YES Problem: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__head(cons(X,XS)) -> mark(X) a__tail(cons(X,XS)) -> mark(XS) mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__nats() -> nats() a__incr(X) -> incr(X) a__pairs() -> pairs() a__odds() -> odds() a__head(X) -> head(X) a__tail(X) -> tail(X) Proof: Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [0] [tail](x0) = [0 0 1]x0 + [1] [0 0 1] [0], [1 0 1] [1] [head](x0) = [0 1 0]x0 + [0] [1 1 0] [1], [1] [pairs] = [0] [0], [1 1 0] [0] [a__tail](x0) = [0 0 1]x0 + [1] [0 0 1] [0], [1 0 1] [1] [a__head](x0) = [0 1 0]x0 + [0] [1 1 0] [1], [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [mark](x0) = x0 , [1 0 1] [a__incr](x0) = [0 0 0]x0 [0 0 0] , [1] [a__odds] = [1] [0], [1] [odds] = [1] [0], [1] [a__pairs] = [0] [0], [1 0 0] [1 0 0] [cons](x0, x1) = [0 1 1]x0 + [0 0 1]x1 [0 0 1] [0 1 1] , [1 0 1] [incr](x0) = [0 0 0]x0 [0 0 0] , [1] [nats] = [0] [0], [0] [0] = [0] [0], [1] [a__nats] = [0] [0] orientation: [1] [1] a__nats() = [0] >= [0] = cons(0(),incr(nats())) [0] [0] [1] [1] a__pairs() = [0] >= [0] = cons(0(),incr(odds())) [0] [0] [1] [1] a__odds() = [1] >= [0] = a__incr(a__pairs()) [0] [0] [1 0 1] [1 1 1] [1 0 0] [1 0 1] a__incr(cons(X,XS)) = [0 0 0]X + [0 0 0]XS >= [0 0 0]X + [0 0 0]XS = cons(s(mark(X)),incr(XS)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 1 1] [1] a__head(cons(X,XS)) = [0 1 1]X + [0 0 1]XS + [0] >= X = mark(X) [1 1 1] [1 0 1] [1] [1 1 1] [1 0 1] [0] a__tail(cons(X,XS)) = [0 0 1]X + [0 1 1]XS + [1] >= XS = mark(XS) [0 0 1] [0 1 1] [0] [1] [1] mark(nats()) = [0] >= [0] = a__nats() [0] [0] [1 0 1] [1 0 1] mark(incr(X)) = [0 0 0]X >= [0 0 0]X = a__incr(mark(X)) [0 0 0] [0 0 0] [1] [1] mark(pairs()) = [0] >= [0] = a__pairs() [0] [0] [1] [1] mark(odds()) = [1] >= [1] = a__odds() [0] [0] [1 0 1] [1] [1 0 1] [1] mark(head(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = a__head(mark(X)) [1 1 0] [1] [1 1 0] [1] [1 1 0] [0] [1 1 0] [0] mark(tail(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = a__tail(mark(X)) [0 0 1] [0] [0 0 1] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] mark(cons(X1,X2)) = [0 1 1]X1 + [0 0 1]X2 >= [0 1 1]X1 + [0 0 1]X2 = cons(mark(X1),X2) [0 0 1] [0 1 1] [0 0 1] [0 1 1] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 0 0] [1 0 0] mark(s(X)) = [0 0 0]X >= [0 0 0]X = s(mark(X)) [0 0 0] [0 0 0] [1] [1] a__nats() = [0] >= [0] = nats() [0] [0] [1 0 1] [1 0 1] a__incr(X) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 0] [0 0 0] [1] [1] a__pairs() = [0] >= [0] = pairs() [0] [0] [1] [1] a__odds() = [1] >= [1] = odds() [0] [0] [1 0 1] [1] [1 0 1] [1] a__head(X) = [0 1 0]X + [0] >= [0 1 0]X + [0] = head(X) [1 1 0] [1] [1 1 0] [1] [1 1 0] [0] [1 1 0] [0] a__tail(X) = [0 0 1]X + [1] >= [0 0 1]X + [1] = tail(X) [0 0 1] [0] [0 0 1] [0] problem: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__tail(cons(X,XS)) -> mark(XS) mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__nats() -> nats() a__incr(X) -> incr(X) a__pairs() -> pairs() a__odds() -> odds() a__head(X) -> head(X) a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1] [tail](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [head](x0) = [0 0 0]x0 [0 0 1] , [0] [pairs] = [0] [0], [1 0 0] [1] [a__tail](x0) = [0 0 0]x0 + [0] [0 1 1] [0], [a__head](x0) = x0 , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [mark](x0) = [0 0 0]x0 [1 0 0] , [1 1 0] [a__incr](x0) = [0 0 0]x0 [0 0 1] , [0] [a__odds] = [0] [0], [0] [odds] = [0] [0], [0] [a__pairs] = [0] [0], [1 0 0] [1 0 0] [cons](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 0] [1 0 0] , [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [0] [nats] = [0] [0], [0] [0] = [0] [0], [0] [a__nats] = [0] [0] orientation: [0] [0] a__nats() = [0] >= [0] = cons(0(),incr(nats())) [0] [0] [0] [0] a__pairs() = [0] >= [0] = cons(0(),incr(odds())) [0] [0] [0] [0] a__odds() = [0] >= [0] = a__incr(a__pairs()) [0] [0] [1 1 0] [1 0 0] [1 0 0] [1 0 0] a__incr(cons(X,XS)) = [0 0 0]X + [0 0 0]XS >= [0 0 0]X + [0 0 0]XS = cons(s(mark(X)),incr(XS)) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] a__tail(cons(X,XS)) = [0 0 0]X + [0 0 0]XS + [0] >= [0 0 0]XS = mark(XS) [0 1 0] [1 0 0] [0] [1 0 0] [0] [0] mark(nats()) = [0] >= [0] = a__nats() [0] [0] [1 0 0] [1 0 0] mark(incr(X)) = [0 0 0]X >= [0 0 0]X = a__incr(mark(X)) [1 0 0] [1 0 0] [0] [0] mark(pairs()) = [0] >= [0] = a__pairs() [0] [0] [0] [0] mark(odds()) = [0] >= [0] = a__odds() [0] [0] [1 0 0] [1 0 0] mark(head(X)) = [0 0 0]X >= [0 0 0]X = a__head(mark(X)) [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1] mark(tail(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__tail(mark(X)) [1 0 0] [1] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(mark(X1),X2) [1 0 0] [1 0 0] [0 0 0] [1 0 0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 0 0] [1 0 0] mark(s(X)) = [0 0 0]X >= [0 0 0]X = s(mark(X)) [1 0 0] [0 0 0] [0] [0] a__nats() = [0] >= [0] = nats() [0] [0] [1 1 0] [1 0 0] a__incr(X) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 1] [0 0 0] [0] [0] a__pairs() = [0] >= [0] = pairs() [0] [0] [0] [0] a__odds() = [0] >= [0] = odds() [0] [0] [1 0 0] a__head(X) = X >= [0 0 0]X = head(X) [0 0 1] [1 0 0] [1] [1 0 0] [1] a__tail(X) = [0 0 0]X + [0] >= [0 0 0]X + [0] = tail(X) [0 1 1] [0] [0 0 0] [0] problem: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__nats() -> nats() a__incr(X) -> incr(X) a__pairs() -> pairs() a__odds() -> odds() a__head(X) -> head(X) a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [tail](x0) = [0 0 1]x0 [0 1 0] , [0] [head](x0) = x0 + [1] [1], [1] [pairs] = [0] [0], [1 0 1] [a__tail](x0) = [0 0 1]x0 [0 1 0] , [0] [a__head](x0) = x0 + [1] [1], [s](x0) = x0 , [1 0 1] [mark](x0) = [0 0 1]x0 [0 1 0] , [1 0 1] [a__incr](x0) = [0 1 1]x0 [0 1 1] , [1] [a__odds] = [0] [0], [1] [odds] = [0] [0], [1] [a__pairs] = [0] [0], [1 0 1] [1 0 0] [cons](x0, x1) = [0 0 1]x0 + [0 0 0]x1 [0 1 0] [0 0 0] , [1 0 0] [incr](x0) = [0 1 1]x0 [0 1 1] , [0] [nats] = [0] [0], [0] [0] = [0] [0], [0] [a__nats] = [0] [0] orientation: [0] [0] a__nats() = [0] >= [0] = cons(0(),incr(nats())) [0] [0] [1] [1] a__pairs() = [0] >= [0] = cons(0(),incr(odds())) [0] [0] [1] [1] a__odds() = [0] >= [0] = a__incr(a__pairs()) [0] [0] [1 1 1] [1 0 0] [1 1 1] [1 0 0] a__incr(cons(X,XS)) = [0 1 1]X + [0 0 0]XS >= [0 1 0]X + [0 0 0]XS = cons(s(mark(X)),incr(XS)) [0 1 1] [0 0 0] [0 0 1] [0 0 0] [0] [0] mark(nats()) = [0] >= [0] = a__nats() [0] [0] [1 1 1] [1 1 1] mark(incr(X)) = [0 1 1]X >= [0 1 1]X = a__incr(mark(X)) [0 1 1] [0 1 1] [1] [1] mark(pairs()) = [0] >= [0] = a__pairs() [0] [0] [1] [1] mark(odds()) = [0] >= [0] = a__odds() [0] [0] [1 0 1] [1] [1 0 1] [0] mark(head(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = a__head(mark(X)) [0 1 0] [1] [0 1 0] [1] [1 1 1] [1 1 1] mark(tail(X)) = [0 1 0]X >= [0 1 0]X = a__tail(mark(X)) [0 0 1] [0 0 1] [1 1 1] [1 0 0] [1 1 1] [1 0 0] mark(cons(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = cons(mark(X1),X2) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 0 1] [1 0 1] mark(s(X)) = [0 0 1]X >= [0 0 1]X = s(mark(X)) [0 1 0] [0 1 0] [0] [0] a__nats() = [0] >= [0] = nats() [0] [0] [1 0 1] [1 0 0] a__incr(X) = [0 1 1]X >= [0 1 1]X = incr(X) [0 1 1] [0 1 1] [1] [1] a__pairs() = [0] >= [0] = pairs() [0] [0] [1] [1] a__odds() = [0] >= [0] = odds() [0] [0] [0] [0] a__head(X) = X + [1] >= X + [1] = head(X) [1] [1] [1 0 1] [1 0 1] a__tail(X) = [0 0 1]X >= [0 0 1]X = tail(X) [0 1 0] [0 1 0] problem: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__nats() -> nats() a__incr(X) -> incr(X) a__pairs() -> pairs() a__odds() -> odds() a__head(X) -> head(X) a__tail(X) -> tail(X) DP Processor: DPs: a__odds#() -> a__pairs#() a__odds#() -> a__incr#(a__pairs()) a__incr#(cons(X,XS)) -> mark#(X) mark#(nats()) -> a__nats#() mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(pairs()) -> a__pairs#() mark#(odds()) -> a__odds#() mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> a__tail#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__nats() -> nats() a__incr(X) -> incr(X) a__pairs() -> pairs() a__odds() -> odds() a__head(X) -> head(X) a__tail(X) -> tail(X) TDG Processor: DPs: a__odds#() -> a__pairs#() a__odds#() -> a__incr#(a__pairs()) a__incr#(cons(X,XS)) -> mark#(X) mark#(nats()) -> a__nats#() mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(pairs()) -> a__pairs#() mark#(odds()) -> a__odds#() mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> a__tail#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__nats() -> nats() a__incr(X) -> incr(X) a__pairs() -> pairs() a__odds() -> odds() a__head(X) -> head(X) a__tail(X) -> tail(X) graph: mark#(tail(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(odds()) -> a__odds#() mark#(tail(X)) -> mark#(X) -> mark#(pairs()) -> a__pairs#() mark#(tail(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(nats()) -> a__nats#() 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#(tail(X)) -> a__tail#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(odds()) -> a__odds#() mark#(s(X)) -> mark#(X) -> mark#(pairs()) -> a__pairs#() mark#(s(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(nats()) -> a__nats#() mark#(odds()) -> a__odds#() -> a__odds#() -> a__incr#(a__pairs()) mark#(odds()) -> a__odds#() -> a__odds#() -> a__pairs#() 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#(tail(X)) -> a__tail#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(odds()) -> a__odds#() mark#(cons(X1,X2)) -> mark#(X1) -> mark#(pairs()) -> a__pairs#() mark#(cons(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(nats()) -> a__nats#() mark#(incr(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(odds()) -> a__odds#() mark#(incr(X)) -> mark#(X) -> mark#(pairs()) -> a__pairs#() mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(nats()) -> a__nats#() mark#(incr(X)) -> a__incr#(mark(X)) -> a__incr#(cons(X,XS)) -> mark#(X) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(odds()) -> a__odds#() a__incr#(cons(X,XS)) -> mark#(X) -> mark#(pairs()) -> a__pairs#() a__incr#(cons(X,XS)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(nats()) -> a__nats#() a__odds#() -> a__incr#(a__pairs()) -> a__incr#(cons(X,XS)) -> mark#(X) SCC Processor: #sccs: 1 #rules: 8 #arcs: 49/144 DPs: mark#(tail(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) mark#(odds()) -> a__odds#() a__odds#() -> a__incr#(a__pairs()) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__nats() -> nats() a__incr(X) -> incr(X) a__pairs() -> pairs() a__odds() -> odds() a__head(X) -> head(X) a__tail(X) -> tail(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 0, [a__incr#](x0) = x0 + 0, [a__odds#] = 0, [tail](x0) = x0 + 0, [head](x0) = 2x0, [pairs] = 0, [a__tail](x0) = x0 + 0, [a__head](x0) = 2x0 + 0, [s](x0) = x0, [mark](x0) = 1x0 + 0, [a__incr](x0) = 1x0 + 0, [a__odds] = 2, [odds] = 1, [a__pairs] = 0, [cons](x0, x1) = x0, [incr](x0) = 1x0 + 0, [nats] = 4, [0] = 0, [a__nats] = 4 orientation: mark#(tail(X)) = X + 0 >= X + 0 = mark#(X) mark#(incr(X)) = 1X + 0 >= X + 0 = mark#(X) mark#(incr(X)) = 1X + 0 >= 1X + 0 = a__incr#(mark(X)) a__incr#(cons(X,XS)) = X + 0 >= X + 0 = mark#(X) mark#(odds()) = 1 >= 0 = a__odds#() a__odds#() = 0 >= 0 = a__incr#(a__pairs()) mark#(cons(X1,X2)) = X1 + 0 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) a__nats() = 4 >= 0 = cons(0(),incr(nats())) a__pairs() = 0 >= 0 = cons(0(),incr(odds())) a__odds() = 2 >= 1 = a__incr(a__pairs()) a__incr(cons(X,XS)) = 1X + 0 >= 1X + 0 = cons(s(mark(X)),incr(XS)) mark(nats()) = 5 >= 4 = a__nats() mark(incr(X)) = 2X + 1 >= 2X + 1 = a__incr(mark(X)) mark(pairs()) = 1 >= 0 = a__pairs() mark(odds()) = 2 >= 2 = a__odds() mark(tail(X)) = 1X + 1 >= 1X + 0 = a__tail(mark(X)) mark(cons(X1,X2)) = 1X1 + 0 >= 1X1 + 0 = cons(mark(X1),X2) mark(0()) = 1 >= 0 = 0() mark(s(X)) = 1X + 0 >= 1X + 0 = s(mark(X)) a__nats() = 4 >= 4 = nats() a__incr(X) = 1X + 0 >= 1X + 0 = incr(X) a__pairs() = 0 >= 0 = pairs() a__odds() = 2 >= 1 = odds() a__head(X) = 2X + 0 >= 2X = head(X) a__tail(X) = X + 0 >= X + 0 = tail(X) problem: DPs: mark#(tail(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) a__odds#() -> a__incr#(a__pairs()) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__nats() -> nats() a__incr(X) -> incr(X) a__pairs() -> pairs() a__odds() -> odds() a__head(X) -> head(X) a__tail(X) -> tail(X) SCC Processor: #sccs: 1 #rules: 6 #arcs: 33/49 DPs: mark#(tail(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__nats() -> nats() a__incr(X) -> incr(X) a__pairs() -> pairs() a__odds() -> odds() a__head(X) -> head(X) a__tail(X) -> tail(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 0, [a__incr#](x0) = x0 + 0, [tail](x0) = 5x0, [head](x0) = x0 + 1, [pairs] = 3, [a__tail](x0) = 5x0, [a__head](x0) = x0 + 3, [s](x0) = x0 + 0, [mark](x0) = x0, [a__incr](x0) = 1x0 + 4, [a__odds] = 4, [odds] = 4, [a__pairs] = 3, [cons](x0, x1) = 3x0 + 0, [incr](x0) = 1x0 + 4, [nats] = 3, [0] = 0, [a__nats] = 3 orientation: mark#(tail(X)) = 5X + 0 >= X + 0 = mark#(X) mark#(incr(X)) = 1X + 4 >= X + 0 = mark#(X) mark#(incr(X)) = 1X + 4 >= X + 0 = a__incr#(mark(X)) a__incr#(cons(X,XS)) = 3X + 0 >= X + 0 = mark#(X) mark#(cons(X1,X2)) = 3X1 + 0 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) a__nats() = 3 >= 3 = cons(0(),incr(nats())) a__pairs() = 3 >= 3 = cons(0(),incr(odds())) a__odds() = 4 >= 4 = a__incr(a__pairs()) a__incr(cons(X,XS)) = 4X + 4 >= 3X + 3 = cons(s(mark(X)),incr(XS)) mark(nats()) = 3 >= 3 = a__nats() mark(incr(X)) = 1X + 4 >= 1X + 4 = a__incr(mark(X)) mark(pairs()) = 3 >= 3 = a__pairs() mark(odds()) = 4 >= 4 = a__odds() mark(tail(X)) = 5X >= 5X = a__tail(mark(X)) mark(cons(X1,X2)) = 3X1 + 0 >= 3X1 + 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) a__nats() = 3 >= 3 = nats() a__incr(X) = 1X + 4 >= 1X + 4 = incr(X) a__pairs() = 3 >= 3 = pairs() a__odds() = 4 >= 4 = odds() a__head(X) = X + 3 >= X + 1 = head(X) a__tail(X) = 5X >= 5X = tail(X) problem: DPs: mark#(tail(X)) -> mark#(X) a__incr#(cons(X,XS)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__nats() -> nats() a__incr(X) -> incr(X) a__pairs() -> pairs() a__odds() -> odds() a__head(X) -> head(X) a__tail(X) -> tail(X) SCC Processor: #sccs: 1 #rules: 3 #arcs: 26/16 DPs: mark#(tail(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__nats() -> nats() a__incr(X) -> incr(X) a__pairs() -> pairs() a__odds() -> odds() a__head(X) -> head(X) a__tail(X) -> tail(X) Subterm Criterion Processor: simple projection: pi(mark#) = 0 problem: DPs: TRS: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__nats() -> nats() a__incr(X) -> incr(X) a__pairs() -> pairs() a__odds() -> odds() a__head(X) -> head(X) a__tail(X) -> tail(X) Qed