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:

active(f(X)) → mark(g(h(f(X))))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(g(X)) → g(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(f(X)) → mark(g(h(f(X))))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
proper(f(X)) → f(proper(X))
proper(g(X)) → g(proper(X))
proper(h(X)) → h(proper(X))

(2) 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(ok(X)) → ok(g(X))
top(ok(X)) → top(active(X))
f(mark(X)) → mark(f(X))
f(ok(X)) → ok(f(X))
top(mark(X)) → top(proper(X))
h(mark(X)) → mark(h(X))
h(ok(X)) → ok(h(X))

Rewrite Strategy: INNERMOST

(3) 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: 5
Accept states: [6]
Transitions:
5→6[g_1|0, top_1|0, f_1|0, h_1|0]
5→7[ok_1|1]
5→8[top_1|1]
5→9[top_1|1]
5→10[mark_1|1]
5→11[ok_1|1]
5→12[mark_1|1]
5→13[ok_1|1]
6→6[ok_1|0, active_1|0, mark_1|0, proper_1|0]
7→6[g_1|1]
7→7[ok_1|1]
8→6[active_1|1]
9→6[proper_1|1]
10→6[f_1|1]
10→10[mark_1|1]
10→11[ok_1|1]
11→6[f_1|1]
11→10[mark_1|1]
11→11[ok_1|1]
12→6[h_1|1]
12→12[mark_1|1]
12→13[ok_1|1]
13→6[h_1|1]
13→12[mark_1|1]
13→13[ok_1|1]

(4) BOUNDS(1, n^1)

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

Converted Cpx (relative) TRS to CDT

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(ok(z0)) → ok(g(z0))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))

Tuples:

G(ok(z0)) → c(G(z0))
TOP(ok(z0)) → c1(TOP(active(z0)))
TOP(mark(z0)) → c2(TOP(proper(z0)))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

S tuples:

G(ok(z0)) → c(G(z0))
TOP(ok(z0)) → c1(TOP(active(z0)))
TOP(mark(z0)) → c2(TOP(proper(z0)))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

K tuples:none
Defined Rule Symbols:

g, top, f, h

Defined Pair Symbols:

G, TOP, F, H

Compound Symbols:

c, c1, c2, c3, c4, c5, c6

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

Removed 2 trailing nodes:

TOP(ok(z0)) → c1(TOP(active(z0)))
TOP(mark(z0)) → c2(TOP(proper(z0)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(ok(z0)) → ok(g(z0))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))

Tuples:

G(ok(z0)) → c(G(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

S tuples:

G(ok(z0)) → c(G(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

K tuples:none
Defined Rule Symbols:

g, top, f, h

Defined Pair Symbols:

G, F, H

Compound Symbols:

c, c3, c4, c5, c6

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

g(ok(z0)) → ok(g(z0))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

G(ok(z0)) → c(G(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

S tuples:

G(ok(z0)) → c(G(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

G, F, H

Compound Symbols:

c, c3, c4, c5, c6

(11) 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(ok(z0)) → c(G(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

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

G(ok(z0)) → c(G(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

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

POL(F(x₁)) = [2]x₁²   
POL(G(x₁)) = [2]x₁   
POL(H(x₁)) = x₁²   
POL(c(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(mark(x₁)) = [2] + x₁   
POL(ok(x₁)) = [2] + x₁   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

G(ok(z0)) → c(G(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

S tuples:none
K tuples:

G(ok(z0)) → c(G(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

G, F, H

Compound Symbols:

c, c3, c4, c5, c6

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

The set S is empty

(14) BOUNDS(1, 1)