(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(x1) → x1
a(b(x1)) → b(b(a(a(c(a(x1))))))
c(c(a(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) → z0
a(b(z0)) → b(b(a(a(c(a(z0))))))
c(c(a(z0))) → z0
Tuples:
A(b(z0)) → c2(A(a(c(a(z0)))), A(c(a(z0))), C(a(z0)), A(z0))
S tuples:
A(b(z0)) → c2(A(a(c(a(z0)))), A(c(a(z0))), C(a(z0)), A(z0))
K tuples:none
Defined Rule Symbols:
a, c
Defined Pair Symbols:
A
Compound Symbols:
c2
(3) CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(z0)) → b(b(a(a(c(a(z0))))))
c(c(a(z0))) → z0
Tuples:
A(b(z0)) → c2(A(a(c(a(z0)))), A(z0))
S tuples:
A(b(z0)) → c2(A(a(c(a(z0)))), A(z0))
K tuples:none
Defined Rule Symbols:
a, c
Defined Pair Symbols:
A
Compound Symbols:
c2
(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
A(b(z0)) → c2(A(a(c(a(z0)))), A(z0))
We considered the (Usable) Rules:
a(z0) → z0
a(b(z0)) → b(b(a(a(c(a(z0))))))
And the Tuples:
A(b(z0)) → c2(A(a(c(a(z0)))), A(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(A(x1)) = [3] + [4]x1
POL(a(x1)) = [2]x1
POL(b(x1)) = [4] + x1
POL(c(x1)) = 0
POL(c2(x1, x2)) = x1 + x2
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(b(z0)) → b(b(a(a(c(a(z0))))))
c(c(a(z0))) → z0
Tuples:
A(b(z0)) → c2(A(a(c(a(z0)))), A(z0))
S tuples:none
K tuples:
A(b(z0)) → c2(A(a(c(a(z0)))), A(z0))
Defined Rule Symbols:
a, c
Defined Pair Symbols:
A
Compound Symbols:
c2
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))