Runtime Complexity (full) proof of /tmp/tmpK0aXlv/Ex1_GL02a

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

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 291 ms)

↳2 BOUNDS(n^1, 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 NoRewriteLemmaProof (LOWER BOUND(ID), 36 ms)

    ↳10 typed CpxTrs

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

    ↳12 typed CpxTrs

    ↳13 RewriteLemmaProof (LOWER BOUND(ID), 344 ms)

    ↳14 BEST

        ↳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:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(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
eq(n__s(X), n__s(Y)) →⁺ eq(X, Y)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [X / n__s(X), Y / n__s(Y)].
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:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(s(X)))
take(0', X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0'
length(cons(X, L)) → s(n__length(activate(L)))
0' → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0'
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(s(X)))
take(0', X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0'
length(cons(X, L)) → s(n__length(activate(L)))
0' → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0'
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

Types:
eq :: n__0:n__s:n__inf:cons:nil:n__take:n__length → n__0:n__s:n__inf:cons:nil:n__take:n__length → true:false
n__0 :: n__0:n__s:n__inf:cons:nil:n__take:n__length
true :: true:false
n__s :: n__0:n__s:n__inf:cons:nil:n__take:n__length → n__0:n__s:n__inf:cons:nil:n__take:n__length
activate :: n__0:n__s:n__inf:cons:nil:n__take:n__length → n__0:n__s:n__inf:cons:nil:n__take:n__length
false :: true:false
inf :: n__0:n__s:n__inf:cons:nil:n__take:n__length → n__0:n__s:n__inf:cons:nil:n__take:n__length
cons :: n__0:n__s:n__inf:cons:nil:n__take:n__length → n__0:n__s:n__inf:cons:nil:n__take:n__length → n__0:n__s:n__inf:cons:nil:n__take:n__length
n__inf :: n__0:n__s:n__inf:cons:nil:n__take:n__length → n__0:n__s:n__inf:cons:nil:n__take:n__length
s :: n__0:n__s:n__inf:cons:nil:n__take:n__length → n__0:n__s:n__inf:cons:nil:n__take:n__length
take :: n__0:n__s:n__inf:cons:nil:n__take:n__length → n__0:n__s:n__inf:cons:nil:n__take:n__length → n__0:n__s:n__inf:cons:nil:n__take:n__length
0' :: n__0:n__s:n__inf:cons:nil:n__take:n__length
nil :: n__0:n__s:n__inf:cons:nil:n__take:n__length
n__take :: n__0:n__s:n__inf:cons:nil:n__take:n__length → n__0:n__s:n__inf:cons:nil:n__take:n__length → n__0:n__s:n__inf:cons:nil:n__take:n__length
length :: n__0:n__s:n__inf:cons:nil:n__take:n__length → n__0:n__s:n__inf:cons:nil:n__take:n__length
n__length :: n__0:n__s:n__inf:cons:nil:n__take:n__length → n__0:n__s:n__inf:cons:nil:n__take:n__length
hole_true:false1_3 :: true:false
hole_n__0:n__s:n__inf:cons:nil:n__take:n__length2_3 :: n__0:n__s:n__inf:cons:nil:n__take:n__length
gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3 :: Nat → n__0:n__s:n__inf:cons:nil:n__take:n__length

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

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

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

(8) Obligation:

Generator Equations:
gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(0) ⇔ n__0
gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(+(x, 1)) ⇔ n__s(gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(x))

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

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

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

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

(10) Obligation:

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

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

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

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

(12) Obligation:

The following defined symbols remain to be analysed:
eq

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

Proved the following rewrite lemma:
eq(gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(+(1, n122_3)), gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(n122_3)) → false, rt ∈ Ω(1 + n122₃)

Induction Base:
eq(gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(+(1, 0)), gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(0)) →_R^Ω(1)
false

Induction Step:
eq(gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(+(1, +(n122_3, 1))), gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(+(n122_3, 1))) →_R^Ω(1)
eq(activate(gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(+(1, n122_3))), activate(gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(n122_3))) →_R^Ω(1)
eq(gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(+(1, n122_3)), activate(gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(n122_3))) →_R^Ω(1)
eq(gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(+(1, n122_3)), gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(n122_3)) →_IH
false

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

(14) Complex Obligation (BEST)

(15) Obligation:

Lemmas:
eq(gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(+(1, n122_3)), gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(n122_3)) → false, rt ∈ Ω(1 + n122₃)

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq(gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(+(1, n122_3)), gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(n122_3)) → false, rt ∈ Ω(1 + n122₃)

(17) BOUNDS(n^1, INF)

(18) Obligation:

Lemmas:
eq(gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(+(1, n122_3)), gen_n__0:n__s:n__inf:cons:nil:n__take:n__length3_3(n122_3)) → false, rt ∈ Ω(1 + n122₃)

No more defined symbols left to analyse.

(0) Obligation:

(1) DecreasingLoopProof (EQUIVALENT transformation)

(2) BOUNDS(n^1, 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) LowerBoundsProof (EQUIVALENT transformation)

(17) BOUNDS(n^1, INF)

(18) Obligation:

(19) LowerBoundsProof (EQUIVALENT transformation)

(20) BOUNDS(n^1, INF)