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