(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(X) → g(n__h(n__f(X)))
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0) → g(n__h(n__f(z0)))
f(z0) → n__f(z0)
h(z0) → n__h(z0)
activate(n__h(z0)) → h(activate(z0))
activate(n__f(z0)) → f(activate(z0))
activate(z0) → z0
Tuples:
ACTIVATE(n__h(z0)) → c3(H(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c4(F(activate(z0)), ACTIVATE(z0))
S tuples:
ACTIVATE(n__h(z0)) → c3(H(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c4(F(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:
f, h, activate
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c3, c4
(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.
ACTIVATE(n__f(z0)) → c4(F(activate(z0)), ACTIVATE(z0))
We considered the (Usable) Rules:
activate(n__h(z0)) → h(activate(z0))
activate(n__f(z0)) → f(activate(z0))
activate(z0) → z0
f(z0) → g(n__h(n__f(z0)))
f(z0) → n__f(z0)
h(z0) → n__h(z0)
And the Tuples:
ACTIVATE(n__h(z0)) → c3(H(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c4(F(activate(z0)), ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ACTIVATE(x1)) = [2]x1
POL(F(x1)) = 0
POL(H(x1)) = [2]
POL(activate(x1)) = [4]x1
POL(c3(x1, x2)) = x1 + x2
POL(c4(x1, x2)) = x1 + x2
POL(f(x1)) = [4] + x1
POL(g(x1)) = [1]
POL(h(x1)) = [4] + x1
POL(n__f(x1)) = [2] + x1
POL(n__h(x1)) = [1] + x1
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0) → g(n__h(n__f(z0)))
f(z0) → n__f(z0)
h(z0) → n__h(z0)
activate(n__h(z0)) → h(activate(z0))
activate(n__f(z0)) → f(activate(z0))
activate(z0) → z0
Tuples:
ACTIVATE(n__h(z0)) → c3(H(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c4(F(activate(z0)), ACTIVATE(z0))
S tuples:
ACTIVATE(n__h(z0)) → c3(H(activate(z0)), ACTIVATE(z0))
K tuples:
ACTIVATE(n__f(z0)) → c4(F(activate(z0)), ACTIVATE(z0))
Defined Rule Symbols:
f, h, activate
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c3, c4
(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.
ACTIVATE(n__h(z0)) → c3(H(activate(z0)), ACTIVATE(z0))
We considered the (Usable) Rules:
activate(n__h(z0)) → h(activate(z0))
activate(n__f(z0)) → f(activate(z0))
activate(z0) → z0
f(z0) → g(n__h(n__f(z0)))
f(z0) → n__f(z0)
h(z0) → n__h(z0)
And the Tuples:
ACTIVATE(n__h(z0)) → c3(H(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c4(F(activate(z0)), ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ACTIVATE(x1)) = [4]x1
POL(F(x1)) = [3]
POL(H(x1)) = 0
POL(activate(x1)) = [3]
POL(c3(x1, x2)) = x1 + x2
POL(c4(x1, x2)) = x1 + x2
POL(f(x1)) = [3] + [2]x1
POL(g(x1)) = [4]
POL(h(x1)) = [5] + [2]x1
POL(n__f(x1)) = [1] + x1
POL(n__h(x1)) = [4] + x1
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0) → g(n__h(n__f(z0)))
f(z0) → n__f(z0)
h(z0) → n__h(z0)
activate(n__h(z0)) → h(activate(z0))
activate(n__f(z0)) → f(activate(z0))
activate(z0) → z0
Tuples:
ACTIVATE(n__h(z0)) → c3(H(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c4(F(activate(z0)), ACTIVATE(z0))
S tuples:none
K tuples:
ACTIVATE(n__f(z0)) → c4(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__h(z0)) → c3(H(activate(z0)), ACTIVATE(z0))
Defined Rule Symbols:
f, h, activate
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c3, c4
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))