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

g(s(x)) → f(x)
f(0) → s(0)
f(s(x)) → s(s(g(x)))
g(0) → 0

Rewrite Strategy: INNERMOST

(1) CpxTrsMatchBoundsProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1.
The certificate found is represented by the following graph.

Start state: 1
Accept states: [2]
Transitions:
1→2[g_1|0, f_1|0, f_1|1, 0|1]
1→3[s_1|1]
1→4[s_1|1]
2→2[s_1|0, 0|0]
3→2[0|1]
4→5[s_1|1]
5→2[g_1|1, f_1|1, 0|1]
5→3[s_1|1]
5→4[s_1|1]

(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:

g(s(z0)) → f(z0)
g(0) → 0
f(0) → s(0)
f(s(z0)) → s(s(g(z0)))

Tuples:

G(s(z0)) → c(F(z0))
G(0) → c1
F(0) → c2
F(s(z0)) → c3(G(z0))

S tuples:

G(s(z0)) → c(F(z0))
G(0) → c1
F(0) → c2
F(s(z0)) → c3(G(z0))

K tuples:none
Defined Rule Symbols:

g, f

Defined Pair Symbols:

G, F

Compound Symbols:

c, c1, c2, c3

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

Removed 2 trailing nodes:

G(0) → c1
F(0) → c2

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(s(z0)) → f(z0)
g(0) → 0
f(0) → s(0)
f(s(z0)) → s(s(g(z0)))

Tuples:

G(s(z0)) → c(F(z0))
F(s(z0)) → c3(G(z0))

S tuples:

G(s(z0)) → c(F(z0))
F(s(z0)) → c3(G(z0))

K tuples:none
Defined Rule Symbols:

g, f

Defined Pair Symbols:

G, F

Compound Symbols:

c, c3

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

g(s(z0)) → f(z0)
g(0) → 0
f(0) → s(0)
f(s(z0)) → s(s(g(z0)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

G(s(z0)) → c(F(z0))
F(s(z0)) → c3(G(z0))

S tuples:

G(s(z0)) → c(F(z0))
F(s(z0)) → c3(G(z0))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

G, F

Compound Symbols:

c, c3

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

G(s(z0)) → c(F(z0))
F(s(z0)) → c3(G(z0))

We considered the (Usable) Rules:none
And the Tuples:

G(s(z0)) → c(F(z0))
F(s(z0)) → c3(G(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x₁)) = [2]x₁ + [2]x₁²   
POL(G(x₁)) = [2] + [2]x₁²   
POL(c(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(s(x₁)) = [2] + x₁   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

G(s(z0)) → c(F(z0))
F(s(z0)) → c3(G(z0))

S tuples:none
K tuples:

G(s(z0)) → c(F(z0))
F(s(z0)) → c3(G(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

G, F

Compound Symbols:

c, c3

(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(12) BOUNDS(1, 1)