(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(x1) → x1
a(a(b(x1))) → c(c(b(a(x1))))
b(c(x1)) → a(b(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(a(b(z0))) → c(c(b(a(z0))))
b(c(z0)) → a(b(z0))
Tuples:
A(a(b(z0))) → c2(B(a(z0)), A(z0))
B(c(z0)) → c3(A(b(z0)), B(z0))
S tuples:
A(a(b(z0))) → c2(B(a(z0)), A(z0))
B(c(z0)) → c3(A(b(z0)), B(z0))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c2, c3
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
A(a(b(z0))) → c2(B(a(z0)), A(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(a(b(z0))) → c(c(b(a(z0))))
b(c(z0)) → a(b(z0))
Tuples:
B(c(z0)) → c3(A(b(z0)), B(z0))
S tuples:
B(c(z0)) → c3(A(b(z0)), B(z0))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
B
Compound Symbols:
c3
(5) CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(a(b(z0))) → c(c(b(a(z0))))
b(c(z0)) → a(b(z0))
Tuples:
B(c(z0)) → c3(B(z0))
S tuples:
B(c(z0)) → c3(B(z0))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
B
Compound Symbols:
c3
(7) 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.
B(c(z0)) → c3(B(z0))
We considered the (Usable) Rules:none
And the Tuples:
B(c(z0)) → c3(B(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(B(x1)) = [5]x1
POL(c(x1)) = [1] + x1
POL(c3(x1)) = x1
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(a(b(z0))) → c(c(b(a(z0))))
b(c(z0)) → a(b(z0))
Tuples:
B(c(z0)) → c3(B(z0))
S tuples:none
K tuples:
B(c(z0)) → c3(B(z0))
Defined Rule Symbols:
a, b
Defined Pair Symbols:
B
Compound Symbols:
c3
(9) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(10) BOUNDS(O(1), O(1))