(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(f(X)) → f(a(b(f(X))))
f(a(g(X))) → b(X)
b(X) → a(X)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0)) → f(a(b(f(z0))))
f(a(g(z0))) → b(z0)
b(z0) → a(z0)
Tuples:
F(f(z0)) → c(F(a(b(f(z0)))), B(f(z0)), F(z0))
F(a(g(z0))) → c1(B(z0))
S tuples:
F(f(z0)) → c(F(a(b(f(z0)))), B(f(z0)), F(z0))
F(a(g(z0))) → c1(B(z0))
K tuples:none
Defined Rule Symbols:
f, b
Defined Pair Symbols:
F
Compound Symbols:
c, c1
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
F(a(g(z0))) → c1(B(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0)) → f(a(b(f(z0))))
f(a(g(z0))) → b(z0)
b(z0) → a(z0)
Tuples:
F(f(z0)) → c(F(a(b(f(z0)))), B(f(z0)), F(z0))
S tuples:
F(f(z0)) → c(F(a(b(f(z0)))), B(f(z0)), F(z0))
K tuples:none
Defined Rule Symbols:
f, b
Defined Pair Symbols:
F
Compound Symbols:
c
(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.
F(f(z0)) → c(F(a(b(f(z0)))), B(f(z0)), F(z0))
We considered the (Usable) Rules:
b(z0) → a(z0)
And the Tuples:
F(f(z0)) → c(F(a(b(f(z0)))), B(f(z0)), F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(B(x1)) = [5] + x1
POL(F(x1)) = [2] + [4]x1
POL(a(x1)) = 0
POL(b(x1)) = 0
POL(c(x1, x2, x3)) = x1 + x2 + x3
POL(f(x1)) = [4] + [4]x1
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0)) → f(a(b(f(z0))))
f(a(g(z0))) → b(z0)
b(z0) → a(z0)
Tuples:
F(f(z0)) → c(F(a(b(f(z0)))), B(f(z0)), F(z0))
S tuples:none
K tuples:
F(f(z0)) → c(F(a(b(f(z0)))), B(f(z0)), F(z0))
Defined Rule Symbols:
f, b
Defined Pair Symbols:
F
Compound Symbols:
c
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))