(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(f(a, x), y) → f(y, f(x, f(a, f(h(a), a))))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(a, z0), z1) → f(z1, f(z0, f(a, f(h(a), a))))
Tuples:
F(f(a, z0), z1) → c(F(z1, f(z0, f(a, f(h(a), a)))), F(z0, f(a, f(h(a), a))), F(a, f(h(a), a)), F(h(a), a))
S tuples:
F(f(a, z0), z1) → c(F(z1, f(z0, f(a, f(h(a), a)))), F(z0, f(a, f(h(a), a))), F(a, f(h(a), a)), F(h(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(
a,
z0),
z1) →
c(
F(
z1,
f(
z0,
f(
a,
f(
h(
a),
a)))),
F(
z0,
f(
a,
f(
h(
a),
a))),
F(
a,
f(
h(
a),
a)),
F(
h(
a),
a)) by
F(f(a, x0), x1) → c(F(x1, f(x0, f(a, f(h(a), a)))), F(x0, f(a, f(h(a), a))))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(a, z0), z1) → f(z1, f(z0, f(a, f(h(a), a))))
Tuples:
F(f(a, x0), x1) → c(F(x1, f(x0, f(a, f(h(a), a)))), F(x0, f(a, f(h(a), a))))
S tuples:
F(f(a, x0), x1) → c(F(x1, f(x0, f(a, f(h(a), a)))), F(x0, f(a, f(h(a), a))))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c
(5) 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
h0(0) → 0
f0(0, 0) → 1
(6) BOUNDS(O(1), O(n^1))