Runtime Complexity (innermost) proof of /tmp/tmpf7wIak/PEANO_nosorts_noand

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

0 CpxTRS

↳1 RenamingProof (⇔, 0 ms)

↳2 CpxRelTRS

↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)

↳4 typed CpxTrs

↳5 OrderProof (LOWER BOUND(ID), 0 ms)

↳6 typed CpxTrs

↳7 NoRewriteLemmaProof (LOWER BOUND(ID), 401 ms)

↳8 typed CpxTrs

↳9 NoRewriteLemmaProof (LOWER BOUND(ID), 91 ms)

↳10 typed CpxTrs

↳11 RewriteLemmaProof (LOWER BOUND(ID), 57 ms)

↳12 BEST

    ↳13 typed CpxTrs

        ↳14 RewriteLemmaProof (LOWER BOUND(ID), 200 ms)

        ↳15 BEST

            ↳16 typed CpxTrs

                ↳17 RewriteLemmaProof (LOWER BOUND(ID), 218 ms)

                ↳18 BEST

                    ↳19 typed CpxTrs

                        ↳20 RewriteLemmaProof (LOWER BOUND(ID), 190 ms)

                        ↳21 BEST

                            ↳22 typed CpxTrs

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

                                ↳24 BEST

                                    ↳25 typed CpxTrs

                                        ↳26 RewriteLemmaProof (LOWER BOUND(ID), 172 ms)

                                        ↳27 BEST

                                            ↳28 typed CpxTrs

                                                ↳29 RewriteLemmaProof (LOWER BOUND(ID), 91 ms)

                                                ↳30 BEST

                                                    ↳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^2, INF)

                            ↳43 typed CpxTrs

                                ↳44 LowerBoundsProof (⇔, 0 ms)

                                ↳45 BOUNDS(n^2, INF)

                    ↳46 typed CpxTrs

                        ↳47 LowerBoundsProof (⇔, 0 ms)

                        ↳48 BOUNDS(n^2, INF)

            ↳49 typed CpxTrs

                ↳50 LowerBoundsProof (⇔, 0 ms)

                ↳51 BOUNDS(n^2, INF)

    ↳52 typed CpxTrs

        ↳53 LowerBoundsProof (⇔, 0 ms)

        ↳54 BOUNDS(n^1, INF)

(0) Obligation:

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

a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__plus(N, 0') → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0') → 0'
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__plus(N, 0') → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0') → 0'
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

Types:
a__U11 :: tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus
tt :: tt:s:0':U11:U12:plus
a__U12 :: tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus
s :: tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus
a__plus :: tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus
mark :: tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus
0' :: tt:s:0':U11:U12:plus
U11 :: tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus
U12 :: tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus
plus :: tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus
hole_tt:s:0':U11:U12:plus1_0 :: tt:s:0':U11:U12:plus
gen_tt:s:0':U11:U12:plus2_0 :: Nat → tt:s:0':U11:U12:plus

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
a__U11, a__U12, a__plus, mark

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

(6) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

The following defined symbols remain to be analysed:
a__U12, a__U11, a__plus, mark

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

(7) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__U12.

(8) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

The following defined symbols remain to be analysed:
a__plus, a__U11, mark

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

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

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

(10) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

The following defined symbols remain to be analysed:
mark, a__U11

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

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

Proved the following rewrite lemma:
mark(gen_tt:s:0':U11:U12:plus2_0(n7802_0)) → gen_tt:s:0':U11:U12:plus2_0(n7802_0), rt ∈ Ω(1 + n7802₀)

Induction Base:
mark(gen_tt:s:0':U11:U12:plus2_0(0)) →_R^Ω(1)
tt

Induction Step:
mark(gen_tt:s:0':U11:U12:plus2_0(+(n7802_0, 1))) →_R^Ω(1)
s(mark(gen_tt:s:0':U11:U12:plus2_0(n7802_0))) →_IH
s(gen_tt:s:0':U11:U12:plus2_0(c7803_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:s:0':U11:U12:plus2_0(n7802_0)) → gen_tt:s:0':U11:U12:plus2_0(n7802_0), rt ∈ Ω(1 + n7802₀)

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

The following defined symbols remain to be analysed:
a__U11, a__U12, a__plus

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

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

Proved the following rewrite lemma:
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n8550_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n8550₀ + n8550₀ + n8550₀²)

Induction Base:
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(c))

Induction Step:
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(+(n8550_0, 1)), gen_tt:s:0':U11:U12:plus2_0(c)) →_R^Ω(1)
a__U12(tt, gen_tt:s:0':U11:U12:plus2_0(+(n8550_0, 1)), gen_tt:s:0':U11:U12:plus2_0(c)) →_R^Ω(1)
s(a__plus(mark(gen_tt:s:0':U11:U12:plus2_0(c)), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n8550_0))))) →_L^{Ω(1 + c)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n8550_0))))) →_L^{Ω(2 + n8550₀)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), gen_tt:s:0':U11:U12:plus2_0(+(1, n8550_0)))) →_R^Ω(1)
s(a__U11(tt, gen_tt:s:0':U11:U12:plus2_0(n8550_0), gen_tt:s:0':U11:U12:plus2_0(c))) →_IH
s(*3_0)

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:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

The following defined symbols remain to be analysed:
a__U12, a__plus, mark

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

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

Proved the following rewrite lemma:
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n11303_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n11303₀ + n11303₀ + n11303₀²)

Induction Base:
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(c))

Induction Step:
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(+(n11303_0, 1)), gen_tt:s:0':U11:U12:plus2_0(c)) →_R^Ω(1)
s(a__plus(mark(gen_tt:s:0':U11:U12:plus2_0(c)), mark(gen_tt:s:0':U11:U12:plus2_0(+(n11303_0, 1))))) →_L^{Ω(1 + c)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n11303_0))))) →_L^{Ω(2 + n11303₀)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), gen_tt:s:0':U11:U12:plus2_0(+(1, n11303_0)))) →_R^Ω(1)
s(a__U11(tt, gen_tt:s:0':U11:U12:plus2_0(n11303_0), gen_tt:s:0':U11:U12:plus2_0(c))) →_R^Ω(1)
s(a__U12(tt, gen_tt:s:0':U11:U12:plus2_0(n11303_0), gen_tt:s:0':U11:U12:plus2_0(c))) →_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:s:0':U11:U12:plus2_0(n7802_0)) → gen_tt:s:0':U11:U12:plus2_0(n7802_0), rt ∈ Ω(1 + n7802₀)
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n8550_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n8550₀ + n8550₀ + n8550₀²)
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n11303_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n11303₀ + n11303₀ + n11303₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

The following defined symbols remain to be analysed:
a__plus, a__U11, mark

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

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

Proved the following rewrite lemma:
a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, n14266_0))) → *3_0, rt ∈ Ω(a·n14266₀ + n14266₀ + n14266₀²)

Induction Base:
a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, 0)))

Induction Step:
a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, +(n14266_0, 1)))) →_R^Ω(1)
a__U11(tt, gen_tt:s:0':U11:U12:plus2_0(+(1, n14266_0)), gen_tt:s:0':U11:U12:plus2_0(a)) →_R^Ω(1)
a__U12(tt, gen_tt:s:0':U11:U12:plus2_0(+(1, n14266_0)), gen_tt:s:0':U11:U12:plus2_0(a)) →_R^Ω(1)
s(a__plus(mark(gen_tt:s:0':U11:U12:plus2_0(a)), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n14266_0))))) →_L^{Ω(1 + a)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(a), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n14266_0))))) →_L^{Ω(2 + n14266₀)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, n14266_0)))) →_IH
s(*3_0)

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

(21) Complex Obligation (BEST)

(22) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

The following defined symbols remain to be analysed:
mark, a__U11, a__U12

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

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

Proved the following rewrite lemma:
mark(gen_tt:s:0':U11:U12:plus2_0(n17964_0)) → gen_tt:s:0':U11:U12:plus2_0(n17964_0), rt ∈ Ω(1 + n17964₀)

Induction Base:
mark(gen_tt:s:0':U11:U12:plus2_0(0)) →_R^Ω(1)
tt

Induction Step:
mark(gen_tt:s:0':U11:U12:plus2_0(+(n17964_0, 1))) →_R^Ω(1)
s(mark(gen_tt:s:0':U11:U12:plus2_0(n17964_0))) →_IH
s(gen_tt:s:0':U11:U12:plus2_0(c17965_0))

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

(24) Complex Obligation (BEST)

(25) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus2_0(n17964_0)) → gen_tt:s:0':U11:U12:plus2_0(n17964_0), rt ∈ Ω(1 + n17964₀)
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n8550_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n8550₀ + n8550₀ + n8550₀²)
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n11303_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n11303₀ + n11303₀ + n11303₀²)
a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, n14266_0))) → *3_0, rt ∈ Ω(a·n14266₀ + n14266₀ + n14266₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

The following defined symbols remain to be analysed:
a__U11, a__U12

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

(26) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n19021_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n19021₀ + n19021₀ + n19021₀²)

Induction Base:
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(c))

Induction Step:
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(+(n19021_0, 1)), gen_tt:s:0':U11:U12:plus2_0(c)) →_R^Ω(1)
a__U12(tt, gen_tt:s:0':U11:U12:plus2_0(+(n19021_0, 1)), gen_tt:s:0':U11:U12:plus2_0(c)) →_R^Ω(1)
s(a__plus(mark(gen_tt:s:0':U11:U12:plus2_0(c)), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n19021_0))))) →_L^{Ω(1 + c)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n19021_0))))) →_L^{Ω(2 + n19021₀)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), gen_tt:s:0':U11:U12:plus2_0(+(1, n19021_0)))) →_R^Ω(1)
s(a__U11(tt, gen_tt:s:0':U11:U12:plus2_0(n19021_0), gen_tt:s:0':U11:U12:plus2_0(c))) →_IH
s(*3_0)

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

(27) Complex Obligation (BEST)

(28) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus2_0(n17964_0)) → gen_tt:s:0':U11:U12:plus2_0(n17964_0), rt ∈ Ω(1 + n17964₀)
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n19021_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n19021₀ + n19021₀ + n19021₀²)
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n11303_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n11303₀ + n11303₀ + n11303₀²)
a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, n14266_0))) → *3_0, rt ∈ Ω(a·n14266₀ + n14266₀ + n14266₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

The following defined symbols remain to be analysed:
a__U12

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

(29) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n22989_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n22989₀ + n22989₀ + n22989₀²)

Induction Base:
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(c))

Induction Step:
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(+(n22989_0, 1)), gen_tt:s:0':U11:U12:plus2_0(c)) →_R^Ω(1)
s(a__plus(mark(gen_tt:s:0':U11:U12:plus2_0(c)), mark(gen_tt:s:0':U11:U12:plus2_0(+(n22989_0, 1))))) →_L^{Ω(1 + c)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n22989_0))))) →_L^{Ω(2 + n22989₀)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), gen_tt:s:0':U11:U12:plus2_0(+(1, n22989_0)))) →_R^Ω(1)
s(a__U11(tt, gen_tt:s:0':U11:U12:plus2_0(n22989_0), gen_tt:s:0':U11:U12:plus2_0(c))) →_R^Ω(1)
s(a__U12(tt, gen_tt:s:0':U11:U12:plus2_0(n22989_0), gen_tt:s:0':U11:U12:plus2_0(c))) →_IH
s(*3_0)

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

(30) Complex Obligation (BEST)

(31) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus2_0(n17964_0)) → gen_tt:s:0':U11:U12:plus2_0(n17964_0), rt ∈ Ω(1 + n17964₀)
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n19021_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n19021₀ + n19021₀ + n19021₀²)
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n22989_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n22989₀ + n22989₀ + n22989₀²)
a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, n14266_0))) → *3_0, rt ∈ Ω(a·n14266₀ + n14266₀ + n14266₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

No more defined symbols left to analyse.

(32) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n19021_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n19021₀ + n19021₀ + n19021₀²)

(33) BOUNDS(n^2, INF)

(34) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus2_0(n17964_0)) → gen_tt:s:0':U11:U12:plus2_0(n17964_0), rt ∈ Ω(1 + n17964₀)
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n19021_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n19021₀ + n19021₀ + n19021₀²)
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n22989_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n22989₀ + n22989₀ + n22989₀²)
a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, n14266_0))) → *3_0, rt ∈ Ω(a·n14266₀ + n14266₀ + n14266₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

No more defined symbols left to analyse.

(35) LowerBoundsProof (EQUIVALENT transformation)

(36) BOUNDS(n^2, INF)

(37) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus2_0(n17964_0)) → gen_tt:s:0':U11:U12:plus2_0(n17964_0), rt ∈ Ω(1 + n17964₀)
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n19021_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n19021₀ + n19021₀ + n19021₀²)
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n11303_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n11303₀ + n11303₀ + n11303₀²)
a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, n14266_0))) → *3_0, rt ∈ Ω(a·n14266₀ + n14266₀ + n14266₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

No more defined symbols left to analyse.

(38) LowerBoundsProof (EQUIVALENT transformation)

(39) BOUNDS(n^2, INF)

(40) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

No more defined symbols left to analyse.

(41) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n8550_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n8550₀ + n8550₀ + n8550₀²)

(42) BOUNDS(n^2, INF)

(43) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

No more defined symbols left to analyse.

(44) LowerBoundsProof (EQUIVALENT transformation)

(45) BOUNDS(n^2, INF)

(46) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

No more defined symbols left to analyse.

(47) LowerBoundsProof (EQUIVALENT transformation)

(48) BOUNDS(n^2, INF)

(49) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

No more defined symbols left to analyse.

(50) LowerBoundsProof (EQUIVALENT transformation)

(51) BOUNDS(n^2, INF)

(52) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus2_0(n7802_0)) → gen_tt:s:0':U11:U12:plus2_0(n7802_0), rt ∈ Ω(1 + n7802₀)

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

No more defined symbols left to analyse.

(53) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_tt:s:0':U11:U12:plus2_0(n7802_0)) → gen_tt:s:0':U11:U12:plus2_0(n7802_0), rt ∈ Ω(1 + n7802₀)

(0) Obligation:

(1) RenamingProof (EQUIVALENT transformation)

(2) Obligation:

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

(4) Obligation:

(5) OrderProof (LOWER BOUND(ID) transformation)

(6) Obligation:

(7) NoRewriteLemmaProof (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) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(27) Complex Obligation (BEST)

(28) Obligation:

(29) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(30) Complex Obligation (BEST)

(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^2, INF)

(43) Obligation:

(44) LowerBoundsProof (EQUIVALENT transformation)

(45) BOUNDS(n^2, INF)

(46) Obligation:

(47) LowerBoundsProof (EQUIVALENT transformation)

(48) BOUNDS(n^2, INF)

(49) Obligation:

(50) LowerBoundsProof (EQUIVALENT transformation)

(51) BOUNDS(n^2, INF)

(52) Obligation:

(53) LowerBoundsProof (EQUIVALENT transformation)

(54) BOUNDS(n^1, INF)