`Compfiles/Usa1983P1.lean`

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7 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 1983, Problem 1

If six points are chosen sequentially at random on the circumference of a circle, what is the probability that the triangle formed by the first three is disjoint from that formed by the second three?

Imports (3)

Imported by 1 file

Compfiles.lean

abbrev Usa1983P1.Ordering Documented declaration: Ordering line 32 source
Source documentation
The cyclic order in which the six chosen points appear around the circle: σ i is the position of the i-th chosen point among the six positions 0, …, 5 listed in clockwise order. Only the cyclic order of the six points matters for this problem, and all 6! orders are equally likely, so we take the uniform distribution on Equiv.Perm (Fin 6) as the sample space.

no axioms what this means
Proof uses: Equiv.Perm, Fin, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.
```
abbrev Ordering := Equiv.Perm (Fin 6)
```
def Usa1983P1.Favorable The event that the triangle formed by the first three points is disjoint from the triangle formed by the last three points line 39 source
Source documentation
The event that the triangle formed by the first three points is disjoint from the triangle formed by the last three points. With six points on a circle, the two triangles are disjoint iff the two vertex sets do not interleave around the circle, which for 3 + 3 points means that the first three points occupy three (cyclically) consecutive positions.

standard axioms: Quot.sound, propext what this means
Statement uses: Usa1983P1.Ordering
Proof uses: Exists, Fin, OfNat.ofNat, Nat, Eq, Finset, Insert.insert, Singleton.singleton, HAdd.hAdd
11 instances, coercions and auxiliary constants left out.
```
def Favorable (σ : Ordering) : Prop :=
…
```
Mentions: Usa1983P1.Ordering
instance Usa1983P1.(instance at line 42) line 42 source
determine Usa1983P1.solution line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: ENNReal
Proof uses: HDiv.hDiv, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.
lemma Usa1983P1.card_favorable Documented declaration: card_favorable line 54 source
Source documentation
The number of cyclic orders in which the first three points occupy three consecutive positions: 6 choices for the starting position of the block, times 3! orders of the first three points within the block, times 3! orders of the remaining three points. Verified by exhaustive enumeration.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Finset.card, Usa1983P1.Ordering, Finset.filter, Usa1983P1.Favorable, Finset.univ, Fin, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
11 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma card_favorable :
    (Finset.univ.filter Favorable).card = 216 := by
…
```
Mentions: Usa1983P1.Favorable
lemma Usa1983P1.card_orderings The total number of cyclic orders of the six points line 60 source
Source documentation
The total number of cyclic orders of the six points.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Fintype.card, Usa1983P1.Ordering, Fin, OfNat.ofNat
Proof uses: Eq.mpr, Nat.factorial, id, congrArg, Equiv.Perm, Fintype.card_perm, Fintype.card_fin, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl
9 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma card_orderings : Fintype.card Ordering = 720 := by
…
```
Mentions: Usa1983P1.Ordering
problem Usa1983P1.usa1983_p1 line 66 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, ENNReal, MeasureTheory.OuterMeasure, Usa1983P1.Ordering, Set, PMF.toOuterMeasure, PMF.uniformOfFintype, Fin, OfNat.ofNat, Nat, Set.ofPred, Usa1983P1.Favorable, Usa1983P1.solution
Proof uses: Eq.mpr, HDiv.hDiv, Fintype.card, Membership.mem, id, congrArg, PMF.toOuterMeasure_uniformOfFintype_apply, Finset.card, Finset.filter, Finset.univ, Subtype, Fintype.card_subtype, congrFun', Finset, Finset.filter_congr, Iff.of_eq, Eq.refl, Usa1983P1.card_favorable, Usa1983P1.card_orderings, Usa1983P1.solution.eq_1 and 25 more
44 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.