0 QTRS
↳1 QTRSToCSRProof (⇔)
↳2 CSR
↳3 CSRRRRProof (⇔)
↳4 CSR
↳5 ContextSensitiveLoopProof (⇔)
↳6 NO
active(zeros) → mark(cons(0, zeros))
active(and(tt, X)) → mark(X)
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(s(length(L)))
active(cons(X1, X2)) → cons(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(length(X)) → length(active(X))
active(s(X)) → s(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
length(mark(X)) → mark(length(X))
s(mark(X)) → mark(s(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
length(ok(X)) → ok(length(X))
s(ok(X)) → ok(s(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(zeros) → mark(cons(0, zeros))
active(and(tt, X)) → mark(X)
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(s(length(L)))
active(cons(X1, X2)) → cons(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(length(X)) → length(active(X))
active(s(X)) → s(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
length(mark(X)) → mark(length(X))
s(mark(X)) → mark(s(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
length(ok(X)) → ok(length(X))
s(ok(X)) → ok(s(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
zeros: empty set
cons: {1}
0: empty set
and: {1}
tt: empty set
length: {1}
nil: empty set
s: {1}
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).
zeros → cons(0, zeros)
and(tt, X) → X
length(nil) → 0
length(cons(N, L)) → s(length(L))
zeros: empty set
cons: {1}
0: empty set
and: {1}
tt: empty set
length: {1}
nil: empty set
s: {1}
zeros → cons(0, zeros)
and(tt, X) → X
length(nil) → 0
length(cons(N, L)) → s(length(L))
zeros: empty set
cons: {1}
0: empty set
and: {1}
tt: empty set
length: {1}
nil: empty set
s: {1}
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(and(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = x1 + x2
POL(length(x1)) = 1 + x1
POL(nil) = 1
POL(s(x1)) = x1
POL(tt) = 1
POL(zeros) = 0
and(tt, X) → X
length(nil) → 0
zeros → cons(0, zeros)
length(cons(N, L)) → s(length(L))
zeros: empty set
cons: {1}
0: empty set
length: {1}
s: {1}
zeros → cons(0, zeros)
length(cons(N, L)) → s(length(L))