(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(a, f(a, f(a, f(x, b)))) → f(f(a, f(a, f(a, x))), b)
f(f(f(a, x), b), b) → f(f(a, f(f(x, b), b)), b)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(a, f(a, f(z0, b)))) → f(f(a, f(a, f(a, z0))), b)
f(f(f(a, z0), b), b) → f(f(a, f(f(z0, b), b)), b)
Tuples:
F(a, f(a, f(a, f(z0, b)))) → c(F(f(a, f(a, f(a, z0))), b), F(a, f(a, f(a, z0))), F(a, f(a, z0)), F(a, z0))
F(f(f(a, z0), b), b) → c1(F(f(a, f(f(z0, b), b)), b), F(a, f(f(z0, b), b)), F(f(z0, b), b), F(z0, b))
S tuples:
F(a, f(a, f(a, f(z0, b)))) → c(F(f(a, f(a, f(a, z0))), b), F(a, f(a, f(a, z0))), F(a, f(a, z0)), F(a, z0))
F(f(f(a, z0), b), b) → c1(F(f(a, f(f(z0, b), b)), b), F(a, f(f(z0, b), b)), F(f(z0, b), b), F(z0, b))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c, c1
(3) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
a,
f(
a,
f(
a,
f(
z0,
b)))) →
c(
F(
f(
a,
f(
a,
f(
a,
z0))),
b),
F(
a,
f(
a,
f(
a,
z0))),
F(
a,
f(
a,
z0)),
F(
a,
z0)) by
F(a, f(a, f(a, f(f(z0, b), b)))) → c(F(f(f(a, f(a, f(a, z0))), b), b), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))), F(a, f(z0, b)))
F(a, f(a, f(a, f(f(a, f(z0, b)), b)))) → c(F(f(a, f(f(a, f(a, f(a, z0))), b)), b), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))))
F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b), F(a, f(a, f(a, f(a, f(a, f(z0, b)))))), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))))
F(a, f(a, f(a, f(x0, b)))) → c
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(a, f(a, f(z0, b)))) → f(f(a, f(a, f(a, z0))), b)
f(f(f(a, z0), b), b) → f(f(a, f(f(z0, b), b)), b)
Tuples:
F(f(f(a, z0), b), b) → c1(F(f(a, f(f(z0, b), b)), b), F(a, f(f(z0, b), b)), F(f(z0, b), b), F(z0, b))
F(a, f(a, f(a, f(f(z0, b), b)))) → c(F(f(f(a, f(a, f(a, z0))), b), b), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))), F(a, f(z0, b)))
F(a, f(a, f(a, f(f(a, f(z0, b)), b)))) → c(F(f(a, f(f(a, f(a, f(a, z0))), b)), b), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))))
F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b), F(a, f(a, f(a, f(a, f(a, f(z0, b)))))), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))))
F(a, f(a, f(a, f(x0, b)))) → c
S tuples:
F(f(f(a, z0), b), b) → c1(F(f(a, f(f(z0, b), b)), b), F(a, f(f(z0, b), b)), F(f(z0, b), b), F(z0, b))
F(a, f(a, f(a, f(f(z0, b), b)))) → c(F(f(f(a, f(a, f(a, z0))), b), b), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))), F(a, f(z0, b)))
F(a, f(a, f(a, f(f(a, f(z0, b)), b)))) → c(F(f(a, f(f(a, f(a, f(a, z0))), b)), b), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))))
F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b), F(a, f(a, f(a, f(a, f(a, f(z0, b)))))), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))))
F(a, f(a, f(a, f(x0, b)))) → c
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c1, c, c
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
F(a, f(a, f(a, f(x0, b)))) → c
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(a, f(a, f(z0, b)))) → f(f(a, f(a, f(a, z0))), b)
f(f(f(a, z0), b), b) → f(f(a, f(f(z0, b), b)), b)
Tuples:
F(f(f(a, z0), b), b) → c1(F(f(a, f(f(z0, b), b)), b), F(a, f(f(z0, b), b)), F(f(z0, b), b), F(z0, b))
F(a, f(a, f(a, f(f(z0, b), b)))) → c(F(f(f(a, f(a, f(a, z0))), b), b), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))), F(a, f(z0, b)))
F(a, f(a, f(a, f(f(a, f(z0, b)), b)))) → c(F(f(a, f(f(a, f(a, f(a, z0))), b)), b), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))))
F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b), F(a, f(a, f(a, f(a, f(a, f(z0, b)))))), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))))
S tuples:
F(f(f(a, z0), b), b) → c1(F(f(a, f(f(z0, b), b)), b), F(a, f(f(z0, b), b)), F(f(z0, b), b), F(z0, b))
F(a, f(a, f(a, f(f(z0, b), b)))) → c(F(f(f(a, f(a, f(a, z0))), b), b), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))), F(a, f(z0, b)))
F(a, f(a, f(a, f(f(a, f(z0, b)), b)))) → c(F(f(a, f(f(a, f(a, f(a, z0))), b)), b), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))))
F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b), F(a, f(a, f(a, f(a, f(a, f(z0, b)))))), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c1, c
(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
f(
f(
a,
z0),
b),
b) →
c1(
F(
f(
a,
f(
f(
z0,
b),
b)),
b),
F(
a,
f(
f(
z0,
b),
b)),
F(
f(
z0,
b),
b),
F(
z0,
b)) by
F(f(f(a, f(a, z0)), b), b) → c1(F(f(a, f(f(a, f(f(z0, b), b)), b)), b), F(a, f(f(f(a, z0), b), b)), F(f(f(a, z0), b), b), F(f(a, z0), b))
F(f(f(a, f(f(a, z0), b)), b), b) → c1(F(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b), F(a, f(f(f(f(a, z0), b), b), b)), F(f(f(f(a, z0), b), b), b), F(f(f(a, z0), b), b))
F(f(f(a, x0), b), b) → c1
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(a, f(a, f(z0, b)))) → f(f(a, f(a, f(a, z0))), b)
f(f(f(a, z0), b), b) → f(f(a, f(f(z0, b), b)), b)
Tuples:
F(a, f(a, f(a, f(f(z0, b), b)))) → c(F(f(f(a, f(a, f(a, z0))), b), b), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))), F(a, f(z0, b)))
F(a, f(a, f(a, f(f(a, f(z0, b)), b)))) → c(F(f(a, f(f(a, f(a, f(a, z0))), b)), b), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))))
F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b), F(a, f(a, f(a, f(a, f(a, f(z0, b)))))), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))))
F(f(f(a, f(a, z0)), b), b) → c1(F(f(a, f(f(a, f(f(z0, b), b)), b)), b), F(a, f(f(f(a, z0), b), b)), F(f(f(a, z0), b), b), F(f(a, z0), b))
F(f(f(a, f(f(a, z0), b)), b), b) → c1(F(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b), F(a, f(f(f(f(a, z0), b), b), b)), F(f(f(f(a, z0), b), b), b), F(f(f(a, z0), b), b))
F(f(f(a, x0), b), b) → c1
S tuples:
F(a, f(a, f(a, f(f(z0, b), b)))) → c(F(f(f(a, f(a, f(a, z0))), b), b), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))), F(a, f(z0, b)))
F(a, f(a, f(a, f(f(a, f(z0, b)), b)))) → c(F(f(a, f(f(a, f(a, f(a, z0))), b)), b), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))))
F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b), F(a, f(a, f(a, f(a, f(a, f(z0, b)))))), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))))
F(f(f(a, f(a, z0)), b), b) → c1(F(f(a, f(f(a, f(f(z0, b), b)), b)), b), F(a, f(f(f(a, z0), b), b)), F(f(f(a, z0), b), b), F(f(a, z0), b))
F(f(f(a, f(f(a, z0), b)), b), b) → c1(F(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b), F(a, f(f(f(f(a, z0), b), b), b)), F(f(f(f(a, z0), b), b), b), F(f(f(a, z0), b), b))
F(f(f(a, x0), b), b) → c1
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c, c1, c1
(9) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
F(f(f(a, x0), b), b) → c1
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(a, f(a, f(z0, b)))) → f(f(a, f(a, f(a, z0))), b)
f(f(f(a, z0), b), b) → f(f(a, f(f(z0, b), b)), b)
Tuples:
F(a, f(a, f(a, f(f(z0, b), b)))) → c(F(f(f(a, f(a, f(a, z0))), b), b), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))), F(a, f(z0, b)))
F(a, f(a, f(a, f(f(a, f(z0, b)), b)))) → c(F(f(a, f(f(a, f(a, f(a, z0))), b)), b), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))))
F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b), F(a, f(a, f(a, f(a, f(a, f(z0, b)))))), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))))
F(f(f(a, f(a, z0)), b), b) → c1(F(f(a, f(f(a, f(f(z0, b), b)), b)), b), F(a, f(f(f(a, z0), b), b)), F(f(f(a, z0), b), b), F(f(a, z0), b))
F(f(f(a, f(f(a, z0), b)), b), b) → c1(F(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b), F(a, f(f(f(f(a, z0), b), b), b)), F(f(f(f(a, z0), b), b), b), F(f(f(a, z0), b), b))
S tuples:
F(a, f(a, f(a, f(f(z0, b), b)))) → c(F(f(f(a, f(a, f(a, z0))), b), b), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))), F(a, f(z0, b)))
F(a, f(a, f(a, f(f(a, f(z0, b)), b)))) → c(F(f(a, f(f(a, f(a, f(a, z0))), b)), b), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))))
F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b), F(a, f(a, f(a, f(a, f(a, f(z0, b)))))), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))))
F(f(f(a, f(a, z0)), b), b) → c1(F(f(a, f(f(a, f(f(z0, b), b)), b)), b), F(a, f(f(f(a, z0), b), b)), F(f(f(a, z0), b), b), F(f(a, z0), b))
F(f(f(a, f(f(a, z0), b)), b), b) → c1(F(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b), F(a, f(f(f(f(a, z0), b), b), b)), F(f(f(f(a, z0), b), b), b), F(f(f(a, z0), b), b))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c, c1
(11) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
a,
f(
a,
f(
a,
f(
f(
z0,
b),
b)))) →
c(
F(
f(
f(
a,
f(
a,
f(
a,
z0))),
b),
b),
F(
a,
f(
a,
f(
a,
f(
z0,
b)))),
F(
a,
f(
a,
f(
z0,
b))),
F(
a,
f(
z0,
b))) by
F(a, f(a, f(a, f(f(x0, b), b)))) → c(F(f(f(a, f(a, f(a, x0))), b), b), F(a, f(a, f(a, f(x0, b)))))
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(a, f(a, f(z0, b)))) → f(f(a, f(a, f(a, z0))), b)
f(f(f(a, z0), b), b) → f(f(a, f(f(z0, b), b)), b)
Tuples:
F(a, f(a, f(a, f(f(a, f(z0, b)), b)))) → c(F(f(a, f(f(a, f(a, f(a, z0))), b)), b), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))))
F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b), F(a, f(a, f(a, f(a, f(a, f(z0, b)))))), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))))
F(f(f(a, f(a, z0)), b), b) → c1(F(f(a, f(f(a, f(f(z0, b), b)), b)), b), F(a, f(f(f(a, z0), b), b)), F(f(f(a, z0), b), b), F(f(a, z0), b))
F(f(f(a, f(f(a, z0), b)), b), b) → c1(F(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b), F(a, f(f(f(f(a, z0), b), b), b)), F(f(f(f(a, z0), b), b), b), F(f(f(a, z0), b), b))
F(a, f(a, f(a, f(f(x0, b), b)))) → c(F(f(f(a, f(a, f(a, x0))), b), b), F(a, f(a, f(a, f(x0, b)))))
S tuples:
F(a, f(a, f(a, f(f(a, f(z0, b)), b)))) → c(F(f(a, f(f(a, f(a, f(a, z0))), b)), b), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))), F(a, f(a, f(z0, b))))
F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b), F(a, f(a, f(a, f(a, f(a, f(z0, b)))))), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))))
F(f(f(a, f(a, z0)), b), b) → c1(F(f(a, f(f(a, f(f(z0, b), b)), b)), b), F(a, f(f(f(a, z0), b), b)), F(f(f(a, z0), b), b), F(f(a, z0), b))
F(f(f(a, f(f(a, z0), b)), b), b) → c1(F(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b), F(a, f(f(f(f(a, z0), b), b), b)), F(f(f(f(a, z0), b), b), b), F(f(f(a, z0), b), b))
F(a, f(a, f(a, f(f(x0, b), b)))) → c(F(f(f(a, f(a, f(a, x0))), b), b), F(a, f(a, f(a, f(x0, b)))))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c, c1, c
(13) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
a,
f(
a,
f(
a,
f(
f(
a,
f(
z0,
b)),
b)))) →
c(
F(
f(
a,
f(
f(
a,
f(
a,
f(
a,
z0))),
b)),
b),
F(
a,
f(
a,
f(
a,
f(
a,
f(
z0,
b))))),
F(
a,
f(
a,
f(
a,
f(
z0,
b)))),
F(
a,
f(
a,
f(
z0,
b)))) by
F(a, f(a, f(a, f(f(a, f(f(z0, b), b)), b)))) → c(F(f(a, f(f(f(a, f(a, f(a, z0))), b), b)), b), F(a, f(a, f(a, f(a, f(f(z0, b), b))))), F(a, f(a, f(a, f(f(z0, b), b)))), F(a, f(a, f(f(z0, b), b))))
F(a, f(a, f(a, f(f(a, f(f(a, f(z0, b)), b)), b)))) → c(F(f(a, f(f(a, f(f(a, f(a, f(a, z0))), b)), b)), b), F(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b))))), F(a, f(a, f(a, f(f(a, f(z0, b)), b)))), F(a, f(a, f(f(a, f(z0, b)), b))))
F(a, f(a, f(a, f(f(a, f(f(a, f(a, f(z0, b))), b)), b)))) → c(F(f(a, f(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b)), b), F(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b))))), F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))), F(a, f(a, f(f(a, f(a, f(z0, b))), b))))
F(a, f(a, f(a, f(f(a, f(x0, b)), b)))) → c(F(a, f(a, f(a, f(a, f(x0, b))))))
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(a, f(a, f(z0, b)))) → f(f(a, f(a, f(a, z0))), b)
f(f(f(a, z0), b), b) → f(f(a, f(f(z0, b), b)), b)
Tuples:
F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b), F(a, f(a, f(a, f(a, f(a, f(z0, b)))))), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))))
F(f(f(a, f(a, z0)), b), b) → c1(F(f(a, f(f(a, f(f(z0, b), b)), b)), b), F(a, f(f(f(a, z0), b), b)), F(f(f(a, z0), b), b), F(f(a, z0), b))
F(f(f(a, f(f(a, z0), b)), b), b) → c1(F(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b), F(a, f(f(f(f(a, z0), b), b), b)), F(f(f(f(a, z0), b), b), b), F(f(f(a, z0), b), b))
F(a, f(a, f(a, f(f(x0, b), b)))) → c(F(f(f(a, f(a, f(a, x0))), b), b), F(a, f(a, f(a, f(x0, b)))))
F(a, f(a, f(a, f(f(a, f(f(z0, b), b)), b)))) → c(F(f(a, f(f(f(a, f(a, f(a, z0))), b), b)), b), F(a, f(a, f(a, f(a, f(f(z0, b), b))))), F(a, f(a, f(a, f(f(z0, b), b)))), F(a, f(a, f(f(z0, b), b))))
F(a, f(a, f(a, f(f(a, f(f(a, f(z0, b)), b)), b)))) → c(F(f(a, f(f(a, f(f(a, f(a, f(a, z0))), b)), b)), b), F(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b))))), F(a, f(a, f(a, f(f(a, f(z0, b)), b)))), F(a, f(a, f(f(a, f(z0, b)), b))))
F(a, f(a, f(a, f(f(a, f(f(a, f(a, f(z0, b))), b)), b)))) → c(F(f(a, f(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b)), b), F(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b))))), F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))), F(a, f(a, f(f(a, f(a, f(z0, b))), b))))
F(a, f(a, f(a, f(f(a, f(x0, b)), b)))) → c(F(a, f(a, f(a, f(a, f(x0, b))))))
S tuples:
F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b), F(a, f(a, f(a, f(a, f(a, f(z0, b)))))), F(a, f(a, f(a, f(a, f(z0, b))))), F(a, f(a, f(a, f(z0, b)))))
F(f(f(a, f(a, z0)), b), b) → c1(F(f(a, f(f(a, f(f(z0, b), b)), b)), b), F(a, f(f(f(a, z0), b), b)), F(f(f(a, z0), b), b), F(f(a, z0), b))
F(f(f(a, f(f(a, z0), b)), b), b) → c1(F(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b), F(a, f(f(f(f(a, z0), b), b), b)), F(f(f(f(a, z0), b), b), b), F(f(f(a, z0), b), b))
F(a, f(a, f(a, f(f(x0, b), b)))) → c(F(f(f(a, f(a, f(a, x0))), b), b), F(a, f(a, f(a, f(x0, b)))))
F(a, f(a, f(a, f(f(a, f(f(z0, b), b)), b)))) → c(F(f(a, f(f(f(a, f(a, f(a, z0))), b), b)), b), F(a, f(a, f(a, f(a, f(f(z0, b), b))))), F(a, f(a, f(a, f(f(z0, b), b)))), F(a, f(a, f(f(z0, b), b))))
F(a, f(a, f(a, f(f(a, f(f(a, f(z0, b)), b)), b)))) → c(F(f(a, f(f(a, f(f(a, f(a, f(a, z0))), b)), b)), b), F(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b))))), F(a, f(a, f(a, f(f(a, f(z0, b)), b)))), F(a, f(a, f(f(a, f(z0, b)), b))))
F(a, f(a, f(a, f(f(a, f(f(a, f(a, f(z0, b))), b)), b)))) → c(F(f(a, f(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b)), b), F(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b))))), F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))), F(a, f(a, f(f(a, f(a, f(z0, b))), b))))
F(a, f(a, f(a, f(f(a, f(x0, b)), b)))) → c(F(a, f(a, f(a, f(a, f(x0, b))))))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c, c1, c, c
(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
a,
f(
a,
f(
a,
f(
f(
a,
f(
a,
f(
z0,
b))),
b)))) →
c(
F(
f(
a,
f(
a,
f(
f(
a,
f(
a,
f(
a,
z0))),
b))),
b),
F(
a,
f(
a,
f(
a,
f(
a,
f(
a,
f(
z0,
b)))))),
F(
a,
f(
a,
f(
a,
f(
a,
f(
z0,
b))))),
F(
a,
f(
a,
f(
a,
f(
z0,
b))))) by
F(a, f(a, f(a, f(f(a, f(a, f(f(z0, b), b))), b)))) → c(F(f(a, f(a, f(f(f(a, f(a, f(a, z0))), b), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(z0, b), b)))))), F(a, f(a, f(a, f(a, f(f(z0, b), b))))), F(a, f(a, f(a, f(f(z0, b), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(f(a, f(z0, b)), b))), b)))) → c(F(f(a, f(a, f(f(a, f(f(a, f(a, f(a, z0))), b)), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b)))))), F(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b))))), F(a, f(a, f(a, f(f(a, f(z0, b)), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(f(a, f(a, f(z0, b))), b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))))), F(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b))))), F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(x0, b))), b)))) → c(F(a, f(a, f(a, f(a, f(a, f(x0, b)))))), F(a, f(a, f(a, f(a, f(x0, b))))))
(16) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(a, f(a, f(z0, b)))) → f(f(a, f(a, f(a, z0))), b)
f(f(f(a, z0), b), b) → f(f(a, f(f(z0, b), b)), b)
Tuples:
F(f(f(a, f(a, z0)), b), b) → c1(F(f(a, f(f(a, f(f(z0, b), b)), b)), b), F(a, f(f(f(a, z0), b), b)), F(f(f(a, z0), b), b), F(f(a, z0), b))
F(f(f(a, f(f(a, z0), b)), b), b) → c1(F(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b), F(a, f(f(f(f(a, z0), b), b), b)), F(f(f(f(a, z0), b), b), b), F(f(f(a, z0), b), b))
F(a, f(a, f(a, f(f(x0, b), b)))) → c(F(f(f(a, f(a, f(a, x0))), b), b), F(a, f(a, f(a, f(x0, b)))))
F(a, f(a, f(a, f(f(a, f(f(z0, b), b)), b)))) → c(F(f(a, f(f(f(a, f(a, f(a, z0))), b), b)), b), F(a, f(a, f(a, f(a, f(f(z0, b), b))))), F(a, f(a, f(a, f(f(z0, b), b)))), F(a, f(a, f(f(z0, b), b))))
F(a, f(a, f(a, f(f(a, f(f(a, f(z0, b)), b)), b)))) → c(F(f(a, f(f(a, f(f(a, f(a, f(a, z0))), b)), b)), b), F(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b))))), F(a, f(a, f(a, f(f(a, f(z0, b)), b)))), F(a, f(a, f(f(a, f(z0, b)), b))))
F(a, f(a, f(a, f(f(a, f(f(a, f(a, f(z0, b))), b)), b)))) → c(F(f(a, f(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b)), b), F(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b))))), F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))), F(a, f(a, f(f(a, f(a, f(z0, b))), b))))
F(a, f(a, f(a, f(f(a, f(x0, b)), b)))) → c(F(a, f(a, f(a, f(a, f(x0, b))))))
F(a, f(a, f(a, f(f(a, f(a, f(f(z0, b), b))), b)))) → c(F(f(a, f(a, f(f(f(a, f(a, f(a, z0))), b), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(z0, b), b)))))), F(a, f(a, f(a, f(a, f(f(z0, b), b))))), F(a, f(a, f(a, f(f(z0, b), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(f(a, f(z0, b)), b))), b)))) → c(F(f(a, f(a, f(f(a, f(f(a, f(a, f(a, z0))), b)), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b)))))), F(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b))))), F(a, f(a, f(a, f(f(a, f(z0, b)), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(f(a, f(a, f(z0, b))), b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))))), F(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b))))), F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(x0, b))), b)))) → c(F(a, f(a, f(a, f(a, f(a, f(x0, b)))))), F(a, f(a, f(a, f(a, f(x0, b))))))
S tuples:
F(f(f(a, f(a, z0)), b), b) → c1(F(f(a, f(f(a, f(f(z0, b), b)), b)), b), F(a, f(f(f(a, z0), b), b)), F(f(f(a, z0), b), b), F(f(a, z0), b))
F(f(f(a, f(f(a, z0), b)), b), b) → c1(F(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b), F(a, f(f(f(f(a, z0), b), b), b)), F(f(f(f(a, z0), b), b), b), F(f(f(a, z0), b), b))
F(a, f(a, f(a, f(f(x0, b), b)))) → c(F(f(f(a, f(a, f(a, x0))), b), b), F(a, f(a, f(a, f(x0, b)))))
F(a, f(a, f(a, f(f(a, f(f(z0, b), b)), b)))) → c(F(f(a, f(f(f(a, f(a, f(a, z0))), b), b)), b), F(a, f(a, f(a, f(a, f(f(z0, b), b))))), F(a, f(a, f(a, f(f(z0, b), b)))), F(a, f(a, f(f(z0, b), b))))
F(a, f(a, f(a, f(f(a, f(f(a, f(z0, b)), b)), b)))) → c(F(f(a, f(f(a, f(f(a, f(a, f(a, z0))), b)), b)), b), F(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b))))), F(a, f(a, f(a, f(f(a, f(z0, b)), b)))), F(a, f(a, f(f(a, f(z0, b)), b))))
F(a, f(a, f(a, f(f(a, f(f(a, f(a, f(z0, b))), b)), b)))) → c(F(f(a, f(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b)), b), F(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b))))), F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))), F(a, f(a, f(f(a, f(a, f(z0, b))), b))))
F(a, f(a, f(a, f(f(a, f(x0, b)), b)))) → c(F(a, f(a, f(a, f(a, f(x0, b))))))
F(a, f(a, f(a, f(f(a, f(a, f(f(z0, b), b))), b)))) → c(F(f(a, f(a, f(f(f(a, f(a, f(a, z0))), b), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(z0, b), b)))))), F(a, f(a, f(a, f(a, f(f(z0, b), b))))), F(a, f(a, f(a, f(f(z0, b), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(f(a, f(z0, b)), b))), b)))) → c(F(f(a, f(a, f(f(a, f(f(a, f(a, f(a, z0))), b)), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b)))))), F(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b))))), F(a, f(a, f(a, f(f(a, f(z0, b)), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(f(a, f(a, f(z0, b))), b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))))), F(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b))))), F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(x0, b))), b)))) → c(F(a, f(a, f(a, f(a, f(a, f(x0, b)))))), F(a, f(a, f(a, f(a, f(x0, b))))))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c1, c, c, c
(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
f(
f(
a,
f(
a,
z0)),
b),
b) →
c1(
F(
f(
a,
f(
f(
a,
f(
f(
z0,
b),
b)),
b)),
b),
F(
a,
f(
f(
f(
a,
z0),
b),
b)),
F(
f(
f(
a,
z0),
b),
b),
F(
f(
a,
z0),
b)) by
F(f(f(a, f(a, f(a, z0))), b), b) → c1(F(f(a, f(f(a, f(f(a, f(f(z0, b), b)), b)), b)), b), F(a, f(f(f(a, f(a, z0)), b), b)), F(f(f(a, f(a, z0)), b), b), F(f(a, f(a, z0)), b))
F(f(f(a, f(a, f(f(a, z0), b))), b), b) → c1(F(f(a, f(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b)), b), F(a, f(f(f(a, f(f(a, z0), b)), b), b)), F(f(f(a, f(f(a, z0), b)), b), b), F(f(a, f(f(a, z0), b)), b))
F(f(f(a, f(a, x0)), b), b) → c1(F(a, f(f(f(a, x0), b), b)))
(18) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(a, f(a, f(z0, b)))) → f(f(a, f(a, f(a, z0))), b)
f(f(f(a, z0), b), b) → f(f(a, f(f(z0, b), b)), b)
Tuples:
F(f(f(a, f(f(a, z0), b)), b), b) → c1(F(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b), F(a, f(f(f(f(a, z0), b), b), b)), F(f(f(f(a, z0), b), b), b), F(f(f(a, z0), b), b))
F(a, f(a, f(a, f(f(x0, b), b)))) → c(F(f(f(a, f(a, f(a, x0))), b), b), F(a, f(a, f(a, f(x0, b)))))
F(a, f(a, f(a, f(f(a, f(f(z0, b), b)), b)))) → c(F(f(a, f(f(f(a, f(a, f(a, z0))), b), b)), b), F(a, f(a, f(a, f(a, f(f(z0, b), b))))), F(a, f(a, f(a, f(f(z0, b), b)))), F(a, f(a, f(f(z0, b), b))))
F(a, f(a, f(a, f(f(a, f(f(a, f(z0, b)), b)), b)))) → c(F(f(a, f(f(a, f(f(a, f(a, f(a, z0))), b)), b)), b), F(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b))))), F(a, f(a, f(a, f(f(a, f(z0, b)), b)))), F(a, f(a, f(f(a, f(z0, b)), b))))
F(a, f(a, f(a, f(f(a, f(f(a, f(a, f(z0, b))), b)), b)))) → c(F(f(a, f(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b)), b), F(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b))))), F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))), F(a, f(a, f(f(a, f(a, f(z0, b))), b))))
F(a, f(a, f(a, f(f(a, f(x0, b)), b)))) → c(F(a, f(a, f(a, f(a, f(x0, b))))))
F(a, f(a, f(a, f(f(a, f(a, f(f(z0, b), b))), b)))) → c(F(f(a, f(a, f(f(f(a, f(a, f(a, z0))), b), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(z0, b), b)))))), F(a, f(a, f(a, f(a, f(f(z0, b), b))))), F(a, f(a, f(a, f(f(z0, b), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(f(a, f(z0, b)), b))), b)))) → c(F(f(a, f(a, f(f(a, f(f(a, f(a, f(a, z0))), b)), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b)))))), F(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b))))), F(a, f(a, f(a, f(f(a, f(z0, b)), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(f(a, f(a, f(z0, b))), b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))))), F(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b))))), F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(x0, b))), b)))) → c(F(a, f(a, f(a, f(a, f(a, f(x0, b)))))), F(a, f(a, f(a, f(a, f(x0, b))))))
F(f(f(a, f(a, f(a, z0))), b), b) → c1(F(f(a, f(f(a, f(f(a, f(f(z0, b), b)), b)), b)), b), F(a, f(f(f(a, f(a, z0)), b), b)), F(f(f(a, f(a, z0)), b), b), F(f(a, f(a, z0)), b))
F(f(f(a, f(a, f(f(a, z0), b))), b), b) → c1(F(f(a, f(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b)), b), F(a, f(f(f(a, f(f(a, z0), b)), b), b)), F(f(f(a, f(f(a, z0), b)), b), b), F(f(a, f(f(a, z0), b)), b))
F(f(f(a, f(a, x0)), b), b) → c1(F(a, f(f(f(a, x0), b), b)))
S tuples:
F(f(f(a, f(f(a, z0), b)), b), b) → c1(F(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b), F(a, f(f(f(f(a, z0), b), b), b)), F(f(f(f(a, z0), b), b), b), F(f(f(a, z0), b), b))
F(a, f(a, f(a, f(f(x0, b), b)))) → c(F(f(f(a, f(a, f(a, x0))), b), b), F(a, f(a, f(a, f(x0, b)))))
F(a, f(a, f(a, f(f(a, f(f(z0, b), b)), b)))) → c(F(f(a, f(f(f(a, f(a, f(a, z0))), b), b)), b), F(a, f(a, f(a, f(a, f(f(z0, b), b))))), F(a, f(a, f(a, f(f(z0, b), b)))), F(a, f(a, f(f(z0, b), b))))
F(a, f(a, f(a, f(f(a, f(f(a, f(z0, b)), b)), b)))) → c(F(f(a, f(f(a, f(f(a, f(a, f(a, z0))), b)), b)), b), F(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b))))), F(a, f(a, f(a, f(f(a, f(z0, b)), b)))), F(a, f(a, f(f(a, f(z0, b)), b))))
F(a, f(a, f(a, f(f(a, f(f(a, f(a, f(z0, b))), b)), b)))) → c(F(f(a, f(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b)), b), F(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b))))), F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))), F(a, f(a, f(f(a, f(a, f(z0, b))), b))))
F(a, f(a, f(a, f(f(a, f(x0, b)), b)))) → c(F(a, f(a, f(a, f(a, f(x0, b))))))
F(a, f(a, f(a, f(f(a, f(a, f(f(z0, b), b))), b)))) → c(F(f(a, f(a, f(f(f(a, f(a, f(a, z0))), b), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(z0, b), b)))))), F(a, f(a, f(a, f(a, f(f(z0, b), b))))), F(a, f(a, f(a, f(f(z0, b), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(f(a, f(z0, b)), b))), b)))) → c(F(f(a, f(a, f(f(a, f(f(a, f(a, f(a, z0))), b)), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b)))))), F(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b))))), F(a, f(a, f(a, f(f(a, f(z0, b)), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(f(a, f(a, f(z0, b))), b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))))), F(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b))))), F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(x0, b))), b)))) → c(F(a, f(a, f(a, f(a, f(a, f(x0, b)))))), F(a, f(a, f(a, f(a, f(x0, b))))))
F(f(f(a, f(a, f(a, z0))), b), b) → c1(F(f(a, f(f(a, f(f(a, f(f(z0, b), b)), b)), b)), b), F(a, f(f(f(a, f(a, z0)), b), b)), F(f(f(a, f(a, z0)), b), b), F(f(a, f(a, z0)), b))
F(f(f(a, f(a, f(f(a, z0), b))), b), b) → c1(F(f(a, f(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b)), b), F(a, f(f(f(a, f(f(a, z0), b)), b), b)), F(f(f(a, f(f(a, z0), b)), b), b), F(f(a, f(f(a, z0), b)), b))
F(f(f(a, f(a, x0)), b), b) → c1(F(a, f(f(f(a, x0), b), b)))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c1, c, c, c, c1
(19) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
f(
f(
a,
f(
f(
a,
z0),
b)),
b),
b) →
c1(
F(
f(
a,
f(
f(
f(
a,
f(
f(
z0,
b),
b)),
b),
b)),
b),
F(
a,
f(
f(
f(
f(
a,
z0),
b),
b),
b)),
F(
f(
f(
f(
a,
z0),
b),
b),
b),
F(
f(
f(
a,
z0),
b),
b)) by
F(f(f(a, f(f(a, x0), b)), b), b) → c1(F(f(a, f(f(a, f(f(f(f(x0, b), b), b), b)), b)), b), F(a, f(f(f(f(a, x0), b), b), b)), F(f(f(f(a, x0), b), b), b), F(f(f(a, x0), b), b))
F(f(f(a, f(f(a, f(a, z0)), b)), b), b) → c1(F(f(a, f(f(f(a, f(f(a, f(f(z0, b), b)), b)), b), b)), b), F(a, f(f(f(f(a, f(a, z0)), b), b), b)), F(f(f(f(a, f(a, z0)), b), b), b), F(f(f(a, f(a, z0)), b), b))
F(f(f(a, f(f(a, f(f(a, z0), b)), b)), b), b) → c1(F(f(a, f(f(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b), b)), b), F(a, f(f(f(f(a, f(f(a, z0), b)), b), b), b)), F(f(f(f(a, f(f(a, z0), b)), b), b), b), F(f(f(a, f(f(a, z0), b)), b), b))
F(f(f(a, f(f(a, x0), b)), b), b) → c1(F(a, f(f(f(f(a, x0), b), b), b)), F(f(f(f(a, x0), b), b), b))
(20) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(a, f(a, f(z0, b)))) → f(f(a, f(a, f(a, z0))), b)
f(f(f(a, z0), b), b) → f(f(a, f(f(z0, b), b)), b)
Tuples:
F(a, f(a, f(a, f(f(x0, b), b)))) → c(F(f(f(a, f(a, f(a, x0))), b), b), F(a, f(a, f(a, f(x0, b)))))
F(a, f(a, f(a, f(f(a, f(f(z0, b), b)), b)))) → c(F(f(a, f(f(f(a, f(a, f(a, z0))), b), b)), b), F(a, f(a, f(a, f(a, f(f(z0, b), b))))), F(a, f(a, f(a, f(f(z0, b), b)))), F(a, f(a, f(f(z0, b), b))))
F(a, f(a, f(a, f(f(a, f(f(a, f(z0, b)), b)), b)))) → c(F(f(a, f(f(a, f(f(a, f(a, f(a, z0))), b)), b)), b), F(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b))))), F(a, f(a, f(a, f(f(a, f(z0, b)), b)))), F(a, f(a, f(f(a, f(z0, b)), b))))
F(a, f(a, f(a, f(f(a, f(f(a, f(a, f(z0, b))), b)), b)))) → c(F(f(a, f(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b)), b), F(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b))))), F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))), F(a, f(a, f(f(a, f(a, f(z0, b))), b))))
F(a, f(a, f(a, f(f(a, f(x0, b)), b)))) → c(F(a, f(a, f(a, f(a, f(x0, b))))))
F(a, f(a, f(a, f(f(a, f(a, f(f(z0, b), b))), b)))) → c(F(f(a, f(a, f(f(f(a, f(a, f(a, z0))), b), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(z0, b), b)))))), F(a, f(a, f(a, f(a, f(f(z0, b), b))))), F(a, f(a, f(a, f(f(z0, b), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(f(a, f(z0, b)), b))), b)))) → c(F(f(a, f(a, f(f(a, f(f(a, f(a, f(a, z0))), b)), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b)))))), F(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b))))), F(a, f(a, f(a, f(f(a, f(z0, b)), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(f(a, f(a, f(z0, b))), b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))))), F(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b))))), F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(x0, b))), b)))) → c(F(a, f(a, f(a, f(a, f(a, f(x0, b)))))), F(a, f(a, f(a, f(a, f(x0, b))))))
F(f(f(a, f(a, f(a, z0))), b), b) → c1(F(f(a, f(f(a, f(f(a, f(f(z0, b), b)), b)), b)), b), F(a, f(f(f(a, f(a, z0)), b), b)), F(f(f(a, f(a, z0)), b), b), F(f(a, f(a, z0)), b))
F(f(f(a, f(a, f(f(a, z0), b))), b), b) → c1(F(f(a, f(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b)), b), F(a, f(f(f(a, f(f(a, z0), b)), b), b)), F(f(f(a, f(f(a, z0), b)), b), b), F(f(a, f(f(a, z0), b)), b))
F(f(f(a, f(a, x0)), b), b) → c1(F(a, f(f(f(a, x0), b), b)))
F(f(f(a, f(f(a, x0), b)), b), b) → c1(F(f(a, f(f(a, f(f(f(f(x0, b), b), b), b)), b)), b), F(a, f(f(f(f(a, x0), b), b), b)), F(f(f(f(a, x0), b), b), b), F(f(f(a, x0), b), b))
F(f(f(a, f(f(a, f(a, z0)), b)), b), b) → c1(F(f(a, f(f(f(a, f(f(a, f(f(z0, b), b)), b)), b), b)), b), F(a, f(f(f(f(a, f(a, z0)), b), b), b)), F(f(f(f(a, f(a, z0)), b), b), b), F(f(f(a, f(a, z0)), b), b))
F(f(f(a, f(f(a, f(f(a, z0), b)), b)), b), b) → c1(F(f(a, f(f(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b), b)), b), F(a, f(f(f(f(a, f(f(a, z0), b)), b), b), b)), F(f(f(f(a, f(f(a, z0), b)), b), b), b), F(f(f(a, f(f(a, z0), b)), b), b))
F(f(f(a, f(f(a, x0), b)), b), b) → c1(F(a, f(f(f(f(a, x0), b), b), b)), F(f(f(f(a, x0), b), b), b))
S tuples:
F(a, f(a, f(a, f(f(x0, b), b)))) → c(F(f(f(a, f(a, f(a, x0))), b), b), F(a, f(a, f(a, f(x0, b)))))
F(a, f(a, f(a, f(f(a, f(f(z0, b), b)), b)))) → c(F(f(a, f(f(f(a, f(a, f(a, z0))), b), b)), b), F(a, f(a, f(a, f(a, f(f(z0, b), b))))), F(a, f(a, f(a, f(f(z0, b), b)))), F(a, f(a, f(f(z0, b), b))))
F(a, f(a, f(a, f(f(a, f(f(a, f(z0, b)), b)), b)))) → c(F(f(a, f(f(a, f(f(a, f(a, f(a, z0))), b)), b)), b), F(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b))))), F(a, f(a, f(a, f(f(a, f(z0, b)), b)))), F(a, f(a, f(f(a, f(z0, b)), b))))
F(a, f(a, f(a, f(f(a, f(f(a, f(a, f(z0, b))), b)), b)))) → c(F(f(a, f(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b)), b), F(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b))))), F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))), F(a, f(a, f(f(a, f(a, f(z0, b))), b))))
F(a, f(a, f(a, f(f(a, f(x0, b)), b)))) → c(F(a, f(a, f(a, f(a, f(x0, b))))))
F(a, f(a, f(a, f(f(a, f(a, f(f(z0, b), b))), b)))) → c(F(f(a, f(a, f(f(f(a, f(a, f(a, z0))), b), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(z0, b), b)))))), F(a, f(a, f(a, f(a, f(f(z0, b), b))))), F(a, f(a, f(a, f(f(z0, b), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(f(a, f(z0, b)), b))), b)))) → c(F(f(a, f(a, f(f(a, f(f(a, f(a, f(a, z0))), b)), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b)))))), F(a, f(a, f(a, f(a, f(f(a, f(z0, b)), b))))), F(a, f(a, f(a, f(f(a, f(z0, b)), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(f(a, f(a, f(z0, b))), b))), b)))) → c(F(f(a, f(a, f(f(a, f(a, f(f(a, f(a, f(a, z0))), b))), b))), b), F(a, f(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))))), F(a, f(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b))))), F(a, f(a, f(a, f(f(a, f(a, f(z0, b))), b)))))
F(a, f(a, f(a, f(f(a, f(a, f(x0, b))), b)))) → c(F(a, f(a, f(a, f(a, f(a, f(x0, b)))))), F(a, f(a, f(a, f(a, f(x0, b))))))
F(f(f(a, f(a, f(a, z0))), b), b) → c1(F(f(a, f(f(a, f(f(a, f(f(z0, b), b)), b)), b)), b), F(a, f(f(f(a, f(a, z0)), b), b)), F(f(f(a, f(a, z0)), b), b), F(f(a, f(a, z0)), b))
F(f(f(a, f(a, f(f(a, z0), b))), b), b) → c1(F(f(a, f(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b)), b), F(a, f(f(f(a, f(f(a, z0), b)), b), b)), F(f(f(a, f(f(a, z0), b)), b), b), F(f(a, f(f(a, z0), b)), b))
F(f(f(a, f(a, x0)), b), b) → c1(F(a, f(f(f(a, x0), b), b)))
F(f(f(a, f(f(a, x0), b)), b), b) → c1(F(f(a, f(f(a, f(f(f(f(x0, b), b), b), b)), b)), b), F(a, f(f(f(f(a, x0), b), b), b)), F(f(f(f(a, x0), b), b), b), F(f(f(a, x0), b), b))
F(f(f(a, f(f(a, f(a, z0)), b)), b), b) → c1(F(f(a, f(f(f(a, f(f(a, f(f(z0, b), b)), b)), b), b)), b), F(a, f(f(f(f(a, f(a, z0)), b), b), b)), F(f(f(f(a, f(a, z0)), b), b), b), F(f(f(a, f(a, z0)), b), b))
F(f(f(a, f(f(a, f(f(a, z0), b)), b)), b), b) → c1(F(f(a, f(f(f(a, f(f(f(a, f(f(z0, b), b)), b), b)), b), b)), b), F(a, f(f(f(f(a, f(f(a, z0), b)), b), b), b)), F(f(f(f(a, f(f(a, z0), b)), b), b), b), F(f(f(a, f(f(a, z0), b)), b), b))
F(f(f(a, f(f(a, x0), b)), b), b) → c1(F(a, f(f(f(f(a, x0), b), b), b)), F(f(f(f(a, x0), b), b), b))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c, c, c, c1, c1, c1
(21) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 0.
The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1]
transitions:
a0() → 0
b0() → 0
f0(0, 0) → 1
(22) BOUNDS(O(1), O(n^1))