(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

exp(x, 0) → s(0)
exp(x, s(y)) → *(x, exp(x, y))
*(0, y) → 0
*(s(x), y) → +(y, *(x, y))
-(0, y) → 0
-(x, 0) → x
-(s(x), s(y)) → -(x, y)

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

exp(z0, 0) → s(0)
exp(z0, s(z1)) → *(z0, exp(z0, z1))
*(0, z0) → 0
*(s(z0), z1) → +(z1, *(z0, z1))
-(0, z0) → 0
-(z0, 0) → z0
-(s(z0), s(z1)) → -(z0, z1)
Tuples:

EXP(z0, 0) → c
EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
*'(0, z0) → c2
*'(s(z0), z1) → c3(*'(z0, z1))
-'(0, z0) → c4
-'(z0, 0) → c5
-'(s(z0), s(z1)) → c6(-'(z0, z1))
S tuples:

EXP(z0, 0) → c
EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
*'(0, z0) → c2
*'(s(z0), z1) → c3(*'(z0, z1))
-'(0, z0) → c4
-'(z0, 0) → c5
-'(s(z0), s(z1)) → c6(-'(z0, z1))
K tuples:none
Defined Rule Symbols:

exp, *, -

Defined Pair Symbols:

EXP, *', -'

Compound Symbols:

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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing nodes:

*'(0, z0) → c2
EXP(z0, 0) → c
-'(z0, 0) → c5
-'(0, z0) → c4

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

exp(z0, 0) → s(0)
exp(z0, s(z1)) → *(z0, exp(z0, z1))
*(0, z0) → 0
*(s(z0), z1) → +(z1, *(z0, z1))
-(0, z0) → 0
-(z0, 0) → z0
-(s(z0), s(z1)) → -(z0, z1)
Tuples:

EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
*'(s(z0), z1) → c3(*'(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
S tuples:

EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
*'(s(z0), z1) → c3(*'(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
K tuples:none
Defined Rule Symbols:

exp, *, -

Defined Pair Symbols:

EXP, *', -'

Compound Symbols:

c1, c3, c6

(5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

-(0, z0) → 0
-(z0, 0) → z0
-(s(z0), s(z1)) → -(z0, z1)

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

exp(z0, 0) → s(0)
exp(z0, s(z1)) → *(z0, exp(z0, z1))
*(0, z0) → 0
*(s(z0), z1) → +(z1, *(z0, z1))
Tuples:

EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
*'(s(z0), z1) → c3(*'(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
S tuples:

EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
*'(s(z0), z1) → c3(*'(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
K tuples:none
Defined Rule Symbols:

exp, *

Defined Pair Symbols:

EXP, *', -'

Compound Symbols:

c1, c3, c6

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

-'(s(z0), s(z1)) → c6(-'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
*'(s(z0), z1) → c3(*'(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(*(x1, x2)) = [2] + x1   
POL(*'(x1, x2)) = 0   
POL(+(x1, x2)) = [1] + x2   
POL(-'(x1, x2)) = [4]x2   
POL(0) = [5]   
POL(EXP(x1, x2)) = 0   
POL(c1(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c6(x1)) = x1   
POL(exp(x1, x2)) = [1] + [3]x1 + x2   
POL(s(x1)) = [1] + x1   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

exp(z0, 0) → s(0)
exp(z0, s(z1)) → *(z0, exp(z0, z1))
*(0, z0) → 0
*(s(z0), z1) → +(z1, *(z0, z1))
Tuples:

EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
*'(s(z0), z1) → c3(*'(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
S tuples:

EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
*'(s(z0), z1) → c3(*'(z0, z1))
K tuples:

-'(s(z0), s(z1)) → c6(-'(z0, z1))
Defined Rule Symbols:

exp, *

Defined Pair Symbols:

EXP, *', -'

Compound Symbols:

c1, c3, c6

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
*'(s(z0), z1) → c3(*'(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(*(x1, x2)) = [5] + [5]x1 + x2   
POL(*'(x1, x2)) = 0   
POL(+(x1, x2)) = [3]   
POL(-'(x1, x2)) = 0   
POL(0) = [3]   
POL(EXP(x1, x2)) = x2   
POL(c1(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c6(x1)) = x1   
POL(exp(x1, x2)) = [3] + [5]x1 + [3]x2   
POL(s(x1)) = [1] + x1   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

exp(z0, 0) → s(0)
exp(z0, s(z1)) → *(z0, exp(z0, z1))
*(0, z0) → 0
*(s(z0), z1) → +(z1, *(z0, z1))
Tuples:

EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
*'(s(z0), z1) → c3(*'(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
S tuples:

*'(s(z0), z1) → c3(*'(z0, z1))
K tuples:

-'(s(z0), s(z1)) → c6(-'(z0, z1))
EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
Defined Rule Symbols:

exp, *

Defined Pair Symbols:

EXP, *', -'

Compound Symbols:

c1, c3, c6

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

*'(s(z0), z1) → c3(*'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
*'(s(z0), z1) → c3(*'(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(*(x1, x2)) = [2]   
POL(*'(x1, x2)) = [2]x1   
POL(+(x1, x2)) = 0   
POL(-'(x1, x2)) = [2]x1 + [2]x2 + x1·x2 + [2]x12   
POL(0) = [2]   
POL(EXP(x1, x2)) = [3]x1·x2   
POL(c1(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c6(x1)) = x1   
POL(exp(x1, x2)) = [2]x22 + [3]x1·x2 + x12   
POL(s(x1)) = [3] + x1   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

exp(z0, 0) → s(0)
exp(z0, s(z1)) → *(z0, exp(z0, z1))
*(0, z0) → 0
*(s(z0), z1) → +(z1, *(z0, z1))
Tuples:

EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
*'(s(z0), z1) → c3(*'(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
S tuples:none
K tuples:

-'(s(z0), s(z1)) → c6(-'(z0, z1))
EXP(z0, s(z1)) → c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
*'(s(z0), z1) → c3(*'(z0, z1))
Defined Rule Symbols:

exp, *

Defined Pair Symbols:

EXP, *', -'

Compound Symbols:

c1, c3, c6

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

The set S is empty

(14) BOUNDS(1, 1)