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

0 CpxTRS
    ↳1 CpxTrsMatchBoundsTAProof (⇔, 12 ms)
    ↳2 BOUNDS(1, n^1)
    ↳3 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳4 CdtProblem
        ↳5 CdtUsableRulesProof (⇔, 0 ms)
        ↳6 CdtProblem
        ↳7 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 41 ms)
        ↳8 CdtProblem
        ↳9 CdtKnowledgeProof (⇔, 0 ms)
        ↳10 BOUNDS(1, 1)

(0) Obligation:

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


The TRS R consists of the following rules:

f(x, y) → g(x, y)
g(h(x), y) → h(f(x, y))
g(h(x), y) → h(g(x, y))

Rewrite Strategy: INNERMOST

(1) 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, 2]
transitions:
h0(0) → 0
f0(0, 0) → 1
g0(0, 0) → 2
g1(0, 0) → 1
f1(0, 0) → 3
h1(3) → 2
g1(0, 0) → 4
h1(4) → 2
g2(0, 0) → 3
h1(3) → 1
h1(3) → 4
h1(4) → 1
h1(4) → 4
h1(3) → 3
h1(4) → 3

(2) BOUNDS(1, n^1)

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

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z1) → g(z0, z1)
g(h(z0), z1) → h(f(z0, z1))
g(h(z0), z1) → h(g(z0, z1))
Tuples:

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))
S tuples:

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))
K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F, G

Compound Symbols:

c, c1, c2

(5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

f(z0, z1) → g(z0, z1)
g(h(z0), z1) → h(f(z0, z1))
g(h(z0), z1) → h(g(z0, z1))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))
S tuples:

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

F, G

Compound Symbols:

c, c1, c2

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1, x2)) = x1   
POL(G(x1, x2)) = x1   
POL(c(x1)) = x1   
POL(c1(x1)) = x1   
POL(c2(x1)) = x1   
POL(h(x1)) = [1] + x1   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))
S tuples:

F(z0, z1) → c(G(z0, z1))
K tuples:

G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:

F, G

Compound Symbols:

c, c1, c2

(9) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))
Now S is empty

(10) BOUNDS(1, 1)