(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(f(x, a), y) → f(f(a, y), f(a, x))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0, a), z1) → f(f(a, z1), f(a, z0))
Tuples:
F(f(z0, a), z1) → c(F(f(a, z1), f(a, z0)), F(a, z1), F(a, z0))
S tuples:
F(f(z0, a), z1) → c(F(f(a, z1), f(a, z0)), F(a, z1), F(a, z0))
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(
f(
z0,
a),
z1) →
c(
F(
f(
a,
z1),
f(
a,
z0)),
F(
a,
z1),
F(
a,
z0)) by
F(f(x0, a), x1) → c(F(f(a, x1), f(a, x0)))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0, a), z1) → f(f(a, z1), f(a, z0))
Tuples:
F(f(x0, a), x1) → c(F(f(a, x1), f(a, x0)))
S tuples:
F(f(x0, a), x1) → c(F(f(a, x1), f(a, x0)))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c
(5) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use instantiation to replace
F(
f(
x0,
a),
x1) →
c(
F(
f(
a,
x1),
f(
a,
x0))) by
F(f(a, a), f(a, x0)) → c(F(f(a, f(a, x0)), f(a, a)))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0, a), z1) → f(f(a, z1), f(a, z0))
Tuples:
F(f(a, a), f(a, x0)) → c(F(f(a, f(a, x0)), f(a, a)))
S tuples:
F(f(a, a), f(a, x0)) → c(F(f(a, f(a, x0)), f(a, a)))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c
(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
F(f(a, a), f(a, x0)) → c(F(f(a, f(a, x0)), f(a, a)))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0, a), z1) → f(f(a, z1), f(a, z0))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:none
Compound Symbols:none
(9) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(10) BOUNDS(O(1), O(1))