1.82/2.05 YES 1.82/2.05 1.82/2.05 Problem 1: 1.82/2.05 1.82/2.05 (VAR v_NonEmpty:S X:S X1:S X2:S Y:S) 1.82/2.05 (RULES 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ) 1.82/2.05 (STRATEGY INNERMOST) 1.82/2.05 1.82/2.05 Problem 1: 1.82/2.05 1.82/2.05 Dependency Pairs Processor: 1.82/2.05 -> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> ADX(Y:S) 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> CONS(X:S,adx(Y:S)) 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> INCR(cons(X:S,adx(Y:S))) 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(hd(cons(X:S,Y:S))) -> MARK(X:S) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> CONS(s(X:S),incr(Y:S)) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> INCR(Y:S) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> S(X:S) 1.82/2.05 ACTIVE(tl(cons(X:S,Y:S))) -> MARK(Y:S) 1.82/2.05 ACTIVE(nats) -> MARK(adx(zeros)) 1.82/2.05 ACTIVE(zeros) -> MARK(cons(0,zeros)) 1.82/2.05 ADX(active(X:S)) -> ADX(X:S) 1.82/2.05 ADX(mark(X:S)) -> ADX(X:S) 1.82/2.05 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) 1.82/2.05 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 1.82/2.05 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 1.82/2.05 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 1.82/2.05 HD(active(X:S)) -> HD(X:S) 1.82/2.05 HD(mark(X:S)) -> HD(X:S) 1.82/2.05 INCR(active(X:S)) -> INCR(X:S) 1.82/2.05 INCR(mark(X:S)) -> INCR(X:S) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(adx(X:S)) -> ADX(mark(X:S)) 1.82/2.05 MARK(adx(X:S)) -> MARK(X:S) 1.82/2.05 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(X1:S,X2:S)) 1.82/2.05 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> HD(mark(X:S)) 1.82/2.05 MARK(hd(X:S)) -> MARK(X:S) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> INCR(mark(X:S)) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(s(X:S)) -> ACTIVE(s(X:S)) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> TL(mark(X:S)) 1.82/2.05 MARK(nats) -> ACTIVE(nats) 1.82/2.05 MARK(zeros) -> ACTIVE(zeros) 1.82/2.05 S(active(X:S)) -> S(X:S) 1.82/2.05 S(mark(X:S)) -> S(X:S) 1.82/2.05 TL(active(X:S)) -> TL(X:S) 1.82/2.05 TL(mark(X:S)) -> TL(X:S) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 1.82/2.05 Problem 1: 1.82/2.05 1.82/2.05 SCC Processor: 1.82/2.05 -> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> ADX(Y:S) 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> CONS(X:S,adx(Y:S)) 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> INCR(cons(X:S,adx(Y:S))) 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(hd(cons(X:S,Y:S))) -> MARK(X:S) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> CONS(s(X:S),incr(Y:S)) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> INCR(Y:S) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> S(X:S) 1.82/2.05 ACTIVE(tl(cons(X:S,Y:S))) -> MARK(Y:S) 1.82/2.05 ACTIVE(nats) -> MARK(adx(zeros)) 1.82/2.05 ACTIVE(zeros) -> MARK(cons(0,zeros)) 1.82/2.05 ADX(active(X:S)) -> ADX(X:S) 1.82/2.05 ADX(mark(X:S)) -> ADX(X:S) 1.82/2.05 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) 1.82/2.05 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 1.82/2.05 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 1.82/2.05 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 1.82/2.05 HD(active(X:S)) -> HD(X:S) 1.82/2.05 HD(mark(X:S)) -> HD(X:S) 1.82/2.05 INCR(active(X:S)) -> INCR(X:S) 1.82/2.05 INCR(mark(X:S)) -> INCR(X:S) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(adx(X:S)) -> ADX(mark(X:S)) 1.82/2.05 MARK(adx(X:S)) -> MARK(X:S) 1.82/2.05 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(X1:S,X2:S)) 1.82/2.05 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> HD(mark(X:S)) 1.82/2.05 MARK(hd(X:S)) -> MARK(X:S) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> INCR(mark(X:S)) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(s(X:S)) -> ACTIVE(s(X:S)) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> TL(mark(X:S)) 1.82/2.05 MARK(nats) -> ACTIVE(nats) 1.82/2.05 MARK(zeros) -> ACTIVE(zeros) 1.82/2.05 S(active(X:S)) -> S(X:S) 1.82/2.05 S(mark(X:S)) -> S(X:S) 1.82/2.05 TL(active(X:S)) -> TL(X:S) 1.82/2.05 TL(mark(X:S)) -> TL(X:S) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Strongly Connected Components: 1.82/2.05 ->->Cycle: 1.82/2.05 ->->-> Pairs: 1.82/2.05 TL(active(X:S)) -> TL(X:S) 1.82/2.05 TL(mark(X:S)) -> TL(X:S) 1.82/2.05 ->->-> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->->Cycle: 1.82/2.05 ->->-> Pairs: 1.82/2.05 S(active(X:S)) -> S(X:S) 1.82/2.05 S(mark(X:S)) -> S(X:S) 1.82/2.05 ->->-> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->->Cycle: 1.82/2.05 ->->-> Pairs: 1.82/2.05 INCR(active(X:S)) -> INCR(X:S) 1.82/2.05 INCR(mark(X:S)) -> INCR(X:S) 1.82/2.05 ->->-> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->->Cycle: 1.82/2.05 ->->-> Pairs: 1.82/2.05 HD(active(X:S)) -> HD(X:S) 1.82/2.05 HD(mark(X:S)) -> HD(X:S) 1.82/2.05 ->->-> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->->Cycle: 1.82/2.05 ->->-> Pairs: 1.82/2.05 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) 1.82/2.05 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 1.82/2.05 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 1.82/2.05 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 1.82/2.05 ->->-> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->->Cycle: 1.82/2.05 ->->-> Pairs: 1.82/2.05 ADX(active(X:S)) -> ADX(X:S) 1.82/2.05 ADX(mark(X:S)) -> ADX(X:S) 1.82/2.05 ->->-> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->->Cycle: 1.82/2.05 ->->-> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(hd(cons(X:S,Y:S))) -> MARK(X:S) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 ACTIVE(tl(cons(X:S,Y:S))) -> MARK(Y:S) 1.82/2.05 ACTIVE(nats) -> MARK(adx(zeros)) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(adx(X:S)) -> MARK(X:S) 1.82/2.05 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> MARK(X:S) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 MARK(nats) -> ACTIVE(nats) 1.82/2.05 ->->-> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 1.82/2.05 1.82/2.05 The problem is decomposed in 7 subproblems. 1.82/2.05 1.82/2.05 Problem 1.1: 1.82/2.05 1.82/2.05 Subterm Processor: 1.82/2.05 -> Pairs: 1.82/2.05 TL(active(X:S)) -> TL(X:S) 1.82/2.05 TL(mark(X:S)) -> TL(X:S) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Projection: 1.82/2.05 pi(TL) = 1 1.82/2.05 1.82/2.05 Problem 1.1: 1.82/2.05 1.82/2.05 SCC Processor: 1.82/2.05 -> Pairs: 1.82/2.05 Empty 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Strongly Connected Components: 1.82/2.05 There is no strongly connected component 1.82/2.05 1.82/2.05 The problem is finite. 1.82/2.05 1.82/2.05 Problem 1.2: 1.82/2.05 1.82/2.05 Subterm Processor: 1.82/2.05 -> Pairs: 1.82/2.05 S(active(X:S)) -> S(X:S) 1.82/2.05 S(mark(X:S)) -> S(X:S) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Projection: 1.82/2.05 pi(S) = 1 1.82/2.05 1.82/2.05 Problem 1.2: 1.82/2.05 1.82/2.05 SCC Processor: 1.82/2.05 -> Pairs: 1.82/2.05 Empty 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Strongly Connected Components: 1.82/2.05 There is no strongly connected component 1.82/2.05 1.82/2.05 The problem is finite. 1.82/2.05 1.82/2.05 Problem 1.3: 1.82/2.05 1.82/2.05 Subterm Processor: 1.82/2.05 -> Pairs: 1.82/2.05 INCR(active(X:S)) -> INCR(X:S) 1.82/2.05 INCR(mark(X:S)) -> INCR(X:S) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Projection: 1.82/2.05 pi(INCR) = 1 1.82/2.05 1.82/2.05 Problem 1.3: 1.82/2.05 1.82/2.05 SCC Processor: 1.82/2.05 -> Pairs: 1.82/2.05 Empty 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Strongly Connected Components: 1.82/2.05 There is no strongly connected component 1.82/2.05 1.82/2.05 The problem is finite. 1.82/2.05 1.82/2.05 Problem 1.4: 1.82/2.05 1.82/2.05 Subterm Processor: 1.82/2.05 -> Pairs: 1.82/2.05 HD(active(X:S)) -> HD(X:S) 1.82/2.05 HD(mark(X:S)) -> HD(X:S) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Projection: 1.82/2.05 pi(HD) = 1 1.82/2.05 1.82/2.05 Problem 1.4: 1.82/2.05 1.82/2.05 SCC Processor: 1.82/2.05 -> Pairs: 1.82/2.05 Empty 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Strongly Connected Components: 1.82/2.05 There is no strongly connected component 1.82/2.05 1.82/2.05 The problem is finite. 1.82/2.05 1.82/2.05 Problem 1.5: 1.82/2.05 1.82/2.05 Subterm Processor: 1.82/2.05 -> Pairs: 1.82/2.05 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) 1.82/2.05 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 1.82/2.05 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 1.82/2.05 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Projection: 1.82/2.05 pi(CONS) = 1 1.82/2.05 1.82/2.05 Problem 1.5: 1.82/2.05 1.82/2.05 SCC Processor: 1.82/2.05 -> Pairs: 1.82/2.05 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 1.82/2.05 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Strongly Connected Components: 1.82/2.05 ->->Cycle: 1.82/2.05 ->->-> Pairs: 1.82/2.05 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 1.82/2.05 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 1.82/2.05 ->->-> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 1.82/2.05 Problem 1.5: 1.82/2.05 1.82/2.05 Subterm Processor: 1.82/2.05 -> Pairs: 1.82/2.05 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 1.82/2.05 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Projection: 1.82/2.05 pi(CONS) = 2 1.82/2.05 1.82/2.05 Problem 1.5: 1.82/2.05 1.82/2.05 SCC Processor: 1.82/2.05 -> Pairs: 1.82/2.05 Empty 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Strongly Connected Components: 1.82/2.05 There is no strongly connected component 1.82/2.05 1.82/2.05 The problem is finite. 1.82/2.05 1.82/2.05 Problem 1.6: 1.82/2.05 1.82/2.05 Subterm Processor: 1.82/2.05 -> Pairs: 1.82/2.05 ADX(active(X:S)) -> ADX(X:S) 1.82/2.05 ADX(mark(X:S)) -> ADX(X:S) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Projection: 1.82/2.05 pi(ADX) = 1 1.82/2.05 1.82/2.05 Problem 1.6: 1.82/2.05 1.82/2.05 SCC Processor: 1.82/2.05 -> Pairs: 1.82/2.05 Empty 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Strongly Connected Components: 1.82/2.05 There is no strongly connected component 1.82/2.05 1.82/2.05 The problem is finite. 1.82/2.05 1.82/2.05 Problem 1.7: 1.82/2.05 1.82/2.05 Reduction Pairs Processor: 1.82/2.05 -> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(hd(cons(X:S,Y:S))) -> MARK(X:S) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 ACTIVE(tl(cons(X:S,Y:S))) -> MARK(Y:S) 1.82/2.05 ACTIVE(nats) -> MARK(adx(zeros)) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(adx(X:S)) -> MARK(X:S) 1.82/2.05 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> MARK(X:S) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 MARK(nats) -> ACTIVE(nats) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 -> Usable rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Interpretation type: 1.82/2.05 Linear 1.82/2.05 ->Coefficients: 1.82/2.05 Natural Numbers 1.82/2.05 ->Dimension: 1.82/2.05 1 1.82/2.05 ->Bound: 1.82/2.05 2 1.82/2.05 ->Interpretation: 1.82/2.05 1.82/2.05 [active](X) = X 1.82/2.05 [adx](X) = X 1.82/2.05 [cons](X1,X2) = 2.X1 + 2.X2 1.82/2.05 [hd](X) = 2.X + 2 1.82/2.05 [incr](X) = X 1.82/2.05 [mark](X) = X 1.82/2.05 [s](X) = X 1.82/2.05 [tl](X) = 2.X + 2 1.82/2.05 [0] = 0 1.82/2.05 [fSNonEmpty] = 0 1.82/2.05 [nats] = 2 1.82/2.05 [zeros] = 0 1.82/2.05 [ACTIVE](X) = 2.X + 2 1.82/2.05 [ADX](X) = 0 1.82/2.05 [CONS](X1,X2) = 0 1.82/2.05 [HD](X) = 0 1.82/2.05 [INCR](X) = 0 1.82/2.05 [MARK](X) = 2.X + 2 1.82/2.05 [S](X) = 0 1.82/2.05 [TL](X) = 0 1.82/2.05 1.82/2.05 Problem 1.7: 1.82/2.05 1.82/2.05 SCC Processor: 1.82/2.05 -> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 ACTIVE(tl(cons(X:S,Y:S))) -> MARK(Y:S) 1.82/2.05 ACTIVE(nats) -> MARK(adx(zeros)) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(adx(X:S)) -> MARK(X:S) 1.82/2.05 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> MARK(X:S) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 MARK(nats) -> ACTIVE(nats) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Strongly Connected Components: 1.82/2.05 ->->Cycle: 1.82/2.05 ->->-> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 ACTIVE(tl(cons(X:S,Y:S))) -> MARK(Y:S) 1.82/2.05 ACTIVE(nats) -> MARK(adx(zeros)) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(adx(X:S)) -> MARK(X:S) 1.82/2.05 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> MARK(X:S) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 MARK(nats) -> ACTIVE(nats) 1.82/2.05 ->->-> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 1.82/2.05 Problem 1.7: 1.82/2.05 1.82/2.05 Reduction Pairs Processor: 1.82/2.05 -> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 ACTIVE(tl(cons(X:S,Y:S))) -> MARK(Y:S) 1.82/2.05 ACTIVE(nats) -> MARK(adx(zeros)) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(adx(X:S)) -> MARK(X:S) 1.82/2.05 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> MARK(X:S) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 MARK(nats) -> ACTIVE(nats) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 -> Usable rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Interpretation type: 1.82/2.05 Linear 1.82/2.05 ->Coefficients: 1.82/2.05 Natural Numbers 1.82/2.05 ->Dimension: 1.82/2.05 1 1.82/2.05 ->Bound: 1.82/2.05 2 1.82/2.05 ->Interpretation: 1.82/2.05 1.82/2.05 [active](X) = X 1.82/2.05 [adx](X) = 2.X 1.82/2.05 [cons](X1,X2) = 2.X1 + 2.X2 1.82/2.05 [hd](X) = X 1.82/2.05 [incr](X) = X 1.82/2.05 [mark](X) = X 1.82/2.05 [s](X) = 0 1.82/2.05 [tl](X) = 2.X + 2 1.82/2.05 [0] = 0 1.82/2.05 [fSNonEmpty] = 0 1.82/2.05 [nats] = 2 1.82/2.05 [zeros] = 0 1.82/2.05 [ACTIVE](X) = X 1.82/2.05 [ADX](X) = 0 1.82/2.05 [CONS](X1,X2) = 0 1.82/2.05 [HD](X) = 0 1.82/2.05 [INCR](X) = 0 1.82/2.05 [MARK](X) = X 1.82/2.05 [S](X) = 0 1.82/2.05 [TL](X) = 0 1.82/2.05 1.82/2.05 Problem 1.7: 1.82/2.05 1.82/2.05 SCC Processor: 1.82/2.05 -> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 ACTIVE(nats) -> MARK(adx(zeros)) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(adx(X:S)) -> MARK(X:S) 1.82/2.05 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> MARK(X:S) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 MARK(nats) -> ACTIVE(nats) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Strongly Connected Components: 1.82/2.05 ->->Cycle: 1.82/2.05 ->->-> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 ACTIVE(nats) -> MARK(adx(zeros)) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(adx(X:S)) -> MARK(X:S) 1.82/2.05 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> MARK(X:S) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 MARK(nats) -> ACTIVE(nats) 1.82/2.05 ->->-> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 1.82/2.05 Problem 1.7: 1.82/2.05 1.82/2.05 Reduction Pairs Processor: 1.82/2.05 -> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 ACTIVE(nats) -> MARK(adx(zeros)) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(adx(X:S)) -> MARK(X:S) 1.82/2.05 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> MARK(X:S) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 MARK(nats) -> ACTIVE(nats) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 -> Usable rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Interpretation type: 1.82/2.05 Linear 1.82/2.05 ->Coefficients: 1.82/2.05 Natural Numbers 1.82/2.05 ->Dimension: 1.82/2.05 1 1.82/2.05 ->Bound: 1.82/2.05 2 1.82/2.05 ->Interpretation: 1.82/2.05 1.82/2.05 [active](X) = X 1.82/2.05 [adx](X) = 2.X + 1 1.82/2.05 [cons](X1,X2) = 2.X1 + X2 1.82/2.05 [hd](X) = 2.X 1.82/2.05 [incr](X) = X 1.82/2.05 [mark](X) = X 1.82/2.05 [s](X) = X 1.82/2.05 [tl](X) = 2.X 1.82/2.05 [0] = 0 1.82/2.05 [fSNonEmpty] = 0 1.82/2.05 [nats] = 2 1.82/2.05 [zeros] = 0 1.82/2.05 [ACTIVE](X) = 2.X + 2 1.82/2.05 [ADX](X) = 0 1.82/2.05 [CONS](X1,X2) = 0 1.82/2.05 [HD](X) = 0 1.82/2.05 [INCR](X) = 0 1.82/2.05 [MARK](X) = 2.X + 2 1.82/2.05 [S](X) = 0 1.82/2.05 [TL](X) = 0 1.82/2.05 1.82/2.05 Problem 1.7: 1.82/2.05 1.82/2.05 SCC Processor: 1.82/2.05 -> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(adx(X:S)) -> MARK(X:S) 1.82/2.05 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> MARK(X:S) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 MARK(nats) -> ACTIVE(nats) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Strongly Connected Components: 1.82/2.05 ->->Cycle: 1.82/2.05 ->->-> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(adx(X:S)) -> MARK(X:S) 1.82/2.05 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> MARK(X:S) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 ->->-> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 1.82/2.05 Problem 1.7: 1.82/2.05 1.82/2.05 Reduction Pairs Processor: 1.82/2.05 -> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(adx(X:S)) -> MARK(X:S) 1.82/2.05 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> MARK(X:S) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 -> Usable rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Interpretation type: 1.82/2.05 Linear 1.82/2.05 ->Coefficients: 1.82/2.05 Natural Numbers 1.82/2.05 ->Dimension: 1.82/2.05 1 1.82/2.05 ->Bound: 1.82/2.05 2 1.82/2.05 ->Interpretation: 1.82/2.05 1.82/2.05 [active](X) = X 1.82/2.05 [adx](X) = 2.X + 2 1.82/2.05 [cons](X1,X2) = X1 + X2 1.82/2.05 [hd](X) = 2.X + 1 1.82/2.05 [incr](X) = X 1.82/2.05 [mark](X) = X 1.82/2.05 [s](X) = X 1.82/2.05 [tl](X) = X + 2 1.82/2.05 [0] = 0 1.82/2.05 [fSNonEmpty] = 0 1.82/2.05 [nats] = 2 1.82/2.05 [zeros] = 0 1.82/2.05 [ACTIVE](X) = 2.X + 2 1.82/2.05 [ADX](X) = 0 1.82/2.05 [CONS](X1,X2) = 0 1.82/2.05 [HD](X) = 0 1.82/2.05 [INCR](X) = 0 1.82/2.05 [MARK](X) = 2.X + 2 1.82/2.05 [S](X) = 0 1.82/2.05 [TL](X) = 0 1.82/2.05 1.82/2.05 Problem 1.7: 1.82/2.05 1.82/2.05 SCC Processor: 1.82/2.05 -> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> MARK(X:S) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Strongly Connected Components: 1.82/2.05 ->->Cycle: 1.82/2.05 ->->-> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> MARK(X:S) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 ->->-> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 1.82/2.05 Problem 1.7: 1.82/2.05 1.82/2.05 Reduction Pairs Processor: 1.82/2.05 -> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> MARK(X:S) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 -> Usable rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Interpretation type: 1.82/2.05 Linear 1.82/2.05 ->Coefficients: 1.82/2.05 Natural Numbers 1.82/2.05 ->Dimension: 1.82/2.05 1 1.82/2.05 ->Bound: 1.82/2.05 2 1.82/2.05 ->Interpretation: 1.82/2.05 1.82/2.05 [active](X) = 0 1.82/2.05 [adx](X) = 1 1.82/2.05 [cons](X1,X2) = 1 1.82/2.05 [hd](X) = 0 1.82/2.05 [incr](X) = 1 1.82/2.05 [mark](X) = 0 1.82/2.05 [s](X) = 2 1.82/2.05 [tl](X) = 1 1.82/2.05 [0] = 0 1.82/2.05 [fSNonEmpty] = 0 1.82/2.05 [nats] = 2 1.82/2.05 [zeros] = 0 1.82/2.05 [ACTIVE](X) = 2.X 1.82/2.05 [ADX](X) = 0 1.82/2.05 [CONS](X1,X2) = 0 1.82/2.05 [HD](X) = 0 1.82/2.05 [INCR](X) = 0 1.82/2.05 [MARK](X) = 2 1.82/2.05 [S](X) = 0 1.82/2.05 [TL](X) = 0 1.82/2.05 1.82/2.05 Problem 1.7: 1.82/2.05 1.82/2.05 SCC Processor: 1.82/2.05 -> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> MARK(X:S) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Strongly Connected Components: 1.82/2.05 ->->Cycle: 1.82/2.05 ->->-> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> MARK(X:S) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 ->->-> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 1.82/2.05 Problem 1.7: 1.82/2.05 1.82/2.05 Reduction Pairs Processor: 1.82/2.05 -> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(hd(X:S)) -> MARK(X:S) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 -> Usable rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.05 hd(active(X:S)) -> hd(X:S) 1.82/2.05 hd(mark(X:S)) -> hd(X:S) 1.82/2.05 incr(active(X:S)) -> incr(X:S) 1.82/2.05 incr(mark(X:S)) -> incr(X:S) 1.82/2.05 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.05 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.05 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.05 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.05 mark(s(X:S)) -> active(s(X:S)) 1.82/2.05 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.05 mark(0) -> active(0) 1.82/2.05 mark(nats) -> active(nats) 1.82/2.05 mark(zeros) -> active(zeros) 1.82/2.05 s(active(X:S)) -> s(X:S) 1.82/2.05 s(mark(X:S)) -> s(X:S) 1.82/2.05 tl(active(X:S)) -> tl(X:S) 1.82/2.05 tl(mark(X:S)) -> tl(X:S) 1.82/2.05 ->Interpretation type: 1.82/2.05 Linear 1.82/2.05 ->Coefficients: 1.82/2.05 Natural Numbers 1.82/2.05 ->Dimension: 1.82/2.05 1 1.82/2.05 ->Bound: 1.82/2.05 2 1.82/2.05 ->Interpretation: 1.82/2.05 1.82/2.05 [active](X) = X 1.82/2.05 [adx](X) = 2.X + 2 1.82/2.05 [cons](X1,X2) = 2.X1 + X2 1.82/2.05 [hd](X) = X + 2 1.82/2.05 [incr](X) = X 1.82/2.05 [mark](X) = X 1.82/2.05 [s](X) = X 1.82/2.05 [tl](X) = 2.X + 1 1.82/2.05 [0] = 0 1.82/2.05 [fSNonEmpty] = 0 1.82/2.05 [nats] = 2 1.82/2.05 [zeros] = 0 1.82/2.05 [ACTIVE](X) = 2.X + 2 1.82/2.05 [ADX](X) = 0 1.82/2.05 [CONS](X1,X2) = 0 1.82/2.05 [HD](X) = 0 1.82/2.05 [INCR](X) = 0 1.82/2.05 [MARK](X) = 2.X + 2 1.82/2.05 [S](X) = 0 1.82/2.05 [TL](X) = 0 1.82/2.05 1.82/2.05 Problem 1.7: 1.82/2.05 1.82/2.05 SCC Processor: 1.82/2.05 -> Pairs: 1.82/2.05 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.05 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.05 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.05 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.05 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.05 -> Rules: 1.82/2.05 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.05 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.05 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.05 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.05 active(nats) -> mark(adx(zeros)) 1.82/2.05 active(zeros) -> mark(cons(0,zeros)) 1.82/2.05 adx(active(X:S)) -> adx(X:S) 1.82/2.05 adx(mark(X:S)) -> adx(X:S) 1.82/2.05 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.05 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 ->Strongly Connected Components: 1.82/2.06 ->->Cycle: 1.82/2.06 ->->-> Pairs: 1.82/2.06 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.06 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.06 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.06 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.06 ->->-> Rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 1.82/2.06 Problem 1.7: 1.82/2.06 1.82/2.06 Reduction Pairs Processor: 1.82/2.06 -> Pairs: 1.82/2.06 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.06 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.06 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S))) 1.82/2.06 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.06 -> Rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 -> Usable rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 ->Interpretation type: 1.82/2.06 Linear 1.82/2.06 ->Coefficients: 1.82/2.06 Natural Numbers 1.82/2.06 ->Dimension: 1.82/2.06 1 1.82/2.06 ->Bound: 1.82/2.06 2 1.82/2.06 ->Interpretation: 1.82/2.06 1.82/2.06 [active](X) = 2 1.82/2.06 [adx](X) = 2 1.82/2.06 [cons](X1,X2) = 0 1.82/2.06 [hd](X) = 2 1.82/2.06 [incr](X) = 2 1.82/2.06 [mark](X) = 2 1.82/2.06 [s](X) = 1 1.82/2.06 [tl](X) = 1 1.82/2.06 [0] = 1 1.82/2.06 [fSNonEmpty] = 0 1.82/2.06 [nats] = 2 1.82/2.06 [zeros] = 2 1.82/2.06 [ACTIVE](X) = X 1.82/2.06 [ADX](X) = 0 1.82/2.06 [CONS](X1,X2) = 0 1.82/2.06 [HD](X) = 0 1.82/2.06 [INCR](X) = 0 1.82/2.06 [MARK](X) = 2 1.82/2.06 [S](X) = 0 1.82/2.06 [TL](X) = 0 1.82/2.06 1.82/2.06 Problem 1.7: 1.82/2.06 1.82/2.06 SCC Processor: 1.82/2.06 -> Pairs: 1.82/2.06 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.06 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.06 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.06 -> Rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 ->Strongly Connected Components: 1.82/2.06 ->->Cycle: 1.82/2.06 ->->-> Pairs: 1.82/2.06 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.06 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.06 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.06 ->->-> Rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 1.82/2.06 Problem 1.7: 1.82/2.06 1.82/2.06 Reduction Pairs Processor: 1.82/2.06 -> Pairs: 1.82/2.06 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.06 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.06 MARK(tl(X:S)) -> MARK(X:S) 1.82/2.06 -> Rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 -> Usable rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 ->Interpretation type: 1.82/2.06 Linear 1.82/2.06 ->Coefficients: 1.82/2.06 Natural Numbers 1.82/2.06 ->Dimension: 1.82/2.06 1 1.82/2.06 ->Bound: 1.82/2.06 2 1.82/2.06 ->Interpretation: 1.82/2.06 1.82/2.06 [active](X) = X 1.82/2.06 [adx](X) = 2.X 1.82/2.06 [cons](X1,X2) = 2.X1 + 2.X2 1.82/2.06 [hd](X) = 2.X + 1 1.82/2.06 [incr](X) = X 1.82/2.06 [mark](X) = X 1.82/2.06 [s](X) = X 1.82/2.06 [tl](X) = X + 2 1.82/2.06 [0] = 0 1.82/2.06 [fSNonEmpty] = 0 1.82/2.06 [nats] = 1 1.82/2.06 [zeros] = 0 1.82/2.06 [ACTIVE](X) = X + 2 1.82/2.06 [ADX](X) = 0 1.82/2.06 [CONS](X1,X2) = 0 1.82/2.06 [HD](X) = 0 1.82/2.06 [INCR](X) = 0 1.82/2.06 [MARK](X) = X + 2 1.82/2.06 [S](X) = 0 1.82/2.06 [TL](X) = 0 1.82/2.06 1.82/2.06 Problem 1.7: 1.82/2.06 1.82/2.06 SCC Processor: 1.82/2.06 -> Pairs: 1.82/2.06 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.06 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.06 -> Rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 ->Strongly Connected Components: 1.82/2.06 ->->Cycle: 1.82/2.06 ->->-> Pairs: 1.82/2.06 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.06 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.06 ->->-> Rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 1.82/2.06 Problem 1.7: 1.82/2.06 1.82/2.06 Reduction Pairs Processor: 1.82/2.06 -> Pairs: 1.82/2.06 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.06 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.06 -> Rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 -> Usable rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 ->Interpretation type: 1.82/2.06 Linear 1.82/2.06 ->Coefficients: 1.82/2.06 All rationals 1.82/2.06 ->Dimension: 1.82/2.06 1 1.82/2.06 ->Bound: 1.82/2.06 2 1.82/2.06 ->Interpretation: 1.82/2.06 1.82/2.06 [active](X) = X 1.82/2.06 [adx](X) = X + 1 1.82/2.06 [cons](X1,X2) = 2.X1 + 1/2.X2 + 1/2 1.82/2.06 [hd](X) = 1/2.X 1.82/2.06 [incr](X) = X 1.82/2.06 [mark](X) = X 1.82/2.06 [s](X) = 1/2.X 1.82/2.06 [tl](X) = 2.X 1.82/2.06 [0] = 0 1.82/2.06 [fSNonEmpty] = 0 1.82/2.06 [nats] = 2 1.82/2.06 [zeros] = 1 1.82/2.06 [ACTIVE](X) = X + 2 1.82/2.06 [ADX](X) = 0 1.82/2.06 [CONS](X1,X2) = 0 1.82/2.06 [HD](X) = 0 1.82/2.06 [INCR](X) = 0 1.82/2.06 [MARK](X) = X + 2 1.82/2.06 [S](X) = 0 1.82/2.06 [TL](X) = 0 1.82/2.06 1.82/2.06 Problem 1.7: 1.82/2.06 1.82/2.06 SCC Processor: 1.82/2.06 -> Pairs: 1.82/2.06 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.06 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.06 -> Rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 ->Strongly Connected Components: 1.82/2.06 ->->Cycle: 1.82/2.06 ->->-> Pairs: 1.82/2.06 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.06 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.06 ->->-> Rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 1.82/2.06 Problem 1.7: 1.82/2.06 1.82/2.06 Reduction Pairs Processor: 1.82/2.06 -> Pairs: 1.82/2.06 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.06 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.06 -> Rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 -> Usable rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 ->Interpretation type: 1.82/2.06 Linear 1.82/2.06 ->Coefficients: 1.82/2.06 Natural Numbers 1.82/2.06 ->Dimension: 1.82/2.06 1 1.82/2.06 ->Bound: 1.82/2.06 2 1.82/2.06 ->Interpretation: 1.82/2.06 1.82/2.06 [active](X) = 2 1.82/2.06 [adx](X) = 0 1.82/2.06 [cons](X1,X2) = 0 1.82/2.06 [hd](X) = 2 1.82/2.06 [incr](X) = 2 1.82/2.06 [mark](X) = 2 1.82/2.06 [s](X) = 2 1.82/2.06 [tl](X) = 2 1.82/2.06 [0] = 2 1.82/2.06 [fSNonEmpty] = 0 1.82/2.06 [nats] = 2 1.82/2.06 [zeros] = 2 1.82/2.06 [ACTIVE](X) = X 1.82/2.06 [ADX](X) = 0 1.82/2.06 [CONS](X1,X2) = 0 1.82/2.06 [HD](X) = 0 1.82/2.06 [INCR](X) = 0 1.82/2.06 [MARK](X) = 2 1.82/2.06 [S](X) = 0 1.82/2.06 [TL](X) = 0 1.82/2.06 1.82/2.06 Problem 1.7: 1.82/2.06 1.82/2.06 SCC Processor: 1.82/2.06 -> Pairs: 1.82/2.06 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.06 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.06 -> Rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 ->Strongly Connected Components: 1.82/2.06 ->->Cycle: 1.82/2.06 ->->-> Pairs: 1.82/2.06 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.06 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.06 ->->-> Rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 1.82/2.06 Problem 1.7: 1.82/2.06 1.82/2.06 Reduction Pairs Processor: 1.82/2.06 -> Pairs: 1.82/2.06 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S))) 1.82/2.06 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.06 -> Rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 -> Usable rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 ->Interpretation type: 1.82/2.06 Linear 1.82/2.06 ->Coefficients: 1.82/2.06 All rationals 1.82/2.06 ->Dimension: 1.82/2.06 1 1.82/2.06 ->Bound: 1.82/2.06 2 1.82/2.06 ->Interpretation: 1.82/2.06 1.82/2.06 [active](X) = X 1.82/2.06 [adx](X) = 2.X + 2 1.82/2.06 [cons](X1,X2) = 2.X1 + 1/2.X2 1.82/2.06 [hd](X) = X + 2 1.82/2.06 [incr](X) = X + 1 1.82/2.06 [mark](X) = X 1.82/2.06 [s](X) = 1/2.X 1.82/2.06 [tl](X) = 2.X + 2 1.82/2.06 [0] = 0 1.82/2.06 [fSNonEmpty] = 0 1.82/2.06 [nats] = 2 1.82/2.06 [zeros] = 0 1.82/2.06 [ACTIVE](X) = X + 1 1.82/2.06 [ADX](X) = 0 1.82/2.06 [CONS](X1,X2) = 0 1.82/2.06 [HD](X) = 0 1.82/2.06 [INCR](X) = 0 1.82/2.06 [MARK](X) = X + 1 1.82/2.06 [S](X) = 0 1.82/2.06 [TL](X) = 0 1.82/2.06 1.82/2.06 Problem 1.7: 1.82/2.06 1.82/2.06 SCC Processor: 1.82/2.06 -> Pairs: 1.82/2.06 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) 1.82/2.06 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.06 -> Rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 ->Strongly Connected Components: 1.82/2.06 ->->Cycle: 1.82/2.06 ->->-> Pairs: 1.82/2.06 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.06 ->->-> Rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 1.82/2.06 Problem 1.7: 1.82/2.06 1.82/2.06 Subterm Processor: 1.82/2.06 -> Pairs: 1.82/2.06 MARK(incr(X:S)) -> MARK(X:S) 1.82/2.06 -> Rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 ->Projection: 1.82/2.06 pi(MARK) = 1 1.82/2.06 1.82/2.06 Problem 1.7: 1.82/2.06 1.82/2.06 SCC Processor: 1.82/2.06 -> Pairs: 1.82/2.06 Empty 1.82/2.06 -> Rules: 1.82/2.06 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 1.82/2.06 active(hd(cons(X:S,Y:S))) -> mark(X:S) 1.82/2.06 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 1.82/2.06 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 1.82/2.06 active(nats) -> mark(adx(zeros)) 1.82/2.06 active(zeros) -> mark(cons(0,zeros)) 1.82/2.06 adx(active(X:S)) -> adx(X:S) 1.82/2.06 adx(mark(X:S)) -> adx(X:S) 1.82/2.06 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 1.82/2.06 hd(active(X:S)) -> hd(X:S) 1.82/2.06 hd(mark(X:S)) -> hd(X:S) 1.82/2.06 incr(active(X:S)) -> incr(X:S) 1.82/2.06 incr(mark(X:S)) -> incr(X:S) 1.82/2.06 mark(adx(X:S)) -> active(adx(mark(X:S))) 1.82/2.06 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) 1.82/2.06 mark(hd(X:S)) -> active(hd(mark(X:S))) 1.82/2.06 mark(incr(X:S)) -> active(incr(mark(X:S))) 1.82/2.06 mark(s(X:S)) -> active(s(X:S)) 1.82/2.06 mark(tl(X:S)) -> active(tl(mark(X:S))) 1.82/2.06 mark(0) -> active(0) 1.82/2.06 mark(nats) -> active(nats) 1.82/2.06 mark(zeros) -> active(zeros) 1.82/2.06 s(active(X:S)) -> s(X:S) 1.82/2.06 s(mark(X:S)) -> s(X:S) 1.82/2.06 tl(active(X:S)) -> tl(X:S) 1.82/2.06 tl(mark(X:S)) -> tl(X:S) 1.82/2.06 ->Strongly Connected Components: 1.82/2.06 There is no strongly connected component 1.82/2.06 1.82/2.06 The problem is finite. 1.82/2.06 EOF