Runtime Complexity (innermost) proof of /tmp/tmpdyrowW/MYNAT_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), 1 ms)

↳6 typed CpxTrs

↳7 NoRewriteLemmaProof (LOWER BOUND(ID), 789 ms)

↳8 typed CpxTrs

↳9 NoRewriteLemmaProof (LOWER BOUND(ID), 345 ms)

↳10 typed CpxTrs

↳11 RewriteLemmaProof (LOWER BOUND(ID), 207 ms)

↳12 BEST

    ↳13 typed CpxTrs

        ↳14 RewriteLemmaProof (LOWER BOUND(ID), 389 ms)

        ↳15 BEST

            ↳16 typed CpxTrs

                ↳17 RewriteLemmaProof (LOWER BOUND(ID), 391 ms)

                ↳18 BEST

                    ↳19 typed CpxTrs

                        ↳20 RewriteLemmaProof (LOWER BOUND(ID), 252 ms)

                        ↳21 BEST

                            ↳22 typed CpxTrs

                                ↳23 RewriteLemmaProof (LOWER BOUND(ID), 340 ms)

                                ↳24 BEST

                                    ↳25 typed CpxTrs

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

                                        ↳27 BEST

                                            ↳28 typed CpxTrs

                                                ↳29 RewriteLemmaProof (LOWER BOUND(ID), 100 ms)

                                                ↳30 BEST

                                                    ↳31 typed CpxTrs

                                                        ↳32 RewriteLemmaProof (LOWER BOUND(ID), 30 ms)

                                                        ↳33 BEST

                                                            ↳34 typed CpxTrs

                                                                ↳35 RewriteLemmaProof (LOWER BOUND(ID), 155 ms)

                                                                ↳36 BEST

                                                                    ↳37 typed CpxTrs

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

                                                                        ↳39 BEST

                                                                            ↳40 typed CpxTrs

                                                                                ↳41 RewriteLemmaProof (LOWER BOUND(ID), 199 ms)

                                                                                ↳42 BEST

                                                                                    ↳43 typed CpxTrs

                                                                                        ↳44 RewriteLemmaProof (LOWER BOUND(ID), 234 ms)

                                                                                        ↳45 BEST

                                                                                            ↳46 typed CpxTrs

                                                                                                ↳47 RewriteLemmaProof (LOWER BOUND(ID), 69 ms)

                                                                                                ↳48 BEST

                                                                                                    ↳49 typed CpxTrs

                                                                                                        ↳50 LowerBoundsProof (⇔, 0 ms)

                                                                                                        ↳51 BOUNDS(n^2, INF)

                                                                                                    ↳52 typed CpxTrs

                                                                                                        ↳53 LowerBoundsProof (⇔, 0 ms)

                                                                                                        ↳54 BOUNDS(n^2, INF)

                                                                                            ↳55 typed CpxTrs

                                                                                                ↳56 LowerBoundsProof (⇔, 0 ms)

                                                                                                ↳57 BOUNDS(n^2, INF)

                                                                                    ↳58 typed CpxTrs

                                                                                        ↳59 LowerBoundsProof (⇔, 0 ms)

                                                                                        ↳60 BOUNDS(n^2, INF)

                                                                            ↳61 typed CpxTrs

                                                                                ↳62 LowerBoundsProof (⇔, 0 ms)

                                                                                ↳63 BOUNDS(n^2, INF)

                                                                    ↳64 typed CpxTrs

                                                                        ↳65 LowerBoundsProof (⇔, 0 ms)

                                                                        ↳66 BOUNDS(n^2, INF)

                                                            ↳67 typed CpxTrs

                                                                ↳68 LowerBoundsProof (⇔, 0 ms)

                                                                ↳69 BOUNDS(n^2, INF)

                                                    ↳70 typed CpxTrs

                                                        ↳71 LowerBoundsProof (⇔, 0 ms)

                                                        ↳72 BOUNDS(n^2, INF)

                                            ↳73 typed CpxTrs

                                                ↳74 LowerBoundsProof (⇔, 0 ms)

                                                ↳75 BOUNDS(n^2, INF)

                                    ↳76 typed CpxTrs

                                        ↳77 LowerBoundsProof (⇔, 0 ms)

                                        ↳78 BOUNDS(n^2, INF)

                            ↳79 typed CpxTrs

                                ↳80 LowerBoundsProof (⇔, 0 ms)

                                ↳81 BOUNDS(n^2, INF)

                    ↳82 typed CpxTrs

                        ↳83 LowerBoundsProof (⇔, 0 ms)

                        ↳84 BOUNDS(n^2, INF)

            ↳85 typed CpxTrs

                ↳86 LowerBoundsProof (⇔, 0 ms)

                ↳87 BOUNDS(n^2, INF)

    ↳88 typed CpxTrs

        ↳89 LowerBoundsProof (⇔, 0 ms)

        ↳90 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__U21(tt, M, N) → a__U22(tt, M, N)
a__U22(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
a__x(N, 0) → 0
a__x(N, s(M)) → a__U21(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(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(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)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__x(X1, X2) → x(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__U21(tt, M, N) → a__U22(tt, M, N)
a__U22(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__plus(N, 0') → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
a__x(N, 0') → 0'
a__x(N, s(M)) → a__U21(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(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(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)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__x(X1, X2) → x(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__U21(tt, M, N) → a__U22(tt, M, N)
a__U22(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__plus(N, 0') → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
a__x(N, 0') → 0'
a__x(N, s(M)) → a__U21(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(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(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)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Types:
a__U11 :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
tt :: tt:s:0':U11:U12:plus:U21:U22:x
a__U12 :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
s :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
a__plus :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
mark :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
a__U21 :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
a__U22 :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
a__x :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
0' :: tt:s:0':U11:U12:plus:U21:U22:x
U11 :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
U12 :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
plus :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
U21 :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
U22 :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
x :: tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x → tt:s:0':U11:U12:plus:U21:U22:x
hole_tt:s:0':U11:U12:plus:U21:U22:x1_0 :: tt:s:0':U11:U12:plus:U21:U22:x
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0 :: Nat → tt:s:0':U11:U12:plus:U21:U22:x

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

Heuristically decided to analyse the following defined symbols:
a__U11, a__U12, a__plus, mark, a__U21, a__U22, a__x

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U11 = a__U21
a__U11 = a__U22
a__U11 = a__x
a__U12 = a__plus
a__U12 = mark
a__U12 = a__U21
a__U12 = a__U22
a__U12 = a__x
a__plus = mark
a__plus = a__U21
a__plus = a__U22
a__plus = a__x
mark = a__U21
mark = a__U22
mark = a__x
a__U21 = a__U22
a__U21 = a__x
a__U22 = a__x

(6) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__U12, a__U11, a__plus, mark, a__U21, a__U22, a__x

(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:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__plus, a__U11, mark, a__U21, a__U22, 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:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
mark, a__U11, a__U21, a__U22, a__x

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

Proved the following rewrite lemma:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85400_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85400_0), rt ∈ Ω(1 + n85400₀)

Induction Base:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0)) →_R^Ω(1)
tt

Induction Step:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(n85400_0, 1))) →_R^Ω(1)
s(mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85400_0))) →_IH
s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c85401_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:plus:U21:U22:x2_0(n85400_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85400_0), rt ∈ Ω(1 + n85400₀)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__U11, a__U12, a__plus, a__U21, a__U22, a__x

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

Proved the following rewrite lemma:
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n92708_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n92708₀ + n92708₀ + n92708₀²)

Induction Base:
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))

Induction Step:
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(n92708_0, 1)), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) →_R^Ω(1)
a__U12(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(n92708_0, 1)), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) →_R^Ω(1)
s(a__plus(mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n92708_0))))) →_L^{Ω(1 + c)}
s(a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n92708_0))))) →_L^{Ω(2 + n92708₀)}
s(a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n92708_0)))) →_R^Ω(1)
s(a__U11(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n92708_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_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:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__U21, a__U12, a__plus, mark, a__U22, a__x

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

Proved the following rewrite lemma:
a__U21(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n97142_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n97142₀ + n97142₀ + n97142₀²)

Induction Base:
a__U21(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))

Induction Step:
a__U21(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(n97142_0, 1)), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) →_R^Ω(1)
a__U22(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(n97142_0, 1)), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) →_R^Ω(1)
a__plus(a__x(mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n97142_0)))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_L^{Ω(1 + c)}
a__plus(a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n97142_0)))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_L^{Ω(2 + n97142₀)}
a__plus(a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n97142_0))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_R^Ω(1)
a__plus(a__U21(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n97142_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_IH
a__plus(*3_0, mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_L^{Ω(1 + c)}
a__plus(*3_0, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))

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

(18) Complex Obligation (BEST)

(19) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__U22, a__U11, a__U12, a__plus, mark, a__x

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

Proved the following rewrite lemma:
a__U22(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n108627_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n108627₀ + n108627₀ + n108627₀²)

Induction Base:
a__U22(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))

Induction Step:
a__U22(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(n108627_0, 1)), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) →_R^Ω(1)
a__plus(a__x(mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(n108627_0, 1)))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_L^{Ω(1 + c)}
a__plus(a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n108627_0)))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_L^{Ω(2 + n108627₀)}
a__plus(a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n108627_0))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_R^Ω(1)
a__plus(a__U21(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n108627_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_R^Ω(1)
a__plus(a__U22(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n108627_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_IH
a__plus(*3_0, mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_L^{Ω(1 + c)}
a__plus(*3_0, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))

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:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__x, a__U11, a__U12, a__plus, mark, a__U21

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

Proved the following rewrite lemma:
a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n120359_0))) → *3_0, rt ∈ Ω(a·n120359₀ + n120359₀ + n120359₀²)

Induction Base:
a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, 0)))

Induction Step:
a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, +(n120359_0, 1)))) →_R^Ω(1)
a__U21(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n120359_0)), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a)) →_R^Ω(1)
a__U22(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n120359_0)), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a)) →_R^Ω(1)
a__plus(a__x(mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a)), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n120359_0)))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a))) →_L^{Ω(1 + a)}
a__plus(a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n120359_0)))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a))) →_L^{Ω(2 + n120359₀)}
a__plus(a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n120359_0))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a))) →_IH
a__plus(*3_0, mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a))) →_L^{Ω(1 + a)}
a__plus(*3_0, gen_tt:s:0':U11:U12:plus:U21:U22: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:s:0':U11:U12:plus:U21:U22:x2_0(n85400_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85400_0), rt ∈ Ω(1 + n85400₀)
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n92708_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n92708₀ + n92708₀ + n92708₀²)
a__U21(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n97142_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n97142₀ + n97142₀ + n97142₀²)
a__U22(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n108627_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n108627₀ + n108627₀ + n108627₀²)
a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n120359_0))) → *3_0, rt ∈ Ω(a·n120359₀ + n120359₀ + n120359₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__U12, a__U11, a__plus, mark, a__U21, a__U22

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

Proved the following rewrite lemma:
a__U12(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n133193_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n133193₀ + n133193₀ + n133193₀²)

Induction Base:
a__U12(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))

Induction Step:
a__U12(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(n133193_0, 1)), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) →_R^Ω(1)
s(a__plus(mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(n133193_0, 1))))) →_L^{Ω(1 + c)}
s(a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n133193_0))))) →_L^{Ω(2 + n133193₀)}
s(a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n133193_0)))) →_R^Ω(1)
s(a__U11(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n133193_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_R^Ω(1)
s(a__U12(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n133193_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_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:

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__plus, a__U11, mark, a__U21, a__U22, a__x

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

Proved the following rewrite lemma:
a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n138905_0))) → *3_0, rt ∈ Ω(a·n138905₀ + n138905₀ + n138905₀²)

Induction Base:
a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, 0)))

Induction Step:
a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, +(n138905_0, 1)))) →_R^Ω(1)
a__U11(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n138905_0)), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a)) →_R^Ω(1)
a__U12(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n138905_0)), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a)) →_R^Ω(1)
s(a__plus(mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a)), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n138905_0))))) →_L^{Ω(1 + a)}
s(a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n138905_0))))) →_L^{Ω(2 + n138905₀)}
s(a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n138905_0)))) →_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:

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
mark, a__U11, a__U12, a__U21, a__U22, a__x

(32) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0), rt ∈ Ω(1 + n145591₀)

Induction Base:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0)) →_R^Ω(1)
tt

Induction Step:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(n145591_0, 1))) →_R^Ω(1)
s(mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0))) →_IH
s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c145592_0))

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

(33) Complex Obligation (BEST)

(34) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0), rt ∈ Ω(1 + n145591₀)
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n92708_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n92708₀ + n92708₀ + n92708₀²)
a__U21(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n97142_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n97142₀ + n97142₀ + n97142₀²)
a__U22(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n108627_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n108627₀ + n108627₀ + n108627₀²)
a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n120359_0))) → *3_0, rt ∈ Ω(a·n120359₀ + n120359₀ + n120359₀²)
a__U12(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n133193_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n133193₀ + n133193₀ + n133193₀²)
a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n138905_0))) → *3_0, rt ∈ Ω(a·n138905₀ + n138905₀ + n138905₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__U11, a__U12, a__U21, a__U22, a__x

(35) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n153517_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n153517₀ + n153517₀ + n153517₀²)

Induction Base:
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))

Induction Step:
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(n153517_0, 1)), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) →_R^Ω(1)
a__U12(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(n153517_0, 1)), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) →_R^Ω(1)
s(a__plus(mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n153517_0))))) →_L^{Ω(1 + c)}
s(a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n153517_0))))) →_L^{Ω(2 + n153517₀)}
s(a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n153517_0)))) →_R^Ω(1)
s(a__U11(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n153517_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_IH
s(*3_0)

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

(36) Complex Obligation (BEST)

(37) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0), rt ∈ Ω(1 + n145591₀)
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n153517_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n153517₀ + n153517₀ + n153517₀²)
a__U21(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n97142_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n97142₀ + n97142₀ + n97142₀²)
a__U22(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n108627_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n108627₀ + n108627₀ + n108627₀²)
a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n120359_0))) → *3_0, rt ∈ Ω(a·n120359₀ + n120359₀ + n120359₀²)
a__U12(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n133193_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n133193₀ + n133193₀ + n133193₀²)
a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n138905_0))) → *3_0, rt ∈ Ω(a·n138905₀ + n138905₀ + n138905₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__U21, a__U12, a__U22, a__x

(38) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__U21(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n160381_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n160381₀ + n160381₀ + n160381₀²)

Induction Base:
a__U21(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))

Induction Step:
a__U21(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(n160381_0, 1)), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) →_R^Ω(1)
a__U22(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(n160381_0, 1)), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) →_R^Ω(1)
a__plus(a__x(mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n160381_0)))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_L^{Ω(1 + c)}
a__plus(a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n160381_0)))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_L^{Ω(2 + n160381₀)}
a__plus(a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n160381_0))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_R^Ω(1)
a__plus(a__U21(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n160381_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_IH
a__plus(*3_0, mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_L^{Ω(1 + c)}
a__plus(*3_0, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))

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

(39) Complex Obligation (BEST)

(40) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0), rt ∈ Ω(1 + n145591₀)
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n153517_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n153517₀ + n153517₀ + n153517₀²)
a__U21(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n160381_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n160381₀ + n160381₀ + n160381₀²)
a__U22(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n108627_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n108627₀ + n108627₀ + n108627₀²)
a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n120359_0))) → *3_0, rt ∈ Ω(a·n120359₀ + n120359₀ + n120359₀²)
a__U12(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n133193_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n133193₀ + n133193₀ + n133193₀²)
a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n138905_0))) → *3_0, rt ∈ Ω(a·n138905₀ + n138905₀ + n138905₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__U22, a__U12, a__x

(41) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__U22(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n174691_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n174691₀ + n174691₀ + n174691₀²)

Induction Base:
a__U22(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))

Induction Step:
a__U22(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(n174691_0, 1)), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) →_R^Ω(1)
a__plus(a__x(mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(n174691_0, 1)))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_L^{Ω(1 + c)}
a__plus(a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n174691_0)))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_L^{Ω(2 + n174691₀)}
a__plus(a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n174691_0))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_R^Ω(1)
a__plus(a__U21(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n174691_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_R^Ω(1)
a__plus(a__U22(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n174691_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_IH
a__plus(*3_0, mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_L^{Ω(1 + c)}
a__plus(*3_0, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))

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

(42) Complex Obligation (BEST)

(43) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0), rt ∈ Ω(1 + n145591₀)
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n153517_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n153517₀ + n153517₀ + n153517₀²)
a__U21(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n160381_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n160381₀ + n160381₀ + n160381₀²)
a__U22(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n174691_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n174691₀ + n174691₀ + n174691₀²)
a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n120359_0))) → *3_0, rt ∈ Ω(a·n120359₀ + n120359₀ + n120359₀²)
a__U12(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n133193_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n133193₀ + n133193₀ + n133193₀²)
a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n138905_0))) → *3_0, rt ∈ Ω(a·n138905₀ + n138905₀ + n138905₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

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

(44) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n188647_0))) → *3_0, rt ∈ Ω(a·n188647₀ + n188647₀ + n188647₀²)

Induction Base:
a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, 0)))

Induction Step:
a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, +(n188647_0, 1)))) →_R^Ω(1)
a__U21(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n188647_0)), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a)) →_R^Ω(1)
a__U22(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n188647_0)), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a)) →_R^Ω(1)
a__plus(a__x(mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a)), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n188647_0)))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a))) →_L^{Ω(1 + a)}
a__plus(a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n188647_0)))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a))) →_L^{Ω(2 + n188647₀)}
a__plus(a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n188647_0))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a))) →_IH
a__plus(*3_0, mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a))) →_L^{Ω(1 + a)}
a__plus(*3_0, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a))

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

(45) Complex Obligation (BEST)

(46) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0), rt ∈ Ω(1 + n145591₀)
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n153517_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n153517₀ + n153517₀ + n153517₀²)
a__U21(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n160381_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n160381₀ + n160381₀ + n160381₀²)
a__U22(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n174691_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n174691₀ + n174691₀ + n174691₀²)
a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n188647_0))) → *3_0, rt ∈ Ω(a·n188647₀ + n188647₀ + n188647₀²)
a__U12(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n133193_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n133193₀ + n133193₀ + n133193₀²)
a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n138905_0))) → *3_0, rt ∈ Ω(a·n138905₀ + n138905₀ + n138905₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

The following defined symbols remain to be analysed:
a__U12

(47) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__U12(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n203188_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n203188₀ + n203188₀ + n203188₀²)

Induction Base:
a__U12(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))

Induction Step:
a__U12(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(n203188_0, 1)), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) →_R^Ω(1)
s(a__plus(mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(n203188_0, 1))))) →_L^{Ω(1 + c)}
s(a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n203188_0))))) →_L^{Ω(2 + n203188₀)}
s(a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n203188_0)))) →_R^Ω(1)
s(a__U11(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n203188_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_R^Ω(1)
s(a__U12(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n203188_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_IH
s(*3_0)

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

(48) Complex Obligation (BEST)

(49) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0), rt ∈ Ω(1 + n145591₀)
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n153517_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n153517₀ + n153517₀ + n153517₀²)
a__U21(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n160381_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n160381₀ + n160381₀ + n160381₀²)
a__U22(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n174691_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n174691₀ + n174691₀ + n174691₀²)
a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n188647_0))) → *3_0, rt ∈ Ω(a·n188647₀ + n188647₀ + n188647₀²)
a__U12(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n203188_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n203188₀ + n203188₀ + n203188₀²)
a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n138905_0))) → *3_0, rt ∈ Ω(a·n138905₀ + n138905₀ + n138905₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

No more defined symbols left to analyse.

(50) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n153517_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n153517₀ + n153517₀ + n153517₀²)

(51) BOUNDS(n^2, INF)

(52) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0), rt ∈ Ω(1 + n145591₀)
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n153517_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n153517₀ + n153517₀ + n153517₀²)
a__U21(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n160381_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n160381₀ + n160381₀ + n160381₀²)
a__U22(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n174691_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n174691₀ + n174691₀ + n174691₀²)
a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n188647_0))) → *3_0, rt ∈ Ω(a·n188647₀ + n188647₀ + n188647₀²)
a__U12(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n203188_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n203188₀ + n203188₀ + n203188₀²)
a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n138905_0))) → *3_0, rt ∈ Ω(a·n138905₀ + n138905₀ + n138905₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

No more defined symbols left to analyse.

(53) LowerBoundsProof (EQUIVALENT transformation)

(54) BOUNDS(n^2, INF)

(55) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0), rt ∈ Ω(1 + n145591₀)
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n153517_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n153517₀ + n153517₀ + n153517₀²)
a__U21(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n160381_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n160381₀ + n160381₀ + n160381₀²)
a__U22(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n174691_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n174691₀ + n174691₀ + n174691₀²)
a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n188647_0))) → *3_0, rt ∈ Ω(a·n188647₀ + n188647₀ + n188647₀²)
a__U12(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n133193_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n133193₀ + n133193₀ + n133193₀²)
a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n138905_0))) → *3_0, rt ∈ Ω(a·n138905₀ + n138905₀ + n138905₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

No more defined symbols left to analyse.

(56) LowerBoundsProof (EQUIVALENT transformation)

(57) BOUNDS(n^2, INF)

(58) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0), rt ∈ Ω(1 + n145591₀)
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n153517_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n153517₀ + n153517₀ + n153517₀²)
a__U21(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n160381_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n160381₀ + n160381₀ + n160381₀²)
a__U22(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n174691_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n174691₀ + n174691₀ + n174691₀²)
a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n120359_0))) → *3_0, rt ∈ Ω(a·n120359₀ + n120359₀ + n120359₀²)
a__U12(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n133193_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n133193₀ + n133193₀ + n133193₀²)
a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n138905_0))) → *3_0, rt ∈ Ω(a·n138905₀ + n138905₀ + n138905₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

No more defined symbols left to analyse.

(59) LowerBoundsProof (EQUIVALENT transformation)

(60) BOUNDS(n^2, INF)

(61) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0), rt ∈ Ω(1 + n145591₀)
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n153517_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n153517₀ + n153517₀ + n153517₀²)
a__U21(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n160381_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n160381₀ + n160381₀ + n160381₀²)
a__U22(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n108627_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n108627₀ + n108627₀ + n108627₀²)
a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n120359_0))) → *3_0, rt ∈ Ω(a·n120359₀ + n120359₀ + n120359₀²)
a__U12(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n133193_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n133193₀ + n133193₀ + n133193₀²)
a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n138905_0))) → *3_0, rt ∈ Ω(a·n138905₀ + n138905₀ + n138905₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

No more defined symbols left to analyse.

(62) LowerBoundsProof (EQUIVALENT transformation)

(63) BOUNDS(n^2, INF)

(64) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n145591_0), rt ∈ Ω(1 + n145591₀)
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n153517_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n153517₀ + n153517₀ + n153517₀²)
a__U21(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n97142_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n97142₀ + n97142₀ + n97142₀²)
a__U22(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n108627_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n108627₀ + n108627₀ + n108627₀²)
a__x(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n120359_0))) → *3_0, rt ∈ Ω(a·n120359₀ + n120359₀ + n120359₀²)
a__U12(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n133193_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n133193₀ + n133193₀ + n133193₀²)
a__plus(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n138905_0))) → *3_0, rt ∈ Ω(a·n138905₀ + n138905₀ + n138905₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

No more defined symbols left to analyse.

(65) LowerBoundsProof (EQUIVALENT transformation)

(66) BOUNDS(n^2, INF)

(67) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

No more defined symbols left to analyse.

(68) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
a__U11(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n92708_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n92708₀ + n92708₀ + n92708₀²)

(69) BOUNDS(n^2, INF)

(70) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

No more defined symbols left to analyse.

(71) LowerBoundsProof (EQUIVALENT transformation)

(72) BOUNDS(n^2, INF)

(73) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

No more defined symbols left to analyse.

(74) LowerBoundsProof (EQUIVALENT transformation)

(75) BOUNDS(n^2, INF)

(76) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

No more defined symbols left to analyse.

(77) LowerBoundsProof (EQUIVALENT transformation)

(78) BOUNDS(n^2, INF)

(79) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

No more defined symbols left to analyse.

(80) LowerBoundsProof (EQUIVALENT transformation)

(81) BOUNDS(n^2, INF)

(82) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

No more defined symbols left to analyse.

(83) LowerBoundsProof (EQUIVALENT transformation)

(84) BOUNDS(n^2, INF)

(85) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

No more defined symbols left to analyse.

(86) LowerBoundsProof (EQUIVALENT transformation)

(87) BOUNDS(n^2, INF)

(88) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85400_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85400_0), rt ∈ Ω(1 + n85400₀)

Generator Equations:
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(x))

No more defined symbols left to analyse.

(89) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85400_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n85400_0), rt ∈ Ω(1 + n85400₀)