`Compfiles/Imo1963P6.lean`

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4 declarations, 0 contain sorry, 0 declared as axioms

International Mathematical Olympiad 1963, Problem 6

Five students A, B, C, D, E were placed 1 to 5 in a contest with no ties. One prediction was that the result would be the order A, B, C, D, E. But no student finished in the position predicted, and no two students predicted to finish consecutively did so. For example, the outcome for C and D was not 1, 2 (respectively), or 2, 3, or 3, 4, or 4, 5. Another prediction was the order D, A, E, C, B. Exactly two students finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively did so. Determine the outcome.

Imports (4)

Imported by 1 file

Compfiles.lean

def Imo1963P6.secondPrediction The second prediction "D, A, E, C, B", given as the function sending each student to the place in which it was predicted to finish line 42 source
Source documentation
The second prediction "D, A, E, C, B", given as the function sending each student to the place in which it was predicted to finish.

standard axioms: propext what this means
Statement uses: Fin, OfNat.ofNat, Nat
Proof uses: Matrix.vecCons, Matrix.vecEmpty
4 instances, coercions and auxiliary constants left out.
```
def secondPrediction : Fin 5 → Fin 5 := ![1, 4, 3, 0, 2]
```
determine Imo1963P6.answer line 44 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Equiv.Perm, Fin, OfNat.ofNat, Nat
Proof uses: Equiv.mk, Matrix.vecCons, Matrix.vecEmpty
6 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo1963P6.classification The conditions of the problem determine the outcome uniquely: there are only `5! line 55 source
Source comment
The conditions of the problem determine the outcome uniquely: there are only `5! = 120` possible outcomes, so they can be checked exhaustively.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Equiv.Perm, Fin, OfNat.ofNat, Nat, Ne, Fin.succ, HAdd.hAdd, Fin.castSucc, Eq, Finset.card, Finset.filter, Imo1963P6.secondPrediction, Finset.univ, Or, And, Imo1963P6.answer
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma classification : ∀ π : Equiv.Perm (Fin 5),
    (∀ s : Fin 5, π s ≠ s) →
    (∀ i : Fin 4, (π i.succ : ℕ) ≠ (π i.castSucc : ℕ) + 1) →
    ((Finset.univ.filter (fun s => π s = secondPrediction s)).card = 2) →
    (((π 0 : ℕ) = (π 3 : ℕ) + 1 ∧ (π 2 : ℕ) = (π 4 : ℕ) + 1) ∨
     ((π 0 : ℕ) = (π 3 : ℕ) + 1 ∧ (π 1 : ℕ) = (π 2 : ℕ) + 1) ∨
     ((π 4 : ℕ) = (π 0 : ℕ) + 1 ∧ (π 1 : ℕ) = (π 2 : ℕ) + 1)) →
    π = answer := by
…
```
Mentions: Imo1963P6.secondPrediction, Imo1963P6.answer
problem Imo1963P6.imo1963_p6 line 67 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Equiv.Perm, Fin, OfNat.ofNat, Nat, Ne, Fin.succ, HAdd.hAdd, Fin.castSucc, Eq, Finset.card, Finset.filter, Imo1963P6.secondPrediction, Finset.univ, Or, And, Imo1963P6.answer
Proof uses: Imo1963P6.classification
22 instances, coercions and auxiliary constants left out.