0 QTRS
↳1 QTRSToCSRProof (⇔)
↳2 CSR
↳3 PoloCSRProof (⇔)
↳4 CSR
↳5 PoloCSRProof (⇔)
↳6 CSR
↳7 RisEmptyProof (⇔)
↳8 TRUE
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U12: {1}
isNePal: {1}
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).
__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → U12(tt)
U12(tt) → tt
isNePal(__(I, __(P, I))) → U11(tt)
__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U12: {1}
isNePal: {1}
Used ordering:
__(X, nil) → X
__(nil, X) → X
isNePal(__(I, __(P, I))) → U11(tt)
POL(U11(x1)) = 2·x1
POL(U12(x1)) = x1
POL(__(x1, x2)) = x1 + x2
POL(isNePal(x1)) = 1 + 2·x1
POL(nil) = 1
POL(tt) = 0
__(__(X, Y), Z) → __(X, __(Y, Z))
U11(tt) → U12(tt)
U12(tt) → tt
__: {1, 2}
U11: {1}
tt: empty set
U12: {1}
Used ordering:
__(__(X, Y), Z) → __(X, __(Y, Z))
U11(tt) → U12(tt)
U12(tt) → tt
POL(U11(x1)) = 2 + 2·x1
POL(U12(x1)) = 1 + 2·x1
POL(__(x1, x2)) = 1 + 2·x1 + x2
POL(tt) = 2