The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

h(f(x, y)) → f(y, f(h(h(x)), a))

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(f(z0, z1)) → f(z1, f(h(h(z0)), a))

Tuples:

H(f(z0, z1)) → c(H(h(z0)), H(z0))

S tuples:

H(f(z0, z1)) → c(H(h(z0)), H(z0))

K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

H

Compound Symbols:

c

(3) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace H(f(z0, z1)) → c(H(h(z0)), H(z0)) by

H(f(f(z0, z1), x1)) → c(H(f(z1, f(h(h(z0)), a))), H(f(z0, z1)))
H(f(x0, x1)) → c

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(f(z0, z1)) → f(z1, f(h(h(z0)), a))

Tuples:

H(f(f(z0, z1), x1)) → c(H(f(z1, f(h(h(z0)), a))), H(f(z0, z1)))
H(f(x0, x1)) → c

S tuples:

H(f(f(z0, z1), x1)) → c(H(f(z1, f(h(h(z0)), a))), H(f(z0, z1)))
H(f(x0, x1)) → c

K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

H

Compound Symbols:

c, c

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

H(f(x0, x1)) → c

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(f(z0, z1)) → f(z1, f(h(h(z0)), a))

Tuples:

H(f(f(z0, z1), x1)) → c(H(f(z1, f(h(h(z0)), a))), H(f(z0, z1)))

S tuples:

H(f(f(z0, z1), x1)) → c(H(f(z1, f(h(h(z0)), a))), H(f(z0, z1)))

K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

H

Compound Symbols:

c

(7) 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 2.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1]
transitions:
f0(0, 0) → 0
a0() → 0
h0(0) → 1
h1(0) → 4
h1(4) → 3
a1() → 5
f1(3, 5) → 2
f1(0, 2) → 1
f1(0, 2) → 4
h2(0) → 8
h2(8) → 7
a2() → 9
f2(7, 9) → 6
f2(2, 6) → 3
f1(0, 2) → 8
f2(2, 6) → 7

(8) BOUNDS(O(1), O(n^1))