YES(O(1), O(n^2)) 11.85/3.46 YES(O(1), O(n^2)) 11.85/3.49 11.85/3.49 11.85/3.49
11.85/3.49 11.85/3.490 CpxTRS11.85/3.49
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))11.85/3.49
↳2 CdtProblem11.85/3.49
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))11.85/3.49
↳4 CdtProblem11.85/3.49
↳5 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))11.85/3.49
↳6 CdtProblem11.85/3.49
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))11.85/3.49
↳8 CdtProblem11.85/3.49
↳9 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))11.85/3.49
↳10 CdtProblem11.85/3.49
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))11.85/3.49
↳12 CdtProblem11.85/3.49
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))11.85/3.49
↳14 CdtProblem11.85/3.49
↳15 SIsEmptyProof (BOTH BOUNDS(ID, ID))11.85/3.49
↳16 BOUNDS(O(1), O(1))11.85/3.49
minus(x, 0) → x 11.85/3.49
minus(s(x), s(y)) → minus(x, y) 11.85/3.49
quot(0, s(y)) → 0 11.85/3.49
quot(s(x), s(y)) → s(quot(minus(x, y), s(y))) 11.85/3.49
plus(0, y) → y 11.85/3.49
plus(s(x), y) → s(plus(x, y)) 11.85/3.49
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))
Tuples:
minus(z0, 0) → z0 11.85/3.49
minus(s(z0), s(z1)) → minus(z0, z1) 11.85/3.49
quot(0, s(z0)) → 0 11.85/3.49
quot(s(z0), s(z1)) → s(quot(minus(z0, z1), s(z1))) 11.85/3.49
plus(0, z0) → z0 11.85/3.49
plus(s(z0), z1) → s(plus(z0, z1)) 11.85/3.49
plus(minus(z0, s(0)), minus(z1, s(s(z2)))) → plus(minus(z1, s(s(z2))), minus(z0, s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 12.30/3.52
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 12.30/3.52
PLUS(s(z0), z1) → c5(PLUS(z0, z1)) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c6(PLUS(minus(z1, s(s(z2))), minus(z0, s(0))), MINUS(z1, s(s(z2))), MINUS(z0, s(0)))
K tuples:none
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 12.30/3.52
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 12.30/3.52
PLUS(s(z0), z1) → c5(PLUS(z0, z1)) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c6(PLUS(minus(z1, s(s(z2))), minus(z0, s(0))), MINUS(z1, s(s(z2))), MINUS(z0, s(0)))
minus, quot, plus
MINUS, QUOT, PLUS
c1, c3, c5, c6
Tuples:
minus(z0, 0) → z0 12.30/3.52
minus(s(z0), s(z1)) → minus(z0, z1) 12.30/3.52
quot(0, s(z0)) → 0 12.30/3.52
quot(s(z0), s(z1)) → s(quot(minus(z0, z1), s(z1))) 12.30/3.52
plus(0, z0) → z0 12.30/3.52
plus(s(z0), z1) → s(plus(z0, z1)) 12.30/3.52
plus(minus(z0, s(0)), minus(z1, s(s(z2)))) → plus(minus(z1, s(s(z2))), minus(z0, s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 12.30/3.52
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 12.30/3.52
PLUS(s(z0), z1) → c5(PLUS(z0, z1)) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c6(MINUS(z1, s(s(z2))), MINUS(z0, s(0)))
K tuples:none
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 12.30/3.52
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 12.30/3.52
PLUS(s(z0), z1) → c5(PLUS(z0, z1)) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c6(MINUS(z1, s(s(z2))), MINUS(z0, s(0)))
minus, quot, plus
MINUS, QUOT, PLUS
c1, c3, c5, c6
Tuples:
minus(z0, 0) → z0 12.30/3.52
minus(s(z0), s(z1)) → minus(z0, z1) 12.30/3.52
quot(0, s(z0)) → 0 12.30/3.52
quot(s(z0), s(z1)) → s(quot(minus(z0, z1), s(z1))) 12.30/3.52
plus(0, z0) → z0 12.30/3.52
plus(s(z0), z1) → s(plus(z0, z1)) 12.30/3.52
plus(minus(z0, s(0)), minus(z1, s(s(z2)))) → plus(minus(z1, s(s(z2))), minus(z0, s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 12.30/3.52
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 12.30/3.52
PLUS(s(z0), z1) → c5(PLUS(z0, z1)) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z1, s(s(z2)))) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z0, s(0)))
K tuples:none
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 12.30/3.52
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 12.30/3.52
PLUS(s(z0), z1) → c5(PLUS(z0, z1)) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z1, s(s(z2)))) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z0, s(0)))
minus, quot, plus
MINUS, QUOT, PLUS
c1, c3, c5, c
We considered the (Usable) Rules:
PLUS(s(z0), z1) → c5(PLUS(z0, z1))
And the Tuples:
minus(z0, 0) → z0 12.30/3.52
minus(s(z0), s(z1)) → minus(z0, z1)
The order we found is given by the following interpretation:
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 12.30/3.52
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 12.30/3.52
PLUS(s(z0), z1) → c5(PLUS(z0, z1)) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z1, s(s(z2)))) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z0, s(0)))
POL(0) = 0 12.30/3.52
POL(MINUS(x1, x2)) = 0 12.30/3.52
POL(PLUS(x1, x2)) = [3]x1 + [3]x2 12.30/3.52
POL(QUOT(x1, x2)) = 0 12.30/3.52
POL(c(x1)) = x1 12.30/3.52
POL(c1(x1)) = x1 12.30/3.52
POL(c3(x1, x2)) = x1 + x2 12.30/3.52
POL(c5(x1)) = x1 12.30/3.52
POL(minus(x1, x2)) = [2]x1 12.30/3.52
POL(s(x1)) = [1] + x1
Tuples:
minus(z0, 0) → z0 12.30/3.52
minus(s(z0), s(z1)) → minus(z0, z1) 12.30/3.52
quot(0, s(z0)) → 0 12.30/3.52
quot(s(z0), s(z1)) → s(quot(minus(z0, z1), s(z1))) 12.30/3.52
plus(0, z0) → z0 12.30/3.52
plus(s(z0), z1) → s(plus(z0, z1)) 12.30/3.52
plus(minus(z0, s(0)), minus(z1, s(s(z2)))) → plus(minus(z1, s(s(z2))), minus(z0, s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 12.30/3.52
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 12.30/3.52
PLUS(s(z0), z1) → c5(PLUS(z0, z1)) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z1, s(s(z2)))) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z0, s(0)))
K tuples:
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 12.30/3.52
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z1, s(s(z2)))) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z0, s(0)))
Defined Rule Symbols:
PLUS(s(z0), z1) → c5(PLUS(z0, z1))
minus, quot, plus
MINUS, QUOT, PLUS
c1, c3, c5, c
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z1, s(s(z2)))) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z0, s(0)))
Tuples:
minus(z0, 0) → z0 12.30/3.52
minus(s(z0), s(z1)) → minus(z0, z1) 12.30/3.52
quot(0, s(z0)) → 0 12.30/3.52
quot(s(z0), s(z1)) → s(quot(minus(z0, z1), s(z1))) 12.30/3.52
plus(0, z0) → z0 12.30/3.52
plus(s(z0), z1) → s(plus(z0, z1)) 12.30/3.52
plus(minus(z0, s(0)), minus(z1, s(s(z2)))) → plus(minus(z1, s(s(z2))), minus(z0, s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 12.30/3.52
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 12.30/3.52
PLUS(s(z0), z1) → c5(PLUS(z0, z1)) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z1, s(s(z2)))) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z0, s(0)))
K tuples:
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 12.30/3.52
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
Defined Rule Symbols:
PLUS(s(z0), z1) → c5(PLUS(z0, z1)) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z1, s(s(z2)))) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z0, s(0)))
minus, quot, plus
MINUS, QUOT, PLUS
c1, c3, c5, c
We considered the (Usable) Rules:
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
And the Tuples:
minus(z0, 0) → z0 12.30/3.52
minus(s(z0), s(z1)) → minus(z0, z1)
The order we found is given by the following interpretation:
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 12.30/3.52
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 12.30/3.52
PLUS(s(z0), z1) → c5(PLUS(z0, z1)) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z1, s(s(z2)))) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z0, s(0)))
POL(0) = [1] 12.30/3.52
POL(MINUS(x1, x2)) = 0 12.30/3.52
POL(PLUS(x1, x2)) = 0 12.30/3.52
POL(QUOT(x1, x2)) = x1 12.30/3.52
POL(c(x1)) = x1 12.30/3.52
POL(c1(x1)) = x1 12.30/3.52
POL(c3(x1, x2)) = x1 + x2 12.30/3.52
POL(c5(x1)) = x1 12.30/3.52
POL(minus(x1, x2)) = x1 12.30/3.52
POL(s(x1)) = [1] + x1
Tuples:
minus(z0, 0) → z0 12.30/3.52
minus(s(z0), s(z1)) → minus(z0, z1) 12.30/3.52
quot(0, s(z0)) → 0 12.30/3.52
quot(s(z0), s(z1)) → s(quot(minus(z0, z1), s(z1))) 12.30/3.52
plus(0, z0) → z0 12.30/3.52
plus(s(z0), z1) → s(plus(z0, z1)) 12.30/3.52
plus(minus(z0, s(0)), minus(z1, s(s(z2)))) → plus(minus(z1, s(s(z2))), minus(z0, s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 12.30/3.52
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 12.30/3.52
PLUS(s(z0), z1) → c5(PLUS(z0, z1)) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z1, s(s(z2)))) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z0, s(0)))
K tuples:
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
Defined Rule Symbols:
PLUS(s(z0), z1) → c5(PLUS(z0, z1)) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z1, s(s(z2)))) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z0, s(0))) 12.30/3.52
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
minus, quot, plus
MINUS, QUOT, PLUS
c1, c3, c5, c
We considered the (Usable) Rules:
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
And the Tuples:
minus(z0, 0) → z0 12.30/3.52
minus(s(z0), s(z1)) → minus(z0, z1)
The order we found is given by the following interpretation:
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 12.30/3.52
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 12.30/3.52
PLUS(s(z0), z1) → c5(PLUS(z0, z1)) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z1, s(s(z2)))) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z0, s(0)))
POL(0) = 0 12.30/3.52
POL(MINUS(x1, x2)) = x1 12.30/3.52
POL(PLUS(x1, x2)) = x1 + [2]x2 12.30/3.52
POL(QUOT(x1, x2)) = [2]x12 12.30/3.52
POL(c(x1)) = x1 12.30/3.52
POL(c1(x1)) = x1 12.30/3.52
POL(c3(x1, x2)) = x1 + x2 12.30/3.52
POL(c5(x1)) = x1 12.30/3.52
POL(minus(x1, x2)) = x1 12.30/3.52
POL(s(x1)) = [2] + x1
Tuples:
minus(z0, 0) → z0 12.30/3.52
minus(s(z0), s(z1)) → minus(z0, z1) 12.30/3.52
quot(0, s(z0)) → 0 12.30/3.52
quot(s(z0), s(z1)) → s(quot(minus(z0, z1), s(z1))) 12.30/3.52
plus(0, z0) → z0 12.30/3.52
plus(s(z0), z1) → s(plus(z0, z1)) 12.30/3.52
plus(minus(z0, s(0)), minus(z1, s(s(z2)))) → plus(minus(z1, s(s(z2))), minus(z0, s(0)))
S tuples:none
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 12.30/3.52
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 12.30/3.52
PLUS(s(z0), z1) → c5(PLUS(z0, z1)) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z1, s(s(z2)))) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z0, s(0)))
Defined Rule Symbols:
PLUS(s(z0), z1) → c5(PLUS(z0, z1)) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z1, s(s(z2)))) 12.30/3.52
PLUS(minus(z0, s(0)), minus(z1, s(s(z2)))) → c(MINUS(z0, s(0))) 12.30/3.52
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 12.30/3.52
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
minus, quot, plus
MINUS, QUOT, PLUS
c1, c3, c5, c