(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(d(x1)) → d(b(x1))
a(x1) → b(b(b(x1)))
d(x1) → x1
a(x1) → x1
b(d(b(x1))) → a(d(x1))
b(c(x1)) → c(d(d(x1)))
a(c(x1)) → b(b(c(d(x1))))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(d(z0)) → d(b(z0))
a(z0) → b(b(b(z0)))
a(z0) → z0
a(c(z0)) → b(b(c(d(z0))))
d(z0) → z0
b(d(b(z0))) → a(d(z0))
b(c(z0)) → c(d(d(z0)))
Tuples:
A(d(z0)) → c1(D(b(z0)), B(z0))
A(z0) → c2(B(b(b(z0))), B(b(z0)), B(z0))
A(c(z0)) → c4(B(b(c(d(z0)))), B(c(d(z0))), D(z0))
B(d(b(z0))) → c6(A(d(z0)), D(z0))
B(c(z0)) → c7(D(d(z0)), D(z0))
S tuples:
A(d(z0)) → c1(D(b(z0)), B(z0))
A(z0) → c2(B(b(b(z0))), B(b(z0)), B(z0))
A(c(z0)) → c4(B(b(c(d(z0)))), B(c(d(z0))), D(z0))
B(d(b(z0))) → c6(A(d(z0)), D(z0))
B(c(z0)) → c7(D(d(z0)), D(z0))
K tuples:none
Defined Rule Symbols:
a, d, b
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c2, c4, c6, c7
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
A(d(z0)) → c1(D(b(z0)), B(z0))
B(d(b(z0))) → c6(A(d(z0)), D(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(d(z0)) → d(b(z0))
a(z0) → b(b(b(z0)))
a(z0) → z0
a(c(z0)) → b(b(c(d(z0))))
d(z0) → z0
b(d(b(z0))) → a(d(z0))
b(c(z0)) → c(d(d(z0)))
Tuples:
A(z0) → c2(B(b(b(z0))), B(b(z0)), B(z0))
A(c(z0)) → c4(B(b(c(d(z0)))), B(c(d(z0))), D(z0))
B(c(z0)) → c7(D(d(z0)), D(z0))
S tuples:
A(z0) → c2(B(b(b(z0))), B(b(z0)), B(z0))
A(c(z0)) → c4(B(b(c(d(z0)))), B(c(d(z0))), D(z0))
B(c(z0)) → c7(D(d(z0)), D(z0))
K tuples:none
Defined Rule Symbols:
a, d, b
Defined Pair Symbols:
A, B
Compound Symbols:
c2, c4, c7
(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)
Removed 3 of 3 dangling nodes:
A(z0) → c2(B(b(b(z0))), B(b(z0)), B(z0))
A(c(z0)) → c4(B(b(c(d(z0)))), B(c(d(z0))), D(z0))
B(c(z0)) → c7(D(d(z0)), D(z0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(d(z0)) → d(b(z0))
a(z0) → b(b(b(z0)))
a(z0) → z0
a(c(z0)) → b(b(c(d(z0))))
d(z0) → z0
b(d(b(z0))) → a(d(z0))
b(c(z0)) → c(d(d(z0)))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a, d, b
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))