(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:
*'(z0, 0) → c1
*'(i(z0), z0) → c
*'(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)