Runtime Complexity (full) proof of /tmp/tmpjNORDu/MYNAT_nosorts_noand

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

0 CpxTRS

↳1 RenamingProof (⇔, 0 ms)

↳2 CpxRelTRS

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

↳4 typed CpxTrs

↳5 OrderProof (LOWER BOUND(ID), 0 ms)

↳6 typed CpxTrs

↳7 NoRewriteLemmaProof (LOWER BOUND(ID), 696 ms)

↳8 typed CpxTrs

↳9 NoRewriteLemmaProof (LOWER BOUND(ID), 389 ms)

↳10 typed CpxTrs

↳11 RewriteLemmaProof (LOWER BOUND(ID), 146 ms)

↳12 BEST

    ↳13 typed CpxTrs

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

        ↳15 BEST

            ↳16 typed CpxTrs

                ↳17 RewriteLemmaProof (LOWER BOUND(ID), 310 ms)

                ↳18 BEST

                    ↳19 typed CpxTrs

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

                        ↳21 BEST

                            ↳22 typed CpxTrs

                                ↳23 RewriteLemmaProof (LOWER BOUND(ID), 312 ms)

                                ↳24 BEST

                                    ↳25 typed CpxTrs

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

                                        ↳27 BEST

                                            ↳28 typed CpxTrs

                                                ↳29 RewriteLemmaProof (LOWER BOUND(ID), 71 ms)

                                                ↳30 BEST

                                                    ↳31 typed CpxTrs

                                                        ↳32 RewriteLemmaProof (LOWER BOUND(ID), 36 ms)

                                                        ↳33 BEST

                                                            ↳34 typed CpxTrs

                                                                ↳35 RewriteLemmaProof (LOWER BOUND(ID), 56 ms)

                                                                ↳36 BEST

                                                                    ↳37 typed CpxTrs

                                                                        ↳38 RewriteLemmaProof (LOWER BOUND(ID), 208 ms)

                                                                        ↳39 BEST

                                                                            ↳40 typed CpxTrs

                                                                                ↳41 RewriteLemmaProof (LOWER BOUND(ID), 152 ms)

                                                                                ↳42 BEST

                                                                                    ↳43 typed CpxTrs

                                                                                        ↳44 RewriteLemmaProof (LOWER BOUND(ID), 214 ms)

                                                                                        ↳45 BEST

                                                                                            ↳46 typed CpxTrs

                                                                                                ↳47 RewriteLemmaProof (LOWER BOUND(ID), 85 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: FULL

(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: FULL

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

Infered types.

(4) Obligation:

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(n92616_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n92616₀ + n92616₀ + n92616₀²)

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(+(n92616_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(+(n92616_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, n92616_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, n92616_0))))) →_L^{Ω(2 + n92616₀)}
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, n92616_0)))) →_R^Ω(1)
s(a__U11(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n92616_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(n95831_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n95831₀ + n95831₀ + n95831₀²)

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(+(n95831_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(+(n95831_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, n95831_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, n95831_0)))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_L^{Ω(2 + n95831₀)}
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, n95831_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(n95831_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(n104303_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n104303₀ + n104303₀ + n104303₀²)

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(+(n104303_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(+(n104303_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, n104303_0)))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_L^{Ω(2 + n104303₀)}
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, n104303_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(n104303_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(n104303_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, n113114_0))) → *3_0, rt ∈ Ω(a·n113114₀ + n113114₀ + n113114₀²)

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, +(n113114_0, 1)))) →_R^Ω(1)
a__U21(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n113114_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, n113114_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, n113114_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, n113114_0)))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a))) →_L^{Ω(2 + n113114₀)}
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, n113114_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(n92616_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n92616₀ + n92616₀ + n92616₀²)
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(n95831_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n95831₀ + n95831₀ + n95831₀²)
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(n104303_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n104303₀ + n104303₀ + n104303₀²)
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, n113114_0))) → *3_0, rt ∈ Ω(a·n113114₀ + n113114₀ + n113114₀²)

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(n122889_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n122889₀ + n122889₀ + n122889₀²)

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(+(n122889_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(+(n122889_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, n122889_0))))) →_L^{Ω(2 + n122889₀)}
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, n122889_0)))) →_R^Ω(1)
s(a__U11(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n122889_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(n122889_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, n127474_0))) → *3_0, rt ∈ Ω(a·n127474₀ + n127474₀ + n127474₀²)

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, +(n127474_0, 1)))) →_R^Ω(1)
a__U11(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n127474_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, n127474_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, n127474_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, n127474_0))))) →_L^{Ω(2 + n127474₀)}
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, n127474_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(n132895_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n132895_0), rt ∈ Ω(1 + n132895₀)

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(+(n132895_0, 1))) →_R^Ω(1)
s(mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n132895_0))) →_IH
s(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c132896_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(n132895_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n132895_0), rt ∈ Ω(1 + n132895₀)
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(n92616_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n92616₀ + n92616₀ + n92616₀²)
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(n95831_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n95831₀ + n95831₀ + n95831₀²)
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(n104303_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n104303₀ + n104303₀ + n104303₀²)
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, n113114_0))) → *3_0, rt ∈ Ω(a·n113114₀ + n113114₀ + n113114₀²)
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(n122889_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n122889₀ + n122889₀ + n122889₀²)
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, n127474_0))) → *3_0, rt ∈ Ω(a·n127474₀ + n127474₀ + n127474₀²)

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(n140729_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n140729₀ + n140729₀ + n140729₀²)

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(+(n140729_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(+(n140729_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, n140729_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, n140729_0))))) →_L^{Ω(2 + n140729₀)}
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, n140729_0)))) →_R^Ω(1)
s(a__U11(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n140729_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(n132895_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n132895_0), rt ∈ Ω(1 + n132895₀)
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(n140729_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n140729₀ + n140729₀ + n140729₀²)
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(n95831_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n95831₀ + n95831₀ + n95831₀²)
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(n104303_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n104303₀ + n104303₀ + n104303₀²)
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, n113114_0))) → *3_0, rt ∈ Ω(a·n113114₀ + n113114₀ + n113114₀²)
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(n122889_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n122889₀ + n122889₀ + n122889₀²)
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, n127474_0))) → *3_0, rt ∈ Ω(a·n127474₀ + n127474₀ + n127474₀²)

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(n146374_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n146374₀ + n146374₀ + n146374₀²)

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(+(n146374_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(+(n146374_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, n146374_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, n146374_0)))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_L^{Ω(2 + n146374₀)}
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, n146374_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(n146374_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(n132895_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n132895_0), rt ∈ Ω(1 + n132895₀)
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(n140729_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n140729₀ + n140729₀ + n140729₀²)
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(n146374_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n146374₀ + n146374₀ + n146374₀²)
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(n104303_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n104303₀ + n104303₀ + n104303₀²)
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, n113114_0))) → *3_0, rt ∈ Ω(a·n113114₀ + n113114₀ + n113114₀²)
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(n122889_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n122889₀ + n122889₀ + n122889₀²)
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, n127474_0))) → *3_0, rt ∈ Ω(a·n127474₀ + n127474₀ + n127474₀²)

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(n157671_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n157671₀ + n157671₀ + n157671₀²)

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(+(n157671_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(+(n157671_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, n157671_0)))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c))) →_L^{Ω(2 + n157671₀)}
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, n157671_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(n157671_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(n157671_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(n132895_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n132895_0), rt ∈ Ω(1 + n132895₀)
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(n140729_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n140729₀ + n140729₀ + n140729₀²)
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(n146374_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n146374₀ + n146374₀ + n146374₀²)
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(n157671_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n157671₀ + n157671₀ + n157671₀²)
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, n113114_0))) → *3_0, rt ∈ Ω(a·n113114₀ + n113114₀ + n113114₀²)
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(n122889_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n122889₀ + n122889₀ + n122889₀²)
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, n127474_0))) → *3_0, rt ∈ Ω(a·n127474₀ + n127474₀ + n127474₀²)

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, n168706_0))) → *3_0, rt ∈ Ω(a·n168706₀ + n168706₀ + n168706₀²)

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, +(n168706_0, 1)))) →_R^Ω(1)
a__U21(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(+(1, n168706_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, n168706_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, n168706_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, n168706_0)))), mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(a))) →_L^{Ω(2 + n168706₀)}
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, n168706_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(n132895_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n132895_0), rt ∈ Ω(1 + n132895₀)
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(n140729_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n140729₀ + n140729₀ + n140729₀²)
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(n146374_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n146374₀ + n146374₀ + n146374₀²)
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(n157671_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n157671₀ + n157671₀ + n157671₀²)
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, n168706_0))) → *3_0, rt ∈ Ω(a·n168706₀ + n168706₀ + n168706₀²)
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(n122889_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n122889₀ + n122889₀ + n122889₀²)
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, n127474_0))) → *3_0, rt ∈ Ω(a·n127474₀ + n127474₀ + n127474₀²)

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(n180188_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n180188₀ + n180188₀ + n180188₀²)

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(+(n180188_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(+(n180188_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, n180188_0))))) →_L^{Ω(2 + n180188₀)}
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, n180188_0)))) →_R^Ω(1)
s(a__U11(tt, gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n180188_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(n180188_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(n132895_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n132895_0), rt ∈ Ω(1 + n132895₀)
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(n140729_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n140729₀ + n140729₀ + n140729₀²)
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(n146374_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n146374₀ + n146374₀ + n146374₀²)
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(n157671_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n157671₀ + n157671₀ + n157671₀²)
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, n168706_0))) → *3_0, rt ∈ Ω(a·n168706₀ + n168706₀ + n168706₀²)
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(n180188_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n180188₀ + n180188₀ + n180188₀²)
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, n127474_0))) → *3_0, rt ∈ Ω(a·n127474₀ + n127474₀ + n127474₀²)

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(n140729_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n140729₀ + n140729₀ + n140729₀²)

(51) BOUNDS(n^2, INF)

(52) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n132895_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n132895_0), rt ∈ Ω(1 + n132895₀)
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(n140729_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n140729₀ + n140729₀ + n140729₀²)
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(n146374_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n146374₀ + n146374₀ + n146374₀²)
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(n157671_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n157671₀ + n157671₀ + n157671₀²)
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, n168706_0))) → *3_0, rt ∈ Ω(a·n168706₀ + n168706₀ + n168706₀²)
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(n180188_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n180188₀ + n180188₀ + n180188₀²)
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, n127474_0))) → *3_0, rt ∈ Ω(a·n127474₀ + n127474₀ + n127474₀²)

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(n132895_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n132895_0), rt ∈ Ω(1 + n132895₀)
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(n140729_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n140729₀ + n140729₀ + n140729₀²)
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(n146374_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n146374₀ + n146374₀ + n146374₀²)
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(n157671_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n157671₀ + n157671₀ + n157671₀²)
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, n168706_0))) → *3_0, rt ∈ Ω(a·n168706₀ + n168706₀ + n168706₀²)
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(n122889_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n122889₀ + n122889₀ + n122889₀²)
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, n127474_0))) → *3_0, rt ∈ Ω(a·n127474₀ + n127474₀ + n127474₀²)

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(n132895_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n132895_0), rt ∈ Ω(1 + n132895₀)
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(n140729_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n140729₀ + n140729₀ + n140729₀²)
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(n146374_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n146374₀ + n146374₀ + n146374₀²)
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(n157671_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n157671₀ + n157671₀ + n157671₀²)
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, n113114_0))) → *3_0, rt ∈ Ω(a·n113114₀ + n113114₀ + n113114₀²)
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(n122889_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n122889₀ + n122889₀ + n122889₀²)
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, n127474_0))) → *3_0, rt ∈ Ω(a·n127474₀ + n127474₀ + n127474₀²)

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(n132895_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n132895_0), rt ∈ Ω(1 + n132895₀)
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(n140729_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n140729₀ + n140729₀ + n140729₀²)
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(n146374_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n146374₀ + n146374₀ + n146374₀²)
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(n104303_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n104303₀ + n104303₀ + n104303₀²)
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, n113114_0))) → *3_0, rt ∈ Ω(a·n113114₀ + n113114₀ + n113114₀²)
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(n122889_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n122889₀ + n122889₀ + n122889₀²)
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, n127474_0))) → *3_0, rt ∈ Ω(a·n127474₀ + n127474₀ + n127474₀²)

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(n132895_0)) → gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(n132895_0), rt ∈ Ω(1 + n132895₀)
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(n140729_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n140729₀ + n140729₀ + n140729₀²)
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(n95831_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n95831₀ + n95831₀ + n95831₀²)
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(n104303_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n104303₀ + n104303₀ + n104303₀²)
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, n113114_0))) → *3_0, rt ∈ Ω(a·n113114₀ + n113114₀ + n113114₀²)
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(n122889_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n122889₀ + n122889₀ + n122889₀²)
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, n127474_0))) → *3_0, rt ∈ Ω(a·n127474₀ + n127474₀ + n127474₀²)

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(n92616_0), gen_tt:s:0':U11:U12:plus:U21:U22:x2_0(c)) → *3_0, rt ∈ Ω(c·n92616₀ + n92616₀ + n92616₀²)

(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)