(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(b(x)) → b(a(x))
a(c(x)) → x
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(b(z0)) → b(a(z0))
a(c(z0)) → z0
Tuples:
A(b(z0)) → c1(A(z0))
A(c(z0)) → c2
S tuples:
A(b(z0)) → c1(A(z0))
A(c(z0)) → c2
K tuples:none
Defined Rule Symbols:
a
Defined Pair Symbols:
A
Compound Symbols:
c1, c2
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
A(c(z0)) → c2
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(b(z0)) → b(a(z0))
a(c(z0)) → z0
Tuples:
A(b(z0)) → c1(A(z0))
S tuples:
A(b(z0)) → c1(A(z0))
K tuples:none
Defined Rule Symbols:
a
Defined Pair Symbols:
A
Compound Symbols:
c1
(5) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
a(b(z0)) → b(a(z0))
a(c(z0)) → z0
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
A(b(z0)) → c1(A(z0))
S tuples:
A(b(z0)) → c1(A(z0))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
A
Compound Symbols:
c1
(7) CdtRuleRemovalProof (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)) → c1(A(z0))
We considered the (Usable) Rules:none
And the Tuples:
A(b(z0)) → c1(A(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(A(x1)) = [5]x1
POL(b(x1)) = [1] + x1
POL(c1(x1)) = x1
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
A(b(z0)) → c1(A(z0))
S tuples:none
K tuples:
A(b(z0)) → c1(A(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
A
Compound Symbols:
c1
(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(10) BOUNDS(O(1), O(1))