(0) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, 1).
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 Cpx (relative) TRS 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(0) → c
F(s(0)) → c1(F(p(s(0))), P(s(0)))
P(s(z0)) → c2
S tuples:
F(0) → c
F(s(0)) → c1(F(p(s(0))), P(s(0)))
P(s(z0)) → c2
K tuples:none
Defined Rule Symbols:
f, p
Defined Pair Symbols:
F, P
Compound Symbols:
c, c1, c2
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
P(s(z0)) → c2
F(0) → c
(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))), 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
(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(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(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
(7) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
f(0) → cons(0)
f(s(0)) → f(p(s(0)))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
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:
p
Defined Pair Symbols:
F
Compound Symbols:
c1
(9) 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))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
Tuples:
F(s(0)) → c1(F(0))
S tuples:
F(s(0)) → c1(F(0))
K tuples:none
Defined Rule Symbols:
p
Defined Pair Symbols:
F
Compound Symbols:
c1
(11) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
F(s(0)) → c1(F(0))
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
p
Defined Pair Symbols:none
Compound Symbols:none
(13) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(14) BOUNDS(1, 1)