/export/starexec/sandbox/solver/bin/starexec_run_ttt2 /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES Problem: a__2nd(cons1(X,cons(Y,Z))) -> mark(Y) a__2nd(cons(X,X1)) -> a__2nd(cons1(mark(X),mark(X1))) a__from(X) -> cons(mark(X),from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1,X2)) -> cons1(mark(X1),mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) Proof: DP Processor: DPs: a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) a__2nd#(cons(X,X1)) -> mark#(X1) a__2nd#(cons(X,X1)) -> mark#(X) a__2nd#(cons(X,X1)) -> a__2nd#(cons1(mark(X),mark(X1))) a__from#(X) -> mark#(X) mark#(2nd(X)) -> mark#(X) mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(cons1(X1,X2)) -> mark#(X2) mark#(cons1(X1,X2)) -> mark#(X1) TRS: a__2nd(cons1(X,cons(Y,Z))) -> mark(Y) a__2nd(cons(X,X1)) -> a__2nd(cons1(mark(X),mark(X1))) a__from(X) -> cons(mark(X),from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1,X2)) -> cons1(mark(X1),mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) TDG Processor: DPs: a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) a__2nd#(cons(X,X1)) -> mark#(X1) a__2nd#(cons(X,X1)) -> mark#(X) a__2nd#(cons(X,X1)) -> a__2nd#(cons1(mark(X),mark(X1))) a__from#(X) -> mark#(X) mark#(2nd(X)) -> mark#(X) mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(cons1(X1,X2)) -> mark#(X2) mark#(cons1(X1,X2)) -> mark#(X1) TRS: a__2nd(cons1(X,cons(Y,Z))) -> mark(Y) a__2nd(cons(X,X1)) -> a__2nd(cons1(mark(X),mark(X1))) a__from(X) -> cons(mark(X),from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1,X2)) -> cons1(mark(X1),mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) graph: a__from#(X) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X2) a__from#(X) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(2nd(X)) -> a__2nd#(mark(X)) a__from#(X) -> mark#(X) -> mark#(2nd(X)) -> mark#(X) mark#(2nd(X)) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X1) mark#(2nd(X)) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X2) mark#(2nd(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(2nd(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(2nd(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(2nd(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(2nd(X)) -> mark#(X) -> mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(2nd(X)) -> mark#(X) -> mark#(2nd(X)) -> mark#(X) mark#(2nd(X)) -> a__2nd#(mark(X)) -> a__2nd#(cons(X,X1)) -> a__2nd#(cons1(mark(X),mark(X1))) mark#(2nd(X)) -> a__2nd#(mark(X)) -> a__2nd#(cons(X,X1)) -> mark#(X) mark#(2nd(X)) -> a__2nd#(mark(X)) -> a__2nd#(cons(X,X1)) -> mark#(X1) mark#(2nd(X)) -> a__2nd#(mark(X)) -> a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) mark#(from(X)) -> a__from#(mark(X)) -> a__from#(X) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X2) mark#(from(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(from(X)) -> mark#(X) -> mark#(2nd(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(2nd(X)) -> mark#(X) mark#(cons1(X1,X2)) -> mark#(X2) -> mark#(cons1(X1,X2)) -> mark#(X1) mark#(cons1(X1,X2)) -> mark#(X2) -> mark#(cons1(X1,X2)) -> mark#(X2) mark#(cons1(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(cons1(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons1(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> a__from#(mark(X)) mark#(cons1(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> mark#(X) mark#(cons1(X1,X2)) -> mark#(X2) -> mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(cons1(X1,X2)) -> mark#(X2) -> mark#(2nd(X)) -> mark#(X) mark#(cons1(X1,X2)) -> mark#(X1) -> mark#(cons1(X1,X2)) -> mark#(X1) mark#(cons1(X1,X2)) -> mark#(X1) -> mark#(cons1(X1,X2)) -> mark#(X2) mark#(cons1(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons1(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons1(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(cons1(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(cons1(X1,X2)) -> mark#(X1) -> mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(cons1(X1,X2)) -> mark#(X1) -> mark#(2nd(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons1(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons1(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(2nd(X)) -> mark#(X) a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) -> mark#(cons1(X1,X2)) -> mark#(X1) a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) -> mark#(cons1(X1,X2)) -> mark#(X2) a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) -> mark#(s(X)) -> mark#(X) a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) -> mark#(cons(X1,X2)) -> mark#(X1) a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) -> mark#(from(X)) -> a__from#(mark(X)) a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) -> mark#(from(X)) -> mark#(X) a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) -> mark#(2nd(X)) -> a__2nd#(mark(X)) a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) -> mark#(2nd(X)) -> mark#(X) a__2nd#(cons(X,X1)) -> mark#(X1) -> mark#(cons1(X1,X2)) -> mark#(X1) a__2nd#(cons(X,X1)) -> mark#(X1) -> mark#(cons1(X1,X2)) -> mark#(X2) a__2nd#(cons(X,X1)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) a__2nd#(cons(X,X1)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) a__2nd#(cons(X,X1)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) a__2nd#(cons(X,X1)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) a__2nd#(cons(X,X1)) -> mark#(X1) -> mark#(2nd(X)) -> a__2nd#(mark(X)) a__2nd#(cons(X,X1)) -> mark#(X1) -> mark#(2nd(X)) -> mark#(X) a__2nd#(cons(X,X1)) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X1) a__2nd#(cons(X,X1)) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X2) a__2nd#(cons(X,X1)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__2nd#(cons(X,X1)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__2nd#(cons(X,X1)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__2nd#(cons(X,X1)) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__2nd#(cons(X,X1)) -> mark#(X) -> mark#(2nd(X)) -> a__2nd#(mark(X)) a__2nd#(cons(X,X1)) -> mark#(X) -> mark#(2nd(X)) -> mark#(X) a__2nd#(cons(X,X1)) -> a__2nd#(cons1(mark(X),mark(X1))) -> a__2nd#(cons(X,X1)) -> a__2nd#(cons1(mark(X),mark(X1))) a__2nd#(cons(X,X1)) -> a__2nd#(cons1(mark(X),mark(X1))) -> a__2nd#(cons(X,X1)) -> mark#(X) a__2nd#(cons(X,X1)) -> a__2nd#(cons1(mark(X),mark(X1))) -> a__2nd#(cons(X,X1)) -> mark#(X1) a__2nd#(cons(X,X1)) -> a__2nd#(cons1(mark(X),mark(X1))) -> a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) EDG Processor: DPs: a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) a__2nd#(cons(X,X1)) -> mark#(X1) a__2nd#(cons(X,X1)) -> mark#(X) a__2nd#(cons(X,X1)) -> a__2nd#(cons1(mark(X),mark(X1))) a__from#(X) -> mark#(X) mark#(2nd(X)) -> mark#(X) mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(cons1(X1,X2)) -> mark#(X2) mark#(cons1(X1,X2)) -> mark#(X1) TRS: a__2nd(cons1(X,cons(Y,Z))) -> mark(Y) a__2nd(cons(X,X1)) -> a__2nd(cons1(mark(X),mark(X1))) a__from(X) -> cons(mark(X),from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1,X2)) -> cons1(mark(X1),mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) graph: a__from#(X) -> mark#(X) -> mark#(2nd(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(2nd(X)) -> a__2nd#(mark(X)) a__from#(X) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X2) a__from#(X) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X1) mark#(2nd(X)) -> mark#(X) -> mark#(2nd(X)) -> mark#(X) mark#(2nd(X)) -> mark#(X) -> mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(2nd(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(2nd(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(2nd(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(2nd(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(2nd(X)) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X2) mark#(2nd(X)) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X1) mark#(2nd(X)) -> a__2nd#(mark(X)) -> a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) mark#(2nd(X)) -> a__2nd#(mark(X)) -> a__2nd#(cons(X,X1)) -> mark#(X1) mark#(2nd(X)) -> a__2nd#(mark(X)) -> a__2nd#(cons(X,X1)) -> mark#(X) mark#(2nd(X)) -> a__2nd#(mark(X)) -> a__2nd#(cons(X,X1)) -> a__2nd#(cons1(mark(X),mark(X1))) mark#(from(X)) -> a__from#(mark(X)) -> a__from#(X) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(2nd(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(from(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X2) mark#(from(X)) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(2nd(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X1) mark#(cons1(X1,X2)) -> mark#(X2) -> mark#(2nd(X)) -> mark#(X) mark#(cons1(X1,X2)) -> mark#(X2) -> mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(cons1(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> mark#(X) mark#(cons1(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> a__from#(mark(X)) mark#(cons1(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons1(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(cons1(X1,X2)) -> mark#(X2) -> mark#(cons1(X1,X2)) -> mark#(X2) mark#(cons1(X1,X2)) -> mark#(X2) -> mark#(cons1(X1,X2)) -> mark#(X1) mark#(cons1(X1,X2)) -> mark#(X1) -> mark#(2nd(X)) -> mark#(X) mark#(cons1(X1,X2)) -> mark#(X1) -> mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(cons1(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(cons1(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(cons1(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons1(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons1(X1,X2)) -> mark#(X1) -> mark#(cons1(X1,X2)) -> mark#(X2) mark#(cons1(X1,X2)) -> mark#(X1) -> mark#(cons1(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(2nd(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons1(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons1(X1,X2)) -> mark#(X1) a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) -> mark#(2nd(X)) -> mark#(X) a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) -> mark#(2nd(X)) -> a__2nd#(mark(X)) a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) -> mark#(from(X)) -> mark#(X) a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) -> mark#(from(X)) -> a__from#(mark(X)) a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) -> mark#(cons(X1,X2)) -> mark#(X1) a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) -> mark#(s(X)) -> mark#(X) a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) -> mark#(cons1(X1,X2)) -> mark#(X2) a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) -> mark#(cons1(X1,X2)) -> mark#(X1) a__2nd#(cons(X,X1)) -> mark#(X1) -> mark#(2nd(X)) -> mark#(X) a__2nd#(cons(X,X1)) -> mark#(X1) -> mark#(2nd(X)) -> a__2nd#(mark(X)) a__2nd#(cons(X,X1)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) a__2nd#(cons(X,X1)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) a__2nd#(cons(X,X1)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) a__2nd#(cons(X,X1)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) a__2nd#(cons(X,X1)) -> mark#(X1) -> mark#(cons1(X1,X2)) -> mark#(X2) a__2nd#(cons(X,X1)) -> mark#(X1) -> mark#(cons1(X1,X2)) -> mark#(X1) a__2nd#(cons(X,X1)) -> mark#(X) -> mark#(2nd(X)) -> mark#(X) a__2nd#(cons(X,X1)) -> mark#(X) -> mark#(2nd(X)) -> a__2nd#(mark(X)) a__2nd#(cons(X,X1)) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__2nd#(cons(X,X1)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__2nd#(cons(X,X1)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__2nd#(cons(X,X1)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__2nd#(cons(X,X1)) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X2) a__2nd#(cons(X,X1)) -> mark#(X) -> mark#(cons1(X1,X2)) -> mark#(X1) a__2nd#(cons(X,X1)) -> a__2nd#(cons1(mark(X),mark(X1))) -> a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) Arctic Interpretation Processor: dimension: 1 usable rules: a__2nd(cons1(X,cons(Y,Z))) -> mark(Y) a__2nd(cons(X,X1)) -> a__2nd(cons1(mark(X),mark(X1))) a__from(X) -> cons(mark(X),from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1,X2)) -> cons1(mark(X1),mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) interpretation: [a__from](x0) = 4x0 + 1, [a__2nd#](x0) = x0 + 0, [cons1](x0, x1) = x0 + x1, [mark](x0) = x0, [s](x0) = x0, [a__from#](x0) = 3x0 + 0, [cons](x0, x1) = x0 + x1 + 0, [2nd](x0) = 6x0, [mark#](x0) = x0 + 0, [from](x0) = 4x0 + 1, [a__2nd](x0) = 6x0 orientation: a__2nd#(cons1(X,cons(Y,Z))) = X + Y + Z + 0 >= Y + 0 = mark#(Y) a__2nd#(cons(X,X1)) = X + X1 + 0 >= X1 + 0 = mark#(X1) a__2nd#(cons(X,X1)) = X + X1 + 0 >= X + 0 = mark#(X) a__2nd#(cons(X,X1)) = X + X1 + 0 >= X + X1 + 0 = a__2nd#(cons1(mark(X),mark(X1))) a__from#(X) = 3X + 0 >= X + 0 = mark#(X) mark#(2nd(X)) = 6X + 0 >= X + 0 = mark#(X) mark#(2nd(X)) = 6X + 0 >= X + 0 = a__2nd#(mark(X)) mark#(from(X)) = 4X + 1 >= X + 0 = mark#(X) mark#(from(X)) = 4X + 1 >= 3X + 0 = a__from#(mark(X)) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(cons1(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(cons1(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) a__2nd(cons1(X,cons(Y,Z))) = 6X + 6Y + 6Z + 6 >= Y = mark(Y) a__2nd(cons(X,X1)) = 6X + 6X1 + 6 >= 6X + 6X1 = a__2nd(cons1(mark(X),mark(X1))) a__from(X) = 4X + 1 >= 4X + 1 = cons(mark(X),from(s(X))) mark(2nd(X)) = 6X >= 6X = a__2nd(mark(X)) mark(from(X)) = 4X + 1 >= 4X + 1 = a__from(mark(X)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(s(X)) = X >= X = s(mark(X)) mark(cons1(X1,X2)) = X1 + X2 >= X1 + X2 = cons1(mark(X1),mark(X2)) a__2nd(X) = 6X >= 6X = 2nd(X) a__from(X) = 4X + 1 >= 4X + 1 = from(X) problem: DPs: a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) a__2nd#(cons(X,X1)) -> mark#(X1) a__2nd#(cons(X,X1)) -> mark#(X) a__2nd#(cons(X,X1)) -> a__2nd#(cons1(mark(X),mark(X1))) a__from#(X) -> mark#(X) mark#(2nd(X)) -> mark#(X) mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(cons1(X1,X2)) -> mark#(X2) mark#(cons1(X1,X2)) -> mark#(X1) TRS: a__2nd(cons1(X,cons(Y,Z))) -> mark(Y) a__2nd(cons(X,X1)) -> a__2nd(cons1(mark(X),mark(X1))) a__from(X) -> cons(mark(X),from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1,X2)) -> cons1(mark(X1),mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) Restore Modifier: DPs: a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) a__2nd#(cons(X,X1)) -> mark#(X1) a__2nd#(cons(X,X1)) -> mark#(X) a__2nd#(cons(X,X1)) -> a__2nd#(cons1(mark(X),mark(X1))) a__from#(X) -> mark#(X) mark#(2nd(X)) -> mark#(X) mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(cons1(X1,X2)) -> mark#(X2) mark#(cons1(X1,X2)) -> mark#(X1) TRS: a__2nd(cons1(X,cons(Y,Z))) -> mark(Y) a__2nd(cons(X,X1)) -> a__2nd(cons1(mark(X),mark(X1))) a__from(X) -> cons(mark(X),from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1,X2)) -> cons1(mark(X1),mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) SCC Processor: #sccs: 1 #rules: 10 #arcs: 86/121 DPs: mark#(cons1(X1,X2)) -> mark#(X1) mark#(cons1(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(2nd(X)) -> a__2nd#(mark(X)) a__2nd#(cons(X,X1)) -> a__2nd#(cons1(mark(X),mark(X1))) a__2nd#(cons1(X,cons(Y,Z))) -> mark#(Y) mark#(2nd(X)) -> mark#(X) a__2nd#(cons(X,X1)) -> mark#(X) a__2nd#(cons(X,X1)) -> mark#(X1) TRS: a__2nd(cons1(X,cons(Y,Z))) -> mark(Y) a__2nd(cons(X,X1)) -> a__2nd(cons1(mark(X),mark(X1))) a__from(X) -> cons(mark(X),from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1,X2)) -> cons1(mark(X1),mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) Arctic Interpretation Processor: dimension: 1 usable rules: a__2nd(cons1(X,cons(Y,Z))) -> mark(Y) a__2nd(cons(X,X1)) -> a__2nd(cons1(mark(X),mark(X1))) a__from(X) -> cons(mark(X),from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1,X2)) -> cons1(mark(X1),mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) interpretation: [a__from](x0) = 7x0 + 7, [a__2nd#](x0) = x0 + 5, [cons1](x0, x1) = 4x0 + x1 + 5, [mark](x0) = x0, [s](x0) = x0 + 0, [cons](x0, x1) = 4x0 + x1, [2nd](x0) = x0 + 5, [mark#](x0) = x0, [from](x0) = 7x0 + 7, [a__2nd](x0) = x0 + 5 orientation: mark#(cons1(X1,X2)) = 4X1 + X2 + 5 >= X1 = mark#(X1) mark#(cons1(X1,X2)) = 4X1 + X2 + 5 >= X2 = mark#(X2) mark#(s(X)) = X + 0 >= X = mark#(X) mark#(cons(X1,X2)) = 4X1 + X2 >= X1 = mark#(X1) mark#(2nd(X)) = X + 5 >= X + 5 = a__2nd#(mark(X)) a__2nd#(cons(X,X1)) = 4X + X1 + 5 >= 4X + X1 + 5 = a__2nd#(cons1(mark(X),mark(X1))) a__2nd#(cons1(X,cons(Y,Z))) = 4X + 4Y + Z + 5 >= Y = mark#(Y) mark#(2nd(X)) = X + 5 >= X = mark#(X) a__2nd#(cons(X,X1)) = 4X + X1 + 5 >= X = mark#(X) a__2nd#(cons(X,X1)) = 4X + X1 + 5 >= X1 = mark#(X1) a__2nd(cons1(X,cons(Y,Z))) = 4X + 4Y + Z + 5 >= Y = mark(Y) a__2nd(cons(X,X1)) = 4X + X1 + 5 >= 4X + X1 + 5 = a__2nd(cons1(mark(X),mark(X1))) a__from(X) = 7X + 7 >= 7X + 7 = cons(mark(X),from(s(X))) mark(2nd(X)) = X + 5 >= X + 5 = a__2nd(mark(X)) mark(from(X)) = 7X + 7 >= 7X + 7 = a__from(mark(X)) mark(cons(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = cons(mark(X1),X2) mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(cons1(X1,X2)) = 4X1 + X2 + 5 >= 4X1 + X2 + 5 = cons1(mark(X1),mark(X2)) a__2nd(X) = X + 5 >= X + 5 = 2nd(X) a__from(X) = 7X + 7 >= 7X + 7 = from(X) problem: DPs: mark#(cons1(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) mark#(2nd(X)) -> a__2nd#(mark(X)) a__2nd#(cons(X,X1)) -> a__2nd#(cons1(mark(X),mark(X1))) mark#(2nd(X)) -> mark#(X) a__2nd#(cons(X,X1)) -> mark#(X1) TRS: a__2nd(cons1(X,cons(Y,Z))) -> mark(Y) a__2nd(cons(X,X1)) -> a__2nd(cons1(mark(X),mark(X1))) a__from(X) -> cons(mark(X),from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1,X2)) -> cons1(mark(X1),mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) Restore Modifier: DPs: mark#(cons1(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) mark#(2nd(X)) -> a__2nd#(mark(X)) a__2nd#(cons(X,X1)) -> a__2nd#(cons1(mark(X),mark(X1))) mark#(2nd(X)) -> mark#(X) a__2nd#(cons(X,X1)) -> mark#(X1) TRS: a__2nd(cons1(X,cons(Y,Z))) -> mark(Y) a__2nd(cons(X,X1)) -> a__2nd(cons1(mark(X),mark(X1))) a__from(X) -> cons(mark(X),from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1,X2)) -> cons1(mark(X1),mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) SCC Processor: #sccs: 1 #rules: 5 #arcs: 53/36 DPs: mark#(2nd(X)) -> mark#(X) mark#(cons1(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) mark#(2nd(X)) -> a__2nd#(mark(X)) a__2nd#(cons(X,X1)) -> mark#(X1) TRS: a__2nd(cons1(X,cons(Y,Z))) -> mark(Y) a__2nd(cons(X,X1)) -> a__2nd(cons1(mark(X),mark(X1))) a__from(X) -> cons(mark(X),from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1,X2)) -> cons1(mark(X1),mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) Arctic Interpretation Processor: dimension: 1 usable rules: a__2nd(cons1(X,cons(Y,Z))) -> mark(Y) a__2nd(cons(X,X1)) -> a__2nd(cons1(mark(X),mark(X1))) a__from(X) -> cons(mark(X),from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1,X2)) -> cons1(mark(X1),mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) interpretation: [a__from](x0) = x0 + 2, [a__2nd#](x0) = 2x0 + 4, [cons1](x0, x1) = x1 + 0, [mark](x0) = x0, [s](x0) = x0, [cons](x0, x1) = x0 + x1 + 2, [2nd](x0) = 2x0 + 4, [mark#](x0) = x0 + 0, [from](x0) = x0 + 2, [a__2nd](x0) = 2x0 + 4 orientation: mark#(2nd(X)) = 2X + 4 >= X + 0 = mark#(X) mark#(cons1(X1,X2)) = X2 + 0 >= X2 + 0 = mark#(X2) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(2nd(X)) = 2X + 4 >= 2X + 4 = a__2nd#(mark(X)) a__2nd#(cons(X,X1)) = 2X + 2X1 + 4 >= X1 + 0 = mark#(X1) a__2nd(cons1(X,cons(Y,Z))) = 2Y + 2Z + 4 >= Y = mark(Y) a__2nd(cons(X,X1)) = 2X + 2X1 + 4 >= 2X1 + 4 = a__2nd(cons1(mark(X),mark(X1))) a__from(X) = X + 2 >= X + 2 = cons(mark(X),from(s(X))) mark(2nd(X)) = 2X + 4 >= 2X + 4 = a__2nd(mark(X)) mark(from(X)) = X + 2 >= X + 2 = a__from(mark(X)) mark(cons(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = cons(mark(X1),X2) mark(s(X)) = X >= X = s(mark(X)) mark(cons1(X1,X2)) = X2 + 0 >= X2 + 0 = cons1(mark(X1),mark(X2)) a__2nd(X) = 2X + 4 >= 2X + 4 = 2nd(X) a__from(X) = X + 2 >= X + 2 = from(X) problem: DPs: mark#(cons1(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) mark#(2nd(X)) -> a__2nd#(mark(X)) TRS: a__2nd(cons1(X,cons(Y,Z))) -> mark(Y) a__2nd(cons(X,X1)) -> a__2nd(cons1(mark(X),mark(X1))) a__from(X) -> cons(mark(X),from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1,X2)) -> cons1(mark(X1),mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) Restore Modifier: DPs: mark#(cons1(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) mark#(2nd(X)) -> a__2nd#(mark(X)) TRS: a__2nd(cons1(X,cons(Y,Z))) -> mark(Y) a__2nd(cons(X,X1)) -> a__2nd(cons1(mark(X),mark(X1))) a__from(X) -> cons(mark(X),from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1,X2)) -> cons1(mark(X1),mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) SCC Processor: #sccs: 1 #rules: 2 #arcs: 17/9 DPs: mark#(s(X)) -> mark#(X) mark#(cons1(X1,X2)) -> mark#(X2) TRS: a__2nd(cons1(X,cons(Y,Z))) -> mark(Y) a__2nd(cons(X,X1)) -> a__2nd(cons1(mark(X),mark(X1))) a__from(X) -> cons(mark(X),from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1,X2)) -> cons1(mark(X1),mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) Size-Change Termination Processor: DPs: TRS: a__2nd(cons1(X,cons(Y,Z))) -> mark(Y) a__2nd(cons(X,X1)) -> a__2nd(cons1(mark(X),mark(X1))) a__from(X) -> cons(mark(X),from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1,X2)) -> cons1(mark(X1),mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) The DP: mark#(s(X)) -> mark#(X) has the edges: 0 > 0 The DP: mark#(cons1(X1,X2)) -> mark#(X2) has the edges: 0 > 0 Qed