Runtime Complexity (innermost) proof of /tmp/tmp2GNTZi/MYNAT_nosorts_noand

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

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 689 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), 733 ms)

    ↳10 typed CpxTrs

    ↳11 NoRewriteLemmaProof (LOWER BOUND(ID), 267 ms)

    ↳12 typed CpxTrs

    ↳13 RewriteLemmaProof (LOWER BOUND(ID), 139 ms)

    ↳14 BEST

        ↳15 typed CpxTrs

            ↳16 RewriteLemmaProof (LOWER BOUND(ID), 300 ms)

            ↳17 BEST

                ↳18 typed CpxTrs

                    ↳19 RewriteLemmaProof (LOWER BOUND(ID), 559 ms)

                    ↳20 BEST

                        ↳21 typed CpxTrs

                            ↳22 RewriteLemmaProof (LOWER BOUND(ID), 532 ms)

                            ↳23 BEST

                                ↳24 typed CpxTrs

                                    ↳25 RewriteLemmaProof (LOWER BOUND(ID), 467 ms)

                                    ↳26 BEST

                                        ↳27 typed CpxTrs

                                            ↳28 RewriteLemmaProof (LOWER BOUND(ID), 166 ms)

                                            ↳29 BEST

                                                ↳30 typed CpxTrs

                                                    ↳31 RewriteLemmaProof (LOWER BOUND(ID), 84 ms)

                                                    ↳32 BEST

                                                        ↳33 typed CpxTrs

                                                            ↳34 RewriteLemmaProof (LOWER BOUND(ID), 5 ms)

                                                            ↳35 BEST

                                                                ↳36 typed CpxTrs

                                                                    ↳37 RewriteLemmaProof (LOWER BOUND(ID), 115 ms)

                                                                    ↳38 BEST

                                                                        ↳39 typed CpxTrs

                                                                            ↳40 RewriteLemmaProof (LOWER BOUND(ID), 253 ms)

                                                                            ↳41 BEST

                                                                                ↳42 typed CpxTrs

                                                                                    ↳43 RewriteLemmaProof (LOWER BOUND(ID), 255 ms)

                                                                                    ↳44 BEST

                                                                                        ↳45 typed CpxTrs

                                                                                            ↳46 RewriteLemmaProof (LOWER BOUND(ID), 256 ms)

                                                                                            ↳47 BEST

                                                                                                ↳48 typed CpxTrs

                                                                                                    ↳49 RewriteLemmaProof (LOWER BOUND(ID), 63 ms)

                                                                                                    ↳50 BEST

                                                                                                        ↳51 typed CpxTrs

                                                                                                            ↳52 LowerBoundsProof (⇔, 0 ms)

                                                                                                            ↳53 BOUNDS(n^2, INF)

                                                                                                        ↳54 typed CpxTrs

                                                                                                            ↳55 LowerBoundsProof (⇔, 0 ms)

                                                                                                            ↳56 BOUNDS(n^2, INF)

                                                                                                ↳57 typed CpxTrs

                                                                                                    ↳58 LowerBoundsProof (⇔, 0 ms)

                                                                                                    ↳59 BOUNDS(n^2, INF)

                                                                                        ↳60 typed CpxTrs

                                                                                            ↳61 LowerBoundsProof (⇔, 0 ms)

                                                                                            ↳62 BOUNDS(n^2, INF)

                                                                                ↳63 typed CpxTrs

                                                                                    ↳64 LowerBoundsProof (⇔, 0 ms)

                                                                                    ↳65 BOUNDS(n^2, INF)

                                                                        ↳66 typed CpxTrs

                                                                            ↳67 LowerBoundsProof (⇔, 0 ms)

                                                                            ↳68 BOUNDS(n^2, INF)

                                                                ↳69 typed CpxTrs

                                                                    ↳70 LowerBoundsProof (⇔, 0 ms)

                                                                    ↳71 BOUNDS(n^2, INF)

                                                        ↳72 typed CpxTrs

                                                            ↳73 LowerBoundsProof (⇔, 0 ms)

                                                            ↳74 BOUNDS(n^2, INF)

                                                ↳75 typed CpxTrs

                                                    ↳76 LowerBoundsProof (⇔, 0 ms)

                                                    ↳77 BOUNDS(n^2, INF)

                                        ↳78 typed CpxTrs

                                            ↳79 LowerBoundsProof (⇔, 0 ms)

                                            ↳80 BOUNDS(n^2, INF)

                                ↳81 typed CpxTrs

                                    ↳82 LowerBoundsProof (⇔, 0 ms)

                                    ↳83 BOUNDS(n^2, INF)

                        ↳84 typed CpxTrs

                            ↳85 LowerBoundsProof (⇔, 0 ms)

                            ↳86 BOUNDS(n^2, INF)

                ↳87 typed CpxTrs

                    ↳88 LowerBoundsProof (⇔, 0 ms)

                    ↳89 BOUNDS(n^2, INF)

        ↳90 typed CpxTrs

            ↳91 LowerBoundsProof (⇔, 0 ms)

            ↳92 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) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
mark(U11(U22(tt, X229731_4, X329732_4), X2, X3)) →⁺ a__U11(a__plus(a__x(mark(X329732_4), mark(X229731_4)), mark(X329732_4)), X2, X3)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
The pumping substitution is [X329732_4 / U11(U22(tt, X229731_4, X329732_4), X2, X3)].
The result substitution is [ ].

The rewrite sequence
mark(U11(U22(tt, X229731_4, X329732_4), X2, X3)) →⁺ a__U11(a__plus(a__x(mark(X329732_4), mark(X229731_4)), mark(X329732_4)), X2, X3)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1].
The pumping substitution is [X329732_4 / U11(U22(tt, X229731_4, X329732_4), X2, X3)].
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__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

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

Infered types.

(6) 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

(7) 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

(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__U12, a__U11, a__plus, mark, a__U21, a__U22, a__x

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

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

(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:
a__plus, a__U11, mark, a__U21, a__U22, a__x

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(12) 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

(13) 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).

(14) Complex Obligation (BEST)

(15) 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

(16) 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).

(17) Complex Obligation (BEST)

(18) 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

(19) 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).

(20) Complex Obligation (BEST)

(21) 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

(22) 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).

(23) Complex Obligation (BEST)

(24) 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

(25) 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).

(26) Complex Obligation (BEST)

(27) 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

(28) 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).

(29) Complex Obligation (BEST)

(30) 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

(31) 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).

(32) Complex Obligation (BEST)

(33) 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

(34) 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).

(35) Complex Obligation (BEST)

(36) 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

(37) 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).

(38) Complex Obligation (BEST)

(39) 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

(40) 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).

(41) Complex Obligation (BEST)

(42) 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

(43) 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).

(44) Complex Obligation (BEST)

(45) 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

(46) 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).

(47) Complex Obligation (BEST)

(48) 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

(49) 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).

(50) Complex Obligation (BEST)

(51) 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.

(52) 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)

(53) BOUNDS(n^2, INF)

(54) 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.

(55) LowerBoundsProof (EQUIVALENT transformation)

(56) BOUNDS(n^2, INF)

(57) 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.

(58) LowerBoundsProof (EQUIVALENT transformation)

(59) BOUNDS(n^2, INF)

(60) 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.

(61) LowerBoundsProof (EQUIVALENT transformation)

(62) BOUNDS(n^2, INF)

(63) 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.

(64) LowerBoundsProof (EQUIVALENT transformation)

(65) BOUNDS(n^2, INF)

(66) 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.

(67) LowerBoundsProof (EQUIVALENT transformation)

(68) BOUNDS(n^2, INF)

(69) 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.

(70) 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)

(71) BOUNDS(n^2, INF)

(72) 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.

(73) LowerBoundsProof (EQUIVALENT transformation)

(74) BOUNDS(n^2, INF)

(75) 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.

(76) LowerBoundsProof (EQUIVALENT transformation)

(77) BOUNDS(n^2, INF)

(78) 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.

(79) LowerBoundsProof (EQUIVALENT transformation)

(80) BOUNDS(n^2, INF)

(81) 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.

(82) LowerBoundsProof (EQUIVALENT transformation)

(83) BOUNDS(n^2, INF)

(84) 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.

(85) LowerBoundsProof (EQUIVALENT transformation)

(86) BOUNDS(n^2, INF)

(87) 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.

(88) LowerBoundsProof (EQUIVALENT transformation)

(89) BOUNDS(n^2, INF)

(90) 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.

(91) 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)