(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(c(x1)) → a(x1)
d(a(x1)) → a(c(b(c(d(x1)))))
a(c(b(c(x1)))) → c(b(c(c(x1))))
c(x1) → b(a(a(x1)))
d(c(x1)) → a(c(d(a(x1))))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(c(z0)) → a(z0)
a(c(b(c(z0)))) → c(b(c(c(z0))))
d(a(z0)) → a(c(b(c(d(z0)))))
d(c(z0)) → a(c(d(a(z0))))
c(z0) → b(a(a(z0)))
Tuples:
A(c(z0)) → c1(A(z0))
A(c(b(c(z0)))) → c2(C(b(c(c(z0)))), C(c(z0)), C(z0))
D(a(z0)) → c3(A(c(b(c(d(z0))))), C(b(c(d(z0)))), C(d(z0)), D(z0))
D(c(z0)) → c4(A(c(d(a(z0)))), C(d(a(z0))), D(a(z0)), A(z0))
C(z0) → c5(A(a(z0)), A(z0))
S tuples:
A(c(z0)) → c1(A(z0))
A(c(b(c(z0)))) → c2(C(b(c(c(z0)))), C(c(z0)), C(z0))
D(a(z0)) → c3(A(c(b(c(d(z0))))), C(b(c(d(z0)))), C(d(z0)), D(z0))
D(c(z0)) → c4(A(c(d(a(z0)))), C(d(a(z0))), D(a(z0)), A(z0))
C(z0) → c5(A(a(z0)), A(z0))
K tuples:none
Defined Rule Symbols:
a, d, c
Defined Pair Symbols:
A, D, C
Compound Symbols:
c1, c2, c3, c4, c5
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
A(c(z0)) → c1(A(z0))
A(c(b(c(z0)))) → c2(C(b(c(c(z0)))), C(c(z0)), C(z0))
D(a(z0)) → c3(A(c(b(c(d(z0))))), C(b(c(d(z0)))), C(d(z0)), D(z0))
D(c(z0)) → c4(A(c(d(a(z0)))), C(d(a(z0))), D(a(z0)), A(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(c(z0)) → a(z0)
a(c(b(c(z0)))) → c(b(c(c(z0))))
d(a(z0)) → a(c(b(c(d(z0)))))
d(c(z0)) → a(c(d(a(z0))))
c(z0) → b(a(a(z0)))
Tuples:
C(z0) → c5(A(a(z0)), A(z0))
S tuples:
C(z0) → c5(A(a(z0)), A(z0))
K tuples:none
Defined Rule Symbols:
a, d, c
Defined Pair Symbols:
C
Compound Symbols:
c5
(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)
Removed 1 of 1 dangling nodes:
C(z0) → c5(A(a(z0)), A(z0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(c(z0)) → a(z0)
a(c(b(c(z0)))) → c(b(c(c(z0))))
d(a(z0)) → a(c(b(c(d(z0)))))
d(c(z0)) → a(c(d(a(z0))))
c(z0) → b(a(a(z0)))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a, d, c
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))