Runtime Complexity (full) proof of /tmp/tmpIiAMPy/MYNAT_nosorts

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), 4 ms)

↳6 typed CpxTrs

↳7 NoRewriteLemmaProof (LOWER BOUND(ID), 493 ms)

↳8 typed CpxTrs

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

↳10 BEST

    ↳11 typed CpxTrs

        ↳12 RewriteLemmaProof (LOWER BOUND(ID), 794 ms)

        ↳13 BEST

            ↳14 typed CpxTrs

                ↳15 RewriteLemmaProof (LOWER BOUND(ID), 330 ms)

                ↳16 BEST

                    ↳17 typed CpxTrs

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

                        ↳19 BEST

                            ↳20 typed CpxTrs

                                ↳21 RewriteLemmaProof (LOWER BOUND(ID), 365 ms)

                                ↳22 BEST

                                    ↳23 typed CpxTrs

                                        ↳24 LowerBoundsProof (⇔, 0 ms)

                                        ↳25 BOUNDS(n^2, INF)

                                    ↳26 typed CpxTrs

                                        ↳27 LowerBoundsProof (⇔, 0 ms)

                                        ↳28 BOUNDS(n^2, INF)

                            ↳29 typed CpxTrs

                                ↳30 LowerBoundsProof (⇔, 0 ms)

                                ↳31 BOUNDS(n^2, INF)

                    ↳32 typed CpxTrs

                        ↳33 LowerBoundsProof (⇔, 0 ms)

                        ↳34 BOUNDS(n^2, INF)

            ↳35 typed CpxTrs

                ↳36 LowerBoundsProof (⇔, 0 ms)

                ↳37 BOUNDS(n^2, INF)

    ↳38 typed CpxTrs

        ↳39 LowerBoundsProof (⇔, 0 ms)

        ↳40 BOUNDS(n^1, INF)

(0) Obligation:

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

a__and(tt, X) → mark(X)
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
a__x(N, 0) → 0
a__x(N, s(M)) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
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__and(tt, X) → mark(X)
a__plus(N, 0') → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
a__x(N, 0') → 0'
a__x(N, s(M)) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
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__and(tt, X) → mark(X)
a__plus(N, 0') → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
a__x(N, 0') → 0'
a__x(N, s(M)) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__x(X1, X2) → x(X1, X2)

Types:
a__and :: tt:0':s:and:plus:x → tt:0':s:and:plus:x → tt:0':s:and:plus:x
tt :: tt:0':s:and:plus:x
mark :: tt:0':s:and:plus:x → tt:0':s:and:plus:x
a__plus :: tt:0':s:and:plus:x → tt:0':s:and:plus:x → tt:0':s:and:plus:x
0' :: tt:0':s:and:plus:x
s :: tt:0':s:and:plus:x → tt:0':s:and:plus:x
a__x :: tt:0':s:and:plus:x → tt:0':s:and:plus:x → tt:0':s:and:plus:x
and :: tt:0':s:and:plus:x → tt:0':s:and:plus:x → tt:0':s:and:plus:x
plus :: tt:0':s:and:plus:x → tt:0':s:and:plus:x → tt:0':s:and:plus:x
x :: tt:0':s:and:plus:x → tt:0':s:and:plus:x → tt:0':s:and:plus:x
hole_tt:0':s:and:plus:x1_0 :: tt:0':s:and:plus:x
gen_tt:0':s:and:plus:x2_0 :: Nat → tt:0':s:and:plus:x

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

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

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

(6) Obligation:

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

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

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

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

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

(8) Obligation:

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

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

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

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_tt:0':s:and:plus:x2_0(n11447_0)) → gen_tt:0':s:and:plus:x2_0(n11447_0), rt ∈ Ω(1 + n11447₀)

Induction Base:
mark(gen_tt:0':s:and:plus:x2_0(0)) →_R^Ω(1)
tt

Induction Step:
mark(gen_tt:0':s:and:plus:x2_0(+(n11447_0, 1))) →_R^Ω(1)
s(mark(gen_tt:0':s:and:plus:x2_0(n11447_0))) →_IH
s(gen_tt:0':s:and:plus:x2_0(c11448_0))

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

(10) Complex Obligation (BEST)

(11) Obligation:

Lemmas:
mark(gen_tt:0':s:and:plus:x2_0(n11447_0)) → gen_tt:0':s:and:plus:x2_0(n11447_0), rt ∈ Ω(1 + n11447₀)

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

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

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

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n12552_0))) → *3_0, rt ∈ Ω(a·n12552₀ + n12552₀ + n12552₀²)

Induction Base:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, 0)))

Induction Step:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, +(n12552_0, 1)))) →_R^Ω(1)
a__plus(a__x(mark(gen_tt:0':s:and:plus:x2_0(a)), mark(gen_tt:0':s:and:plus:x2_0(+(1, n12552_0)))), mark(gen_tt:0':s:and:plus:x2_0(a))) →_L^{Ω(1 + a)}
a__plus(a__x(gen_tt:0':s:and:plus:x2_0(a), mark(gen_tt:0':s:and:plus:x2_0(+(1, n12552_0)))), mark(gen_tt:0':s:and:plus:x2_0(a))) →_L^{Ω(2 + n12552₀)}
a__plus(a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n12552_0))), mark(gen_tt:0':s:and:plus:x2_0(a))) →_IH
a__plus(*3_0, mark(gen_tt:0':s:and:plus:x2_0(a))) →_L^{Ω(1 + a)}
a__plus(*3_0, gen_tt:0':s:and:plus:x2_0(a))

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

(13) Complex Obligation (BEST)

(14) Obligation:

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

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

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

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__plus(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n15954_0))) → *3_0, rt ∈ Ω(a·n15954₀ + n15954₀ + n15954₀²)

Induction Base:
a__plus(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, 0)))

Induction Step:
a__plus(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, +(n15954_0, 1)))) →_R^Ω(1)
s(a__plus(mark(gen_tt:0':s:and:plus:x2_0(a)), mark(gen_tt:0':s:and:plus:x2_0(+(1, n15954_0))))) →_L^{Ω(1 + a)}
s(a__plus(gen_tt:0':s:and:plus:x2_0(a), mark(gen_tt:0':s:and:plus:x2_0(+(1, n15954_0))))) →_L^{Ω(2 + n15954₀)}
s(a__plus(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n15954_0)))) →_IH
s(*3_0)

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

(16) Complex Obligation (BEST)

(17) Obligation:

Lemmas:
mark(gen_tt:0':s:and:plus:x2_0(n11447_0)) → gen_tt:0':s:and:plus:x2_0(n11447_0), rt ∈ Ω(1 + n11447₀)
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n12552_0))) → *3_0, rt ∈ Ω(a·n12552₀ + n12552₀ + n12552₀²)
a__plus(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n15954_0))) → *3_0, rt ∈ Ω(a·n15954₀ + n15954₀ + n15954₀²)

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

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

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

(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_tt:0':s:and:plus:x2_0(n17693_0)) → gen_tt:0':s:and:plus:x2_0(n17693_0), rt ∈ Ω(1 + n17693₀)

Induction Base:
mark(gen_tt:0':s:and:plus:x2_0(0)) →_R^Ω(1)
tt

Induction Step:
mark(gen_tt:0':s:and:plus:x2_0(+(n17693_0, 1))) →_R^Ω(1)
s(mark(gen_tt:0':s:and:plus:x2_0(n17693_0))) →_IH
s(gen_tt:0':s:and:plus:x2_0(c17694_0))

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

(19) Complex Obligation (BEST)

(20) Obligation:

Lemmas:
mark(gen_tt:0':s:and:plus:x2_0(n17693_0)) → gen_tt:0':s:and:plus:x2_0(n17693_0), rt ∈ Ω(1 + n17693₀)
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n12552_0))) → *3_0, rt ∈ Ω(a·n12552₀ + n12552₀ + n12552₀²)
a__plus(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n15954_0))) → *3_0, rt ∈ Ω(a·n15954₀ + n15954₀ + n15954₀²)

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

The following defined symbols remain to be analysed:
a__x

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

(21) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n19004_0))) → *3_0, rt ∈ Ω(a·n19004₀ + n19004₀ + n19004₀²)

Induction Base:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, 0)))

Induction Step:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, +(n19004_0, 1)))) →_R^Ω(1)
a__plus(a__x(mark(gen_tt:0':s:and:plus:x2_0(a)), mark(gen_tt:0':s:and:plus:x2_0(+(1, n19004_0)))), mark(gen_tt:0':s:and:plus:x2_0(a))) →_L^{Ω(1 + a)}
a__plus(a__x(gen_tt:0':s:and:plus:x2_0(a), mark(gen_tt:0':s:and:plus:x2_0(+(1, n19004_0)))), mark(gen_tt:0':s:and:plus:x2_0(a))) →_L^{Ω(2 + n19004₀)}
a__plus(a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n19004_0))), mark(gen_tt:0':s:and:plus:x2_0(a))) →_IH
a__plus(*3_0, mark(gen_tt:0':s:and:plus:x2_0(a))) →_L^{Ω(1 + a)}
a__plus(*3_0, gen_tt:0':s:and:plus:x2_0(a))

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

(22) Complex Obligation (BEST)

(23) Obligation:

Lemmas:
mark(gen_tt:0':s:and:plus:x2_0(n17693_0)) → gen_tt:0':s:and:plus:x2_0(n17693_0), rt ∈ Ω(1 + n17693₀)
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n19004_0))) → *3_0, rt ∈ Ω(a·n19004₀ + n19004₀ + n19004₀²)
a__plus(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n15954_0))) → *3_0, rt ∈ Ω(a·n15954₀ + n15954₀ + n15954₀²)

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n19004_0))) → *3_0, rt ∈ Ω(a·n19004₀ + n19004₀ + n19004₀²)

(25) BOUNDS(n^2, INF)

(26) Obligation:

Lemmas:
mark(gen_tt:0':s:and:plus:x2_0(n17693_0)) → gen_tt:0':s:and:plus:x2_0(n17693_0), rt ∈ Ω(1 + n17693₀)
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n19004_0))) → *3_0, rt ∈ Ω(a·n19004₀ + n19004₀ + n19004₀²)
a__plus(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n15954_0))) → *3_0, rt ∈ Ω(a·n15954₀ + n15954₀ + n15954₀²)

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n19004_0))) → *3_0, rt ∈ Ω(a·n19004₀ + n19004₀ + n19004₀²)

(28) BOUNDS(n^2, INF)

(29) Obligation:

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

No more defined symbols left to analyse.

(30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n12552_0))) → *3_0, rt ∈ Ω(a·n12552₀ + n12552₀ + n12552₀²)

(31) BOUNDS(n^2, INF)

(32) Obligation:

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

No more defined symbols left to analyse.

(33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n12552_0))) → *3_0, rt ∈ Ω(a·n12552₀ + n12552₀ + n12552₀²)

(34) BOUNDS(n^2, INF)

(35) Obligation:

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

No more defined symbols left to analyse.

(36) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
a__x(gen_tt:0':s:and:plus:x2_0(a), gen_tt:0':s:and:plus:x2_0(+(1, n12552_0))) → *3_0, rt ∈ Ω(a·n12552₀ + n12552₀ + n12552₀²)

(37) BOUNDS(n^2, INF)

(38) Obligation:

Lemmas:
mark(gen_tt:0':s:and:plus:x2_0(n11447_0)) → gen_tt:0':s:and:plus:x2_0(n11447_0), rt ∈ Ω(1 + n11447₀)

Generator Equations:
gen_tt:0':s:and:plus:x2_0(0) ⇔ tt
gen_tt:0':s:and:plus:x2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus:x2_0(x))

No more defined symbols left to analyse.

(39) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_tt:0':s:and:plus:x2_0(n11447_0)) → gen_tt:0':s:and:plus:x2_0(n11447_0), rt ∈ Ω(1 + n11447₀)

(0) Obligation:

(1) RenamingProof (EQUIVALENT transformation)

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

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

(8) Obligation:

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(10) Complex Obligation (BEST)

(11) Obligation:

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(13) Complex Obligation (BEST)

(14) Obligation:

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(16) Complex Obligation (BEST)

(17) Obligation:

(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(19) Complex Obligation (BEST)

(20) Obligation:

(21) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(22) Complex Obligation (BEST)

(23) Obligation:

(24) LowerBoundsProof (EQUIVALENT transformation)

(25) BOUNDS(n^2, INF)

(26) Obligation:

(27) LowerBoundsProof (EQUIVALENT transformation)

(28) BOUNDS(n^2, INF)

(29) Obligation:

(30) LowerBoundsProof (EQUIVALENT transformation)

(31) BOUNDS(n^2, INF)

(32) Obligation:

(33) LowerBoundsProof (EQUIVALENT transformation)

(34) BOUNDS(n^2, INF)

(35) Obligation:

(36) LowerBoundsProof (EQUIVALENT transformation)

(37) BOUNDS(n^2, INF)

(38) Obligation:

(39) LowerBoundsProof (EQUIVALENT transformation)

(40) BOUNDS(n^1, INF)