Runtime Complexity (full) proof of /tmp/tmp2ea8ba/Ex1_2_Luc02c

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(2^n, INF).

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 94 ms)

↳2 BOUNDS(2^n, INF)

↳3 RenamingProof (⇔, 0 ms)

↳4 CpxRelTRS

    ↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳6 typed CpxTrs

    ↳7 OrderProof (LOWER BOUND(ID), 0 ms)

    ↳8 typed CpxTrs

    ↳9 RewriteLemmaProof (LOWER BOUND(ID), 370 ms)

    ↳10 BEST

        ↳11 typed CpxTrs

            ↳12 LowerBoundsProof (⇔, 0 ms)

            ↳13 BOUNDS(n^1, INF)

        ↳14 typed CpxTrs

            ↳15 LowerBoundsProof (⇔, 0 ms)

            ↳16 BOUNDS(n^1, INF)

(0) Obligation:

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

2nd(cons(X, n__cons(Y, Z))) → activate(Y)
from(X) → cons(X, n__from(n__s(X)))
cons(X1, X2) → n__cons(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
activate(n__cons(n__from(X128_0), X2)) →⁺ cons(cons(activate(X128_0), n__from(n__s(activate(X128_0)))), X2)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [X128_0 / n__cons(n__from(X128_0), X2)].
The result substitution is [ ].

The rewrite sequence
activate(n__cons(n__from(X128_0), X2)) →⁺ cons(cons(activate(X128_0), n__from(n__s(activate(X128_0)))), X2)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0,0].
The pumping substitution is [X128_0 / n__cons(n__from(X128_0), X2)].
The result substitution is [ ].

(2) BOUNDS(2^n, 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:

2nd(cons(X, n__cons(Y, Z))) → activate(Y)
from(X) → cons(X, n__from(n__s(X)))
cons(X1, X2) → n__cons(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
2nd(cons(X, n__cons(Y, Z))) → activate(Y)
from(X) → cons(X, n__from(n__s(X)))
cons(X1, X2) → n__cons(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Types:
2nd :: n__cons:n__s:n__from → n__cons:n__s:n__from
cons :: n__cons:n__s:n__from → n__cons:n__s:n__from → n__cons:n__s:n__from
n__cons :: n__cons:n__s:n__from → n__cons:n__s:n__from → n__cons:n__s:n__from
activate :: n__cons:n__s:n__from → n__cons:n__s:n__from
from :: n__cons:n__s:n__from → n__cons:n__s:n__from
n__from :: n__cons:n__s:n__from → n__cons:n__s:n__from
n__s :: n__cons:n__s:n__from → n__cons:n__s:n__from
s :: n__cons:n__s:n__from → n__cons:n__s:n__from
hole_n__cons:n__s:n__from1_0 :: n__cons:n__s:n__from
gen_n__cons:n__s:n__from2_0 :: Nat → n__cons:n__s:n__from

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
activate

(8) Obligation:

Generator Equations:
gen_n__cons:n__s:n__from2_0(0) ⇔ hole_n__cons:n__s:n__from1_0
gen_n__cons:n__s:n__from2_0(+(x, 1)) ⇔ n__cons(gen_n__cons:n__s:n__from2_0(x), hole_n__cons:n__s:n__from1_0)

The following defined symbols remain to be analysed:
activate

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
activate(gen_n__cons:n__s:n__from2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
activate(gen_n__cons:n__s:n__from2_0(+(1, 0)))

Induction Step:
activate(gen_n__cons:n__s:n__from2_0(+(1, +(n4_0, 1)))) →_R^Ω(1)
cons(activate(gen_n__cons:n__s:n__from2_0(+(1, n4_0))), hole_n__cons:n__s:n__from1_0) →_IH
cons(*3_0, hole_n__cons:n__s:n__from1_0)

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

(10) Complex Obligation (BEST)

(11) Obligation:

Lemmas:
activate(gen_n__cons:n__s:n__from2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_n__cons:n__s:n__from2_0(0) ⇔ hole_n__cons:n__s:n__from1_0
gen_n__cons:n__s:n__from2_0(+(x, 1)) ⇔ n__cons(gen_n__cons:n__s:n__from2_0(x), hole_n__cons:n__s:n__from1_0)

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_n__cons:n__s:n__from2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Lemmas:
activate(gen_n__cons:n__s:n__from2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_n__cons:n__s:n__from2_0(0) ⇔ hole_n__cons:n__s:n__from1_0
gen_n__cons:n__s:n__from2_0(+(x, 1)) ⇔ n__cons(gen_n__cons:n__s:n__from2_0(x), hole_n__cons:n__s:n__from1_0)

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_n__cons:n__s:n__from2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(0) Obligation:

(1) DecreasingLoopProof (EQUIVALENT transformation)

(2) BOUNDS(2^n, INF)

(3) RenamingProof (EQUIVALENT transformation)

(4) Obligation:

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

(6) Obligation:

(7) OrderProof (LOWER BOUND(ID) transformation)

(8) Obligation:

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(10) Complex Obligation (BEST)

(11) Obligation:

(12) LowerBoundsProof (EQUIVALENT transformation)

(13) BOUNDS(n^1, INF)

(14) Obligation:

(15) LowerBoundsProof (EQUIVALENT transformation)

(16) BOUNDS(n^1, INF)