YES(O(1), O(n^2)) 9.57/3.06 YES(O(1), O(n^2)) 9.57/3.08 9.57/3.08 9.57/3.08
9.57/3.08 9.57/3.080 CpxTRS9.57/3.08
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))9.57/3.08
↳2 CdtProblem9.57/3.08
↳3 CdtInstantiationProof (BOTH BOUNDS(ID, ID))9.57/3.08
↳4 CdtProblem9.57/3.08
↳5 CdtInstantiationProof (BOTH BOUNDS(ID, ID))9.57/3.08
↳6 CdtProblem9.57/3.08
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))9.57/3.08
↳8 CdtProblem9.57/3.08
↳9 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))9.57/3.08
↳10 CdtProblem9.57/3.08
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))9.57/3.08
↳12 CdtProblem9.57/3.08
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))9.57/3.08
↳14 CdtProblem9.57/3.08
↳15 SIsEmptyProof (BOTH BOUNDS(ID, ID))9.57/3.08
↳16 BOUNDS(O(1), O(1))9.57/3.08
r(xs, ys, zs, nil) → xs 9.57/3.08
r(xs, nil, zs, cons(w, ws)) → r(xs, xs, cons(succ(zero), zs), ws) 9.57/3.08
r(xs, cons(y, ys), nil, cons(w, ws)) → r(xs, xs, cons(succ(zero), nil), ws) 9.57/3.08
r(xs, cons(y, ys), cons(z, zs), cons(w, ws)) → r(ys, cons(y, ys), zs, cons(succ(zero), cons(w, ws)))
Tuples:
r(z0, z1, z2, nil) → z0 9.57/3.08
r(z0, nil, z1, cons(z2, z3)) → r(z0, z0, cons(succ(zero), z1), z3) 9.57/3.08
r(z0, cons(z1, z2), nil, cons(z3, z4)) → r(z0, z0, cons(succ(zero), nil), z4) 9.57/3.08
r(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → r(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))
S tuples:
R(z0, nil, z1, cons(z2, z3)) → c1(R(z0, z0, cons(succ(zero), z1), z3)) 9.57/3.08
R(z0, cons(z1, z2), nil, cons(z3, z4)) → c2(R(z0, z0, cons(succ(zero), nil), z4)) 9.57/3.08
R(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → c3(R(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6))))
K tuples:none
R(z0, nil, z1, cons(z2, z3)) → c1(R(z0, z0, cons(succ(zero), z1), z3)) 9.57/3.08
R(z0, cons(z1, z2), nil, cons(z3, z4)) → c2(R(z0, z0, cons(succ(zero), nil), z4)) 9.57/3.08
R(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → c3(R(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6))))
r
R
c1, c2, c3
R(nil, nil, cons(succ(zero), x1), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), x1)), z3)) 9.57/3.08
R(nil, nil, cons(succ(zero), nil), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), nil)), z3))
Tuples:
r(z0, z1, z2, nil) → z0 9.57/3.08
r(z0, nil, z1, cons(z2, z3)) → r(z0, z0, cons(succ(zero), z1), z3) 9.57/3.08
r(z0, cons(z1, z2), nil, cons(z3, z4)) → r(z0, z0, cons(succ(zero), nil), z4) 9.57/3.08
r(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → r(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))
S tuples:
R(z0, cons(z1, z2), nil, cons(z3, z4)) → c2(R(z0, z0, cons(succ(zero), nil), z4)) 9.57/3.09
R(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → c3(R(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))) 9.57/3.09
R(nil, nil, cons(succ(zero), x1), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), x1)), z3)) 9.57/3.09
R(nil, nil, cons(succ(zero), nil), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), nil)), z3))
K tuples:none
R(z0, cons(z1, z2), nil, cons(z3, z4)) → c2(R(z0, z0, cons(succ(zero), nil), z4)) 9.57/3.09
R(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → c3(R(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))) 9.57/3.09
R(nil, nil, cons(succ(zero), x1), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), x1)), z3)) 9.57/3.09
R(nil, nil, cons(succ(zero), nil), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), nil)), z3))
r
R
c2, c3, c1
R(x2, cons(x1, x2), nil, cons(succ(zero), cons(x5, x6))) → c2(R(x2, x2, cons(succ(zero), nil), cons(x5, x6)))
Tuples:
r(z0, z1, z2, nil) → z0 9.57/3.09
r(z0, nil, z1, cons(z2, z3)) → r(z0, z0, cons(succ(zero), z1), z3) 9.57/3.09
r(z0, cons(z1, z2), nil, cons(z3, z4)) → r(z0, z0, cons(succ(zero), nil), z4) 9.57/3.09
r(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → r(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))
S tuples:
R(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → c3(R(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))) 9.57/3.09
R(nil, nil, cons(succ(zero), x1), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), x1)), z3)) 9.57/3.09
R(nil, nil, cons(succ(zero), nil), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), nil)), z3)) 9.57/3.09
R(x2, cons(x1, x2), nil, cons(succ(zero), cons(x5, x6))) → c2(R(x2, x2, cons(succ(zero), nil), cons(x5, x6)))
K tuples:none
R(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → c3(R(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))) 9.57/3.09
R(nil, nil, cons(succ(zero), x1), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), x1)), z3)) 9.57/3.09
R(nil, nil, cons(succ(zero), nil), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), nil)), z3)) 9.57/3.09
R(x2, cons(x1, x2), nil, cons(succ(zero), cons(x5, x6))) → c2(R(x2, x2, cons(succ(zero), nil), cons(x5, x6)))
r
R
c3, c1, c2
We considered the (Usable) Rules:none
R(x2, cons(x1, x2), nil, cons(succ(zero), cons(x5, x6))) → c2(R(x2, x2, cons(succ(zero), nil), cons(x5, x6)))
The order we found is given by the following interpretation:
R(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → c3(R(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))) 9.57/3.09
R(nil, nil, cons(succ(zero), x1), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), x1)), z3)) 9.57/3.09
R(nil, nil, cons(succ(zero), nil), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), nil)), z3)) 9.57/3.09
R(x2, cons(x1, x2), nil, cons(succ(zero), cons(x5, x6))) → c2(R(x2, x2, cons(succ(zero), nil), cons(x5, x6)))
POL(R(x1, x2, x3, x4)) = x2 9.57/3.09
POL(c1(x1)) = x1 9.57/3.09
POL(c2(x1)) = x1 9.57/3.09
POL(c3(x1)) = x1 9.57/3.09
POL(cons(x1, x2)) = [4] + x2 9.57/3.09
POL(nil) = [1] 9.57/3.09
POL(succ(x1)) = x1 9.57/3.09
POL(zero) = 0
Tuples:
r(z0, z1, z2, nil) → z0 9.57/3.09
r(z0, nil, z1, cons(z2, z3)) → r(z0, z0, cons(succ(zero), z1), z3) 9.57/3.09
r(z0, cons(z1, z2), nil, cons(z3, z4)) → r(z0, z0, cons(succ(zero), nil), z4) 9.57/3.09
r(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → r(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))
S tuples:
R(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → c3(R(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))) 9.57/3.09
R(nil, nil, cons(succ(zero), x1), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), x1)), z3)) 9.57/3.09
R(nil, nil, cons(succ(zero), nil), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), nil)), z3)) 9.57/3.09
R(x2, cons(x1, x2), nil, cons(succ(zero), cons(x5, x6))) → c2(R(x2, x2, cons(succ(zero), nil), cons(x5, x6)))
K tuples:
R(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → c3(R(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))) 9.57/3.09
R(nil, nil, cons(succ(zero), x1), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), x1)), z3)) 9.57/3.09
R(nil, nil, cons(succ(zero), nil), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), nil)), z3))
Defined Rule Symbols:
R(x2, cons(x1, x2), nil, cons(succ(zero), cons(x5, x6))) → c2(R(x2, x2, cons(succ(zero), nil), cons(x5, x6)))
r
R
c3, c1, c2
R(nil, nil, cons(succ(zero), nil), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), nil)), z3))
Tuples:
r(z0, z1, z2, nil) → z0 9.57/3.09
r(z0, nil, z1, cons(z2, z3)) → r(z0, z0, cons(succ(zero), z1), z3) 9.57/3.09
r(z0, cons(z1, z2), nil, cons(z3, z4)) → r(z0, z0, cons(succ(zero), nil), z4) 9.57/3.09
r(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → r(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))
S tuples:
R(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → c3(R(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))) 9.57/3.09
R(nil, nil, cons(succ(zero), x1), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), x1)), z3)) 9.57/3.09
R(nil, nil, cons(succ(zero), nil), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), nil)), z3)) 9.57/3.09
R(x2, cons(x1, x2), nil, cons(succ(zero), cons(x5, x6))) → c2(R(x2, x2, cons(succ(zero), nil), cons(x5, x6)))
K tuples:
R(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → c3(R(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))) 9.57/3.09
R(nil, nil, cons(succ(zero), x1), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), x1)), z3))
Defined Rule Symbols:
R(x2, cons(x1, x2), nil, cons(succ(zero), cons(x5, x6))) → c2(R(x2, x2, cons(succ(zero), nil), cons(x5, x6))) 9.57/3.09
R(nil, nil, cons(succ(zero), nil), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), nil)), z3))
r
R
c3, c1, c2
We considered the (Usable) Rules:none
R(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → c3(R(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6))))
The order we found is given by the following interpretation:
R(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → c3(R(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))) 9.57/3.09
R(nil, nil, cons(succ(zero), x1), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), x1)), z3)) 9.57/3.09
R(nil, nil, cons(succ(zero), nil), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), nil)), z3)) 9.57/3.09
R(x2, cons(x1, x2), nil, cons(succ(zero), cons(x5, x6))) → c2(R(x2, x2, cons(succ(zero), nil), cons(x5, x6)))
POL(R(x1, x2, x3, x4)) = x2·x3 + x22 9.57/3.09
POL(c1(x1)) = x1 9.57/3.09
POL(c2(x1)) = x1 9.57/3.09
POL(c3(x1)) = x1 9.57/3.09
POL(cons(x1, x2)) = [2] + x2 9.57/3.09
POL(nil) = 0 9.57/3.09
POL(succ(x1)) = 0 9.57/3.09
POL(zero) = 0
Tuples:
r(z0, z1, z2, nil) → z0 9.57/3.09
r(z0, nil, z1, cons(z2, z3)) → r(z0, z0, cons(succ(zero), z1), z3) 9.57/3.09
r(z0, cons(z1, z2), nil, cons(z3, z4)) → r(z0, z0, cons(succ(zero), nil), z4) 9.57/3.09
r(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → r(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))
S tuples:
R(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → c3(R(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))) 9.57/3.09
R(nil, nil, cons(succ(zero), x1), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), x1)), z3)) 9.57/3.09
R(nil, nil, cons(succ(zero), nil), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), nil)), z3)) 9.57/3.09
R(x2, cons(x1, x2), nil, cons(succ(zero), cons(x5, x6))) → c2(R(x2, x2, cons(succ(zero), nil), cons(x5, x6)))
K tuples:
R(nil, nil, cons(succ(zero), x1), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), x1)), z3))
Defined Rule Symbols:
R(x2, cons(x1, x2), nil, cons(succ(zero), cons(x5, x6))) → c2(R(x2, x2, cons(succ(zero), nil), cons(x5, x6))) 9.57/3.09
R(nil, nil, cons(succ(zero), nil), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), nil)), z3)) 9.57/3.09
R(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → c3(R(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6))))
r
R
c3, c1, c2
We considered the (Usable) Rules:none
R(nil, nil, cons(succ(zero), x1), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), x1)), z3))
The order we found is given by the following interpretation:
R(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → c3(R(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))) 9.57/3.09
R(nil, nil, cons(succ(zero), x1), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), x1)), z3)) 9.57/3.09
R(nil, nil, cons(succ(zero), nil), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), nil)), z3)) 9.57/3.09
R(x2, cons(x1, x2), nil, cons(succ(zero), cons(x5, x6))) → c2(R(x2, x2, cons(succ(zero), nil), cons(x5, x6)))
POL(R(x1, x2, x3, x4)) = [2]x2 + [2]x4 + [2]x2·x3 + [3]x22 9.57/3.09
POL(c1(x1)) = x1 9.57/3.09
POL(c2(x1)) = x1 9.57/3.09
POL(c3(x1)) = x1 9.57/3.09
POL(cons(x1, x2)) = [1] + x2 9.57/3.09
POL(nil) = 0 9.57/3.09
POL(succ(x1)) = 0 9.57/3.09
POL(zero) = 0
Tuples:
r(z0, z1, z2, nil) → z0 9.57/3.09
r(z0, nil, z1, cons(z2, z3)) → r(z0, z0, cons(succ(zero), z1), z3) 9.57/3.09
r(z0, cons(z1, z2), nil, cons(z3, z4)) → r(z0, z0, cons(succ(zero), nil), z4) 9.57/3.09
r(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → r(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))
S tuples:none
R(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → c3(R(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))) 9.57/3.09
R(nil, nil, cons(succ(zero), x1), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), x1)), z3)) 9.57/3.09
R(nil, nil, cons(succ(zero), nil), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), nil)), z3)) 9.57/3.09
R(x2, cons(x1, x2), nil, cons(succ(zero), cons(x5, x6))) → c2(R(x2, x2, cons(succ(zero), nil), cons(x5, x6)))
Defined Rule Symbols:
R(x2, cons(x1, x2), nil, cons(succ(zero), cons(x5, x6))) → c2(R(x2, x2, cons(succ(zero), nil), cons(x5, x6))) 9.57/3.09
R(nil, nil, cons(succ(zero), nil), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), nil)), z3)) 9.57/3.09
R(z0, cons(z1, z2), cons(z3, z4), cons(z5, z6)) → c3(R(z2, cons(z1, z2), z4, cons(succ(zero), cons(z5, z6)))) 9.57/3.09
R(nil, nil, cons(succ(zero), x1), cons(z2, z3)) → c1(R(nil, nil, cons(succ(zero), cons(succ(zero), x1)), z3))
r
R
c3, c1, c2