(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)))
b(d(b(x1))) → a(c(x1))
c(x1) → 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)))
b(d(b(z0))) → a(c(z0))
c(z0) → d(z0)
Tuples:
A(d(z0)) → c1(B(z0))
A(z0) → c2(B(b(b(z0))), B(b(z0)), B(z0))
B(d(b(z0))) → c3(A(c(z0)), C(z0))
S tuples:
A(d(z0)) → c1(B(z0))
A(z0) → c2(B(b(b(z0))), B(b(z0)), B(z0))
B(d(b(z0))) → c3(A(c(z0)), C(z0))
K tuples:none
Defined Rule Symbols:
a, b, c
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c2, c3
(3) CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(d(z0)) → d(b(z0))
a(z0) → b(b(b(z0)))
b(d(b(z0))) → a(c(z0))
c(z0) → d(z0)
Tuples:
A(d(z0)) → c1(B(z0))
A(z0) → c2(B(b(b(z0))), B(b(z0)), B(z0))
B(d(b(z0))) → c3(A(c(z0)))
S tuples:
A(d(z0)) → c1(B(z0))
A(z0) → c2(B(b(b(z0))), B(b(z0)), B(z0))
B(d(b(z0))) → c3(A(c(z0)))
K tuples:none
Defined Rule Symbols:
a, b, c
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c2, c3
(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
A(
z0) →
c2(
B(
b(
b(
z0))),
B(
b(
z0)),
B(
z0)) by
A(d(b(z0))) → c2(B(b(a(c(z0)))), B(b(d(b(z0)))), B(d(b(z0))))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(d(z0)) → d(b(z0))
a(z0) → b(b(b(z0)))
b(d(b(z0))) → a(c(z0))
c(z0) → d(z0)
Tuples:
A(d(z0)) → c1(B(z0))
B(d(b(z0))) → c3(A(c(z0)))
A(d(b(z0))) → c2(B(b(a(c(z0)))), B(b(d(b(z0)))), B(d(b(z0))))
S tuples:
A(d(z0)) → c1(B(z0))
B(d(b(z0))) → c3(A(c(z0)))
A(d(b(z0))) → c2(B(b(a(c(z0)))), B(b(d(b(z0)))), B(d(b(z0))))
K tuples:none
Defined Rule Symbols:
a, b, c
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c3, c2
(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
B(
d(
b(
z0))) →
c3(
A(
c(
z0))) by
B(d(b(z0))) → c3(A(d(z0)))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(d(z0)) → d(b(z0))
a(z0) → b(b(b(z0)))
b(d(b(z0))) → a(c(z0))
c(z0) → d(z0)
Tuples:
A(d(z0)) → c1(B(z0))
A(d(b(z0))) → c2(B(b(a(c(z0)))), B(b(d(b(z0)))), B(d(b(z0))))
B(d(b(z0))) → c3(A(d(z0)))
S tuples:
A(d(z0)) → c1(B(z0))
A(d(b(z0))) → c2(B(b(a(c(z0)))), B(b(d(b(z0)))), B(d(b(z0))))
B(d(b(z0))) → c3(A(d(z0)))
K tuples:none
Defined Rule Symbols:
a, b, c
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c2, c3
(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
A(
d(
b(
z0))) →
c2(
B(
b(
a(
c(
z0)))),
B(
b(
d(
b(
z0)))),
B(
d(
b(
z0)))) by
A(d(b(x0))) → c2(B(b(b(b(b(c(x0)))))), B(b(d(b(x0)))), B(d(b(x0))))
A(d(b(z0))) → c2(B(b(a(d(z0)))), B(b(d(b(z0)))), B(d(b(z0))))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(d(z0)) → d(b(z0))
a(z0) → b(b(b(z0)))
b(d(b(z0))) → a(c(z0))
c(z0) → d(z0)
Tuples:
A(d(z0)) → c1(B(z0))
B(d(b(z0))) → c3(A(d(z0)))
A(d(b(x0))) → c2(B(b(b(b(b(c(x0)))))), B(b(d(b(x0)))), B(d(b(x0))))
A(d(b(z0))) → c2(B(b(a(d(z0)))), B(b(d(b(z0)))), B(d(b(z0))))
S tuples:
A(d(z0)) → c1(B(z0))
B(d(b(z0))) → c3(A(d(z0)))
A(d(b(x0))) → c2(B(b(b(b(b(c(x0)))))), B(b(d(b(x0)))), B(d(b(x0))))
A(d(b(z0))) → c2(B(b(a(d(z0)))), B(b(d(b(z0)))), B(d(b(z0))))
K tuples:none
Defined Rule Symbols:
a, b, c
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c3, c2
(11) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace A(d(b(x0))) → c2(B(b(b(b(b(c(x0)))))), B(b(d(b(x0)))), B(d(b(x0)))) by A(d(b(z0))) → c2(B(b(b(b(b(d(z0)))))), B(b(d(b(z0)))), B(d(b(z0))))
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(d(z0)) → d(b(z0))
a(z0) → b(b(b(z0)))
b(d(b(z0))) → a(c(z0))
c(z0) → d(z0)
Tuples:
A(d(z0)) → c1(B(z0))
B(d(b(z0))) → c3(A(d(z0)))
A(d(b(z0))) → c2(B(b(a(d(z0)))), B(b(d(b(z0)))), B(d(b(z0))))
A(d(b(z0))) → c2(B(b(b(b(b(d(z0)))))), B(b(d(b(z0)))), B(d(b(z0))))
S tuples:
A(d(z0)) → c1(B(z0))
B(d(b(z0))) → c3(A(d(z0)))
A(d(b(z0))) → c2(B(b(a(d(z0)))), B(b(d(b(z0)))), B(d(b(z0))))
A(d(b(z0))) → c2(B(b(b(b(b(d(z0)))))), B(b(d(b(z0)))), B(d(b(z0))))
K tuples:none
Defined Rule Symbols:
a, b, c
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c3, c2
(13) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use forward instantiation to replace
A(
d(
z0)) →
c1(
B(
z0)) by
A(d(d(b(y0)))) → c1(B(d(b(y0))))
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(d(z0)) → d(b(z0))
a(z0) → b(b(b(z0)))
b(d(b(z0))) → a(c(z0))
c(z0) → d(z0)
Tuples:
B(d(b(z0))) → c3(A(d(z0)))
A(d(b(z0))) → c2(B(b(a(d(z0)))), B(b(d(b(z0)))), B(d(b(z0))))
A(d(b(z0))) → c2(B(b(b(b(b(d(z0)))))), B(b(d(b(z0)))), B(d(b(z0))))
A(d(d(b(y0)))) → c1(B(d(b(y0))))
S tuples:
B(d(b(z0))) → c3(A(d(z0)))
A(d(b(z0))) → c2(B(b(a(d(z0)))), B(b(d(b(z0)))), B(d(b(z0))))
A(d(b(z0))) → c2(B(b(b(b(b(d(z0)))))), B(b(d(b(z0)))), B(d(b(z0))))
A(d(d(b(y0)))) → c1(B(d(b(y0))))
K tuples:none
Defined Rule Symbols:
a, b, c
Defined Pair Symbols:
B, A
Compound Symbols:
c3, c2, c1
(15) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
B(d(b(z0))) → c3(A(d(z0)))
A(d(b(z0))) → c2(B(b(a(d(z0)))), B(b(d(b(z0)))), B(d(b(z0))))
A(d(b(z0))) → c2(B(b(b(b(b(d(z0)))))), B(b(d(b(z0)))), B(d(b(z0))))
A(d(d(b(y0)))) → c1(B(d(b(y0))))
(16) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(d(z0)) → d(b(z0))
a(z0) → b(b(b(z0)))
b(d(b(z0))) → a(c(z0))
c(z0) → d(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a, b, c
Defined Pair Symbols:none
Compound Symbols:none
(17) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(18) BOUNDS(O(1), O(1))