64313.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

8 declarations, 1 contain sorry, 0 declared as axioms

Integer part of area of a regular polygon with $n$ sides each of length 1

The area of a regular $n$-gon with side length 1 is given by $\frac{n}{4} \cot(\pi / n) = \frac{n}{4 \tan(\pi / n)}$.

References:

A064313

Imports (1)

FormalConjecturesUtil

def OeisA64313.area The exact area of a regular $n$-gon with side length 1 line 35 source
Source documentation
The exact area of a regular $n$-gon with side length 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: HDiv.hDiv, HMul.hMul, OfNat.ofNat, Real.tan, Real.pi
9 instances, coercions and auxiliary constants left out.
```
noncomputable def area (n : ℕ) : ℝ :=
…
```
def OeisA64313.a Integer part of area of a regular polygon with $n$ sides each of length 1 line 39 source
Source documentation
Integer part of area of a regular polygon with $n$ sides each of length 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Int.toNat, Int.floor, Real, OeisA64313.area
3 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA64313.a_1 line 43 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA64313.a, OfNat.ofNat
Proof uses: id, Real, HDiv.hDiv, Real.pi, of_eq_true, Eq.trans, True, congrFun', congrArg, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_natCast, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Nat.cast_one, div_one, eq_self, eq_true, True.intro, Eq.mpr, Int.toNat and 16 more
40 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem OeisA64313.a_2 line 50 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA64313.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Int.toNat, Int.floor, Real, HDiv.hDiv, HMul.hMul, Real.tan, Real.pi, congrArg, Real.tan_pi_div_two, of_eq_true, Eq.trans, True, congrFun', Int, Mathlib.Meta.NormNum.IsNat.to_eq, inferInstance, AddMonoidWithOne, Rat.isNat_intFloor and 13 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA64313.a_3 line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA64313.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Int.toNat, Int.floor, Real, HDiv.hDiv, HMul.hMul, Real.tan, Real.pi, Real.sqrt, congrArg, Real.tan_pi_div_three, Int, And, LE.le, LT.lt, HAdd.hAdd, propext, Int.floor_eq_iff, congr and 81 more
96 instances, coercions and auxiliary constants left out.

Tactic trace: 17 steps
Open to load the checked before-and-after goal states.
theorem OeisA64313.a_4 line 71 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA64313.a, OfNat.ofNat
Proof uses: id, Real, HDiv.hDiv, HMul.hMul, Real.tan, Real.pi, Eq.mpr, congrArg, Real.tan_pi_div_four, Mathlib.Tactic.Ring.of_eq, HAdd.hAdd, Nat.rawCast, Mathlib.Meta.NormNum.instAddMonoidWithOne', Mathlib.Tactic.Ring.Common.div_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Mathlib.Tactic.Ring.Common.mul_congr, Nat.cast_one and 36 more
62 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
theorem OeisA64313.a_6 line 84 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA64313.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Int.toNat, Int.floor, Real, HDiv.hDiv, HMul.hMul, Real.tan, Real.pi, Real.sqrt, congrArg, Real.tan_pi_div_six, LT.lt, Iff.mpr, Real.sqrt_pos, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_zero, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo and 134 more
111 instances, coercions and auxiliary constants left out.

Tactic trace: 27 steps
Open to load the checked before-and-after goal states.
theorem OeisA64313.conjecture "Usually (perhaps always?) $\lfloor n^2 / (4\pi) - \pi / 12 \rfloor$ for a polygon of circumference $n$ sorry line 110 source
Source documentation
"Usually (perhaps always?) $\lfloor n^2 / (4\pi) - \pi / 12 \rfloor$ for a polygon of circumference $n$. Note that the area of a circle with circumference $C$ is $C^2 / (4\pi)$."

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Eq, OeisA64313.a, Int.toNat, Int.floor, Real, HSub.hSub, HDiv.hDiv, HPow.hPow, HMul.hMul, Real.pi
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture (n : ℕ) (hn : 3 ≤ n) :
    a n = (Int.floor ((n : ℝ) ^ 2 / (4 * Real.pi) - Real.pi / 12)).toNat := by
…
```
Mentions: OeisA64313.a