/export/starexec/sandbox/solver/bin/starexec_run_ttt2 /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES Problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = x0, [ok](x0) = x0, [s](x0) = 2x0, [active](x0) = x0, [cons](x0, x1) = 2x0 + 3x1, [top](x0) = x0, [tt] = 0, [isNat](x0) = x0, [zeros] = 0, [length](x0) = 2x0, [proper](x0) = x0, [isNatIList](x0) = 4x0 + 1, [isNatList](x0) = x0, [and](x0, x1) = 2x0 + x1, [nil] = 0, [U11](x0, x1) = x0 + 4x1, [0] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = 4L >= 4L = mark(s(length(L))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 2V1 >= V1 = mark(isNatList(V1)) active(isNat(s(V1))) = 2V1 >= V1 = mark(isNat(V1)) active(isNatIList(V)) = 4V + 1 >= V = mark(isNatList(V)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 8V1 + 12V2 + 1 >= 2V1 + 4V2 + 1 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 2V1 + 3V2 >= 2V1 + V2 = mark(and(isNat(V1),isNatList(V2))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 6L + 4N >= 6L + N = mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) = 2X1 + 3X2 >= 2X1 + 3X2 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = U11(active(X1),X2) active(s(X)) = 2X >= 2X = s(active(X)) active(length(X)) = 2X >= 2X = length(active(X)) active(and(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = and(active(X1),X2) cons(mark(X1),X2) = 2X1 + 3X2 >= 2X1 + 3X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 4X2 >= X1 + 4X2 = mark(U11(X1,X2)) s(mark(X)) = 2X >= 2X = mark(s(X)) length(mark(X)) = 2X >= 2X = mark(length(X)) and(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 2X1 + 3X2 >= 2X1 + 3X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = 2X >= 2X = s(proper(X)) proper(length(X)) = 2X >= 2X = length(proper(X)) proper(and(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = and(proper(X1),proper(X2)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(isNatList(X)) = X >= X = isNatList(proper(X)) proper(isNatIList(X)) = 4X + 1 >= 4X + 1 = isNatIList(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 2X1 + 3X2 >= 2X1 + 3X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 4X2 >= X1 + 4X2 = ok(U11(X1,X2)) s(ok(X)) = 2X >= 2X = ok(s(X)) length(ok(X)) = 2X >= 2X = ok(length(X)) and(ok(X1),ok(X2)) = 2X1 + X2 >= 2X1 + X2 = ok(and(X1,X2)) isNat(ok(X)) = X >= X = ok(isNat(X)) isNatList(ok(X)) = X >= X = ok(isNatList(X)) isNatIList(ok(X)) = 4X + 1 >= 4X + 1 = ok(isNatIList(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = x0, [ok](x0) = x0, [s](x0) = x0, [active](x0) = x0, [cons](x0, x1) = 4x0 + 5x1, [top](x0) = x0, [tt] = 0, [isNat](x0) = x0, [zeros] = 0, [length](x0) = 4x0, [proper](x0) = x0, [isNatIList](x0) = x0, [isNatList](x0) = x0, [and](x0, x1) = 4x0 + 2x1, [nil] = 2, [U11](x0, x1) = 4x0 + 4x1, [0] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = 4L >= 4L = mark(s(length(L))) active(and(tt(),X)) = 2X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 4V1 >= V1 = mark(isNatList(V1)) active(isNat(s(V1))) = V1 >= V1 = mark(isNat(V1)) active(isNatIList(cons(V1,V2))) = 4V1 + 5V2 >= 4V1 + 2V2 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 2 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 4V1 + 5V2 >= 4V1 + 2V2 = mark(and(isNat(V1),isNatList(V2))) active(length(nil())) = 8 >= 0 = mark(0()) active(length(cons(N,L))) = 20L + 16N >= 20L + 8N = mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) = 4X1 + 5X2 >= 4X1 + 5X2 = cons(active(X1),X2) active(U11(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = U11(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = 4X >= 4X = length(active(X)) active(and(X1,X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = and(active(X1),X2) cons(mark(X1),X2) = 4X1 + 5X2 >= 4X1 + 5X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = 4X1 + 4X2 >= 4X1 + 4X2 = mark(U11(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = 4X >= 4X = mark(length(X)) and(mark(X1),X2) = 4X1 + 2X2 >= 4X1 + 2X2 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 4X1 + 5X2 >= 4X1 + 5X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = 4X >= 4X = length(proper(X)) proper(and(X1,X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = and(proper(X1),proper(X2)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(isNatList(X)) = X >= X = isNatList(proper(X)) proper(isNatIList(X)) = X >= X = isNatIList(proper(X)) proper(nil()) = 2 >= 2 = ok(nil()) cons(ok(X1),ok(X2)) = 4X1 + 5X2 >= 4X1 + 5X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = ok(U11(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = 4X >= 4X = ok(length(X)) and(ok(X1),ok(X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = ok(and(X1,X2)) isNat(ok(X)) = X >= X = ok(isNat(X)) isNatList(ok(X)) = X >= X = ok(isNatList(X)) isNatIList(ok(X)) = X >= X = ok(isNatIList(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = x0, [ok](x0) = x0, [s](x0) = x0, [active](x0) = x0, [cons](x0, x1) = 2x0 + 4x1, [top](x0) = x0, [tt] = 0, [isNat](x0) = 2x0, [zeros] = 0, [length](x0) = 4x0 + 2, [proper](x0) = x0, [isNatIList](x0) = x0, [isNatList](x0) = 7x0, [and](x0, x1) = x0 + 2x1, [nil] = 0, [U11](x0, x1) = x0 + 5x1 + 2, [0] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = 5L + 2 >= 4L + 2 = mark(s(length(L))) active(and(tt(),X)) = 2X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 8V1 + 4 >= 7V1 = mark(isNatList(V1)) active(isNat(s(V1))) = 2V1 >= 2V1 = mark(isNat(V1)) active(isNatIList(cons(V1,V2))) = 2V1 + 4V2 >= 2V1 + 2V2 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(cons(V1,V2))) = 14V1 + 28V2 >= 2V1 + 14V2 = mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) = 16L + 8N + 2 >= 12L + 4N + 2 = mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 5X2 + 2 >= X1 + 5X2 + 2 = U11(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = 4X + 2 >= 4X + 2 = length(active(X)) active(and(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = and(active(X1),X2) cons(mark(X1),X2) = 2X1 + 4X2 >= 2X1 + 4X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 5X2 + 2 >= X1 + 5X2 + 2 = mark(U11(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = 4X + 2 >= 4X + 2 = mark(length(X)) and(mark(X1),X2) = X1 + 2X2 >= X1 + 2X2 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = X1 + 5X2 + 2 >= X1 + 5X2 + 2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = 4X + 2 >= 4X + 2 = length(proper(X)) proper(and(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = and(proper(X1),proper(X2)) proper(isNat(X)) = 2X >= 2X = isNat(proper(X)) proper(isNatList(X)) = 7X >= 7X = isNatList(proper(X)) proper(isNatIList(X)) = X >= X = isNatIList(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 5X2 + 2 >= X1 + 5X2 + 2 = ok(U11(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = 4X + 2 >= 4X + 2 = ok(length(X)) and(ok(X1),ok(X2)) = X1 + 2X2 >= X1 + 2X2 = ok(and(X1,X2)) isNat(ok(X)) = 2X >= 2X = ok(isNat(X)) isNatList(ok(X)) = 7X >= 7X = ok(isNatList(X)) isNatIList(ok(X)) = X >= X = ok(isNatIList(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=2 interpretation: [1 0] [mark](x0) = [0 0]x0, [ok](x0) = x0, [1 0] [s](x0) = [0 0]x0, [1 0] [0] [active](x0) = [0 3]x0 + [1], [2 0] [2 2] [0] [cons](x0, x1) = [0 0]x0 + [0 0]x1 + [2], [2 0] [top](x0) = [2 0]x0, [0] [tt] = [0], [1 0] [isNat](x0) = [1 0]x0, [0] [zeros] = [0], [2 0] [0] [length](x0) = [0 0]x0 + [2], [proper](x0) = x0, [1 2] [isNatIList](x0) = [0 0]x0, [1 1] [0] [isNatList](x0) = [0 0]x0 + [1], [1 0] [1 2] [and](x0, x1) = [0 0]x0 + [0 0]x1, [2] [nil] = [1], [1 0] [2 2] [U11](x0, x1) = [0 0]x0 + [0 0]x1, [0] [0] = [0] orientation: [0] [0] active(zeros()) = [1] >= [0] = mark(cons(0(),zeros())) [2 2] [0] [2 0] active(U11(tt(),L)) = [0 0]L + [1] >= [0 0]L = mark(s(length(L))) [1 2] [0] [1 0] active(and(tt(),X)) = [0 0]X + [1] >= [0 0]X = mark(X) [0] [0] active(isNat(0())) = [1] >= [0] = mark(tt()) [1 0] [0] [1 0] active(isNat(s(V1))) = [3 0]V1 + [1] >= [0 0]V1 = mark(isNat(V1)) [2 0] [2 2] [4] [1 0] [1 2] active(isNatIList(cons(V1,V2))) = [0 0]V1 + [0 0]V2 + [1] >= [0 0]V1 + [0 0]V2 = mark(and(isNat(V1),isNatIList(V2))) [2 0] [2 2] [2] [1 0] [1 1] [2] active(isNatList(cons(V1,V2))) = [0 0]V1 + [0 0]V2 + [4] >= [0 0]V1 + [0 0]V2 + [0] = mark(and(isNat(V1),isNatList(V2))) [4 4] [4 0] [0] [3 3] [3 0] active(length(cons(N,L))) = [0 0]L + [0 0]N + [7] >= [0 0]L + [0 0]N = mark(U11(and(isNatList(L),isNat(N)),L)) [2 0] [2 2] [0] [2 0] [2 2] [0] active(cons(X1,X2)) = [0 0]X1 + [0 0]X2 + [7] >= [0 0]X1 + [0 0]X2 + [2] = cons(active(X1),X2) [1 0] [2 2] [0] [1 0] [2 2] active(U11(X1,X2)) = [0 0]X1 + [0 0]X2 + [1] >= [0 0]X1 + [0 0]X2 = U11(active(X1),X2) [1 0] [0] [1 0] active(s(X)) = [0 0]X + [1] >= [0 0]X = s(active(X)) [2 0] [0] [2 0] [0] active(length(X)) = [0 0]X + [7] >= [0 0]X + [2] = length(active(X)) [1 0] [1 2] [0] [1 0] [1 2] active(and(X1,X2)) = [0 0]X1 + [0 0]X2 + [1] >= [0 0]X1 + [0 0]X2 = and(active(X1),X2) [2 0] [2 2] [0] [2 0] [2 2] cons(mark(X1),X2) = [0 0]X1 + [0 0]X2 + [2] >= [0 0]X1 + [0 0]X2 = mark(cons(X1,X2)) [1 0] [2 2] [1 0] [2 2] U11(mark(X1),X2) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = mark(U11(X1,X2)) [1 0] [1 0] s(mark(X)) = [0 0]X >= [0 0]X = mark(s(X)) [2 0] [0] [2 0] length(mark(X)) = [0 0]X + [2] >= [0 0]X = mark(length(X)) [1 0] [1 2] [1 0] [1 2] and(mark(X1),X2) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = mark(and(X1,X2)) [0] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [2 0] [2 2] [0] [2 0] [2 2] [0] proper(cons(X1,X2)) = [0 0]X1 + [0 0]X2 + [2] >= [0 0]X1 + [0 0]X2 + [2] = cons(proper(X1),proper(X2)) [0] [0] proper(0()) = [0] >= [0] = ok(0()) [1 0] [2 2] [1 0] [2 2] proper(U11(X1,X2)) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = U11(proper(X1),proper(X2)) [0] [0] proper(tt()) = [0] >= [0] = ok(tt()) [1 0] [1 0] proper(s(X)) = [0 0]X >= [0 0]X = s(proper(X)) [2 0] [0] [2 0] [0] proper(length(X)) = [0 0]X + [2] >= [0 0]X + [2] = length(proper(X)) [1 0] [1 2] [1 0] [1 2] proper(and(X1,X2)) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = and(proper(X1),proper(X2)) [1 0] [1 0] proper(isNat(X)) = [1 0]X >= [1 0]X = isNat(proper(X)) [1 1] [0] [1 1] [0] proper(isNatList(X)) = [0 0]X + [1] >= [0 0]X + [1] = isNatList(proper(X)) [1 2] [1 2] proper(isNatIList(X)) = [0 0]X >= [0 0]X = isNatIList(proper(X)) [2] [2] proper(nil()) = [1] >= [1] = ok(nil()) [2 0] [2 2] [0] [2 0] [2 2] [0] cons(ok(X1),ok(X2)) = [0 0]X1 + [0 0]X2 + [2] >= [0 0]X1 + [0 0]X2 + [2] = ok(cons(X1,X2)) [1 0] [2 2] [1 0] [2 2] U11(ok(X1),ok(X2)) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = ok(U11(X1,X2)) [1 0] [1 0] s(ok(X)) = [0 0]X >= [0 0]X = ok(s(X)) [2 0] [0] [2 0] [0] length(ok(X)) = [0 0]X + [2] >= [0 0]X + [2] = ok(length(X)) [1 0] [1 2] [1 0] [1 2] and(ok(X1),ok(X2)) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = ok(and(X1,X2)) [1 0] [1 0] isNat(ok(X)) = [1 0]X >= [1 0]X = ok(isNat(X)) [1 1] [0] [1 1] [0] isNatList(ok(X)) = [0 0]X + [1] >= [0 0]X + [1] = ok(isNatList(X)) [1 2] [1 2] isNatIList(ok(X)) = [0 0]X >= [0 0]X = ok(isNatIList(X)) [2 0] [2 0] top(mark(X)) = [2 0]X >= [2 0]X = top(proper(X)) [2 0] [2 0] top(ok(X)) = [2 0]X >= [2 0]X = top(active(X)) problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [mark](x0) = [0 1 0]x0 [0 0 0] , [ok](x0) = x0 , [1 0 0] [s](x0) = [0 0 0]x0 [1 0 0] , [active](x0) = x0 , [1 0 0] [1 1 1] [0] [cons](x0, x1) = [0 0 0]x0 + [1 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [1 0 0] [0] [top](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [0] [tt] = [0] [0], [1 0 0] [isNat](x0) = [0 0 0]x0 [0 0 0] , [0] [zeros] = [0] [0], [1 1 0] [1] [length](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [proper](x0) = x0 , [1 0 0] [1] [isNatIList](x0) = [1 0 0]x0 + [0] [0 0 0] [0], [1 0 1] [0] [isNatList](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [1 0 0] [and](x0, x1) = [0 0 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [0] [nil] = [0] [0], [1 0 0] [1 1 0] [1] [U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [1 0 0] [0], [0] [0] = [0] [0] orientation: [0] [0] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [0] [0] [1 1 0] [1] [1 1 0] [1] active(U11(tt(),L)) = [0 0 0]L + [0] >= [0 0 0]L + [0] = mark(s(length(L))) [1 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] active(and(tt(),X)) = [0 1 0]X >= [0 1 0]X = mark(X) [0 0 0] [0 0 0] [0] [0] active(isNat(0())) = [0] >= [0] = mark(tt()) [0] [0] [1 0 0] [1 0 0] active(isNat(s(V1))) = [0 0 0]V1 >= [0 0 0]V1 = mark(isNat(V1)) [0 0 0] [0 0 0] [1 0 0] [1 1 1] [1] [1 0 0] [1 0 1] [0] active(isNatList(cons(V1,V2))) = [0 0 0]V1 + [0 0 0]V2 + [1] >= [0 0 0]V1 + [0 0 0]V2 + [1] = mark(and(isNat(V1),isNatList(V2))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [2 1 1] [1 0 0] [1] [2 1 1] [1 0 0] [1] active(length(cons(N,L))) = [0 0 0]L + [0 0 0]N + [0] >= [0 0 0]L + [0 0 0]N + [0] = mark(U11(and(isNatList(L),isNat(N)),L)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] active(cons(X1,X2)) = [0 0 0]X1 + [1 0 0]X2 + [0] >= [0 0 0]X1 + [1 0 0]X2 + [0] = cons(active(X1),X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 1 0] [1] [1 0 0] [1 1 0] [1] active(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = U11(active(X1),X2) [0 0 0] [1 0 0] [0] [0 0 0] [1 0 0] [0] [1 0 0] [1 0 0] active(s(X)) = [0 0 0]X >= [0 0 0]X = s(active(X)) [1 0 0] [1 0 0] [1 1 0] [1] [1 1 0] [1] active(length(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(active(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(and(X1,X2)) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = and(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] cons(mark(X1),X2) = [0 0 0]X1 + [1 0 0]X2 + [0] >= [0 0 0]X1 + [1 0 0]X2 = mark(cons(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1 0 0] [1 1 0] [1] [1 0 0] [1 1 0] [1] U11(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(U11(X1,X2)) [0 0 0] [1 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(s(X)) [1 0 0] [0 0 0] [1 1 0] [1] [1 1 0] [1] length(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(length(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] and(mark(X1),X2) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = mark(and(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [0] [0] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] proper(cons(X1,X2)) = [0 0 0]X1 + [1 0 0]X2 + [0] >= [0 0 0]X1 + [1 0 0]X2 + [0] = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 0 0] [1 1 0] [1] [1 0 0] [1 1 0] [1] proper(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = U11(proper(X1),proper(X2)) [0 0 0] [1 0 0] [0] [0 0 0] [1 0 0] [0] [0] [0] proper(tt()) = [0] >= [0] = ok(tt()) [0] [0] [1 0 0] [1 0 0] proper(s(X)) = [0 0 0]X >= [0 0 0]X = s(proper(X)) [1 0 0] [1 0 0] [1 1 0] [1] [1 1 0] [1] proper(length(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(and(X1,X2)) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = and(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(isNat(X)) = [0 0 0]X >= [0 0 0]X = isNat(proper(X)) [0 0 0] [0 0 0] [1 0 1] [0] [1 0 1] [0] proper(isNatList(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = isNatList(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] proper(isNatIList(X)) = [1 0 0]X + [0] >= [1 0 0]X + [0] = isNatIList(proper(X)) [0 0 0] [0] [0 0 0] [0] [0] [0] proper(nil()) = [0] >= [0] = ok(nil()) [0] [0] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [1 0 0]X2 + [0] >= [0 0 0]X1 + [1 0 0]X2 + [0] = ok(cons(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 1 0] [1] [1 0 0] [1 1 0] [1] U11(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = ok(U11(X1,X2)) [0 0 0] [1 0 0] [0] [0 0 0] [1 0 0] [0] [1 0 0] [1 0 0] s(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(s(X)) [1 0 0] [1 0 0] [1 1 0] [1] [1 1 0] [1] length(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(length(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] and(ok(X1),ok(X2)) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = ok(and(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] isNat(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(isNat(X)) [0 0 0] [0 0 0] [1 0 1] [0] [1 0 1] [0] isNatList(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(isNatList(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] isNatIList(ok(X)) = [1 0 0]X + [0] >= [1 0 0]X + [0] = ok(isNatIList(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] top(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = top(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] top(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = top(active(X)) [0 0 0] [0] [0 0 0] [0] problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(isNat(V1)) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = x0, [ok](x0) = x0, [s](x0) = x0, [active](x0) = x0, [cons](x0, x1) = 4x0 + 4x1, [top](x0) = x0, [tt] = 1, [isNat](x0) = 2x0 + 1, [zeros] = 0, [length](x0) = 2x0 + 4, [proper](x0) = x0, [isNatIList](x0) = 4x0, [isNatList](x0) = x0, [and](x0, x1) = x0 + x1, [nil] = 0, [U11](x0, x1) = 4x0 + 2x1, [0] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = 2L + 4 >= 2L + 4 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= X = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(s(V1))) = 2V1 + 1 >= 2V1 + 1 = mark(isNat(V1)) active(length(cons(N,L))) = 8L + 8N + 4 >= 6L + 8N + 4 = mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = cons(active(X1),X2) active(U11(X1,X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = U11(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = 2X + 4 >= 2X + 4 = length(active(X)) active(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(active(X1),X2) cons(mark(X1),X2) = 4X1 + 4X2 >= 4X1 + 4X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = 4X1 + 2X2 >= 4X1 + 2X2 = mark(U11(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = 2X + 4 >= 2X + 4 = mark(length(X)) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = 2X + 4 >= 2X + 4 = length(proper(X)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(isNat(X)) = 2X + 1 >= 2X + 1 = isNat(proper(X)) proper(isNatList(X)) = X >= X = isNatList(proper(X)) proper(isNatIList(X)) = 4X >= 4X = isNatIList(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = ok(U11(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = 2X + 4 >= 2X + 4 = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNat(ok(X)) = 2X + 1 >= 2X + 1 = ok(isNat(X)) isNatList(ok(X)) = X >= X = ok(isNatList(X)) isNatIList(ok(X)) = 4X >= 4X = ok(isNatIList(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(isNat(V1)) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [active](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [1 0 1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [1 0 1] [1 0 0] , [1 0 0] [0] [top](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [0] [tt] = [0] [1], [1 0 0] [0] [isNat](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [0] [zeros] = [0] [0], [1 0 1] [length](x0) = [0 0 0]x0 [1 0 1] , [1 0 0] [proper](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [isNatIList](x0) = [0 0 0]x0 [1 0 0] , [1 0 0] [isNatList](x0) = [0 0 0]x0 [1 0 1] , [1 0 0] [1 0 0] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [nil] = [0] [0], [1 0 1] [1 0 1] [U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [1 0 0] [1 0 0] , [0] [0] = [0] [0] orientation: [0] [0] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [0] [0] [1 0 1] [1] [1 0 1] active(U11(tt(),L)) = [0 0 0]L + [0] >= [0 0 0]L = mark(s(length(L))) [1 0 0] [0] [0 0 0] [0] [0] active(isNat(0())) = [0] >= [0] = mark(tt()) [1] [1] [1 0 0] [0] [1 0 0] [0] active(isNat(s(V1))) = [0 0 0]V1 + [0] >= [0 0 0]V1 + [0] = mark(isNat(V1)) [0 0 0] [1] [0 0 0] [1] [2 0 1] [2 0 1] [2 0 1] [1 0 0] active(length(cons(N,L))) = [0 0 0]L + [0 0 0]N >= [0 0 0]L + [0 0 0]N = mark(U11(and(isNatList(L),isNat(N)),L)) [2 0 1] [2 0 1] [2 0 0] [1 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] active(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(active(X1),X2) [1 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] active(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U11(active(X1),X2) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(s(X)) = [0 0 0]X >= [0 0 0]X = s(active(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] active(length(X)) = [0 0 0]X >= [0 0 0]X = length(active(X)) [1 0 1] [1 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(cons(X1,X2)) [1 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] U11(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(U11(X1,X2)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(s(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] length(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(length(X)) [1 0 1] [1 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] and(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(and(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [0] [0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] proper(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [1 0 1] [1 0 0] [1 0 1] [1 0 0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] proper(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U11(proper(X1),proper(X2)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [0] [0] proper(tt()) = [0] >= [0] = ok(tt()) [1] [1] [1 0 0] [1 0 0] proper(s(X)) = [0 0 0]X >= [0 0 0]X = s(proper(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] proper(length(X)) = [0 0 0]X >= [0 0 0]X = length(proper(X)) [1 0 1] [1 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] proper(isNat(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = isNat(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] proper(isNatList(X)) = [0 0 0]X >= [0 0 0]X = isNatList(proper(X)) [1 0 1] [1 0 1] [1 0 0] [1 0 0] proper(isNatIList(X)) = [0 0 0]X >= [0 0 0]X = isNatIList(proper(X)) [1 0 0] [1 0 0] [0] [0] proper(nil()) = [0] >= [0] = ok(nil()) [0] [0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [1 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] U11(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(U11(X1,X2)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] s(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(s(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] length(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(length(X)) [1 0 1] [1 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] and(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(and(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] isNat(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(isNat(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] isNatList(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(isNatList(X)) [1 0 1] [1 0 1] [1 0 0] [1 0 0] isNatIList(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(isNatIList(X)) [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [0] top(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] top(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(active(X)) [0 0 0] [1] [0 0 0] [1] problem: active(zeros()) -> mark(cons(0(),zeros())) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(isNat(V1)) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = x0, [ok](x0) = x0, [s](x0) = x0 + 1, [active](x0) = x0, [cons](x0, x1) = x0 + 2x1, [top](x0) = x0, [tt] = 0, [isNat](x0) = x0 + 1, [zeros] = 0, [length](x0) = 4x0 + 4, [proper](x0) = x0, [isNatIList](x0) = x0 + 3, [isNatList](x0) = 6x0, [and](x0, x1) = x0 + x1 + 1, [nil] = 0, [U11](x0, x1) = x0 + 2x1 + 2, [0] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(s(V1))) = V1 + 2 >= V1 + 1 = mark(isNat(V1)) active(length(cons(N,L))) = 8L + 4N + 4 >= 8L + N + 4 = mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = U11(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = 4X + 4 >= 4X + 4 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + 2X2 >= X1 + 2X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = mark(U11(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = 4X + 4 >= 4X + 4 = mark(length(X)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = 4X + 4 >= 4X + 4 = length(proper(X)) proper(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(isNatList(X)) = 6X >= 6X = isNatList(proper(X)) proper(isNatIList(X)) = X + 3 >= X + 3 = isNatIList(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 2X2 >= X1 + 2X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = ok(U11(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = 4X + 4 >= 4X + 4 = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(and(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) isNatList(ok(X)) = 6X >= 6X = ok(isNatList(X)) isNatIList(ok(X)) = X + 3 >= X + 3 = ok(isNatIList(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(zeros()) -> mark(cons(0(),zeros())) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = x0, [ok](x0) = x0, [s](x0) = 4x0 + 2, [active](x0) = x0, [cons](x0, x1) = 6x0 + 2x1, [top](x0) = x0 + 7, [tt] = 0, [isNat](x0) = x0, [zeros] = 0, [length](x0) = 4x0 + 1, [proper](x0) = x0, [isNatIList](x0) = 4x0 + 6, [isNatList](x0) = 2x0, [and](x0, x1) = x0 + x1, [nil] = 2, [U11](x0, x1) = 2x0 + x1, [0] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(length(cons(N,L))) = 8L + 24N + 1 >= 5L + 2N = mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) = 6X1 + 2X2 >= 6X1 + 2X2 = cons(active(X1),X2) active(U11(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = U11(active(X1),X2) active(s(X)) = 4X + 2 >= 4X + 2 = s(active(X)) active(length(X)) = 4X + 1 >= 4X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(active(X1),X2) cons(mark(X1),X2) = 6X1 + 2X2 >= 6X1 + 2X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = mark(U11(X1,X2)) s(mark(X)) = 4X + 2 >= 4X + 2 = mark(s(X)) length(mark(X)) = 4X + 1 >= 4X + 1 = mark(length(X)) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 6X1 + 2X2 >= 6X1 + 2X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = 4X + 2 >= 4X + 2 = s(proper(X)) proper(length(X)) = 4X + 1 >= 4X + 1 = length(proper(X)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(isNatList(X)) = 2X >= 2X = isNatList(proper(X)) proper(isNatIList(X)) = 4X + 6 >= 4X + 6 = isNatIList(proper(X)) proper(nil()) = 2 >= 2 = ok(nil()) cons(ok(X1),ok(X2)) = 6X1 + 2X2 >= 6X1 + 2X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 2X1 + X2 >= 2X1 + X2 = ok(U11(X1,X2)) s(ok(X)) = 4X + 2 >= 4X + 2 = ok(s(X)) length(ok(X)) = 4X + 1 >= 4X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNat(ok(X)) = X >= X = ok(isNat(X)) isNatList(ok(X)) = 2X >= 2X = ok(isNatList(X)) isNatIList(ok(X)) = 4X + 6 >= 4X + 6 = ok(isNatIList(X)) top(mark(X)) = X + 7 >= X + 7 = top(proper(X)) top(ok(X)) = X + 7 >= X + 7 = top(active(X)) problem: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [mark](x0) = [0 1 1]x0 + [1] [0 0 0] [0], [1 0 0] [ok](x0) = [0 0 1]x0 [0 1 0] , [s](x0) = x0 , [1 0 0] [active](x0) = [0 1 0]x0 [0 0 0] , [1 1 1] [1 0 0] [cons](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] , [1 0 0] [0] [top](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1] [tt] = [0] [0], [1 0 0] [0] [isNat](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [0] [zeros] = [1] [0], [1 1 1] [length](x0) = [0 1 0]x0 [0 0 1] , [1 0 0] [proper](x0) = [0 0 1]x0 [0 1 0] , [1 0 0] [isNatIList](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [isNatList](x0) = [1 0 0]x0 [1 0 0] , [1 1 1] [1 0 0] [and](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] , [0] [nil] = [1] [0], [1 0 0] [U11](x0, x1) = x0 + [0 0 0]x1 [0 0 0] , [0] [0] = [0] [0] orientation: [0] [0] active(zeros()) = [1] >= [1] = mark(cons(0(),zeros())) [0] [0] [1 1 1] [1 0 0] [1 1 0] [1 0 0] active(cons(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = cons(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(U11(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = U11(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(s(X)) = [0 1 0]X >= [0 1 0]X = s(active(X)) [0 0 0] [0 0 0] [1 1 1] [1 1 0] active(length(X)) = [0 1 0]X >= [0 1 0]X = length(active(X)) [0 0 0] [0 0 0] [1 1 1] [1 0 0] [1 1 0] [1 0 0] active(and(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = and(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 0 0] [1] [1 1 1] [1 0 0] [0] cons(mark(X1),X2) = [0 1 1]X1 + [0 0 0]X2 + [1] >= [0 1 1]X1 + [0 0 0]X2 + [1] = mark(cons(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] U11(mark(X1),X2) = [0 1 1]X1 + [0 0 0]X2 + [1] >= [0 1 1]X1 + [0 0 0]X2 + [1] = mark(U11(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] s(mark(X)) = [0 1 1]X + [1] >= [0 1 1]X + [1] = mark(s(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1] [1 1 1] [0] length(mark(X)) = [0 1 1]X + [1] >= [0 1 1]X + [1] = mark(length(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 0 0] [1] [1 1 1] [1 0 0] [0] and(mark(X1),X2) = [0 1 1]X1 + [0 0 0]X2 + [1] >= [0 1 1]X1 + [0 0 0]X2 + [1] = mark(and(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [1] [1] [1 1 1] [1 0 0] [1 1 1] [1 0 0] proper(cons(X1,X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 1]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [0 1 0] [0 0 0] [0 1 0] [0 0 0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(U11(X1,X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 1]X1 + [0 0 0]X2 = U11(proper(X1),proper(X2)) [0 1 0] [0 0 0] [0 1 0] [0 0 0] [1] [1] proper(tt()) = [0] >= [0] = ok(tt()) [0] [0] [1 0 0] [1 0 0] proper(s(X)) = [0 0 1]X >= [0 0 1]X = s(proper(X)) [0 1 0] [0 1 0] [1 1 1] [1 1 1] proper(length(X)) = [0 0 1]X >= [0 0 1]X = length(proper(X)) [0 1 0] [0 1 0] [1 1 1] [1 0 0] [1 1 1] [1 0 0] proper(and(X1,X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 1]X1 + [0 0 0]X2 = and(proper(X1),proper(X2)) [0 1 0] [0 0 0] [0 1 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] proper(isNat(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = isNat(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] proper(isNatList(X)) = [1 0 0]X >= [1 0 0]X = isNatList(proper(X)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(isNatIList(X)) = [0 0 0]X >= [0 0 0]X = isNatIList(proper(X)) [0 0 0] [0 0 0] [0] [0] proper(nil()) = [0] >= [0] = ok(nil()) [1] [1] [1 1 1] [1 0 0] [1 1 1] [1 0 0] cons(ok(X1),ok(X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 1]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 1 0] [0 0 0] [0 1 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U11(ok(X1),ok(X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 1]X1 + [0 0 0]X2 = ok(U11(X1,X2)) [0 1 0] [0 0 0] [0 1 0] [0 0 0] [1 0 0] [1 0 0] s(ok(X)) = [0 0 1]X >= [0 0 1]X = ok(s(X)) [0 1 0] [0 1 0] [1 1 1] [1 1 1] length(ok(X)) = [0 0 1]X >= [0 0 1]X = ok(length(X)) [0 1 0] [0 1 0] [1 1 1] [1 0 0] [1 1 1] [1 0 0] and(ok(X1),ok(X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 1]X1 + [0 0 0]X2 = ok(and(X1,X2)) [0 1 0] [0 0 0] [0 1 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] isNat(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(isNat(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] isNatList(ok(X)) = [1 0 0]X >= [1 0 0]X = ok(isNatList(X)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] isNatIList(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(isNatIList(X)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] top(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] top(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(active(X)) [0 0 0] [1] [0 0 0] [1] problem: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [mark](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [ok](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [s](x0) = [1 1 0]x0 [0 0 0] , [active](x0) = x0 , [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [1 0 0]x1 [0 0 0] [0 1 0] , [1 0 0] [top](x0) = [0 0 0]x0 [0 0 0] , [0] [tt] = [0] [0], [1 0 0] [isNat](x0) = [0 0 0]x0 [0 0 0] , [0] [zeros] = [1] [0], [1 0 0] [0] [length](x0) = [1 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [proper](x0) = [0 1 0]x0 [0 1 1] , [1 0 0] [isNatIList](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [isNatList](x0) = [0 0 0]x0 [1 0 0] , [1 0 0] [1 0 0] [and](x0, x1) = [0 0 0]x0 + [1 0 0]x1 [0 0 0] [0 0 0] , [0] [nil] = [1] [0], [1 1 0] [1 0 0] [U11](x0, x1) = [1 1 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [0] = [0] [0] orientation: [0] [0] active(zeros()) = [1] >= [1] = mark(cons(0(),zeros())) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(cons(X1,X2)) = [0 0 0]X1 + [1 0 0]X2 >= [0 0 0]X1 + [1 0 0]X2 = cons(active(X1),X2) [0 0 0] [0 1 0] [0 0 0] [0 1 0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] active(U11(X1,X2)) = [1 1 0]X1 + [0 0 0]X2 >= [1 1 0]X1 + [0 0 0]X2 = U11(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(s(X)) = [1 1 0]X >= [1 1 0]X = s(active(X)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] active(length(X)) = [1 0 0]X + [0] >= [1 0 0]X + [0] = length(active(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(and(X1,X2)) = [0 0 0]X1 + [1 0 0]X2 >= [0 0 0]X1 + [1 0 0]X2 = and(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [0] U11(mark(X1),X2) = [1 1 0]X1 + [0 0 0]X2 + [1] >= [1 1 0]X1 + [0 0 0]X2 + [1] = mark(U11(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] s(mark(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = mark(s(X)) [0 0 0] [0] [0 0 0] [0] [0] [0] proper(zeros()) = [1] >= [1] = ok(zeros()) [1] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(cons(X1,X2)) = [0 0 0]X1 + [1 0 0]X2 >= [0 0 0]X1 + [1 0 0]X2 = cons(proper(X1),proper(X2)) [0 0 0] [1 1 0] [0 0 0] [0 1 0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] proper(U11(X1,X2)) = [1 1 0]X1 + [0 0 0]X2 >= [1 1 0]X1 + [0 0 0]X2 = U11(proper(X1),proper(X2)) [1 1 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(tt()) = [0] >= [0] = ok(tt()) [0] [0] [1 0 0] [1 0 0] proper(s(X)) = [1 1 0]X >= [1 1 0]X = s(proper(X)) [1 1 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] proper(length(X)) = [1 0 0]X + [0] >= [1 0 0]X + [0] = length(proper(X)) [1 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(and(X1,X2)) = [0 0 0]X1 + [1 0 0]X2 >= [0 0 0]X1 + [1 0 0]X2 = and(proper(X1),proper(X2)) [0 0 0] [1 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(isNat(X)) = [0 0 0]X >= [0 0 0]X = isNat(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(isNatList(X)) = [0 0 0]X >= [0 0 0]X = isNatList(proper(X)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(isNatIList(X)) = [0 0 0]X >= [0 0 0]X = isNatIList(proper(X)) [0 0 0] [0 0 0] [0] [0] proper(nil()) = [1] >= [1] = ok(nil()) [1] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [1 0 0]X2 >= [0 0 0]X1 + [1 0 0]X2 = ok(cons(X1,X2)) [0 0 0] [0 1 0] [0 0 0] [0 0 0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] U11(ok(X1),ok(X2)) = [1 1 0]X1 + [0 0 0]X2 >= [1 1 0]X1 + [0 0 0]X2 = ok(U11(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] s(ok(X)) = [1 1 0]X >= [1 1 0]X = ok(s(X)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] length(ok(X)) = [1 0 0]X + [0] >= [1 0 0]X = ok(length(X)) [0 0 0] [1] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] and(ok(X1),ok(X2)) = [0 0 0]X1 + [1 0 0]X2 >= [0 0 0]X1 + [1 0 0]X2 = ok(and(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] isNat(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(isNat(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] isNatList(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(isNatList(X)) [1 0 0] [0 0 0] [1 0 0] [1 0 0] isNatIList(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(isNatIList(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X)) [0 0 0] [0 0 0] problem: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) s(mark(X)) -> mark(s(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [mark](x0) = [0 0 0]x0 [0 1 1] , [1 0 0] [ok](x0) = [0 0 0]x0 [0 1 1] , [0] [s](x0) = x0 + [0] [1], [1 1 0] [active](x0) = [0 0 1]x0 [0 0 1] , [1 0 0] [cons](x0, x1) = x0 + [0 0 0]x1 [0 0 0] , [1 0 1] [top](x0) = [0 0 0]x0 [0 0 1] , [0] [tt] = [1] [0], [1 0 0] [isNat](x0) = [0 0 0]x0 [0 0 0] , [0] [zeros] = [1] [0], [length](x0) = x0 , [1 0 0] [0] [proper](x0) = [1 0 0]x0 + [1] [0 1 1] [0], [isNatIList](x0) = x0 , [0] [isNatList](x0) = x0 + [0] [1], [1 0 0] [and](x0, x1) = x0 + [0 0 0]x1 [0 0 0] , [0] [nil] = [1] [0], [1 0 0] [U11](x0, x1) = x0 + [0 0 0]x1 [0 0 0] , [0] [0] = [0] [0] orientation: [1] [0] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [0] [0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] active(cons(X1,X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 1]X1 + [0 0 0]X2 = cons(active(X1),X2) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] active(U11(X1,X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 1]X1 + [0 0 0]X2 = U11(active(X1),X2) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 1 0] [0] [1 1 0] [0] active(s(X)) = [0 0 1]X + [1] >= [0 0 1]X + [0] = s(active(X)) [0 0 1] [1] [0 0 1] [1] [1 1 0] [1 1 0] active(length(X)) = [0 0 1]X >= [0 0 1]X = length(active(X)) [0 0 1] [0 0 1] [1 1 0] [1 0 0] [1 1 0] [1 0 0] active(and(X1,X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 1]X1 + [0 0 0]X2 = and(active(X1),X2) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 0 0] [0] [1 0 0] [0] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(s(X)) [0 1 1] [1] [0 1 1] [1] [0] [0] proper(zeros()) = [1] >= [0] = ok(zeros()) [1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] proper(cons(X1,X2)) = [1 0 0]X1 + [1 0 0]X2 + [1] >= [1 0 0]X1 + [0 0 0]X2 + [1] = cons(proper(X1),proper(X2)) [0 1 1] [0 0 0] [0] [0 1 1] [0 0 0] [0] [0] [0] proper(0()) = [1] >= [0] = ok(0()) [0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] proper(U11(X1,X2)) = [1 0 0]X1 + [1 0 0]X2 + [1] >= [1 0 0]X1 + [0 0 0]X2 + [1] = U11(proper(X1),proper(X2)) [0 1 1] [0 0 0] [0] [0 1 1] [0 0 0] [0] [0] [0] proper(tt()) = [1] >= [0] = ok(tt()) [1] [1] [1 0 0] [0] [1 0 0] [0] proper(s(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = s(proper(X)) [0 1 1] [1] [0 1 1] [1] [1 0 0] [0] [1 0 0] [0] proper(length(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = length(proper(X)) [0 1 1] [0] [0 1 1] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] proper(and(X1,X2)) = [1 0 0]X1 + [1 0 0]X2 + [1] >= [1 0 0]X1 + [0 0 0]X2 + [1] = and(proper(X1),proper(X2)) [0 1 1] [0 0 0] [0] [0 1 1] [0 0 0] [0] [1 0 0] [0] [1 0 0] proper(isNat(X)) = [1 0 0]X + [1] >= [0 0 0]X = isNat(proper(X)) [0 0 0] [0] [0 0 0] [1 0 0] [0] [1 0 0] [0] proper(isNatList(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = isNatList(proper(X)) [0 1 1] [1] [0 1 1] [1] [1 0 0] [0] [1 0 0] [0] proper(isNatIList(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = isNatIList(proper(X)) [0 1 1] [0] [0 1 1] [0] [0] [0] proper(nil()) = [1] >= [0] = ok(nil()) [1] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 1 1] [0 0 0] [0 1 1] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U11(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(U11(X1,X2)) [0 1 1] [0 0 0] [0 1 1] [0 0 0] [1 0 0] [0] [1 0 0] [0] s(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(s(X)) [0 1 1] [1] [0 1 1] [1] [1 0 0] [1 0 0] length(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(length(X)) [0 1 1] [0 1 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] and(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(and(X1,X2)) [0 1 1] [0 0 0] [0 1 1] [0 0 0] [1 0 0] [1 0 0] isNat(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(isNat(X)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] isNatList(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(isNatList(X)) [0 1 1] [1] [0 1 1] [1] [1 0 0] [1 0 0] isNatIList(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(isNatIList(X)) [0 1 1] [0 1 1] [1 1 1] [1 1 1] top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X)) [0 1 1] [0 1 1] [1 1 1] [1 1 1] top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X)) [0 1 1] [0 0 1] problem: active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) s(mark(X)) -> mark(s(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = 4x0 + 4, [ok](x0) = x0 + 3, [s](x0) = x0, [active](x0) = x0, [cons](x0, x1) = x0 + x1 + 4, [top](x0) = x0 + 2, [tt] = 2, [isNat](x0) = x0 + 1, [zeros] = 5, [length](x0) = 2x0 + 1, [proper](x0) = 2x0 + 1, [isNatIList](x0) = x0 + 2, [isNatList](x0) = 4x0 + 5, [and](x0, x1) = 2x0 + x1 + 6, [nil] = 2, [U11](x0, x1) = x0 + 6x1 + 6, [0] = 4 orientation: active(cons(X1,X2)) = X1 + X2 + 4 >= X1 + X2 + 4 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 6X2 + 6 >= X1 + 6X2 + 6 = U11(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = 2X + 1 >= 2X + 1 = length(active(X)) active(and(X1,X2)) = 2X1 + X2 + 6 >= 2X1 + X2 + 6 = and(active(X1),X2) s(mark(X)) = 4X + 4 >= 4X + 4 = mark(s(X)) proper(zeros()) = 11 >= 8 = ok(zeros()) proper(cons(X1,X2)) = 2X1 + 2X2 + 9 >= 2X1 + 2X2 + 6 = cons(proper(X1),proper(X2)) proper(0()) = 9 >= 7 = ok(0()) proper(U11(X1,X2)) = 2X1 + 12X2 + 13 >= 2X1 + 12X2 + 13 = U11(proper(X1),proper(X2)) proper(tt()) = 5 >= 5 = ok(tt()) proper(s(X)) = 2X + 1 >= 2X + 1 = s(proper(X)) proper(length(X)) = 4X + 3 >= 4X + 3 = length(proper(X)) proper(and(X1,X2)) = 4X1 + 2X2 + 13 >= 4X1 + 2X2 + 9 = and(proper(X1),proper(X2)) proper(isNat(X)) = 2X + 3 >= 2X + 2 = isNat(proper(X)) proper(isNatList(X)) = 8X + 11 >= 8X + 9 = isNatList(proper(X)) proper(isNatIList(X)) = 2X + 5 >= 2X + 3 = isNatIList(proper(X)) proper(nil()) = 5 >= 5 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 + 10 >= X1 + X2 + 7 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 6X2 + 27 >= X1 + 6X2 + 9 = ok(U11(X1,X2)) s(ok(X)) = X + 3 >= X + 3 = ok(s(X)) length(ok(X)) = 2X + 7 >= 2X + 4 = ok(length(X)) and(ok(X1),ok(X2)) = 2X1 + X2 + 15 >= 2X1 + X2 + 9 = ok(and(X1,X2)) isNat(ok(X)) = X + 4 >= X + 4 = ok(isNat(X)) isNatList(ok(X)) = 4X + 17 >= 4X + 8 = ok(isNatList(X)) isNatIList(ok(X)) = X + 5 >= X + 5 = ok(isNatIList(X)) top(mark(X)) = 4X + 6 >= 2X + 3 = top(proper(X)) top(ok(X)) = X + 5 >= X + 2 = top(active(X)) problem: active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) s(mark(X)) -> mark(s(X)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) s(ok(X)) -> ok(s(X)) isNat(ok(X)) -> ok(isNat(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = 2x0 + 2, [ok](x0) = x0 + 4, [s](x0) = x0, [active](x0) = 4x0, [cons](x0, x1) = x0 + x1, [tt] = 4, [isNat](x0) = 2x0 + 4, [length](x0) = 2x0, [proper](x0) = 2x0, [isNatIList](x0) = 4x0, [and](x0, x1) = 4x0 + 3x1 + 7, [nil] = 4, [U11](x0, x1) = x0 + x1 orientation: active(cons(X1,X2)) = 4X1 + 4X2 >= 4X1 + X2 = cons(active(X1),X2) active(U11(X1,X2)) = 4X1 + 4X2 >= 4X1 + X2 = U11(active(X1),X2) active(s(X)) = 4X >= 4X = s(active(X)) active(length(X)) = 8X >= 8X = length(active(X)) active(and(X1,X2)) = 16X1 + 12X2 + 28 >= 16X1 + 3X2 + 7 = and(active(X1),X2) s(mark(X)) = 2X + 2 >= 2X + 2 = mark(s(X)) proper(U11(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = U11(proper(X1),proper(X2)) proper(tt()) = 8 >= 8 = ok(tt()) proper(s(X)) = 2X >= 2X = s(proper(X)) proper(length(X)) = 4X >= 4X = length(proper(X)) proper(nil()) = 8 >= 8 = ok(nil()) s(ok(X)) = X + 4 >= X + 4 = ok(s(X)) isNat(ok(X)) = 2X + 12 >= 2X + 8 = ok(isNat(X)) isNatIList(ok(X)) = 4X + 16 >= 4X + 4 = ok(isNatIList(X)) problem: active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) s(mark(X)) -> mark(s(X)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) s(ok(X)) -> ok(s(X)) DP Processor: DPs: active#(cons(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(length(X)) -> active#(X) s#(mark(X)) -> s#(X) proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) s#(ok(X)) -> s#(X) TRS: active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) s(mark(X)) -> mark(s(X)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) s(ok(X)) -> ok(s(X)) TDG Processor: DPs: active#(cons(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(length(X)) -> active#(X) s#(mark(X)) -> s#(X) proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) s#(ok(X)) -> s#(X) TRS: active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) s(mark(X)) -> mark(s(X)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) s(ok(X)) -> ok(s(X)) graph: proper#(s(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> proper#(X2) proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X) proper#(s(X)) -> s#(proper(X)) -> s#(mark(X)) -> s#(X) proper#(length(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) proper#(length(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(length(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2)) -> proper#(X2) s#(ok(X)) -> s#(X) -> s#(ok(X)) -> s#(X) s#(ok(X)) -> s#(X) -> s#(mark(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(ok(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(mark(X)) -> s#(X) active#(s(X)) -> s#(active(X)) -> s#(ok(X)) -> s#(X) active#(s(X)) -> s#(active(X)) -> s#(mark(X)) -> s#(X) active#(s(X)) -> active#(X) -> active#(length(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(s(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(U11(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(length(X)) -> active#(X) -> active#(length(X)) -> active#(X) active#(length(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(length(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(length(X)) -> active#(X) -> active#(U11(X1,X2)) -> active#(X1) active#(length(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> active#(X1) -> active#(length(X)) -> active#(X) active#(U11(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(U11(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(U11(X1,X2)) -> active#(X1) -> active#(U11(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(length(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(U11(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) SCC Processor: #sccs: 3 #rules: 10 #arcs: 48/144 DPs: active#(s(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> active#(X1) active#(length(X)) -> active#(X) TRS: active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) s(mark(X)) -> mark(s(X)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) s(ok(X)) -> ok(s(X)) Subterm Criterion Processor: simple projection: pi(active#) = 0 problem: DPs: TRS: active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) s(mark(X)) -> mark(s(X)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) s(ok(X)) -> ok(s(X)) Qed DPs: proper#(s(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) TRS: active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) s(mark(X)) -> mark(s(X)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) s(ok(X)) -> ok(s(X)) Subterm Criterion Processor: simple projection: pi(proper#) = 0 problem: DPs: TRS: active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) s(mark(X)) -> mark(s(X)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) s(ok(X)) -> ok(s(X)) Qed DPs: s#(mark(X)) -> s#(X) s#(ok(X)) -> s#(X) TRS: active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) s(mark(X)) -> mark(s(X)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) s(ok(X)) -> ok(s(X)) Subterm Criterion Processor: simple projection: pi(s#) = 0 problem: DPs: TRS: active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) s(mark(X)) -> mark(s(X)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) s(ok(X)) -> ok(s(X)) Qed