0.00/0.01 YES 0.00/0.01 0.00/0.01 Problem 1: 0.00/0.01 0.00/0.01 (VAR v_NonEmpty:S N:S X:S Y:S) 0.00/0.01 (RULES 0.00/0.01 add(0,X:S) -> X:S 0.00/0.01 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.01 dbl(0) -> 0 0.00/0.01 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.01 first(0,X:S) -> nil 0.00/0.01 first(s(X:S),cons(Y:S)) -> cons(Y:S) 0.00/0.01 half(dbl(X:S)) -> X:S 0.00/0.01 half(0) -> 0 0.00/0.01 half(s(0)) -> 0 0.00/0.01 half(s(s(X:S))) -> s(half(X:S)) 0.00/0.01 sqr(0) -> 0 0.00/0.01 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.01 terms(N:S) -> cons(recip(sqr(N:S))) 0.00/0.01 ) 0.00/0.01 (STRATEGY INNERMOST) 0.00/0.01 0.00/0.01 Problem 1: 0.00/0.01 0.00/0.01 Dependency Pairs Processor: 0.00/0.01 -> Pairs: 0.00/0.01 ADD(s(X:S),Y:S) -> ADD(X:S,Y:S) 0.00/0.01 DBL(s(X:S)) -> DBL(X:S) 0.00/0.01 HALF(s(s(X:S))) -> HALF(X:S) 0.00/0.01 SQR(s(X:S)) -> ADD(sqr(X:S),dbl(X:S)) 0.00/0.01 SQR(s(X:S)) -> DBL(X:S) 0.00/0.01 SQR(s(X:S)) -> SQR(X:S) 0.00/0.01 TERMS(N:S) -> SQR(N:S) 0.00/0.01 -> Rules: 0.00/0.01 add(0,X:S) -> X:S 0.00/0.01 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.01 dbl(0) -> 0 0.00/0.01 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.01 first(0,X:S) -> nil 0.00/0.01 first(s(X:S),cons(Y:S)) -> cons(Y:S) 0.00/0.01 half(dbl(X:S)) -> X:S 0.00/0.01 half(0) -> 0 0.00/0.01 half(s(0)) -> 0 0.00/0.01 half(s(s(X:S))) -> s(half(X:S)) 0.00/0.01 sqr(0) -> 0 0.00/0.01 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.01 terms(N:S) -> cons(recip(sqr(N:S))) 0.00/0.01 0.00/0.01 Problem 1: 0.00/0.01 0.00/0.01 SCC Processor: 0.00/0.01 -> Pairs: 0.00/0.01 ADD(s(X:S),Y:S) -> ADD(X:S,Y:S) 0.00/0.01 DBL(s(X:S)) -> DBL(X:S) 0.00/0.01 HALF(s(s(X:S))) -> HALF(X:S) 0.00/0.01 SQR(s(X:S)) -> ADD(sqr(X:S),dbl(X:S)) 0.00/0.01 SQR(s(X:S)) -> DBL(X:S) 0.00/0.01 SQR(s(X:S)) -> SQR(X:S) 0.00/0.01 TERMS(N:S) -> SQR(N:S) 0.00/0.01 -> Rules: 0.00/0.01 add(0,X:S) -> X:S 0.00/0.01 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.01 dbl(0) -> 0 0.00/0.01 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.01 first(0,X:S) -> nil 0.00/0.01 first(s(X:S),cons(Y:S)) -> cons(Y:S) 0.00/0.01 half(dbl(X:S)) -> X:S 0.00/0.01 half(0) -> 0 0.00/0.01 half(s(0)) -> 0 0.00/0.01 half(s(s(X:S))) -> s(half(X:S)) 0.00/0.01 sqr(0) -> 0 0.00/0.01 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.01 terms(N:S) -> cons(recip(sqr(N:S))) 0.00/0.01 ->Strongly Connected Components: 0.00/0.01 ->->Cycle: 0.00/0.01 ->->-> Pairs: 0.00/0.01 HALF(s(s(X:S))) -> HALF(X:S) 0.00/0.01 ->->-> Rules: 0.00/0.01 add(0,X:S) -> X:S 0.00/0.01 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.01 dbl(0) -> 0 0.00/0.01 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.01 first(0,X:S) -> nil 0.00/0.01 first(s(X:S),cons(Y:S)) -> cons(Y:S) 0.00/0.01 half(dbl(X:S)) -> X:S 0.00/0.01 half(0) -> 0 0.00/0.01 half(s(0)) -> 0 0.00/0.01 half(s(s(X:S))) -> s(half(X:S)) 0.00/0.01 sqr(0) -> 0 0.00/0.01 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.01 terms(N:S) -> cons(recip(sqr(N:S))) 0.00/0.01 ->->Cycle: 0.00/0.01 ->->-> Pairs: 0.00/0.01 DBL(s(X:S)) -> DBL(X:S) 0.00/0.01 ->->-> Rules: 0.00/0.01 add(0,X:S) -> X:S 0.00/0.01 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.01 dbl(0) -> 0 0.00/0.01 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.01 first(0,X:S) -> nil 0.00/0.01 first(s(X:S),cons(Y:S)) -> cons(Y:S) 0.00/0.01 half(dbl(X:S)) -> X:S 0.00/0.01 half(0) -> 0 0.00/0.01 half(s(0)) -> 0 0.00/0.01 half(s(s(X:S))) -> s(half(X:S)) 0.00/0.01 sqr(0) -> 0 0.00/0.01 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.01 terms(N:S) -> cons(recip(sqr(N:S))) 0.00/0.01 ->->Cycle: 0.00/0.01 ->->-> Pairs: 0.00/0.01 ADD(s(X:S),Y:S) -> ADD(X:S,Y:S) 0.00/0.01 ->->-> Rules: 0.00/0.01 add(0,X:S) -> X:S 0.00/0.01 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.01 dbl(0) -> 0 0.00/0.01 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.01 first(0,X:S) -> nil 0.00/0.01 first(s(X:S),cons(Y:S)) -> cons(Y:S) 0.00/0.01 half(dbl(X:S)) -> X:S 0.00/0.01 half(0) -> 0 0.00/0.01 half(s(0)) -> 0 0.00/0.01 half(s(s(X:S))) -> s(half(X:S)) 0.00/0.01 sqr(0) -> 0 0.00/0.01 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.01 terms(N:S) -> cons(recip(sqr(N:S))) 0.00/0.01 ->->Cycle: 0.00/0.01 ->->-> Pairs: 0.00/0.01 SQR(s(X:S)) -> SQR(X:S) 0.00/0.01 ->->-> Rules: 0.00/0.01 add(0,X:S) -> X:S 0.00/0.01 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.01 dbl(0) -> 0 0.00/0.01 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.01 first(0,X:S) -> nil 0.00/0.01 first(s(X:S),cons(Y:S)) -> cons(Y:S) 0.00/0.01 half(dbl(X:S)) -> X:S 0.00/0.01 half(0) -> 0 0.00/0.01 half(s(0)) -> 0 0.00/0.01 half(s(s(X:S))) -> s(half(X:S)) 0.00/0.01 sqr(0) -> 0 0.00/0.01 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.01 terms(N:S) -> cons(recip(sqr(N:S))) 0.00/0.01 0.00/0.01 0.00/0.01 The problem is decomposed in 4 subproblems. 0.00/0.01 0.00/0.01 Problem 1.1: 0.00/0.01 0.00/0.01 Subterm Processor: 0.00/0.01 -> Pairs: 0.00/0.01 HALF(s(s(X:S))) -> HALF(X:S) 0.00/0.01 -> Rules: 0.00/0.01 add(0,X:S) -> X:S 0.00/0.01 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.01 dbl(0) -> 0 0.00/0.01 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.01 first(0,X:S) -> nil 0.00/0.01 first(s(X:S),cons(Y:S)) -> cons(Y:S) 0.00/0.01 half(dbl(X:S)) -> X:S 0.00/0.01 half(0) -> 0 0.00/0.01 half(s(0)) -> 0 0.00/0.01 half(s(s(X:S))) -> s(half(X:S)) 0.00/0.01 sqr(0) -> 0 0.00/0.01 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.01 terms(N:S) -> cons(recip(sqr(N:S))) 0.00/0.01 ->Projection: 0.00/0.01 pi(HALF) = 1 0.00/0.01 0.00/0.01 Problem 1.1: 0.00/0.01 0.00/0.01 SCC Processor: 0.00/0.01 -> Pairs: 0.00/0.01 Empty 0.00/0.01 -> Rules: 0.00/0.01 add(0,X:S) -> X:S 0.00/0.01 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.01 dbl(0) -> 0 0.00/0.01 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.01 first(0,X:S) -> nil 0.00/0.01 first(s(X:S),cons(Y:S)) -> cons(Y:S) 0.00/0.01 half(dbl(X:S)) -> X:S 0.00/0.01 half(0) -> 0 0.00/0.01 half(s(0)) -> 0 0.00/0.01 half(s(s(X:S))) -> s(half(X:S)) 0.00/0.01 sqr(0) -> 0 0.00/0.01 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.01 terms(N:S) -> cons(recip(sqr(N:S))) 0.00/0.01 ->Strongly Connected Components: 0.00/0.01 There is no strongly connected component 0.00/0.01 0.00/0.01 The problem is finite. 0.00/0.01 0.00/0.01 Problem 1.2: 0.00/0.01 0.00/0.01 Subterm Processor: 0.00/0.01 -> Pairs: 0.00/0.01 DBL(s(X:S)) -> DBL(X:S) 0.00/0.01 -> Rules: 0.00/0.01 add(0,X:S) -> X:S 0.00/0.01 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.01 dbl(0) -> 0 0.00/0.01 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.01 first(0,X:S) -> nil 0.00/0.01 first(s(X:S),cons(Y:S)) -> cons(Y:S) 0.00/0.01 half(dbl(X:S)) -> X:S 0.00/0.01 half(0) -> 0 0.00/0.01 half(s(0)) -> 0 0.00/0.01 half(s(s(X:S))) -> s(half(X:S)) 0.00/0.01 sqr(0) -> 0 0.00/0.01 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.01 terms(N:S) -> cons(recip(sqr(N:S))) 0.00/0.01 ->Projection: 0.00/0.01 pi(DBL) = 1 0.00/0.01 0.00/0.01 Problem 1.2: 0.00/0.01 0.00/0.01 SCC Processor: 0.00/0.01 -> Pairs: 0.00/0.01 Empty 0.00/0.01 -> Rules: 0.00/0.01 add(0,X:S) -> X:S 0.00/0.01 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.01 dbl(0) -> 0 0.00/0.01 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.01 first(0,X:S) -> nil 0.00/0.01 first(s(X:S),cons(Y:S)) -> cons(Y:S) 0.00/0.01 half(dbl(X:S)) -> X:S 0.00/0.01 half(0) -> 0 0.00/0.01 half(s(0)) -> 0 0.00/0.01 half(s(s(X:S))) -> s(half(X:S)) 0.00/0.01 sqr(0) -> 0 0.00/0.01 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.01 terms(N:S) -> cons(recip(sqr(N:S))) 0.00/0.01 ->Strongly Connected Components: 0.00/0.01 There is no strongly connected component 0.00/0.01 0.00/0.01 The problem is finite. 0.00/0.01 0.00/0.01 Problem 1.3: 0.00/0.01 0.00/0.01 Subterm Processor: 0.00/0.01 -> Pairs: 0.00/0.01 ADD(s(X:S),Y:S) -> ADD(X:S,Y:S) 0.00/0.01 -> Rules: 0.00/0.01 add(0,X:S) -> X:S 0.00/0.01 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.01 dbl(0) -> 0 0.00/0.01 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.01 first(0,X:S) -> nil 0.00/0.01 first(s(X:S),cons(Y:S)) -> cons(Y:S) 0.00/0.01 half(dbl(X:S)) -> X:S 0.00/0.01 half(0) -> 0 0.00/0.01 half(s(0)) -> 0 0.00/0.01 half(s(s(X:S))) -> s(half(X:S)) 0.00/0.01 sqr(0) -> 0 0.00/0.01 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.01 terms(N:S) -> cons(recip(sqr(N:S))) 0.00/0.01 ->Projection: 0.00/0.01 pi(ADD) = 1 0.00/0.01 0.00/0.01 Problem 1.3: 0.00/0.01 0.00/0.01 SCC Processor: 0.00/0.01 -> Pairs: 0.00/0.01 Empty 0.00/0.01 -> Rules: 0.00/0.01 add(0,X:S) -> X:S 0.00/0.01 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.01 dbl(0) -> 0 0.00/0.01 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.01 first(0,X:S) -> nil 0.00/0.01 first(s(X:S),cons(Y:S)) -> cons(Y:S) 0.00/0.01 half(dbl(X:S)) -> X:S 0.00/0.01 half(0) -> 0 0.00/0.01 half(s(0)) -> 0 0.00/0.01 half(s(s(X:S))) -> s(half(X:S)) 0.00/0.01 sqr(0) -> 0 0.00/0.01 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.01 terms(N:S) -> cons(recip(sqr(N:S))) 0.00/0.01 ->Strongly Connected Components: 0.00/0.01 There is no strongly connected component 0.00/0.01 0.00/0.01 The problem is finite. 0.00/0.01 0.00/0.01 Problem 1.4: 0.00/0.01 0.00/0.01 Subterm Processor: 0.00/0.01 -> Pairs: 0.00/0.01 SQR(s(X:S)) -> SQR(X:S) 0.00/0.01 -> Rules: 0.00/0.01 add(0,X:S) -> X:S 0.00/0.01 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.01 dbl(0) -> 0 0.00/0.01 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.01 first(0,X:S) -> nil 0.00/0.01 first(s(X:S),cons(Y:S)) -> cons(Y:S) 0.00/0.01 half(dbl(X:S)) -> X:S 0.00/0.01 half(0) -> 0 0.00/0.01 half(s(0)) -> 0 0.00/0.01 half(s(s(X:S))) -> s(half(X:S)) 0.00/0.01 sqr(0) -> 0 0.00/0.01 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.01 terms(N:S) -> cons(recip(sqr(N:S))) 0.00/0.01 ->Projection: 0.00/0.01 pi(SQR) = 1 0.00/0.01 0.00/0.01 Problem 1.4: 0.00/0.01 0.00/0.01 SCC Processor: 0.00/0.01 -> Pairs: 0.00/0.01 Empty 0.00/0.01 -> Rules: 0.00/0.01 add(0,X:S) -> X:S 0.00/0.01 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.01 dbl(0) -> 0 0.00/0.01 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.01 first(0,X:S) -> nil 0.00/0.01 first(s(X:S),cons(Y:S)) -> cons(Y:S) 0.00/0.01 half(dbl(X:S)) -> X:S 0.00/0.01 half(0) -> 0 0.00/0.01 half(s(0)) -> 0 0.00/0.01 half(s(s(X:S))) -> s(half(X:S)) 0.00/0.01 sqr(0) -> 0 0.00/0.01 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.01 terms(N:S) -> cons(recip(sqr(N:S))) 0.00/0.01 ->Strongly Connected Components: 0.00/0.01 There is no strongly connected component 0.00/0.01 0.00/0.01 The problem is finite. 0.00/0.01 EOF