`Compfiles/Imo1987P1.lean`

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10 declarations, 0 contain sorry, 0 declared as axioms

International Mathematical Olympiad 1987, Problem 1

Let $p_{n, k}$ be the number of permutations of a set of cardinality n ≥ 1 that fix exactly k elements. Prove that $∑_{k=0}^n k p_{n,k}=n!$.

Imports (4)

Imported by 1 file

Compfiles.lean

def Imo1987P1.fiber Given α : Type* and k : ℕ, fiber α k is the set of permutations of α with exactly k fixed points line 33 source
Source documentation
Given α : Type* and k : ℕ, fiber α k is the set of permutations of α with exactly k fixed points.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Nat, Set, Equiv.Perm
Proof uses: Set.ofPred, Eq, Fintype.card, Function.fixedPoints, Membership.mem
7 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
def fiber (α : Type*) [Fintype α] [DecidableEq α] (k : ℕ) : Set (Equiv.Perm α) :=
…
```
instance Imo1987P1.(instance at line 36) line 36 source
def Imo1987P1.p p α k is the number of permutations of α with exactly k fixed points line 40 source
Source documentation
p α k is the number of permutations of α with exactly k fixed points.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Nat
Proof uses: Fintype.card, Equiv.Perm, Imo1987P1.fiber
2 instances, coercions and auxiliary constants left out.
```
def p (α : Type*) [Fintype α] [DecidableEq α] (k : ℕ) : ℕ := Fintype.card (fiber α k)
…
```
Mentions: Imo1987P1.fiber
def Imo1987P1.fixedPointsEquiv The set of pairs (x : α, σ : Perm α) such that σ x = x is equivalent to the set of pairs (x : α, σ : Perm {x}ᶜ) line 68 source
Source documentation
The set of pairs (x : α, σ : Perm α) such that σ x = x is equivalent to the set of pairs (x : α, σ : Perm {x}ᶜ).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: DecidableEq, Equiv, Subtype, Prod, Equiv.Perm, Eq, Prod.snd, Prod.fst, Sigma, Compl.compl, Set, Singleton.singleton
Proof uses: Trans.trans, Membership.mem, Equiv.refl, Equiv.setProdEquivSigma, Equiv.sigmaCongrRight, Set.equivOfEq, Equiv.Set.compl
17 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
def fixedPointsEquiv : { σx : α × Perm α // σx.2 σx.1 = σx.1 } ≃ Σ x : α, Perm ({x}ᶜ : Set α) :=
…
```
theorem Imo1987P1.card_fixed_points line 76 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Eq, Nat, Fintype.card, Subtype, Prod, Equiv.Perm, Prod.snd, Prod.fst, HMul.hMul, Nat.factorial, HSub.hSub, OfNat.ofNat
Proof uses: Eq.mpr, Finset.sum, Finset.univ, Compl.compl, Set, Singleton.singleton, Membership.mem, id, congrFun', congrArg, Eq.trans, Sigma, Fintype.card_congr, Imo1987P1.fixedPointsEquiv, Fintype.card_sigma, Finset.sum_congr, Eq.refl, Finset, Fintype.card_perm, of_eq_true and 6 more
38 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem Imo1987P1.mem_fiber line 83 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Equiv.Perm, Nat, Iff, Membership.mem, Set, Imo1987P1.fiber, Eq, Fintype.card, Function.fixedPoints
Proof uses: Iff.rfl
8 instances, coercions and auxiliary constants left out.
def Imo1987P1.fixedPointsEquiv' The set of triples (k ≤ card α, σ ∈ fiber α k, x ∈ fixedPoints σ) is equivalent to the set of pairs (x : α, σ : Perm α) such that σ x = x line 92 source
Source documentation
The set of triples (k ≤ card α, σ ∈ fiber α k, x ∈ fixedPoints σ) is equivalent to the set of pairs (x : α, σ : Perm α) such that σ x = x. The equivalence sends (k, σ, x) to (x, σ) and (x, σ) to (card (fixedPoints σ), σ, x).
It is easy to see that the cardinality of the LHS is given by ∑ k : Fin (card α + 1), k * p α k.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Equiv, Sigma, Fin, HAdd.hAdd, Nat, Fintype.card, OfNat.ofNat, Equiv.Perm, Imo1987P1.fiber, Function.fixedPoints, Membership.mem, Set, Subtype, Prod, Eq, Prod.snd, Prod.fst
Proof uses: Equiv.mk, Subtype.mk, Prod.mk, Sigma.fst, Sigma.snd, Sigma.mk, Fin.mk
28 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
def fixedPointsEquiv' :
    (Σ (k : Fin (card α + 1)) (σ : fiber α k), fixedPoints σ.1) ≃
      { σx : α × Perm α // σx.2 σx.1 = σx.1 } where
…
```
Mentions: Imo1987P1.fiber
theorem Imo1987P1.main_fintype Main statement for any (α : Type*) [Fintype α] line 105 source
Source documentation
Main statement for any (α : Type*) [Fintype α].

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Eq, Nat, Finset.sum, Finset.range, HAdd.hAdd, Fintype.card, OfNat.ofNat, HMul.hMul, Imo1987P1.p, Nat.factorial, HSub.hSub
Proof uses: Equiv.Perm, Imo1987P1.fiber, Function.fixedPoints, Membership.mem, Set, Subtype.property, Eq.mpr, Fin, Finset.univ, id, congrFun', congrArg, Eq.mp, Sigma, Subtype, Prod, Prod.snd, Prod.fst, congr, Eq.trans and 11 more
40 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem main_fintype :
    ∑ k ∈ Finset.range (card α + 1), k * p α k = card α * (card α - 1)! := by
…
```
Mentions: Imo1987P1.p
theorem Imo1987P1.main₀ Main statement for permutations of Fin n, a version that works for n = 0 line 112 source
Source documentation
Main statement for permutations of Fin n, a version that works for n = 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HMul.hMul, Imo1987P1.p, Fin, Nat.factorial, HSub.hSub
Proof uses: Eq.mp, Fintype.card, congr, congrArg, Finset.sum_congr, Finset, congrFun', Fintype.card_fin, Membership.mem, Eq.refl, Imo1987P1.main_fintype
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem main₀ (n : ℕ) : ∑ k ∈ Finset.range (n + 1), k * p (Fin n) k = n * (n - 1)! := by
…
```
Mentions: Imo1987P1.p
problem Imo1987P1.imo1987_p1 line 118 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Eq, Finset.sum, Finset.range, HAdd.hAdd, HMul.hMul, Imo1987P1.p, Fin, Nat.factorial
Proof uses: Eq.mpr, HSub.hSub, id, congrArg, Imo1987P1.main₀, Nat.mul_factorial_pred, Iff.mp, Ne, Nat.one_le_iff_ne_zero, Eq.refl
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.