`Compfiles/Usa1983P5.lean`

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25 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 1983, Problem 5

Consider an open interval of length 1/n on the real number line, where n is a positive integer. Prove that the number of irreducible fractions p/q, with 1 ≤ q ≤ n, contained in the given interval is at most (n + 1) / 2.

Imports (6)

Imported by 1 file

Compfiles.lean

def Usa1983P5.fracs Rationals in the interval with reduced denominator at most n line 34 source
Source comment
Rationals in the interval with reduced denominator at most `n`.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat, Set, Rat
Proof uses: Set.ofPred, And, LE.le, Rat.den, Membership.mem, Set.Ioo, HAdd.hAdd, HDiv.hDiv, OfNat.ofNat
14 instances, coercions and auxiliary constants left out.
```
def fracs (x : ℝ) (n : ℕ) : Set ℚ :=
…
```
def Usa1983P5.OccursDen line 37 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat
Proof uses: Exists, Rat, And, Membership.mem, Set, Usa1983P5.fracs, Eq, Rat.den
1 instances, coercions and auxiliary constants left out.
def Usa1983P5.denoms line 40 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat, Finset
Proof uses: Finset.filter, Usa1983P5.OccursDen, Classical.propDecidable, Finset.Icc, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Usa1983P5.dist_lt_interval_length Openness of the interval gives strict spacing line 45 source
Source comment
Openness of the interval gives strict spacing.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat, LT.lt, OfNat.ofNat, Rat, Membership.mem, Set, Usa1983P5.fracs, abs, HSub.hSub, HDiv.hDiv
Proof uses: HAdd.hAdd, Set.Ioo, And.right, LE.le, Rat.den, Eq.mpr, And, id, Eq, congrArg, propext, abs_sub_lt_iff, And.intro, lt_of_not_ge, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, HMul.hMul, inferInstance, AddMonoidWithOne, Mathlib.Tactic.Ring.of_eq and 63 more
110 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
lemma dist_lt_interval_length
    {x : ℝ} {n : ℕ} (_hn : 0 < n) {r s : ℚ}
    (hr : r ∈ fracs x n) (hs : s ∈ fracs x n) :
    |(r : ℝ) - (s : ℝ)| < 1 / (n : ℝ) := by
…
```
Mentions: Usa1983P5.fracs
lemma Usa1983P5.one_div_den_le_abs_rat line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat, Ne, OfNat.ofNat, LE.le, Real, HDiv.hDiv, Rat.den, abs
Proof uses: Rat.num, cast, Int, Eq.trans, Nat, congrFun', congrArg, Eq.symm, Nat.cast_one, Int.cast_one, Int.one_le_abs, Iff.mpr, Rat.num_ne_zero, LT.lt, Nat.cast_zero, Rat.den_pos, Eq.mpr, id, Eq, Rat.cast_def and 3 more
89 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Usa1983P5.one_div_den_le_dist_of_dvd_den Spacing for denominators related by divisibility line 64 source
Source comment
Spacing for denominators related by divisibility.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat, Nat, Eq, Rat.den, Dvd.dvd, Ne, LE.le, Real, HDiv.hDiv, OfNat.ofNat, abs, HSub.hSub
Proof uses: LT.lt, Eq.mpr, id, congrArg, Eq.symm, Rat.den_pos, Nat.lcm, Rat.sub_den_dvd_lcm, Nat.lcm_eq_right, Eq.mp, Nat.le_of_dvd, Usa1983P5.one_div_den_le_abs_rat, Iff.mpr, sub_ne_zero, one_div_le_one_div_of_le, cast, Eq.trans, congrFun', Nat.cast_zero, LE.le.trans and 4 more
80 instances, coercions and auxiliary constants left out.

Tactic trace: 17 steps
Open to load the checked before-and-after goal states.
```
lemma one_div_den_le_dist_of_dvd_den
    {r s : ℚ} {d e : ℕ}
    (hrd : r.den = d) (hse : s.den = e)
    (hdiv : d ∣ e) (hne : r ≠ s) :
    1 / (e : ℝ) ≤ |(r : ℝ) - (s : ℝ)| := by
…
```
lemma Usa1983P5.same_den_unique line 87 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat, LT.lt, OfNat.ofNat, Rat, Membership.mem, Set, Usa1983P5.fracs, Eq, Rat.den
Proof uses: Decidable.byContradiction, Not, LE.le, HDiv.hDiv, abs, HSub.hSub, Usa1983P5.one_div_den_le_dist_of_dvd_den, Eq.refl, Eq.mpr, Dvd.dvd, id, congrArg, dvd_refl, Usa1983P5.dist_lt_interval_length, Rat.den_pos, one_div_le_one_div_of_le, cast, Eq.symm, Eq.trans, congrFun' and 70 more
106 instances, coercions and auxiliary constants left out.

Tactic trace: 18 steps
Open to load the checked before-and-after goal states.
lemma Usa1983P5.den_mem_denoms line 106 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat, Rat, Membership.mem, Set, Usa1983P5.fracs, Finset, Usa1983P5.denoms, Rat.den
Proof uses: Eq.mpr, Finset.filter, Usa1983P5.OccursDen, Classical.propDecidable, Finset.Icc, OfNat.ofNat, id, Eq, congrArg, Usa1983P5.denoms.eq_1, And, LE.le, Eq.trans, congrFun', And.intro, Rat.den_pos, And.left, Set.Ioo, HAdd.hAdd, HDiv.hDiv and 2 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Usa1983P5.no_dvd_between_occurring_denoms Occurring denominators form a divisibility antichain line 113 source
Source comment
Occurring denominators form a divisibility antichain.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat, LT.lt, OfNat.ofNat, Membership.mem, Finset, Usa1983P5.denoms, Ne, Not, Dvd.dvd
Proof uses: Rat, And, Set, Usa1983P5.fracs, Eq, Rat.den, Exists, False, And.right, LE.le, Eq.mp, Usa1983P5.OccursDen, congrArg, Usa1983P5.OccursDen.eq_1, Finset.filter, Classical.propDecidable, Finset.Icc, Eq.trans, congrFun', Usa1983P5.denoms.eq_1 and 82 more
114 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
```
lemma no_dvd_between_occurring_denoms
    {x : ℝ} {n d e : ℕ} (hn : 0 < n)
    (hd : d ∈ denoms x n) (he : e ∈ denoms x n) (hne : d ≠ e) :
    ¬ d ∣ e := by
…
```
Mentions: Usa1983P5.denoms
lemma Usa1983P5.den_bijOn line 139 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat, LT.lt, OfNat.ofNat, Set.BijOn, Rat, Rat.den, Usa1983P5.fracs, Finset, Usa1983P5.denoms
Proof uses: And.intro, Set.MapsTo, And, Set.InjOn, Set.SurjOn, Membership.mem, Set, Usa1983P5.den_mem_denoms, Eq, Usa1983P5.same_den_unique, Exists, Set.image, And.right, LE.le, Eq.mp, Usa1983P5.OccursDen, congrArg, Usa1983P5.OccursDen.eq_1, Finset.filter, Classical.propDecidable and 5 more
18 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
lemma Usa1983P5.fracs_ncard_eq_denoms_card line 153 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat, LT.lt, OfNat.ofNat, Eq, Set.ncard, Rat, Usa1983P5.fracs, Finset.card, Usa1983P5.denoms
Proof uses: Eq.mp, Finset, congrArg, Set.ncard_coe_finset, Set.BijOn.ncard_eq, Rat.den, Usa1983P5.den_bijOn
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Usa1983P5.fracs_finite line 157 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat, LT.lt, OfNat.ofNat, Set.Finite, Rat, Usa1983P5.fracs
Proof uses: Iff.mpr, Finset, Usa1983P5.denoms, Set.BijOn.finite_iff_finite, Rat.den, Usa1983P5.den_bijOn, Finset.finite_toSet
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def Usa1983P5.twoPart Decompose a into its power-of-two factor and the remaining odd factor line 163 source
Source comment
Decompose `a` into its power-of-two factor and the remaining odd factor. For positive `a`, `oddPart a` is the largest odd divisor.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HPow.hPow, OfNat.ofNat, Finsupp, Nat.factorization
9 instances, coercions and auxiliary constants left out.
```
def twoPart (a : ℕ) : ℕ :=
…
```
def Usa1983P5.oddPart line 166 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HDiv.hDiv, HPow.hPow, OfNat.ofNat, Finsupp, Nat.factorization
11 instances, coercions and auxiliary constants left out.
lemma Usa1983P5.twoPart_mul_oddPart line 169 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, HMul.hMul, Usa1983P5.twoPart, Usa1983P5.oddPart
Proof uses: id, Nat.ordProj_mul_ordCompl_eq_self, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Usa1983P5.oddPart_odd line 172 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Ne, OfNat.ofNat, Odd, Usa1983P5.oddPart
Proof uses: Iff.mp, Not, Even, Nat.not_even_iff_odd, Eq.mpr, Dvd.dvd, id, Eq, congrArg, propext, even_iff_two_dvd, HMod.hMod, HDiv.hDiv, HPow.hPow, Finsupp, Nat.factorization, Eq.mp, Nat.not_dvd_ordCompl, Nat.prime_two
32 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Usa1983P5.oddPart_le_self line 178 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, Usa1983P5.oddPart
Proof uses: id, Nat.ordCompl_le, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Usa1983P5.pos_of_mem_Icc_one line 182 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Membership.mem, Finset, Finset.Icc, OfNat.ofNat, LT.lt
Proof uses: Iff.mp, LE.le, Nat.succ, Nat.succ_le_iff, And.left, And, Finset.mem_Icc
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Usa1983P5.odd_eq_of_half_eq line 186 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Odd, Eq, HDiv.hDiv, OfNat.ofNat
Proof uses: Trans.trans, HAdd.hAdd, HMul.hMul, Eq.symm, Nat.two_mul_div_two_add_one_of_odd, Eq.mpr, id, congrArg, Eq.refl
13 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Usa1983P5.oddPart_index_lt_half line 193 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Membership.mem, Finset, Finset.Icc, OfNat.ofNat, LT.lt, HDiv.hDiv, Usa1983P5.oddPart, HAdd.hAdd
Proof uses: LE.le, And.right, Iff.mp, And, Finset.mem_Icc, Ne, LT.lt.ne', Usa1983P5.pos_of_mem_Icc_one, Odd, Usa1983P5.oddPart_odd, LE.le.trans, Usa1983P5.oddPart_le_self, Eq, HMul.hMul, Nat.two_mul_div_two_add_one_of_odd
25 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Usa1983P5.dvd_or_dvd_of_oddPart_eq line 203 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, Usa1983P5.oddPart, Or, Dvd.dvd
Proof uses: dite, LE.le, Finsupp, Nat.factorization, OfNat.ofNat, Or.inl, HPow.hPow, pow_dvd_pow, HMul.hMul, Exists.intro, Eq.mpr, Usa1983P5.twoPart, id, congrArg, Eq.symm, Usa1983P5.twoPart_mul_oddPart, Usa1983P5.twoPart.eq_1, congr, Lean.Data.AC.Context.eq_of_norm, Lean.Data.AC.Context.mk and 19 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
lemma Usa1983P5.oddIndex_injective_of_no_dvd line 220 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Nat, LE.le, Finset.Icc, OfNat.ofNat, Membership.mem, Ne, Not, Dvd.dvd, Function.Injective, Subtype, Fin, HDiv.hDiv, HAdd.hAdd, Fin.mk, Usa1983P5.oddPart, Usa1983P5.oddPart_index_lt_half, Subtype.property
Proof uses: Eq, Subtype.ext, dite, False.elim, LT.lt.ne', Usa1983P5.pos_of_mem_Icc_one, congrArg, Usa1983P5.odd_eq_of_half_eq, Usa1983P5.oddPart_odd, Or, False, Usa1983P5.dvd_or_dvd_of_oddPart_eq, Ne.symm
31 instances, coercions and auxiliary constants left out.

Tactic trace: 17 steps
Open to load the checked before-and-after goal states.
lemma Usa1983P5.card_le_half_of_no_dvd Largest-odd-divisor pigeonhole step line 243 source
Source comment
Largest-odd-divisor pigeonhole step.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Nat, LE.le, Finset.Icc, OfNat.ofNat, Membership.mem, Ne, Not, Dvd.dvd, Finset.card, HDiv.hDiv, HAdd.hAdd
Proof uses: Subtype, Fin, Fin.mk, Usa1983P5.oddPart, Usa1983P5.oddPart_index_lt_half, Subtype.property, Function.Injective, Usa1983P5.oddIndex_injective_of_no_dvd, Eq.mpr, id, Eq, Eq.mp, Fintype.card, congr, congrArg, Fintype.card_coe, Fintype.card_fin, Fintype.card_le_of_injective
32 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
lemma card_le_half_of_no_dvd
    {s : Finset ℕ} {n : ℕ}
    (hs : s ⊆ Finset.Icc 1 n)
    (hantichain : ∀ a ∈ s, ∀ b ∈ s, a ≠ b → ¬ a ∣ b) :
    s.card ≤ (n + 1) / 2 := by
…
```
lemma Usa1983P5.denoms_subset_Icc line 254 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat, LE.le, Finset, Usa1983P5.denoms, Finset.Icc, OfNat.ofNat
Proof uses: Membership.mem, Eq.mpr, And, id, Eq, Eq.mp, And.left, Usa1983P5.OccursDen, Finset.filter, Classical.propDecidable, congrArg, Usa1983P5.denoms.eq_1
14 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
problem Usa1983P5.usa1983_p5 line 263 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat, LT.lt, OfNat.ofNat, Set, Rat, Set.ofPred, And, LE.le, Rat.den, Membership.mem, Set.Ioo, HAdd.hAdd, HDiv.hDiv, Set.Finite, Set.ncard
Proof uses: id, And.intro, Usa1983P5.fracs, Usa1983P5.fracs_finite, Trans.trans, Eq, Finset.card, Usa1983P5.denoms, Usa1983P5.fracs_ncard_eq_denoms_card, Usa1983P5.card_le_half_of_no_dvd, Usa1983P5.denoms_subset_Icc, Finset, Ne, Usa1983P5.no_dvd_between_occurring_denoms
39 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.