(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(a, f(x, a)) → f(a, f(a, f(f(a, a), x)))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(z0, a)) → f(a, f(a, f(f(a, a), z0)))
Tuples:
F(a, f(z0, a)) → c(F(a, f(a, f(f(a, a), z0))), F(a, f(f(a, a), z0)), F(f(a, a), z0), F(a, a))
S tuples:
F(a, f(z0, a)) → c(F(a, f(a, f(f(a, a), z0))), F(a, f(f(a, a), z0)), F(f(a, a), z0), F(a, a))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c
(3) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
a,
f(
z0,
a)) →
c(
F(
a,
f(
a,
f(
f(
a,
a),
z0))),
F(
a,
f(
f(
a,
a),
z0)),
F(
f(
a,
a),
z0),
F(
a,
a)) by
F(a, f(a, a)) → c(F(a, f(a, f(a, f(f(a, a), f(a, a))))), F(a, f(f(a, a), a)), F(f(a, a), a), F(a, a))
F(a, f(x0, a)) → c
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(z0, a)) → f(a, f(a, f(f(a, a), z0)))
Tuples:
F(a, f(a, a)) → c(F(a, f(a, f(a, f(f(a, a), f(a, a))))), F(a, f(f(a, a), a)), F(f(a, a), a), F(a, a))
F(a, f(x0, a)) → c
S tuples:
F(a, f(a, a)) → c(F(a, f(a, f(a, f(f(a, a), f(a, a))))), F(a, f(f(a, a), a)), F(f(a, a), a), F(a, a))
F(a, f(x0, a)) → c
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c, c
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
F(a, f(a, a)) → c(F(a, f(a, f(a, f(f(a, a), f(a, a))))), F(a, f(f(a, a), a)), F(f(a, a), a), F(a, a))
F(a, f(x0, a)) → c
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(z0, a)) → f(a, f(a, f(f(a, a), z0)))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))