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

0 CpxTRS
↳1 RenamingProof (⇔, 0 ms)
↳2 CpxRelTRS
↳3 SlicingProof (LOWER BOUND(ID), 0 ms)
↳4 CpxRelTRS
↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 typed CpxTrs
↳7 OrderProof (LOWER BOUND(ID), 5 ms)
↳8 typed CpxTrs
↳9 NoRewriteLemmaProof (LOWER BOUND(ID), 590 ms)
↳10 typed CpxTrs
↳11 RewriteLemmaProof (LOWER BOUND(ID), 0 ms)
↳12 BEST
    ↳13 typed CpxTrs
        ↳14 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)
        ↳15 typed CpxTrs
        ↳16 NoRewriteLemmaProof (LOWER BOUND(ID), 6 ms)
        ↳17 typed CpxTrs
        ↳18 LowerBoundsProof (⇔, 0 ms)
        ↳19 BOUNDS(n^1, INF)
    ↳20 typed CpxTrs
        ↳21 LowerBoundsProof (⇔, 0 ms)
        ↳22 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) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) 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

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
n__g/0
g/0

(4) 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

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

Infered types.

(6) 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

(7) 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

(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

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

(9) 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))

(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:
activate, f

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

(11) 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).

(12) Complex Obligation (BEST)

(13) 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

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(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:
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 c.

(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))

No more defined symbols left to analyse.

(18) 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)

(19) BOUNDS(n^1, INF)

(20) 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.

(21) 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)

(22) BOUNDS(n^1, INF)