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