YES
0 QTRS
↳1 QTRSToCSRProof (⇔, 0 ms)
↳2 CSR
↳3 CSRRRRProof (⇔, 34 ms)
↳4 CSR
↳5 CSRRRRProof (⇔, 0 ms)
↳6 CSR
↳7 RisEmptyProof (⇔, 0 ms)
↳8 YES
active(f(b, X, c)) → mark(f(X, c, X))
active(c) → mark(b)
active(f(X1, X2, X3)) → f(X1, active(X2), X3)
f(X1, mark(X2), X3) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(f(b, X, c)) → mark(f(X, c, X))
active(c) → mark(b)
active(f(X1, X2, X3)) → f(X1, active(X2), X3)
f(X1, mark(X2), X3) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
f: {2}
b: empty set
c: empty set
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).
f(b, X, c) → f(X, c, X)
c → b
f: {2}
b: empty set
c: empty set
f(b, X, c) → f(X, c, X)
c → b
f: {2}
b: empty set
c: 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(b) = 1
POL(c) = 2
POL(f(x1, x2, x3)) = 2·x2 + 2·x3
c → b
f(b, X, c) → f(X, c, X)
f: {2}
b: empty set
c: empty set
f(b, X, c) → f(X, c, X)
f: {2}
b: empty set
c: 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(b) = 1
POL(c) = 0
POL(f(x1, x2, x3)) = x1 + x2
f(b, X, c) → f(X, c, X)