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

0 CpxTRS
↳1 DecreasingLoopProof (⇔, 194 ms)
↳2 BOUNDS(n^1, INF)
↳3 RenamingProof (⇔, 0 ms)
↳4 CpxRelTRS
    ↳5 SlicingProof (LOWER BOUND(ID), 0 ms)
    ↳6 CpxRelTRS
    ↳7 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳8 typed CpxTrs
    ↳9 OrderProof (LOWER BOUND(ID), 0 ms)
    ↳10 typed CpxTrs
    ↳11 NoRewriteLemmaProof (LOWER BOUND(ID), 531 ms)
    ↳12 typed CpxTrs
    ↳13 RewriteLemmaProof (LOWER BOUND(ID), 58 ms)
    ↳14 BEST
        ↳15 typed CpxTrs
            ↳16 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)
            ↳17 typed CpxTrs
            ↳18 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)
            ↳19 typed CpxTrs
            ↳20 LowerBoundsProof (⇔, 0 ms)
            ↳21 BOUNDS(n^1, INF)
        ↳22 typed CpxTrs
            ↳23 LowerBoundsProof (⇔, 0 ms)
            ↳24 BOUNDS(n^1, INF)

(0) Obligation:

Runtime Complexity TRS:
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: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
activate(n__f(X)) →+ f(activate(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / n__f(X)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative TRS:
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

S is empty.
Rewrite Strategy: FULL

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
n__g/0
g/0

(6) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

TRS:
Rules:
f(f(X)) → c(n__f(n__g))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
g → n__g
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g) → g
activate(n__d(X)) → d(X)
activate(X) → X

Types:
f :: n__g:n__f:n__d → n__g:n__f:n__d
c :: n__g:n__f:n__d → n__g:n__f:n__d
n__f :: n__g:n__f:n__d → n__g:n__f:n__d
n__g :: n__g:n__f:n__d
d :: n__g:n__f:n__d → n__g:n__f:n__d
activate :: n__g:n__f:n__d → n__g:n__f:n__d
h :: n__g:n__f:n__d → n__g:n__f:n__d
n__d :: n__g:n__f:n__d → n__g:n__f:n__d
g :: n__g:n__f:n__d
hole_n__g:n__f:n__d1_0 :: n__g:n__f:n__d
gen_n__g:n__f:n__d2_0 :: Nat → n__g:n__f:n__d

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
f, c, activate

They will be analysed ascendingly in the following order:
f = c
f = activate
c = activate

(10) Obligation:

TRS:
Rules:
f(f(X)) → c(n__f(n__g))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
g → n__g
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g) → g
activate(n__d(X)) → d(X)
activate(X) → X

Types:
f :: n__g:n__f:n__d → n__g:n__f:n__d
c :: n__g:n__f:n__d → n__g:n__f:n__d
n__f :: n__g:n__f:n__d → n__g:n__f:n__d
n__g :: n__g:n__f:n__d
d :: n__g:n__f:n__d → n__g:n__f:n__d
activate :: n__g:n__f:n__d → n__g:n__f:n__d
h :: n__g:n__f:n__d → n__g:n__f:n__d
n__d :: n__g:n__f:n__d → n__g:n__f:n__d
g :: n__g:n__f:n__d
hole_n__g:n__f:n__d1_0 :: n__g:n__f:n__d
gen_n__g:n__f:n__d2_0 :: Nat → n__g:n__f:n__d

Generator Equations:
gen_n__g:n__f:n__d2_0(0) ⇔ n__g
gen_n__g:n__f:n__d2_0(+(x, 1)) ⇔ n__f(gen_n__g:n__f:n__d2_0(x))

The following defined symbols remain to be analysed:
c, f, activate

They will be analysed ascendingly in the following order:
f = c
f = activate
c = activate

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol c.

The following conjecture could not be proven:

c(gen_n__g:n__f:n__d2_0(n4_0)) →? n__d(gen_n__g:n__f:n__d2_0(n4_0))

(12) Obligation:

TRS:
Rules:
f(f(X)) → c(n__f(n__g))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
g → n__g
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g) → g
activate(n__d(X)) → d(X)
activate(X) → X

Types:
f :: n__g:n__f:n__d → n__g:n__f:n__d
c :: n__g:n__f:n__d → n__g:n__f:n__d
n__f :: n__g:n__f:n__d → n__g:n__f:n__d
n__g :: n__g:n__f:n__d
d :: n__g:n__f:n__d → n__g:n__f:n__d
activate :: n__g:n__f:n__d → n__g:n__f:n__d
h :: n__g:n__f:n__d → n__g:n__f:n__d
n__d :: n__g:n__f:n__d → n__g:n__f:n__d
g :: n__g:n__f:n__d
hole_n__g:n__f:n__d1_0 :: n__g:n__f:n__d
gen_n__g:n__f:n__d2_0 :: Nat → n__g:n__f:n__d

Generator Equations:
gen_n__g:n__f:n__d2_0(0) ⇔ n__g
gen_n__g:n__f:n__d2_0(+(x, 1)) ⇔ n__f(gen_n__g:n__f:n__d2_0(x))

The following defined symbols remain to be analysed:
activate, f

They will be analysed ascendingly in the following order:
f = c
f = activate
c = activate

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
activate(gen_n__g:n__f:n__d2_0(n1429_0)) → gen_n__g:n__f:n__d2_0(n1429_0), rt ∈ Ω(1 + n14290)

Induction Base:
activate(gen_n__g:n__f:n__d2_0(0)) →RΩ(1)
gen_n__g:n__f:n__d2_0(0)

Induction Step:
activate(gen_n__g:n__f:n__d2_0(+(n1429_0, 1))) →RΩ(1)
f(activate(gen_n__g:n__f:n__d2_0(n1429_0))) →IH
f(gen_n__g:n__f:n__d2_0(c1430_0)) →RΩ(1)
n__f(gen_n__g:n__f:n__d2_0(n1429_0))

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(14) Complex Obligation (BEST)

(15) Obligation:

TRS:
Rules:
f(f(X)) → c(n__f(n__g))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
g → n__g
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g) → g
activate(n__d(X)) → d(X)
activate(X) → X

Types:
f :: n__g:n__f:n__d → n__g:n__f:n__d
c :: n__g:n__f:n__d → n__g:n__f:n__d
n__f :: n__g:n__f:n__d → n__g:n__f:n__d
n__g :: n__g:n__f:n__d
d :: n__g:n__f:n__d → n__g:n__f:n__d
activate :: n__g:n__f:n__d → n__g:n__f:n__d
h :: n__g:n__f:n__d → n__g:n__f:n__d
n__d :: n__g:n__f:n__d → n__g:n__f:n__d
g :: n__g:n__f:n__d
hole_n__g:n__f:n__d1_0 :: n__g:n__f:n__d
gen_n__g:n__f:n__d2_0 :: Nat → n__g:n__f:n__d

Lemmas:
activate(gen_n__g:n__f:n__d2_0(n1429_0)) → gen_n__g:n__f:n__d2_0(n1429_0), rt ∈ Ω(1 + n14290)

Generator Equations:
gen_n__g:n__f:n__d2_0(0) ⇔ n__g
gen_n__g:n__f:n__d2_0(+(x, 1)) ⇔ n__f(gen_n__g:n__f:n__d2_0(x))

The following defined symbols remain to be analysed:
f, c

They will be analysed ascendingly in the following order:
f = c
f = activate
c = activate

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol f.

(17) Obligation:

TRS:
Rules:
f(f(X)) → c(n__f(n__g))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
g → n__g
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g) → g
activate(n__d(X)) → d(X)
activate(X) → X

Types:
f :: n__g:n__f:n__d → n__g:n__f:n__d
c :: n__g:n__f:n__d → n__g:n__f:n__d
n__f :: n__g:n__f:n__d → n__g:n__f:n__d
n__g :: n__g:n__f:n__d
d :: n__g:n__f:n__d → n__g:n__f:n__d
activate :: n__g:n__f:n__d → n__g:n__f:n__d
h :: n__g:n__f:n__d → n__g:n__f:n__d
n__d :: n__g:n__f:n__d → n__g:n__f:n__d
g :: n__g:n__f:n__d
hole_n__g:n__f:n__d1_0 :: n__g:n__f:n__d
gen_n__g:n__f:n__d2_0 :: Nat → n__g:n__f:n__d

Lemmas:
activate(gen_n__g:n__f:n__d2_0(n1429_0)) → gen_n__g:n__f:n__d2_0(n1429_0), rt ∈ Ω(1 + n14290)

Generator Equations:
gen_n__g:n__f:n__d2_0(0) ⇔ n__g
gen_n__g:n__f:n__d2_0(+(x, 1)) ⇔ n__f(gen_n__g:n__f:n__d2_0(x))

The following defined symbols remain to be analysed:
c

They will be analysed ascendingly in the following order:
f = c
f = activate
c = activate

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol c.

(19) Obligation:

TRS:
Rules:
f(f(X)) → c(n__f(n__g))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
g → n__g
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g) → g
activate(n__d(X)) → d(X)
activate(X) → X

Types:
f :: n__g:n__f:n__d → n__g:n__f:n__d
c :: n__g:n__f:n__d → n__g:n__f:n__d
n__f :: n__g:n__f:n__d → n__g:n__f:n__d
n__g :: n__g:n__f:n__d
d :: n__g:n__f:n__d → n__g:n__f:n__d
activate :: n__g:n__f:n__d → n__g:n__f:n__d
h :: n__g:n__f:n__d → n__g:n__f:n__d
n__d :: n__g:n__f:n__d → n__g:n__f:n__d
g :: n__g:n__f:n__d
hole_n__g:n__f:n__d1_0 :: n__g:n__f:n__d
gen_n__g:n__f:n__d2_0 :: Nat → n__g:n__f:n__d

Lemmas:
activate(gen_n__g:n__f:n__d2_0(n1429_0)) → gen_n__g:n__f:n__d2_0(n1429_0), rt ∈ Ω(1 + n14290)

Generator Equations:
gen_n__g:n__f:n__d2_0(0) ⇔ n__g
gen_n__g:n__f:n__d2_0(+(x, 1)) ⇔ n__f(gen_n__g:n__f:n__d2_0(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
activate(gen_n__g:n__f:n__d2_0(n1429_0)) → gen_n__g:n__f:n__d2_0(n1429_0), rt ∈ Ω(1 + n14290)

(21) BOUNDS(n^1, INF)

(22) Obligation:

TRS:
Rules:
f(f(X)) → c(n__f(n__g))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
g → n__g
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g) → g
activate(n__d(X)) → d(X)
activate(X) → X

Types:
f :: n__g:n__f:n__d → n__g:n__f:n__d
c :: n__g:n__f:n__d → n__g:n__f:n__d
n__f :: n__g:n__f:n__d → n__g:n__f:n__d
n__g :: n__g:n__f:n__d
d :: n__g:n__f:n__d → n__g:n__f:n__d
activate :: n__g:n__f:n__d → n__g:n__f:n__d
h :: n__g:n__f:n__d → n__g:n__f:n__d
n__d :: n__g:n__f:n__d → n__g:n__f:n__d
g :: n__g:n__f:n__d
hole_n__g:n__f:n__d1_0 :: n__g:n__f:n__d
gen_n__g:n__f:n__d2_0 :: Nat → n__g:n__f:n__d

Lemmas:
activate(gen_n__g:n__f:n__d2_0(n1429_0)) → gen_n__g:n__f:n__d2_0(n1429_0), rt ∈ Ω(1 + n14290)

Generator Equations:
gen_n__g:n__f:n__d2_0(0) ⇔ n__g
gen_n__g:n__f:n__d2_0(+(x, 1)) ⇔ n__f(gen_n__g:n__f:n__d2_0(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
activate(gen_n__g:n__f:n__d2_0(n1429_0)) → gen_n__g:n__f:n__d2_0(n1429_0), rt ∈ Ω(1 + n14290)

(24) BOUNDS(n^1, INF)