`Compfiles/Imo1978P5.lean`

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4 declarations, 0 contain sorry, 0 declared as axioms

International Mathematical Olympiad 1978, Problem 5

Let a_k be a sequence of distinct positive integers for k = 1,2,3, ...

Prove that for all natural numbers n, we have:

sum_{k=1}^{n} a(k)/(k^2) >= sum_{k=1}^{n} (1/k).

Imports (2)

Imported by 1 file

Compfiles.lean

lemma Imo1978P5.triangular_le_sum A finset of m distinct positive integers sums to at least 1 + 2 + ⋯ + m line 42 source
Source documentation
A finset of m distinct positive integers sums to at least 1 + 2 + ⋯ + m.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, Eq, Finset.card, Membership.mem, LE.le, OfNat.ofNat, Finset.sum, Finset.Icc
Proof uses: Nat.recAux, of_eq_true, Eq.trans, True, congrFun', congrArg, Finset.sum_congr, EmptyCollection.emptyCollection, Finset.Icc_eq_empty_of_lt, LT.lt, eq_self, Eq.refl, HAdd.hAdd, Finset.Nonempty, Iff.mp, Finset.card_pos, Finset.max', Finset.max'_mem, Finset.card_le_card, Iff.mpr and 16 more
52 instances, coercions and auxiliary constants left out.

Tactic trace: 18 steps
Open to load the checked before-and-after goal states.
```
lemma triangular_le_sum :
    ∀ (m : ℕ) (T : Finset ℕ), T.card = m → (∀ x ∈ T, 1 ≤ x) →
    ∑ k ∈ Icc 1 m, k ≤ ∑ x ∈ T, x := by
…
```
lemma Imo1978P5.triangular_le_partial_sum The partial sums of a sequence of distinct positive integers dominate the partial sums of the identity line 66 source
Source documentation
The partial sums of a sequence of distinct positive integers dominate the partial sums of the identity.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Ne, LE.le, Finset.sum, Finset.Icc
Proof uses: Set.InjOn, Finset, Membership.mem, Set, Eq, Decidable.byContradiction, Not, Eq.mp, Set.Icc, And, congrArg, propext, Set.mem_Icc, Finset.coe_Icc, Finset.card, Finset.image, Eq.mpr, id, Finset.card_image_of_injOn, HSub.hSub and 10 more
28 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
```
lemma triangular_le_partial_sum (f : ℕ → ℕ)
    (h₀ : ∀ m : ℕ, 0 < m → 0 < f m)
    (h₁ : ∀ p q : ℕ, 0 < p → 0 < q → p ≠ q → f p ≠ f q)
    (m : ℕ) :
    ∑ k ∈ Icc 1 m, k ≤ ∑ k ∈ Icc 1 m, f k := by
…
```
Mentions: f
lemma Imo1978P5.abel_compare Abel summation comparison: if the partial sums of h are dominated by the partial sums of g, then the same holds for sums weighted by any nonnegative decreasing sequence w line 90 source
Source documentation
Abel summation comparison: if the partial sums of h are dominated by the partial sums of g, then the same holds for sums weighted by any nonnegative decreasing sequence w.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Membership.mem, Finset, Finset.Icc, OfNat.ofNat, LE.le, Finset.sum, HMul.hMul
Proof uses: Nat.recAux, of_eq_true, Eq.trans, True, congr, congrArg, Finset.sum_congr, EmptyCollection.emptyCollection, Finset.Icc_eq_empty_of_lt, LT.lt, Eq, eq_self, Eq.refl, HAdd.hAdd, HSub.hSub, Eq.mpr, id, Finset.sum_Icc_succ_top, sub_mul, Finset.sum_sub_distrib and 62 more
114 instances, coercions and auxiliary constants left out.

Tactic trace: 28 steps
Open to load the checked before-and-after goal states.
```
lemma abel_compare (n : ℕ) (w g h : ℕ → ℝ)
    (hw : ∀ k ∈ Icc 1 n, 0 ≤ w k)
    (hanti : ∀ j k : ℕ, 1 ≤ j → j ≤ k → k ≤ n → w k ≤ w j)
    (hps : ∀ m : ℕ, 1 ≤ m → m ≤ n → ∑ k ∈ Icc 1 m, h k ≤ ∑ k ∈ Icc 1 m, g k) :
    ∑ k ∈ Icc 1 n, w k * h k ≤ ∑ k ∈ Icc 1 n, w k * g k := by
…
```
Mentions: h
problem Imo1978P5.imo1978_p5 line 122 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Ne, LE.le, Real, Finset.sum, Finset.Icc, HDiv.hDiv, HPow.hPow
Proof uses: HMul.hMul, Imo1978P5.abel_compare, Membership.mem, Finset, Mathlib.Meta.Positivity.div_nonneg_of_pos_of_nonneg, Mathlib.Meta.Positivity.pos_of_isNat, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_one, Eq.refl, Bool, Nat.ble, Even.pow_nonneg, even_two_mul, cast, Eq.symm, Eq.trans, congrFun', congrArg, Nat.cast_zero, div_le_div₀ and 58 more
143 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.