YES(O(1), O(n^2)) 0.00/0.89 YES(O(1), O(n^2)) 0.00/0.90 0.00/0.90 0.00/0.90
0.00/0.90 0.00/0.900 CpxTRS0.00/0.90
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))0.00/0.90
↳2 CdtProblem0.00/0.90
↳3 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))0.00/0.90
↳4 CdtProblem0.00/0.90
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))0.00/0.90
↳6 CdtProblem0.00/0.90
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))0.00/0.90
↳8 CdtProblem0.00/0.90
↳9 SIsEmptyProof (BOTH BOUNDS(ID, ID))0.00/0.90
↳10 BOUNDS(O(1), O(1))0.00/0.90
g(S(x), y) → g(x, S(y)) 0.00/0.90
f(y, S(x)) → f(S(y), x) 0.00/0.90
g(0, x2) → x2 0.00/0.90
f(x1, 0) → g(x1, 0)
Tuples:
g(S(z0), z1) → g(z0, S(z1)) 0.00/0.90
g(0, z0) → z0 0.00/0.90
f(z0, S(z1)) → f(S(z0), z1) 0.00/0.90
f(z0, 0) → g(z0, 0)
S tuples:
G(S(z0), z1) → c(G(z0, S(z1))) 0.00/0.90
F(z0, S(z1)) → c2(F(S(z0), z1)) 0.00/0.90
F(z0, 0) → c3(G(z0, 0))
K tuples:none
G(S(z0), z1) → c(G(z0, S(z1))) 0.00/0.90
F(z0, S(z1)) → c2(F(S(z0), z1)) 0.00/0.90
F(z0, 0) → c3(G(z0, 0))
g, f
G, F
c, c2, c3
We considered the (Usable) Rules:none
F(z0, 0) → c3(G(z0, 0))
The order we found is given by the following interpretation:
G(S(z0), z1) → c(G(z0, S(z1))) 0.00/0.90
F(z0, S(z1)) → c2(F(S(z0), z1)) 0.00/0.91
F(z0, 0) → c3(G(z0, 0))
POL(0) = [4] 0.00/0.91
POL(F(x1, x2)) = [3] + [4]x1 + [4]x2 0.00/0.91
POL(G(x1, x2)) = [3] + [2]x2 0.00/0.91
POL(S(x1)) = x1 0.00/0.91
POL(c(x1)) = x1 0.00/0.91
POL(c2(x1)) = x1 0.00/0.91
POL(c3(x1)) = x1
Tuples:
g(S(z0), z1) → g(z0, S(z1)) 0.00/0.91
g(0, z0) → z0 0.00/0.91
f(z0, S(z1)) → f(S(z0), z1) 0.00/0.91
f(z0, 0) → g(z0, 0)
S tuples:
G(S(z0), z1) → c(G(z0, S(z1))) 0.00/0.91
F(z0, S(z1)) → c2(F(S(z0), z1)) 0.00/0.91
F(z0, 0) → c3(G(z0, 0))
K tuples:
G(S(z0), z1) → c(G(z0, S(z1))) 0.00/0.91
F(z0, S(z1)) → c2(F(S(z0), z1))
Defined Rule Symbols:
F(z0, 0) → c3(G(z0, 0))
g, f
G, F
c, c2, c3
We considered the (Usable) Rules:none
G(S(z0), z1) → c(G(z0, S(z1)))
The order we found is given by the following interpretation:
G(S(z0), z1) → c(G(z0, S(z1))) 0.00/0.91
F(z0, S(z1)) → c2(F(S(z0), z1)) 0.00/0.91
F(z0, 0) → c3(G(z0, 0))
POL(0) = [4] 0.00/0.91
POL(F(x1, x2)) = [1] + [4]x1 + [4]x2 0.00/0.91
POL(G(x1, x2)) = [5] + [4]x1 + [2]x2 0.00/0.91
POL(S(x1)) = [4] + x1 0.00/0.91
POL(c(x1)) = x1 0.00/0.91
POL(c2(x1)) = x1 0.00/0.91
POL(c3(x1)) = x1
Tuples:
g(S(z0), z1) → g(z0, S(z1)) 0.00/0.91
g(0, z0) → z0 0.00/0.91
f(z0, S(z1)) → f(S(z0), z1) 0.00/0.91
f(z0, 0) → g(z0, 0)
S tuples:
G(S(z0), z1) → c(G(z0, S(z1))) 0.00/0.91
F(z0, S(z1)) → c2(F(S(z0), z1)) 0.00/0.91
F(z0, 0) → c3(G(z0, 0))
K tuples:
F(z0, S(z1)) → c2(F(S(z0), z1))
Defined Rule Symbols:
F(z0, 0) → c3(G(z0, 0)) 0.00/0.91
G(S(z0), z1) → c(G(z0, S(z1)))
g, f
G, F
c, c2, c3
We considered the (Usable) Rules:none
F(z0, S(z1)) → c2(F(S(z0), z1))
The order we found is given by the following interpretation:
G(S(z0), z1) → c(G(z0, S(z1))) 0.00/0.91
F(z0, S(z1)) → c2(F(S(z0), z1)) 0.00/0.91
F(z0, 0) → c3(G(z0, 0))
POL(0) = 0 0.00/0.91
POL(F(x1, x2)) = [2]x22 0.00/0.91
POL(G(x1, x2)) = 0 0.00/0.91
POL(S(x1)) = [2] + x1 0.00/0.91
POL(c(x1)) = x1 0.00/0.91
POL(c2(x1)) = x1 0.00/0.91
POL(c3(x1)) = x1
Tuples:
g(S(z0), z1) → g(z0, S(z1)) 0.00/0.91
g(0, z0) → z0 0.00/0.91
f(z0, S(z1)) → f(S(z0), z1) 0.00/0.91
f(z0, 0) → g(z0, 0)
S tuples:none
G(S(z0), z1) → c(G(z0, S(z1))) 0.00/0.91
F(z0, S(z1)) → c2(F(S(z0), z1)) 0.00/0.91
F(z0, 0) → c3(G(z0, 0))
Defined Rule Symbols:
F(z0, 0) → c3(G(z0, 0)) 0.00/0.91
G(S(z0), z1) → c(G(z0, S(z1))) 0.00/0.91
F(z0, S(z1)) → c2(F(S(z0), z1))
g, f
G, F
c, c2, c3