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

0 CpxTRS
    ↳1 CpxTrsMatchBoundsProof (⇔, 0 ms)
    ↳2 BOUNDS(1, n^1)
    ↳3 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳4 CdtProblem
        ↳5 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
        ↳6 CdtProblem
        ↳7 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳8 CdtProblem
        ↳9 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
        ↳10 CdtProblem
        ↳11 CdtUsableRulesProof (⇔, 0 ms)
        ↳12 CdtProblem
        ↳13 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 3 ms)
        ↳14 CdtProblem
        ↳15 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳16 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(f(X)) → c(n__f(n__g(n__f(X))))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
g(X) → n__g(X)
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(X)
activate(n__d(X)) → d(X)
activate(X) → X

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 4.
The certificate found is represented by the following graph.
Start state: 13
Accept states: [14]
Transitions:
13→14[f_1|0, c_1|0, h_1|0, g_1|0, d_1|0, activate_1|0, n__f_1|1, n__g_1|1, n__d_1|1, g_1|1, d_1|1, n__g_1|2, n__d_1|2]
13→15[d_1|1, n__d_1|2]
13→16[c_1|1]
13→17[f_1|1, n__f_1|2]
13→18[d_1|2, n__d_1|3]
13→19[c_1|2]
13→22[d_1|3, n__d_1|4]
14→14[n__f_1|0, n__g_1|0, n__d_1|0]
15→14[activate_1|1, n__f_1|1, g_1|1, n__g_1|1, d_1|1, n__d_1|1, n__g_1|2, n__d_1|2]
15→17[f_1|1, n__f_1|2]
15→19[c_1|2]
15→22[d_1|3, n__d_1|4]
16→14[n__d_1|1]
17→14[activate_1|1, n__f_1|1, g_1|1, n__g_1|1, d_1|1, n__d_1|1, n__g_1|2, n__d_1|2]
17→17[f_1|1, n__f_1|2]
17→19[c_1|2]
17→22[d_1|3, n__d_1|4]
18→16[activate_1|2]
18→14[d_1|2, n__d_1|2, n__d_1|3]
19→20[n__f_1|2]
20→21[n__g_1|2]
21→17[n__f_1|2]
22→19[activate_1|3]
22→23[f_1|3, n__f_1|4]
22→20[n__f_1|3]
23→20[activate_1|3]
23→21[g_1|3, n__g_1|3, n__g_1|4]

(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(f(z0)) → c(n__f(n__g(n__f(z0))))
f(z0) → n__f(z0)
c(z0) → d(activate(z0))
h(z0) → c(n__d(z0))
g(z0) → n__g(z0)
d(z0) → n__d(z0)
activate(n__f(z0)) → f(activate(z0))
activate(n__g(z0)) → g(z0)
activate(n__d(z0)) → d(z0)
activate(z0) → z0
Tuples:

F(f(z0)) → c1(C(n__f(n__g(n__f(z0)))))
F(z0) → c2
C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
H(z0) → c4(C(n__d(z0)))
G(z0) → c5
D(z0) → c6
ACTIVATE(n__f(z0)) → c7(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c8(G(z0))
ACTIVATE(n__d(z0)) → c9(D(z0))
ACTIVATE(z0) → c10
S tuples:

F(f(z0)) → c1(C(n__f(n__g(n__f(z0)))))
F(z0) → c2
C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
H(z0) → c4(C(n__d(z0)))
G(z0) → c5
D(z0) → c6
ACTIVATE(n__f(z0)) → c7(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c8(G(z0))
ACTIVATE(n__d(z0)) → c9(D(z0))
ACTIVATE(z0) → c10
K tuples:none
Defined Rule Symbols:

f, c, h, g, d, activate

Defined Pair Symbols:

F, C, H, G, D, ACTIVATE

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 2 leading nodes:

F(f(z0)) → c1(C(n__f(n__g(n__f(z0)))))
H(z0) → c4(C(n__d(z0)))
Removed 6 trailing nodes:

D(z0) → c6
ACTIVATE(n__d(z0)) → c9(D(z0))
G(z0) → c5
ACTIVATE(z0) → c10
F(z0) → c2
ACTIVATE(n__g(z0)) → c8(G(z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → c(n__f(n__g(n__f(z0))))
f(z0) → n__f(z0)
c(z0) → d(activate(z0))
h(z0) → c(n__d(z0))
g(z0) → n__g(z0)
d(z0) → n__d(z0)
activate(n__f(z0)) → f(activate(z0))
activate(n__g(z0)) → g(z0)
activate(n__d(z0)) → d(z0)
activate(z0) → z0
Tuples:

C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c7(F(activate(z0)), ACTIVATE(z0))
S tuples:

C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c7(F(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

f, c, h, g, d, activate

Defined Pair Symbols:

C, ACTIVATE

Compound Symbols:

c3, c7

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

Removed 2 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → c(n__f(n__g(n__f(z0))))
f(z0) → n__f(z0)
c(z0) → d(activate(z0))
h(z0) → c(n__d(z0))
g(z0) → n__g(z0)
d(z0) → n__d(z0)
activate(n__f(z0)) → f(activate(z0))
activate(n__g(z0)) → g(z0)
activate(n__d(z0)) → d(z0)
activate(z0) → z0
Tuples:

C(z0) → c3(ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c7(ACTIVATE(z0))
S tuples:

C(z0) → c3(ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c7(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

f, c, h, g, d, activate

Defined Pair Symbols:

C, ACTIVATE

Compound Symbols:

c3, c7

(9) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

C(z0) → c3(ACTIVATE(z0))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → c(n__f(n__g(n__f(z0))))
f(z0) → n__f(z0)
c(z0) → d(activate(z0))
h(z0) → c(n__d(z0))
g(z0) → n__g(z0)
d(z0) → n__d(z0)
activate(n__f(z0)) → f(activate(z0))
activate(n__g(z0)) → g(z0)
activate(n__d(z0)) → d(z0)
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c7(ACTIVATE(z0))
S tuples:

ACTIVATE(n__f(z0)) → c7(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

f, c, h, g, d, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c7

(11) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

f(f(z0)) → c(n__f(n__g(n__f(z0))))
f(z0) → n__f(z0)
c(z0) → d(activate(z0))
h(z0) → c(n__d(z0))
g(z0) → n__g(z0)
d(z0) → n__d(z0)
activate(n__f(z0)) → f(activate(z0))
activate(n__g(z0)) → g(z0)
activate(n__d(z0)) → d(z0)
activate(z0) → z0

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__f(z0)) → c7(ACTIVATE(z0))
S tuples:

ACTIVATE(n__f(z0)) → c7(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c7

(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.

ACTIVATE(n__f(z0)) → c7(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__f(z0)) → c7(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = x1   
POL(c7(x1)) = x1   
POL(n__f(x1)) = [1] + x1   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__f(z0)) → c7(ACTIVATE(z0))
S tuples:none
K tuples:

ACTIVATE(n__f(z0)) → c7(ACTIVATE(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c7

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

The set S is empty

(16) BOUNDS(1, 1)