(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(s(x)) → s(s(f(p(s(x)))))
f(0) → 0
p(s(x)) → x
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(s(z0)) → s(s(f(p(s(z0)))))
f(0) → 0
p(s(z0)) → z0
Tuples:
F(s(z0)) → c(F(p(s(z0))), P(s(z0)))
S tuples:
F(s(z0)) → c(F(p(s(z0))), P(s(z0)))
K tuples:none
Defined Rule Symbols:
f, p
Defined Pair Symbols:
F
Compound Symbols:
c
(3) CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(s(z0)) → s(s(f(p(s(z0)))))
f(0) → 0
p(s(z0)) → z0
Tuples:
F(s(z0)) → c(F(p(s(z0))))
S tuples:
F(s(z0)) → c(F(p(s(z0))))
K tuples:none
Defined Rule Symbols:
f, p
Defined Pair Symbols:
F
Compound Symbols:
c
(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
s(
z0)) →
c(
F(
p(
s(
z0)))) by
F(s(z0)) → c(F(z0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(s(z0)) → s(s(f(p(s(z0)))))
f(0) → 0
p(s(z0)) → z0
Tuples:
F(s(z0)) → c(F(z0))
S tuples:
F(s(z0)) → c(F(z0))
K tuples:none
Defined Rule Symbols:
f, p
Defined Pair Symbols:
F
Compound Symbols:
c
(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
F(s(z0)) → c(F(z0))
We considered the (Usable) Rules:none
And the Tuples:
F(s(z0)) → c(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(F(x1)) = x1
POL(c(x1)) = x1
POL(s(x1)) = [2] + x1
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(s(z0)) → s(s(f(p(s(z0)))))
f(0) → 0
p(s(z0)) → z0
Tuples:
F(s(z0)) → c(F(z0))
S tuples:none
K tuples:
F(s(z0)) → c(F(z0))
Defined Rule Symbols:
f, p
Defined Pair Symbols:
F
Compound Symbols:
c
(9) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(10) BOUNDS(O(1), O(1))