YES(O(1), O(n^1)) 77.09/27.14 YES(O(1), O(n^1)) 77.09/27.16 77.09/27.16 77.09/27.16
77.09/27.16 77.09/27.160 CpxTRS77.09/27.16
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))77.09/27.16
↳2 CdtProblem77.09/27.16
↳3 CdtNarrowingProof (BOTH BOUNDS(ID, ID))77.09/27.16
↳4 CdtProblem77.09/27.16
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))77.09/27.16
↳6 CdtProblem77.09/27.16
↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID))77.09/27.16
↳8 CdtProblem77.09/27.16
↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))77.09/27.16
↳10 CdtProblem77.09/27.16
↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID))77.09/27.16
↳12 CdtProblem77.09/27.16
↳13 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))77.09/27.16
↳14 CdtProblem77.09/27.16
↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))77.09/27.16
↳16 CdtProblem77.09/27.16
↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))77.09/27.16
↳18 CdtProblem77.09/27.16
↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))77.09/27.16
↳20 CdtProblem77.09/27.16
↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID))77.09/27.16
↳22 CdtProblem77.09/27.16
↳23 CdtUnreachableProof (⇔)77.09/27.16
↳24 CdtProblem77.09/27.16
↳25 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))77.09/27.16
↳26 CdtProblem77.09/27.16
↳27 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))77.09/27.16
↳28 CdtProblem77.09/27.16
↳29 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))77.09/27.16
↳30 CdtProblem77.09/27.16
↳31 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))77.09/27.16
↳32 CdtProblem77.09/27.16
↳33 SIsEmptyProof (BOTH BOUNDS(ID, ID))77.09/27.16
↳34 BOUNDS(O(1), O(1))77.09/27.16
active(f(a, b, X)) → mark(f(X, X, X)) 77.09/27.16
active(c) → mark(a) 77.09/27.16
active(c) → mark(b) 77.09/27.16
active(f(X1, X2, X3)) → f(X1, X2, active(X3)) 77.09/27.16
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3)) 77.09/27.16
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3)) 77.09/27.16
proper(a) → ok(a) 77.09/27.16
proper(b) → ok(b) 77.09/27.16
proper(c) → ok(c) 77.09/27.16
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3)) 77.09/27.16
top(mark(X)) → top(proper(X)) 77.09/27.16
top(ok(X)) → top(active(X))
Tuples:
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.09/27.19
active(c) → mark(a) 77.09/27.19
active(c) → mark(b) 77.09/27.19
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.09/27.19
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.09/27.19
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.09/27.19
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.09/27.19
proper(a) → ok(a) 77.09/27.19
proper(b) → ok(b) 77.09/27.19
proper(c) → ok(c) 77.09/27.19
top(mark(z0)) → top(proper(z0)) 77.09/27.19
top(ok(z0)) → top(active(z0))
S tuples:
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.09/27.19
ACTIVE(f(z0, z1, z2)) → c4(F(z0, z1, active(z2)), ACTIVE(z2)) 77.09/27.19
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.09/27.19
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.09/27.19
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 77.09/27.19
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.09/27.19
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
K tuples:none
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.09/27.19
ACTIVE(f(z0, z1, z2)) → c4(F(z0, z1, active(z2)), ACTIVE(z2)) 77.09/27.19
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.09/27.19
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.09/27.19
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 77.09/27.19
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.09/27.19
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
active, f, proper, top
ACTIVE, F, PROPER, TOP
c1, c4, c5, c6, c7, c11, c12
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a)), ACTIVE(c)) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b)), ACTIVE(c)) 77.09/27.19
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
Tuples:
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.09/27.19
active(c) → mark(a) 77.09/27.19
active(c) → mark(b) 77.09/27.19
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.09/27.19
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.09/27.19
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.09/27.19
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.09/27.19
proper(a) → ok(a) 77.09/27.19
proper(b) → ok(b) 77.09/27.19
proper(c) → ok(c) 77.09/27.19
top(mark(z0)) → top(proper(z0)) 77.09/27.19
top(ok(z0)) → top(active(z0))
S tuples:
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.09/27.19
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.09/27.19
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.09/27.19
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 77.09/27.19
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.09/27.19
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.09/27.19
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a)), ACTIVE(c)) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b)), ACTIVE(c)) 77.09/27.19
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
K tuples:none
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.09/27.19
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.09/27.19
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.09/27.19
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 77.09/27.19
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.09/27.19
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.09/27.19
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a)), ACTIVE(c)) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b)), ACTIVE(c)) 77.09/27.19
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
active, f, proper, top
ACTIVE, F, PROPER, TOP
c1, c5, c6, c7, c11, c12, c4
Tuples:
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.09/27.19
active(c) → mark(a) 77.09/27.19
active(c) → mark(b) 77.09/27.19
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.09/27.19
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.09/27.19
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.09/27.19
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.09/27.19
proper(a) → ok(a) 77.09/27.19
proper(b) → ok(b) 77.09/27.19
proper(c) → ok(c) 77.09/27.19
top(mark(z0)) → top(proper(z0)) 77.09/27.19
top(ok(z0)) → top(active(z0))
S tuples:
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.09/27.19
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.09/27.19
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.09/27.19
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 77.09/27.19
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.09/27.19
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.09/27.19
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.09/27.19
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b)))
K tuples:none
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.09/27.19
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.09/27.19
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.09/27.19
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 77.09/27.19
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.09/27.19
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.09/27.19
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.09/27.19
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.09/27.19
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b)))
active, f, proper, top
ACTIVE, F, PROPER, TOP
c1, c5, c6, c7, c11, c12, c4, c4
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.22
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1), PROPER(a)) 77.40/27.22
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b)) 77.40/27.22
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c)) 77.40/27.22
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.22
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(a), PROPER(x2)) 77.40/27.22
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2)) 77.40/27.22
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2)) 77.40/27.22
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(a), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
Tuples:
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.22
active(c) → mark(a) 77.40/27.22
active(c) → mark(b) 77.40/27.22
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.22
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.22
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.22
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.22
proper(a) → ok(a) 77.40/27.22
proper(b) → ok(b) 77.40/27.22
proper(c) → ok(c) 77.40/27.22
top(mark(z0)) → top(proper(z0)) 77.40/27.22
top(ok(z0)) → top(active(z0))
S tuples:
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.22
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.22
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.22
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.40/27.22
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.22
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.22
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.22
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.22
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1), PROPER(a)) 77.40/27.22
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b)) 77.40/27.22
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c)) 77.40/27.22
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.22
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(a), PROPER(x2)) 77.40/27.22
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2)) 77.40/27.22
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2)) 77.40/27.22
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(a), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
K tuples:none
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.22
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.22
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.22
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.40/27.22
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.22
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.22
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.22
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.22
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1), PROPER(a)) 77.40/27.22
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b)) 77.40/27.22
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c)) 77.40/27.22
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.22
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(a), PROPER(x2)) 77.40/27.22
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2)) 77.40/27.22
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2)) 77.40/27.22
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(a), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
active, f, proper, top
ACTIVE, F, TOP, PROPER
c1, c5, c6, c11, c12, c4, c4, c7
Tuples:
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.22
active(c) → mark(a) 77.40/27.22
active(c) → mark(b) 77.40/27.22
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.22
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.22
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.22
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.22
proper(a) → ok(a) 77.40/27.22
proper(b) → ok(b) 77.40/27.22
proper(c) → ok(c) 77.40/27.22
top(mark(z0)) → top(proper(z0)) 77.40/27.22
top(ok(z0)) → top(active(z0))
S tuples:
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.22
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.22
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.22
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.40/27.22
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.22
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.22
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.22
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.22
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.22
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.22
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.22
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.22
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.22
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.22
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.22
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:none
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.22
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.22
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.22
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) 77.40/27.22
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.22
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.22
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.22
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.22
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.22
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.22
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.22
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.22
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.22
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.22
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.22
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
active, f, proper, top
ACTIVE, F, TOP, PROPER
c1, c5, c6, c11, c12, c4, c4, c7, c7
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.22
TOP(mark(a)) → c11(TOP(ok(a)), PROPER(a)) 77.40/27.22
TOP(mark(b)) → c11(TOP(ok(b)), PROPER(b)) 77.40/27.22
TOP(mark(c)) → c11(TOP(ok(c)), PROPER(c))
Tuples:
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.22
active(c) → mark(a) 77.40/27.22
active(c) → mark(b) 77.40/27.22
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.22
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.22
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.22
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.22
proper(a) → ok(a) 77.40/27.22
proper(b) → ok(b) 77.40/27.22
proper(c) → ok(c) 77.40/27.22
top(mark(z0)) → top(proper(z0)) 77.40/27.22
top(ok(z0)) → top(active(z0))
S tuples:
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.22
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.22
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.22
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.22
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.22
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.22
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.22
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.22
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.22
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.22
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.22
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.22
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.22
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.22
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.22
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.22
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.22
TOP(mark(a)) → c11(TOP(ok(a)), PROPER(a)) 77.40/27.22
TOP(mark(b)) → c11(TOP(ok(b)), PROPER(b)) 77.40/27.22
TOP(mark(c)) → c11(TOP(ok(c)), PROPER(c))
K tuples:none
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.22
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.22
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.22
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.22
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.22
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.22
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.22
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.22
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.22
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a)), PROPER(a)) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b)), PROPER(b)) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)), PROPER(c))
active, f, proper, top
ACTIVE, F, TOP, PROPER
c1, c5, c6, c12, c4, c4, c7, c7, c11
Tuples:
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
S tuples:
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
K tuples:none
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
active, f, proper, top
ACTIVE, F, TOP, PROPER
c1, c5, c6, c12, c4, c4, c7, c7, c11, c11
We considered the (Usable) Rules:
TOP(mark(b)) → c11(TOP(ok(b)))
And the Tuples:
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
The order we found is given by the following interpretation:
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
POL(ACTIVE(x1)) = 0 77.40/27.23
POL(F(x1, x2, x3)) = 0 77.40/27.23
POL(PROPER(x1)) = 0 77.40/27.23
POL(TOP(x1)) = x1 77.40/27.23
POL(a) = [2] 77.40/27.23
POL(active(x1)) = x1 77.40/27.23
POL(b) = 0 77.40/27.23
POL(c) = [2] 77.40/27.23
POL(c1(x1)) = x1 77.40/27.23
POL(c11(x1)) = x1 77.40/27.23
POL(c11(x1, x2)) = x1 + x2 77.40/27.23
POL(c12(x1, x2)) = x1 + x2 77.40/27.23
POL(c4(x1)) = x1 77.40/27.23
POL(c4(x1, x2)) = x1 + x2 77.40/27.23
POL(c5(x1)) = x1 77.40/27.23
POL(c6(x1)) = x1 77.40/27.23
POL(c7(x1, x2, x3)) = x1 + x2 + x3 77.40/27.23
POL(c7(x1, x2, x3, x4)) = x1 + x2 + x3 + x4 77.40/27.23
POL(f(x1, x2, x3)) = [2] 77.40/27.23
POL(mark(x1)) = [2] 77.40/27.23
POL(ok(x1)) = x1 77.40/27.23
POL(proper(x1)) = [2]x1
Tuples:
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
S tuples:
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
K tuples:
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
Defined Rule Symbols:
TOP(mark(b)) → c11(TOP(ok(b)))
active, f, proper, top
ACTIVE, F, TOP, PROPER
c1, c5, c6, c12, c4, c4, c7, c7, c11, c11
We considered the (Usable) Rules:
TOP(mark(c)) → c11(TOP(ok(c)))
And the Tuples:
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
The order we found is given by the following interpretation:
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
POL(ACTIVE(x1)) = 0 77.40/27.23
POL(F(x1, x2, x3)) = 0 77.40/27.23
POL(PROPER(x1)) = 0 77.40/27.23
POL(TOP(x1)) = [2]x1 77.40/27.23
POL(a) = 0 77.40/27.23
POL(active(x1)) = 0 77.40/27.23
POL(b) = 0 77.40/27.23
POL(c) = [2] 77.40/27.23
POL(c1(x1)) = x1 77.40/27.23
POL(c11(x1)) = x1 77.40/27.23
POL(c11(x1, x2)) = x1 + x2 77.40/27.23
POL(c12(x1, x2)) = x1 + x2 77.40/27.23
POL(c4(x1)) = x1 77.40/27.23
POL(c4(x1, x2)) = x1 + x2 77.40/27.23
POL(c5(x1)) = x1 77.40/27.23
POL(c6(x1)) = x1 77.40/27.23
POL(c7(x1, x2, x3)) = x1 + x2 + x3 77.40/27.23
POL(c7(x1, x2, x3, x4)) = x1 + x2 + x3 + x4 77.40/27.23
POL(f(x1, x2, x3)) = 0 77.40/27.23
POL(mark(x1)) = x1 77.40/27.23
POL(ok(x1)) = 0 77.40/27.23
POL(proper(x1)) = 0
Tuples:
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
S tuples:
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
K tuples:
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a)))
Defined Rule Symbols:
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
active, f, proper, top
ACTIVE, F, TOP, PROPER
c1, c5, c6, c12, c4, c4, c7, c7, c11, c11
We considered the (Usable) Rules:
TOP(mark(a)) → c11(TOP(ok(a)))
And the Tuples:
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2))
The order we found is given by the following interpretation:
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
POL(ACTIVE(x1)) = 0 77.40/27.23
POL(F(x1, x2, x3)) = 0 77.40/27.23
POL(PROPER(x1)) = 0 77.40/27.23
POL(TOP(x1)) = [4]x1 77.40/27.23
POL(a) = 0 77.40/27.23
POL(active(x1)) = x1 77.40/27.23
POL(b) = 0 77.40/27.23
POL(c) = [2] 77.40/27.23
POL(c1(x1)) = x1 77.40/27.23
POL(c11(x1)) = x1 77.40/27.23
POL(c11(x1, x2)) = x1 + x2 77.40/27.23
POL(c12(x1, x2)) = x1 + x2 77.40/27.23
POL(c4(x1)) = x1 77.40/27.23
POL(c4(x1, x2)) = x1 + x2 77.40/27.23
POL(c5(x1)) = x1 77.40/27.23
POL(c6(x1)) = x1 77.40/27.23
POL(c7(x1, x2, x3)) = x1 + x2 + x3 77.40/27.23
POL(c7(x1, x2, x3, x4)) = x1 + x2 + x3 + x4 77.40/27.23
POL(f(x1, x2, x3)) = [2] 77.40/27.23
POL(mark(x1)) = [2] 77.40/27.23
POL(ok(x1)) = x1 77.40/27.23
POL(proper(x1)) = 0
Tuples:
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
S tuples:
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c)))
K tuples:
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
Defined Rule Symbols:
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a)))
active, f, proper, top
ACTIVE, F, TOP, PROPER
c1, c5, c6, c12, c4, c4, c7, c7, c11, c11
TOP(ok(f(a, b, z0))) → c12(TOP(mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(a)), ACTIVE(c)) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(b)), ACTIVE(c)) 77.40/27.23
TOP(ok(f(z0, z1, z2))) → c12(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
Tuples:
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
S tuples:
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c))) 77.40/27.23
TOP(ok(f(a, b, z0))) → c12(TOP(mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(a)), ACTIVE(c)) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(b)), ACTIVE(c)) 77.40/27.23
TOP(ok(f(z0, z1, z2))) → c12(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
K tuples:
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(ok(f(a, b, z0))) → c12(TOP(mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(a)), ACTIVE(c)) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(b)), ACTIVE(c)) 77.40/27.23
TOP(ok(f(z0, z1, z2))) → c12(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
Defined Rule Symbols:
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a)))
active, f, proper, top
ACTIVE, F, PROPER, TOP
c1, c5, c6, c4, c4, c7, c7, c11, c11, c12
ACTIVE(f(a, b, z0)) → c1(F(z0, z0, z0)) 77.40/27.23
ACTIVE(f(x0, x1, f(a, b, z0))) → c4(F(x0, x1, mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
ACTIVE(f(x0, x1, f(z0, z1, z2))) → c4(F(x0, x1, f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(a))) 77.40/27.23
ACTIVE(f(x0, x1, c)) → c4(F(x0, x1, mark(b))) 77.40/27.23
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2))) 77.40/27.23
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2)) 77.40/27.23
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(x0, x1, a)) → c7(F(proper(x0), proper(x1), ok(a)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 77.40/27.23
PROPER(f(x0, a, x2)) → c7(F(proper(x0), ok(a), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 77.40/27.23
PROPER(f(a, x1, x2)) → c7(F(ok(a), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 77.40/27.23
TOP(mark(f(z0, z1, z2))) → c11(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2))) 77.40/27.23
TOP(ok(f(a, b, z0))) → c12(TOP(mark(f(z0, z0, z0))), ACTIVE(f(a, b, z0))) 77.40/27.23
TOP(ok(f(z0, z1, z2))) → c12(TOP(f(z0, z1, active(z2))), ACTIVE(f(z0, z1, z2)))
Tuples:
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
S tuples:
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a))) 77.40/27.23
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c))) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(a)), ACTIVE(c)) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(b)), ACTIVE(c))
K tuples:
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(a)), ACTIVE(c)) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(b)), ACTIVE(c))
Defined Rule Symbols:
TOP(mark(b)) → c11(TOP(ok(b))) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c))) 77.40/27.23
TOP(mark(a)) → c11(TOP(ok(a)))
active, f, proper, top
F, TOP
c5, c6, c11, c12
Tuples:
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
S tuples:
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c))) 77.40/27.23
TOP(mark(a)) → c11 77.40/27.23
TOP(mark(b)) → c11 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(a))) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(b)))
K tuples:
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2)) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(a))) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(b)))
Defined Rule Symbols:
TOP(mark(c)) → c11(TOP(ok(c))) 77.40/27.23
TOP(mark(a)) → c11 77.40/27.23
TOP(mark(b)) → c11
active, f, proper, top
F, TOP
c5, c6, c11, c11, c12
TOP(ok(c)) → c12(TOP(mark(b))) 77.40/27.23
TOP(mark(b)) → c11 77.40/27.23
TOP(mark(a)) → c11 77.40/27.23
TOP(mark(c)) → c11(TOP(ok(c))) 77.40/27.23
TOP(ok(c)) → c12(TOP(mark(a)))
Tuples:
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
S tuples:
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
K tuples:none
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
active, f, proper, top
F
c5, c6
We considered the (Usable) Rules:none
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
The order we found is given by the following interpretation:
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
POL(F(x1, x2, x3)) = x1 + x2 + [2]x3 77.40/27.23
POL(c5(x1)) = x1 77.40/27.23
POL(c6(x1)) = x1 77.40/27.23
POL(mark(x1)) = [1] + x1 77.40/27.23
POL(ok(x1)) = x1
Tuples:
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
S tuples:
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
K tuples:
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
Defined Rule Symbols:
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2))
active, f, proper, top
F
c5, c6
We considered the (Usable) Rules:none
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
The order we found is given by the following interpretation:
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
POL(F(x1, x2, x3)) = x1 77.40/27.23
POL(c5(x1)) = x1 77.40/27.23
POL(c6(x1)) = x1 77.40/27.23
POL(mark(x1)) = x1 77.40/27.23
POL(ok(x1)) = [1] + x1
Tuples:
active(f(a, b, z0)) → mark(f(z0, z0, z0)) 77.40/27.23
active(c) → mark(a) 77.40/27.23
active(c) → mark(b) 77.40/27.23
active(f(z0, z1, z2)) → f(z0, z1, active(z2)) 77.40/27.23
f(z0, z1, mark(z2)) → mark(f(z0, z1, z2)) 77.40/27.23
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2)) 77.40/27.23
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2)) 77.40/27.23
proper(a) → ok(a) 77.40/27.23
proper(b) → ok(b) 77.40/27.23
proper(c) → ok(c) 77.40/27.23
top(mark(z0)) → top(proper(z0)) 77.40/27.23
top(ok(z0)) → top(active(z0))
S tuples:none
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
Defined Rule Symbols:
F(z0, z1, mark(z2)) → c5(F(z0, z1, z2)) 77.40/27.23
F(ok(z0), ok(z1), ok(z2)) → c6(F(z0, z1, z2))
active, f, proper, top
F
c5, c6