`Compfiles/Usa1995P2.lean`

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10 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 1995, Problem 2

A trigonometric map is any one of sin, cos, tan, arcsin, arccos and arctan. Show that given any positive rational number x, one can find a finite sequence of trigonometric maps which take 0 to x. [So we need to show that we can always find a sequence of trigonometric maps tᵢ so that: x₁ = t₀(0), x₂ = t₁(x₁), ..., xₙ = tₙ₋₁(xₙ₋₁), x = tₙ(xₙ).]

Imports (2)

Imported by 1 file

Compfiles.lean

inductive Usa1995P2.TrigMap The six trigonometric maps allowed by the problem line 29 source
Source documentation
The six trigonometric maps allowed by the problem.

no axioms what this means
```
inductive TrigMap where
…
```
def Usa1995P2.TrigMap.apply Evaluate a trigonometric map at a real number line 33 source
Source documentation
Evaluate a trigonometric map at a real number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Usa1995P2.TrigMap, Real
Proof uses: Usa1995P2.TrigMap.apply.match_1, Unit, Real.sin, Real.cos, Real.tan, Real.arcsin, Real.arccos, Real.arctan
```
noncomputable def TrigMap.apply : TrigMap → ℝ → ℝ
  | .sin => Real.sin
  | .cos => Real.cos
  | .tan => Real.tan
  | .arcsin => Real.arcsin
  | .arccos => Real.arccos
  | .arctan => Real.arctan

/-- Apply a finite sequence of trigonometric maps to a starting value. -/
```
Mentions: Usa1995P2.TrigMap
def Usa1995P2.run Apply a finite sequence of trigonometric maps to a starting value line 42 source
Source documentation
Apply a finite sequence of trigonometric maps to a starting value.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: List, Usa1995P2.TrigMap, Real
Proof uses: List.foldl, Usa1995P2.TrigMap.apply
```
noncomputable def run (l : List TrigMap) (x : ℝ) : ℝ := l.foldl (fun a f => f.apply a) x
…
```
Mentions: Usa1995P2.TrigMap, f
lemma Usa1995P2.run_append line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: List, Usa1995P2.TrigMap, Real, Eq, Usa1995P2.run, HAppend.hAppend
Proof uses: of_eq_true, List.foldl, Usa1995P2.TrigMap.apply, Eq.trans, True, congrFun', congrArg, List.foldl_append, eq_self
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def Usa1995P2.chainInv The chain arctan, sin, arccos, tan of trigonometric maps line 52 source
Source documentation
The chain arctan, sin, arccos, tan of trigonometric maps. We show in run_chainInv that it sends any positive x to 1 / x.

no axioms what this means
Statement uses: List, Usa1995P2.TrigMap
Proof uses: List.cons, Usa1995P2.TrigMap.arctan, Usa1995P2.TrigMap.sin, Usa1995P2.TrigMap.arccos, Usa1995P2.TrigMap.tan, List.nil
```
def chainInv : List TrigMap := [.arctan, .sin, .arccos, .tan]
```
Mentions: Usa1995P2.TrigMap
def Usa1995P2.chainSqrt The chain arctan, cos, arctan, sin, arccos, tan of trigonometric maps line 56 source
Source documentation
The chain arctan, cos, arctan, sin, arccos, tan of trigonometric maps. We show in run_chainSqrt that it sends any real x to √(1 + x²).

no axioms what this means
Statement uses: List, Usa1995P2.TrigMap
Proof uses: List.cons, Usa1995P2.TrigMap.arctan, Usa1995P2.TrigMap.cos, Usa1995P2.TrigMap.sin, Usa1995P2.TrigMap.arccos, Usa1995P2.TrigMap.tan, List.nil
```
def chainSqrt : List TrigMap := [.arctan, .cos, .arctan, .sin, .arccos, .tan]
```
Mentions: Usa1995P2.TrigMap
lemma Usa1995P2.run_chainInv Documented declaration: run_chainInv line 61 source
Source documentation
The chain arctan, sin, arccos, tan sends x to 1 / x: with t = arctan x we have sin t = x / √(1 + x²), and if u = arccos (sin t) then tan u = sin u / cos u = √(1 - x²/(1+x²)) / (x/√(1+x²)) = 1 / x.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Eq, Usa1995P2.run, Usa1995P2.chainInv, HDiv.hDiv
Proof uses: Real.sqrt, HAdd.hAdd, HPow.hPow, Nat, Iff.mpr, Real.sqrt_pos, add_pos', Mathlib.Meta.Positivity.pos_of_isNat, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_one, Eq.refl, Bool, Nat.ble, pow_pos, Eq.mpr, id, congrArg, propext, Real.lt_sqrt, LT.lt.le and 138 more
125 instances, coercions and auxiliary constants left out.

Tactic trace: 20 steps
Open to load the checked before-and-after goal states.
```
lemma run_chainInv (x : ℝ) (hx : 0 < x) : run chainInv x = 1 / x := by
…
```
Mentions: Usa1995P2.run, Usa1995P2.chainInv
lemma Usa1995P2.run_chainSqrt The chain arctan, cos, arctan, sin, arccos, tan sends x to √(1 + x²): it is the previous chain applied to cos (arctan x) = 1 / √(1 + x²) line 83 source
Source documentation
The chain arctan, cos, arctan, sin, arccos, tan sends x to √(1 + x²): it is the previous chain applied to cos (arctan x) = 1 / √(1 + x²).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, Usa1995P2.run, Usa1995P2.chainSqrt, Real.sqrt, HAdd.hAdd, OfNat.ofNat, HPow.hPow, Nat
Proof uses: Usa1995P2.chainInv, Real.cos, Real.arctan, rfl, Eq.mpr, id, congrArg, HDiv.hDiv, Usa1995P2.run_chainInv, Real.cos_arctan_pos, Real.cos_arctan, one_div_one_div, Eq.refl
31 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma run_chainSqrt (x : ℝ) : run chainSqrt x = √(1 + x ^ 2) := by
…
```
Mentions: Usa1995P2.run, Usa1995P2.chainSqrt
lemma Usa1995P2.reachable_sqrt_div Every number of the form √(m / n) with m, n positive naturals can be reached from 0 by a finite sequence of trigonometric maps line 94 source
Source documentation
Every number of the form √(m / n) with m, n positive naturals can be reached from 0 by a finite sequence of trigonometric maps. Proof by well-founded induction on m + n (Euclidean algorithm):
- if m = n then √(m/n) = 1 = cos 0;
- if m > n then √(m/n) = √(1 + (√((m-n)/n))²) and (m-n) + n < m + n;
- if m < n then √(m/n) = 1 / √(n/m) and √(n/m) = √(1 + (√((n-m)/m))²) with (n-m) + m < m + n.
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Exists, List, Usa1995P2.TrigMap, Eq, Real, Usa1995P2.run, Real.sqrt, HDiv.hDiv
Proof uses: PSigma.mk, PSigma
12 instances, coercions and auxiliary constants left out.

Tactic trace: 38 steps
Open to load the checked before-and-after goal states.
```
lemma reachable_sqrt_div (m n : ℕ) (hm : 0 < m) (hn : 0 < n) :
    ∃ l : List TrigMap, run l 0 = √((m : ℝ) / n) := by
…
```
Mentions: Usa1995P2.TrigMap, Usa1995P2.run
problem Usa1995P2.usa1995_p2 line 135 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat, LT.lt, OfNat.ofNat, Exists, List, Usa1995P2.TrigMap, Eq, Real, Usa1995P2.run
Proof uses: Int, Rat.num, Iff.mpr, Rat.num_pos, Nat, HPow.hPow, Int.toNat, pow_pos, Real.sqrt, HDiv.hDiv, Rat.den, Usa1995P2.reachable_sqrt_div, Rat.den_pos, Exists.intro, Eq.mpr, id, congrArg, Eq.trans, Eq.symm, Int.cast_natCast and 13 more
86 instances, coercions and auxiliary constants left out.

Tactic trace: 17 steps
Open to load the checked before-and-after goal states.