(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(y, f(x, f(a, x))) → f(f(a, f(x, a)), f(a, y))
f(x, f(x, y)) → f(f(f(x, a), a), a)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0, f(z1, f(a, z1))) → f(f(a, f(z1, a)), f(a, z0))
f(z0, f(z0, z1)) → f(f(f(z0, a), a), a)
Tuples:
F(z0, f(z1, f(a, z1))) → c(F(f(a, f(z1, a)), f(a, z0)), F(a, f(z1, a)), F(z1, a), F(a, z0))
F(z0, f(z0, z1)) → c1(F(f(f(z0, a), a), a), F(f(z0, a), a), F(z0, a))
S tuples:
F(z0, f(z1, f(a, z1))) → c(F(f(a, f(z1, a)), f(a, z0)), F(a, f(z1, a)), F(z1, a), F(a, z0))
F(z0, f(z0, z1)) → c1(F(f(f(z0, a), a), a), F(f(z0, a), a), F(z0, a))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c, c1
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
F(z0, f(z0, z1)) → c1(F(f(f(z0, a), a), a), F(f(z0, a), a), F(z0, a))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0, f(z1, f(a, z1))) → f(f(a, f(z1, a)), f(a, z0))
f(z0, f(z0, z1)) → f(f(f(z0, a), a), a)
Tuples:
F(z0, f(z1, f(a, z1))) → c(F(f(a, f(z1, a)), f(a, z0)), F(a, f(z1, a)), F(z1, a), F(a, z0))
S tuples:
F(z0, f(z1, f(a, z1))) → c(F(f(a, f(z1, a)), f(a, z0)), F(a, f(z1, a)), F(z1, a), F(a, z0))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c
(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
z0,
f(
z1,
f(
a,
z1))) →
c(
F(
f(
a,
f(
z1,
a)),
f(
a,
z0)),
F(
a,
f(
z1,
a)),
F(
z1,
a),
F(
a,
z0)) by
F(f(z1, f(a, z1)), f(x1, f(a, x1))) → c(F(f(a, f(x1, a)), f(f(a, f(z1, a)), f(a, a))), F(a, f(x1, a)), F(x1, a), F(a, f(z1, f(a, z1))))
F(f(a, z1), f(x1, f(a, x1))) → c(F(f(a, f(x1, a)), f(f(f(a, a), a), a)), F(a, f(x1, a)), F(x1, a), F(a, f(a, z1)))
F(x0, f(x1, f(a, x1))) → c
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0, f(z1, f(a, z1))) → f(f(a, f(z1, a)), f(a, z0))
f(z0, f(z0, z1)) → f(f(f(z0, a), a), a)
Tuples:
F(f(z1, f(a, z1)), f(x1, f(a, x1))) → c(F(f(a, f(x1, a)), f(f(a, f(z1, a)), f(a, a))), F(a, f(x1, a)), F(x1, a), F(a, f(z1, f(a, z1))))
F(f(a, z1), f(x1, f(a, x1))) → c(F(f(a, f(x1, a)), f(f(f(a, a), a), a)), F(a, f(x1, a)), F(x1, a), F(a, f(a, z1)))
F(x0, f(x1, f(a, x1))) → c
S tuples:
F(f(z1, f(a, z1)), f(x1, f(a, x1))) → c(F(f(a, f(x1, a)), f(f(a, f(z1, a)), f(a, a))), F(a, f(x1, a)), F(x1, a), F(a, f(z1, f(a, z1))))
F(f(a, z1), f(x1, f(a, x1))) → c(F(f(a, f(x1, a)), f(f(f(a, a), a), a)), F(a, f(x1, a)), F(x1, a), F(a, f(a, z1)))
F(x0, f(x1, f(a, x1))) → c
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c, c
(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
F(f(a, z1), f(x1, f(a, x1))) → c(F(f(a, f(x1, a)), f(f(f(a, a), a), a)), F(a, f(x1, a)), F(x1, a), F(a, f(a, z1)))
F(f(z1, f(a, z1)), f(x1, f(a, x1))) → c(F(f(a, f(x1, a)), f(f(a, f(z1, a)), f(a, a))), F(a, f(x1, a)), F(x1, a), F(a, f(z1, f(a, z1))))
F(x0, f(x1, f(a, x1))) → c
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0, f(z1, f(a, z1))) → f(f(a, f(z1, a)), f(a, z0))
f(z0, f(z0, z1)) → f(f(f(z0, a), a), a)
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))