(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(f(y, z), f(x, f(a, x))) → f(f(f(a, z), f(x, a)), f(a, y))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0, z1), f(z2, f(a, z2))) → f(f(f(a, z1), f(z2, a)), f(a, z0))
Tuples:
F(f(z0, z1), f(z2, f(a, z2))) → c(F(f(f(a, z1), f(z2, a)), f(a, z0)), F(f(a, z1), f(z2, a)), F(a, z1), F(z2, a), F(a, z0))
S tuples:
F(f(z0, z1), f(z2, f(a, z2))) → c(F(f(f(a, z1), f(z2, a)), f(a, z0)), F(f(a, z1), f(z2, a)), F(a, z1), F(z2, a), 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,
z1),
f(
z2,
f(
a,
z2))) →
c(
F(
f(
f(
a,
z1),
f(
z2,
a)),
f(
a,
z0)),
F(
f(
a,
z1),
f(
z2,
a)),
F(
a,
z1),
F(
z2,
a),
F(
a,
z0)) by
F(f(x0, x1), f(x2, f(a, x2))) → c(F(f(f(a, x1), f(x2, a)), f(a, x0)))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0, z1), f(z2, f(a, z2))) → f(f(f(a, z1), f(z2, a)), f(a, z0))
Tuples:
F(f(x0, x1), f(x2, f(a, x2))) → c(F(f(f(a, x1), f(x2, a)), f(a, x0)))
S tuples:
F(f(x0, x1), f(x2, f(a, x2))) → c(F(f(f(a, x1), f(x2, a)), 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,
x1),
f(
x2,
f(
a,
x2))) →
c(
F(
f(
f(
a,
x1),
f(
x2,
a)),
f(
a,
x0))) by
F(f(f(a, x1), f(x2, a)), f(a, f(a, a))) → c(F(f(f(a, f(x2, a)), f(a, a)), f(a, f(a, x1))))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0, z1), f(z2, f(a, z2))) → f(f(f(a, z1), f(z2, a)), f(a, z0))
Tuples:
F(f(f(a, x1), f(x2, a)), f(a, f(a, a))) → c(F(f(f(a, f(x2, a)), f(a, a)), f(a, f(a, x1))))
S tuples:
F(f(f(a, x1), f(x2, a)), f(a, f(a, a))) → c(F(f(f(a, f(x2, a)), f(a, a)), f(a, f(a, x1))))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c
(7) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use instantiation to replace
F(
f(
f(
a,
x1),
f(
x2,
a)),
f(
a,
f(
a,
a))) →
c(
F(
f(
f(
a,
f(
x2,
a)),
f(
a,
a)),
f(
a,
f(
a,
x1)))) by
F(f(f(a, f(x1, a)), f(a, a)), f(a, f(a, a))) → c(F(f(f(a, f(a, a)), f(a, a)), f(a, f(a, f(x1, a)))))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0, z1), f(z2, f(a, z2))) → f(f(f(a, z1), f(z2, a)), f(a, z0))
Tuples:
F(f(f(a, f(x1, a)), f(a, a)), f(a, f(a, a))) → c(F(f(f(a, f(a, a)), f(a, a)), f(a, f(a, f(x1, a)))))
S tuples:
F(f(f(a, f(x1, a)), f(a, a)), f(a, f(a, a))) → c(F(f(f(a, f(a, a)), f(a, a)), f(a, f(a, f(x1, a)))))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c
(9) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
F(f(f(a, f(x1, a)), f(a, a)), f(a, f(a, a))) → c(F(f(f(a, f(a, a)), f(a, a)), f(a, f(a, f(x1, a)))))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0, z1), f(z2, f(a, z2))) → f(f(f(a, z1), f(z2, a)), f(a, z0))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:none
Compound Symbols:none
(11) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(12) BOUNDS(O(1), O(1))