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 (BOTH BOUNDS(ID, ID), 0 ms)
        ↳6 CdtProblem
        ↳7 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳8 CdtProblem
        ↳9 CdtUsableRulesProof (⇔, 0 ms)
        ↳10 CdtProblem
        ↳11 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 45 ms)
        ↳12 CdtProblem
        ↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳14 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(a)) → c(n__f(n__g(n__f(n__a))))
f(X) → n__f(X)
g(X) → n__g(X)
a → n__a
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(n__a) → a
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 2.
The certificate found is represented by the following graph.
Start state: 9
Accept states: [10]
Transitions:
9→10[f_1|0, g_1|0, a|0, activate_1|0, n__f_1|1, n__g_1|1, n__a|1, a|1, c_1|1, n__a|2]
9→11[f_1|1, n__f_1|2]
9→12[g_1|1, n__g_1|2]
9→13[c_1|2]
10→10[c_1|0, n__f_1|0, n__g_1|0, n__a|0]
11→10[activate_1|1, n__f_1|1, n__g_1|1, a|1, n__a|1, c_1|1, n__a|2]
11→11[f_1|1, n__f_1|2]
11→12[g_1|1, n__g_1|2]
11→13[c_1|2]
12→10[activate_1|1, n__f_1|1, n__g_1|1, a|1, n__a|1, c_1|1, n__a|2]
12→11[f_1|1, n__f_1|2]
12→12[g_1|1, n__g_1|2]
12→13[c_1|2]
13→14[n__f_1|2]
14→15[n__g_1|2]
15→16[n__f_1|2]
16→10[n__a|2]

(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(a)) → c(n__f(n__g(n__f(n__a))))
f(z0) → n__f(z0)
g(z0) → n__g(z0)
a → n__a
activate(n__f(z0)) → f(activate(z0))
activate(n__g(z0)) → g(activate(z0))
activate(n__a) → a
activate(z0) → z0
Tuples:

F(f(a)) → c1
F(z0) → c2
G(z0) → c3
A → c4
ACTIVATE(n__f(z0)) → c5(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c6(G(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__a) → c7(A)
ACTIVATE(z0) → c8
S tuples:

F(f(a)) → c1
F(z0) → c2
G(z0) → c3
A → c4
ACTIVATE(n__f(z0)) → c5(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c6(G(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__a) → c7(A)
ACTIVATE(z0) → c8
K tuples:none
Defined Rule Symbols:

f, g, a, activate

Defined Pair Symbols:

F, G, A, ACTIVATE

Compound Symbols:

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

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

Removed 6 trailing nodes:

F(f(a)) → c1
G(z0) → c3
F(z0) → c2
ACTIVATE(z0) → c8
A → c4
ACTIVATE(n__a) → c7(A)

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(a)) → c(n__f(n__g(n__f(n__a))))
f(z0) → n__f(z0)
g(z0) → n__g(z0)
a → n__a
activate(n__f(z0)) → f(activate(z0))
activate(n__g(z0)) → g(activate(z0))
activate(n__a) → a
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c5(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c6(G(activate(z0)), ACTIVATE(z0))
S tuples:

ACTIVATE(n__f(z0)) → c5(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c6(G(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

f, g, a, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6

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

Removed 2 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(a)) → c(n__f(n__g(n__f(n__a))))
f(z0) → n__f(z0)
g(z0) → n__g(z0)
a → n__a
activate(n__f(z0)) → f(activate(z0))
activate(n__g(z0)) → g(activate(z0))
activate(n__a) → a
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0))
S tuples:

ACTIVATE(n__f(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

f, g, a, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

f(f(a)) → c(n__f(n__g(n__f(n__a))))
f(z0) → n__f(z0)
g(z0) → n__g(z0)
a → n__a
activate(n__f(z0)) → f(activate(z0))
activate(n__g(z0)) → g(activate(z0))
activate(n__a) → a
activate(z0) → z0

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__f(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0))
S tuples:

ACTIVATE(n__f(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6

(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.

ACTIVATE(n__f(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__f(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = [2]x1   
POL(c5(x1)) = x1   
POL(c6(x1)) = x1   
POL(n__f(x1)) = [1] + x1   
POL(n__g(x1)) = [1] + x1   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__f(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0))
S tuples:none
K tuples:

ACTIVATE(n__f(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6

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

The set S is empty

(14) BOUNDS(1, 1)