(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
minus(0) → 0
+(x, 0) → x
+(0, y) → y
+(minus(1), 1) → 0
minus(minus(x)) → x
+(x, minus(y)) → minus(+(minus(x), y))
+(x, +(y, z)) → +(+(x, y), z)
+(minus(+(x, 1)), 1) → minus(x)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
minus(0) → 0
minus(minus(z0)) → z0
+(z0, 0) → z0
+(0, z0) → z0
+(minus(1), 1) → 0
+(z0, minus(z1)) → minus(+(minus(z0), z1))
+(z0, +(z1, z2)) → +(+(z0, z1), z2)
+(minus(+(z0, 1)), 1) → minus(z0)
Tuples:
+'(z0, minus(z1)) → c5(MINUS(+(minus(z0), z1)), +'(minus(z0), z1), MINUS(z0))
+'(z0, +(z1, z2)) → c6(+'(+(z0, z1), z2), +'(z0, z1))
+'(minus(+(z0, 1)), 1) → c7(MINUS(z0))
S tuples:
+'(z0, minus(z1)) → c5(MINUS(+(minus(z0), z1)), +'(minus(z0), z1), MINUS(z0))
+'(z0, +(z1, z2)) → c6(+'(+(z0, z1), z2), +'(z0, z1))
+'(minus(+(z0, 1)), 1) → c7(MINUS(z0))
K tuples:none
Defined Rule Symbols:
minus, +
Defined Pair Symbols:
+'
Compound Symbols:
c5, c6, c7
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
+'(z0, minus(z1)) → c5(MINUS(+(minus(z0), z1)), +'(minus(z0), z1), MINUS(z0))
+'(z0, +(z1, z2)) → c6(+'(+(z0, z1), z2), +'(z0, z1))
+'(minus(+(z0, 1)), 1) → c7(MINUS(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
minus(0) → 0
minus(minus(z0)) → z0
+(z0, 0) → z0
+(0, z0) → z0
+(minus(1), 1) → 0
+(z0, minus(z1)) → minus(+(minus(z0), z1))
+(z0, +(z1, z2)) → +(+(z0, z1), z2)
+(minus(+(z0, 1)), 1) → minus(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
minus, +
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(6) BOUNDS(O(1), O(1))