YES(O(1), O(n^1)) 0.00/0.82 YES(O(1), O(n^1)) 0.00/0.85 0.00/0.85 0.00/0.85
0.00/0.85 0.00/0.850 CpxTRS0.00/0.85
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))0.00/0.85
↳2 CdtProblem0.00/0.85
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))0.00/0.85
↳4 CdtProblem0.00/0.85
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))0.00/0.85
↳6 CdtProblem0.00/0.85
↳7 CdtKnowledgeProof (⇔)0.00/0.85
↳8 BOUNDS(O(1), O(1))0.00/0.85
U11(tt, M, N) → U12(tt, activate(M), activate(N)) 0.00/0.85
U12(tt, M, N) → s(plus(activate(N), activate(M))) 0.00/0.85
plus(N, 0) → N 0.00/0.85
plus(N, s(M)) → U11(tt, M, N) 0.00/0.85
activate(X) → X
Tuples:
U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1)) 0.00/0.85
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 0.00/0.85
plus(z0, 0) → z0 0.00/0.85
plus(z0, s(z1)) → U11(tt, z1, z0) 0.00/0.85
activate(z0) → z0
S tuples:
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 0.00/0.85
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 0.00/0.85
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
K tuples:none
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 0.00/0.85
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 0.00/0.85
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
U11, U12, plus, activate
U11', U12', PLUS
c, c1, c3
Tuples:
U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1)) 0.00/0.85
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 0.00/0.85
plus(z0, 0) → z0 0.00/0.85
plus(z0, s(z1)) → U11(tt, z1, z0) 0.00/0.85
activate(z0) → z0
S tuples:
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0)) 0.00/0.85
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 0.00/0.85
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
K tuples:none
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0)) 0.00/0.85
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 0.00/0.85
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U11, U12, plus, activate
PLUS, U11', U12'
c3, c, c1
We considered the (Usable) Rules:
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
And the Tuples:
activate(z0) → z0
The order we found is given by the following interpretation:
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0)) 0.00/0.85
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 0.00/0.85
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
POL(PLUS(x1, x2)) = x2 0.00/0.85
POL(U11'(x1, x2, x3)) = x1 + x2 0.00/0.85
POL(U12'(x1, x2, x3)) = x1 + x2 0.00/0.85
POL(activate(x1)) = x1 0.00/0.85
POL(c(x1)) = x1 0.00/0.85
POL(c1(x1)) = x1 0.00/0.85
POL(c3(x1)) = x1 0.00/0.85
POL(s(x1)) = [2] + x1 0.00/0.85
POL(tt) = 0
Tuples:
U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1)) 0.00/0.85
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 0.00/0.85
plus(z0, 0) → z0 0.00/0.85
plus(z0, s(z1)) → U11(tt, z1, z0) 0.00/0.85
activate(z0) → z0
S tuples:
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0)) 0.00/0.85
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 0.00/0.85
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
K tuples:
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 0.00/0.85
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
Defined Rule Symbols:
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
U11, U12, plus, activate
PLUS, U11', U12'
c3, c, c1
Now S is empty
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 0.00/0.85
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0))) 0.00/0.85
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0))) 0.00/0.85
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))