### (0) Obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

The TRS R consists of the following rules:

D(t) → 1
D(constant) → 0
D(+(x, y)) → +(D(x), D(y))
D(*(x, y)) → +(*(y, D(x)), *(x, D(y)))
D(-(x, y)) → -(D(x), D(y))
D(minus(x)) → minus(D(x))
D(div(x, y)) → -(div(D(x), y), div(*(x, D(y)), pow(y, 2)))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y)))

Rewrite Strategy: FULL

### (1) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.

### (2) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

The TRS R consists of the following rules:

D(t) → 1
D(constant) → 0
D(+(x, y)) → +(D(x), D(y))
D(*(x, y)) → +(*(y, D(x)), *(x, D(y)))
D(-(x, y)) → -(D(x), D(y))
D(minus(x)) → minus(D(x))
D(div(x, y)) → -(div(D(x), y), div(*(x, D(y)), pow(y, 2)))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y)))

Rewrite Strategy: INNERMOST

### (3) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

D(t) → 1
D(constant) → 0
D(+(z0, z1)) → +(D(z0), D(z1))
D(*(z0, z1)) → +(*(z1, D(z0)), *(z0, D(z1)))
D(-(z0, z1)) → -(D(z0), D(z1))
D(minus(z0)) → minus(D(z0))
D(div(z0, z1)) → -(div(D(z0), z1), div(*(z0, D(z1)), pow(z1, 2)))
D(ln(z0)) → div(D(z0), z0)
D(pow(z0, z1)) → +(*(*(z1, pow(z0, -(z1, 1))), D(z0)), *(*(pow(z0, z1), ln(z0)), D(z1)))
Tuples:

D'(t) → c
D'(constant) → c1
D'(+(z0, z1)) → c2(D'(z0), D'(z1))
D'(*(z0, z1)) → c3(D'(z0), D'(z1))
D'(-(z0, z1)) → c4(D'(z0), D'(z1))
D'(minus(z0)) → c5(D'(z0))
D'(div(z0, z1)) → c6(D'(z0), D'(z1))
D'(ln(z0)) → c7(D'(z0))
D'(pow(z0, z1)) → c8(D'(z0), D'(z1))
S tuples:

D'(t) → c
D'(constant) → c1
D'(+(z0, z1)) → c2(D'(z0), D'(z1))
D'(*(z0, z1)) → c3(D'(z0), D'(z1))
D'(-(z0, z1)) → c4(D'(z0), D'(z1))
D'(minus(z0)) → c5(D'(z0))
D'(div(z0, z1)) → c6(D'(z0), D'(z1))
D'(ln(z0)) → c7(D'(z0))
D'(pow(z0, z1)) → c8(D'(z0), D'(z1))
K tuples:none
Defined Rule Symbols:

D

Defined Pair Symbols:

D'

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8

### (5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

D'(t) → c
D'(constant) → c1

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

D(t) → 1
D(constant) → 0
D(+(z0, z1)) → +(D(z0), D(z1))
D(*(z0, z1)) → +(*(z1, D(z0)), *(z0, D(z1)))
D(-(z0, z1)) → -(D(z0), D(z1))
D(minus(z0)) → minus(D(z0))
D(div(z0, z1)) → -(div(D(z0), z1), div(*(z0, D(z1)), pow(z1, 2)))
D(ln(z0)) → div(D(z0), z0)
D(pow(z0, z1)) → +(*(*(z1, pow(z0, -(z1, 1))), D(z0)), *(*(pow(z0, z1), ln(z0)), D(z1)))
Tuples:

D'(+(z0, z1)) → c2(D'(z0), D'(z1))
D'(*(z0, z1)) → c3(D'(z0), D'(z1))
D'(-(z0, z1)) → c4(D'(z0), D'(z1))
D'(minus(z0)) → c5(D'(z0))
D'(div(z0, z1)) → c6(D'(z0), D'(z1))
D'(ln(z0)) → c7(D'(z0))
D'(pow(z0, z1)) → c8(D'(z0), D'(z1))
S tuples:

D'(+(z0, z1)) → c2(D'(z0), D'(z1))
D'(*(z0, z1)) → c3(D'(z0), D'(z1))
D'(-(z0, z1)) → c4(D'(z0), D'(z1))
D'(minus(z0)) → c5(D'(z0))
D'(div(z0, z1)) → c6(D'(z0), D'(z1))
D'(ln(z0)) → c7(D'(z0))
D'(pow(z0, z1)) → c8(D'(z0), D'(z1))
K tuples:none
Defined Rule Symbols:

D

Defined Pair Symbols:

D'

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

D(t) → 1
D(constant) → 0
D(+(z0, z1)) → +(D(z0), D(z1))
D(*(z0, z1)) → +(*(z1, D(z0)), *(z0, D(z1)))
D(-(z0, z1)) → -(D(z0), D(z1))
D(minus(z0)) → minus(D(z0))
D(div(z0, z1)) → -(div(D(z0), z1), div(*(z0, D(z1)), pow(z1, 2)))
D(ln(z0)) → div(D(z0), z0)
D(pow(z0, z1)) → +(*(*(z1, pow(z0, -(z1, 1))), D(z0)), *(*(pow(z0, z1), ln(z0)), D(z1)))

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

D'(+(z0, z1)) → c2(D'(z0), D'(z1))
D'(*(z0, z1)) → c3(D'(z0), D'(z1))
D'(-(z0, z1)) → c4(D'(z0), D'(z1))
D'(minus(z0)) → c5(D'(z0))
D'(div(z0, z1)) → c6(D'(z0), D'(z1))
D'(ln(z0)) → c7(D'(z0))
D'(pow(z0, z1)) → c8(D'(z0), D'(z1))
S tuples:

D'(+(z0, z1)) → c2(D'(z0), D'(z1))
D'(*(z0, z1)) → c3(D'(z0), D'(z1))
D'(-(z0, z1)) → c4(D'(z0), D'(z1))
D'(minus(z0)) → c5(D'(z0))
D'(div(z0, z1)) → c6(D'(z0), D'(z1))
D'(ln(z0)) → c7(D'(z0))
D'(pow(z0, z1)) → c8(D'(z0), D'(z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

D'

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8

### (9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

D'(+(z0, z1)) → c2(D'(z0), D'(z1))
D'(*(z0, z1)) → c3(D'(z0), D'(z1))
D'(minus(z0)) → c5(D'(z0))
D'(div(z0, z1)) → c6(D'(z0), D'(z1))
D'(pow(z0, z1)) → c8(D'(z0), D'(z1))
We considered the (Usable) Rules:none
And the Tuples:

D'(+(z0, z1)) → c2(D'(z0), D'(z1))
D'(*(z0, z1)) → c3(D'(z0), D'(z1))
D'(-(z0, z1)) → c4(D'(z0), D'(z1))
D'(minus(z0)) → c5(D'(z0))
D'(div(z0, z1)) → c6(D'(z0), D'(z1))
D'(ln(z0)) → c7(D'(z0))
D'(pow(z0, z1)) → c8(D'(z0), D'(z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(*(x1, x2)) = [2] + x1 + x2
POL(+(x1, x2)) = [1] + x1 + x2
POL(-(x1, x2)) = x1 + x2
POL(D'(x1)) = [2]x1 + [2]x12
POL(c2(x1, x2)) = x1 + x2
POL(c3(x1, x2)) = x1 + x2
POL(c4(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(c6(x1, x2)) = x1 + x2
POL(c7(x1)) = x1
POL(c8(x1, x2)) = x1 + x2
POL(div(x1, x2)) = [2] + x1 + x2
POL(ln(x1)) = x1
POL(minus(x1)) = [2] + x1
POL(pow(x1, x2)) = [2] + x1 + x2

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

D'(+(z0, z1)) → c2(D'(z0), D'(z1))
D'(*(z0, z1)) → c3(D'(z0), D'(z1))
D'(-(z0, z1)) → c4(D'(z0), D'(z1))
D'(minus(z0)) → c5(D'(z0))
D'(div(z0, z1)) → c6(D'(z0), D'(z1))
D'(ln(z0)) → c7(D'(z0))
D'(pow(z0, z1)) → c8(D'(z0), D'(z1))
S tuples:

D'(-(z0, z1)) → c4(D'(z0), D'(z1))
D'(ln(z0)) → c7(D'(z0))
K tuples:

D'(+(z0, z1)) → c2(D'(z0), D'(z1))
D'(*(z0, z1)) → c3(D'(z0), D'(z1))
D'(minus(z0)) → c5(D'(z0))
D'(div(z0, z1)) → c6(D'(z0), D'(z1))
D'(pow(z0, z1)) → c8(D'(z0), D'(z1))
Defined Rule Symbols:none

Defined Pair Symbols:

D'

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8

### (11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

D'(-(z0, z1)) → c4(D'(z0), D'(z1))
D'(ln(z0)) → c7(D'(z0))
We considered the (Usable) Rules:none
And the Tuples:

D'(+(z0, z1)) → c2(D'(z0), D'(z1))
D'(*(z0, z1)) → c3(D'(z0), D'(z1))
D'(-(z0, z1)) → c4(D'(z0), D'(z1))
D'(minus(z0)) → c5(D'(z0))
D'(div(z0, z1)) → c6(D'(z0), D'(z1))
D'(ln(z0)) → c7(D'(z0))
D'(pow(z0, z1)) → c8(D'(z0), D'(z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(*(x1, x2)) = [1] + x1 + x2
POL(+(x1, x2)) = [1] + x1 + x2
POL(-(x1, x2)) = [3] + x1 + x2
POL(D'(x1)) = [2] + [3]x1
POL(c2(x1, x2)) = x1 + x2
POL(c3(x1, x2)) = x1 + x2
POL(c4(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(c6(x1, x2)) = x1 + x2
POL(c7(x1)) = x1
POL(c8(x1, x2)) = x1 + x2
POL(div(x1, x2)) = [3] + x1 + x2
POL(ln(x1)) = [2] + x1
POL(minus(x1)) = x1
POL(pow(x1, x2)) = [2] + x1 + x2

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

D'(+(z0, z1)) → c2(D'(z0), D'(z1))
D'(*(z0, z1)) → c3(D'(z0), D'(z1))
D'(-(z0, z1)) → c4(D'(z0), D'(z1))
D'(minus(z0)) → c5(D'(z0))
D'(div(z0, z1)) → c6(D'(z0), D'(z1))
D'(ln(z0)) → c7(D'(z0))
D'(pow(z0, z1)) → c8(D'(z0), D'(z1))
S tuples:none
K tuples:

D'(+(z0, z1)) → c2(D'(z0), D'(z1))
D'(*(z0, z1)) → c3(D'(z0), D'(z1))
D'(minus(z0)) → c5(D'(z0))
D'(div(z0, z1)) → c6(D'(z0), D'(z1))
D'(pow(z0, z1)) → c8(D'(z0), D'(z1))
D'(-(z0, z1)) → c4(D'(z0), D'(z1))
D'(ln(z0)) → c7(D'(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

D'

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8

### (13) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty