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