(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(x1) → x1
a(a(b(x1))) → b(b(a(a(x1))))
b(x1) → a(c(a(x1)))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(a(b(z0))) → b(b(a(a(z0))))
b(z0) → a(c(a(z0)))
Tuples:
A(a(b(z0))) → c2(B(b(a(a(z0)))), B(a(a(z0))), A(a(z0)), A(z0))
B(z0) → c3(A(c(a(z0))), A(z0))
S tuples:
A(a(b(z0))) → c2(B(b(a(a(z0)))), B(a(a(z0))), A(a(z0)), A(z0))
B(z0) → c3(A(c(a(z0))), A(z0))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c2, c3
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
A(a(b(z0))) → c2(B(b(a(a(z0)))), B(a(a(z0))), A(a(z0)), A(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(a(b(z0))) → b(b(a(a(z0))))
b(z0) → a(c(a(z0)))
Tuples:
B(z0) → c3(A(c(a(z0))), A(z0))
S tuples:
B(z0) → c3(A(c(a(z0))), A(z0))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
B
Compound Symbols:
c3
(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)
Removed 1 of 1 dangling nodes:
B(z0) → c3(A(c(a(z0))), A(z0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(a(b(z0))) → b(b(a(a(z0))))
b(z0) → a(c(a(z0)))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))