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