Runtime Complexity (full) proof of /tmp/tmp2u6j0B/Ex3_3_25_Bor03

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

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 692 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), 428 ms)

    ↳10 BEST

        ↳11 typed CpxTrs

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

a__app(nil, YS) → mark(YS)
a__app(cons(X, XS), YS) → cons(mark(X), app(XS, YS))
a__from(X) → cons(mark(X), from(s(X)))
a__zWadr(nil, YS) → nil
a__zWadr(XS, nil) → nil
a__zWadr(cons(X, XS), cons(Y, YS)) → cons(a__app(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
a__prefix(L) → cons(nil, zWadr(L, prefix(L)))
mark(app(X1, X2)) → a__app(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(zWadr(X1, X2)) → a__zWadr(mark(X1), mark(X2))
mark(prefix(X)) → a__prefix(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__app(X1, X2) → app(X1, X2)
a__from(X) → from(X)
a__zWadr(X1, X2) → zWadr(X1, X2)
a__prefix(X) → prefix(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(app(from(X36060_0), X2)) →⁺ a__app(cons(mark(mark(X36060_0)), from(s(mark(X36060_0)))), mark(X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
The pumping substitution is [X36060_0 / app(from(X36060_0), X2)].
The result substitution is [ ].

The rewrite sequence
mark(app(from(X36060_0), X2)) →⁺ a__app(cons(mark(mark(X36060_0)), from(s(mark(X36060_0)))), mark(X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0,0].
The pumping substitution is [X36060_0 / app(from(X36060_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:

a__app(nil, YS) → mark(YS)
a__app(cons(X, XS), YS) → cons(mark(X), app(XS, YS))
a__from(X) → cons(mark(X), from(s(X)))
a__zWadr(nil, YS) → nil
a__zWadr(XS, nil) → nil
a__zWadr(cons(X, XS), cons(Y, YS)) → cons(a__app(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
a__prefix(L) → cons(nil, zWadr(L, prefix(L)))
mark(app(X1, X2)) → a__app(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(zWadr(X1, X2)) → a__zWadr(mark(X1), mark(X2))
mark(prefix(X)) → a__prefix(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__app(X1, X2) → app(X1, X2)
a__from(X) → from(X)
a__zWadr(X1, X2) → zWadr(X1, X2)
a__prefix(X) → prefix(X)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
a__app(nil, YS) → mark(YS)
a__app(cons(X, XS), YS) → cons(mark(X), app(XS, YS))
a__from(X) → cons(mark(X), from(s(X)))
a__zWadr(nil, YS) → nil
a__zWadr(XS, nil) → nil
a__zWadr(cons(X, XS), cons(Y, YS)) → cons(a__app(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
a__prefix(L) → cons(nil, zWadr(L, prefix(L)))
mark(app(X1, X2)) → a__app(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(zWadr(X1, X2)) → a__zWadr(mark(X1), mark(X2))
mark(prefix(X)) → a__prefix(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__app(X1, X2) → app(X1, X2)
a__from(X) → from(X)
a__zWadr(X1, X2) → zWadr(X1, X2)
a__prefix(X) → prefix(X)

Types:
a__app :: nil:cons:app:s:from:zWadr:prefix → nil:cons:app:s:from:zWadr:prefix → nil:cons:app:s:from:zWadr:prefix
nil :: nil:cons:app:s:from:zWadr:prefix
mark :: nil:cons:app:s:from:zWadr:prefix → nil:cons:app:s:from:zWadr:prefix
cons :: nil:cons:app:s:from:zWadr:prefix → nil:cons:app:s:from:zWadr:prefix → nil:cons:app:s:from:zWadr:prefix
app :: nil:cons:app:s:from:zWadr:prefix → nil:cons:app:s:from:zWadr:prefix → nil:cons:app:s:from:zWadr:prefix
a__from :: nil:cons:app:s:from:zWadr:prefix → nil:cons:app:s:from:zWadr:prefix
from :: nil:cons:app:s:from:zWadr:prefix → nil:cons:app:s:from:zWadr:prefix
s :: nil:cons:app:s:from:zWadr:prefix → nil:cons:app:s:from:zWadr:prefix
a__zWadr :: nil:cons:app:s:from:zWadr:prefix → nil:cons:app:s:from:zWadr:prefix → nil:cons:app:s:from:zWadr:prefix
zWadr :: nil:cons:app:s:from:zWadr:prefix → nil:cons:app:s:from:zWadr:prefix → nil:cons:app:s:from:zWadr:prefix
a__prefix :: nil:cons:app:s:from:zWadr:prefix → nil:cons:app:s:from:zWadr:prefix
prefix :: nil:cons:app:s:from:zWadr:prefix → nil:cons:app:s:from:zWadr:prefix
hole_nil:cons:app:s:from:zWadr:prefix1_0 :: nil:cons:app:s:from:zWadr:prefix
gen_nil:cons:app:s:from:zWadr:prefix2_0 :: Nat → nil:cons:app:s:from:zWadr:prefix

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

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

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

(8) Obligation:

Generator Equations:
gen_nil:cons:app:s:from:zWadr:prefix2_0(0) ⇔ nil
gen_nil:cons:app:s:from:zWadr:prefix2_0(+(x, 1)) ⇔ cons(gen_nil:cons:app:s:from:zWadr:prefix2_0(x), nil)

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

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

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

Proved the following rewrite lemma:
mark(gen_nil:cons:app:s:from:zWadr:prefix2_0(n4_0)) → gen_nil:cons:app:s:from:zWadr:prefix2_0(n4_0), rt ∈ Ω(1 + n4₀)

Induction Base:
mark(gen_nil:cons:app:s:from:zWadr:prefix2_0(0)) →_R^Ω(1)
nil

Induction Step:
mark(gen_nil:cons:app:s:from:zWadr:prefix2_0(+(n4_0, 1))) →_R^Ω(1)
cons(mark(gen_nil:cons:app:s:from:zWadr:prefix2_0(n4_0)), nil) →_IH
cons(gen_nil:cons:app:s:from:zWadr:prefix2_0(c5_0), nil)

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_nil:cons:app:s:from:zWadr:prefix2_0(n4_0)) → gen_nil:cons:app:s:from:zWadr:prefix2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_nil:cons:app:s:from:zWadr:prefix2_0(0) ⇔ nil
gen_nil:cons:app:s:from:zWadr:prefix2_0(+(x, 1)) ⇔ cons(gen_nil:cons:app:s:from:zWadr:prefix2_0(x), nil)

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

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

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

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

(13) Obligation:

Lemmas:
mark(gen_nil:cons:app:s:from:zWadr:prefix2_0(n4_0)) → gen_nil:cons:app:s:from:zWadr:prefix2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_nil:cons:app:s:from:zWadr:prefix2_0(0) ⇔ nil
gen_nil:cons:app:s:from:zWadr:prefix2_0(+(x, 1)) ⇔ cons(gen_nil:cons:app:s:from:zWadr:prefix2_0(x), nil)

The following defined symbols remain to be analysed:
a__from

They will be analysed ascendingly in the following order:
a__app = mark
a__app = 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_nil:cons:app:s:from:zWadr:prefix2_0(n4_0)) → gen_nil:cons:app:s:from:zWadr:prefix2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_nil:cons:app:s:from:zWadr:prefix2_0(0) ⇔ nil
gen_nil:cons:app:s:from:zWadr:prefix2_0(+(x, 1)) ⇔ cons(gen_nil:cons:app:s:from:zWadr:prefix2_0(x), nil)

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_nil:cons:app:s:from:zWadr:prefix2_0(n4_0)) → gen_nil:cons:app:s:from:zWadr:prefix2_0(n4_0), rt ∈ Ω(1 + n4₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

Lemmas:
mark(gen_nil:cons:app:s:from:zWadr:prefix2_0(n4_0)) → gen_nil:cons:app:s:from:zWadr:prefix2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_nil:cons:app:s:from:zWadr:prefix2_0(0) ⇔ nil
gen_nil:cons:app:s:from:zWadr:prefix2_0(+(x, 1)) ⇔ cons(gen_nil:cons:app:s:from:zWadr:prefix2_0(x), nil)

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_nil:cons:app:s:from:zWadr:prefix2_0(n4_0)) → gen_nil:cons:app:s:from:zWadr:prefix2_0(n4_0), rt ∈ Ω(1 + 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)