(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(x, f(a, y)) → f(a, f(f(f(a, a), y), 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(a, f(f(f(a, a), z1), z0))
Tuples:
F(z0, f(a, z1)) → c(F(a, f(f(f(a, a), z1), z0)), F(f(f(a, a), z1), z0), F(f(a, a), z1), F(a, a))
S tuples:
F(z0, f(a, z1)) → c(F(a, f(f(f(a, a), z1), z0)), F(f(f(a, a), z1), z0), F(f(a, a), z1), 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(
a,
f(
f(
f(
a,
a),
z1),
z0)),
F(
f(
f(
a,
a),
z1),
z0),
F(
f(
a,
a),
z1),
F(
a,
a)) by
F(f(a, z1), f(a, x1)) → c(F(a, f(a, f(f(f(a, a), z1), f(f(a, a), x1)))), F(f(f(a, a), x1), f(a, z1)), F(f(a, a), x1), F(a, a))
F(x0, f(a, x1)) → c(F(f(a, a), x1))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0, f(a, z1)) → f(a, f(f(f(a, a), z1), z0))
Tuples:
F(f(a, z1), f(a, x1)) → c(F(a, f(a, f(f(f(a, a), z1), f(f(a, a), x1)))), F(f(f(a, a), x1), f(a, z1)), F(f(a, a), x1), F(a, a))
F(x0, f(a, x1)) → c(F(f(a, a), x1))
S tuples:
F(f(a, z1), f(a, x1)) → c(F(a, f(a, f(f(f(a, a), z1), f(f(a, a), x1)))), F(f(f(a, a), x1), f(a, z1)), F(f(a, a), x1), F(a, a))
F(x0, f(a, x1)) → c(F(f(a, a), x1))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c, c
(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
f(
a,
z1),
f(
a,
x1)) →
c(
F(
a,
f(
a,
f(
f(
f(
a,
a),
z1),
f(
f(
a,
a),
x1)))),
F(
f(
f(
a,
a),
x1),
f(
a,
z1)),
F(
f(
a,
a),
x1),
F(
a,
a)) by
F(f(a, x0), f(a, x1)) → c(F(a, f(a, f(f(f(a, a), x0), f(f(a, a), x1)))), F(f(f(a, a), x1), f(a, x0)), F(f(a, a), x1))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0, f(a, z1)) → f(a, f(f(f(a, a), z1), z0))
Tuples:
F(x0, f(a, x1)) → c(F(f(a, a), x1))
F(f(a, x0), f(a, x1)) → c(F(a, f(a, f(f(f(a, a), x0), f(f(a, a), x1)))), F(f(f(a, a), x1), f(a, x0)), F(f(a, a), x1))
S tuples:
F(x0, f(a, x1)) → c(F(f(a, a), x1))
F(f(a, x0), f(a, x1)) → c(F(a, f(a, f(f(f(a, a), x0), f(f(a, a), x1)))), F(f(f(a, a), x1), f(a, x0)), F(f(a, a), x1))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c, c
(7) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use forward instantiation to replace
F(
x0,
f(
a,
x1)) →
c(
F(
f(
a,
a),
x1)) by
F(z0, f(a, f(a, y1))) → c(F(f(a, a), f(a, y1)))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0, f(a, z1)) → f(a, f(f(f(a, a), z1), z0))
Tuples:
F(f(a, x0), f(a, x1)) → c(F(a, f(a, f(f(f(a, a), x0), f(f(a, a), x1)))), F(f(f(a, a), x1), f(a, x0)), F(f(a, a), x1))
F(z0, f(a, f(a, y1))) → c(F(f(a, a), f(a, y1)))
S tuples:
F(f(a, x0), f(a, x1)) → c(F(a, f(a, f(f(f(a, a), x0), f(f(a, a), x1)))), F(f(f(a, a), x1), f(a, x0)), F(f(a, a), x1))
F(z0, f(a, f(a, y1))) → c(F(f(a, a), f(a, y1)))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c, c
(9) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
F(z0, f(a, f(a, y1))) → c(F(f(a, a), f(a, y1)))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0, f(a, z1)) → f(a, f(f(f(a, a), z1), z0))
Tuples:
F(f(a, x0), f(a, x1)) → c(F(a, f(a, f(f(f(a, a), x0), f(f(a, a), x1)))), F(f(f(a, a), x1), f(a, x0)), F(f(a, a), x1))
S tuples:
F(f(a, x0), f(a, x1)) → c(F(a, f(a, f(f(f(a, a), x0), f(f(a, a), x1)))), F(f(f(a, a), x1), f(a, x0)), F(f(a, a), x1))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c
(11) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use instantiation to replace
F(
f(
a,
x0),
f(
a,
x1)) →
c(
F(
a,
f(
a,
f(
f(
f(
a,
a),
x0),
f(
f(
a,
a),
x1)))),
F(
f(
f(
a,
a),
x1),
f(
a,
x0)),
F(
f(
a,
a),
x1)) by
F(f(a, a), f(a, z1)) → c(F(a, f(a, f(f(f(a, a), a), f(f(a, a), z1)))), F(f(f(a, a), z1), f(a, a)), F(f(a, a), z1))
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0, f(a, z1)) → f(a, f(f(f(a, a), z1), z0))
Tuples:
F(f(a, a), f(a, z1)) → c(F(a, f(a, f(f(f(a, a), a), f(f(a, a), z1)))), F(f(f(a, a), z1), f(a, a)), F(f(a, a), z1))
S tuples:
F(f(a, a), f(a, z1)) → c(F(a, f(a, f(f(f(a, a), a), f(f(a, a), z1)))), F(f(f(a, a), z1), f(a, a)), F(f(a, a), z1))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c
(13) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use forward instantiation to replace
F(
f(
a,
a),
f(
a,
z1)) →
c(
F(
a,
f(
a,
f(
f(
f(
a,
a),
a),
f(
f(
a,
a),
z1)))),
F(
f(
f(
a,
a),
z1),
f(
a,
a)),
F(
f(
a,
a),
z1)) by
F(f(a, a), f(a, f(a, y0))) → c(F(a, f(a, f(f(f(a, a), a), f(f(a, a), f(a, y0))))), F(f(f(a, a), f(a, y0)), f(a, a)), F(f(a, a), f(a, y0)))
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0, f(a, z1)) → f(a, f(f(f(a, a), z1), z0))
Tuples:
F(f(a, a), f(a, f(a, y0))) → c(F(a, f(a, f(f(f(a, a), a), f(f(a, a), f(a, y0))))), F(f(f(a, a), f(a, y0)), f(a, a)), F(f(a, a), f(a, y0)))
S tuples:
F(f(a, a), f(a, f(a, y0))) → c(F(a, f(a, f(f(f(a, a), a), f(f(a, a), f(a, y0))))), F(f(f(a, a), f(a, y0)), f(a, a)), F(f(a, a), f(a, y0)))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c
(15) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
F(f(a, a), f(a, f(a, y0))) → c(F(a, f(a, f(f(f(a, a), a), f(f(a, a), f(a, y0))))), F(f(f(a, a), f(a, y0)), f(a, a)), F(f(a, a), f(a, y0)))
(16) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0, f(a, z1)) → f(a, f(f(f(a, a), z1), z0))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:none
Compound Symbols:none
(17) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(18) BOUNDS(O(1), O(1))