(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(0) → cons(0)
f(s(0)) → f(p(s(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(0) → cons(0)
f(s(0)) → f(p(s(0)))
p(s(z0)) → z0
Tuples:
F(s(0)) → c1(F(p(s(0))), P(s(0)))
S tuples:
F(s(0)) → c1(F(p(s(0))), P(s(0)))
K tuples:none
Defined Rule Symbols:
f, p
Defined Pair Symbols:
F
Compound Symbols:
c1
(3) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0) → cons(0)
f(s(0)) → f(p(s(0)))
p(s(z0)) → z0
Tuples:
F(s(0)) → c1(F(p(s(0))))
S tuples:
F(s(0)) → c1(F(p(s(0))))
K tuples:none
Defined Rule Symbols:
f, p
Defined Pair Symbols:
F
Compound Symbols:
c1
(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
s(
0)) →
c1(
F(
p(
s(
0)))) by
F(s(0)) → c1(F(0))
F(s(0)) → c1
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0) → cons(0)
f(s(0)) → f(p(s(0)))
p(s(z0)) → z0
Tuples:
F(s(0)) → c1(F(0))
F(s(0)) → c1
S tuples:
F(s(0)) → c1(F(0))
F(s(0)) → c1
K tuples:none
Defined Rule Symbols:
f, p
Defined Pair Symbols:
F
Compound Symbols:
c1, c1
(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
F(s(0)) → c1
F(s(0)) → c1(F(0))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0) → cons(0)
f(s(0)) → f(p(s(0)))
p(s(z0)) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, p
Defined Pair Symbols:none
Compound Symbols:none
(9) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(10) BOUNDS(O(1), O(1))