Runtime Complexity (full) proof of /tmp/tmpUUWuI3/Ex14_AEGL02

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

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 292 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), 4 ms)

    ↳8 typed CpxTrs

    ↳9 NoRewriteLemmaProof (LOWER BOUND(ID), 94 ms)

    ↳10 typed CpxTrs

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

    ↳12 typed CpxTrs

    ↳13 RewriteLemmaProof (LOWER BOUND(ID), 347 ms)

    ↳14 BEST

        ↳15 typed CpxTrs

            ↳16 NoRewriteLemmaProof (LOWER BOUND(ID), 0 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:

a__from(X) → cons(mark(X), from(s(X)))
a__length(nil) → 0
a__length(cons(X, Y)) → s(a__length1(Y))
a__length1(X) → a__length(X)
mark(from(X)) → a__from(mark(X))
mark(length(X)) → a__length(X)
mark(length1(X)) → a__length1(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(0) → 0
a__from(X) → from(X)
a__length(X) → length(X)
a__length1(X) → length1(X)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

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

a__from(X) → cons(mark(X), from(s(X)))
a__length(nil) → 0'
a__length(cons(X, Y)) → s(a__length1(Y))
a__length1(X) → a__length(X)
mark(from(X)) → a__from(mark(X))
mark(length(X)) → a__length(X)
mark(length1(X)) → a__length1(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(0') → 0'
a__from(X) → from(X)
a__length(X) → length(X)
a__length1(X) → length1(X)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__length(nil) → 0'
a__length(cons(X, Y)) → s(a__length1(Y))
a__length1(X) → a__length(X)
mark(from(X)) → a__from(mark(X))
mark(length(X)) → a__length(X)
mark(length1(X)) → a__length1(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(0') → 0'
a__from(X) → from(X)
a__length(X) → length(X)
a__length1(X) → length1(X)

Types:
a__from :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
cons :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
mark :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
from :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
s :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
a__length :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
nil :: s:from:cons:nil:0':length:length1
0' :: s:from:cons:nil:0':length:length1
a__length1 :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
length :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
length1 :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
hole_s:from:cons:nil:0':length:length11_0 :: s:from:cons:nil:0':length:length1
gen_s:from:cons:nil:0':length:length12_0 :: Nat → s:from:cons:nil:0':length:length1

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

Heuristically decided to analyse the following defined symbols:
a__from, mark, a__length, a__length1

They will be analysed ascendingly in the following order:
a__from = mark
a__length < mark
a__length1 < mark
a__length = a__length1

(8) Obligation:

Generator Equations:
gen_s:from:cons:nil:0':length:length12_0(0) ⇔ nil
gen_s:from:cons:nil:0':length:length12_0(+(x, 1)) ⇔ cons(gen_s:from:cons:nil:0':length:length12_0(x), nil)

The following defined symbols remain to be analysed:
a__length1, a__from, mark, a__length

They will be analysed ascendingly in the following order:
a__from = mark
a__length < mark
a__length1 < mark
a__length = a__length1

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__length1.

(10) Obligation:

Generator Equations:
gen_s:from:cons:nil:0':length:length12_0(0) ⇔ nil
gen_s:from:cons:nil:0':length:length12_0(+(x, 1)) ⇔ cons(gen_s:from:cons:nil:0':length:length12_0(x), nil)

The following defined symbols remain to be analysed:
a__length, a__from, mark

They will be analysed ascendingly in the following order:
a__from = mark
a__length < mark
a__length1 < mark
a__length = a__length1

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

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

(12) Obligation:

Generator Equations:
gen_s:from:cons:nil:0':length:length12_0(0) ⇔ nil
gen_s:from:cons:nil:0':length:length12_0(+(x, 1)) ⇔ cons(gen_s:from:cons:nil:0':length:length12_0(x), nil)

The following defined symbols remain to be analysed:
mark, a__from

They will be analysed ascendingly in the following order:
a__from = mark

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

Proved the following rewrite lemma:
mark(gen_s:from:cons:nil:0':length:length12_0(n58_0)) → gen_s:from:cons:nil:0':length:length12_0(n58_0), rt ∈ Ω(1 + n58₀)

Induction Base:
mark(gen_s:from:cons:nil:0':length:length12_0(0)) →_R^Ω(1)
nil

Induction Step:
mark(gen_s:from:cons:nil:0':length:length12_0(+(n58_0, 1))) →_R^Ω(1)
cons(mark(gen_s:from:cons:nil:0':length:length12_0(n58_0)), nil) →_IH
cons(gen_s:from:cons:nil:0':length:length12_0(c59_0), nil)

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

(14) Complex Obligation (BEST)

(15) Obligation:

Lemmas:
mark(gen_s:from:cons:nil:0':length:length12_0(n58_0)) → gen_s:from:cons:nil:0':length:length12_0(n58_0), rt ∈ Ω(1 + n58₀)

Generator Equations:
gen_s:from:cons:nil:0':length:length12_0(0) ⇔ nil
gen_s:from:cons:nil:0':length:length12_0(+(x, 1)) ⇔ cons(gen_s:from:cons:nil:0':length:length12_0(x), nil)

The following defined symbols remain to be analysed:
a__from

They will be analysed ascendingly in the following order:
a__from = mark

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

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

(17) Obligation:

Lemmas:
mark(gen_s:from:cons:nil:0':length:length12_0(n58_0)) → gen_s:from:cons:nil:0':length:length12_0(n58_0), rt ∈ Ω(1 + n58₀)

Generator Equations:
gen_s:from:cons:nil:0':length:length12_0(0) ⇔ nil
gen_s:from:cons:nil:0':length:length12_0(+(x, 1)) ⇔ cons(gen_s:from:cons:nil:0':length:length12_0(x), nil)

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_s:from:cons:nil:0':length:length12_0(n58_0)) → gen_s:from:cons:nil:0':length:length12_0(n58_0), rt ∈ Ω(1 + n58₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Lemmas:
mark(gen_s:from:cons:nil:0':length:length12_0(n58_0)) → gen_s:from:cons:nil:0':length:length12_0(n58_0), rt ∈ Ω(1 + n58₀)

Generator Equations:
gen_s:from:cons:nil:0':length:length12_0(0) ⇔ nil
gen_s:from:cons:nil:0':length:length12_0(+(x, 1)) ⇔ cons(gen_s:from:cons:nil:0':length:length12_0(x), nil)

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_s:from:cons:nil:0':length:length12_0(n58_0)) → gen_s:from:cons:nil:0':length:length12_0(n58_0), rt ∈ Ω(1 + n58₀)

(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) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(10) Obligation:

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

(12) Obligation:

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

(14) Complex Obligation (BEST)

(15) Obligation:

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

(17) Obligation:

(18) LowerBoundsProof (EQUIVALENT transformation)

(19) BOUNDS(n^1, INF)

(20) Obligation:

(21) LowerBoundsProof (EQUIVALENT transformation)

(22) BOUNDS(n^1, INF)