(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(a, f(f(a, x), a)) → f(f(a, f(a, x)), a)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(f(a, z0), a)) → f(f(a, f(a, z0)), a)
Tuples:
F(a, f(f(a, z0), a)) → c(F(f(a, f(a, z0)), a), F(a, f(a, z0)), F(a, z0))
S tuples:
F(a, f(f(a, z0), a)) → c(F(f(a, f(a, z0)), a), F(a, f(a, z0)), F(a, z0))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c
(3) 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.
F(a, f(f(a, z0), a)) → c(F(f(a, f(a, z0)), a), F(a, f(a, z0)), F(a, z0))
We considered the (Usable) Rules:none
And the Tuples:
F(a, f(f(a, z0), a)) → c(F(f(a, f(a, z0)), a), F(a, f(a, z0)), F(a, z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(F(x1, x2)) = [2] + [5]x2
POL(a) = [4]
POL(c(x1, x2, x3)) = x1 + x2 + x3
POL(f(x1, x2)) = [5] + [4]x1 + [5]x2
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(f(a, z0), a)) → f(f(a, f(a, z0)), a)
Tuples:
F(a, f(f(a, z0), a)) → c(F(f(a, f(a, z0)), a), F(a, f(a, z0)), F(a, z0))
S tuples:none
K tuples:
F(a, f(f(a, z0), a)) → c(F(f(a, f(a, z0)), a), F(a, f(a, z0)), F(a, z0))
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c
(5) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(6) BOUNDS(O(1), O(1))