Runtime Complexity (full) proof of /tmp/tmpGcloFN/Ex6_9_Luc02c

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

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 290 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), 812 ms)

    ↳10 BEST

        ↳11 typed CpxTrs

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

            ↳13 typed CpxTrs

            ↳14 NoRewriteLemmaProof (LOWER BOUND(ID), 491 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:

a__2nd(cons1(X, cons(Y, Z))) → mark(Y)
a__2nd(cons(X, X1)) → a__2nd(cons1(mark(X), mark(X1)))
a__from(X) → cons(mark(X), from(s(X)))
mark(2nd(X)) → a__2nd(mark(X))
mark(from(X)) → a__from(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
a__2nd(X) → 2nd(X)
a__from(X) → from(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(2nd(from(X7523_0))) →⁺ a__2nd(cons(mark(mark(X7523_0)), from(s(mark(X7523_0)))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
The pumping substitution is [X7523_0 / 2nd(from(X7523_0))].
The result substitution is [ ].

The rewrite sequence
mark(2nd(from(X7523_0))) →⁺ a__2nd(cons(mark(mark(X7523_0)), from(s(mark(X7523_0)))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0,0].
The pumping substitution is [X7523_0 / 2nd(from(X7523_0))].
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__2nd(cons1(X, cons(Y, Z))) → mark(Y)
a__2nd(cons(X, X1)) → a__2nd(cons1(mark(X), mark(X1)))
a__from(X) → cons(mark(X), from(s(X)))
mark(2nd(X)) → a__2nd(mark(X))
mark(from(X)) → a__from(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
a__2nd(X) → 2nd(X)
a__from(X) → from(X)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
a__2nd(cons1(X, cons(Y, Z))) → mark(Y)
a__2nd(cons(X, X1)) → a__2nd(cons1(mark(X), mark(X1)))
a__from(X) → cons(mark(X), from(s(X)))
mark(2nd(X)) → a__2nd(mark(X))
mark(from(X)) → a__from(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
a__2nd(X) → 2nd(X)
a__from(X) → from(X)

Types:
a__2nd :: cons:cons1:s:from:2nd → cons:cons1:s:from:2nd
cons1 :: cons:cons1:s:from:2nd → cons:cons1:s:from:2nd → cons:cons1:s:from:2nd
cons :: cons:cons1:s:from:2nd → cons:cons1:s:from:2nd → cons:cons1:s:from:2nd
mark :: cons:cons1:s:from:2nd → cons:cons1:s:from:2nd
a__from :: cons:cons1:s:from:2nd → cons:cons1:s:from:2nd
from :: cons:cons1:s:from:2nd → cons:cons1:s:from:2nd
s :: cons:cons1:s:from:2nd → cons:cons1:s:from:2nd
2nd :: cons:cons1:s:from:2nd → cons:cons1:s:from:2nd
hole_cons:cons1:s:from:2nd1_0 :: cons:cons1:s:from:2nd
gen_cons:cons1:s:from:2nd2_0 :: Nat → cons:cons1:s:from:2nd

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

Heuristically decided to analyse the following defined symbols:
a__2nd, mark, a__from

They will be analysed ascendingly in the following order:
a__2nd = mark
a__2nd = a__from
mark = a__from

(8) Obligation:

Generator Equations:
gen_cons:cons1:s:from:2nd2_0(0) ⇔ hole_cons:cons1:s:from:2nd1_0
gen_cons:cons1:s:from:2nd2_0(+(x, 1)) ⇔ cons1(hole_cons:cons1:s:from:2nd1_0, gen_cons:cons1:s:from:2nd2_0(x))

The following defined symbols remain to be analysed:
mark, a__2nd, a__from

They will be analysed ascendingly in the following order:
a__2nd = mark
a__2nd = a__from
mark = a__from

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

Proved the following rewrite lemma:
mark(gen_cons:cons1:s:from:2nd2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
mark(gen_cons:cons1:s:from:2nd2_0(+(1, 0)))

Induction Step:
mark(gen_cons:cons1:s:from:2nd2_0(+(1, +(n4_0, 1)))) →_R^Ω(1)
cons1(mark(hole_cons:cons1:s:from:2nd1_0), mark(gen_cons:cons1:s:from:2nd2_0(+(1, n4_0)))) →_IH
cons1(mark(hole_cons:cons1:s:from:2nd1_0), *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:
mark(gen_cons:cons1:s:from:2nd2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_cons:cons1:s:from:2nd2_0(0) ⇔ hole_cons:cons1:s:from:2nd1_0
gen_cons:cons1:s:from:2nd2_0(+(x, 1)) ⇔ cons1(hole_cons:cons1:s:from:2nd1_0, gen_cons:cons1:s:from:2nd2_0(x))

The following defined symbols remain to be analysed:
a__2nd, a__from

They will be analysed ascendingly in the following order:
a__2nd = mark
a__2nd = a__from
mark = a__from

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

Could not prove a rewrite lemma for the defined symbol a__2nd.

(13) Obligation:

Lemmas:
mark(gen_cons:cons1:s:from:2nd2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_cons:cons1:s:from:2nd2_0(0) ⇔ hole_cons:cons1:s:from:2nd1_0
gen_cons:cons1:s:from:2nd2_0(+(x, 1)) ⇔ cons1(hole_cons:cons1:s:from:2nd1_0, gen_cons:cons1:s:from:2nd2_0(x))

The following defined symbols remain to be analysed:
a__from

They will be analysed ascendingly in the following order:
a__2nd = mark
a__2nd = a__from
mark = a__from

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

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

(15) Obligation:

Lemmas:
mark(gen_cons:cons1:s:from:2nd2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_cons:cons1:s:from:2nd2_0(0) ⇔ hole_cons:cons1:s:from:2nd1_0
gen_cons:cons1:s:from:2nd2_0(+(x, 1)) ⇔ cons1(hole_cons:cons1:s:from:2nd1_0, gen_cons:cons1:s:from:2nd2_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_cons:cons1:s:from:2nd2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

Lemmas:
mark(gen_cons:cons1:s:from:2nd2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_cons:cons1:s:from:2nd2_0(0) ⇔ hole_cons:cons1:s:from:2nd1_0
gen_cons:cons1:s:from:2nd2_0(+(x, 1)) ⇔ cons1(hole_cons:cons1:s:from:2nd1_0, gen_cons:cons1:s:from:2nd2_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_cons:cons1:s:from:2nd2_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)