0 QTRS
↳1 QTRSToCSRProof (⇔, 0 ms)
↳2 CSR
↳3 CSRRRRProof (⇔, 54 ms)
↳4 CSR
↳5 CSRRRRProof (⇔, 3 ms)
↳6 CSR
↳7 ContextSensitiveLoopProof (⇔, 0 ms)
↳8 NO
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(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(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(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(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
s: {1}
length: {1}
U21: {1}
U22: {1}
U23: {1}
take: {1, 2}
nil: empty set
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).
zeros → cons(0, zeros)
U11(tt, L) → U12(tt, L)
U12(tt, L) → s(length(L))
U21(tt, IL, M, N) → U22(tt, IL, M, N)
U22(tt, IL, M, N) → U23(tt, IL, M, N)
U23(tt, IL, M, N) → cons(N, take(M, IL))
length(nil) → 0
length(cons(N, L)) → U11(tt, L)
take(0, IL) → nil
take(s(M), cons(N, IL)) → U21(tt, IL, M, N)
zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
s: {1}
length: {1}
U21: {1}
U22: {1}
U23: {1}
take: {1, 2}
nil: empty set
zeros → cons(0, zeros)
U11(tt, L) → U12(tt, L)
U12(tt, L) → s(length(L))
U21(tt, IL, M, N) → U22(tt, IL, M, N)
U22(tt, IL, M, N) → U23(tt, IL, M, N)
U23(tt, IL, M, N) → cons(N, take(M, IL))
length(nil) → 0
length(cons(N, L)) → U11(tt, L)
take(0, IL) → nil
take(s(M), cons(N, IL)) → U21(tt, IL, M, N)
zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
s: {1}
length: {1}
U21: {1}
U22: {1}
U23: {1}
take: {1, 2}
nil: empty set
Used ordering:
Polynomial interpretation [POLO]:
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
POL(0) = 0
POL(U11(x1, x2)) = x1 + x2
POL(U12(x1, x2)) = x1 + x2
POL(U21(x1, x2, x3, x4)) = 1 + x1 + x2 + x3 + x4
POL(U22(x1, x2, x3, x4)) = 1 + x1 + x2 + x3 + x4
POL(U23(x1, x2, x3, x4)) = 1 + x1 + x2 + x3 + x4
POL(cons(x1, x2)) = x1 + x2
POL(length(x1)) = x1
POL(nil) = 1
POL(s(x1)) = x1
POL(take(x1, x2)) = 1 + x1 + x2
POL(tt) = 0
POL(zeros) = 0
length(nil) → 0
zeros → cons(0, zeros)
U11(tt, L) → U12(tt, L)
U12(tt, L) → s(length(L))
U21(tt, IL, M, N) → U22(tt, IL, M, N)
U22(tt, IL, M, N) → U23(tt, IL, M, N)
U23(tt, IL, M, N) → cons(N, take(M, IL))
length(cons(N, L)) → U11(tt, L)
take(0, IL) → nil
take(s(M), cons(N, IL)) → U21(tt, IL, M, N)
zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
s: {1}
length: {1}
U21: {1}
U22: {1}
U23: {1}
take: {1, 2}
nil: empty set
zeros → cons(0, zeros)
U11(tt, L) → U12(tt, L)
U12(tt, L) → s(length(L))
U21(tt, IL, M, N) → U22(tt, IL, M, N)
U22(tt, IL, M, N) → U23(tt, IL, M, N)
U23(tt, IL, M, N) → cons(N, take(M, IL))
length(cons(N, L)) → U11(tt, L)
take(0, IL) → nil
take(s(M), cons(N, IL)) → U21(tt, IL, M, N)
zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
s: {1}
length: {1}
U21: {1}
U22: {1}
U23: {1}
take: {1, 2}
nil: empty set
Used ordering:
Polynomial interpretation [POLO]:
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
POL(0) = 0
POL(U11(x1, x2)) = x1 + x2
POL(U12(x1, x2)) = x1 + x2
POL(U21(x1, x2, x3, x4)) = 1 + x1 + x2 + x3 + x4
POL(U22(x1, x2, x3, x4)) = 1 + x1 + x2 + x3 + x4
POL(U23(x1, x2, x3, x4)) = 1 + x1 + x2 + x3 + x4
POL(cons(x1, x2)) = x1 + x2
POL(length(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 1 + x1 + x2
POL(tt) = 0
POL(zeros) = 0
take(0, IL) → nil
zeros → cons(0, zeros)
U11(tt, L) → U12(tt, L)
U12(tt, L) → s(length(L))
U21(tt, IL, M, N) → U22(tt, IL, M, N)
U22(tt, IL, M, N) → U23(tt, IL, M, N)
U23(tt, IL, M, N) → cons(N, take(M, IL))
length(cons(N, L)) → U11(tt, L)
take(s(M), cons(N, IL)) → U21(tt, IL, M, N)
zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
s: {1}
length: {1}
U21: {1}
U22: {1}
U23: {1}
take: {1, 2}
zeros → cons(0, zeros)
U11(tt, L) → U12(tt, L)
U12(tt, L) → s(length(L))
U21(tt, IL, M, N) → U22(tt, IL, M, N)
U22(tt, IL, M, N) → U23(tt, IL, M, N)
U23(tt, IL, M, N) → cons(N, take(M, IL))
length(cons(N, L)) → U11(tt, L)
take(s(M), cons(N, IL)) → U21(tt, IL, M, N)