(0) Obligation:

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


The TRS R consists of the following rules:

f(f(a)) → c(n__f(g(f(a))))
f(X) → n__f(X)
activate(n__f(X)) → f(X)
activate(X) → X

Rewrite Strategy: INNERMOST

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

The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
f(f(a)) → c(n__f(g(f(a))))

(2) Obligation:

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


The TRS R consists of the following rules:

activate(n__f(X)) → f(X)
f(X) → n__f(X)
activate(X) → 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 2.
The certificate found is represented by the following graph.
Start state: 3
Accept states: [4]
Transitions:
3→4[activate_1|0, f_1|0, f_1|1, n__f_1|1, n__f_1|2]
4→4[n__f_1|0]

(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:

activate(n__f(z0)) → f(z0)
activate(z0) → z0
f(z0) → n__f(z0)
Tuples:

ACTIVATE(n__f(z0)) → c(F(z0))
ACTIVATE(z0) → c1
F(z0) → c2
S tuples:

ACTIVATE(n__f(z0)) → c(F(z0))
ACTIVATE(z0) → c1
F(z0) → c2
K tuples:none
Defined Rule Symbols:

activate, f

Defined Pair Symbols:

ACTIVATE, F

Compound Symbols:

c, c1, c2

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

Removed 3 trailing nodes:

ACTIVATE(n__f(z0)) → c(F(z0))
F(z0) → c2
ACTIVATE(z0) → c1

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

activate(n__f(z0)) → f(z0)
activate(z0) → z0
f(z0) → n__f(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

activate, f

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty

(10) BOUNDS(1, 1)