Runtime Complexity (full) proof of /tmp/tmpFsquDe/polo2.xml

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

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 491 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 RewriteLemmaProof (LOWER BOUND(ID), 1373 ms)

    ↳10 BEST

        ↳11 typed CpxTrs

            ↳12 LowerBoundsProof (⇔, 0 ms)

            ↳13 BOUNDS(n^1, INF)

        ↳14 typed CpxTrs

            ↳15 LowerBoundsProof (⇔, 0 ms)

            ↳16 BOUNDS(n^1, INF)

(0) Obligation:

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

dx(X) → one
dx(a) → zero
dx(plus(ALPHA, BETA)) → plus(dx(ALPHA), dx(BETA))
dx(times(ALPHA, BETA)) → plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA)))
dx(minus(ALPHA, BETA)) → minus(dx(ALPHA), dx(BETA))
dx(neg(ALPHA)) → neg(dx(ALPHA))
dx(div(ALPHA, BETA)) → minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two))))
dx(ln(ALPHA)) → div(dx(ALPHA), ALPHA)
dx(exp(ALPHA, BETA)) → plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
dx(plus(ALPHA, BETA)) →⁺ plus(dx(ALPHA), dx(BETA))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [ALPHA / plus(ALPHA, BETA)].
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:

dx(X) → one
dx(a) → zero
dx(plus(ALPHA, BETA)) → plus(dx(ALPHA), dx(BETA))
dx(times(ALPHA, BETA)) → plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA)))
dx(minus(ALPHA, BETA)) → minus(dx(ALPHA), dx(BETA))
dx(neg(ALPHA)) → neg(dx(ALPHA))
dx(div(ALPHA, BETA)) → minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two))))
dx(ln(ALPHA)) → div(dx(ALPHA), ALPHA)
dx(exp(ALPHA, BETA)) → plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
dx(X) → one
dx(a) → zero
dx(plus(ALPHA, BETA)) → plus(dx(ALPHA), dx(BETA))
dx(times(ALPHA, BETA)) → plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA)))
dx(minus(ALPHA, BETA)) → minus(dx(ALPHA), dx(BETA))
dx(neg(ALPHA)) → neg(dx(ALPHA))
dx(div(ALPHA, BETA)) → minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two))))
dx(ln(ALPHA)) → div(dx(ALPHA), ALPHA)
dx(exp(ALPHA, BETA)) → plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))

Types:
dx :: one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln
one :: one:a:zero:plus:times:minus:neg:div:two:exp:ln
a :: one:a:zero:plus:times:minus:neg:div:two:exp:ln
zero :: one:a:zero:plus:times:minus:neg:div:two:exp:ln
plus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln
times :: one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln
minus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln
neg :: one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln
div :: one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln
exp :: one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln
two :: one:a:zero:plus:times:minus:neg:div:two:exp:ln
ln :: one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln
hole_one:a:zero:plus:times:minus:neg:div:two:exp:ln1_0 :: one:a:zero:plus:times:minus:neg:div:two:exp:ln
gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0 :: Nat → one:a:zero:plus:times:minus:neg:div:two:exp:ln

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

Heuristically decided to analyse the following defined symbols:
dx

(8) Obligation:

Generator Equations:
gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(0) ⇔ a
gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(+(x, 1)) ⇔ plus(a, gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(x))

The following defined symbols remain to be analysed:
dx

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

Proved the following rewrite lemma:
dx(gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
dx(gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(0))

Induction Step:
dx(gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(+(n4_0, 1))) →_R^Ω(1)
plus(dx(a), dx(gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(n4_0))) →_R^Ω(1)
plus(one, dx(gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(n4_0))) →_IH
plus(one, *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:
dx(gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
dx(gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)

(13) BOUNDS(n^1, INF)

(14) Obligation: