6.25/6.40 YES 6.25/6.40 6.25/6.40 Problem 1: 6.25/6.40 6.25/6.40 (VAR v_NonEmpty:S X:S X1:S X2:S Y:S) 6.25/6.40 (RULES 6.25/6.40 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.25/6.40 active(adx(X:S)) -> adx(active(X:S)) 6.25/6.40 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.25/6.40 active(hd(X:S)) -> hd(active(X:S)) 6.25/6.40 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.25/6.40 active(incr(X:S)) -> incr(active(X:S)) 6.25/6.40 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.25/6.40 active(tl(X:S)) -> tl(active(X:S)) 6.25/6.40 active(nats) -> mark(adx(zeros)) 6.25/6.40 active(zeros) -> mark(cons(0,zeros)) 6.25/6.40 adx(mark(X:S)) -> mark(adx(X:S)) 6.25/6.40 adx(ok(X:S)) -> ok(adx(X:S)) 6.25/6.40 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.25/6.40 hd(mark(X:S)) -> mark(hd(X:S)) 6.25/6.40 hd(ok(X:S)) -> ok(hd(X:S)) 6.25/6.40 incr(mark(X:S)) -> mark(incr(X:S)) 6.25/6.40 incr(ok(X:S)) -> ok(incr(X:S)) 6.25/6.40 proper(adx(X:S)) -> adx(proper(X:S)) 6.25/6.40 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.25/6.40 proper(hd(X:S)) -> hd(proper(X:S)) 6.25/6.40 proper(incr(X:S)) -> incr(proper(X:S)) 6.25/6.40 proper(s(X:S)) -> s(proper(X:S)) 6.25/6.40 proper(tl(X:S)) -> tl(proper(X:S)) 6.25/6.40 proper(0) -> ok(0) 6.25/6.40 proper(nats) -> ok(nats) 6.25/6.40 proper(zeros) -> ok(zeros) 6.25/6.40 s(ok(X:S)) -> ok(s(X:S)) 6.25/6.40 tl(mark(X:S)) -> mark(tl(X:S)) 6.25/6.40 tl(ok(X:S)) -> ok(tl(X:S)) 6.25/6.40 top(mark(X:S)) -> top(proper(X:S)) 6.25/6.40 top(ok(X:S)) -> top(active(X:S)) 6.25/6.40 ) 6.25/6.40 (STRATEGY INNERMOST) 6.25/6.40 6.25/6.40 Problem 1: 6.25/6.40 6.25/6.40 Dependency Pairs Processor: 6.25/6.40 -> Pairs: 6.25/6.40 ACTIVE(adx(cons(X:S,Y:S))) -> ADX(Y:S) 6.25/6.40 ACTIVE(adx(cons(X:S,Y:S))) -> CONS(X:S,adx(Y:S)) 6.25/6.40 ACTIVE(adx(cons(X:S,Y:S))) -> INCR(cons(X:S,adx(Y:S))) 6.25/6.40 ACTIVE(adx(X:S)) -> ACTIVE(X:S) 6.25/6.40 ACTIVE(adx(X:S)) -> ADX(active(X:S)) 6.25/6.40 ACTIVE(hd(X:S)) -> ACTIVE(X:S) 6.25/6.40 ACTIVE(hd(X:S)) -> HD(active(X:S)) 6.25/6.40 ACTIVE(incr(cons(X:S,Y:S))) -> CONS(s(X:S),incr(Y:S)) 6.25/6.40 ACTIVE(incr(cons(X:S,Y:S))) -> INCR(Y:S) 6.25/6.40 ACTIVE(incr(cons(X:S,Y:S))) -> S(X:S) 6.25/6.40 ACTIVE(incr(X:S)) -> ACTIVE(X:S) 6.25/6.40 ACTIVE(incr(X:S)) -> INCR(active(X:S)) 6.25/6.40 ACTIVE(tl(X:S)) -> ACTIVE(X:S) 6.25/6.40 ACTIVE(tl(X:S)) -> TL(active(X:S)) 6.25/6.40 ADX(mark(X:S)) -> ADX(X:S) 6.25/6.40 ADX(ok(X:S)) -> ADX(X:S) 6.25/6.40 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 6.25/6.40 HD(mark(X:S)) -> HD(X:S) 6.25/6.40 HD(ok(X:S)) -> HD(X:S) 6.25/6.40 INCR(mark(X:S)) -> INCR(X:S) 6.25/6.40 INCR(ok(X:S)) -> INCR(X:S) 6.25/6.40 PROPER(adx(X:S)) -> ADX(proper(X:S)) 6.25/6.40 PROPER(adx(X:S)) -> PROPER(X:S) 6.25/6.40 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) 6.25/6.40 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 6.25/6.40 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 6.25/6.40 PROPER(hd(X:S)) -> HD(proper(X:S)) 6.25/6.40 PROPER(hd(X:S)) -> PROPER(X:S) 6.25/6.40 PROPER(incr(X:S)) -> INCR(proper(X:S)) 6.25/6.40 PROPER(incr(X:S)) -> PROPER(X:S) 6.25/6.40 PROPER(s(X:S)) -> PROPER(X:S) 6.25/6.40 PROPER(s(X:S)) -> S(proper(X:S)) 6.25/6.40 PROPER(tl(X:S)) -> PROPER(X:S) 6.25/6.40 PROPER(tl(X:S)) -> TL(proper(X:S)) 6.25/6.40 S(ok(X:S)) -> S(X:S) 6.25/6.40 TL(mark(X:S)) -> TL(X:S) 6.25/6.40 TL(ok(X:S)) -> TL(X:S) 6.25/6.40 TOP(mark(X:S)) -> PROPER(X:S) 6.25/6.40 TOP(mark(X:S)) -> TOP(proper(X:S)) 6.25/6.40 TOP(ok(X:S)) -> ACTIVE(X:S) 6.25/6.40 TOP(ok(X:S)) -> TOP(active(X:S)) 6.25/6.40 -> Rules: 6.25/6.40 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.25/6.40 active(adx(X:S)) -> adx(active(X:S)) 6.25/6.40 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.25/6.40 active(hd(X:S)) -> hd(active(X:S)) 6.25/6.40 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.25/6.40 active(incr(X:S)) -> incr(active(X:S)) 6.25/6.40 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.25/6.40 active(tl(X:S)) -> tl(active(X:S)) 6.25/6.40 active(nats) -> mark(adx(zeros)) 6.25/6.40 active(zeros) -> mark(cons(0,zeros)) 6.25/6.40 adx(mark(X:S)) -> mark(adx(X:S)) 6.25/6.40 adx(ok(X:S)) -> ok(adx(X:S)) 6.25/6.40 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.25/6.40 hd(mark(X:S)) -> mark(hd(X:S)) 6.25/6.40 hd(ok(X:S)) -> ok(hd(X:S)) 6.25/6.40 incr(mark(X:S)) -> mark(incr(X:S)) 6.25/6.40 incr(ok(X:S)) -> ok(incr(X:S)) 6.25/6.40 proper(adx(X:S)) -> adx(proper(X:S)) 6.25/6.40 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.25/6.40 proper(hd(X:S)) -> hd(proper(X:S)) 6.25/6.40 proper(incr(X:S)) -> incr(proper(X:S)) 6.25/6.40 proper(s(X:S)) -> s(proper(X:S)) 6.25/6.40 proper(tl(X:S)) -> tl(proper(X:S)) 6.25/6.40 proper(0) -> ok(0) 6.25/6.40 proper(nats) -> ok(nats) 6.25/6.40 proper(zeros) -> ok(zeros) 6.25/6.40 s(ok(X:S)) -> ok(s(X:S)) 6.25/6.40 tl(mark(X:S)) -> mark(tl(X:S)) 6.25/6.40 tl(ok(X:S)) -> ok(tl(X:S)) 6.25/6.40 top(mark(X:S)) -> top(proper(X:S)) 6.25/6.40 top(ok(X:S)) -> top(active(X:S)) 6.25/6.40 6.25/6.40 Problem 1: 6.25/6.40 6.25/6.40 SCC Processor: 6.25/6.40 -> Pairs: 6.25/6.40 ACTIVE(adx(cons(X:S,Y:S))) -> ADX(Y:S) 6.25/6.40 ACTIVE(adx(cons(X:S,Y:S))) -> CONS(X:S,adx(Y:S)) 6.25/6.40 ACTIVE(adx(cons(X:S,Y:S))) -> INCR(cons(X:S,adx(Y:S))) 6.25/6.40 ACTIVE(adx(X:S)) -> ACTIVE(X:S) 6.25/6.40 ACTIVE(adx(X:S)) -> ADX(active(X:S)) 6.25/6.40 ACTIVE(hd(X:S)) -> ACTIVE(X:S) 6.25/6.40 ACTIVE(hd(X:S)) -> HD(active(X:S)) 6.25/6.40 ACTIVE(incr(cons(X:S,Y:S))) -> CONS(s(X:S),incr(Y:S)) 6.25/6.40 ACTIVE(incr(cons(X:S,Y:S))) -> INCR(Y:S) 6.25/6.40 ACTIVE(incr(cons(X:S,Y:S))) -> S(X:S) 6.25/6.40 ACTIVE(incr(X:S)) -> ACTIVE(X:S) 6.25/6.40 ACTIVE(incr(X:S)) -> INCR(active(X:S)) 6.25/6.40 ACTIVE(tl(X:S)) -> ACTIVE(X:S) 6.25/6.40 ACTIVE(tl(X:S)) -> TL(active(X:S)) 6.25/6.40 ADX(mark(X:S)) -> ADX(X:S) 6.25/6.40 ADX(ok(X:S)) -> ADX(X:S) 6.25/6.40 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 6.25/6.40 HD(mark(X:S)) -> HD(X:S) 6.25/6.40 HD(ok(X:S)) -> HD(X:S) 6.25/6.40 INCR(mark(X:S)) -> INCR(X:S) 6.25/6.40 INCR(ok(X:S)) -> INCR(X:S) 6.25/6.40 PROPER(adx(X:S)) -> ADX(proper(X:S)) 6.25/6.40 PROPER(adx(X:S)) -> PROPER(X:S) 6.25/6.40 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) 6.25/6.40 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 6.25/6.40 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 6.25/6.40 PROPER(hd(X:S)) -> HD(proper(X:S)) 6.25/6.40 PROPER(hd(X:S)) -> PROPER(X:S) 6.25/6.40 PROPER(incr(X:S)) -> INCR(proper(X:S)) 6.25/6.40 PROPER(incr(X:S)) -> PROPER(X:S) 6.25/6.40 PROPER(s(X:S)) -> PROPER(X:S) 6.25/6.40 PROPER(s(X:S)) -> S(proper(X:S)) 6.25/6.40 PROPER(tl(X:S)) -> PROPER(X:S) 6.25/6.40 PROPER(tl(X:S)) -> TL(proper(X:S)) 6.25/6.41 S(ok(X:S)) -> S(X:S) 6.25/6.41 TL(mark(X:S)) -> TL(X:S) 6.25/6.41 TL(ok(X:S)) -> TL(X:S) 6.25/6.41 TOP(mark(X:S)) -> PROPER(X:S) 6.25/6.41 TOP(mark(X:S)) -> TOP(proper(X:S)) 6.25/6.41 TOP(ok(X:S)) -> ACTIVE(X:S) 6.25/6.41 TOP(ok(X:S)) -> TOP(active(X:S)) 6.25/6.41 -> Rules: 6.25/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.25/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.25/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.25/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.25/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.25/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.25/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.25/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.25/6.41 active(nats) -> mark(adx(zeros)) 6.25/6.41 active(zeros) -> mark(cons(0,zeros)) 6.25/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.25/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.25/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.25/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.25/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.25/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.25/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.25/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.25/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.25/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.25/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.25/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.25/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.25/6.41 proper(0) -> ok(0) 6.25/6.41 proper(nats) -> ok(nats) 6.25/6.41 proper(zeros) -> ok(zeros) 6.25/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.25/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.25/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.25/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.25/6.41 top(ok(X:S)) -> top(active(X:S)) 6.25/6.41 ->Strongly Connected Components: 6.25/6.41 ->->Cycle: 6.25/6.41 ->->-> Pairs: 6.25/6.41 TL(mark(X:S)) -> TL(X:S) 6.25/6.41 TL(ok(X:S)) -> TL(X:S) 6.25/6.41 ->->-> Rules: 6.25/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.25/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.25/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.25/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.25/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.25/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.25/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.25/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.25/6.41 active(nats) -> mark(adx(zeros)) 6.25/6.41 active(zeros) -> mark(cons(0,zeros)) 6.25/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.25/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.25/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.25/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.25/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.25/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.25/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.25/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.25/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.25/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.25/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.25/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.25/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.25/6.41 proper(0) -> ok(0) 6.25/6.41 proper(nats) -> ok(nats) 6.25/6.41 proper(zeros) -> ok(zeros) 6.25/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.25/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.25/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.25/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.25/6.41 top(ok(X:S)) -> top(active(X:S)) 6.25/6.41 ->->Cycle: 6.25/6.41 ->->-> Pairs: 6.25/6.41 S(ok(X:S)) -> S(X:S) 6.25/6.41 ->->-> Rules: 6.25/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.25/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.25/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.25/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.25/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.25/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.25/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.25/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.25/6.41 active(nats) -> mark(adx(zeros)) 6.25/6.41 active(zeros) -> mark(cons(0,zeros)) 6.25/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.25/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.25/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.25/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.25/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.25/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.25/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.25/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.25/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.25/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.25/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.25/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.25/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.25/6.41 proper(0) -> ok(0) 6.25/6.41 proper(nats) -> ok(nats) 6.25/6.41 proper(zeros) -> ok(zeros) 6.25/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.25/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.25/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.25/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.25/6.41 top(ok(X:S)) -> top(active(X:S)) 6.25/6.41 ->->Cycle: 6.25/6.41 ->->-> Pairs: 6.25/6.41 INCR(mark(X:S)) -> INCR(X:S) 6.25/6.41 INCR(ok(X:S)) -> INCR(X:S) 6.25/6.41 ->->-> Rules: 6.25/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.25/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.25/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.25/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.25/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.25/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.25/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.25/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.25/6.41 active(nats) -> mark(adx(zeros)) 6.25/6.41 active(zeros) -> mark(cons(0,zeros)) 6.25/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.25/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.25/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.25/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.25/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.25/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.25/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.25/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.25/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.25/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.25/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.25/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.25/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.25/6.41 proper(0) -> ok(0) 6.25/6.41 proper(nats) -> ok(nats) 6.25/6.41 proper(zeros) -> ok(zeros) 6.25/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.25/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.25/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.25/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.25/6.41 top(ok(X:S)) -> top(active(X:S)) 6.25/6.41 ->->Cycle: 6.25/6.41 ->->-> Pairs: 6.25/6.41 HD(mark(X:S)) -> HD(X:S) 6.25/6.41 HD(ok(X:S)) -> HD(X:S) 6.25/6.41 ->->-> Rules: 6.25/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.25/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.25/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.25/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.25/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.25/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.25/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.25/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.25/6.41 active(nats) -> mark(adx(zeros)) 6.25/6.41 active(zeros) -> mark(cons(0,zeros)) 6.25/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.25/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.25/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.25/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.25/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.25/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.25/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.25/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.25/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.25/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.25/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.25/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.25/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.25/6.41 proper(0) -> ok(0) 6.25/6.41 proper(nats) -> ok(nats) 6.25/6.41 proper(zeros) -> ok(zeros) 6.25/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.25/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.25/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.25/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.25/6.41 top(ok(X:S)) -> top(active(X:S)) 6.25/6.41 ->->Cycle: 6.25/6.41 ->->-> Pairs: 6.25/6.41 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 6.25/6.41 ->->-> Rules: 6.25/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.25/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.25/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.25/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.25/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.25/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.25/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.25/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.25/6.41 active(nats) -> mark(adx(zeros)) 6.25/6.41 active(zeros) -> mark(cons(0,zeros)) 6.25/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.25/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.25/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.25/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.25/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.25/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.25/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.25/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.25/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.25/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.25/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.25/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.25/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.25/6.41 proper(0) -> ok(0) 6.25/6.41 proper(nats) -> ok(nats) 6.25/6.41 proper(zeros) -> ok(zeros) 6.25/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.25/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.25/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.25/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.25/6.41 top(ok(X:S)) -> top(active(X:S)) 6.25/6.41 ->->Cycle: 6.25/6.41 ->->-> Pairs: 6.25/6.41 ADX(mark(X:S)) -> ADX(X:S) 6.25/6.41 ADX(ok(X:S)) -> ADX(X:S) 6.25/6.41 ->->-> Rules: 6.25/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.25/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.25/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.25/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.25/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.25/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.25/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.25/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.25/6.41 active(nats) -> mark(adx(zeros)) 6.25/6.41 active(zeros) -> mark(cons(0,zeros)) 6.25/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.25/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.25/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.25/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.25/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->->Cycle: 6.37/6.41 ->->-> Pairs: 6.37/6.41 PROPER(adx(X:S)) -> PROPER(X:S) 6.37/6.41 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 6.37/6.41 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 6.37/6.41 PROPER(hd(X:S)) -> PROPER(X:S) 6.37/6.41 PROPER(incr(X:S)) -> PROPER(X:S) 6.37/6.41 PROPER(s(X:S)) -> PROPER(X:S) 6.37/6.41 PROPER(tl(X:S)) -> PROPER(X:S) 6.37/6.41 ->->-> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->->Cycle: 6.37/6.41 ->->-> Pairs: 6.37/6.41 ACTIVE(adx(X:S)) -> ACTIVE(X:S) 6.37/6.41 ACTIVE(hd(X:S)) -> ACTIVE(X:S) 6.37/6.41 ACTIVE(incr(X:S)) -> ACTIVE(X:S) 6.37/6.41 ACTIVE(tl(X:S)) -> ACTIVE(X:S) 6.37/6.41 ->->-> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->->Cycle: 6.37/6.41 ->->-> Pairs: 6.37/6.41 TOP(mark(X:S)) -> TOP(proper(X:S)) 6.37/6.41 TOP(ok(X:S)) -> TOP(active(X:S)) 6.37/6.41 ->->-> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 6.37/6.41 6.37/6.41 The problem is decomposed in 9 subproblems. 6.37/6.41 6.37/6.41 Problem 1.1: 6.37/6.41 6.37/6.41 Subterm Processor: 6.37/6.41 -> Pairs: 6.37/6.41 TL(mark(X:S)) -> TL(X:S) 6.37/6.41 TL(ok(X:S)) -> TL(X:S) 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->Projection: 6.37/6.41 pi(TL) = 1 6.37/6.41 6.37/6.41 Problem 1.1: 6.37/6.41 6.37/6.41 SCC Processor: 6.37/6.41 -> Pairs: 6.37/6.41 Empty 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->Strongly Connected Components: 6.37/6.41 There is no strongly connected component 6.37/6.41 6.37/6.41 The problem is finite. 6.37/6.41 6.37/6.41 Problem 1.2: 6.37/6.41 6.37/6.41 Subterm Processor: 6.37/6.41 -> Pairs: 6.37/6.41 S(ok(X:S)) -> S(X:S) 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->Projection: 6.37/6.41 pi(S) = 1 6.37/6.41 6.37/6.41 Problem 1.2: 6.37/6.41 6.37/6.41 SCC Processor: 6.37/6.41 -> Pairs: 6.37/6.41 Empty 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->Strongly Connected Components: 6.37/6.41 There is no strongly connected component 6.37/6.41 6.37/6.41 The problem is finite. 6.37/6.41 6.37/6.41 Problem 1.3: 6.37/6.41 6.37/6.41 Subterm Processor: 6.37/6.41 -> Pairs: 6.37/6.41 INCR(mark(X:S)) -> INCR(X:S) 6.37/6.41 INCR(ok(X:S)) -> INCR(X:S) 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->Projection: 6.37/6.41 pi(INCR) = 1 6.37/6.41 6.37/6.41 Problem 1.3: 6.37/6.41 6.37/6.41 SCC Processor: 6.37/6.41 -> Pairs: 6.37/6.41 Empty 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->Strongly Connected Components: 6.37/6.41 There is no strongly connected component 6.37/6.41 6.37/6.41 The problem is finite. 6.37/6.41 6.37/6.41 Problem 1.4: 6.37/6.41 6.37/6.41 Subterm Processor: 6.37/6.41 -> Pairs: 6.37/6.41 HD(mark(X:S)) -> HD(X:S) 6.37/6.41 HD(ok(X:S)) -> HD(X:S) 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->Projection: 6.37/6.41 pi(HD) = 1 6.37/6.41 6.37/6.41 Problem 1.4: 6.37/6.41 6.37/6.41 SCC Processor: 6.37/6.41 -> Pairs: 6.37/6.41 Empty 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->Strongly Connected Components: 6.37/6.41 There is no strongly connected component 6.37/6.41 6.37/6.41 The problem is finite. 6.37/6.41 6.37/6.41 Problem 1.5: 6.37/6.41 6.37/6.41 Subterm Processor: 6.37/6.41 -> Pairs: 6.37/6.41 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->Projection: 6.37/6.41 pi(CONS) = 1 6.37/6.41 6.37/6.41 Problem 1.5: 6.37/6.41 6.37/6.41 SCC Processor: 6.37/6.41 -> Pairs: 6.37/6.41 Empty 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->Strongly Connected Components: 6.37/6.41 There is no strongly connected component 6.37/6.41 6.37/6.41 The problem is finite. 6.37/6.41 6.37/6.41 Problem 1.6: 6.37/6.41 6.37/6.41 Subterm Processor: 6.37/6.41 -> Pairs: 6.37/6.41 ADX(mark(X:S)) -> ADX(X:S) 6.37/6.41 ADX(ok(X:S)) -> ADX(X:S) 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->Projection: 6.37/6.41 pi(ADX) = 1 6.37/6.41 6.37/6.41 Problem 1.6: 6.37/6.41 6.37/6.41 SCC Processor: 6.37/6.41 -> Pairs: 6.37/6.41 Empty 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->Strongly Connected Components: 6.37/6.41 There is no strongly connected component 6.37/6.41 6.37/6.41 The problem is finite. 6.37/6.41 6.37/6.41 Problem 1.7: 6.37/6.41 6.37/6.41 Subterm Processor: 6.37/6.41 -> Pairs: 6.37/6.41 PROPER(adx(X:S)) -> PROPER(X:S) 6.37/6.41 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 6.37/6.41 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 6.37/6.41 PROPER(hd(X:S)) -> PROPER(X:S) 6.37/6.41 PROPER(incr(X:S)) -> PROPER(X:S) 6.37/6.41 PROPER(s(X:S)) -> PROPER(X:S) 6.37/6.41 PROPER(tl(X:S)) -> PROPER(X:S) 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->Projection: 6.37/6.41 pi(PROPER) = 1 6.37/6.41 6.37/6.41 Problem 1.7: 6.37/6.41 6.37/6.41 SCC Processor: 6.37/6.41 -> Pairs: 6.37/6.41 Empty 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->Strongly Connected Components: 6.37/6.41 There is no strongly connected component 6.37/6.41 6.37/6.41 The problem is finite. 6.37/6.41 6.37/6.41 Problem 1.8: 6.37/6.41 6.37/6.41 Subterm Processor: 6.37/6.41 -> Pairs: 6.37/6.41 ACTIVE(adx(X:S)) -> ACTIVE(X:S) 6.37/6.41 ACTIVE(hd(X:S)) -> ACTIVE(X:S) 6.37/6.41 ACTIVE(incr(X:S)) -> ACTIVE(X:S) 6.37/6.41 ACTIVE(tl(X:S)) -> ACTIVE(X:S) 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->Projection: 6.37/6.41 pi(ACTIVE) = 1 6.37/6.41 6.37/6.41 Problem 1.8: 6.37/6.41 6.37/6.41 SCC Processor: 6.37/6.41 -> Pairs: 6.37/6.41 Empty 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->Strongly Connected Components: 6.37/6.41 There is no strongly connected component 6.37/6.41 6.37/6.41 The problem is finite. 6.37/6.41 6.37/6.41 Problem 1.9: 6.37/6.41 6.37/6.41 Reduction Pairs Processor: 6.37/6.41 -> Pairs: 6.37/6.41 TOP(mark(X:S)) -> TOP(proper(X:S)) 6.37/6.41 TOP(ok(X:S)) -> TOP(active(X:S)) 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 -> Usable rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 ->Interpretation type: 6.37/6.41 Linear 6.37/6.41 ->Coefficients: 6.37/6.41 All rationals 6.37/6.41 ->Dimension: 6.37/6.41 1 6.37/6.41 ->Bound: 6.37/6.41 4 6.37/6.41 ->Interpretation: 6.37/6.41 6.37/6.41 [active](X) = X 6.37/6.41 [adx](X) = 4.X + 1/3 6.37/6.41 [cons](X1,X2) = 3/4.X1 + 1/4.X2 + 1/4 6.37/6.41 [hd](X) = 4/3.X 6.37/6.41 [incr](X) = X + 1/3 6.37/6.41 [proper](X) = X 6.37/6.41 [s](X) = X 6.37/6.41 [tl](X) = 4.X + 3/2 6.37/6.41 [top](X) = 0 6.37/6.41 [0] = 0 6.37/6.41 [fSNonEmpty] = 0 6.37/6.41 [mark](X) = X + 1/4 6.37/6.41 [nats] = 4 6.37/6.41 [ok](X) = X 6.37/6.41 [zeros] = 2/3 6.37/6.41 [ACTIVE](X) = 0 6.37/6.41 [ADX](X) = 0 6.37/6.41 [CONS](X1,X2) = 0 6.37/6.41 [HD](X) = 0 6.37/6.41 [INCR](X) = 0 6.37/6.41 [PROPER](X) = 0 6.37/6.41 [S](X) = 0 6.37/6.41 [TL](X) = 0 6.37/6.41 [TOP](X) = 1/4.X 6.37/6.41 6.37/6.41 Problem 1.9: 6.37/6.41 6.37/6.41 SCC Processor: 6.37/6.41 -> Pairs: 6.37/6.41 TOP(ok(X:S)) -> TOP(active(X:S)) 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->Strongly Connected Components: 6.37/6.41 ->->Cycle: 6.37/6.41 ->->-> Pairs: 6.37/6.41 TOP(ok(X:S)) -> TOP(active(X:S)) 6.37/6.41 ->->-> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 6.37/6.41 Problem 1.9: 6.37/6.41 6.37/6.41 Reduction Pairs Processor: 6.37/6.41 -> Pairs: 6.37/6.41 TOP(ok(X:S)) -> TOP(active(X:S)) 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 -> Usable rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 ->Interpretation type: 6.37/6.41 Linear 6.37/6.41 ->Coefficients: 6.37/6.41 Natural Numbers 6.37/6.41 ->Dimension: 6.37/6.41 1 6.37/6.41 ->Bound: 6.37/6.41 2 6.37/6.41 ->Interpretation: 6.37/6.41 6.37/6.41 [active](X) = 2.X + 1 6.37/6.41 [adx](X) = 2.X + 1 6.37/6.41 [cons](X1,X2) = 2.X1 + 2 6.37/6.41 [hd](X) = 2.X + 2 6.37/6.41 [incr](X) = 2.X + 2 6.37/6.41 [proper](X) = 0 6.37/6.41 [s](X) = X 6.37/6.41 [tl](X) = X 6.37/6.41 [top](X) = 0 6.37/6.41 [0] = 1 6.37/6.41 [fSNonEmpty] = 0 6.37/6.41 [mark](X) = 1 6.37/6.41 [nats] = 0 6.37/6.41 [ok](X) = 2.X + 2 6.37/6.41 [zeros] = 2 6.37/6.41 [ACTIVE](X) = 0 6.37/6.41 [ADX](X) = 0 6.37/6.41 [CONS](X1,X2) = 0 6.37/6.41 [HD](X) = 0 6.37/6.41 [INCR](X) = 0 6.37/6.41 [PROPER](X) = 0 6.37/6.41 [S](X) = 0 6.37/6.41 [TL](X) = 0 6.37/6.41 [TOP](X) = 2.X 6.37/6.41 6.37/6.41 Problem 1.9: 6.37/6.41 6.37/6.41 SCC Processor: 6.37/6.41 -> Pairs: 6.37/6.41 Empty 6.37/6.41 -> Rules: 6.37/6.41 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S)))) 6.37/6.41 active(adx(X:S)) -> adx(active(X:S)) 6.37/6.41 active(hd(cons(X:S,Y:S))) -> mark(X:S) 6.37/6.41 active(hd(X:S)) -> hd(active(X:S)) 6.37/6.41 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S))) 6.37/6.41 active(incr(X:S)) -> incr(active(X:S)) 6.37/6.41 active(tl(cons(X:S,Y:S))) -> mark(Y:S) 6.37/6.41 active(tl(X:S)) -> tl(active(X:S)) 6.37/6.41 active(nats) -> mark(adx(zeros)) 6.37/6.41 active(zeros) -> mark(cons(0,zeros)) 6.37/6.41 adx(mark(X:S)) -> mark(adx(X:S)) 6.37/6.41 adx(ok(X:S)) -> ok(adx(X:S)) 6.37/6.41 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 6.37/6.41 hd(mark(X:S)) -> mark(hd(X:S)) 6.37/6.41 hd(ok(X:S)) -> ok(hd(X:S)) 6.37/6.41 incr(mark(X:S)) -> mark(incr(X:S)) 6.37/6.41 incr(ok(X:S)) -> ok(incr(X:S)) 6.37/6.41 proper(adx(X:S)) -> adx(proper(X:S)) 6.37/6.41 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 6.37/6.41 proper(hd(X:S)) -> hd(proper(X:S)) 6.37/6.41 proper(incr(X:S)) -> incr(proper(X:S)) 6.37/6.41 proper(s(X:S)) -> s(proper(X:S)) 6.37/6.41 proper(tl(X:S)) -> tl(proper(X:S)) 6.37/6.41 proper(0) -> ok(0) 6.37/6.41 proper(nats) -> ok(nats) 6.37/6.41 proper(zeros) -> ok(zeros) 6.37/6.41 s(ok(X:S)) -> ok(s(X:S)) 6.37/6.41 tl(mark(X:S)) -> mark(tl(X:S)) 6.37/6.41 tl(ok(X:S)) -> ok(tl(X:S)) 6.37/6.41 top(mark(X:S)) -> top(proper(X:S)) 6.37/6.41 top(ok(X:S)) -> top(active(X:S)) 6.37/6.41 ->Strongly Connected Components: 6.37/6.41 There is no strongly connected component 6.37/6.41 6.37/6.41 The problem is finite. 6.37/6.41 EOF