0.00/0.30 YES 0.00/0.30 0.00/0.30 Problem 1: 0.00/0.30 0.00/0.30 (VAR v_NonEmpty:S x1:S) 0.00/0.30 (RULES 0.00/0.30 p(p(s(x1:S))) -> p(x1:S) 0.00/0.30 p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) 0.00/0.30 p(s(x1:S)) -> x1:S 0.00/0.30 tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) 0.00/0.30 tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) 0.00/0.30 twice(0(x1:S)) -> 0(x1:S) 0.00/0.30 twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) 0.00/0.30 twoto(0(x1:S)) -> s(0(x1:S)) 0.00/0.30 twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) 0.00/0.30 ) 0.00/0.30 0.00/0.30 Problem 1: 0.00/0.30 0.00/0.30 Dependency Pairs Processor: 0.00/0.30 -> Pairs: 0.00/0.30 P(p(s(x1:S))) -> P(x1:S) 0.00/0.30 TOWER(0(x1:S)) -> P(s(p(s(x1:S)))) 0.00/0.30 TOWER(0(x1:S)) -> P(s(x1:S)) 0.00/0.30 TOWER(s(x1:S)) -> P(s(p(s(tower(p(s(p(s(x1:S))))))))) 0.00/0.30 TOWER(s(x1:S)) -> P(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) 0.00/0.30 TOWER(s(x1:S)) -> P(s(p(s(x1:S)))) 0.00/0.30 TOWER(s(x1:S)) -> P(s(tower(p(s(p(s(x1:S))))))) 0.00/0.30 TOWER(s(x1:S)) -> P(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))) 0.00/0.30 TOWER(s(x1:S)) -> P(s(x1:S)) 0.00/0.30 TOWER(s(x1:S)) -> TOWER(p(s(p(s(x1:S))))) 0.00/0.30 TOWER(s(x1:S)) -> TWOTO(p(s(p(s(tower(p(s(p(s(x1:S)))))))))) 0.00/0.30 TWICE(s(x1:S)) -> P(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) 0.00/0.30 TWICE(s(x1:S)) -> P(p(p(s(s(s(x1:S)))))) 0.00/0.30 TWICE(s(x1:S)) -> P(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S)))))))))))))) 0.00/0.30 TWICE(s(x1:S)) -> P(p(s(s(s(x1:S))))) 0.00/0.30 TWICE(s(x1:S)) -> P(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))) 0.00/0.30 TWICE(s(x1:S)) -> P(s(s(s(x1:S)))) 0.00/0.30 TWICE(s(x1:S)) -> TWICE(p(p(p(s(s(s(x1:S))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))))))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(p(s(x1:S)))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(s(s(twoto(p(s(p(s(x1:S))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(x1:S)) 0.00/0.30 TWOTO(s(x1:S)) -> TWICE(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> TWOTO(p(s(p(s(x1:S))))) 0.00/0.30 -> Rules: 0.00/0.30 p(p(s(x1:S))) -> p(x1:S) 0.00/0.30 p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) 0.00/0.30 p(s(x1:S)) -> x1:S 0.00/0.30 tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) 0.00/0.30 tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) 0.00/0.30 twice(0(x1:S)) -> 0(x1:S) 0.00/0.30 twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) 0.00/0.30 twoto(0(x1:S)) -> s(0(x1:S)) 0.00/0.30 twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) 0.00/0.30 0.00/0.30 Problem 1: 0.00/0.30 0.00/0.30 SCC Processor: 0.00/0.30 -> Pairs: 0.00/0.30 P(p(s(x1:S))) -> P(x1:S) 0.00/0.30 TOWER(0(x1:S)) -> P(s(p(s(x1:S)))) 0.00/0.30 TOWER(0(x1:S)) -> P(s(x1:S)) 0.00/0.30 TOWER(s(x1:S)) -> P(s(p(s(tower(p(s(p(s(x1:S))))))))) 0.00/0.30 TOWER(s(x1:S)) -> P(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) 0.00/0.30 TOWER(s(x1:S)) -> P(s(p(s(x1:S)))) 0.00/0.30 TOWER(s(x1:S)) -> P(s(tower(p(s(p(s(x1:S))))))) 0.00/0.30 TOWER(s(x1:S)) -> P(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))) 0.00/0.30 TOWER(s(x1:S)) -> P(s(x1:S)) 0.00/0.30 TOWER(s(x1:S)) -> TOWER(p(s(p(s(x1:S))))) 0.00/0.30 TOWER(s(x1:S)) -> TWOTO(p(s(p(s(tower(p(s(p(s(x1:S)))))))))) 0.00/0.30 TWICE(s(x1:S)) -> P(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) 0.00/0.30 TWICE(s(x1:S)) -> P(p(p(s(s(s(x1:S)))))) 0.00/0.30 TWICE(s(x1:S)) -> P(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S)))))))))))))) 0.00/0.30 TWICE(s(x1:S)) -> P(p(s(s(s(x1:S))))) 0.00/0.30 TWICE(s(x1:S)) -> P(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))) 0.00/0.30 TWICE(s(x1:S)) -> P(s(s(s(x1:S)))) 0.00/0.30 TWICE(s(x1:S)) -> TWICE(p(p(p(s(s(s(x1:S))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))))))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(p(s(x1:S)))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(s(s(twoto(p(s(p(s(x1:S))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> P(s(x1:S)) 0.00/0.30 TWOTO(s(x1:S)) -> TWICE(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))) 0.00/0.30 TWOTO(s(x1:S)) -> TWOTO(p(s(p(s(x1:S))))) 0.00/0.30 -> Rules: 0.00/0.30 p(p(s(x1:S))) -> p(x1:S) 0.00/0.30 p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) 0.00/0.30 p(s(x1:S)) -> x1:S 0.00/0.30 tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) 0.00/0.30 tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) 0.00/0.30 twice(0(x1:S)) -> 0(x1:S) 0.00/0.30 twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) 0.00/0.30 twoto(0(x1:S)) -> s(0(x1:S)) 0.00/0.30 twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) 0.00/0.30 ->Strongly Connected Components: 0.00/0.30 ->->Cycle: 0.00/0.30 ->->-> Pairs: 0.00/0.30 P(p(s(x1:S))) -> P(x1:S) 0.00/0.30 ->->-> Rules: 0.00/0.30 p(p(s(x1:S))) -> p(x1:S) 0.00/0.30 p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) 0.00/0.30 p(s(x1:S)) -> x1:S 0.00/0.30 tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) 0.00/0.30 tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) 0.00/0.30 twice(0(x1:S)) -> 0(x1:S) 0.00/0.30 twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) 0.00/0.30 twoto(0(x1:S)) -> s(0(x1:S)) 0.00/0.30 twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) 0.00/0.30 ->->Cycle: 0.00/0.30 ->->-> Pairs: 0.00/0.30 TWICE(s(x1:S)) -> TWICE(p(p(p(s(s(s(x1:S))))))) 0.00/0.30 ->->-> Rules: 0.00/0.30 p(p(s(x1:S))) -> p(x1:S) 0.00/0.30 p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) 0.00/0.30 p(s(x1:S)) -> x1:S 0.00/0.30 tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) 0.00/0.30 tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) 0.00/0.30 twice(0(x1:S)) -> 0(x1:S) 0.00/0.30 twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) 0.00/0.30 twoto(0(x1:S)) -> s(0(x1:S)) 0.00/0.30 twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) 0.00/0.30 ->->Cycle: 0.00/0.30 ->->-> Pairs: 0.00/0.30 TWOTO(s(x1:S)) -> TWOTO(p(s(p(s(x1:S))))) 0.00/0.30 ->->-> Rules: 0.00/0.30 p(p(s(x1:S))) -> p(x1:S) 0.00/0.30 p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) 0.00/0.30 p(s(x1:S)) -> x1:S 0.00/0.30 tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) 0.00/0.30 tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) 0.00/0.30 twice(0(x1:S)) -> 0(x1:S) 0.00/0.30 twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) 0.00/0.30 twoto(0(x1:S)) -> s(0(x1:S)) 0.00/0.30 twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) 0.00/0.30 ->->Cycle: 0.00/0.30 ->->-> Pairs: 0.00/0.30 TOWER(s(x1:S)) -> TOWER(p(s(p(s(x1:S))))) 0.00/0.30 ->->-> Rules: 0.00/0.30 p(p(s(x1:S))) -> p(x1:S) 0.00/0.30 p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) 0.00/0.30 p(s(x1:S)) -> x1:S 0.00/0.30 tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) 0.00/0.30 tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) 0.00/0.30 twice(0(x1:S)) -> 0(x1:S) 0.00/0.30 twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) 0.00/0.30 twoto(0(x1:S)) -> s(0(x1:S)) 0.00/0.30 twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) 0.00/0.30 0.00/0.30 0.00/0.30 The problem is decomposed in 4 subproblems. 0.00/0.30 0.00/0.30 Problem 1.1: 0.00/0.30 0.00/0.30 Subterm Processor: 0.00/0.30 -> Pairs: 0.00/0.30 P(p(s(x1:S))) -> P(x1:S) 0.00/0.30 -> Rules: 0.00/0.30 p(p(s(x1:S))) -> p(x1:S) 0.00/0.30 p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) 0.00/0.30 p(s(x1:S)) -> x1:S 0.00/0.30 tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) 0.00/0.30 tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) 0.00/0.30 twice(0(x1:S)) -> 0(x1:S) 0.00/0.30 twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) 0.00/0.30 twoto(0(x1:S)) -> s(0(x1:S)) 0.00/0.30 twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) 0.00/0.30 ->Projection: 0.00/0.30 pi(P) = 1 0.00/0.30 0.00/0.30 Problem 1.1: 0.00/0.30 0.00/0.30 SCC Processor: 0.00/0.30 -> Pairs: 0.00/0.30 Empty 0.00/0.30 -> Rules: 0.00/0.30 p(p(s(x1:S))) -> p(x1:S) 0.00/0.30 p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) 0.00/0.30 p(s(x1:S)) -> x1:S 0.00/0.30 tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) 0.00/0.30 tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) 0.00/0.30 twice(0(x1:S)) -> 0(x1:S) 0.00/0.30 twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) 0.00/0.30 twoto(0(x1:S)) -> s(0(x1:S)) 0.00/0.30 twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) 0.00/0.30 ->Strongly Connected Components: 0.00/0.30 There is no strongly connected component 0.00/0.30 0.00/0.30 The problem is finite. 0.00/0.30 0.00/0.30 Problem 1.2: 0.00/0.30 0.00/0.30 Reduction Pair Processor: 0.00/0.30 -> Pairs: 0.00/0.30 TWICE(s(x1:S)) -> TWICE(p(p(p(s(s(s(x1:S))))))) 0.00/0.30 -> Rules: 0.00/0.30 p(p(s(x1:S))) -> p(x1:S) 0.00/0.30 p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) 0.00/0.30 p(s(x1:S)) -> x1:S 0.00/0.30 tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) 0.00/0.30 tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) 0.00/0.30 twice(0(x1:S)) -> 0(x1:S) 0.00/0.30 twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) 0.00/0.30 twoto(0(x1:S)) -> s(0(x1:S)) 0.00/0.30 twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) 0.00/0.30 -> Usable rules: 0.00/0.30 p(p(s(x1:S))) -> p(x1:S) 0.00/0.30 p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) 0.00/0.30 p(s(x1:S)) -> x1:S 0.00/0.30 ->Interpretation type: 0.00/0.30 Linear 0.00/0.30 ->Coefficients: 0.00/0.30 All rationals 0.00/0.30 ->Dimension: 0.00/0.30 1 0.00/0.30 ->Bound: 0.00/0.30 2 0.00/0.30 ->Interpretation: 0.00/0.30 0.00/0.30 [p](X) = 1/2.X 0.00/0.30 [0](X) = 0 0.00/0.30 [s](X) = 2.X + 1/2 0.00/0.30 [TWICE](X) = 1/2.X 0.00/0.30 0.00/0.30 Problem 1.2: 0.00/0.30 0.00/0.30 SCC Processor: 0.00/0.30 -> Pairs: 0.00/0.30 Empty 0.00/0.30 -> Rules: 0.00/0.30 p(p(s(x1:S))) -> p(x1:S) 0.00/0.30 p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) 0.00/0.30 p(s(x1:S)) -> x1:S 0.00/0.30 tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) 0.00/0.30 tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) 0.00/0.30 twice(0(x1:S)) -> 0(x1:S) 0.00/0.30 twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) 0.00/0.30 twoto(0(x1:S)) -> s(0(x1:S)) 0.00/0.30 twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) 0.00/0.30 ->Strongly Connected Components: 0.00/0.30 There is no strongly connected component 0.00/0.30 0.00/0.30 The problem is finite. 0.00/0.30 0.00/0.30 Problem 1.3: 0.00/0.30 0.00/0.30 Reduction Pair Processor: 0.00/0.30 -> Pairs: 0.00/0.30 TWOTO(s(x1:S)) -> TWOTO(p(s(p(s(x1:S))))) 0.00/0.30 -> Rules: 0.00/0.30 p(p(s(x1:S))) -> p(x1:S) 0.00/0.30 p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) 0.00/0.30 p(s(x1:S)) -> x1:S 0.00/0.30 tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) 0.00/0.30 tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) 0.00/0.30 twice(0(x1:S)) -> 0(x1:S) 0.00/0.30 twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) 0.00/0.30 twoto(0(x1:S)) -> s(0(x1:S)) 0.00/0.30 twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) 0.00/0.30 -> Usable rules: 0.00/0.30 p(p(s(x1:S))) -> p(x1:S) 0.00/0.30 p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) 0.00/0.30 p(s(x1:S)) -> x1:S 0.00/0.30 ->Interpretation type: 0.00/0.30 Linear 0.00/0.30 ->Coefficients: 0.00/0.30 All rationals 0.00/0.30 ->Dimension: 0.00/0.30 1 0.00/0.30 ->Bound: 0.00/0.30 4 0.00/0.30 ->Interpretation: 0.00/0.30 0.00/0.30 [p](X) = 1/4.X 0.00/0.30 [0](X) = 0 0.00/0.30 [s](X) = 4.X + 1 0.00/0.30 [TWOTO](X) = 4.X 0.00/0.30 0.00/0.30 Problem 1.3: 0.00/0.30 0.00/0.30 SCC Processor: 0.00/0.30 -> Pairs: 0.00/0.30 Empty 0.00/0.30 -> Rules: 0.00/0.30 p(p(s(x1:S))) -> p(x1:S) 0.00/0.30 p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) 0.00/0.30 p(s(x1:S)) -> x1:S 0.00/0.30 tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) 0.00/0.30 tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) 0.00/0.30 twice(0(x1:S)) -> 0(x1:S) 0.00/0.30 twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) 0.00/0.30 twoto(0(x1:S)) -> s(0(x1:S)) 0.00/0.30 twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) 0.00/0.30 ->Strongly Connected Components: 0.00/0.30 There is no strongly connected component 0.00/0.30 0.00/0.30 The problem is finite. 0.00/0.30 0.00/0.30 Problem 1.4: 0.00/0.30 0.00/0.30 Reduction Pair Processor: 0.00/0.30 -> Pairs: 0.00/0.30 TOWER(s(x1:S)) -> TOWER(p(s(p(s(x1:S))))) 0.00/0.30 -> Rules: 0.00/0.30 p(p(s(x1:S))) -> p(x1:S) 0.00/0.30 p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) 0.00/0.30 p(s(x1:S)) -> x1:S 0.00/0.30 tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) 0.00/0.30 tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) 0.00/0.30 twice(0(x1:S)) -> 0(x1:S) 0.00/0.30 twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) 0.00/0.30 twoto(0(x1:S)) -> s(0(x1:S)) 0.00/0.30 twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) 0.00/0.30 -> Usable rules: 0.00/0.30 p(p(s(x1:S))) -> p(x1:S) 0.00/0.30 p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) 0.00/0.30 p(s(x1:S)) -> x1:S 0.00/0.30 ->Interpretation type: 0.00/0.30 Linear 0.00/0.30 ->Coefficients: 0.00/0.30 All rationals 0.00/0.30 ->Dimension: 0.00/0.30 1 0.00/0.30 ->Bound: 0.00/0.30 4 0.00/0.30 ->Interpretation: 0.00/0.30 0.00/0.30 [p](X) = 1/4.X 0.00/0.30 [0](X) = 0 0.00/0.30 [s](X) = 4.X + 1 0.00/0.30 [TOWER](X) = 4.X 0.00/0.30 0.00/0.30 Problem 1.4: 0.00/0.30 0.00/0.30 SCC Processor: 0.00/0.30 -> Pairs: 0.00/0.30 Empty 0.00/0.30 -> Rules: 0.00/0.30 p(p(s(x1:S))) -> p(x1:S) 0.00/0.30 p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) 0.00/0.30 p(s(x1:S)) -> x1:S 0.00/0.30 tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) 0.00/0.30 tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) 0.00/0.30 twice(0(x1:S)) -> 0(x1:S) 0.00/0.30 twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) 0.00/0.30 twoto(0(x1:S)) -> s(0(x1:S)) 0.00/0.30 twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) 0.00/0.30 ->Strongly Connected Components: 0.00/0.30 There is no strongly connected component 0.00/0.30 0.00/0.30 The problem is finite. 0.00/0.30 EOF