1.96/2.43 YES 1.96/2.43 1.96/2.43 Problem 1: 1.96/2.43 1.96/2.43 (VAR v_NonEmpty:S L:S X:S X1:S X2:S) 1.96/2.43 (RULES 1.96/2.43 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.43 active(adx(nil)) -> mark(nil) 1.96/2.43 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.43 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.43 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.43 active(head(X:S)) -> head(active(X:S)) 1.96/2.43 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.43 active(incr(nil)) -> mark(nil) 1.96/2.43 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.43 active(s(X:S)) -> s(active(X:S)) 1.96/2.43 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.43 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.43 active(nats) -> mark(adx(zeros)) 1.96/2.43 active(zeros) -> mark(cons(0,zeros)) 1.96/2.43 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.43 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.43 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.43 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.43 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.43 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.43 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.43 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.43 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.43 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.43 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.43 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.43 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.43 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.43 proper(0) -> ok(0) 1.96/2.43 proper(nats) -> ok(nats) 1.96/2.43 proper(nil) -> ok(nil) 1.96/2.43 proper(zeros) -> ok(zeros) 1.96/2.43 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.43 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.43 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.43 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.43 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.43 top(ok(X:S)) -> top(active(X:S)) 1.96/2.43 ) 1.96/2.43 (STRATEGY INNERMOST) 1.96/2.43 1.96/2.43 Problem 1: 1.96/2.43 1.96/2.43 Dependency Pairs Processor: 1.96/2.43 -> Pairs: 1.96/2.43 ACTIVE(adx(cons(X:S,L:S))) -> ADX(L:S) 1.96/2.43 ACTIVE(adx(cons(X:S,L:S))) -> CONS(X:S,adx(L:S)) 1.96/2.43 ACTIVE(adx(cons(X:S,L:S))) -> INCR(cons(X:S,adx(L:S))) 1.96/2.43 ACTIVE(adx(X:S)) -> ACTIVE(X:S) 1.96/2.43 ACTIVE(adx(X:S)) -> ADX(active(X:S)) 1.96/2.43 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 1.96/2.43 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S) 1.96/2.43 ACTIVE(head(X:S)) -> ACTIVE(X:S) 1.96/2.43 ACTIVE(head(X:S)) -> HEAD(active(X:S)) 1.96/2.43 ACTIVE(incr(cons(X:S,L:S))) -> CONS(s(X:S),incr(L:S)) 1.96/2.43 ACTIVE(incr(cons(X:S,L:S))) -> INCR(L:S) 1.96/2.43 ACTIVE(incr(cons(X:S,L:S))) -> S(X:S) 1.96/2.43 ACTIVE(incr(X:S)) -> ACTIVE(X:S) 1.96/2.43 ACTIVE(incr(X:S)) -> INCR(active(X:S)) 1.96/2.43 ACTIVE(s(X:S)) -> ACTIVE(X:S) 1.96/2.43 ACTIVE(s(X:S)) -> S(active(X:S)) 1.96/2.43 ACTIVE(tail(X:S)) -> ACTIVE(X:S) 1.96/2.43 ACTIVE(tail(X:S)) -> TAIL(active(X:S)) 1.96/2.43 ADX(mark(X:S)) -> ADX(X:S) 1.96/2.43 ADX(ok(X:S)) -> ADX(X:S) 1.96/2.43 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 1.96/2.43 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 1.96/2.43 HEAD(mark(X:S)) -> HEAD(X:S) 1.96/2.43 HEAD(ok(X:S)) -> HEAD(X:S) 1.96/2.43 INCR(mark(X:S)) -> INCR(X:S) 1.96/2.43 INCR(ok(X:S)) -> INCR(X:S) 1.96/2.43 PROPER(adx(X:S)) -> ADX(proper(X:S)) 1.96/2.43 PROPER(adx(X:S)) -> PROPER(X:S) 1.96/2.43 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) 1.96/2.43 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 1.96/2.43 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 1.96/2.43 PROPER(head(X:S)) -> HEAD(proper(X:S)) 1.96/2.43 PROPER(head(X:S)) -> PROPER(X:S) 1.96/2.43 PROPER(incr(X:S)) -> INCR(proper(X:S)) 1.96/2.43 PROPER(incr(X:S)) -> PROPER(X:S) 1.96/2.43 PROPER(s(X:S)) -> PROPER(X:S) 1.96/2.43 PROPER(s(X:S)) -> S(proper(X:S)) 1.96/2.43 PROPER(tail(X:S)) -> PROPER(X:S) 1.96/2.43 PROPER(tail(X:S)) -> TAIL(proper(X:S)) 1.96/2.43 S(mark(X:S)) -> S(X:S) 1.96/2.43 S(ok(X:S)) -> S(X:S) 1.96/2.43 TAIL(mark(X:S)) -> TAIL(X:S) 1.96/2.43 TAIL(ok(X:S)) -> TAIL(X:S) 1.96/2.43 TOP(mark(X:S)) -> PROPER(X:S) 1.96/2.43 TOP(mark(X:S)) -> TOP(proper(X:S)) 1.96/2.43 TOP(ok(X:S)) -> ACTIVE(X:S) 1.96/2.43 TOP(ok(X:S)) -> TOP(active(X:S)) 1.96/2.43 -> Rules: 1.96/2.43 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.43 active(adx(nil)) -> mark(nil) 1.96/2.43 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.43 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.43 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.43 active(head(X:S)) -> head(active(X:S)) 1.96/2.43 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.43 active(incr(nil)) -> mark(nil) 1.96/2.43 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.43 active(s(X:S)) -> s(active(X:S)) 1.96/2.43 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.43 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.43 active(nats) -> mark(adx(zeros)) 1.96/2.43 active(zeros) -> mark(cons(0,zeros)) 1.96/2.43 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.43 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.43 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.43 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.43 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.43 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.43 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.43 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.43 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.43 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.43 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.43 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.43 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.43 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.43 proper(0) -> ok(0) 1.96/2.43 proper(nats) -> ok(nats) 1.96/2.43 proper(nil) -> ok(nil) 1.96/2.43 proper(zeros) -> ok(zeros) 1.96/2.43 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.43 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.43 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.43 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.43 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.43 top(ok(X:S)) -> top(active(X:S)) 1.96/2.43 1.96/2.43 Problem 1: 1.96/2.43 1.96/2.43 SCC Processor: 1.96/2.43 -> Pairs: 1.96/2.43 ACTIVE(adx(cons(X:S,L:S))) -> ADX(L:S) 1.96/2.43 ACTIVE(adx(cons(X:S,L:S))) -> CONS(X:S,adx(L:S)) 1.96/2.43 ACTIVE(adx(cons(X:S,L:S))) -> INCR(cons(X:S,adx(L:S))) 1.96/2.43 ACTIVE(adx(X:S)) -> ACTIVE(X:S) 1.96/2.43 ACTIVE(adx(X:S)) -> ADX(active(X:S)) 1.96/2.43 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 1.96/2.43 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S) 1.96/2.43 ACTIVE(head(X:S)) -> ACTIVE(X:S) 1.96/2.43 ACTIVE(head(X:S)) -> HEAD(active(X:S)) 1.96/2.43 ACTIVE(incr(cons(X:S,L:S))) -> CONS(s(X:S),incr(L:S)) 1.96/2.43 ACTIVE(incr(cons(X:S,L:S))) -> INCR(L:S) 1.96/2.43 ACTIVE(incr(cons(X:S,L:S))) -> S(X:S) 1.96/2.43 ACTIVE(incr(X:S)) -> ACTIVE(X:S) 1.96/2.43 ACTIVE(incr(X:S)) -> INCR(active(X:S)) 1.96/2.43 ACTIVE(s(X:S)) -> ACTIVE(X:S) 1.96/2.43 ACTIVE(s(X:S)) -> S(active(X:S)) 1.96/2.43 ACTIVE(tail(X:S)) -> ACTIVE(X:S) 1.96/2.43 ACTIVE(tail(X:S)) -> TAIL(active(X:S)) 1.96/2.43 ADX(mark(X:S)) -> ADX(X:S) 1.96/2.43 ADX(ok(X:S)) -> ADX(X:S) 1.96/2.43 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 1.96/2.43 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 1.96/2.43 HEAD(mark(X:S)) -> HEAD(X:S) 1.96/2.43 HEAD(ok(X:S)) -> HEAD(X:S) 1.96/2.43 INCR(mark(X:S)) -> INCR(X:S) 1.96/2.43 INCR(ok(X:S)) -> INCR(X:S) 1.96/2.43 PROPER(adx(X:S)) -> ADX(proper(X:S)) 1.96/2.43 PROPER(adx(X:S)) -> PROPER(X:S) 1.96/2.43 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) 1.96/2.43 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 1.96/2.43 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 1.96/2.43 PROPER(head(X:S)) -> HEAD(proper(X:S)) 1.96/2.43 PROPER(head(X:S)) -> PROPER(X:S) 1.96/2.43 PROPER(incr(X:S)) -> INCR(proper(X:S)) 1.96/2.43 PROPER(incr(X:S)) -> PROPER(X:S) 1.96/2.43 PROPER(s(X:S)) -> PROPER(X:S) 1.96/2.43 PROPER(s(X:S)) -> S(proper(X:S)) 1.96/2.43 PROPER(tail(X:S)) -> PROPER(X:S) 1.96/2.43 PROPER(tail(X:S)) -> TAIL(proper(X:S)) 1.96/2.43 S(mark(X:S)) -> S(X:S) 1.96/2.43 S(ok(X:S)) -> S(X:S) 1.96/2.43 TAIL(mark(X:S)) -> TAIL(X:S) 1.96/2.43 TAIL(ok(X:S)) -> TAIL(X:S) 1.96/2.43 TOP(mark(X:S)) -> PROPER(X:S) 1.96/2.43 TOP(mark(X:S)) -> TOP(proper(X:S)) 1.96/2.43 TOP(ok(X:S)) -> ACTIVE(X:S) 1.96/2.43 TOP(ok(X:S)) -> TOP(active(X:S)) 1.96/2.43 -> Rules: 1.96/2.43 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.43 active(adx(nil)) -> mark(nil) 1.96/2.43 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.43 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.43 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.43 active(head(X:S)) -> head(active(X:S)) 1.96/2.43 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.43 active(incr(nil)) -> mark(nil) 1.96/2.43 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.43 active(s(X:S)) -> s(active(X:S)) 1.96/2.43 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.43 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.43 active(nats) -> mark(adx(zeros)) 1.96/2.43 active(zeros) -> mark(cons(0,zeros)) 1.96/2.43 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.43 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.43 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.43 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.43 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.43 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.43 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.43 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.43 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.43 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.43 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.43 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.43 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.43 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.43 proper(0) -> ok(0) 1.96/2.43 proper(nats) -> ok(nats) 1.96/2.43 proper(nil) -> ok(nil) 1.96/2.43 proper(zeros) -> ok(zeros) 1.96/2.43 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.43 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.43 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.43 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.43 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.43 top(ok(X:S)) -> top(active(X:S)) 1.96/2.43 ->Strongly Connected Components: 1.96/2.43 ->->Cycle: 1.96/2.43 ->->-> Pairs: 1.96/2.43 TAIL(mark(X:S)) -> TAIL(X:S) 1.96/2.43 TAIL(ok(X:S)) -> TAIL(X:S) 1.96/2.43 ->->-> Rules: 1.96/2.43 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.43 active(adx(nil)) -> mark(nil) 1.96/2.43 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.43 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.43 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.43 active(head(X:S)) -> head(active(X:S)) 1.96/2.43 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.43 active(incr(nil)) -> mark(nil) 1.96/2.43 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.43 active(s(X:S)) -> s(active(X:S)) 1.96/2.43 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.43 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.43 active(nats) -> mark(adx(zeros)) 1.96/2.43 active(zeros) -> mark(cons(0,zeros)) 1.96/2.43 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.43 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.43 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.43 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.43 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.43 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.43 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.43 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.43 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.43 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.43 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.43 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.43 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.43 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.43 proper(0) -> ok(0) 1.96/2.43 proper(nats) -> ok(nats) 1.96/2.43 proper(nil) -> ok(nil) 1.96/2.43 proper(zeros) -> ok(zeros) 1.96/2.43 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.43 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.43 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.43 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.43 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.43 top(ok(X:S)) -> top(active(X:S)) 1.96/2.43 ->->Cycle: 1.96/2.43 ->->-> Pairs: 1.96/2.43 S(mark(X:S)) -> S(X:S) 1.96/2.43 S(ok(X:S)) -> S(X:S) 1.96/2.43 ->->-> Rules: 1.96/2.43 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.43 active(adx(nil)) -> mark(nil) 1.96/2.43 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.43 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.43 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.43 active(head(X:S)) -> head(active(X:S)) 1.96/2.43 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.43 active(incr(nil)) -> mark(nil) 1.96/2.43 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.43 active(s(X:S)) -> s(active(X:S)) 1.96/2.43 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.43 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.43 active(nats) -> mark(adx(zeros)) 1.96/2.43 active(zeros) -> mark(cons(0,zeros)) 1.96/2.43 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.43 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.43 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.43 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.43 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.43 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.43 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.43 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.43 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.43 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.43 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.43 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.43 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.43 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.43 proper(0) -> ok(0) 1.96/2.43 proper(nats) -> ok(nats) 1.96/2.43 proper(nil) -> ok(nil) 1.96/2.43 proper(zeros) -> ok(zeros) 1.96/2.43 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.43 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.43 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.43 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.43 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.43 top(ok(X:S)) -> top(active(X:S)) 1.96/2.43 ->->Cycle: 1.96/2.43 ->->-> Pairs: 1.96/2.43 INCR(mark(X:S)) -> INCR(X:S) 1.96/2.43 INCR(ok(X:S)) -> INCR(X:S) 1.96/2.43 ->->-> Rules: 1.96/2.43 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.43 active(adx(nil)) -> mark(nil) 1.96/2.43 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.43 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.43 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.43 active(head(X:S)) -> head(active(X:S)) 1.96/2.43 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.43 active(incr(nil)) -> mark(nil) 1.96/2.43 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.43 active(s(X:S)) -> s(active(X:S)) 1.96/2.43 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.43 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.43 active(nats) -> mark(adx(zeros)) 1.96/2.43 active(zeros) -> mark(cons(0,zeros)) 1.96/2.43 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.43 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.43 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.43 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.43 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.43 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.43 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.43 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.43 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.43 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.43 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.43 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.43 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.43 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.43 proper(0) -> ok(0) 1.96/2.43 proper(nats) -> ok(nats) 1.96/2.43 proper(nil) -> ok(nil) 1.96/2.43 proper(zeros) -> ok(zeros) 1.96/2.43 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.43 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.43 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.43 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.43 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.43 top(ok(X:S)) -> top(active(X:S)) 1.96/2.43 ->->Cycle: 1.96/2.43 ->->-> Pairs: 1.96/2.43 HEAD(mark(X:S)) -> HEAD(X:S) 1.96/2.43 HEAD(ok(X:S)) -> HEAD(X:S) 1.96/2.43 ->->-> Rules: 1.96/2.43 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.43 active(adx(nil)) -> mark(nil) 1.96/2.43 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.43 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.43 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.43 active(head(X:S)) -> head(active(X:S)) 1.96/2.43 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.43 active(incr(nil)) -> mark(nil) 1.96/2.43 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.43 active(s(X:S)) -> s(active(X:S)) 1.96/2.43 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.43 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.43 active(nats) -> mark(adx(zeros)) 1.96/2.43 active(zeros) -> mark(cons(0,zeros)) 1.96/2.43 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.43 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.43 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.43 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.43 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.43 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.43 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.43 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.43 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.43 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.43 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.43 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.43 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.43 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.43 proper(0) -> ok(0) 1.96/2.43 proper(nats) -> ok(nats) 1.96/2.43 proper(nil) -> ok(nil) 1.96/2.43 proper(zeros) -> ok(zeros) 1.96/2.43 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.43 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.43 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.43 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.43 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.43 top(ok(X:S)) -> top(active(X:S)) 1.96/2.43 ->->Cycle: 1.96/2.43 ->->-> Pairs: 1.96/2.43 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 1.96/2.43 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 1.96/2.43 ->->-> Rules: 1.96/2.43 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.43 active(adx(nil)) -> mark(nil) 1.96/2.43 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.43 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.43 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.43 active(head(X:S)) -> head(active(X:S)) 1.96/2.43 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.43 active(incr(nil)) -> mark(nil) 1.96/2.43 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.43 active(s(X:S)) -> s(active(X:S)) 1.96/2.43 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.43 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.43 active(nats) -> mark(adx(zeros)) 1.96/2.43 active(zeros) -> mark(cons(0,zeros)) 1.96/2.43 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.43 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.43 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.43 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.43 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.43 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.43 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.43 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.43 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.43 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.43 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.43 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.43 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.43 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.43 proper(0) -> ok(0) 1.96/2.43 proper(nats) -> ok(nats) 1.96/2.43 proper(nil) -> ok(nil) 1.96/2.43 proper(zeros) -> ok(zeros) 1.96/2.43 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.43 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.43 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.43 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.43 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.43 top(ok(X:S)) -> top(active(X:S)) 1.96/2.43 ->->Cycle: 1.96/2.43 ->->-> Pairs: 1.96/2.43 ADX(mark(X:S)) -> ADX(X:S) 1.96/2.43 ADX(ok(X:S)) -> ADX(X:S) 1.96/2.43 ->->-> Rules: 1.96/2.43 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.43 active(adx(nil)) -> mark(nil) 1.96/2.43 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.43 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.43 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.43 active(head(X:S)) -> head(active(X:S)) 1.96/2.43 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.43 active(incr(nil)) -> mark(nil) 1.96/2.43 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.43 active(s(X:S)) -> s(active(X:S)) 1.96/2.43 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.43 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.43 active(nats) -> mark(adx(zeros)) 1.96/2.43 active(zeros) -> mark(cons(0,zeros)) 1.96/2.43 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.43 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.43 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.43 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.43 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.43 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.43 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.43 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.43 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.43 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.43 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.43 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.43 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.43 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.43 proper(0) -> ok(0) 1.96/2.43 proper(nats) -> ok(nats) 1.96/2.43 proper(nil) -> ok(nil) 1.96/2.43 proper(zeros) -> ok(zeros) 1.96/2.43 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.43 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.43 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.43 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.43 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.43 top(ok(X:S)) -> top(active(X:S)) 1.96/2.43 ->->Cycle: 1.96/2.43 ->->-> Pairs: 1.96/2.43 PROPER(adx(X:S)) -> PROPER(X:S) 1.96/2.43 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 1.96/2.43 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 1.96/2.43 PROPER(head(X:S)) -> PROPER(X:S) 1.96/2.43 PROPER(incr(X:S)) -> PROPER(X:S) 1.96/2.43 PROPER(s(X:S)) -> PROPER(X:S) 1.96/2.43 PROPER(tail(X:S)) -> PROPER(X:S) 1.96/2.43 ->->-> Rules: 1.96/2.43 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.43 active(adx(nil)) -> mark(nil) 1.96/2.43 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.43 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.43 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.43 active(head(X:S)) -> head(active(X:S)) 1.96/2.43 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.43 active(incr(nil)) -> mark(nil) 1.96/2.43 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.43 active(s(X:S)) -> s(active(X:S)) 1.96/2.43 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.43 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.43 active(nats) -> mark(adx(zeros)) 1.96/2.43 active(zeros) -> mark(cons(0,zeros)) 1.96/2.43 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.43 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.43 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.43 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.43 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.43 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.43 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.43 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.43 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.43 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.43 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.43 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.43 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.43 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.43 proper(0) -> ok(0) 1.96/2.43 proper(nats) -> ok(nats) 1.96/2.43 proper(nil) -> ok(nil) 1.96/2.43 proper(zeros) -> ok(zeros) 1.96/2.43 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.43 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.43 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.43 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.43 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.43 top(ok(X:S)) -> top(active(X:S)) 1.96/2.43 ->->Cycle: 1.96/2.43 ->->-> Pairs: 1.96/2.43 ACTIVE(adx(X:S)) -> ACTIVE(X:S) 1.96/2.43 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 1.96/2.43 ACTIVE(head(X:S)) -> ACTIVE(X:S) 1.96/2.43 ACTIVE(incr(X:S)) -> ACTIVE(X:S) 1.96/2.43 ACTIVE(s(X:S)) -> ACTIVE(X:S) 1.96/2.43 ACTIVE(tail(X:S)) -> ACTIVE(X:S) 1.96/2.43 ->->-> Rules: 1.96/2.43 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.43 active(adx(nil)) -> mark(nil) 1.96/2.43 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.43 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.43 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.43 active(head(X:S)) -> head(active(X:S)) 1.96/2.43 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.43 active(incr(nil)) -> mark(nil) 1.96/2.43 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.43 active(s(X:S)) -> s(active(X:S)) 1.96/2.43 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.43 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.43 active(nats) -> mark(adx(zeros)) 1.96/2.43 active(zeros) -> mark(cons(0,zeros)) 1.96/2.43 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.43 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.43 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.43 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.43 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.43 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.43 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.43 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.43 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.43 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.43 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.43 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.43 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.43 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.43 proper(0) -> ok(0) 1.96/2.43 proper(nats) -> ok(nats) 1.96/2.43 proper(nil) -> ok(nil) 1.96/2.43 proper(zeros) -> ok(zeros) 1.96/2.43 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.43 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.43 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.43 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.43 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.43 top(ok(X:S)) -> top(active(X:S)) 1.96/2.43 ->->Cycle: 1.96/2.43 ->->-> Pairs: 1.96/2.43 TOP(mark(X:S)) -> TOP(proper(X:S)) 1.96/2.43 TOP(ok(X:S)) -> TOP(active(X:S)) 1.96/2.43 ->->-> Rules: 1.96/2.43 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.43 active(adx(nil)) -> mark(nil) 1.96/2.43 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.43 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.43 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.43 active(head(X:S)) -> head(active(X:S)) 1.96/2.43 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.43 active(incr(nil)) -> mark(nil) 1.96/2.43 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.43 active(s(X:S)) -> s(active(X:S)) 1.96/2.43 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.43 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.43 active(nats) -> mark(adx(zeros)) 1.96/2.43 active(zeros) -> mark(cons(0,zeros)) 1.96/2.43 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.43 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.43 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.43 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.43 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.43 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.43 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.43 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.43 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.43 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.43 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.43 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.43 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.43 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.43 proper(0) -> ok(0) 1.96/2.43 proper(nats) -> ok(nats) 1.96/2.43 proper(nil) -> ok(nil) 1.96/2.43 proper(zeros) -> ok(zeros) 1.96/2.43 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.43 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.43 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.43 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.43 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.43 top(ok(X:S)) -> top(active(X:S)) 1.96/2.43 1.96/2.43 1.96/2.43 The problem is decomposed in 9 subproblems. 1.96/2.43 1.96/2.43 Problem 1.1: 1.96/2.43 1.96/2.43 Subterm Processor: 1.96/2.43 -> Pairs: 1.96/2.43 TAIL(mark(X:S)) -> TAIL(X:S) 1.96/2.43 TAIL(ok(X:S)) -> TAIL(X:S) 1.96/2.43 -> Rules: 1.96/2.43 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.43 active(adx(nil)) -> mark(nil) 1.96/2.43 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.43 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.43 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.43 active(head(X:S)) -> head(active(X:S)) 1.96/2.43 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.43 active(incr(nil)) -> mark(nil) 1.96/2.43 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.43 active(s(X:S)) -> s(active(X:S)) 1.96/2.43 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.43 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.43 active(nats) -> mark(adx(zeros)) 1.96/2.43 active(zeros) -> mark(cons(0,zeros)) 1.96/2.43 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.43 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.43 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.43 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.43 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.43 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.43 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.43 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.43 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.43 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.43 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.43 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.43 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.43 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.43 proper(0) -> ok(0) 1.96/2.43 proper(nats) -> ok(nats) 1.96/2.43 proper(nil) -> ok(nil) 1.96/2.43 proper(zeros) -> ok(zeros) 1.96/2.43 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.43 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.43 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.43 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.43 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.43 top(ok(X:S)) -> top(active(X:S)) 1.96/2.43 ->Projection: 1.96/2.43 pi(TAIL) = 1 1.96/2.43 1.96/2.43 Problem 1.1: 1.96/2.43 1.96/2.43 SCC Processor: 1.96/2.43 -> Pairs: 1.96/2.43 Empty 1.96/2.43 -> Rules: 1.96/2.43 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.43 active(adx(nil)) -> mark(nil) 1.96/2.43 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.43 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.43 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.43 active(head(X:S)) -> head(active(X:S)) 1.96/2.43 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.43 active(incr(nil)) -> mark(nil) 1.96/2.43 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.43 active(s(X:S)) -> s(active(X:S)) 1.96/2.43 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.43 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.43 active(nats) -> mark(adx(zeros)) 1.96/2.43 active(zeros) -> mark(cons(0,zeros)) 1.96/2.43 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.43 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.43 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.43 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.43 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.43 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.43 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.43 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.43 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.43 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.43 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.43 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.43 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.43 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.43 proper(0) -> ok(0) 1.96/2.43 proper(nats) -> ok(nats) 1.96/2.43 proper(nil) -> ok(nil) 1.96/2.43 proper(zeros) -> ok(zeros) 1.96/2.43 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.43 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.43 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.43 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.43 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.43 top(ok(X:S)) -> top(active(X:S)) 1.96/2.43 ->Strongly Connected Components: 1.96/2.43 There is no strongly connected component 1.96/2.43 1.96/2.43 The problem is finite. 1.96/2.43 1.96/2.43 Problem 1.2: 1.96/2.43 1.96/2.43 Subterm Processor: 1.96/2.43 -> Pairs: 1.96/2.43 S(mark(X:S)) -> S(X:S) 1.96/2.43 S(ok(X:S)) -> S(X:S) 1.96/2.43 -> Rules: 1.96/2.43 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.43 active(adx(nil)) -> mark(nil) 1.96/2.43 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.43 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.43 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.43 active(head(X:S)) -> head(active(X:S)) 1.96/2.43 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.43 active(incr(nil)) -> mark(nil) 1.96/2.43 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.43 active(s(X:S)) -> s(active(X:S)) 1.96/2.43 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.43 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.43 active(nats) -> mark(adx(zeros)) 1.96/2.43 active(zeros) -> mark(cons(0,zeros)) 1.96/2.43 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.43 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.43 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.43 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.43 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.43 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.43 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.43 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.43 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.43 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.43 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.43 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.43 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.43 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.43 proper(0) -> ok(0) 1.96/2.43 proper(nats) -> ok(nats) 1.96/2.43 proper(nil) -> ok(nil) 1.96/2.43 proper(zeros) -> ok(zeros) 1.96/2.43 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.43 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.43 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.43 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.43 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.43 top(ok(X:S)) -> top(active(X:S)) 1.96/2.43 ->Projection: 1.96/2.43 pi(S) = 1 1.96/2.43 1.96/2.43 Problem 1.2: 1.96/2.43 1.96/2.43 SCC Processor: 1.96/2.43 -> Pairs: 1.96/2.43 Empty 1.96/2.43 -> Rules: 1.96/2.43 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.43 active(adx(nil)) -> mark(nil) 1.96/2.43 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.43 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.43 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.43 active(head(X:S)) -> head(active(X:S)) 1.96/2.43 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.43 active(incr(nil)) -> mark(nil) 1.96/2.43 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.43 active(s(X:S)) -> s(active(X:S)) 1.96/2.43 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.43 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.43 active(nats) -> mark(adx(zeros)) 1.96/2.43 active(zeros) -> mark(cons(0,zeros)) 1.96/2.43 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.43 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.43 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.43 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.43 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.43 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.43 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.43 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.43 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.43 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.43 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.43 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.43 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.43 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.43 proper(0) -> ok(0) 1.96/2.43 proper(nats) -> ok(nats) 1.96/2.43 proper(nil) -> ok(nil) 1.96/2.43 proper(zeros) -> ok(zeros) 1.96/2.43 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.43 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.43 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.43 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.43 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.43 top(ok(X:S)) -> top(active(X:S)) 1.96/2.43 ->Strongly Connected Components: 1.96/2.43 There is no strongly connected component 1.96/2.43 1.96/2.43 The problem is finite. 1.96/2.43 1.96/2.43 Problem 1.3: 1.96/2.43 1.96/2.43 Subterm Processor: 1.96/2.43 -> Pairs: 1.96/2.43 INCR(mark(X:S)) -> INCR(X:S) 1.96/2.43 INCR(ok(X:S)) -> INCR(X:S) 1.96/2.43 -> Rules: 1.96/2.43 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.43 active(adx(nil)) -> mark(nil) 1.96/2.43 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.43 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.43 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.43 active(head(X:S)) -> head(active(X:S)) 1.96/2.43 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.43 active(incr(nil)) -> mark(nil) 1.96/2.43 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.43 active(s(X:S)) -> s(active(X:S)) 1.96/2.43 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.43 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.43 active(nats) -> mark(adx(zeros)) 1.96/2.43 active(zeros) -> mark(cons(0,zeros)) 1.96/2.43 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.43 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.43 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.43 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.43 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.43 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.43 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.43 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.43 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.43 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.43 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.43 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.43 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.43 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.43 proper(0) -> ok(0) 1.96/2.43 proper(nats) -> ok(nats) 1.96/2.43 proper(nil) -> ok(nil) 1.96/2.43 proper(zeros) -> ok(zeros) 1.96/2.43 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.43 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.44 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.44 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.44 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.44 top(ok(X:S)) -> top(active(X:S)) 1.96/2.44 ->Projection: 1.96/2.44 pi(INCR) = 1 1.96/2.44 1.96/2.44 Problem 1.3: 1.96/2.44 1.96/2.44 SCC Processor: 1.96/2.44 -> Pairs: 1.96/2.44 Empty 1.96/2.44 -> Rules: 1.96/2.44 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.44 active(adx(nil)) -> mark(nil) 1.96/2.44 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.44 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.44 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.44 active(head(X:S)) -> head(active(X:S)) 1.96/2.44 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.44 active(incr(nil)) -> mark(nil) 1.96/2.44 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.44 active(s(X:S)) -> s(active(X:S)) 1.96/2.44 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.44 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.44 active(nats) -> mark(adx(zeros)) 1.96/2.44 active(zeros) -> mark(cons(0,zeros)) 1.96/2.44 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.44 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.44 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.44 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.44 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.44 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.44 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.44 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.44 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.44 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.44 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.44 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.44 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.44 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.44 proper(0) -> ok(0) 1.96/2.44 proper(nats) -> ok(nats) 1.96/2.44 proper(nil) -> ok(nil) 1.96/2.44 proper(zeros) -> ok(zeros) 1.96/2.44 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.44 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.44 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.44 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.44 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.44 top(ok(X:S)) -> top(active(X:S)) 1.96/2.44 ->Strongly Connected Components: 1.96/2.44 There is no strongly connected component 1.96/2.44 1.96/2.44 The problem is finite. 1.96/2.44 1.96/2.44 Problem 1.4: 1.96/2.44 1.96/2.44 Subterm Processor: 1.96/2.44 -> Pairs: 1.96/2.44 HEAD(mark(X:S)) -> HEAD(X:S) 1.96/2.44 HEAD(ok(X:S)) -> HEAD(X:S) 1.96/2.44 -> Rules: 1.96/2.44 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.44 active(adx(nil)) -> mark(nil) 1.96/2.44 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.44 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.44 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.44 active(head(X:S)) -> head(active(X:S)) 1.96/2.44 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.44 active(incr(nil)) -> mark(nil) 1.96/2.44 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.44 active(s(X:S)) -> s(active(X:S)) 1.96/2.44 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.44 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.44 active(nats) -> mark(adx(zeros)) 1.96/2.44 active(zeros) -> mark(cons(0,zeros)) 1.96/2.44 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.44 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.44 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.44 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.44 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.44 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.44 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.44 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.44 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.44 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.44 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.44 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.44 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.44 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.44 proper(0) -> ok(0) 1.96/2.44 proper(nats) -> ok(nats) 1.96/2.44 proper(nil) -> ok(nil) 1.96/2.44 proper(zeros) -> ok(zeros) 1.96/2.44 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.44 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.44 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.44 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.44 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.44 top(ok(X:S)) -> top(active(X:S)) 1.96/2.44 ->Projection: 1.96/2.44 pi(HEAD) = 1 1.96/2.44 1.96/2.44 Problem 1.4: 1.96/2.44 1.96/2.44 SCC Processor: 1.96/2.44 -> Pairs: 1.96/2.44 Empty 1.96/2.44 -> Rules: 1.96/2.44 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.44 active(adx(nil)) -> mark(nil) 1.96/2.44 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.44 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.44 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.44 active(head(X:S)) -> head(active(X:S)) 1.96/2.44 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.44 active(incr(nil)) -> mark(nil) 1.96/2.44 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.44 active(s(X:S)) -> s(active(X:S)) 1.96/2.44 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.44 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.44 active(nats) -> mark(adx(zeros)) 1.96/2.44 active(zeros) -> mark(cons(0,zeros)) 1.96/2.44 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.44 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.44 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.44 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.44 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.44 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.44 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.44 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.44 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.44 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.44 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.44 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.44 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.44 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.44 proper(0) -> ok(0) 1.96/2.44 proper(nats) -> ok(nats) 1.96/2.44 proper(nil) -> ok(nil) 1.96/2.44 proper(zeros) -> ok(zeros) 1.96/2.44 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.44 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.44 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.44 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.44 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.44 top(ok(X:S)) -> top(active(X:S)) 1.96/2.44 ->Strongly Connected Components: 1.96/2.44 There is no strongly connected component 1.96/2.44 1.96/2.44 The problem is finite. 1.96/2.44 1.96/2.44 Problem 1.5: 1.96/2.44 1.96/2.44 Subterm Processor: 1.96/2.44 -> Pairs: 1.96/2.44 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 1.96/2.44 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 1.96/2.44 -> Rules: 1.96/2.44 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.44 active(adx(nil)) -> mark(nil) 1.96/2.44 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.44 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.44 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.44 active(head(X:S)) -> head(active(X:S)) 1.96/2.44 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.44 active(incr(nil)) -> mark(nil) 1.96/2.44 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.44 active(s(X:S)) -> s(active(X:S)) 1.96/2.44 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.44 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.44 active(nats) -> mark(adx(zeros)) 1.96/2.44 active(zeros) -> mark(cons(0,zeros)) 1.96/2.44 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.44 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.44 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.44 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.44 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.44 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.44 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.44 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.44 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.44 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.44 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.44 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.44 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.44 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.44 proper(0) -> ok(0) 1.96/2.44 proper(nats) -> ok(nats) 1.96/2.44 proper(nil) -> ok(nil) 1.96/2.44 proper(zeros) -> ok(zeros) 1.96/2.44 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.44 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.44 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.44 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.44 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.44 top(ok(X:S)) -> top(active(X:S)) 1.96/2.44 ->Projection: 1.96/2.44 pi(CONS) = 1 1.96/2.44 1.96/2.44 Problem 1.5: 1.96/2.44 1.96/2.44 SCC Processor: 1.96/2.44 -> Pairs: 1.96/2.44 Empty 1.96/2.44 -> Rules: 1.96/2.44 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.44 active(adx(nil)) -> mark(nil) 1.96/2.44 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.44 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.44 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.44 active(head(X:S)) -> head(active(X:S)) 1.96/2.44 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.44 active(incr(nil)) -> mark(nil) 1.96/2.44 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.44 active(s(X:S)) -> s(active(X:S)) 1.96/2.44 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.44 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.44 active(nats) -> mark(adx(zeros)) 1.96/2.44 active(zeros) -> mark(cons(0,zeros)) 1.96/2.44 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.44 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.44 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.44 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.44 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.44 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.44 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.44 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.44 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.44 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.44 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.44 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.44 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.44 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.44 proper(0) -> ok(0) 1.96/2.44 proper(nats) -> ok(nats) 1.96/2.44 proper(nil) -> ok(nil) 1.96/2.44 proper(zeros) -> ok(zeros) 1.96/2.44 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.44 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.44 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.44 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.44 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.44 top(ok(X:S)) -> top(active(X:S)) 1.96/2.44 ->Strongly Connected Components: 1.96/2.44 There is no strongly connected component 1.96/2.44 1.96/2.44 The problem is finite. 1.96/2.44 1.96/2.44 Problem 1.6: 1.96/2.44 1.96/2.44 Subterm Processor: 1.96/2.44 -> Pairs: 1.96/2.44 ADX(mark(X:S)) -> ADX(X:S) 1.96/2.44 ADX(ok(X:S)) -> ADX(X:S) 1.96/2.44 -> Rules: 1.96/2.44 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.44 active(adx(nil)) -> mark(nil) 1.96/2.44 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.44 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.44 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.44 active(head(X:S)) -> head(active(X:S)) 1.96/2.44 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.44 active(incr(nil)) -> mark(nil) 1.96/2.44 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.44 active(s(X:S)) -> s(active(X:S)) 1.96/2.44 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.44 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.44 active(nats) -> mark(adx(zeros)) 1.96/2.44 active(zeros) -> mark(cons(0,zeros)) 1.96/2.44 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.44 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.44 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.44 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.44 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.44 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.44 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.44 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.44 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.44 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.44 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.44 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.44 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.44 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.44 proper(0) -> ok(0) 1.96/2.44 proper(nats) -> ok(nats) 1.96/2.44 proper(nil) -> ok(nil) 1.96/2.44 proper(zeros) -> ok(zeros) 1.96/2.44 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.44 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.44 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.44 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.44 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.44 top(ok(X:S)) -> top(active(X:S)) 1.96/2.44 ->Projection: 1.96/2.44 pi(ADX) = 1 1.96/2.44 1.96/2.44 Problem 1.6: 1.96/2.44 1.96/2.44 SCC Processor: 1.96/2.44 -> Pairs: 1.96/2.44 Empty 1.96/2.44 -> Rules: 1.96/2.44 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.44 active(adx(nil)) -> mark(nil) 1.96/2.44 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.44 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.44 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.44 active(head(X:S)) -> head(active(X:S)) 1.96/2.44 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.44 active(incr(nil)) -> mark(nil) 1.96/2.44 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.44 active(s(X:S)) -> s(active(X:S)) 1.96/2.44 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.44 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.44 active(nats) -> mark(adx(zeros)) 1.96/2.44 active(zeros) -> mark(cons(0,zeros)) 1.96/2.44 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.44 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.44 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.44 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.44 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.44 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.44 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.44 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.44 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.44 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.44 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.44 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.44 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.44 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.44 proper(0) -> ok(0) 1.96/2.44 proper(nats) -> ok(nats) 1.96/2.44 proper(nil) -> ok(nil) 1.96/2.44 proper(zeros) -> ok(zeros) 1.96/2.44 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.44 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.44 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.44 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.44 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.44 top(ok(X:S)) -> top(active(X:S)) 1.96/2.44 ->Strongly Connected Components: 1.96/2.44 There is no strongly connected component 1.96/2.44 1.96/2.44 The problem is finite. 1.96/2.44 1.96/2.44 Problem 1.7: 1.96/2.44 1.96/2.44 Subterm Processor: 1.96/2.44 -> Pairs: 1.96/2.44 PROPER(adx(X:S)) -> PROPER(X:S) 1.96/2.44 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 1.96/2.44 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 1.96/2.44 PROPER(head(X:S)) -> PROPER(X:S) 1.96/2.44 PROPER(incr(X:S)) -> PROPER(X:S) 1.96/2.44 PROPER(s(X:S)) -> PROPER(X:S) 1.96/2.44 PROPER(tail(X:S)) -> PROPER(X:S) 1.96/2.44 -> Rules: 1.96/2.44 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.44 active(adx(nil)) -> mark(nil) 1.96/2.44 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.44 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.44 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.44 active(head(X:S)) -> head(active(X:S)) 1.96/2.44 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.44 active(incr(nil)) -> mark(nil) 1.96/2.44 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.44 active(s(X:S)) -> s(active(X:S)) 1.96/2.44 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.44 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.44 active(nats) -> mark(adx(zeros)) 1.96/2.44 active(zeros) -> mark(cons(0,zeros)) 1.96/2.44 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.44 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.44 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.44 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.44 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.44 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.44 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.44 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.44 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.44 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.44 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.44 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.44 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.44 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.44 proper(0) -> ok(0) 1.96/2.44 proper(nats) -> ok(nats) 1.96/2.44 proper(nil) -> ok(nil) 1.96/2.44 proper(zeros) -> ok(zeros) 1.96/2.44 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.44 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.44 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.44 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.44 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.44 top(ok(X:S)) -> top(active(X:S)) 1.96/2.44 ->Projection: 1.96/2.44 pi(PROPER) = 1 1.96/2.44 1.96/2.44 Problem 1.7: 1.96/2.44 1.96/2.44 SCC Processor: 1.96/2.44 -> Pairs: 1.96/2.44 Empty 1.96/2.44 -> Rules: 1.96/2.44 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.44 active(adx(nil)) -> mark(nil) 1.96/2.44 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.44 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.44 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.44 active(head(X:S)) -> head(active(X:S)) 1.96/2.44 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.44 active(incr(nil)) -> mark(nil) 1.96/2.44 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.44 active(s(X:S)) -> s(active(X:S)) 1.96/2.44 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.44 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.44 active(nats) -> mark(adx(zeros)) 1.96/2.44 active(zeros) -> mark(cons(0,zeros)) 1.96/2.44 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.44 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.44 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.44 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.44 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.44 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.44 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.44 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.44 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.44 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.44 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.44 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.44 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.44 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.44 proper(0) -> ok(0) 1.96/2.44 proper(nats) -> ok(nats) 1.96/2.44 proper(nil) -> ok(nil) 1.96/2.44 proper(zeros) -> ok(zeros) 1.96/2.44 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.44 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.44 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.44 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.44 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.44 top(ok(X:S)) -> top(active(X:S)) 1.96/2.44 ->Strongly Connected Components: 1.96/2.44 There is no strongly connected component 1.96/2.44 1.96/2.44 The problem is finite. 1.96/2.44 1.96/2.44 Problem 1.8: 1.96/2.44 1.96/2.44 Subterm Processor: 1.96/2.44 -> Pairs: 1.96/2.44 ACTIVE(adx(X:S)) -> ACTIVE(X:S) 1.96/2.44 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 1.96/2.44 ACTIVE(head(X:S)) -> ACTIVE(X:S) 1.96/2.44 ACTIVE(incr(X:S)) -> ACTIVE(X:S) 1.96/2.44 ACTIVE(s(X:S)) -> ACTIVE(X:S) 1.96/2.44 ACTIVE(tail(X:S)) -> ACTIVE(X:S) 1.96/2.44 -> Rules: 1.96/2.44 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.44 active(adx(nil)) -> mark(nil) 1.96/2.44 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.44 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.44 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.44 active(head(X:S)) -> head(active(X:S)) 1.96/2.44 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.44 active(incr(nil)) -> mark(nil) 1.96/2.44 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.44 active(s(X:S)) -> s(active(X:S)) 1.96/2.44 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.44 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.44 active(nats) -> mark(adx(zeros)) 1.96/2.44 active(zeros) -> mark(cons(0,zeros)) 1.96/2.44 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.44 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.44 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.44 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.44 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.44 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.44 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.44 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.44 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.44 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.44 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.44 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.44 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.44 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.44 proper(0) -> ok(0) 1.96/2.44 proper(nats) -> ok(nats) 1.96/2.44 proper(nil) -> ok(nil) 1.96/2.44 proper(zeros) -> ok(zeros) 1.96/2.44 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.44 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.44 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.44 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.44 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.44 top(ok(X:S)) -> top(active(X:S)) 1.96/2.44 ->Projection: 1.96/2.44 pi(ACTIVE) = 1 1.96/2.44 1.96/2.44 Problem 1.8: 1.96/2.44 1.96/2.44 SCC Processor: 1.96/2.44 -> Pairs: 1.96/2.44 Empty 1.96/2.44 -> Rules: 1.96/2.44 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.44 active(adx(nil)) -> mark(nil) 1.96/2.44 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.44 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.44 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.44 active(head(X:S)) -> head(active(X:S)) 1.96/2.44 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.44 active(incr(nil)) -> mark(nil) 1.96/2.44 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.44 active(s(X:S)) -> s(active(X:S)) 1.96/2.44 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.44 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.44 active(nats) -> mark(adx(zeros)) 1.96/2.44 active(zeros) -> mark(cons(0,zeros)) 1.96/2.44 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.44 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.44 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.44 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.44 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.44 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.44 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.44 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.44 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.44 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.44 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.44 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.44 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.44 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.44 proper(0) -> ok(0) 1.96/2.44 proper(nats) -> ok(nats) 1.96/2.44 proper(nil) -> ok(nil) 1.96/2.44 proper(zeros) -> ok(zeros) 1.96/2.44 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.44 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.44 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.44 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.44 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.44 top(ok(X:S)) -> top(active(X:S)) 1.96/2.44 ->Strongly Connected Components: 1.96/2.44 There is no strongly connected component 1.96/2.44 1.96/2.44 The problem is finite. 1.96/2.44 1.96/2.44 Problem 1.9: 1.96/2.44 1.96/2.44 Reduction Pairs Processor: 1.96/2.44 -> Pairs: 1.96/2.44 TOP(mark(X:S)) -> TOP(proper(X:S)) 1.96/2.44 TOP(ok(X:S)) -> TOP(active(X:S)) 1.96/2.44 -> Rules: 1.96/2.44 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.44 active(adx(nil)) -> mark(nil) 1.96/2.44 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.44 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.44 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.44 active(head(X:S)) -> head(active(X:S)) 1.96/2.44 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.44 active(incr(nil)) -> mark(nil) 1.96/2.44 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.44 active(s(X:S)) -> s(active(X:S)) 1.96/2.44 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.44 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.44 active(nats) -> mark(adx(zeros)) 1.96/2.44 active(zeros) -> mark(cons(0,zeros)) 1.96/2.44 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.44 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.44 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.44 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.44 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.44 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.44 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.44 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.44 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.44 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.44 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.44 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.44 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.44 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.44 proper(0) -> ok(0) 1.96/2.44 proper(nats) -> ok(nats) 1.96/2.44 proper(nil) -> ok(nil) 1.96/2.44 proper(zeros) -> ok(zeros) 1.96/2.44 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.44 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.44 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.44 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.44 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.44 top(ok(X:S)) -> top(active(X:S)) 1.96/2.44 -> Usable rules: 1.96/2.44 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.44 active(adx(nil)) -> mark(nil) 1.96/2.44 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.44 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.44 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.44 active(head(X:S)) -> head(active(X:S)) 1.96/2.44 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.44 active(incr(nil)) -> mark(nil) 1.96/2.44 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.44 active(s(X:S)) -> s(active(X:S)) 1.96/2.44 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.44 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.44 active(nats) -> mark(adx(zeros)) 1.96/2.44 active(zeros) -> mark(cons(0,zeros)) 1.96/2.44 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.44 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.44 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.44 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.44 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.44 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.44 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.44 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.44 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.44 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.44 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.44 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.44 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.44 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.44 proper(0) -> ok(0) 1.96/2.44 proper(nats) -> ok(nats) 1.96/2.44 proper(nil) -> ok(nil) 1.96/2.44 proper(zeros) -> ok(zeros) 1.96/2.44 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.44 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.44 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.44 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.44 ->Interpretation type: 1.96/2.44 Linear 1.96/2.44 ->Coefficients: 1.96/2.44 All rationals 1.96/2.44 ->Dimension: 1.96/2.44 1 1.96/2.44 ->Bound: 1.96/2.44 4 1.96/2.44 ->Interpretation: 1.96/2.44 1.96/2.44 [active](X) = X 1.96/2.44 [adx](X) = X + 4/3 1.96/2.44 [cons](X1,X2) = 4/3.X1 + 1/4.X2 1.96/2.44 [head](X) = 4.X + 1/3 1.96/2.44 [incr](X) = X + 2/3 1.96/2.44 [proper](X) = X 1.96/2.44 [s](X) = X 1.96/2.44 [tail](X) = 4.X + 1/3 1.96/2.44 [top](X) = 0 1.96/2.44 [0] = 0 1.96/2.44 [fSNonEmpty] = 0 1.96/2.44 [mark](X) = X + 1/3 1.96/2.44 [nats] = 4 1.96/2.44 [nil] = 1/4 1.96/2.44 [ok](X) = X 1.96/2.44 [zeros] = 2 1.96/2.44 [ACTIVE](X) = 0 1.96/2.44 [ADX](X) = 0 1.96/2.44 [CONS](X1,X2) = 0 1.96/2.44 [HEAD](X) = 0 1.96/2.44 [INCR](X) = 0 1.96/2.44 [PROPER](X) = 0 1.96/2.44 [S](X) = 0 1.96/2.44 [TAIL](X) = 0 1.96/2.44 [TOP](X) = 4.X 1.96/2.44 1.96/2.44 Problem 1.9: 1.96/2.44 1.96/2.44 SCC Processor: 1.96/2.44 -> Pairs: 1.96/2.44 TOP(ok(X:S)) -> TOP(active(X:S)) 1.96/2.44 -> Rules: 1.96/2.44 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.44 active(adx(nil)) -> mark(nil) 1.96/2.44 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.44 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.44 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.44 active(head(X:S)) -> head(active(X:S)) 1.96/2.44 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.44 active(incr(nil)) -> mark(nil) 1.96/2.44 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.44 active(s(X:S)) -> s(active(X:S)) 1.96/2.44 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.44 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.44 active(nats) -> mark(adx(zeros)) 1.96/2.44 active(zeros) -> mark(cons(0,zeros)) 1.96/2.44 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.44 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.44 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.44 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.44 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.44 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.44 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.44 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.44 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.44 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.44 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.44 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.44 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.44 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.44 proper(0) -> ok(0) 1.96/2.44 proper(nats) -> ok(nats) 1.96/2.44 proper(nil) -> ok(nil) 1.96/2.44 proper(zeros) -> ok(zeros) 1.96/2.44 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.44 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.44 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.44 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.44 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.44 top(ok(X:S)) -> top(active(X:S)) 1.96/2.44 ->Strongly Connected Components: 1.96/2.44 ->->Cycle: 1.96/2.44 ->->-> Pairs: 1.96/2.44 TOP(ok(X:S)) -> TOP(active(X:S)) 1.96/2.44 ->->-> Rules: 1.96/2.44 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.44 active(adx(nil)) -> mark(nil) 1.96/2.44 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.44 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.44 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.44 active(head(X:S)) -> head(active(X:S)) 1.96/2.44 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.44 active(incr(nil)) -> mark(nil) 1.96/2.44 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.44 active(s(X:S)) -> s(active(X:S)) 1.96/2.44 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.44 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.44 active(nats) -> mark(adx(zeros)) 1.96/2.44 active(zeros) -> mark(cons(0,zeros)) 1.96/2.44 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.44 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.44 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.44 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.44 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.44 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.44 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.44 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.44 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.44 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.44 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.44 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.44 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.44 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.44 proper(0) -> ok(0) 1.96/2.44 proper(nats) -> ok(nats) 1.96/2.44 proper(nil) -> ok(nil) 1.96/2.44 proper(zeros) -> ok(zeros) 1.96/2.44 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.44 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.44 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.44 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.44 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.44 top(ok(X:S)) -> top(active(X:S)) 1.96/2.44 1.96/2.44 Problem 1.9: 1.96/2.44 1.96/2.44 Reduction Pairs Processor: 1.96/2.44 -> Pairs: 1.96/2.44 TOP(ok(X:S)) -> TOP(active(X:S)) 1.96/2.44 -> Rules: 1.96/2.44 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.44 active(adx(nil)) -> mark(nil) 1.96/2.44 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.44 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.44 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.44 active(head(X:S)) -> head(active(X:S)) 1.96/2.44 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.44 active(incr(nil)) -> mark(nil) 1.96/2.44 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.44 active(s(X:S)) -> s(active(X:S)) 1.96/2.44 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.44 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.44 active(nats) -> mark(adx(zeros)) 1.96/2.44 active(zeros) -> mark(cons(0,zeros)) 1.96/2.44 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.44 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.44 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.44 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.44 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.44 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.44 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.44 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.44 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.44 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.44 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.44 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.44 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.44 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.44 proper(0) -> ok(0) 1.96/2.44 proper(nats) -> ok(nats) 1.96/2.44 proper(nil) -> ok(nil) 1.96/2.44 proper(zeros) -> ok(zeros) 1.96/2.44 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.44 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.44 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.44 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.44 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.44 top(ok(X:S)) -> top(active(X:S)) 1.96/2.44 -> Usable rules: 1.96/2.44 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.44 active(adx(nil)) -> mark(nil) 1.96/2.44 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.44 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.44 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.44 active(head(X:S)) -> head(active(X:S)) 1.96/2.44 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.44 active(incr(nil)) -> mark(nil) 1.96/2.44 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.44 active(s(X:S)) -> s(active(X:S)) 1.96/2.44 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.44 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.44 active(nats) -> mark(adx(zeros)) 1.96/2.44 active(zeros) -> mark(cons(0,zeros)) 1.96/2.44 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.44 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.44 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.44 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.44 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.44 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.44 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.44 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.44 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.44 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.44 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.44 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.44 ->Interpretation type: 1.96/2.44 Linear 1.96/2.44 ->Coefficients: 1.96/2.44 Natural Numbers 1.96/2.44 ->Dimension: 1.96/2.44 1 1.96/2.44 ->Bound: 1.96/2.44 2 1.96/2.44 ->Interpretation: 1.96/2.44 1.96/2.44 [active](X) = 2.X 1.96/2.44 [adx](X) = 2.X 1.96/2.44 [cons](X1,X2) = 2.X1 + 2 1.96/2.44 [head](X) = 2.X + 1 1.96/2.44 [incr](X) = 2.X 1.96/2.44 [proper](X) = 0 1.96/2.44 [s](X) = 2.X + 2 1.96/2.44 [tail](X) = 2.X + 2 1.96/2.44 [top](X) = 0 1.96/2.44 [0] = 2 1.96/2.44 [fSNonEmpty] = 0 1.96/2.44 [mark](X) = 2 1.96/2.44 [nats] = 2 1.96/2.44 [nil] = 2 1.96/2.44 [ok](X) = 2.X + 2 1.96/2.44 [zeros] = 2 1.96/2.44 [ACTIVE](X) = 0 1.96/2.44 [ADX](X) = 0 1.96/2.44 [CONS](X1,X2) = 0 1.96/2.44 [HEAD](X) = 0 1.96/2.44 [INCR](X) = 0 1.96/2.44 [PROPER](X) = 0 1.96/2.44 [S](X) = 0 1.96/2.44 [TAIL](X) = 0 1.96/2.44 [TOP](X) = X 1.96/2.44 1.96/2.44 Problem 1.9: 1.96/2.44 1.96/2.44 SCC Processor: 1.96/2.44 -> Pairs: 1.96/2.44 Empty 1.96/2.44 -> Rules: 1.96/2.44 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) 1.96/2.44 active(adx(nil)) -> mark(nil) 1.96/2.44 active(adx(X:S)) -> adx(active(X:S)) 1.96/2.44 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 1.96/2.44 active(head(cons(X:S,L:S))) -> mark(X:S) 1.96/2.44 active(head(X:S)) -> head(active(X:S)) 1.96/2.44 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) 1.96/2.44 active(incr(nil)) -> mark(nil) 1.96/2.44 active(incr(X:S)) -> incr(active(X:S)) 1.96/2.44 active(s(X:S)) -> s(active(X:S)) 1.96/2.44 active(tail(cons(X:S,L:S))) -> mark(L:S) 1.96/2.44 active(tail(X:S)) -> tail(active(X:S)) 1.96/2.44 active(nats) -> mark(adx(zeros)) 1.96/2.44 active(zeros) -> mark(cons(0,zeros)) 1.96/2.44 adx(mark(X:S)) -> mark(adx(X:S)) 1.96/2.44 adx(ok(X:S)) -> ok(adx(X:S)) 1.96/2.44 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 1.96/2.44 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 1.96/2.44 head(mark(X:S)) -> mark(head(X:S)) 1.96/2.44 head(ok(X:S)) -> ok(head(X:S)) 1.96/2.44 incr(mark(X:S)) -> mark(incr(X:S)) 1.96/2.44 incr(ok(X:S)) -> ok(incr(X:S)) 1.96/2.44 proper(adx(X:S)) -> adx(proper(X:S)) 1.96/2.44 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 1.96/2.44 proper(head(X:S)) -> head(proper(X:S)) 1.96/2.44 proper(incr(X:S)) -> incr(proper(X:S)) 1.96/2.44 proper(s(X:S)) -> s(proper(X:S)) 1.96/2.44 proper(tail(X:S)) -> tail(proper(X:S)) 1.96/2.44 proper(0) -> ok(0) 1.96/2.44 proper(nats) -> ok(nats) 1.96/2.44 proper(nil) -> ok(nil) 1.96/2.44 proper(zeros) -> ok(zeros) 1.96/2.44 s(mark(X:S)) -> mark(s(X:S)) 1.96/2.44 s(ok(X:S)) -> ok(s(X:S)) 1.96/2.44 tail(mark(X:S)) -> mark(tail(X:S)) 1.96/2.44 tail(ok(X:S)) -> ok(tail(X:S)) 1.96/2.44 top(mark(X:S)) -> top(proper(X:S)) 1.96/2.44 top(ok(X:S)) -> top(active(X:S)) 1.96/2.44 ->Strongly Connected Components: 1.96/2.44 There is no strongly connected component 1.96/2.44 1.96/2.44 The problem is finite. 1.96/2.44 EOF