YES Problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) a__isNatIList(X) -> isNatIList(X) Proof: Matrix Interpretation Processor: dim=1 interpretation: [U11](x0, x1) = 2x0 + 2x1, [tt] = 0, [nil] = 0, [s](x0) = x0, [0] = 0, [cons](x0, x1) = x0 + 3x1, [isNatList](x0) = 2x0, [and](x0, x1) = x0 + x1, [isNat](x0) = x0, [a__U11](x0, x1) = 2x0 + 2x1, [a__isNat](x0) = x0, [a__zeros] = 0, [a__length](x0) = 2x0, [isNatIList](x0) = 2x0 + 4, [a__isNatList](x0) = 2x0, [length](x0) = 2x0, [a__and](x0, x1) = x0 + x1, [a__isNatIList](x0) = 2x0 + 4, [mark](x0) = x0, [zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),L) = 2L >= 2L = s(a__length(mark(L))) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 2V1 >= 2V1 = a__isNatList(V1) a__isNat(s(V1)) = V1 >= V1 = a__isNat(V1) a__isNatIList(V) = 2V + 4 >= 2V = a__isNatList(V) a__isNatIList(zeros()) = 4 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 2V1 + 6V2 + 4 >= V1 + 2V2 + 4 = a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 2V1 + 6V2 >= V1 + 2V2 = a__and(a__isNat(V1),isNatList(V2)) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 6L + 2N >= 6L + 2N = a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = a__U11(mark(X1),X2) mark(length(X)) = 2X >= 2X = a__length(mark(X)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNat(X)) = X >= X = a__isNat(X) mark(isNatList(X)) = 2X >= 2X = a__isNatList(X) mark(isNatIList(X)) = 2X + 4 >= 2X + 4 = a__isNatIList(X) mark(cons(X1,X2)) = X1 + 3X2 >= X1 + 3X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = 2X1 + 2X2 >= 2X1 + 2X2 = U11(X1,X2) a__length(X) = 2X >= 2X = length(X) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNat(X) = X >= X = isNat(X) a__isNatList(X) = 2X >= 2X = isNatList(X) a__isNatIList(X) = 2X + 4 >= 2X + 4 = isNatIList(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) a__isNatIList(X) -> isNatIList(X) Matrix Interpretation Processor: dim=1 interpretation: [U11](x0, x1) = x0 + x1, [tt] = 0, [nil] = 2, [s](x0) = x0, [0] = 0, [cons](x0, x1) = 4x0 + 4x1, [isNatList](x0) = x0, [and](x0, x1) = 2x0 + x1, [isNat](x0) = 2x0, [a__U11](x0, x1) = x0 + x1, [a__isNat](x0) = 2x0, [a__zeros] = 0, [a__length](x0) = x0, [isNatIList](x0) = 2x0 + 2, [a__isNatList](x0) = x0, [length](x0) = x0, [a__and](x0, x1) = 2x0 + x1, [a__isNatIList](x0) = 2x0 + 2, [mark](x0) = x0, [zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),L) = L >= L = s(a__length(mark(L))) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 2V1 >= V1 = a__isNatList(V1) a__isNat(s(V1)) = 2V1 >= 2V1 = a__isNat(V1) a__isNatIList(cons(V1,V2)) = 8V1 + 8V2 + 2 >= 4V1 + 2V2 + 2 = a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(nil()) = 2 >= 0 = tt() a__isNatList(cons(V1,V2)) = 4V1 + 4V2 >= 4V1 + V2 = a__and(a__isNat(V1),isNatList(V2)) a__length(nil()) = 2 >= 0 = 0() a__length(cons(N,L)) = 4L + 4N >= 3L + 2N = a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = X1 + X2 >= X1 + X2 = a__U11(mark(X1),X2) mark(length(X)) = X >= X = a__length(mark(X)) mark(and(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = a__and(mark(X1),X2) mark(isNat(X)) = 2X >= 2X = a__isNat(X) mark(isNatList(X)) = X >= X = a__isNatList(X) mark(isNatIList(X)) = 2X + 2 >= 2X + 2 = a__isNatIList(X) mark(cons(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 2 >= 2 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) a__length(X) = X >= X = length(X) a__and(X1,X2) = 2X1 + X2 >= 2X1 + X2 = and(X1,X2) a__isNat(X) = 2X >= 2X = isNat(X) a__isNatList(X) = X >= X = isNatList(X) a__isNatIList(X) = 2X + 2 >= 2X + 2 = isNatIList(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) a__isNatIList(X) -> isNatIList(X) Matrix Interpretation Processor: dim=1 interpretation: [U11](x0, x1) = 4x0 + 4x1 + 2, [tt] = 0, [nil] = 2, [s](x0) = x0, [0] = 0, [cons](x0, x1) = 2x0 + 4x1, [isNatList](x0) = x0, [and](x0, x1) = x0 + x1, [isNat](x0) = x0, [a__U11](x0, x1) = 4x0 + 4x1 + 2, [a__isNat](x0) = x0, [a__zeros] = 0, [a__length](x0) = 4x0 + 2, [isNatIList](x0) = 2x0, [a__isNatList](x0) = x0, [length](x0) = 4x0 + 2, [a__and](x0, x1) = x0 + x1, [a__isNatIList](x0) = 2x0, [mark](x0) = x0, [zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),L) = 4L + 2 >= 4L + 2 = s(a__length(mark(L))) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 4V1 + 2 >= V1 = a__isNatList(V1) a__isNat(s(V1)) = V1 >= V1 = a__isNat(V1) a__isNatIList(cons(V1,V2)) = 4V1 + 8V2 >= V1 + 2V2 = a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(cons(V1,V2)) = 2V1 + 4V2 >= V1 + V2 = a__and(a__isNat(V1),isNatList(V2)) a__length(cons(N,L)) = 16L + 8N + 2 >= 8L + 4N + 2 = a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 4X1 + 4X2 + 2 >= 4X1 + 4X2 + 2 = a__U11(mark(X1),X2) mark(length(X)) = 4X + 2 >= 4X + 2 = a__length(mark(X)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNat(X)) = X >= X = a__isNat(X) mark(isNatList(X)) = X >= X = a__isNatList(X) mark(isNatIList(X)) = 2X >= 2X = a__isNatIList(X) mark(cons(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 2 >= 2 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = 4X1 + 4X2 + 2 >= 4X1 + 4X2 + 2 = U11(X1,X2) a__length(X) = 4X + 2 >= 4X + 2 = length(X) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNat(X) = X >= X = isNat(X) a__isNatList(X) = X >= X = isNatList(X) a__isNatIList(X) = 2X >= 2X = isNatIList(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) a__isNatIList(X) -> isNatIList(X) Matrix Interpretation Processor: dim=1 interpretation: [U11](x0, x1) = x0 + 3x1, [tt] = 0, [nil] = 4, [s](x0) = x0, [0] = 0, [cons](x0, x1) = x0 + 5x1, [isNatList](x0) = 2x0, [and](x0, x1) = x0 + 2x1, [isNat](x0) = x0, [a__U11](x0, x1) = x0 + 6x1, [a__isNat](x0) = 2x0, [a__zeros] = 0, [a__length](x0) = 3x0, [isNatIList](x0) = x0, [a__isNatList](x0) = 2x0, [length](x0) = 3x0, [a__and](x0, x1) = x0 + 2x1, [a__isNatIList](x0) = 2x0, [mark](x0) = 2x0, [zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),L) = 6L >= 6L = s(a__length(mark(L))) a__and(tt(),X) = 2X >= 2X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(V1)) = 2V1 >= 2V1 = a__isNat(V1) a__isNatIList(cons(V1,V2)) = 2V1 + 10V2 >= 2V1 + 2V2 = a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(cons(V1,V2)) = 2V1 + 10V2 >= 2V1 + 4V2 = a__and(a__isNat(V1),isNatList(V2)) a__length(cons(N,L)) = 15L + 3N >= 8L + 2N = a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 2X1 + 6X2 >= 2X1 + 6X2 = a__U11(mark(X1),X2) mark(length(X)) = 6X >= 6X = a__length(mark(X)) mark(and(X1,X2)) = 2X1 + 4X2 >= 2X1 + 2X2 = a__and(mark(X1),X2) mark(isNat(X)) = 2X >= 2X = a__isNat(X) mark(isNatList(X)) = 4X >= 2X = a__isNatList(X) mark(isNatIList(X)) = 2X >= 2X = a__isNatIList(X) mark(cons(X1,X2)) = 2X1 + 10X2 >= 2X1 + 5X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 2X >= 2X = s(mark(X)) mark(nil()) = 8 >= 4 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 + 6X2 >= X1 + 3X2 = U11(X1,X2) a__length(X) = 3X >= 3X = length(X) a__and(X1,X2) = X1 + 2X2 >= X1 + 2X2 = and(X1,X2) a__isNat(X) = 2X >= X = isNat(X) a__isNatList(X) = 2X >= 2X = isNatList(X) a__isNatIList(X) = 2X >= X = isNatIList(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) a__isNatIList(X) -> isNatIList(X) Matrix Interpretation Processor: dim=1 interpretation: [U11](x0, x1) = x0 + 2x1, [tt] = 0, [s](x0) = 2x0, [0] = 0, [cons](x0, x1) = 2x0 + 6x1, [isNatList](x0) = x0, [and](x0, x1) = 2x0 + 2x1, [isNat](x0) = x0, [a__U11](x0, x1) = x0 + 4x1, [a__isNat](x0) = x0, [a__zeros] = 0, [a__length](x0) = x0, [isNatIList](x0) = 4x0 + 2, [a__isNatList](x0) = x0, [length](x0) = x0, [a__and](x0, x1) = 2x0 + 2x1, [a__isNatIList](x0) = 4x0 + 4, [mark](x0) = 2x0, [zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),L) = 4L >= 4L = s(a__length(mark(L))) a__and(tt(),X) = 2X >= 2X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(V1)) = 2V1 >= V1 = a__isNat(V1) a__isNatIList(cons(V1,V2)) = 8V1 + 24V2 + 4 >= 2V1 + 8V2 + 4 = a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(cons(V1,V2)) = 2V1 + 6V2 >= 2V1 + 2V2 = a__and(a__isNat(V1),isNatList(V2)) a__length(cons(N,L)) = 6L + 2N >= 6L + 2N = a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = a__U11(mark(X1),X2) mark(length(X)) = 2X >= 2X = a__length(mark(X)) mark(and(X1,X2)) = 4X1 + 4X2 >= 4X1 + 2X2 = a__and(mark(X1),X2) mark(isNat(X)) = 2X >= X = a__isNat(X) mark(isNatList(X)) = 2X >= X = a__isNatList(X) mark(isNatIList(X)) = 8X + 4 >= 4X + 4 = a__isNatIList(X) mark(cons(X1,X2)) = 4X1 + 12X2 >= 4X1 + 6X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 4X >= 4X = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 + 4X2 >= X1 + 2X2 = U11(X1,X2) a__length(X) = X >= X = length(X) a__and(X1,X2) = 2X1 + 2X2 >= 2X1 + 2X2 = and(X1,X2) a__isNat(X) = X >= X = isNat(X) a__isNatList(X) = X >= X = isNatList(X) a__isNatIList(X) = 4X + 4 >= 4X + 2 = isNatIList(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) Matrix Interpretation Processor: dim=1 interpretation: [U11](x0, x1) = x0 + 2x1 + 1, [tt] = 0, [s](x0) = x0, [0] = 0, [cons](x0, x1) = 2x0 + 3x1, [isNatList](x0) = x0, [and](x0, x1) = x0 + x1, [isNat](x0) = x0, [a__U11](x0, x1) = x0 + 2x1 + 1, [a__isNat](x0) = 2x0, [a__zeros] = 0, [a__length](x0) = x0 + 1, [isNatIList](x0) = 3x0, [a__isNatList](x0) = x0, [length](x0) = x0 + 1, [a__and](x0, x1) = x0 + 2x1, [a__isNatIList](x0) = 2x0, [mark](x0) = 2x0, [zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),L) = 2L + 1 >= 2L + 1 = s(a__length(mark(L))) a__and(tt(),X) = 2X >= 2X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(V1)) = 2V1 >= 2V1 = a__isNat(V1) a__isNatIList(cons(V1,V2)) = 4V1 + 6V2 >= 2V1 + 6V2 = a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(cons(V1,V2)) = 2V1 + 3V2 >= 2V1 + 2V2 = a__and(a__isNat(V1),isNatList(V2)) a__length(cons(N,L)) = 3L + 2N + 1 >= 3L + 2N + 1 = a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 2X1 + 4X2 + 2 >= 2X1 + 2X2 + 1 = a__U11(mark(X1),X2) mark(length(X)) = 2X + 2 >= 2X + 1 = a__length(mark(X)) mark(and(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = a__and(mark(X1),X2) mark(isNat(X)) = 2X >= 2X = a__isNat(X) mark(isNatList(X)) = 2X >= X = a__isNatList(X) mark(isNatIList(X)) = 6X >= 2X = a__isNatIList(X) mark(cons(X1,X2)) = 4X1 + 6X2 >= 4X1 + 3X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 2X >= 2X = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 + 2X2 + 1 >= X1 + 2X2 + 1 = U11(X1,X2) a__length(X) = X + 1 >= X + 1 = length(X) a__and(X1,X2) = X1 + 2X2 >= X1 + X2 = and(X1,X2) a__isNat(X) = 2X >= X = isNat(X) a__isNatList(X) = X >= X = isNatList(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) -> a__zeros() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) Matrix Interpretation Processor: dim=1 interpretation: [U11](x0, x1) = x0 + x1, [tt] = 0, [s](x0) = x0, [0] = 0, [cons](x0, x1) = 2x0 + 5x1, [isNatList](x0) = 2x0, [and](x0, x1) = 4x0 + x1, [isNat](x0) = x0, [a__U11](x0, x1) = x0 + 7x1 + 1, [a__isNat](x0) = x0, [a__zeros] = 0, [a__length](x0) = 3x0 + 1, [isNatIList](x0) = 5x0, [a__isNatList](x0) = 2x0, [length](x0) = 2x0 + 1, [a__and](x0, x1) = 4x0 + 2x1, [a__isNatIList](x0) = 2x0, [mark](x0) = 2x0, [zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),L) = 7L + 1 >= 6L + 1 = s(a__length(mark(L))) a__and(tt(),X) = 2X >= 2X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(V1)) = V1 >= V1 = a__isNat(V1) a__isNatIList(cons(V1,V2)) = 4V1 + 10V2 >= 4V1 + 10V2 = a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(cons(V1,V2)) = 4V1 + 10V2 >= 4V1 + 4V2 = a__and(a__isNat(V1),isNatList(V2)) a__length(cons(N,L)) = 15L + 6N + 1 >= 15L + 2N + 1 = a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(and(X1,X2)) = 8X1 + 2X2 >= 8X1 + 2X2 = a__and(mark(X1),X2) mark(isNat(X)) = 2X >= X = a__isNat(X) mark(isNatList(X)) = 4X >= 2X = a__isNatList(X) mark(isNatIList(X)) = 10X >= 2X = a__isNatIList(X) mark(cons(X1,X2)) = 4X1 + 10X2 >= 4X1 + 5X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 2X >= 2X = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 + 7X2 + 1 >= X1 + X2 = U11(X1,X2) a__length(X) = 3X + 1 >= 2X + 1 = length(X) a__and(X1,X2) = 4X1 + 2X2 >= 4X1 + X2 = and(X1,X2) a__isNat(X) = X >= X = isNat(X) a__isNatList(X) = 2X >= 2X = isNatList(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) -> a__zeros() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) Matrix Interpretation Processor: dim=1 interpretation: [tt] = 0, [s](x0) = x0, [0] = 0, [cons](x0, x1) = 4x0 + 5x1, [isNatList](x0) = x0, [and](x0, x1) = x0 + 4x1, [isNat](x0) = x0, [a__U11](x0, x1) = x0 + 4x1 + 6, [a__isNat](x0) = x0, [a__zeros] = 0, [a__length](x0) = x0 + 6, [isNatIList](x0) = x0, [a__isNatList](x0) = x0, [length](x0) = x0, [a__and](x0, x1) = x0 + 4x1, [a__isNatIList](x0) = x0, [mark](x0) = 4x0, [zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),L) = 4L + 6 >= 4L + 6 = s(a__length(mark(L))) a__and(tt(),X) = 4X >= 4X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(V1)) = V1 >= V1 = a__isNat(V1) a__isNatIList(cons(V1,V2)) = 4V1 + 5V2 >= V1 + 4V2 = a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(cons(V1,V2)) = 4V1 + 5V2 >= V1 + 4V2 = a__and(a__isNat(V1),isNatList(V2)) a__length(cons(N,L)) = 5L + 4N + 6 >= 5L + 4N + 6 = a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(and(X1,X2)) = 4X1 + 16X2 >= 4X1 + 4X2 = a__and(mark(X1),X2) mark(isNat(X)) = 4X >= X = a__isNat(X) mark(isNatList(X)) = 4X >= X = a__isNatList(X) mark(isNatIList(X)) = 4X >= X = a__isNatIList(X) mark(cons(X1,X2)) = 16X1 + 20X2 >= 16X1 + 5X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 4X >= 4X = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__length(X) = X + 6 >= X = length(X) a__and(X1,X2) = X1 + 4X2 >= X1 + 4X2 = and(X1,X2) a__isNat(X) = X >= X = isNat(X) a__isNatList(X) = X >= X = isNatList(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) -> a__zeros() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) Matrix Interpretation Processor: dim=1 interpretation: [tt] = 1, [s](x0) = x0, [0] = 0, [cons](x0, x1) = 4x0 + 6x1 + 1, [isNatList](x0) = x0, [and](x0, x1) = x0 + x1, [isNat](x0) = x0, [a__U11](x0, x1) = x0 + 4x1, [a__isNat](x0) = x0 + 1, [a__zeros] = 1, [a__length](x0) = x0, [isNatIList](x0) = x0, [a__isNatList](x0) = 2x0, [a__and](x0, x1) = x0 + 4x1, [a__isNatIList](x0) = x0, [mark](x0) = 4x0 + 1, [zeros] = 0 orientation: a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt(),L) = 4L + 1 >= 4L + 1 = s(a__length(mark(L))) a__and(tt(),X) = 4X + 1 >= 4X + 1 = mark(X) a__isNat(0()) = 1 >= 1 = tt() a__isNat(s(V1)) = V1 + 1 >= V1 + 1 = a__isNat(V1) a__isNatIList(cons(V1,V2)) = 4V1 + 6V2 + 1 >= V1 + 4V2 + 1 = a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(cons(V1,V2)) = 8V1 + 12V2 + 2 >= V1 + 4V2 + 1 = a__and(a__isNat(V1),isNatList(V2)) a__length(cons(N,L)) = 6L + 4N + 1 >= 6L + 4N = a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) = 1 >= 1 = a__zeros() mark(and(X1,X2)) = 4X1 + 4X2 + 1 >= 4X1 + 4X2 + 1 = a__and(mark(X1),X2) mark(isNat(X)) = 4X + 1 >= X + 1 = a__isNat(X) mark(isNatList(X)) = 4X + 1 >= 2X = a__isNatList(X) mark(isNatIList(X)) = 4X + 1 >= X = a__isNatIList(X) mark(cons(X1,X2)) = 16X1 + 24X2 + 5 >= 16X1 + 6X2 + 5 = cons(mark(X1),X2) mark(0()) = 1 >= 0 = 0() mark(tt()) = 5 >= 1 = tt() mark(s(X)) = 4X + 1 >= 4X + 1 = s(mark(X)) a__zeros() = 1 >= 0 = zeros() a__and(X1,X2) = X1 + 4X2 >= X1 + X2 = and(X1,X2) a__isNat(X) = X + 1 >= X = isNat(X) a__isNatList(X) = 2X >= X = isNatList(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) mark(zeros()) -> a__zeros() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) a__isNatList(X) -> isNatList(X) Matrix Interpretation Processor: dim=1 interpretation: [tt] = 0, [s](x0) = x0, [0] = 0, [cons](x0, x1) = 2x0 + 2x1, [isNatList](x0) = x0, [and](x0, x1) = x0 + x1, [isNat](x0) = x0, [a__U11](x0, x1) = x0 + 4x1, [a__isNat](x0) = 4x0, [a__zeros] = 0, [a__length](x0) = x0, [isNatIList](x0) = x0, [a__isNatList](x0) = x0 + 1, [a__and](x0, x1) = x0 + 4x1, [a__isNatIList](x0) = 2x0, [mark](x0) = 4x0, [zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),L) = 4L >= 4L = s(a__length(mark(L))) a__and(tt(),X) = 4X >= 4X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(V1)) = 4V1 >= 4V1 = a__isNat(V1) a__isNatIList(cons(V1,V2)) = 4V1 + 4V2 >= 4V1 + 4V2 = a__and(a__isNat(V1),isNatIList(V2)) mark(zeros()) = 0 >= 0 = a__zeros() mark(and(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = a__and(mark(X1),X2) mark(isNat(X)) = 4X >= 4X = a__isNat(X) mark(cons(X1,X2)) = 8X1 + 8X2 >= 8X1 + 2X2 = cons(mark(X1),X2) mark(s(X)) = 4X >= 4X = s(mark(X)) a__and(X1,X2) = X1 + 4X2 >= X1 + X2 = and(X1,X2) a__isNatList(X) = X + 1 >= X = isNatList(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) mark(zeros()) -> a__zeros() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) Matrix Interpretation Processor: dim=1 interpretation: [tt] = 1, [s](x0) = x0, [0] = 0, [cons](x0, x1) = x0 + x1 + 2, [and](x0, x1) = x0 + 4x1, [isNat](x0) = 4x0 + 1, [a__U11](x0, x1) = x0 + 3x1 + 4, [a__isNat](x0) = 2x0 + 2, [a__zeros] = 3, [a__length](x0) = x0 + 3, [isNatIList](x0) = x0 + 2, [a__and](x0, x1) = x0 + 4x1, [a__isNatIList](x0) = 4x0 + 2, [mark](x0) = 3x0 + 1, [zeros] = 1 orientation: a__zeros() = 3 >= 3 = cons(0(),zeros()) a__U11(tt(),L) = 3L + 5 >= 3L + 4 = s(a__length(mark(L))) a__and(tt(),X) = 4X + 1 >= 3X + 1 = mark(X) a__isNat(0()) = 2 >= 1 = tt() a__isNat(s(V1)) = 2V1 + 2 >= 2V1 + 2 = a__isNat(V1) a__isNatIList(cons(V1,V2)) = 4V1 + 4V2 + 10 >= 2V1 + 4V2 + 10 = a__and(a__isNat(V1),isNatIList(V2)) mark(zeros()) = 4 >= 3 = a__zeros() mark(and(X1,X2)) = 3X1 + 12X2 + 1 >= 3X1 + 4X2 + 1 = a__and(mark(X1),X2) mark(isNat(X)) = 12X + 4 >= 2X + 2 = a__isNat(X) mark(cons(X1,X2)) = 3X1 + 3X2 + 7 >= 3X1 + X2 + 3 = cons(mark(X1),X2) mark(s(X)) = 3X + 1 >= 3X + 1 = s(mark(X)) a__and(X1,X2) = X1 + 4X2 >= X1 + 4X2 = and(X1,X2) problem: a__zeros() -> cons(0(),zeros()) a__and(tt(),X) -> mark(X) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) Matrix Interpretation Processor: dim=1 interpretation: [tt] = 6, [s](x0) = x0, [0] = 0, [cons](x0, x1) = x0 + 4x1, [and](x0, x1) = x0 + 2x1, [a__isNat](x0) = 2x0, [a__zeros] = 0, [isNatIList](x0) = 3x0, [a__and](x0, x1) = x0 + 4x1, [a__isNatIList](x0) = 3x0 + 1, [mark](x0) = 4x0 + 6, [zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__and(tt(),X) = 4X + 6 >= 4X + 6 = mark(X) a__isNat(s(V1)) = 2V1 >= 2V1 = a__isNat(V1) a__isNatIList(cons(V1,V2)) = 3V1 + 12V2 + 1 >= 2V1 + 12V2 = a__and(a__isNat(V1),isNatIList(V2)) mark(and(X1,X2)) = 4X1 + 8X2 + 6 >= 4X1 + 4X2 + 6 = a__and(mark(X1),X2) mark(s(X)) = 4X + 6 >= 4X + 6 = s(mark(X)) a__and(X1,X2) = X1 + 4X2 >= X1 + 2X2 = and(X1,X2) problem: a__zeros() -> cons(0(),zeros()) a__and(tt(),X) -> mark(X) a__isNat(s(V1)) -> a__isNat(V1) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) Matrix Interpretation Processor: dim=1 interpretation: [tt] = 4, [s](x0) = x0, [0] = 0, [cons](x0, x1) = x0 + x1 + 4, [and](x0, x1) = 2x0 + x1 + 4, [a__isNat](x0) = x0, [a__zeros] = 5, [a__and](x0, x1) = 2x0 + x1 + 4, [mark](x0) = x0, [zeros] = 0 orientation: a__zeros() = 5 >= 4 = cons(0(),zeros()) a__and(tt(),X) = X + 12 >= X = mark(X) a__isNat(s(V1)) = V1 >= V1 = a__isNat(V1) mark(and(X1,X2)) = 2X1 + X2 + 4 >= 2X1 + X2 + 4 = a__and(mark(X1),X2) mark(s(X)) = X >= X = s(mark(X)) a__and(X1,X2) = 2X1 + X2 + 4 >= 2X1 + X2 + 4 = and(X1,X2) problem: a__isNat(s(V1)) -> a__isNat(V1) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) Matrix Interpretation Processor: dim=1 interpretation: [s](x0) = x0 + 5, [and](x0, x1) = 4x0 + x1 + 1, [a__isNat](x0) = x0 + 4, [a__and](x0, x1) = 4x0 + 5x1 + 4, [mark](x0) = 5x0 orientation: a__isNat(s(V1)) = V1 + 9 >= V1 + 4 = a__isNat(V1) mark(and(X1,X2)) = 20X1 + 5X2 + 5 >= 20X1 + 5X2 + 4 = a__and(mark(X1),X2) mark(s(X)) = 5X + 25 >= 5X + 5 = s(mark(X)) a__and(X1,X2) = 4X1 + 5X2 + 4 >= 4X1 + X2 + 1 = and(X1,X2) problem: Qed