YES(O(1), O(n^1)) 0.00/0.80 YES(O(1), O(n^1)) 0.00/0.82 0.00/0.82 0.00/0.82
0.00/0.82 0.00/0.820 CpxTRS0.00/0.82
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))0.00/0.82
↳2 CdtProblem0.00/0.82
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))0.00/0.82
↳4 CdtProblem0.00/0.82
↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))0.00/0.82
↳6 CdtProblem0.00/0.82
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))0.00/0.82
↳8 CdtProblem0.00/0.82
↳9 CdtKnowledgeProof (⇔)0.00/0.82
↳10 BOUNDS(O(1), O(1))0.00/0.82
a__f(f(a)) → a__f(g(f(a))) 0.00/0.82
mark(f(X)) → a__f(mark(X)) 0.00/0.82
mark(a) → a 0.00/0.82
mark(g(X)) → g(X) 0.00/0.82
a__f(X) → f(X)
Tuples:
a__f(f(a)) → a__f(g(f(a))) 0.00/0.82
a__f(z0) → f(z0) 0.00/0.82
mark(f(z0)) → a__f(mark(z0)) 0.00/0.82
mark(a) → a 0.00/0.82
mark(g(z0)) → g(z0)
S tuples:
A__F(f(a)) → c(A__F(g(f(a)))) 0.00/0.82
MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))
K tuples:none
A__F(f(a)) → c(A__F(g(f(a)))) 0.00/0.82
MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))
a__f, mark
A__F, MARK
c, c2
Tuples:
a__f(f(a)) → a__f(g(f(a))) 0.00/0.82
a__f(z0) → f(z0) 0.00/0.82
mark(f(z0)) → a__f(mark(z0)) 0.00/0.82
mark(a) → a 0.00/0.82
mark(g(z0)) → g(z0)
S tuples:
MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0)) 0.00/0.82
A__F(f(a)) → c
K tuples:none
MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0)) 0.00/0.82
A__F(f(a)) → c
a__f, mark
MARK, A__F
c2, c
A__F(f(a)) → c
Tuples:
a__f(f(a)) → a__f(g(f(a))) 0.00/0.82
a__f(z0) → f(z0) 0.00/0.82
mark(f(z0)) → a__f(mark(z0)) 0.00/0.82
mark(a) → a 0.00/0.82
mark(g(z0)) → g(z0)
S tuples:
MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0)) 0.00/0.82
A__F(f(a)) → c
K tuples:none
MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0)) 0.00/0.82
A__F(f(a)) → c
a__f, mark
MARK, A__F
c2, c
We considered the (Usable) Rules:
MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))
And the Tuples:
mark(f(z0)) → a__f(mark(z0)) 0.00/0.82
mark(a) → a 0.00/0.82
mark(g(z0)) → g(z0) 0.00/0.82
a__f(f(a)) → a__f(g(f(a))) 0.00/0.82
a__f(z0) → f(z0)
The order we found is given by the following interpretation:
MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0)) 0.00/0.82
A__F(f(a)) → c
POL(A__F(x1)) = 0 0.00/0.82
POL(MARK(x1)) = [4]x1 0.00/0.82
POL(a) = 0 0.00/0.82
POL(a__f(x1)) = [3] 0.00/0.82
POL(c) = 0 0.00/0.82
POL(c2(x1, x2)) = x1 + x2 0.00/0.82
POL(f(x1)) = [4] + x1 0.00/0.82
POL(g(x1)) = [3] 0.00/0.82
POL(mark(x1)) = 0
Tuples:
a__f(f(a)) → a__f(g(f(a))) 0.00/0.82
a__f(z0) → f(z0) 0.00/0.82
mark(f(z0)) → a__f(mark(z0)) 0.00/0.82
mark(a) → a 0.00/0.82
mark(g(z0)) → g(z0)
S tuples:
MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0)) 0.00/0.82
A__F(f(a)) → c
K tuples:
A__F(f(a)) → c
Defined Rule Symbols:
MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))
a__f, mark
MARK, A__F
c2, c
Now S is empty
A__F(f(a)) → c