(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(a, f(a, x)) → f(a, f(x, f(f(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(a, f(a, z0)) → f(a, f(z0, f(f(a, a), a)))
Tuples:
F(a, f(a, z0)) → c(F(a, f(z0, f(f(a, a), a))), F(z0, f(f(a, a), a)), F(f(a, a), a), F(a, a))
S tuples:
F(a, f(a, z0)) → c(F(a, f(z0, f(f(a, a), a))), F(z0, 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(
a,
f(
a,
z0)) →
c(
F(
a,
f(
z0,
f(
f(
a,
a),
a))),
F(
z0,
f(
f(
a,
a),
a)),
F(
f(
a,
a),
a),
F(
a,
a)) by
F(a, f(a, x0)) → c(F(a, f(x0, f(f(a, a), a))))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(a, z0)) → f(a, f(z0, f(f(a, a), a)))
Tuples:
F(a, f(a, x0)) → c(F(a, f(x0, f(f(a, a), a))))
S tuples:
F(a, f(a, x0)) → c(F(a, f(x0, f(f(a, a), a))))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c
(5) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use forward instantiation to replace
F(
a,
f(
a,
x0)) →
c(
F(
a,
f(
x0,
f(
f(
a,
a),
a)))) by
F(a, f(a, a)) → c(F(a, f(a, f(f(a, a), a))))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(a, z0)) → f(a, f(z0, f(f(a, a), a)))
Tuples:
F(a, f(a, a)) → c(F(a, f(a, f(f(a, a), a))))
S tuples:
F(a, f(a, a)) → c(F(a, f(a, f(f(a, a), a))))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c
(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
F(a, f(a, a)) → c(F(a, f(a, f(f(a, a), a))))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(a, z0)) → f(a, f(z0, f(f(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))