y)) # label(congruence). 0.64/0.75 -arrow_s0(x,y) | arrow_s0(f4(x),f4(y)) # label(congruence). 0.64/0.75 -arrow_s0(x,y) | arrow_s0(f5(x),f5(y)) # label(congruence). 0.64/0.75 -arrow_s0(x,y) | arrow_s0(f10(x),f10(y)) # label(congruence). 0.64/0.75 -arrow_s0(x,y) | arrow_s0(f11(x,z,u),f11(y,z,u)) # label(congruence). 0.64/0.75 -arrow_s0(x,y) | arrow_s0(f11(z,x,u),f11(z,y,u)) # label(congruence). 0.64/0.75 -arrow_s0(x,y) | arrow_s0(f11(z,u,x),f11(z,u,y)) # label(congruence). 0.64/0.75 -arrow_s0(x,y) | arrow_s0(f12(x),f12(y)) # label(congruence). 0.64/0.75 -arrow_s0(x,y) | arrow_s0(f13(x),f13(y)) # label(congruence). 0.64/0.75 arrow_s0(f3(f2(x),y,z),f3(x,y,z)) # label(replacement). 0.64/0.75 arrow_s0(f3(f5(x),y,z),f3(x,y,z)) # label(replacement). 0.64/0.75 arrow_s0(f3(x,f2(y),z),f3(x,y,z)) # label(replacement). 0.64/0.75 arrow_s0(f3(x,f5(y),z),f3(x,y,z)) # label(replacement). 0.64/0.75 arrow_s0(f3(x,y,f2(z)),f3(x,y,z)) # label(replacement). 0.64/0.75 arrow_s0(f3(x,y,f5(z)),f3(x,y,z)) # label(replacement). 0.64/0.75 -arrow_s0(x,y) | gtrsim_s0(x,y) # label(inclusion). 0.64/0.75 sqsupset_s0(f10(f3(x,f4(x),y)),f13(f3(y,y,y))) # label(replacement). 0.64/0.75 succeq_s0(f13(f3(x,y,z)),f10(f3(x,y,z))) # label(replacement). 0.64/0.75 -sqsupset_s0(x,y) | sqsupsetStar_s0(x,y) # label(inclusion). 0.64/0.75 -sqsupset_s0(x,y) | -sqsupsetStar_s0(y,z) | sqsupsetStar_s0(x,z) # label(compatibility). 0.64/0.75 -sqsupsetStar_s0(x,x) # label(wellfoundedness). 0.64/0.75 end_of_list. 0.64/0.75 0.64/0.75 ============================== end of clauses for search ============= 0.64/0.75 0.64/0.75 % There are no natural numbers in the input. 0.64/0.75 0.64/0.75 ============================== DOMAIN SIZE 2 ========================= 0.64/0.75 0.64/0.75 ============================== MODEL ================================= 0.64/0.75 0.64/0.75 interpretation( 2, [number=1, seconds=0], [ 0.64/0.75 0.64/0.75 function(f10(_), [ 0, 1 ]), 0.64/0.75 0.64/0.75 function(f12(_), [ 0, 0 ]), 0.64/0.75 0.64/0.75 function(f13(_), [ 0, 1 ]), 0.64/0.75 0.64/0.75 function(f2(_), [ 0, 1 ]), 0.64/0.75 0.64/0.75 function(f4(_), [ 1, 0 ]), 0.64/0.75 0.64/0.75 function(f5(_), [ 0, 1 ]), 0.64/0.75 0.64/0.75 function(f11(_,_,_), [ 0.64/0.75 0, 0, 0.64/0.75 0, 0, 0.64/0.75 0, 0, 0.64/0.75 0, 0 ]), 0.64/0.75 0.64/0.75 function(f3(_,_,_), [ 0.64/0.75 0, 0, 0.64/0.75 1, 1, 0.64/0.75 1, 1, 0.64/0.75 0, 0 ]), 0.64/0.75 0.64/0.75 relation(arrow_s0(_,_), [ 0.64/0.75 1, 0, 0.64/0.75 0, 1 ]), 0.64/0.75 0.64/0.75 relation(gtrsim_s0(_,_), [ 0.64/0.75 1, 0, 0.64/0.75 0, 1 ]), 0.64/0.75 0.64/0.75 relation(sqsupsetStar_s0(_,_), [ 0.64/0.75 0, 0, 0.64/0.75 1, 0 ]), 0.64/0.75 0.64/0.75 relation(sqsupset_s0(_,_), [ 0.64/0.75 0, 0, 0.64/0.75 1, 0 ]), 0.64/0.75 0.64/0.75 relation(succeq_s0(_,_), [ 0.64/0.75 1, 0, 0.64/0.75 0, 1 ]) 0.64/0.75 ]). 0.64/0.75 0.64/0.75 ============================== end of model ========================== 0.64/0.75 0.64/0.75 ============================== STATISTICS ============================ 0.64/0.75 0.64/0.75 For domain size 2. 0.64/0.75 0.64/0.75 Current CPU time: 0.00 seconds (total CPU time: 0.17 seconds). 0.64/0.75 Ground clauses: seen=222, kept=222. 0.64/0.75 Selections=23126, assignments=46232, propagations=41873, current_models=1. 0.64/0.75 Rewrite_terms=676722, rewrite_bools=621519, indexes=50863. 0.64/0.75 Rules_from_neg_clauses=32, cross_offs=32. 0.64/0.75 0.64/0.75 ============================== end of statistics ===================== 0.64/0.75 0.64/0.75 User_CPU=0.17, System_CPU=0.01, Wall_clock=1. 0.64/0.75 0.64/0.75 Exiting with 1 model. 0.64/0.75 0.64/0.75 Process 52554 exit (max_models) Thu Mar 28 22:31:44 2019 0.64/0.75 The process finished Thu Mar 28 22:31:44 2019 0.64/0.75 0.64/0.75 0.64/0.75 Mace4 cooked interpretation: 0.64/0.75 0.64/0.75 % number = 1 0.64/0.75 % seconds = 0 0.64/0.75 0.64/0.75 % Interpretation of size 2 0.64/0.75 0.64/0.75 f10(0) = 0. 0.64/0.75 f10(1) = 1. 0.64/0.75 0.64/0.75 f12(0) = 0. 0.64/0.75 f12(1) = 0. 0.64/0.75 0.64/0.75 f13(0) = 0. 0.64/0.75 f13(1) = 1. 0.64/0.75 0.64/0.75 f2(0) = 0. 0.64/0.75 f2(1) = 1. 0.64/0.75 0.64/0.75 f4(0) = 1. 0.64/0.75 f4(1) = 0. 0.64/0.75 0.64/0.75 f5(0) = 0. 0.64/0.75 f5(1) = 1. 0.64/0.75 0.64/0.75 f11(0,0,0) = 0. 0.64/0.75 f11(0,0,1) = 0. 0.64/0.75 f11(0,1,0) = 0. 0.64/0.75 f11(0,1,1) = 0. 0.64/0.75 f11(1,0,0) = 0. 0.64/0.75 f11(1,0,1) = 0. 0.64/0.75 f11(1,1,0) = 0. 0.64/0.75 f11(1,1,1) = 0. 0.64/0.75 0.64/0.75 f3(0,0,0) = 0. 0.64/0.75 f3(0,0,1) = 0. 0.64/0.75 f3(0,1,0) = 1. 0.64/0.75 f3(0,1,1) = 1. 0.64/0.75 f3(1,0,0) = 1. 0.64/0.75 f3(1,0,1) = 1. 0.64/0.75 f3(1,1,0) = 0. 0.64/0.75 f3(1,1,1) = 0. 0.64/0.75 0.64/0.75 arrow_s0(0,0). 0.64/0.75 - arrow_s0(0,1). 0.64/0.75 - arrow_s0(1,0). 0.64/0.75 arrow_s0(1,1). 0.64/0.75 0.64/0.75 gtrsim_s0(0,0). 0.64/0.75 - gtrsim_s0(0,1). 0.64/0.75 - gtrsim_s0(1,0). 0.64/0.75 gtrsim_s0(1,1). 0.64/0.75 0.64/0.75 - sqsupsetStar_s0(0,0). 0.64/0.75 - sqsupsetStar_s0(0,1). 0.64/0.75 sqsupsetStar_s0(1,0). 0.64/0.75 - sqsupsetStar_s0(1,1). 0.64/0.75 0.64/0.75 - sqsupset_s0(0,0). 0.64/0.75 - sqsupset_s0(0,1). 0.64/0.75 sqsupset_s0(1,0). 0.64/0.75 - sqsupset_s0(1,1). 0.64/0.75 0.64/0.75 succeq_s0(0,0). 0.64/0.75 - succeq_s0(0,1). 0.64/0.75 - succeq_s0(1,0). 0.64/0.75 succeq_s0(1,1). 0.64/0.75 0.64/0.75 0.64/0.75 Problem 1.3: 0.64/0.75 0.64/0.75 SCC Processor: 0.64/0.75 -> Pairs: 0.64/0.75 MARK(f(X1:S,X2:S,X3:S)) -> ACTIVE(f(X1:S,X2:S,X3:S)) 0.64/0.75 -> Rules: 0.64/0.75 active(f(X:S,g(X:S),Y:S)) -> mark(f(Y:S,Y:S,Y:S)) 0.64/0.75 active(g(b)) -> mark(c) 0.64/0.75 active(b) -> mark(c) 0.64/0.75 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S) 0.64/0.75 g(active(X:S)) -> g(X:S) 0.64/0.75 g(mark(X:S)) -> g(X:S) 0.64/0.75 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,X2:S,X3:S)) 0.64/0.75 mark(g(X:S)) -> active(g(mark(X:S))) 0.64/0.75 mark(b) -> active(b) 0.64/0.75 mark(c) -> active(c) 0.64/0.75 ->Strongly Connected Components: 0.64/0.75 There is no strongly connected component 0.64/0.75 0.64/0.75 The problem is finite. 0.64/0.75 EOF