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

(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(X)) → c(n__f(g(n__f(X))))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
d(X) → n__d(X)
activate(n__f(X)) → f(X)
activate(n__d(X)) → d(X)
activate(X) → X

Rewrite Strategy: INNERMOST

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

The following defined symbols can occur below the 0th argument of d: d, f, activate

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

(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__d(X)) → d(X)
activate(n__f(X)) → f(X)
f(X) → n__f(X)
d(X) → n__d(X)
h(X) → c(n__d(X))
activate(X) → X
c(X) → d(activate(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 3.
The certificate found is represented by the following graph.

Start state: 6
Accept states: [7]
Transitions:
6→7[activate_1|0, f_1|0, d_1|0, h_1|0, c_1|0, d_1|1, n__d_1|1, f_1|1, n__f_1|1, n__f_1|2, n__d_1|2]
6→8[c_1|1]
6→9[d_1|1, n__d_1|2]
6→10[d_1|2, n__d_1|3]
7→7[n__d_1|0, n__f_1|0]
8→7[n__d_1|1]
9→7[activate_1|1, d_1|1, n__d_1|1, f_1|1, n__f_1|1, n__f_1|2, n__d_1|2]
10→8[activate_1|2]
10→7[d_1|2, n__d_1|2, n__d_1|3]

(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__d(z0)) → d(z0)
activate(n__f(z0)) → f(z0)
activate(z0) → z0
f(z0) → n__f(z0)
d(z0) → n__d(z0)
h(z0) → c(n__d(z0))
c(z0) → d(activate(z0))

Tuples:

ACTIVATE(n__d(z0)) → c1(D(z0))
ACTIVATE(n__f(z0)) → c2(F(z0))
ACTIVATE(z0) → c3
F(z0) → c4
D(z0) → c5
H(z0) → c6(C(n__d(z0)))
C(z0) → c7(D(activate(z0)), ACTIVATE(z0))

S tuples:

ACTIVATE(n__d(z0)) → c1(D(z0))
ACTIVATE(n__f(z0)) → c2(F(z0))
ACTIVATE(z0) → c3
F(z0) → c4
D(z0) → c5
H(z0) → c6(C(n__d(z0)))
C(z0) → c7(D(activate(z0)), ACTIVATE(z0))

K tuples:none
Defined Rule Symbols:

activate, f, d, h, c

Defined Pair Symbols:

ACTIVATE, F, D, H, C

Compound Symbols:

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

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

Removed 7 trailing nodes:

C(z0) → c7(D(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c2(F(z0))
H(z0) → c6(C(n__d(z0)))
D(z0) → c5
ACTIVATE(z0) → c3
F(z0) → c4
ACTIVATE(n__d(z0)) → c1(D(z0))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

activate, f, d, h, c

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty

(10) BOUNDS(1, 1)