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

0 CpxTRS
↳1 NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CpxTRS
    ↳3 CpxTrsMatchBoundsProof (⇔, 11 ms)
    ↳4 BOUNDS(1, n^1)
    ↳5 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 CdtProblem
        ↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳8 CdtProblem
        ↳9 CdtUsableRulesProof (⇔, 0 ms)
        ↳10 CdtProblem
        ↳11 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 78 ms)
        ↳12 CdtProblem
        ↳13 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 7 ms)
        ↳14 CdtProblem
        ↳15 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)
        ↳16 CdtProblem
        ↳17 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 5 ms)
        ↳18 CdtProblem
        ↳19 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳20 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:

active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(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(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(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(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(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))
c(ok(X)) → ok(c(X))
f(ok(X)) → ok(f(X))
d(ok(X)) → ok(d(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: 7
Accept states: [8]
Transitions:
7→8[g_1|0, top_1|0, f_1|0, c_1|0, d_1|0, h_1|0]
7→9[ok_1|1]
7→10[top_1|1]
7→11[top_1|1]
7→12[mark_1|1]
7→13[ok_1|1]
7→14[ok_1|1]
7→15[ok_1|1]
7→16[mark_1|1]
7→17[ok_1|1]
8→8[ok_1|0, active_1|0, mark_1|0, proper_1|0]
9→8[g_1|1]
9→9[ok_1|1]
10→8[active_1|1]
11→8[proper_1|1]
12→8[f_1|1]
12→12[mark_1|1]
12→13[ok_1|1]
13→8[f_1|1]
13→12[mark_1|1]
13→13[ok_1|1]
14→8[c_1|1]
14→14[ok_1|1]
15→8[d_1|1]
15→15[ok_1|1]
16→8[h_1|1]
16→16[mark_1|1]
16→17[ok_1|1]
17→8[h_1|1]
17→16[mark_1|1]
17→17[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))
c(ok(z0)) → ok(c(z0))
d(ok(z0)) → ok(d(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
Tuples:

G(ok(z0)) → c1(G(z0))
TOP(ok(z0)) → c2(TOP(active(z0)))
TOP(mark(z0)) → c3(TOP(proper(z0)))
F(mark(z0)) → c4(F(z0))
F(ok(z0)) → c5(F(z0))
C(ok(z0)) → c6(C(z0))
D(ok(z0)) → c7(D(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
S tuples:

G(ok(z0)) → c1(G(z0))
TOP(ok(z0)) → c2(TOP(active(z0)))
TOP(mark(z0)) → c3(TOP(proper(z0)))
F(mark(z0)) → c4(F(z0))
F(ok(z0)) → c5(F(z0))
C(ok(z0)) → c6(C(z0))
D(ok(z0)) → c7(D(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
K tuples:none
Defined Rule Symbols:

g, top, f, c, d, h

Defined Pair Symbols:

G, TOP, F, C, D, H

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9

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

Removed 2 trailing nodes:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TOP(ok(z0)) → c2(TOP(active(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))
c(ok(z0)) → ok(c(z0))
d(ok(z0)) → ok(d(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
Tuples:

G(ok(z0)) → c1(G(z0))
F(mark(z0)) → c4(F(z0))
F(ok(z0)) → c5(F(z0))
C(ok(z0)) → c6(C(z0))
D(ok(z0)) → c7(D(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
S tuples:

G(ok(z0)) → c1(G(z0))
F(mark(z0)) → c4(F(z0))
F(ok(z0)) → c5(F(z0))
C(ok(z0)) → c6(C(z0))
D(ok(z0)) → c7(D(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
K tuples:none
Defined Rule Symbols:

g, top, f, c, d, h

Defined Pair Symbols:

G, F, C, D, H

Compound Symbols:

c1, c4, c5, c6, c7, c8, c9

(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))
c(ok(z0)) → ok(c(z0))
d(ok(z0)) → ok(d(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)) → c1(G(z0))
F(mark(z0)) → c4(F(z0))
F(ok(z0)) → c5(F(z0))
C(ok(z0)) → c6(C(z0))
D(ok(z0)) → c7(D(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
S tuples:

G(ok(z0)) → c1(G(z0))
F(mark(z0)) → c4(F(z0))
F(ok(z0)) → c5(F(z0))
C(ok(z0)) → c6(C(z0))
D(ok(z0)) → c7(D(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

G, F, C, D, H

Compound Symbols:

c1, c4, c5, c6, c7, c8, c9

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

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

F(mark(z0)) → c4(F(z0))
F(ok(z0)) → c5(F(z0))
C(ok(z0)) → c6(C(z0))
We considered the (Usable) Rules:none
And the Tuples:

G(ok(z0)) → c1(G(z0))
F(mark(z0)) → c4(F(z0))
F(ok(z0)) → c5(F(z0))
C(ok(z0)) → c6(C(z0))
D(ok(z0)) → c7(D(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(C(x1)) = x1   
POL(D(x1)) = 0   
POL(F(x1)) = x1   
POL(G(x1)) = 0   
POL(H(x1)) = 0   
POL(c1(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = [1] + x1   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

G(ok(z0)) → c1(G(z0))
F(mark(z0)) → c4(F(z0))
F(ok(z0)) → c5(F(z0))
C(ok(z0)) → c6(C(z0))
D(ok(z0)) → c7(D(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
S tuples:

G(ok(z0)) → c1(G(z0))
D(ok(z0)) → c7(D(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
K tuples:

F(mark(z0)) → c4(F(z0))
F(ok(z0)) → c5(F(z0))
C(ok(z0)) → c6(C(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

G, F, C, D, H

Compound Symbols:

c1, c4, c5, c6, c7, c8, c9

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

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

H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
We considered the (Usable) Rules:none
And the Tuples:

G(ok(z0)) → c1(G(z0))
F(mark(z0)) → c4(F(z0))
F(ok(z0)) → c5(F(z0))
C(ok(z0)) → c6(C(z0))
D(ok(z0)) → c7(D(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(C(x1)) = 0   
POL(D(x1)) = 0   
POL(F(x1)) = 0   
POL(G(x1)) = 0   
POL(H(x1)) = x1   
POL(c1(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = [1] + x1   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

G(ok(z0)) → c1(G(z0))
F(mark(z0)) → c4(F(z0))
F(ok(z0)) → c5(F(z0))
C(ok(z0)) → c6(C(z0))
D(ok(z0)) → c7(D(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
S tuples:

G(ok(z0)) → c1(G(z0))
D(ok(z0)) → c7(D(z0))
K tuples:

F(mark(z0)) → c4(F(z0))
F(ok(z0)) → c5(F(z0))
C(ok(z0)) → c6(C(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

G, F, C, D, H

Compound Symbols:

c1, c4, c5, c6, c7, c8, c9

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

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

D(ok(z0)) → c7(D(z0))
We considered the (Usable) Rules:none
And the Tuples:

G(ok(z0)) → c1(G(z0))
F(mark(z0)) → c4(F(z0))
F(ok(z0)) → c5(F(z0))
C(ok(z0)) → c6(C(z0))
D(ok(z0)) → c7(D(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(C(x1)) = 0   
POL(D(x1)) = x1   
POL(F(x1)) = 0   
POL(G(x1)) = 0   
POL(H(x1)) = 0   
POL(c1(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(mark(x1)) = 0   
POL(ok(x1)) = [1] + x1   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

G(ok(z0)) → c1(G(z0))
F(mark(z0)) → c4(F(z0))
F(ok(z0)) → c5(F(z0))
C(ok(z0)) → c6(C(z0))
D(ok(z0)) → c7(D(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
S tuples:

G(ok(z0)) → c1(G(z0))
K tuples:

F(mark(z0)) → c4(F(z0))
F(ok(z0)) → c5(F(z0))
C(ok(z0)) → c6(C(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
D(ok(z0)) → c7(D(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

G, F, C, D, H

Compound Symbols:

c1, c4, c5, c6, c7, c8, c9

(17) 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(ok(z0)) → c1(G(z0))
We considered the (Usable) Rules:none
And the Tuples:

G(ok(z0)) → c1(G(z0))
F(mark(z0)) → c4(F(z0))
F(ok(z0)) → c5(F(z0))
C(ok(z0)) → c6(C(z0))
D(ok(z0)) → c7(D(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(C(x1)) = 0   
POL(D(x1)) = 0   
POL(F(x1)) = x1   
POL(G(x1)) = x1   
POL(H(x1)) = 0   
POL(c1(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(mark(x1)) = x1   
POL(ok(x1)) = [1] + x1   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

G(ok(z0)) → c1(G(z0))
F(mark(z0)) → c4(F(z0))
F(ok(z0)) → c5(F(z0))
C(ok(z0)) → c6(C(z0))
D(ok(z0)) → c7(D(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
S tuples:none
K tuples:

F(mark(z0)) → c4(F(z0))
F(ok(z0)) → c5(F(z0))
C(ok(z0)) → c6(C(z0))
H(mark(z0)) → c8(H(z0))
H(ok(z0)) → c9(H(z0))
D(ok(z0)) → c7(D(z0))
G(ok(z0)) → c1(G(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

G, F, C, D, H

Compound Symbols:

c1, c4, c5, c6, c7, c8, c9

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

The set S is empty

(20) BOUNDS(1, 1)