/export/starexec/sandbox2/solver/bin/starexec_run_ttt2 /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(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(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Proof: Matrix Interpretation Processor: dim=1 interpretation: [a__length](x0) = 2x0 + 4, [tt] = 0, [a__isNat](x0) = 2x0, [a__U41](x0, x1) = x0 + 4x1, [0] = 0, [cons](x0, x1) = 2x0 + 4x1 + 1, [a__U62](x0, x1) = x0 + 4x1 + 6, [nil] = 2, [U42](x0) = x0, [U62](x0, x1) = x0 + 4x1 + 4, [s](x0) = x0, [mark](x0) = 2x0 + 1, [isNatIList](x0) = 4x0, [U11](x0) = x0 + 1, [U51](x0, x1) = 2x0 + 2x1 + 3, [a__U11](x0) = x0 + 2, [isNat](x0) = x0, [a__U42](x0) = x0, [a__zeros] = 1, [a__U31](x0) = x0, [length](x0) = 2x0 + 4, [U41](x0, x1) = x0 + 4x1, [isNatList](x0) = 4x0, [a__U61](x0, x1, x2) = x0 + 4x1 + 3x2 + 6, [U61](x0, x1, x2) = x0 + 4x1 + 2x2 + 4, [a__isNatList](x0) = 4x0, [U21](x0) = x0, [U31](x0) = x0, [a__U51](x0, x1) = 2x0 + 4x1 + 4, [a__isNatIList](x0) = 4x0, [a__U52](x0) = x0 + 2, [a__U21](x0) = x0, [U52](x0) = x0 + 2, [zeros] = 0 orientation: a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt()) = 2 >= 0 = tt() a__U21(tt()) = 0 >= 0 = tt() a__U31(tt()) = 0 >= 0 = tt() a__U41(tt(),V2) = 4V2 >= 4V2 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 0 >= 0 = tt() a__U51(tt(),V2) = 4V2 + 4 >= 4V2 + 2 = a__U52(a__isNatList(V2)) a__U52(tt()) = 2 >= 0 = tt() a__U61(tt(),L,N) = 4L + 3N + 6 >= 4L + 2N + 6 = a__U62(a__isNat(N),L) a__U62(tt(),L) = 4L + 6 >= 4L + 6 = s(a__length(mark(L))) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 4V1 + 8 >= 4V1 + 2 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = 2V1 >= 2V1 = a__U21(a__isNat(V1)) a__isNatIList(V) = 4V >= 4V = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 8V1 + 16V2 + 4 >= 2V1 + 4V2 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 8 >= 0 = tt() a__isNatList(cons(V1,V2)) = 8V1 + 16V2 + 4 >= 4V1 + 4V2 + 4 = a__U51(a__isNat(V1),V2) a__length(nil()) = 8 >= 0 = 0() a__length(cons(N,L)) = 8L + 4N + 6 >= 8L + 3N + 6 = a__U61(a__isNatList(L),L,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X)) = 2X + 3 >= 2X + 3 = a__U11(mark(X)) mark(U21(X)) = 2X + 1 >= 2X + 1 = a__U21(mark(X)) mark(U31(X)) = 2X + 1 >= 2X + 1 = a__U31(mark(X)) mark(U41(X1,X2)) = 2X1 + 8X2 + 1 >= 2X1 + 4X2 + 1 = a__U41(mark(X1),X2) mark(U42(X)) = 2X + 1 >= 2X + 1 = a__U42(mark(X)) mark(isNatIList(X)) = 8X + 1 >= 4X = a__isNatIList(X) mark(U51(X1,X2)) = 4X1 + 4X2 + 7 >= 4X1 + 4X2 + 6 = a__U51(mark(X1),X2) mark(U52(X)) = 2X + 5 >= 2X + 3 = a__U52(mark(X)) mark(isNatList(X)) = 8X + 1 >= 4X = a__isNatList(X) mark(U61(X1,X2,X3)) = 2X1 + 8X2 + 4X3 + 9 >= 2X1 + 4X2 + 3X3 + 7 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 2X1 + 8X2 + 9 >= 2X1 + 4X2 + 7 = a__U62(mark(X1),X2) mark(isNat(X)) = 2X + 1 >= 2X = a__isNat(X) mark(length(X)) = 4X + 9 >= 4X + 6 = a__length(mark(X)) mark(cons(X1,X2)) = 4X1 + 8X2 + 3 >= 4X1 + 4X2 + 3 = cons(mark(X1),X2) mark(0()) = 1 >= 0 = 0() mark(tt()) = 1 >= 0 = tt() mark(s(X)) = 2X + 1 >= 2X + 1 = s(mark(X)) mark(nil()) = 5 >= 2 = nil() a__zeros() = 1 >= 0 = zeros() a__U11(X) = X + 2 >= X + 1 = U11(X) a__U21(X) = X >= X = U21(X) a__U31(X) = X >= X = U31(X) a__U41(X1,X2) = X1 + 4X2 >= X1 + 4X2 = U41(X1,X2) a__U42(X) = X >= X = U42(X) a__isNatIList(X) = 4X >= 4X = isNatIList(X) a__U51(X1,X2) = 2X1 + 4X2 + 4 >= 2X1 + 2X2 + 3 = U51(X1,X2) a__U52(X) = X + 2 >= X + 2 = U52(X) a__isNatList(X) = 4X >= 4X = isNatList(X) a__U61(X1,X2,X3) = X1 + 4X2 + 3X3 + 6 >= X1 + 4X2 + 2X3 + 4 = U61(X1,X2,X3) a__U62(X1,X2) = X1 + 4X2 + 6 >= X1 + 4X2 + 4 = U62(X1,X2) a__isNat(X) = 2X >= X = isNat(X) a__length(X) = 2X + 4 >= 2X + 4 = length(X) problem: a__zeros() -> cons(0(),zeros()) a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Matrix Interpretation Processor: dim=1 interpretation: [a__length](x0) = x0, [tt] = 0, [a__isNat](x0) = x0, [a__U41](x0, x1) = x0 + 4x1 + 2, [0] = 0, [cons](x0, x1) = x0 + 6x1, [a__U62](x0, x1) = x0 + 4x1, [U42](x0) = 2x0, [s](x0) = 2x0, [mark](x0) = 2x0, [isNatIList](x0) = x0, [U11](x0) = 4x0 + 4, [a__U11](x0) = x0, [isNat](x0) = x0, [a__U42](x0) = 2x0, [a__zeros] = 0, [a__U31](x0) = x0, [length](x0) = x0, [U41](x0, x1) = x0 + 4x1 + 1, [isNatList](x0) = 2x0, [a__U61](x0, x1, x2) = x0 + 4x1 + x2, [a__isNatList](x0) = 2x0, [U21](x0) = x0, [U31](x0) = x0, [a__U51](x0, x1) = x0 + 2x1, [a__isNatIList](x0) = 2x0, [a__U52](x0) = x0 + 4, [a__U21](x0) = x0, [U52](x0) = x0, [zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U21(tt()) = 0 >= 0 = tt() a__U31(tt()) = 0 >= 0 = tt() a__U41(tt(),V2) = 4V2 + 2 >= 4V2 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 0 >= 0 = tt() a__U61(tt(),L,N) = 4L + N >= 4L + N = a__U62(a__isNat(N),L) a__U62(tt(),L) = 4L >= 4L = s(a__length(mark(L))) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(V1)) = 2V1 >= V1 = a__U21(a__isNat(V1)) a__isNatIList(V) = 2V >= 2V = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 2V1 + 12V2 >= V1 + 2V2 = a__U51(a__isNat(V1),V2) a__length(cons(N,L)) = 6L + N >= 6L + N = a__U61(a__isNatList(L),L,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X)) = 8X + 8 >= 2X = a__U11(mark(X)) mark(U21(X)) = 2X >= 2X = a__U21(mark(X)) mark(U31(X)) = 2X >= 2X = a__U31(mark(X)) mark(U41(X1,X2)) = 2X1 + 8X2 + 2 >= 2X1 + 4X2 + 2 = a__U41(mark(X1),X2) mark(U42(X)) = 4X >= 4X = a__U42(mark(X)) mark(cons(X1,X2)) = 2X1 + 12X2 >= 2X1 + 6X2 = cons(mark(X1),X2) mark(s(X)) = 4X >= 4X = s(mark(X)) a__U21(X) = X >= X = U21(X) a__U31(X) = X >= X = U31(X) a__U41(X1,X2) = X1 + 4X2 + 2 >= X1 + 4X2 + 1 = U41(X1,X2) a__U42(X) = 2X >= 2X = U42(X) a__isNatIList(X) = 2X >= X = isNatIList(X) a__U52(X) = X + 4 >= X = U52(X) a__isNatList(X) = 2X >= 2X = isNatList(X) a__isNat(X) = X >= X = isNat(X) a__length(X) = X >= X = length(X) problem: a__zeros() -> cons(0(),zeros()) a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U42(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Matrix Interpretation Processor: dim=1 interpretation: [a__length](x0) = x0 + 2, [tt] = 0, [a__isNat](x0) = x0, [a__U41](x0, x1) = 4x0 + 2x1, [0] = 0, [cons](x0, x1) = x0 + 4x1, [a__U62](x0, x1) = x0 + 2x1 + 2, [U42](x0) = x0 + 2, [s](x0) = x0, [mark](x0) = 2x0, [isNatIList](x0) = 2x0, [isNat](x0) = x0, [a__U42](x0) = x0 + 2, [a__zeros] = 0, [a__U31](x0) = x0, [length](x0) = x0, [U41](x0, x1) = 4x0 + x1, [isNatList](x0) = x0, [a__U61](x0, x1, x2) = x0 + 2x1 + x2 + 2, [a__isNatList](x0) = x0, [U21](x0) = x0, [U31](x0) = x0, [a__U51](x0, x1) = x0 + x1, [a__isNatIList](x0) = 2x0, [a__U21](x0) = x0, [zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U21(tt()) = 0 >= 0 = tt() a__U31(tt()) = 0 >= 0 = tt() a__U42(tt()) = 2 >= 0 = tt() a__U61(tt(),L,N) = 2L + N + 2 >= 2L + N + 2 = a__U62(a__isNat(N),L) a__U62(tt(),L) = 2L + 2 >= 2L + 2 = s(a__length(mark(L))) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(V1)) = V1 >= V1 = a__U21(a__isNat(V1)) a__isNatIList(V) = 2V >= V = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = V1 + 4V2 >= V1 + V2 = a__U51(a__isNat(V1),V2) a__length(cons(N,L)) = 4L + N + 2 >= 3L + N + 2 = a__U61(a__isNatList(L),L,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U21(X)) = 2X >= 2X = a__U21(mark(X)) mark(U31(X)) = 2X >= 2X = a__U31(mark(X)) mark(U41(X1,X2)) = 8X1 + 2X2 >= 8X1 + 2X2 = a__U41(mark(X1),X2) mark(U42(X)) = 2X + 4 >= 2X + 2 = a__U42(mark(X)) mark(cons(X1,X2)) = 2X1 + 8X2 >= 2X1 + 4X2 = cons(mark(X1),X2) mark(s(X)) = 2X >= 2X = s(mark(X)) a__U21(X) = X >= X = U21(X) a__U31(X) = X >= X = U31(X) a__U42(X) = X + 2 >= X + 2 = U42(X) a__isNatIList(X) = 2X >= 2X = isNatIList(X) a__isNatList(X) = X >= X = isNatList(X) a__isNat(X) = X >= X = isNat(X) a__length(X) = X + 2 >= X = length(X) problem: a__zeros() -> cons(0(),zeros()) a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [0] [a__length](x0) = [1 1 0]x0 + [1] [1 1 0] [0], [0] [tt] = [0] [0], [1 0 0] [0] [a__isNat](x0) = [0 0 1]x0 + [1] [0 0 1] [0], [1 0 1] [1 0 0] [1] [a__U41](x0, x1) = [0 0 0]x0 + [1 0 0]x1 + [0] [0 0 0] [0 0 0] [0], [0] [0] = [0] [1], [1 0 0] [1 1 0] [1] [cons](x0, x1) = [1 0 0]x0 + [1 0 0]x1 + [1] [0 1 0] [1 0 0] [0], [1 0 0] [1 1 0] [1] [a__U62](x0, x1) = [0 0 0]x0 + [0 1 0]x1 + [1] [0 0 0] [0 0 0] [1], [1 0 0] [U42](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [s](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [1 0 0] [1] [mark](x0) = [1 0 0]x0 + [1] [0 1 0] [0], [1 1 1] [0] [isNatIList](x0) = [0 1 0]x0 + [1] [1 0 0] [1], [1 0 0] [0] [isNat](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 1 0] [1] [a__U42](x0) = [0 0 1]x0 + [0] [0 0 1] [0], [1] [a__zeros] = [1] [0], [1 0 0] [0] [a__U31](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [1 1 0] [1 0 0] [1] [U41](x0, x1) = [1 0 0]x0 + [1 0 0]x1 + [0] [0 0 0] [0 0 0] [0], [1 0 0] [0] [isNatList](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [1 1 0] [1 0 0] [1] [a__U61](x0, x1, x2) = [0 1 0]x0 + [0 1 0]x1 + [1 0 0]x2 + [1] [0 1 0] [1 0 0] [0 0 0] [1], [1 0 0] [0] [a__isNatList](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [U21](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [U31](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [1 0 0] [1] [a__U51](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [0], [1 1 1] [0] [a__isNatIList](x0) = [0 1 0]x0 + [1] [1 0 0] [1], [1 0 0] [0] [a__U21](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [0] [zeros] = [0] [0] orientation: [1] [1] a__zeros() = [1] >= [1] = cons(0(),zeros()) [0] [0] [0] [0] a__U21(tt()) = [1] >= [0] = tt() [0] [0] [0] [0] a__U31(tt()) = [1] >= [0] = tt() [1] [0] [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [1] a__U61(tt(),L,N) = [0 1 0]L + [1 0 0]N + [1] >= [0 1 0]L + [0 0 0]N + [1] = a__U62(a__isNat(N),L) [1 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 0] [1] [1 1 0] [1] a__U62(tt(),L) = [0 1 0]L + [1] >= [0 0 0]L + [1] = s(a__length(mark(L))) [0 0 0] [1] [0 0 0] [1] [0] [0] a__isNat(0()) = [2] >= [0] = tt() [1] [0] [1 0 0] [0] [1 0 0] [0] a__isNat(s(V1)) = [0 0 0]V1 + [2] >= [0 0 0]V1 + [1] = a__U21(a__isNat(V1)) [0 0 0] [1] [0 0 0] [0] [1 1 1] [0] [1 0 0] [0] a__isNatIList(V) = [0 1 0]V + [1] >= [0 0 0]V + [1] = a__U31(a__isNatList(V)) [1 0 0] [1] [0 0 0] [1] [0] [0] a__isNatIList(zeros()) = [1] >= [0] = tt() [1] [0] [1 0 0] [1 1 0] [1] [1 0 0] [1 0 0] [1] a__isNatList(cons(V1,V2)) = [0 0 0]V1 + [0 0 0]V2 + [1] >= [0 0 0]V1 + [0 0 0]V2 + [0] = a__U51(a__isNat(V1),V2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [2 1 0] [1 1 0] [1] [2 1 0] [1 0 0] [1] a__length(cons(N,L)) = [2 1 0]L + [2 0 0]N + [3] >= [0 1 0]L + [1 0 0]N + [2] = a__U61(a__isNatList(L),L,N) [2 1 0] [2 0 0] [2] [1 0 0] [0 0 0] [2] [1] [1] mark(zeros()) = [1] >= [1] = a__zeros() [0] [0] [1 0 0] [1] [1 0 0] [1] mark(U21(X)) = [1 0 0]X + [1] >= [0 0 0]X + [1] = a__U21(mark(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] mark(U31(X)) = [1 0 0]X + [1] >= [0 0 0]X + [1] = a__U31(mark(X)) [0 0 0] [1] [0 0 0] [1] [1 1 0] [1 0 0] [2] [1 1 0] [1 0 0] [2] mark(U41(X1,X2)) = [1 1 0]X1 + [1 0 0]X2 + [2] >= [0 0 0]X1 + [1 0 0]X2 + [0] = a__U41(mark(X1),X2) [1 0 0] [1 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 1 0] [2] [1 0 0] [1 1 0] [2] mark(cons(X1,X2)) = [1 0 0]X1 + [1 1 0]X2 + [2] >= [1 0 0]X1 + [1 0 0]X2 + [2] = cons(mark(X1),X2) [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1] [1 0 0] [1] mark(s(X)) = [1 0 0]X + [1] >= [0 0 0]X + [1] = s(mark(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] a__U21(X) = [0 0 0]X + [1] >= [0 0 0]X = U21(X) [0 0 0] [0] [0 0 0] [1 0 0] [0] [1 0 0] [0] a__U31(X) = [0 0 0]X + [1] >= [0 0 0]X + [1] = U31(X) [0 0 0] [1] [0 0 0] [0] [1 1 0] [1] [1 0 0] a__U42(X) = [0 0 1]X + [0] >= [0 0 0]X = U42(X) [0 0 1] [0] [0 0 0] [1 1 1] [0] [1 1 1] [0] a__isNatIList(X) = [0 1 0]X + [1] >= [0 1 0]X + [1] = isNatIList(X) [1 0 0] [1] [1 0 0] [1] [1 0 0] [0] [1 0 0] [0] a__isNatList(X) = [0 0 0]X + [1] >= [0 0 0]X + [1] = isNatList(X) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] a__isNat(X) = [0 0 1]X + [1] >= [0 0 0]X + [1] = isNat(X) [0 0 1] [0] [0 0 0] [0] problem: a__zeros() -> cons(0(),zeros()) a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) Matrix Interpretation Processor: dim=1 interpretation: [a__length](x0) = x0, [tt] = 0, [a__isNat](x0) = x0, [a__U41](x0, x1) = 4x0 + x1 + 4, [0] = 0, [cons](x0, x1) = x0 + 6x1, [a__U62](x0, x1) = x0 + 2x1, [s](x0) = 2x0, [mark](x0) = x0, [isNatIList](x0) = x0, [isNat](x0) = x0, [a__zeros] = 0, [a__U31](x0) = x0 + 1, [U41](x0, x1) = 5x0 + x1 + 5, [isNatList](x0) = x0, [a__U61](x0, x1, x2) = 2x0 + 2x1 + x2, [a__isNatList](x0) = 2x0, [U21](x0) = 2x0, [U31](x0) = x0 + 1, [a__U51](x0, x1) = 2x0 + x1, [a__isNatIList](x0) = 7x0 + 1, [a__U21](x0) = 2x0, [zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U21(tt()) = 0 >= 0 = tt() a__U31(tt()) = 1 >= 0 = tt() a__U61(tt(),L,N) = 2L + N >= 2L + N = a__U62(a__isNat(N),L) a__U62(tt(),L) = 2L >= 2L = s(a__length(mark(L))) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(V1)) = 2V1 >= 2V1 = a__U21(a__isNat(V1)) a__isNatIList(V) = 7V + 1 >= 2V + 1 = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 1 >= 0 = tt() a__isNatList(cons(V1,V2)) = 2V1 + 12V2 >= 2V1 + V2 = a__U51(a__isNat(V1),V2) a__length(cons(N,L)) = 6L + N >= 6L + N = a__U61(a__isNatList(L),L,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U21(X)) = 2X >= 2X = a__U21(mark(X)) mark(U31(X)) = X + 1 >= X + 1 = a__U31(mark(X)) mark(U41(X1,X2)) = 5X1 + X2 + 5 >= 4X1 + X2 + 4 = a__U41(mark(X1),X2) mark(cons(X1,X2)) = X1 + 6X2 >= X1 + 6X2 = cons(mark(X1),X2) mark(s(X)) = 2X >= 2X = s(mark(X)) a__U21(X) = 2X >= 2X = U21(X) a__U31(X) = X + 1 >= X + 1 = U31(X) a__isNatIList(X) = 7X + 1 >= X = isNatIList(X) a__isNatList(X) = 2X >= X = isNatList(X) a__isNat(X) = X >= X = isNat(X) problem: a__zeros() -> cons(0(),zeros()) a__U21(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) Matrix Interpretation Processor: dim=1 interpretation: [a__length](x0) = x0 + 1, [tt] = 1, [a__isNat](x0) = x0 + 1, [0] = 0, [cons](x0, x1) = 2x0 + 6x1 + 2, [a__U62](x0, x1) = x0 + 4x1 + 4, [s](x0) = x0 + 2, [mark](x0) = 4x0 + 2, [isNat](x0) = x0 + 1, [a__zeros] = 2, [a__U31](x0) = x0 + 6, [isNatList](x0) = x0, [a__U61](x0, x1, x2) = 2x0 + 4x1 + x2 + 3, [a__isNatList](x0) = x0, [U21](x0) = x0, [U31](x0) = x0 + 2, [a__U51](x0, x1) = 2x0 + 4x1, [a__isNatIList](x0) = 2x0 + 6, [a__U21](x0) = x0, [zeros] = 0 orientation: a__zeros() = 2 >= 2 = cons(0(),zeros()) a__U21(tt()) = 1 >= 1 = tt() a__U61(tt(),L,N) = 4L + N + 5 >= 4L + N + 5 = a__U62(a__isNat(N),L) a__U62(tt(),L) = 4L + 5 >= 4L + 5 = s(a__length(mark(L))) a__isNat(0()) = 1 >= 1 = tt() a__isNat(s(V1)) = V1 + 3 >= V1 + 1 = a__U21(a__isNat(V1)) a__isNatIList(V) = 2V + 6 >= V + 6 = a__U31(a__isNatList(V)) a__isNatList(cons(V1,V2)) = 2V1 + 6V2 + 2 >= 2V1 + 4V2 + 2 = a__U51(a__isNat(V1),V2) a__length(cons(N,L)) = 6L + 2N + 3 >= 6L + N + 3 = a__U61(a__isNatList(L),L,N) mark(zeros()) = 2 >= 2 = a__zeros() mark(U21(X)) = 4X + 2 >= 4X + 2 = a__U21(mark(X)) mark(U31(X)) = 4X + 10 >= 4X + 8 = a__U31(mark(X)) mark(cons(X1,X2)) = 8X1 + 24X2 + 10 >= 8X1 + 6X2 + 6 = cons(mark(X1),X2) mark(s(X)) = 4X + 10 >= 4X + 4 = s(mark(X)) a__U21(X) = X >= X = U21(X) a__U31(X) = X + 6 >= X + 2 = U31(X) a__isNatList(X) = X >= X = isNatList(X) a__isNat(X) = X + 1 >= X + 1 = isNat(X) problem: a__zeros() -> cons(0(),zeros()) a__U21(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U21(X)) -> a__U21(mark(X)) a__U21(X) -> U21(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) Matrix Interpretation Processor: dim=1 interpretation: [a__length](x0) = x0 + 4, [tt] = 4, [a__isNat](x0) = 2x0 + 4, [0] = 0, [cons](x0, x1) = 2x0 + 4x1, [a__U62](x0, x1) = x0 + x1, [s](x0) = x0, [mark](x0) = x0, [isNat](x0) = x0, [a__zeros] = 0, [a__U31](x0) = x0 + 1, [isNatList](x0) = 2x0 + 4, [a__U61](x0, x1, x2) = x0 + 2x1 + 2x2, [a__isNatList](x0) = 2x0 + 4, [U21](x0) = 2x0 + 1, [a__U51](x0, x1) = x0 + 4x1, [a__isNatIList](x0) = 3x0 + 5, [a__U21](x0) = 2x0 + 1, [zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U21(tt()) = 9 >= 4 = tt() a__U61(tt(),L,N) = 2L + 2N + 4 >= L + 2N + 4 = a__U62(a__isNat(N),L) a__U62(tt(),L) = L + 4 >= L + 4 = s(a__length(mark(L))) a__isNat(0()) = 4 >= 4 = tt() a__isNatIList(V) = 3V + 5 >= 2V + 5 = a__U31(a__isNatList(V)) a__isNatList(cons(V1,V2)) = 4V1 + 8V2 + 4 >= 2V1 + 4V2 + 4 = a__U51(a__isNat(V1),V2) a__length(cons(N,L)) = 4L + 2N + 4 >= 4L + 2N + 4 = a__U61(a__isNatList(L),L,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U21(X)) = 2X + 1 >= 2X + 1 = a__U21(mark(X)) a__U21(X) = 2X + 1 >= 2X + 1 = U21(X) a__isNatList(X) = 2X + 4 >= 2X + 4 = isNatList(X) a__isNat(X) = 2X + 4 >= X = isNat(X) problem: a__zeros() -> cons(0(),zeros()) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U21(X)) -> a__U21(mark(X)) a__U21(X) -> U21(X) a__isNatList(X) -> isNatList(X) Matrix Interpretation Processor: dim=1 interpretation: [a__length](x0) = 2x0, [tt] = 5, [a__isNat](x0) = x0 + 5, [0] = 0, [cons](x0, x1) = x0 + 5x1 + 2, [a__U62](x0, x1) = 2x0 + 2x1 + 4, [s](x0) = x0, [mark](x0) = x0 + 7, [a__zeros] = 7, [a__U31](x0) = x0, [isNatList](x0) = 2x0, [a__U61](x0, x1, x2) = 2x0 + 2x1 + 2x2 + 4, [a__isNatList](x0) = 4x0, [U21](x0) = x0 + 5, [a__U51](x0, x1) = x0 + x1 + 2, [a__isNatIList](x0) = 5x0, [a__U21](x0) = x0 + 5, [zeros] = 1 orientation: a__zeros() = 7 >= 7 = cons(0(),zeros()) a__U61(tt(),L,N) = 2L + 2N + 14 >= 2L + 2N + 14 = a__U62(a__isNat(N),L) a__U62(tt(),L) = 2L + 14 >= 2L + 14 = s(a__length(mark(L))) a__isNat(0()) = 5 >= 5 = tt() a__isNatIList(V) = 5V >= 4V = a__U31(a__isNatList(V)) a__isNatList(cons(V1,V2)) = 4V1 + 20V2 + 8 >= V1 + V2 + 7 = a__U51(a__isNat(V1),V2) a__length(cons(N,L)) = 10L + 2N + 4 >= 10L + 2N + 4 = a__U61(a__isNatList(L),L,N) mark(zeros()) = 8 >= 7 = a__zeros() mark(U21(X)) = X + 12 >= X + 12 = a__U21(mark(X)) a__U21(X) = X + 5 >= X + 5 = U21(X) a__isNatList(X) = 4X >= 2X = isNatList(X) problem: a__zeros() -> cons(0(),zeros()) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNatIList(V) -> a__U31(a__isNatList(V)) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(U21(X)) -> a__U21(mark(X)) a__U21(X) -> U21(X) a__isNatList(X) -> isNatList(X) Matrix Interpretation Processor: dim=1 interpretation: [a__length](x0) = x0 + 2, [tt] = 1, [a__isNat](x0) = 2x0 + 1, [0] = 0, [cons](x0, x1) = 2x0 + 3x1 + 5, [a__U62](x0, x1) = x0 + x1 + 3, [s](x0) = x0 + 2, [mark](x0) = x0, [a__zeros] = 6, [a__U31](x0) = x0 + 2, [isNatList](x0) = x0 + 4, [a__U61](x0, x1, x2) = x0 + x1 + 2x2 + 3, [a__isNatList](x0) = 2x0 + 4, [U21](x0) = x0 + 4, [a__isNatIList](x0) = 2x0 + 6, [a__U21](x0) = x0 + 4, [zeros] = 0 orientation: a__zeros() = 6 >= 5 = cons(0(),zeros()) a__U61(tt(),L,N) = L + 2N + 4 >= L + 2N + 4 = a__U62(a__isNat(N),L) a__U62(tt(),L) = L + 4 >= L + 4 = s(a__length(mark(L))) a__isNat(0()) = 1 >= 1 = tt() a__isNatIList(V) = 2V + 6 >= 2V + 6 = a__U31(a__isNatList(V)) a__length(cons(N,L)) = 3L + 2N + 7 >= 3L + 2N + 7 = a__U61(a__isNatList(L),L,N) mark(U21(X)) = X + 4 >= X + 4 = a__U21(mark(X)) a__U21(X) = X + 4 >= X + 4 = U21(X) a__isNatList(X) = 2X + 4 >= X + 4 = isNatList(X) problem: a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNatIList(V) -> a__U31(a__isNatList(V)) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(U21(X)) -> a__U21(mark(X)) a__U21(X) -> U21(X) a__isNatList(X) -> isNatList(X) Matrix Interpretation Processor: dim=1 interpretation: [a__length](x0) = x0, [tt] = 2, [a__isNat](x0) = x0, [0] = 2, [cons](x0, x1) = x0 + 5x1 + 4, [a__U62](x0, x1) = x0 + 3x1 + 4, [s](x0) = x0 + 6, [mark](x0) = 3x0, [a__U31](x0) = x0, [isNatList](x0) = x0, [a__U61](x0, x1, x2) = x0 + 3x1 + x2 + 2, [a__isNatList](x0) = x0, [U21](x0) = 6x0, [a__isNatIList](x0) = 4x0, [a__U21](x0) = 6x0 orientation: a__U61(tt(),L,N) = 3L + N + 4 >= 3L + N + 4 = a__U62(a__isNat(N),L) a__U62(tt(),L) = 3L + 6 >= 3L + 6 = s(a__length(mark(L))) a__isNat(0()) = 2 >= 2 = tt() a__isNatIList(V) = 4V >= V = a__U31(a__isNatList(V)) a__length(cons(N,L)) = 5L + N + 4 >= 4L + N + 2 = a__U61(a__isNatList(L),L,N) mark(U21(X)) = 18X >= 18X = a__U21(mark(X)) a__U21(X) = 6X >= 6X = U21(X) a__isNatList(X) = X >= X = isNatList(X) problem: a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNatIList(V) -> a__U31(a__isNatList(V)) mark(U21(X)) -> a__U21(mark(X)) a__U21(X) -> U21(X) a__isNatList(X) -> isNatList(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [a__length](x0) = [0 1 0]x0 [0 0 0] , [1] [tt] = [0] [0], [1 0 0] [a__isNat](x0) = [0 0 1]x0 [0 0 0] , [1] [0] = [0] [1], [1 0 0] [1 0 0] [a__U62](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [1 0 0] [0 0 1] , [1 0 0] [0] [s](x0) = [0 0 0]x0 + [0] [0 1 0] [1], [1 0 0] [mark](x0) = [0 0 1]x0 [0 0 0] , [1 1 1] [0] [a__U31](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [isNatList](x0) = [0 0 0]x0 [0 1 1] , [1 0 0] [1 0 0] [1 0 0] [a__U61](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 0] [0 0 1] [1 0 0] , [1 0 0] [0] [a__isNatList](x0) = [0 0 0]x0 + [1] [0 1 1] [0], [1 0 0] [U21](x0) = [0 0 0]x0 [0 0 0] , [1 1 1] [1] [a__isNatIList](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 1] [a__U21](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] a__U61(tt(),L,N) = [0 0 0]L + [0 0 0]N + [0] >= [0 0 0]L + [0 0 0]N = a__U62(a__isNat(N),L) [0 0 1] [1 0 0] [0] [0 0 1] [1 0 0] [1 0 0] [1] [1 0 0] [0] a__U62(tt(),L) = [0 0 0]L + [0] >= [0 0 0]L + [0] = s(a__length(mark(L))) [0 0 1] [1] [0 0 1] [1] [1] [1] a__isNat(0()) = [1] >= [0] = tt() [0] [0] [1 1 1] [1] [1 1 1] [1] a__isNatIList(V) = [0 0 0]V + [1] >= [0 0 0]V + [1] = a__U31(a__isNatList(V)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] mark(U21(X)) = [0 0 0]X >= [0 0 0]X = a__U21(mark(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 0] a__U21(X) = [0 0 0]X >= [0 0 0]X = U21(X) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] a__isNatList(X) = [0 0 0]X + [1] >= [0 0 0]X = isNatList(X) [0 1 1] [0] [0 1 1] problem: a__isNat(0()) -> tt() a__isNatIList(V) -> a__U31(a__isNatList(V)) mark(U21(X)) -> a__U21(mark(X)) a__U21(X) -> U21(X) a__isNatList(X) -> isNatList(X) Matrix Interpretation Processor: dim=3 interpretation: [0] [tt] = [0] [0], [1 0 0] [1] [a__isNat](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [0] [0] = [0] [0], [0] [mark](x0) = x0 + [1] [0], [1 0 0] [a__U31](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [isNatList](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [a__isNatList](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [U21](x0) = [0 0 1]x0 [1 0 0] , [1 1 0] [a__isNatIList](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [a__U21](x0) = [0 0 1]x0 + [1] [1 0 0] [0] orientation: [1] [0] a__isNat(0()) = [0] >= [0] = tt() [0] [0] [1 1 0] [1 0 0] a__isNatIList(V) = [0 0 0]V >= [0 0 0]V = a__U31(a__isNatList(V)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] mark(U21(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = a__U21(mark(X)) [1 0 0] [0] [1 0 0] [0] [1 0 0] [0] [1 0 0] a__U21(X) = [0 0 1]X + [1] >= [0 0 1]X = U21(X) [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] a__isNatList(X) = [0 0 0]X >= [0 0 0]X = isNatList(X) [0 0 0] [0 0 0] problem: a__isNatIList(V) -> a__U31(a__isNatList(V)) mark(U21(X)) -> a__U21(mark(X)) a__U21(X) -> U21(X) a__isNatList(X) -> isNatList(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [mark](x0) = [0 0 1]x0 [0 0 1] , [1 0 0] [a__U31](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [isNatList](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [a__isNatList](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [0] [U21](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 0] [0] [a__isNatIList](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [1 0 0] [0] [a__U21](x0) = [0 0 1]x0 + [1] [0 0 1] [1] orientation: [1 0 0] [0] [1 0 0] a__isNatIList(V) = [0 0 0]V + [1] >= [0 0 0]V = a__U31(a__isNatList(V)) [0 0 0] [1] [0 0 0] [1 0 1] [1] [1 0 1] [0] mark(U21(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = a__U21(mark(X)) [0 0 1] [1] [0 0 1] [1] [1 0 0] [0] [1 0 0] [0] a__U21(X) = [0 0 1]X + [1] >= [0 0 0]X + [0] = U21(X) [0 0 1] [1] [0 0 1] [1] [1 0 0] [0] [1 0 0] a__isNatList(X) = [0 0 0]X + [1] >= [0 0 0]X = isNatList(X) [0 0 0] [0] [0 0 0] problem: a__isNatIList(V) -> a__U31(a__isNatList(V)) a__U21(X) -> U21(X) a__isNatList(X) -> isNatList(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [a__U31](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [isNatList](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [a__isNatList](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [U21](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [a__isNatIList](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1] [a__U21](x0) = [0 0 0]x0 + [0] [0 0 0] [0] orientation: [1 0 0] [1 0 0] a__isNatIList(V) = [0 0 0]V >= [0 0 0]V = a__U31(a__isNatList(V)) [0 0 0] [0 0 0] [1 0 0] [1] [1 0 0] a__U21(X) = [0 0 0]X + [0] >= [0 0 0]X = U21(X) [0 0 0] [0] [0 0 0] [1 0 0] [1 0 0] a__isNatList(X) = [0 0 0]X >= [0 0 0]X = isNatList(X) [0 0 0] [0 0 0] problem: a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatList(X) -> isNatList(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [a__U31](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [isNatList](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [a__isNatList](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [1] [a__isNatIList](x0) = [1 0 0]x0 + [0] [0 0 0] [0] orientation: [1 1 0] [1] [1 0 0] a__isNatIList(V) = [1 0 0]V + [0] >= [0 0 0]V = a__U31(a__isNatList(V)) [0 0 0] [0] [0 0 0] [1 0 0] [1 0 0] a__isNatList(X) = [0 0 0]X >= [0 0 0]X = isNatList(X) [0 0 0] [0 0 0] problem: a__isNatList(X) -> isNatList(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [isNatList](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1] [a__isNatList](x0) = [0 0 0]x0 + [0] [0 0 1] [0] orientation: [1 0 0] [1] [1 0 0] a__isNatList(X) = [0 0 0]X + [0] >= [0 0 0]X = isNatList(X) [0 0 1] [0] [0 0 0] problem: Qed