(0) Obligation:

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


The TRS R consists of the following rules:

*(i(x), x) → 1
*(1, y) → y
*(x, 0) → 0
*(*(x, y), z) → *(x, *(y, z))

Rewrite Strategy: INNERMOST

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

The following rules are not reachable from basic terms in the dependency graph and can be removed:
*(*(x, y), z) → *(x, *(y, z))

(2) Obligation:

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


The TRS R consists of the following rules:

*(i(x), x) → 1
*(x, 0) → 0
*(1, y) → y

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

*(i(z0), z0) → 1
*(z0, 0) → 0
*(1, z0) → z0
Tuples:

*'(i(z0), z0) → c
*'(z0, 0) → c1
*'(1, z0) → c2
S tuples:

*'(i(z0), z0) → c
*'(z0, 0) → c1
*'(1, z0) → c2
K tuples:none
Defined Rule Symbols:

*

Defined Pair Symbols:

*'

Compound Symbols:

c, c1, c2

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

Removed 3 trailing nodes:

*'(i(z0), z0) → c
*'(z0, 0) → c1
*'(1, z0) → c2

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

*(i(z0), z0) → 1
*(z0, 0) → 0
*(1, z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

*

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty

(8) BOUNDS(1, 1)