Runtime Complexity (full) proof of /tmp/tmpi4Uyze/PEANO_nosorts

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

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 191 ms)

↳2 BOUNDS(n^1, 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), 402 ms)

    ↳10 typed CpxTrs

    ↳11 RewriteLemmaProof (LOWER BOUND(ID), 96 ms)

    ↳12 BEST

        ↳13 typed CpxTrs

            ↳14 RewriteLemmaProof (LOWER BOUND(ID), 569 ms)

            ↳15 BEST

                ↳16 typed CpxTrs

                    ↳17 RewriteLemmaProof (LOWER BOUND(ID), 0 ms)

                    ↳18 BEST

                        ↳19 typed CpxTrs

                            ↳20 LowerBoundsProof (⇔, 0 ms)

                            ↳21 BOUNDS(n^2, INF)

                        ↳22 typed CpxTrs

                            ↳23 LowerBoundsProof (⇔, 0 ms)

                            ↳24 BOUNDS(n^2, INF)

                ↳25 typed CpxTrs

                    ↳26 LowerBoundsProof (⇔, 0 ms)

                    ↳27 BOUNDS(n^2, INF)

        ↳28 typed CpxTrs

            ↳29 LowerBoundsProof (⇔, 0 ms)

            ↳30 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)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(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)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
a__and(tt, and(tt, X26_3)) →⁺ a__and(tt, X26_3)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [X26_3 / and(tt, X26_3)].
The result substitution is [ ].

(2) BOUNDS(n^1, 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__and(tt, X) → mark(X)
a__plus(N, 0') → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(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)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) 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)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(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)

Types:
a__and :: tt:0':s:and:plus → tt:0':s:and:plus → tt:0':s:and:plus
tt :: tt:0':s:and:plus
mark :: tt:0':s:and:plus → tt:0':s:and:plus
a__plus :: tt:0':s:and:plus → tt:0':s:and:plus → tt:0':s:and:plus
0' :: tt:0':s:and:plus
s :: tt:0':s:and:plus → tt:0':s:and:plus
and :: tt:0':s:and:plus → tt:0':s:and:plus → tt:0':s:and:plus
plus :: tt:0':s:and:plus → tt:0':s:and:plus → tt:0':s:and:plus
hole_tt:0':s:and:plus1_0 :: tt:0':s:and:plus
gen_tt:0':s:and:plus2_0 :: Nat → tt:0':s:and:plus

(7) OrderProof (LOWER BOUND(ID) transformation)

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

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

(8) Obligation:

Generator Equations:
gen_tt:0':s:and:plus2_0(0) ⇔ tt
gen_tt:0':s:and:plus2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus2_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

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

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

(10) Obligation:

Generator Equations:
gen_tt:0':s:and:plus2_0(0) ⇔ tt
gen_tt:0':s:and:plus2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus2_0(x))

The following defined symbols remain to be analysed:
mark

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

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

Proved the following rewrite lemma:
mark(gen_tt:0':s:and:plus2_0(n5360_0)) → gen_tt:0':s:and:plus2_0(n5360_0), rt ∈ Ω(1 + n5360₀)

Induction Base:
mark(gen_tt:0':s:and:plus2_0(0)) →_R^Ω(1)
tt

Induction Step:
mark(gen_tt:0':s:and:plus2_0(+(n5360_0, 1))) →_R^Ω(1)
s(mark(gen_tt:0':s:and:plus2_0(n5360_0))) →_IH
s(gen_tt:0':s:and:plus2_0(c5361_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:0':s:and:plus2_0(n5360_0)) → gen_tt:0':s:and:plus2_0(n5360_0), rt ∈ Ω(1 + n5360₀)

Generator Equations:
gen_tt:0':s:and:plus2_0(0) ⇔ tt
gen_tt:0':s:and:plus2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus2_0(x))

The following defined symbols remain to be analysed:
a__plus

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

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

Proved the following rewrite lemma:
a__plus(gen_tt:0':s:and:plus2_0(a), gen_tt:0':s:and:plus2_0(+(1, n5915_0))) → *3_0, rt ∈ Ω(a·n5915₀ + n5915₀ + n5915₀²)

Induction Base:
a__plus(gen_tt:0':s:and:plus2_0(a), gen_tt:0':s:and:plus2_0(+(1, 0)))

Induction Step:
a__plus(gen_tt:0':s:and:plus2_0(a), gen_tt:0':s:and:plus2_0(+(1, +(n5915_0, 1)))) →_R^Ω(1)
s(a__plus(mark(gen_tt:0':s:and:plus2_0(a)), mark(gen_tt:0':s:and:plus2_0(+(1, n5915_0))))) →_L^{Ω(1 + a)}
s(a__plus(gen_tt:0':s:and:plus2_0(a), mark(gen_tt:0':s:and:plus2_0(+(1, n5915_0))))) →_L^{Ω(2 + n5915₀)}
s(a__plus(gen_tt:0':s:and:plus2_0(a), gen_tt:0':s:and:plus2_0(+(1, n5915_0)))) →_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:

Lemmas:
mark(gen_tt:0':s:and:plus2_0(n5360_0)) → gen_tt:0':s:and:plus2_0(n5360_0), rt ∈ Ω(1 + n5360₀)
a__plus(gen_tt:0':s:and:plus2_0(a), gen_tt:0':s:and:plus2_0(+(1, n5915_0))) → *3_0, rt ∈ Ω(a·n5915₀ + n5915₀ + n5915₀²)

Generator Equations:
gen_tt:0':s:and:plus2_0(0) ⇔ tt
gen_tt:0':s:and:plus2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus2_0(x))

The following defined symbols remain to be analysed:
mark

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

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

Proved the following rewrite lemma:
mark(gen_tt:0':s:and:plus2_0(n7074_0)) → gen_tt:0':s:and:plus2_0(n7074_0), rt ∈ Ω(1 + n7074₀)

Induction Base:
mark(gen_tt:0':s:and:plus2_0(0)) →_R^Ω(1)
tt

Induction Step:
mark(gen_tt:0':s:and:plus2_0(+(n7074_0, 1))) →_R^Ω(1)
s(mark(gen_tt:0':s:and:plus2_0(n7074_0))) →_IH
s(gen_tt:0':s:and:plus2_0(c7075_0))

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

(18) Complex Obligation (BEST)

(19) Obligation:

Lemmas:
mark(gen_tt:0':s:and:plus2_0(n7074_0)) → gen_tt:0':s:and:plus2_0(n7074_0), rt ∈ Ω(1 + n7074₀)
a__plus(gen_tt:0':s:and:plus2_0(a), gen_tt:0':s:and:plus2_0(+(1, n5915_0))) → *3_0, rt ∈ Ω(a·n5915₀ + n5915₀ + n5915₀²)

Generator Equations:
gen_tt:0':s:and:plus2_0(0) ⇔ tt
gen_tt:0':s:and:plus2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus2_0(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
a__plus(gen_tt:0':s:and:plus2_0(a), gen_tt:0':s:and:plus2_0(+(1, n5915_0))) → *3_0, rt ∈ Ω(a·n5915₀ + n5915₀ + n5915₀²)

(21) BOUNDS(n^2, INF)

(22) Obligation:

Generator Equations:
gen_tt:0':s:and:plus2_0(0) ⇔ tt
gen_tt:0':s:and:plus2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus2_0(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
a__plus(gen_tt:0':s:and:plus2_0(a), gen_tt:0':s:and:plus2_0(+(1, n5915_0))) → *3_0, rt ∈ Ω(a·n5915₀ + n5915₀ + n5915₀²)

(24) BOUNDS(n^2, INF)

(25) Obligation:

Generator Equations:
gen_tt:0':s:and:plus2_0(0) ⇔ tt
gen_tt:0':s:and:plus2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus2_0(x))

No more defined symbols left to analyse.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
a__plus(gen_tt:0':s:and:plus2_0(a), gen_tt:0':s:and:plus2_0(+(1, n5915_0))) → *3_0, rt ∈ Ω(a·n5915₀ + n5915₀ + n5915₀²)

(27) BOUNDS(n^2, INF)

(28) Obligation:

Lemmas:
mark(gen_tt:0':s:and:plus2_0(n5360_0)) → gen_tt:0':s:and:plus2_0(n5360_0), rt ∈ Ω(1 + n5360₀)

Generator Equations:
gen_tt:0':s:and:plus2_0(0) ⇔ tt
gen_tt:0':s:and:plus2_0(+(x, 1)) ⇔ s(gen_tt:0':s:and:plus2_0(x))

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_tt:0':s:and:plus2_0(n5360_0)) → gen_tt:0':s:and:plus2_0(n5360_0), rt ∈ Ω(1 + n5360₀)

(0) Obligation:

(1) DecreasingLoopProof (EQUIVALENT transformation)

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

(4) Obligation:

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

(6) Obligation:

(7) OrderProof (LOWER BOUND(ID) transformation)

(8) Obligation:

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

(10) Obligation:

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

(12) Complex Obligation (BEST)

(13) Obligation:

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

(15) Complex Obligation (BEST)

(16) Obligation:

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

(18) Complex Obligation (BEST)

(19) Obligation:

(20) LowerBoundsProof (EQUIVALENT transformation)

(21) BOUNDS(n^2, INF)

(22) Obligation:

(23) LowerBoundsProof (EQUIVALENT transformation)

(24) BOUNDS(n^2, INF)

(25) Obligation:

(26) LowerBoundsProof (EQUIVALENT transformation)

(27) BOUNDS(n^2, INF)

(28) Obligation:

(29) LowerBoundsProof (EQUIVALENT transformation)

(30) BOUNDS(n^1, INF)