(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
+(x, 0) → x
+(x, i(x)) → 0
+(+(x, y), z) → +(x, +(y, z))
*(x, +(y, z)) → +(*(x, y), *(x, z))
*(+(x, y), z) → +(*(x, z), *(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:
+(z0, 0) → z0
+(z0, i(z0)) → 0
+(+(z0, z1), z2) → +(z0, +(z1, z2))
*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, z2))
*(+(z0, z1), z2) → +(*(z0, z2), *(z1, z2))
Tuples:
+'(z0, 0) → c
+'(z0, i(z0)) → c1
+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2))
*'(z0, +(z1, z2)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))
S tuples:
+'(z0, 0) → c
+'(z0, i(z0)) → c1
+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2))
*'(z0, +(z1, z2)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))
K tuples:none
Defined Rule Symbols:
+, *
Defined Pair Symbols:
+', *'
Compound Symbols:
c, c1, c2, c3, c4
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
+'(z0, 0) → c
+'(z0, i(z0)) → c1
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
+(z0, 0) → z0
+(z0, i(z0)) → 0
+(+(z0, z1), z2) → +(z0, +(z1, z2))
*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, z2))
*(+(z0, z1), z2) → +(*(z0, z2), *(z1, z2))
Tuples:
+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2))
*'(z0, +(z1, z2)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))
S tuples:
+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2))
*'(z0, +(z1, z2)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))
K tuples:none
Defined Rule Symbols:
+, *
Defined Pair Symbols:
+', *'
Compound Symbols:
c2, c3, c4
(5) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2))
*'(z0, +(z1, z2)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
+(z0, 0) → z0
+(z0, i(z0)) → 0
+(+(z0, z1), z2) → +(z0, +(z1, z2))
*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, z2))
*(+(z0, z1), z2) → +(*(z0, z2), *(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)