YES(O(1), O(n^2)) 3.19/1.21 YES(O(1), O(n^2)) 3.19/1.23 3.19/1.23 3.19/1.23
3.19/1.23 3.19/1.230 CpxTRS3.19/1.23
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))3.19/1.23
↳2 CdtProblem3.19/1.23
↳3 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))3.19/1.23
↳4 CdtProblem3.19/1.23
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))3.19/1.23
↳6 CdtProblem3.19/1.23
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))3.19/1.23
↳8 CdtProblem3.19/1.23
↳9 SIsEmptyProof (BOTH BOUNDS(ID, ID))3.19/1.23
↳10 BOUNDS(O(1), O(1))3.19/1.23
exp(x, 0) → s(0) 3.19/1.23
exp(x, s(y)) → *(x, exp(x, y)) 3.19/1.23
*(0, y) → 0 3.19/1.23
*(s(x), y) → +(y, *(x, y)) 3.19/1.23
-(0, y) → 0 3.19/1.23
-(x, 0) → x 3.19/1.23
-(s(x), s(y)) → -(x, y)
Tuples:
exp(z0, 0) → s(0) 3.19/1.23
exp(z0, s(z1)) → *(z0, exp(z0, z1)) 3.19/1.23
*(0, z0) → 0 3.19/1.23
*(s(z0), z1) → +(z1, *(z0, z1)) 3.19/1.23
-(0, z0) → 0 3.19/1.23
-(z0, 0) → z0 3.19/1.23
-(s(z0), s(z1)) → -(z0, z1)
S tuples:
EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1)) 3.19/1.23
*'(s(z0), z1) → c3(*'(z0, z1)) 3.19/1.23
-'(s(z0), s(z1)) → c6(-'(z0, z1))
K tuples:none
EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1)) 3.19/1.23
*'(s(z0), z1) → c3(*'(z0, z1)) 3.19/1.23
-'(s(z0), s(z1)) → c6(-'(z0, z1))
exp, *, -
EXP, *', -'
c1, c3, c6
We considered the (Usable) Rules:
EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
And the Tuples:
exp(z0, 0) → s(0) 3.19/1.23
exp(z0, s(z1)) → *(z0, exp(z0, z1)) 3.19/1.23
*(0, z0) → 0 3.19/1.23
*(s(z0), z1) → +(z1, *(z0, z1))
The order we found is given by the following interpretation:
EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1)) 3.19/1.23
*'(s(z0), z1) → c3(*'(z0, z1)) 3.19/1.23
-'(s(z0), s(z1)) → c6(-'(z0, z1))
POL(*(x1, x2)) = [2] + [4]x1 3.19/1.23
POL(*'(x1, x2)) = [5] 3.19/1.23
POL(+(x1, x2)) = [4] 3.19/1.23
POL(-'(x1, x2)) = 0 3.19/1.23
POL(0) = 0 3.19/1.23
POL(EXP(x1, x2)) = [4]x2 3.19/1.23
POL(c1(x1, x2)) = x1 + x2 3.19/1.23
POL(c3(x1)) = x1 3.19/1.23
POL(c6(x1)) = x1 3.19/1.23
POL(exp(x1, x2)) = [4] + [4]x1 + [2]x2 3.19/1.23
POL(s(x1)) = [4] + x1
Tuples:
exp(z0, 0) → s(0) 3.19/1.23
exp(z0, s(z1)) → *(z0, exp(z0, z1)) 3.19/1.23
*(0, z0) → 0 3.19/1.23
*(s(z0), z1) → +(z1, *(z0, z1)) 3.19/1.23
-(0, z0) → 0 3.19/1.23
-(z0, 0) → z0 3.19/1.23
-(s(z0), s(z1)) → -(z0, z1)
S tuples:
EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1)) 3.19/1.23
*'(s(z0), z1) → c3(*'(z0, z1)) 3.19/1.23
-'(s(z0), s(z1)) → c6(-'(z0, z1))
K tuples:
*'(s(z0), z1) → c3(*'(z0, z1)) 3.19/1.23
-'(s(z0), s(z1)) → c6(-'(z0, z1))
Defined Rule Symbols:
EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
exp, *, -
EXP, *', -'
c1, c3, c6
We considered the (Usable) Rules:
-'(s(z0), s(z1)) → c6(-'(z0, z1))
And the Tuples:
exp(z0, 0) → s(0) 3.19/1.24
exp(z0, s(z1)) → *(z0, exp(z0, z1)) 3.19/1.24
*(0, z0) → 0 3.19/1.24
*(s(z0), z1) → +(z1, *(z0, z1))
The order we found is given by the following interpretation:
EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1)) 3.19/1.24
*'(s(z0), z1) → c3(*'(z0, z1)) 3.19/1.24
-'(s(z0), s(z1)) → c6(-'(z0, z1))
POL(*(x1, x2)) = 0 3.19/1.24
POL(*'(x1, x2)) = [1] 3.19/1.24
POL(+(x1, x2)) = 0 3.19/1.24
POL(-'(x1, x2)) = x1 3.19/1.24
POL(0) = 0 3.19/1.24
POL(EXP(x1, x2)) = [2]x2 3.19/1.24
POL(c1(x1, x2)) = x1 + x2 3.19/1.24
POL(c3(x1)) = x1 3.19/1.24
POL(c6(x1)) = x1 3.19/1.24
POL(exp(x1, x2)) = [4] + [2]x2 3.19/1.24
POL(s(x1)) = [2] + x1
Tuples:
exp(z0, 0) → s(0) 3.19/1.24
exp(z0, s(z1)) → *(z0, exp(z0, z1)) 3.19/1.24
*(0, z0) → 0 3.19/1.24
*(s(z0), z1) → +(z1, *(z0, z1)) 3.19/1.24
-(0, z0) → 0 3.19/1.24
-(z0, 0) → z0 3.19/1.24
-(s(z0), s(z1)) → -(z0, z1)
S tuples:
EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1)) 3.19/1.24
*'(s(z0), z1) → c3(*'(z0, z1)) 3.19/1.24
-'(s(z0), s(z1)) → c6(-'(z0, z1))
K tuples:
*'(s(z0), z1) → c3(*'(z0, z1))
Defined Rule Symbols:
EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1)) 3.19/1.24
-'(s(z0), s(z1)) → c6(-'(z0, z1))
exp, *, -
EXP, *', -'
c1, c3, c6
We considered the (Usable) Rules:
*'(s(z0), z1) → c3(*'(z0, z1))
And the Tuples:
exp(z0, 0) → s(0) 3.19/1.24
exp(z0, s(z1)) → *(z0, exp(z0, z1)) 3.19/1.24
*(0, z0) → 0 3.19/1.24
*(s(z0), z1) → +(z1, *(z0, z1))
The order we found is given by the following interpretation:
EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1)) 3.19/1.24
*'(s(z0), z1) → c3(*'(z0, z1)) 3.19/1.24
-'(s(z0), s(z1)) → c6(-'(z0, z1))
POL(*(x1, x2)) = [1] + [3]x1 3.19/1.24
POL(*'(x1, x2)) = [2]x1 3.19/1.24
POL(+(x1, x2)) = [3] + x2 3.19/1.24
POL(-'(x1, x2)) = x1 + x2 3.19/1.24
POL(0) = [1] 3.19/1.24
POL(EXP(x1, x2)) = x1·x2 3.19/1.24
POL(c1(x1, x2)) = x1 + x2 3.19/1.24
POL(c3(x1)) = x1 3.19/1.24
POL(c6(x1)) = x1 3.19/1.24
POL(exp(x1, x2)) = [3] + [3]x1 + x2 + x22 3.19/1.24
POL(s(x1)) = [3] + x1
Tuples:
exp(z0, 0) → s(0) 3.19/1.24
exp(z0, s(z1)) → *(z0, exp(z0, z1)) 3.19/1.24
*(0, z0) → 0 3.19/1.24
*(s(z0), z1) → +(z1, *(z0, z1)) 3.19/1.24
-(0, z0) → 0 3.19/1.24
-(z0, 0) → z0 3.19/1.24
-(s(z0), s(z1)) → -(z0, z1)
S tuples:none
EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1)) 3.19/1.24
*'(s(z0), z1) → c3(*'(z0, z1)) 3.19/1.24
-'(s(z0), s(z1)) → c6(-'(z0, z1))
Defined Rule Symbols:
EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1)) 3.19/1.24
-'(s(z0), s(z1)) → c6(-'(z0, z1)) 3.19/1.24
*'(s(z0), z1) → c3(*'(z0, z1))
exp, *, -
EXP, *', -'
c1, c3, c6