(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(a(x1)) → b(b(b(x1)))
b(x1) → c(c(d(x1)))
c(x1) → d(d(d(x1)))
b(c(x1)) → c(b(x1))
b(c(d(x1))) → a(x1)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(z0)) → b(b(b(z0)))
b(z0) → c(c(d(z0)))
b(c(z0)) → c(b(z0))
b(c(d(z0))) → a(z0)
c(z0) → d(d(d(z0)))
Tuples:
A(a(z0)) → c1(B(b(b(z0))), B(b(z0)), B(z0))
B(z0) → c2(C(c(d(z0))), C(d(z0)))
B(c(z0)) → c3(C(b(z0)), B(z0))
B(c(d(z0))) → c4(A(z0))
S tuples:
A(a(z0)) → c1(B(b(b(z0))), B(b(z0)), B(z0))
B(z0) → c2(C(c(d(z0))), C(d(z0)))
B(c(z0)) → c3(C(b(z0)), B(z0))
B(c(d(z0))) → c4(A(z0))
K tuples:none
Defined Rule Symbols:
a, b, c
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c2, c3, c4
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
A(a(z0)) → c1(B(b(b(z0))), B(b(z0)), B(z0))
B(c(z0)) → c3(C(b(z0)), B(z0))
B(c(d(z0))) → c4(A(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(z0)) → b(b(b(z0)))
b(z0) → c(c(d(z0)))
b(c(z0)) → c(b(z0))
b(c(d(z0))) → a(z0)
c(z0) → d(d(d(z0)))
Tuples:
B(z0) → c2(C(c(d(z0))), C(d(z0)))
S tuples:
B(z0) → c2(C(c(d(z0))), C(d(z0)))
K tuples:none
Defined Rule Symbols:
a, b, c
Defined Pair Symbols:
B
Compound Symbols:
c2
(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)
Removed 1 of 1 dangling nodes:
B(z0) → c2(C(c(d(z0))), C(d(z0)))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(z0)) → b(b(b(z0)))
b(z0) → c(c(d(z0)))
b(c(z0)) → c(b(z0))
b(c(d(z0))) → a(z0)
c(z0) → d(d(d(z0)))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a, b, c
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))