(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(x1) → b(c(x1))
a(c(b(x1))) → b(a(a(c(x1))))
b(b(x1)) → x1
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → b(c(z0))
a(c(b(z0))) → b(a(a(c(z0))))
b(b(z0)) → z0
Tuples:
A(z0) → c1(B(c(z0)))
A(c(b(z0))) → c2(B(a(a(c(z0)))), A(a(c(z0))), A(c(z0)))
S tuples:
A(z0) → c1(B(c(z0)))
A(c(b(z0))) → c2(B(a(a(c(z0)))), A(a(c(z0))), A(c(z0)))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A
Compound Symbols:
c1, c2
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
A(c(b(z0))) → c2(B(a(a(c(z0)))), A(a(c(z0))), A(c(z0)))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → b(c(z0))
a(c(b(z0))) → b(a(a(c(z0))))
b(b(z0)) → z0
Tuples:
A(z0) → c1(B(c(z0)))
S tuples:
A(z0) → c1(B(c(z0)))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A
Compound Symbols:
c1
(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)
Removed 1 of 1 dangling nodes:
A(z0) → c1(B(c(z0)))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → b(c(z0))
a(c(b(z0))) → b(a(a(c(z0))))
b(b(z0)) → 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))