Problem:
a__zeros() -> cons(0(),zeros())
a__U11(tt(),L) -> a__U12(tt(),L)
a__U12(tt(),L) -> s(a__length(mark(L)))
a__length(nil()) -> 0()
a__length(cons(N,L)) -> a__U11(tt(),L)
mark(zeros()) -> a__zeros()
mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
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(X1,X2) -> U11(X1,X2)
a__U12(X1,X2) -> U12(X1,X2)
a__length(X) -> length(X)
Proof:
Matrix Interpretation Processor: dim=1
interpretation:
[length](x0) = x0 + 1,
[U12](x0, x1) = x0 + x1,
[U11](x0, x1) = x0 + x1,
[nil] = 4,
[s](x0) = x0,
[a__length](x0) = x0 + 1,
[mark](x0) = x0 + 1,
[a__U12](x0, x1) = x0 + x1,
[a__U11](x0, x1) = x0 + x1,
[tt] = 2,
[cons](x0, x1) = x0 + 4x1 + 1,
[zeros] = 0,
[0] = 0,
[a__zeros] = 1
orientation:
a__zeros() = 1 >= 1 = cons(0(),zeros())
a__U11(tt(),L) = L + 2 >= L + 2 = a__U12(tt(),L)
a__U12(tt(),L) = L + 2 >= L + 2 = s(a__length(mark(L)))
a__length(nil()) = 5 >= 0 = 0()
a__length(cons(N,L)) = 4L + N + 2 >= L + 2 = a__U11(tt(),L)
mark(zeros()) = 1 >= 1 = a__zeros()
mark(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U11(mark(X1),X2)
mark(U12(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U12(mark(X1),X2)
mark(length(X)) = X + 2 >= X + 2 = a__length(mark(X))
mark(cons(X1,X2)) = X1 + 4X2 + 2 >= X1 + 4X2 + 2 = cons(mark(X1),X2)
mark(0()) = 1 >= 0 = 0()
mark(tt()) = 3 >= 2 = tt()
mark(s(X)) = X + 1 >= X + 1 = s(mark(X))
mark(nil()) = 5 >= 4 = nil()
a__zeros() = 1 >= 0 = zeros()
a__U11(X1,X2) = X1 + X2 >= X1 + X2 = U11(X1,X2)
a__U12(X1,X2) = X1 + X2 >= X1 + X2 = U12(X1,X2)
a__length(X) = X + 1 >= X + 1 = length(X)
problem:
a__zeros() -> cons(0(),zeros())
a__U11(tt(),L) -> a__U12(tt(),L)
a__U12(tt(),L) -> s(a__length(mark(L)))
a__length(cons(N,L)) -> a__U11(tt(),L)
mark(zeros()) -> a__zeros()
mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
mark(length(X)) -> a__length(mark(X))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(s(X)) -> s(mark(X))
a__U11(X1,X2) -> U11(X1,X2)
a__U12(X1,X2) -> U12(X1,X2)
a__length(X) -> length(X)
Matrix Interpretation Processor: dim=3
interpretation:
[1 1 0] [1]
[length](x0) = [0 0 1]x0 + [0]
[1 1 0] [1],
[1 1 0] [1 0 1] [1]
[U12](x0, x1) = [0 0 1]x0 + [0 1 0]x1 + [0]
[1 1 0] [1 1 1] [1],
[1 1 0] [1 0 1] [0]
[U11](x0, x1) = [0 0 1]x0 + [0 1 0]x1 + [1]
[1 1 0] [1 1 1] [1],
[s](x0) = x0
,
[1 1 0] [1]
[a__length](x0) = [0 0 1]x0 + [1]
[1 1 0] [1],
[1 1 0]
[mark](x0) = [0 0 1]x0
[1 1 0] ,
[1 1 0] [1 1 1] [1]
[a__U12](x0, x1) = [0 0 1]x0 + [1 1 0]x1 + [1]
[1 1 0] [1 1 1] [1],
[1 1 0] [1 1 1] [1]
[a__U11](x0, x1) = [0 0 1]x0 + [1 1 0]x1 + [1]
[1 1 0] [1 1 1] [1],
[0]
[tt] = [0]
[0],
[1 0 1]
[cons](x0, x1) = x0 + [0 1 0]x1
[1 1 0] ,
[0]
[zeros] = [1]
[1],
[0]
[0] = [0]
[0],
[1]
[a__zeros] = [1]
[1]
orientation:
[1] [1]
a__zeros() = [1] >= [1] = cons(0(),zeros())
[1] [1]
[1 1 1] [1] [1 1 1] [1]
a__U11(tt(),L) = [1 1 0]L + [1] >= [1 1 0]L + [1] = a__U12(tt(),L)
[1 1 1] [1] [1 1 1] [1]
[1 1 1] [1] [1 1 1] [1]
a__U12(tt(),L) = [1 1 0]L + [1] >= [1 1 0]L + [1] = s(a__length(mark(L)))
[1 1 1] [1] [1 1 1] [1]
[1 1 1] [1 1 0] [1] [1 1 1] [1]
a__length(cons(N,L)) = [1 1 0]L + [0 0 1]N + [1] >= [1 1 0]L + [1] = a__U11(tt(),L)
[1 1 1] [1 1 0] [1] [1 1 1] [1]
[1] [1]
mark(zeros()) = [1] >= [1] = a__zeros()
[1] [1]
[1 1 1] [1 1 1] [1] [1 1 1] [1 1 1] [1]
mark(U11(X1,X2)) = [1 1 0]X1 + [1 1 1]X2 + [1] >= [1 1 0]X1 + [1 1 0]X2 + [1] = a__U11(mark(X1),X2)
[1 1 1] [1 1 1] [1] [1 1 1] [1 1 1] [1]
[1 1 1] [1 1 1] [1] [1 1 1] [1 1 1] [1]
mark(U12(X1,X2)) = [1 1 0]X1 + [1 1 1]X2 + [1] >= [1 1 0]X1 + [1 1 0]X2 + [1] = a__U12(mark(X1),X2)
[1 1 1] [1 1 1] [1] [1 1 1] [1 1 1] [1]
[1 1 1] [1] [1 1 1] [1]
mark(length(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = a__length(mark(X))
[1 1 1] [1] [1 1 1] [1]
[1 1 0] [1 1 1] [1 1 0] [1 0 1]
mark(cons(X1,X2)) = [0 0 1]X1 + [1 1 0]X2 >= [0 0 1]X1 + [0 1 0]X2 = cons(mark(X1),X2)
[1 1 0] [1 1 1] [1 1 0] [1 1 0]
[1 1 0] [1 1 0]
mark(s(X)) = [0 0 1]X >= [0 0 1]X = s(mark(X))
[1 1 0] [1 1 0]
[1 1 0] [1 1 1] [1] [1 1 0] [1 0 1] [0]
a__U11(X1,X2) = [0 0 1]X1 + [1 1 0]X2 + [1] >= [0 0 1]X1 + [0 1 0]X2 + [1] = U11(X1,X2)
[1 1 0] [1 1 1] [1] [1 1 0] [1 1 1] [1]
[1 1 0] [1 1 1] [1] [1 1 0] [1 0 1] [1]
a__U12(X1,X2) = [0 0 1]X1 + [1 1 0]X2 + [1] >= [0 0 1]X1 + [0 1 0]X2 + [0] = U12(X1,X2)
[1 1 0] [1 1 1] [1] [1 1 0] [1 1 1] [1]
[1 1 0] [1] [1 1 0] [1]
a__length(X) = [0 0 1]X + [1] >= [0 0 1]X + [0] = length(X)
[1 1 0] [1] [1 1 0] [1]
problem:
a__zeros() -> cons(0(),zeros())
a__U11(tt(),L) -> a__U12(tt(),L)
a__U12(tt(),L) -> s(a__length(mark(L)))
a__length(cons(N,L)) -> a__U11(tt(),L)
mark(zeros()) -> a__zeros()
mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
mark(length(X)) -> a__length(mark(X))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(s(X)) -> s(mark(X))
a__U12(X1,X2) -> U12(X1,X2)
a__length(X) -> length(X)
Matrix Interpretation Processor: dim=3
interpretation:
[1 0 0] [1]
[length](x0) = [0 0 0]x0 + [0]
[0 0 0] [0],
[1 0 0] [1 0 0] [1]
[U12](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0]
[0 0 0] [0 0 0] [0],
[1 0 0] [1 0 1] [1]
[U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0]
[0 0 0] [0 0 0] [0],
[1 0 0]
[s](x0) = [0 0 0]x0
[0 0 0] ,
[1 0 0] [1]
[a__length](x0) = [0 0 0]x0 + [0]
[0 0 0] [0],
[1 0 0]
[mark](x0) = [0 0 0]x0
[0 0 0] ,
[1 0 1] [1 0 0] [1]
[a__U12](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0]
[0 0 1] [0 0 0] [0],
[1 1 0] [1 0 0]
[a__U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1
[0 0 0] [0 0 0] ,
[0]
[tt] = [1]
[0],
[1 0 1] [1 1 0]
[cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1
[0 0 0] [0 0 0] ,
[1]
[zeros] = [0]
[0],
[0]
[0] = [0]
[0],
[1]
[a__zeros] = [0]
[0]
orientation:
[1] [1]
a__zeros() = [0] >= [0] = cons(0(),zeros())
[0] [0]
[1 0 0] [1] [1 0 0] [1]
a__U11(tt(),L) = [0 0 0]L + [0] >= [0 0 0]L + [0] = a__U12(tt(),L)
[0 0 0] [0] [0 0 0] [0]
[1 0 0] [1] [1 0 0] [1]
a__U12(tt(),L) = [0 0 0]L + [0] >= [0 0 0]L + [0] = s(a__length(mark(L)))
[0 0 0] [0] [0 0 0] [0]
[1 1 0] [1 0 1] [1] [1 0 0] [1]
a__length(cons(N,L)) = [0 0 0]L + [0 0 0]N + [0] >= [0 0 0]L + [0] = a__U11(tt(),L)
[0 0 0] [0 0 0] [0] [0 0 0] [0]
[1] [1]
mark(zeros()) = [0] >= [0] = a__zeros()
[0] [0]
[1 0 0] [1 0 1] [1] [1 0 0] [1 0 0]
mark(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = a__U11(mark(X1),X2)
[0 0 0] [0 0 0] [0] [0 0 0] [0 0 0]
[1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1]
mark(U12(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = a__U12(mark(X1),X2)
[0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0]
[1 0 0] [1] [1 0 0] [1]
mark(length(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__length(mark(X))
[0 0 0] [0] [0 0 0] [0]
[1 0 1] [1 1 0] [1 0 0] [1 1 0]
mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(mark(X1),X2)
[0 0 0] [0 0 0] [0 0 0] [0 0 0]
[1 0 0] [1 0 0]
mark(s(X)) = [0 0 0]X >= [0 0 0]X = s(mark(X))
[0 0 0] [0 0 0]
[1 0 1] [1 0 0] [1] [1 0 0] [1 0 0] [1]
a__U12(X1,X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = U12(X1,X2)
[0 0 1] [0 0 0] [0] [0 0 0] [0 0 0] [0]
[1 0 0] [1] [1 0 0] [1]
a__length(X) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(X)
[0 0 0] [0] [0 0 0] [0]
problem:
a__zeros() -> cons(0(),zeros())
a__U11(tt(),L) -> a__U12(tt(),L)
a__U12(tt(),L) -> s(a__length(mark(L)))
a__length(cons(N,L)) -> a__U11(tt(),L)
mark(zeros()) -> a__zeros()
mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
mark(length(X)) -> a__length(mark(X))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(s(X)) -> s(mark(X))
a__U12(X1,X2) -> U12(X1,X2)
a__length(X) -> length(X)
Matrix Interpretation Processor: dim=1
interpretation:
[length](x0) = 2x0 + 2,
[U12](x0, x1) = x0 + 2x1 + 1,
[s](x0) = x0,
[a__length](x0) = 2x0 + 2,
[mark](x0) = 2x0,
[a__U12](x0, x1) = x0 + 4x1 + 2,
[a__U11](x0, x1) = 4x0 + 5x1 + 2,
[tt] = 0,
[cons](x0, x1) = 4x0 + 4x1,
[zeros] = 0,
[0] = 0,
[a__zeros] = 0
orientation:
a__zeros() = 0 >= 0 = cons(0(),zeros())
a__U11(tt(),L) = 5L + 2 >= 4L + 2 = a__U12(tt(),L)
a__U12(tt(),L) = 4L + 2 >= 4L + 2 = s(a__length(mark(L)))
a__length(cons(N,L)) = 8L + 8N + 2 >= 5L + 2 = a__U11(tt(),L)
mark(zeros()) = 0 >= 0 = a__zeros()
mark(U12(X1,X2)) = 2X1 + 4X2 + 2 >= 2X1 + 4X2 + 2 = a__U12(mark(X1),X2)
mark(length(X)) = 4X + 4 >= 4X + 2 = a__length(mark(X))
mark(cons(X1,X2)) = 8X1 + 8X2 >= 8X1 + 4X2 = cons(mark(X1),X2)
mark(s(X)) = 2X >= 2X = s(mark(X))
a__U12(X1,X2) = X1 + 4X2 + 2 >= X1 + 2X2 + 1 = U12(X1,X2)
a__length(X) = 2X + 2 >= 2X + 2 = length(X)
problem:
a__zeros() -> cons(0(),zeros())
a__U11(tt(),L) -> a__U12(tt(),L)
a__U12(tt(),L) -> s(a__length(mark(L)))
a__length(cons(N,L)) -> a__U11(tt(),L)
mark(zeros()) -> a__zeros()
mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(s(X)) -> s(mark(X))
a__length(X) -> length(X)
Matrix Interpretation Processor: dim=1
interpretation:
[length](x0) = x0,
[U12](x0, x1) = 4x0 + 4x1 + 2,
[s](x0) = x0,
[a__length](x0) = x0,
[mark](x0) = 2x0,
[a__U12](x0, x1) = x0 + 2x1,
[a__U11](x0, x1) = x0 + 2x1,
[tt] = 0,
[cons](x0, x1) = 4x0 + 2x1,
[zeros] = 2,
[0] = 0,
[a__zeros] = 4
orientation:
a__zeros() = 4 >= 4 = cons(0(),zeros())
a__U11(tt(),L) = 2L >= 2L = a__U12(tt(),L)
a__U12(tt(),L) = 2L >= 2L = s(a__length(mark(L)))
a__length(cons(N,L)) = 2L + 4N >= 2L = a__U11(tt(),L)
mark(zeros()) = 4 >= 4 = a__zeros()
mark(U12(X1,X2)) = 8X1 + 8X2 + 4 >= 2X1 + 2X2 = a__U12(mark(X1),X2)
mark(cons(X1,X2)) = 8X1 + 4X2 >= 8X1 + 2X2 = cons(mark(X1),X2)
mark(s(X)) = 2X >= 2X = s(mark(X))
a__length(X) = X >= X = length(X)
problem:
a__zeros() -> cons(0(),zeros())
a__U11(tt(),L) -> a__U12(tt(),L)
a__U12(tt(),L) -> s(a__length(mark(L)))
a__length(cons(N,L)) -> a__U11(tt(),L)
mark(zeros()) -> a__zeros()
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(s(X)) -> s(mark(X))
a__length(X) -> length(X)
Unfolding Processor:
loop length: 5
terms:
a__U11(tt(),zeros())
a__U12(tt(),zeros())
s(a__length(mark(zeros())))
s(a__length(a__zeros()))
s(a__length(cons(0(),zeros())))
context: s([])
substitution:
Qed