Runtime Complexity (full) proof of /tmp/tmpmSYFZz/MYNAT_nosorts

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

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 392 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 NoRewriteLemmaProof (LOWER BOUND(ID), 426 ms)

    ↳10 typed CpxTrs

    ↳11 RewriteLemmaProof (LOWER BOUND(ID), 94 ms)

    ↳12 BEST

        ↳13 typed CpxTrs

            ↳14 RewriteLemmaProof (LOWER BOUND(ID), 863 ms)

            ↳15 BEST

                ↳16 typed CpxTrs

                    ↳17 RewriteLemmaProof (LOWER BOUND(ID), 268 ms)

                    ↳18 BEST

                        ↳19 typed CpxTrs

                            ↳20 RewriteLemmaProof (LOWER BOUND(ID), 43 ms)

                            ↳21 BEST

                                ↳22 typed CpxTrs

                                    ↳23 RewriteLemmaProof (LOWER BOUND(ID), 497 ms)

                                    ↳24 BEST

                                        ↳25 typed CpxTrs

                                            ↳26 LowerBoundsProof (⇔, 0 ms)

                                            ↳27 BOUNDS(n^2, INF)

                                        ↳28 typed CpxTrs

                                            ↳29 LowerBoundsProof (⇔, 0 ms)

                                            ↳30 BOUNDS(n^2, INF)

                                ↳31 typed CpxTrs

                                    ↳32 LowerBoundsProof (⇔, 0 ms)

                                    ↳33 BOUNDS(n^2, INF)

                        ↳34 typed CpxTrs

                            ↳35 LowerBoundsProof (⇔, 0 ms)

                            ↳36 BOUNDS(n^2, INF)

                ↳37 typed CpxTrs

                    ↳38 LowerBoundsProof (⇔, 0 ms)

                    ↳39 BOUNDS(n^2, INF)

        ↳40 typed CpxTrs

            ↳41 LowerBoundsProof (⇔, 0 ms)

            ↳42 BOUNDS(n^1, INF)

(0) Obligation:

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

a__and(tt, X) → mark(X)
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
a__x(N, 0) → 0
a__x(N, s(M)) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__x(X1, X2) → x(X1, X2)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
mark(and(x(X110212_3, s(X10414_3)), X2)) →⁺ a__and(a__plus(a__x(mark(mark(X110212_3)), mark(mark(X10414_3))), mark(mark(X110212_3))), X2)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0].
The pumping substitution is [X110212_3 / and(x(X110212_3, s(X10414_3)), X2)].
The result substitution is [ ].

The rewrite sequence
mark(and(x(X110212_3, s(X10414_3)), X2)) →⁺ a__and(a__plus(a__x(mark(mark(X110212_3)), mark(mark(X10414_3))), mark(mark(X110212_3))), X2)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0].
The pumping substitution is [X110212_3 / and(x(X110212_3, s(X10414_3)), 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__and(tt, X) → mark(X)
a__plus(N, 0') → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
a__x(N, 0') → 0'
a__x(N, s(M)) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__x(X1, X2) → x(X1, X2)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
a__and(tt, X) → mark(X)
a__plus(N, 0') → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
a__x(N, 0') → 0'
a__x(N, s(M)) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__x(X1, X2) → x(X1, X2)

Types:
a__and :: tt:0':s:and:plus:x → tt:0':s:and:plus:x → tt:0':s:and:plus:x
tt :: tt:0':s:and:plus:x
mark :: tt:0':s:and:plus:x → tt:0':s:and:plus:x
a__plus :: tt:0':s:and:plus:x → tt:0':s:and:plus:x → tt:0':s:and:plus:x
0' :: tt:0':s:and:plus:x
s :: tt:0':s:and:plus:x → tt:0':s:and:plus:x
a__x :: tt:0':s:and:plus:x → tt:0':s:and:plus:x → tt:0':s:and:plus:x
and :: tt:0':s:and:plus:x → tt:0':s:and:plus:x → tt:0':s:and:plus:x
plus :: tt:0':s:and:plus:x → tt:0':s:and:plus:x → tt:0':s:and:plus:x
x :: tt:0':s:and:plus:x → tt:0':s:and:plus:x → tt:0':s:and:plus:x
hole_tt:0':s:and:plus:x1_0 :: tt:0':s:and:plus:x
gen_tt:0':s:and:plus:x2_0 :: Nat → tt:0':s:and:plus:x

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

Heuristically decided to analyse the following defined symbols:
mark, a__plus, a__x

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

(8) Obligation:

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

The following defined symbols remain to be analysed:
a__plus, mark, a__x

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

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

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

(10) Obligation:

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_tt:0':s:and:plus:x2_0(n11447_0)) → gen_tt:0':s:and:plus:x2_0(n11447_0), rt ∈ Ω(1 + n11447₀)

Induction Base:
mark(gen_tt:0':s:and:plus:x2_0(0)) →_R^Ω(1)
tt

Induction Step:
mark(gen_tt:0':s:and:plus:x2_0(+(n11447_0, 1))) →_R^Ω(1)
s(mark(gen_tt:0':s:and:plus:x2_0(n11447_0))) →_IH
s(gen_tt:0':s:and:plus:x2_0(c11448_0))

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

(12) Complex Obligation (BEST)

(13) Obligation:

Lemmas:
mark(gen_tt:0':s:and:plus:x2_0(n11447_0)) → gen_tt:0':s:and:plus:x2_0(n11447_0), rt ∈ Ω(1 + n11447₀)

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

The following defined symbols remain to be analysed:
a__x, a__plus

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

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n12552_0))) → *3_0, rt ∈ Ω(a·n12552₀ + n12552₀ + n12552₀²)

Induction Base:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, 0)))

Induction Step:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, +(n12552_0, 1)))) →_R^Ω(1)
a__plus(a__x(mark(gen_tt:0':s:and:plus:x2_0(a)), mark(gen_tt:0':s:and:plus:x2_0(+(1, n12552_0)))), mark(gen_tt:0':s:and:plus:x2_0(a))) →_L^{Ω(1 + a)}
a__plus(a__x(gen_tt:0':s:and:plus:x2_0(a), mark(gen_tt:0':s:and:plus:x2_0(+(1, n12552_0)))), mark(gen_tt:0':s:and:plus:x2_0(a))) →_L^{Ω(2 + n12552₀)}
a__plus(a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n12552_0))), mark(gen_tt:0':s:and:plus:x2_0(a))) →_IH
a__plus(*3_0, mark(gen_tt:0':s:and:plus:x2_0(a))) →_L^{Ω(1 + a)}
a__plus(*3_0, gen_tt:0':s:and:plus:x2_0(a))

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(15) Complex Obligation (BEST)

(16) Obligation:

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

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

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

(17) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__plus(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n15954_0))) → *3_0, rt ∈ Ω(a·n15954₀ + n15954₀ + n15954₀²)

Induction Base:
a__plus(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, 0)))

Induction Step:
a__plus(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, +(n15954_0, 1)))) →_R^Ω(1)
s(a__plus(mark(gen_tt:0':s:and:plus:x2_0(a)), mark(gen_tt:0':s:and:plus:x2_0(+(1, n15954_0))))) →_L^{Ω(1 + a)}
s(a__plus(gen_tt:0':s:and:plus:x2_0(a), mark(gen_tt:0':s:and:plus:x2_0(+(1, n15954_0))))) →_L^{Ω(2 + n15954₀)}
s(a__plus(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n15954_0)))) →_IH
s(*3_0)

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(18) Complex Obligation (BEST)

(19) Obligation:

Lemmas:
mark(gen_tt:0':s:and:plus:x2_0(n11447_0)) → gen_tt:0':s:and:plus:x2_0(n11447_0), rt ∈ Ω(1 + n11447₀)
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n12552_0))) → *3_0, rt ∈ Ω(a·n12552₀ + n12552₀ + n12552₀²)
a__plus(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n15954_0))) → *3_0, rt ∈ Ω(a·n15954₀ + n15954₀ + n15954₀²)

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

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

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

(20) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_tt:0':s:and:plus:x2_0(n17693_0)) → gen_tt:0':s:and:plus:x2_0(n17693_0), rt ∈ Ω(1 + n17693₀)

Induction Base:
mark(gen_tt:0':s:and:plus:x2_0(0)) →_R^Ω(1)
tt

Induction Step:
mark(gen_tt:0':s:and:plus:x2_0(+(n17693_0, 1))) →_R^Ω(1)
s(mark(gen_tt:0':s:and:plus:x2_0(n17693_0))) →_IH
s(gen_tt:0':s:and:plus:x2_0(c17694_0))

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

(21) Complex Obligation (BEST)

(22) Obligation:

Lemmas:
mark(gen_tt:0':s:and:plus:x2_0(n17693_0)) → gen_tt:0':s:and:plus:x2_0(n17693_0), rt ∈ Ω(1 + n17693₀)
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n12552_0))) → *3_0, rt ∈ Ω(a·n12552₀ + n12552₀ + n12552₀²)
a__plus(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n15954_0))) → *3_0, rt ∈ Ω(a·n15954₀ + n15954₀ + n15954₀²)

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

The following defined symbols remain to be analysed:
a__x

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

(23) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n19004_0))) → *3_0, rt ∈ Ω(a·n19004₀ + n19004₀ + n19004₀²)

Induction Base:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, 0)))

Induction Step:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, +(n19004_0, 1)))) →_R^Ω(1)
a__plus(a__x(mark(gen_tt:0':s:and:plus:x2_0(a)), mark(gen_tt:0':s:and:plus:x2_0(+(1, n19004_0)))), mark(gen_tt:0':s:and:plus:x2_0(a))) →_L^{Ω(1 + a)}
a__plus(a__x(gen_tt:0':s:and:plus:x2_0(a), mark(gen_tt:0':s:and:plus:x2_0(+(1, n19004_0)))), mark(gen_tt:0':s:and:plus:x2_0(a))) →_L^{Ω(2 + n19004₀)}
a__plus(a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n19004_0))), mark(gen_tt:0':s:and:plus:x2_0(a))) →_IH
a__plus(*3_0, mark(gen_tt:0':s:and:plus:x2_0(a))) →_L^{Ω(1 + a)}
a__plus(*3_0, gen_tt:0':s:and:plus:x2_0(a))

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(24) Complex Obligation (BEST)

(25) Obligation:

Lemmas:
mark(gen_tt:0':s:and:plus:x2_0(n17693_0)) → gen_tt:0':s:and:plus:x2_0(n17693_0), rt ∈ Ω(1 + n17693₀)
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n19004_0))) → *3_0, rt ∈ Ω(a·n19004₀ + n19004₀ + n19004₀²)
a__plus(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n15954_0))) → *3_0, rt ∈ Ω(a·n15954₀ + n15954₀ + n15954₀²)

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

No more defined symbols left to analyse.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n19004_0))) → *3_0, rt ∈ Ω(a·n19004₀ + n19004₀ + n19004₀²)

(27) BOUNDS(n^2, INF)

(28) Obligation:

Lemmas:
mark(gen_tt:0':s:and:plus:x2_0(n17693_0)) → gen_tt:0':s:and:plus:x2_0(n17693_0), rt ∈ Ω(1 + n17693₀)
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n19004_0))) → *3_0, rt ∈ Ω(a·n19004₀ + n19004₀ + n19004₀²)
a__plus(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n15954_0))) → *3_0, rt ∈ Ω(a·n15954₀ + n15954₀ + n15954₀²)

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n19004_0))) → *3_0, rt ∈ Ω(a·n19004₀ + n19004₀ + n19004₀²)

(30) BOUNDS(n^2, INF)

(31) Obligation:

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

No more defined symbols left to analyse.

(32) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n12552_0))) → *3_0, rt ∈ Ω(a·n12552₀ + n12552₀ + n12552₀²)

(33) BOUNDS(n^2, INF)

(34) Obligation:

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

No more defined symbols left to analyse.

(35) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n12552_0))) → *3_0, rt ∈ Ω(a·n12552₀ + n12552₀ + n12552₀²)

(36) BOUNDS(n^2, INF)

(37) Obligation:

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

No more defined symbols left to analyse.

(38) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n12552_0))) → *3_0, rt ∈ Ω(a·n12552₀ + n12552₀ + n12552₀²)

(39) BOUNDS(n^2, INF)

(40) Obligation:

Lemmas:
mark(gen_tt:0':s:and:plus:x2_0(n11447_0)) → gen_tt:0':s:and:plus:x2_0(n11447_0), rt ∈ Ω(1 + n11447₀)

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

No more defined symbols left to analyse.

(41) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_tt:0':s:and:plus:x2_0(n11447_0)) → gen_tt:0':s:and:plus:x2_0(n11447_0), rt ∈ Ω(1 + n11447₀)

(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) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(12) Complex Obligation (BEST)

(13) Obligation:

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(15) Complex Obligation (BEST)

(16) Obligation:

(17) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(18) Complex Obligation (BEST)

(19) Obligation:

(20) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(21) Complex Obligation (BEST)

(22) Obligation:

(23) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(24) Complex Obligation (BEST)

(25) Obligation:

(26) LowerBoundsProof (EQUIVALENT transformation)

(27) BOUNDS(n^2, INF)

(28) Obligation:

(29) LowerBoundsProof (EQUIVALENT transformation)

(30) BOUNDS(n^2, INF)

(31) Obligation:

(32) LowerBoundsProof (EQUIVALENT transformation)

(33) BOUNDS(n^2, INF)

(34) Obligation:

(35) LowerBoundsProof (EQUIVALENT transformation)

(36) BOUNDS(n^2, INF)

(37) Obligation:

(38) LowerBoundsProof (EQUIVALENT transformation)

(39) BOUNDS(n^2, INF)

(40) Obligation:

(41) LowerBoundsProof (EQUIVALENT transformation)

(42) BOUNDS(n^1, INF)