Runtime Complexity (full) proof of /tmp/tmpCOJb17/Ex14_AEGL02

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

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 192 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), 7172 ms)

    ↳10 BEST

        ↳11 typed CpxTrs

            ↳12 NoRewriteLemmaProof (LOWER BOUND(ID), 5308 ms)

            ↳13 typed CpxTrs

            ↳14 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)

            ↳15 typed CpxTrs

            ↳16 LowerBoundsProof (⇔, 0 ms)

            ↳17 BOUNDS(n^1, INF)

        ↳18 typed CpxTrs

            ↳19 LowerBoundsProof (⇔, 0 ms)

            ↳20 BOUNDS(n^1, INF)

(0) Obligation:

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

from(X) → cons(X, n__from(n__s(X)))
length(n__nil) → 0
length(n__cons(X, Y)) → s(length1(activate(Y)))
length1(X) → length(activate(X))
from(X) → n__from(X)
s(X) → n__s(X)
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
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__from(X)) →⁺ cons(activate(X), n__from(n__s(activate(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / n__from(X)].
The result substitution is [ ].

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

from(X) → cons(X, n__from(n__s(X)))
length(n__nil) → 0'
length(n__cons(X, Y)) → s(length1(activate(Y)))
length1(X) → length(activate(X))
from(X) → n__from(X)
s(X) → n__s(X)
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(X) → X

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
from(X) → cons(X, n__from(n__s(X)))
length(n__nil) → 0'
length(n__cons(X, Y)) → s(length1(activate(Y)))
length1(X) → length(activate(X))
from(X) → n__from(X)
s(X) → n__s(X)
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(X) → X

Types:
from :: n__s:n__from:n__nil:0':n__cons → n__s:n__from:n__nil:0':n__cons
cons :: n__s:n__from:n__nil:0':n__cons → n__s:n__from:n__nil:0':n__cons → n__s:n__from:n__nil:0':n__cons
n__from :: n__s:n__from:n__nil:0':n__cons → n__s:n__from:n__nil:0':n__cons
n__s :: n__s:n__from:n__nil:0':n__cons → n__s:n__from:n__nil:0':n__cons
length :: n__s:n__from:n__nil:0':n__cons → n__s:n__from:n__nil:0':n__cons
n__nil :: n__s:n__from:n__nil:0':n__cons
0' :: n__s:n__from:n__nil:0':n__cons
n__cons :: n__s:n__from:n__nil:0':n__cons → n__s:n__from:n__nil:0':n__cons → n__s:n__from:n__nil:0':n__cons
s :: n__s:n__from:n__nil:0':n__cons → n__s:n__from:n__nil:0':n__cons
length1 :: n__s:n__from:n__nil:0':n__cons → n__s:n__from:n__nil:0':n__cons
activate :: n__s:n__from:n__nil:0':n__cons → n__s:n__from:n__nil:0':n__cons
nil :: n__s:n__from:n__nil:0':n__cons
hole_n__s:n__from:n__nil:0':n__cons1_0 :: n__s:n__from:n__nil:0':n__cons
gen_n__s:n__from:n__nil:0':n__cons2_0 :: Nat → n__s:n__from:n__nil:0':n__cons

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

Heuristically decided to analyse the following defined symbols:
length, length1, activate

They will be analysed ascendingly in the following order:
length = length1
activate < length
activate < length1

(8) Obligation:

Generator Equations:
gen_n__s:n__from:n__nil:0':n__cons2_0(0) ⇔ n__nil
gen_n__s:n__from:n__nil:0':n__cons2_0(+(x, 1)) ⇔ n__from(gen_n__s:n__from:n__nil:0':n__cons2_0(x))

The following defined symbols remain to be analysed:
activate, length, length1

They will be analysed ascendingly in the following order:
length = length1
activate < length
activate < length1

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

Proved the following rewrite lemma:
activate(gen_n__s:n__from:n__nil:0':n__cons2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
activate(gen_n__s:n__from:n__nil:0':n__cons2_0(+(1, 0)))

Induction Step:
activate(gen_n__s:n__from:n__nil:0':n__cons2_0(+(1, +(n4_0, 1)))) →_R^Ω(1)
from(activate(gen_n__s:n__from:n__nil:0':n__cons2_0(+(1, n4_0)))) →_IH
from(*3_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__s:n__from:n__nil:0':n__cons2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_n__s:n__from:n__nil:0':n__cons2_0(0) ⇔ n__nil
gen_n__s:n__from:n__nil:0':n__cons2_0(+(x, 1)) ⇔ n__from(gen_n__s:n__from:n__nil:0':n__cons2_0(x))

The following defined symbols remain to be analysed:
length1, length

They will be analysed ascendingly in the following order:
length = length1

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol length1.

(13) Obligation:

Lemmas:
activate(gen_n__s:n__from:n__nil:0':n__cons2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_n__s:n__from:n__nil:0':n__cons2_0(0) ⇔ n__nil
gen_n__s:n__from:n__nil:0':n__cons2_0(+(x, 1)) ⇔ n__from(gen_n__s:n__from:n__nil:0':n__cons2_0(x))

The following defined symbols remain to be analysed:
length

They will be analysed ascendingly in the following order:
length = length1

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

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

(15) Obligation:

Lemmas:
activate(gen_n__s:n__from:n__nil:0':n__cons2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_n__s:n__from:n__nil:0':n__cons2_0(0) ⇔ n__nil
gen_n__s:n__from:n__nil:0':n__cons2_0(+(x, 1)) ⇔ n__from(gen_n__s:n__from:n__nil:0':n__cons2_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_n__s:n__from:n__nil:0':n__cons2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

Lemmas:
activate(gen_n__s:n__from:n__nil:0':n__cons2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_n__s:n__from:n__nil:0':n__cons2_0(0) ⇔ n__nil
gen_n__s:n__from:n__nil:0':n__cons2_0(+(x, 1)) ⇔ n__from(gen_n__s:n__from:n__nil:0':n__cons2_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_n__s:n__from:n__nil:0':n__cons2_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) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(13) Obligation:

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

(15) Obligation:

(16) LowerBoundsProof (EQUIVALENT transformation)

(17) BOUNDS(n^1, INF)

(18) Obligation:

(19) LowerBoundsProof (EQUIVALENT transformation)

(20) BOUNDS(n^1, INF)