(0) Obligation:
Runtime Complexity TRS:
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) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
*(i(z0), z0) → 1
*(1, z0) → z0
*(z0, 0) → 0
*(*(z0, z1), z2) → *(z0, *(z1, z2))
Tuples:
*'(i(z0), z0) → c
*'(1, z0) → c1
*'(z0, 0) → c2
*'(*(z0, z1), z2) → c3(*'(z0, *(z1, z2)), *'(z1, z2))
S tuples:
*'(i(z0), z0) → c
*'(1, z0) → c1
*'(z0, 0) → c2
*'(*(z0, z1), z2) → c3(*'(z0, *(z1, z2)), *'(z1, z2))
K tuples:none
Defined Rule Symbols:
*
Defined Pair Symbols:
*'
Compound Symbols:
c, c1, c2, c3
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
*'(z0, 0) → c2
*'(1, z0) → c1
*'(i(z0), z0) → c
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
*(i(z0), z0) → 1
*(1, z0) → z0
*(z0, 0) → 0
*(*(z0, z1), z2) → *(z0, *(z1, z2))
Tuples:
*'(*(z0, z1), z2) → c3(*'(z0, *(z1, z2)), *'(z1, z2))
S tuples:
*'(*(z0, z1), z2) → c3(*'(z0, *(z1, z2)), *'(z1, z2))
K tuples:none
Defined Rule Symbols:
*
Defined Pair Symbols:
*'
Compound Symbols:
c3
(5) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
*'(*(z0, z1), z2) → c3(*'(z0, *(z1, z2)), *'(z1, z2))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
*(i(z0), z0) → 1
*(1, z0) → z0
*(z0, 0) → 0
*(*(z0, z1), z2) → *(z0, *(z1, z2))
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)