(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
*(x, +(y, z)) → +(*(x, y), *(x, z))
*(+(y, z), x) → +(*(x, y), *(x, z))
*(*(x, y), z) → *(x, *(y, z))
+(+(x, y), z) → +(x, +(y, z))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, z2))
*(+(z0, z1), z2) → +(*(z2, z0), *(z2, z1))
*(*(z0, z1), z2) → *(z0, *(z1, z2))
+(+(z0, z1), z2) → +(z0, +(z1, z2))
Tuples:
*'(z0, +(z1, z2)) → c(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c1(+'(*(z2, z0), *(z2, z1)), *'(z2, z0), *'(z2, z1))
*'(*(z0, z1), z2) → c2(*'(z0, *(z1, z2)), *'(z1, z2))
+'(+(z0, z1), z2) → c3(+'(z0, +(z1, z2)), +'(z1, z2))
S tuples:
*'(z0, +(z1, z2)) → c(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c1(+'(*(z2, z0), *(z2, z1)), *'(z2, z0), *'(z2, z1))
*'(*(z0, z1), z2) → c2(*'(z0, *(z1, z2)), *'(z1, z2))
+'(+(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) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
*'(z0, +(z1, z2)) → c(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c1(+'(*(z2, z0), *(z2, z1)), *'(z2, z0), *'(z2, z1))
*'(*(z0, z1), z2) → c2(*'(z0, *(z1, z2)), *'(z1, z2))
+'(+(z0, z1), z2) → c3(+'(z0, +(z1, z2)), +'(z1, z2))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, z2))
*(+(z0, z1), z2) → +(*(z2, z0), *(z2, z1))
*(*(z0, z1), z2) → *(z0, *(z1, z2))
+(+(z0, z1), z2) → +(z0, +(z1, z2))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
*, +
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(6) BOUNDS(O(1), O(1))