DedekindNumber.lean`

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36 declarations, 3 contain sorry, 0 declared as axioms

Dedekind Numbers

A Dedekind number M(n) counts the number of monotone Boolean functions on n variables, or equivalently, the number of antichains (Sperner families) in the Boolean lattice 2^[n].

For example, $$M ( 0 ) = 2 , M ( 1 ) = 3 , M ( 2 ) = 6 , and M ( 3 ) = 20 .$$ The first few values grew slowly: $$M ( 4 ) = 168 , M ( 5 ) = 7581$$, but then rapidly: $$M ( 6 ) = 7828354 , M ( 7 ) = 2414682040998 , M ( 8 ) = 56130437228687557907788$$, and $$M ( 9 ) = 286386577668298411128469151667598498812366$$ (computed in 2023).

We formalize two definitions:

M n: the number of monotone Boolean functions (Fin n → Bool) → Bool
M' n: the number of antichains (Sperner families) of Finset (Fin n)

We prove their values for small n and show that the two definitions agree for all n.

The problem is to determine the exact values of $M(n)$ for $n ≥ 10$. In particular, the value of $M(10)$ is currently unknown.

References:

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/ErdosProblems/497.lean

instance DedekindNumber.piFinBoolDecidableLE line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, DecidableRel, Fin, Bool, LE.le
Proof uses: Decidable, Eq.mpr
8 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
def DedekindNumber.M $M(n)$ is the number of monotone Boolean functions on $n$ variables line 63 source
Source documentation
$M(n)$ is the number of monotone Boolean functions on $n$ variables.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Fintype.card, Subtype, Fin, Bool, Monotone, LE.le
16 instances, coercions and auxiliary constants left out.
```
def M (n : ℕ) : ℕ :=
…
```
def DedekindNumber.IsSperner A Sperner family (antichain) of subsets of Fin n: a family of sets such that no member is a subset of another line 68 source
Source documentation
A Sperner family (antichain) of subsets of Fin n: a family of sets such that no member is a subset of another.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, Fin
Proof uses: IsAntichain, LE.le
5 instances, coercions and auxiliary constants left out.
```
def IsSperner {n : ℕ} (A : Finset (Finset (Fin n))) : Prop :=
…
```
instance DedekindNumber.isSpernerDecidable line 71 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, DecidablePred, Finset, Fin, DedekindNumber.IsSperner
Proof uses: id, Decidable, IsAntichain, LE.le, Set.Pairwise, Compl.compl, Eq.mpr, Membership.mem, Set, Ne, Not, inferInstance, Eq
19 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
def DedekindNumber.M' $M'(n)$ is the number of antichains (Sperner families) of subsets of Fin n line 79 source
Source documentation
$M'(n)$ is the number of antichains (Sperner families) of subsets of Fin n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Fintype.card, Subtype, Finset, Fin, DedekindNumber.IsSperner
4 instances, coercions and auxiliary constants left out.
```
def M' (n : ℕ) : ℕ :=
…
```
theorem DedekindNumber.M_zero Values for small n line 84 source
Source documentation
Values for small n

extra axioms: _private.0.DedekindNumber.M_zero._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, DedekindNumber.M, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem M_zero : M 0 = 2 := by native_decide
…
```
Mentions: DedekindNumber.M
theorem DedekindNumber.M_one line 87 source
extra axioms: _private.0.DedekindNumber.M_one._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, DedekindNumber.M, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem DedekindNumber.M_two line 90 source
extra axioms: _private.0.DedekindNumber.M_two._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, DedekindNumber.M, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem DedekindNumber.M_three line 93 source
extra axioms: _private.0.DedekindNumber.M_three._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, DedekindNumber.M, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem DedekindNumber.M'_zero line 96 source
extra axioms: _private.0.DedekindNumber.M'_zero._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, DedekindNumber.M', OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem DedekindNumber.M'_one line 99 source
extra axioms: _private.0.DedekindNumber.M'_one._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, DedekindNumber.M', OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem DedekindNumber.M'_two line 102 source
extra axioms: _private.0.DedekindNumber.M'_two._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, DedekindNumber.M', OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem DedekindNumber.M'_three line 105 source
extra axioms: _private.0.DedekindNumber.M'_three._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, DedekindNumber.M', OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def DedekindNumber.χ The indicator function of a finset: χ s i = true ↔ i ∈ s line 111 source
Source documentation
The indicator function of a finset: χ s i = true ↔ i ∈ s.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, Fin, Bool
Proof uses: Decidable.decide, Membership.mem
4 instances, coercions and auxiliary constants left out.
```
def χ {n : ℕ} (s : Finset (Fin n)) : Fin n → Bool :=
…
```
def DedekindNumber.supp The support of a Boolean-valued function: supp v = {i | v i = true} line 115 source
Source documentation
The support of a Boolean-valued function: supp v = {i | v i = true}.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Bool, Finset
Proof uses: Finset.filter, Eq, Bool.true, Finset.univ
2 instances, coercions and auxiliary constants left out.
```
def supp {n : ℕ} (v : Fin n → Bool) : Finset (Fin n) :=
…
```
def DedekindNumber.toSperner Forward map: monotone function → Sperner family (the minimal true sets) line 119 source
Source documentation
Forward map: monotone function → Sperner family (the minimal true sets).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Bool, Finset
Proof uses: Finset.filter, And, Eq, DedekindNumber.χ, Bool.true, LE.le, Finset.univ
11 instances, coercions and auxiliary constants left out.
```
def toSperner {n : ℕ} (f : (Fin n → Bool) → Bool) : Finset (Finset (Fin n)) :=
…
```
def DedekindNumber.fromSperner Backward map: Sperner family → monotone Boolean function line 124 source
Source documentation
Backward map: Sperner family → monotone Boolean function.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, Fin, Bool
Proof uses: Decidable.decide, Exists, And, Membership.mem, Eq, Bool.true
8 instances, coercions and auxiliary constants left out.
```
def fromSperner {n : ℕ} (A : Finset (Finset (Fin n))) (v : Fin n → Bool) : Bool :=
…
```
lemma DedekindNumber.χ_supp line 130 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Bool, Eq, DedekindNumber.χ, DedekindNumber.supp
Proof uses: funext, of_eq_true, Eq.trans, True, congrFun', congrArg, Decidable.decide, Membership.mem, Finset, Bool.true, Decidable.decide.congr_simp, Finset.filter, Finset.univ, And, true_and, Bool.decide_eq_true, eq_self
8 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma DedekindNumber.supp_χ line 134 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, Fin, Eq, DedekindNumber.supp, DedekindNumber.χ
Proof uses: Finset.ext, of_eq_true, Iff, Membership.mem, Eq.trans, True, congrFun', congrArg, Finset.filter, Bool, Bool.true, Finset.univ, Finset.filter.congr_simp, funext, Decidable.decide, decide_eq_true_eq, Eq.refl, Finset.filter_mem_eq_of_subset, LE.le, iff_self
10 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma DedekindNumber.χ_le_iff line 139 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, Fin, Iff, LE.le, Bool, DedekindNumber.χ
Proof uses: of_eq_true, Eq.trans, Membership.mem, True, congr, congrArg, forall_congr, Decidable.decide, Eq, Bool.true, implies_congr, decide_eq_true_eq, iff_self
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma DedekindNumber.mem_supp_iff line 143 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Bool, Iff, Membership.mem, Finset, DedekindNumber.supp, Eq, Bool.true
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Finset.filter, Finset.univ, And, true_and, iff_self
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma DedekindNumber.toSperner_isSperner line 147 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Bool, Monotone, DedekindNumber.IsSperner, DedekindNumber.toSperner
Proof uses: Finset, Membership.mem, Set, Ne, Eq.mpr, Compl.compl, LE.le, Not, id, Eq, Mathlib.Tactic.Contrapose.contrapose₃, Exists, Iff, implies_congr, Eq.refl, Eq.trans, binderNameHint, forall_congr, DedekindNumber.χ, Bool.true and 11 more
31 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma DedekindNumber.fromSperner_monotone line 153 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, Fin, DedekindNumber.IsSperner, Monotone, Bool, DedekindNumber.fromSperner
Proof uses: LE.le, Eq, Bool.true, And, Membership.mem, DedekindNumber.supp, Exists, of_eq_true, Eq.trans, True, congrArg, funext, forall_congr, implies_congr_ctx, Eq.refl, eq_true, id, Eq.mp, Decidable.decide, decide_eq_true_eq and 11 more
21 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma DedekindNumber.exists_minimal_true_subset Every true set of a monotone Boolean function contains a minimal true set line 168 source
Source documentation
Every true set of a monotone Boolean function contains a minimal true set.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Bool, Monotone, Finset, Eq, DedekindNumber.χ, Bool.true, Exists, And, LE.le
Proof uses: Membership.mem, Set, Set.ofPred, Finset.card, Set.exists_min_image, Iff.mpr, Set.Finite, BddAbove, Set.finite_iff_bddAbove, Exists.intro, upperBounds, And.left, And.intro, Finset.Subset.refl, And.right, Iff.mp, Not, Classical.not_not, not_lt_of_ge, LE.le.trans and 14 more
27 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
lemma exists_minimal_true_subset {n : ℕ} {f : (Fin n → Bool) → Bool} (_ : Monotone f)
    {s : Finset (Fin n)} (hs : f (χ s) = true) :
    ∃ t, t ⊆ s ∧ f (χ t) = true ∧ ∀ u, u ⊆ t → f (χ u) = true → t ⊆ u := by
…
```
Mentions: DedekindNumber.χ
lemma DedekindNumber.fromSperner_toSperner Converting a monotone function to a Sperner family and back yields the same function line 184 source
Source documentation
Converting a monotone function to a Sperner family and back yields the same function.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Bool, Monotone, Eq, DedekindNumber.fromSperner, DedekindNumber.toSperner
Proof uses: funext, dite, Bool.true, Eq.mpr, Exists, Finset, And, Membership.mem, id, Eq.trans, Decidable.decide, congrArg, decide_eq_true_eq, LE.le, Finset.filter, Finset.univ, DedekindNumber.χ, DedekindNumber.exists_minimal_true_subset, eq_of_heq, HEq and 36 more
38 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
```
lemma fromSperner_toSperner {n : ℕ} (f : (Fin n → Bool) → Bool) (hf : Monotone f) :
    fromSperner (toSperner f) = f := by
…
```
Mentions: DedekindNumber.fromSperner, DedekindNumber.toSperner
lemma DedekindNumber.toSperner_fromSperner Converting a Sperner family to a monotone function and back yields the same family line 203 source
Source documentation
Converting a Sperner family to a monotone function and back yields the same family.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, Fin, DedekindNumber.IsSperner, Eq, DedekindNumber.toSperner, DedekindNumber.fromSperner
Proof uses: Finset.ext, Eq.mpr, Iff, Membership.mem, And, Exists, Bool, DedekindNumber.χ, Bool.true, LE.le, id, congrFun', congrArg, Eq.trans, Finset.univ, Finset.filter, Finset.filter_congr, Iff.of_eq, congr, decide_eq_true_eq and 32 more
36 instances, coercions and auxiliary constants left out.

Tactic trace: 27 steps
Open to load the checked before-and-after goal states.
```
lemma toSperner_fromSperner {n : ℕ} (A : Finset (Finset (Fin n))) (hA : IsSperner A) :
    toSperner (fromSperner A) = A := by
…
```
Mentions: DedekindNumber.IsSperner, DedekindNumber.toSperner, DedekindNumber.fromSperner
def DedekindNumber.equivMonotoneSperner The set of monotone Boolean functions on n variables is in bijection with the set of Sperner families of subsets of Fin n line 219 source
Source documentation
The set of monotone Boolean functions on n variables is in bijection with the set of Sperner families of subsets of Fin n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Equiv, Subtype, Fin, Bool, Monotone, Finset, DedekindNumber.IsSperner
Proof uses: Equiv.mk, DedekindNumber.equivMonotoneSperner.match_1, Subtype.mk, DedekindNumber.toSperner, DedekindNumber.toSperner_isSperner, DedekindNumber.equivMonotoneSperner.match_2, DedekindNumber.fromSperner, DedekindNumber.fromSperner_monotone
8 instances, coercions and auxiliary constants left out.
```
def equivMonotoneSperner (n : ℕ) :
    {f : (Fin n → Bool) → Bool // Monotone f} ≃
    {A : Finset (Finset (Fin n)) // IsSperner A} where
…
```
Mentions: DedekindNumber.IsSperner
theorem DedekindNumber.M_eq_M' line 228 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, DedekindNumber.M, DedekindNumber.M'
Proof uses: funext, Fintype.card_congr, Subtype, Fin, Bool, Monotone, Finset, DedekindNumber.IsSperner, LE.le, DedekindNumber.equivMonotoneSperner
18 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
def DedekindNumber.kisielewiczFormula A closed formula for the Dedekind numbers as found by Kisielewicz (1998) line 238 source
Source documentation
A closed formula for the Dedekind numbers as found by Kisielewicz (1998): $$ M(n) = \sum_{k=0}^{2^{2^n}}\prod_{j = 1}^{2 ^ n - 1}\prod_{i = 0}^{j - 1} \left( 1 - b_i^kb_j^k \prod_{m = 0}^{\log_2 i} (1 - b_m^i + b_m^ib_m^j)\right), $$, where $b_i^k$ is the $i$-th bit of $k$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.range, HPow.hPow, OfNat.ofNat, Finset.prod, Finset.Icc, HSub.hSub, HMul.hMul, Bool.toNat, Nat.testBit, HAdd.hAdd, Nat.log2
15 instances, coercions and auxiliary constants left out.
```
def kisielewiczFormula (n : ℕ) : ℕ :=
…
```
theorem DedekindNumber.kisielewiczFormula_zero line 243 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, DedekindNumber.kisielewiczFormula, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem DedekindNumber.kisielewiczFormula_one line 244 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, DedekindNumber.kisielewiczFormula, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem DedekindNumber.kisielewiczFormula_two line 245 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, DedekindNumber.kisielewiczFormula, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem DedekindNumber.kisielewiczFormula_three line 246 source
extra axioms: _private.0.DedekindNumber.kisielewiczFormula_three._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, DedekindNumber.kisielewiczFormula, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem DedekindNumber.M_eq_kisielewiczFormula Kisielewicz (1988) proved the following arithmetic formula for the Dedekind numbers sorry line 259 source
Source documentation
Kisielewicz (1988) proved the following arithmetic formula for the Dedekind numbers: $$ M(n) = \sum_{k=0}^{2^{2^n}}\prod_{j = 1}^{2 ^ n - 1}\prod_{i = 0}^{j - 1} \left( 1 - b_i^kb_j^k \prod_{m = 0}^{\log_2 i} (1 - b_m^i + b_m^ib_m^j)\right), $$ where $b_i^k$ is the $i$-th bit of $k$. However, this formula is not computationally efficient for large $n$.

depends on sorry what this means
Statement uses: Eq, Nat, DedekindNumber.M, DedekindNumber.kisielewiczFormula
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem M_eq_kisielewiczFormula : M = kisielewiczFormula := by
…
```
Mentions: DedekindNumber.M, DedekindNumber.kisielewiczFormula
theorem DedekindNumber.M_eq No closed-form expression that allows efficient computation of Dedekind numbers is currently known sorry line 267 source
Source documentation
No closed-form expression that allows efficient computation of Dedekind numbers is currently known.

depends on sorry what this means
Statement uses: Eq, Nat, DedekindNumber.M, sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem M_eq : M = answer(sorry) := by
…
```
Mentions: DedekindNumber.M
theorem DedekindNumber.Dedekind_10 In particular, the Dedekind number for n = 10 is currently unknown sorry line 274 source
Source documentation
In particular, the Dedekind number for n = 10 is currently unknown.

depends on sorry what this means
Statement uses: Eq, Nat, DedekindNumber.M, OfNat.ofNat, sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem Dedekind_10 : M 10 = answer(sorry) := by sorry
…
```
Mentions: DedekindNumber.M