Runtime Complexity (full) proof of /tmp/tmpA6OeE4/MYNAT_nosorts_noand

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

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 791 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), 802 ms)

    ↳10 typed CpxTrs

    ↳11 NoRewriteLemmaProof (LOWER BOUND(ID), 284 ms)

    ↳12 typed CpxTrs

    ↳13 RewriteLemmaProof (LOWER BOUND(ID), 128 ms)

    ↳14 BEST

        ↳15 typed CpxTrs

            ↳16 RewriteLemmaProof (LOWER BOUND(ID), 221 ms)

            ↳17 BEST

                ↳18 typed CpxTrs

                    ↳19 RewriteLemmaProof (LOWER BOUND(ID), 471 ms)

                    ↳20 BEST

                        ↳21 typed CpxTrs

                            ↳22 RewriteLemmaProof (LOWER BOUND(ID), 366 ms)

                            ↳23 BEST

                                ↳24 typed CpxTrs

                                    ↳25 RewriteLemmaProof (LOWER BOUND(ID), 376 ms)

                                    ↳26 BEST

                                        ↳27 typed CpxTrs

                                            ↳28 RewriteLemmaProof (LOWER BOUND(ID), 122 ms)

                                            ↳29 BEST

                                                ↳30 typed CpxTrs

                                                    ↳31 RewriteLemmaProof (LOWER BOUND(ID), 148 ms)

                                                    ↳32 BEST

                                                        ↳33 typed CpxTrs

                                                            ↳34 RewriteLemmaProof (LOWER BOUND(ID), 58 ms)

                                                            ↳35 BEST

                                                                ↳36 typed CpxTrs

                                                                    ↳37 RewriteLemmaProof (LOWER BOUND(ID), 72 ms)

                                                                    ↳38 BEST

                                                                        ↳39 typed CpxTrs

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

                                                                            ↳41 BEST

                                                                                ↳42 typed CpxTrs

                                                                                    ↳43 RewriteLemmaProof (LOWER BOUND(ID), 238 ms)

                                                                                    ↳44 BEST

                                                                                        ↳45 typed CpxTrs

                                                                                            ↳46 RewriteLemmaProof (LOWER BOUND(ID), 198 ms)

                                                                                            ↳47 BEST

                                                                                                ↳48 typed CpxTrs

                                                                                                    ↳49 RewriteLemmaProof (LOWER BOUND(ID), 52 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: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

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

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

Infered types.

(6) 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

(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(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).

(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(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).

(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(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).

(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, 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).

(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(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

(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(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).

(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, 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).

(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(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).

(35) Complex Obligation (BEST)

(36) 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

(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(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).

(38) Complex Obligation (BEST)

(39) 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

(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(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).

(41) Complex Obligation (BEST)

(42) 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

(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(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).

(44) Complex Obligation (BEST)

(45) 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

(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, 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).

(47) Complex Obligation (BEST)

(48) 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

(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(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).

(50) Complex Obligation (BEST)

(51) 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.

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

(53) BOUNDS(n^2, INF)

(54) 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.

(55) LowerBoundsProof (EQUIVALENT transformation)

(56) BOUNDS(n^2, INF)

(57) 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.

(58) LowerBoundsProof (EQUIVALENT transformation)

(59) BOUNDS(n^2, INF)

(60) 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.

(61) LowerBoundsProof (EQUIVALENT transformation)

(62) BOUNDS(n^2, INF)

(63) 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.

(64) LowerBoundsProof (EQUIVALENT transformation)

(65) BOUNDS(n^2, INF)

(66) 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.

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

(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)