(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(0) → cons(0, n__f(s(0)))
f(s(0)) → f(p(s(0)))
p(s(X)) → X
f(X) → n__f(X)
activate(n__f(X)) → f(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(0) → cons(0, n__f(s(0)))
f(s(0)) → f(p(s(0)))
f(z0) → n__f(z0)
p(s(z0)) → z0
activate(n__f(z0)) → f(z0)
activate(z0) → z0
Tuples:
F(s(0)) → c1(F(p(s(0))), P(s(0)))
ACTIVATE(n__f(z0)) → c4(F(z0))
S tuples:
F(s(0)) → c1(F(p(s(0))), P(s(0)))
ACTIVATE(n__f(z0)) → c4(F(z0))
K tuples:none
Defined Rule Symbols:
f, p, activate
Defined Pair Symbols:
F, ACTIVATE
Compound Symbols:
c1, c4
(3) CdtGraphRemoveDanglingProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 of 2 dangling nodes:
ACTIVATE(n__f(z0)) → c4(F(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0) → cons(0, n__f(s(0)))
f(s(0)) → f(p(s(0)))
f(z0) → n__f(z0)
p(s(z0)) → z0
activate(n__f(z0)) → f(z0)
activate(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, activate
Defined Pair Symbols:
F
Compound Symbols:
c1
(5) CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0) → cons(0, n__f(s(0)))
f(s(0)) → f(p(s(0)))
f(z0) → n__f(z0)
p(s(z0)) → z0
activate(n__f(z0)) → f(z0)
activate(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, activate
Defined Pair Symbols:
F
Compound Symbols:
c1
(7) 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))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0) → cons(0, n__f(s(0)))
f(s(0)) → f(p(s(0)))
f(z0) → n__f(z0)
p(s(z0)) → z0
activate(n__f(z0)) → f(z0)
activate(z0) → z0
Tuples:
F(s(0)) → c1(F(0))
S tuples:
F(s(0)) → c1(F(0))
K tuples:none
Defined Rule Symbols:
f, p, activate
Defined Pair Symbols:
F
Compound Symbols:
c1
(9) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)
Removed 1 of 1 dangling nodes:
F(s(0)) → c1(F(0))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0) → cons(0, n__f(s(0)))
f(s(0)) → f(p(s(0)))
f(z0) → n__f(z0)
p(s(z0)) → z0
activate(n__f(z0)) → f(z0)
activate(z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, p, activate
Defined Pair Symbols:none
Compound Symbols:none
(11) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(12) BOUNDS(O(1), O(1))