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Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

10 declarations, 3 contain sorry, 0 declared as axioms

Lam--Litt conjecture

A conjecture of Lam and Litt on algebraic solutions of algebraic ODEs.

Let $g \in \mathbb{Q}(z, y_0, \dots, y_{n-1})$ be a rational function in $n + 1$ variables. Let $f$ be a power series over $\mathbb{Q}$ such that $f^{(n)}(z) = g(z, f(z), f'(z), \dots, f^{(n-1)}(z))$. Also, assume that $g(0, f(0), f'(0), \dots, f^{(n-1)}(0))$ is defined. Then the following are equivalent:

$f$ is algebraic over $\mathbb{Q}[z]$.
There exists $N$ such that for all $n$, the $n$-th coefficient of $f$ is in $\mathbb{Z}[1/N]$.
There exists an integer-valued function $\omega$ on the set of primes with $\lim_{p \to \infty} \omega(p) / p = \infty$ such that, for each prime $p$, the rational numbers $a_0, a_1, \dots, a_{\omega(p)}$ are in $\mathbb{Z}_{(p)}$.

The implication 1) => 2) is due to Eisenstein, and 2) => 3) is trivial.

References:

Litt's problem 1
Yeuk Hay Joshua Lam, Daniel Litt, "Algebraicity and integrality of solutions to differential equations", arxiv/2501.13175
Gotthold Eisenstein. "Über eine allgemeine Eigenschaft der Reihen-Entwicklungen aller algebraischen Funktionen", Bericht der Königl. Preuss. Akademie der Wissenschaften zu Berlin, 1852

TODO:

Lam-Litt conjecture implies Grothendieck p-curvature conjecture.
Examples in Remark 1.1.3 and 1.1.5 on the conditions of the conjecture.

Imports (1)

FormalConjecturesUtil

def LamLitt.IsSolutionOfAlgebraicODE A power series line 67 source
Source documentation
A power series $f$ is a solution of an algebraic ODE defined by the rational function $g \in \mathbb{Q}(z, y_0, \dots, y_{n-1})$ if $f^{(n)}(z) = g(z, f(z), f'(z), \dots, f^{(n-1)}(z))$ and $g(0, f(0), f'(0), \dots, f^{(n-1)}(0))$ is defined. Concretely, $g = p / q$ for polynomials $p, q$ with $q(0, f(0), \dots, f^{(n-1)}(0)) \neq 0$, so that $q(z, f(z), \dots, f^{(n-1)}(z))$ is an invertible power series, and $f^{(n)}(z) \cdot q(z, f(z), \dots, f^{(n-1)}(z)) = p(z, f(z), \dots, f^{(n-1)}(z))$. The variable indexed by 0 : Fin (n + 1) corresponds to $z$, and the variable indexed by i.succ corresponds to $y_i = f^{(i)}(z)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, PowerSeries, Rat, MvRatFunc, Fin, HAdd.hAdd, OfNat.ofNat
Proof uses: Fin.cases, PowerSeries.X, Nat.iterate, Derivation, Unit, PowerSeries.derivative, Exists, MvPolynomial, And, Eq, HDiv.hDiv, Finsupp, nonZeroDivisors, RingHom, Algebra.algebraMap, Ne, MvPolynomial.eval, Function.comp, PowerSeries.constantCoeff, HMul.hMul and 2 more
56 instances, coercions and auxiliary constants left out.
```
def IsSolutionOfAlgebraicODE (n : ℕ) (f : PowerSeries ℚ) (g : MvRatFunc (Fin (n + 1)) ℚ) : Prop :=
…
```
Mentions: MvRatFunc
def LamLitt.ℤAdjoinInvNat line 74 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Subalgebra, Int, Rat
Proof uses: Algebra.adjoin, Singleton.singleton, Set, HDiv.hDiv, OfNat.ofNat
16 instances, coercions and auxiliary constants left out.
def LamLitt.IsCoeffIntegralAdjointInvNat There exists $N$ such that for all $n$, the $n$-th coefficient of $f$ is in $\mathbb{Z}[1/N]$ line 79 source
Source documentation
There exists $N$ such that for all $n$, the $n$-th coefficient of $f$ is in $\mathbb{Z}[1/N]$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PowerSeries, Rat, Nat
Proof uses: Membership.mem, Subalgebra, Int, LamLitt.ℤAdjoinInvNat, LinearMap, RingHom.id, Unit, PowerSeries.coeff
14 instances, coercions and auxiliary constants left out.
```
def IsCoeffIntegralAdjointInvNat (f : PowerSeries ℚ) (N : ℕ) : Prop :=
…
```
def LamLitt.omegaIntegral For an integer-valued function line 88 source
Source documentation
For an integer-valued function $\omega$ on the set of primes and a sequence $a_n$ of rational numbers, the condition $\omega$-integrality means that for each prime $p$, the rational numbers $a_0, a_1, \dots, a_{\omega(p)}$ are in $\mathbb{Z}_{(p)}$, i.e. their denominators are not divisible by $p$. When $\omega(p) < 0$ the constraint at $p$ is vacuous.

standard axioms: propext what this means
Statement uses: Nat.Primes, Int, Nat, Rat
Proof uses: LE.le, Nat.Coprime, Rat.den, Nat.Prime
4 instances, coercions and auxiliary constants left out.
```
def omegaIntegral (ω : Nat.Primes → ℤ) (a : ℕ → ℚ) : Prop :=
…
```
def LamLitt.omegaSuperlinear The growth condition on $\omega$: the ratio $\omega(p) / p$ tends to infinity as the prime $p$ tends to infinity, i.e. $\lim_{p \to \infty} \omega(p) / p = \infty$ line 97 source
Source documentation
The growth condition on $\omega$: the ratio $\omega(p) / p$ tends to infinity as the prime $p$ tends to infinity, i.e. $\lim_{p \to \infty} \omega(p) / p = \infty$. Here the source filter is Filter.atTop on the primes, obtained by pulling back Filter.atTop on $\mathbb{N}$ along the coercion Nat.Primes → ℕ.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat.Primes, Int
Proof uses: Filter.Tendsto, Real, HDiv.hDiv, Nat, Nat.Prime, Filter.comap, Filter.atTop
10 instances, coercions and auxiliary constants left out.
```
def omegaSuperlinear (ω : Nat.Primes → ℤ) : Prop :=
…
```
theorem LamLitt.lam_litt.variants.eisenstein Eisenstein's theorem (1852): an algebraic power series over $\mathbb{Q}[z]$ has bounded denominators, i.e., there exists $N$ such that all coefficients lie in $\mathbb{Z}[1/N]$ sorry line 106 source
Source documentation
Eisenstein's theorem (1852): an algebraic power series over $\mathbb{Q}[z]$ has bounded denominators, i.e., there exists $N$ such that all coefficients lie in $\mathbb{Z}[1/N]$.

depends on sorry what this means
Statement uses: PowerSeries, Rat, IsAlgebraic, Polynomial, Unit, Exists, Nat, LamLitt.IsCoeffIntegralAdjointInvNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem lam_litt.variants.eisenstein (f : PowerSeries ℚ) (hAlg : IsAlgebraic (Polynomial ℚ) f) :
    ∃ N : ℕ, IsCoeffIntegralAdjointInvNat f N := by
…
```
Mentions: LamLitt.IsCoeffIntegralAdjointInvNat
lemma LamLitt.den_coprime_of_mem_adjoinInvNat Every element of $\mathbb{Z}[1/N]$ has denominator coprime to any prime $p$ not in the prime factor set of $N$ (vacuously, the case $N = 0$ gives $\mathbb{Z}[1/N] = \mathbb{Z}$) line 113 source
Source documentation
Every element of $\mathbb{Z}[1/N]$ has denominator coprime to any prime $p$ not in the prime factor set of $N$ (vacuously, the case $N = 0$ gives $\mathbb{Z}[1/N] = \mathbb{Z}$).
```
private lemma den_coprime_of_mem_adjoinInvNat {N p : ℕ} (hp : p.Prime)
    (hpN : p ∉ N.primeFactors) {q : ℚ} (hq : q ∈ ℤAdjoinInvNat N) :
    Nat.Coprime q.den p := by
…
```
Mentions: LamLitt.ℤAdjoinInvNat
theorem LamLitt.lam_litt.variants.integrality_implies_omega_integrality Textbook implication: integrality (2) trivially implies ω(p)-integrality (3) line 131 source
Source documentation
Textbook implication: integrality (2) trivially implies ω(p)-integrality (3).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PowerSeries, Rat, Nat, LamLitt.IsCoeffIntegralAdjointInvNat, Exists, Nat.Primes, Int, And, LamLitt.omegaSuperlinear, LamLitt.omegaIntegral, LinearMap, RingHom.id, Unit, PowerSeries.coeff
Proof uses: Exists.intro, ite, Membership.mem, Finset, Nat.primeFactors, Nat.Prime, Neg.neg, OfNat.ofNat, HPow.hPow, And.intro, Filter.Tendsto.congr', Real, Function.comp, HDiv.hDiv, Filter.comap, Filter.atTop, Filter.mp_mem, Set.ofPred, Subtype, LT.lt and 43 more
97 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
```
theorem lam_litt.variants.integrality_implies_omega_integrality
    (f : PowerSeries ℚ) (N : ℕ) (hN : IsCoeffIntegralAdjointInvNat f N) :
    ∃ ω : Nat.Primes → ℤ, omegaSuperlinear ω ∧ omegaIntegral ω (PowerSeries.coeff · f) := by
…
```
Mentions: LamLitt.IsCoeffIntegralAdjointInvNat, LamLitt.omegaSuperlinear, LamLitt.omegaIntegral
theorem LamLitt.lam_litt.variants.omega_integrality_implies_algebraicity Documented declaration: omega_integrality_implies_algebraicity sorry line 150 source
Source documentation
1. implies 2): if the coefficients of $f$ satisfy the $\omega$-integrality condition for some superlinear $\omega$, then there exists $N$ such that for all $n$, the $n$-th coefficient of $f$ is in $\mathbb{Z}[1/N]$.
depends on sorry what this means
Statement uses: Nat, PowerSeries, Rat, MvRatFunc, Fin, HAdd.hAdd, OfNat.ofNat, LamLitt.IsSolutionOfAlgebraicODE, Nat.Primes, Int, And, LamLitt.omegaSuperlinear, LamLitt.omegaIntegral, LinearMap, RingHom.id, Unit, PowerSeries.coeff, Exists, LamLitt.IsCoeffIntegralAdjointInvNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem lam_litt.variants.omega_integrality_implies_algebraicity {n : ℕ} (f : PowerSeries ℚ)
    (g : MvRatFunc (Fin (n + 1)) ℚ) (hODE : IsSolutionOfAlgebraicODE n f g)
    (ω : Nat.Primes → ℤ) (hω : omegaSuperlinear ω ∧ omegaIntegral ω (PowerSeries.coeff · f)) :
    ∃ N : ℕ, IsCoeffIntegralAdjointInvNat f N := by
…
```
Mentions: MvRatFunc, LamLitt.IsSolutionOfAlgebraicODE, LamLitt.omegaSuperlinear, LamLitt.omegaIntegral, LamLitt.IsCoeffIntegralAdjointInvNat
theorem LamLitt.lam_litt.variants.integrality_implies_algebraicity 2) implies 1): if the coefficients of $f$ are in $\mathbb{Z}[1/N]$ for some $N$, then $f$ is algebraic over $\mathbb{Q}[z]$ sorry line 161 source
Source documentation
1. implies 1): if the coefficients of $f$ are in $\mathbb{Z}[1/N]$ for some $N$, then $f$ is algebraic over $\mathbb{Q}[z]$. Also the version of conjecture of Litt's problem 1 on his website.
depends on sorry what this means
Statement uses: Nat, PowerSeries, Rat, MvRatFunc, Fin, HAdd.hAdd, OfNat.ofNat, LamLitt.IsSolutionOfAlgebraicODE, LamLitt.IsCoeffIntegralAdjointInvNat, IsAlgebraic, Polynomial, Unit
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem lam_litt.variants.integrality_implies_algebraicity {n : ℕ} (f : PowerSeries ℚ)
    (g : MvRatFunc (Fin (n + 1)) ℚ) (hODE : IsSolutionOfAlgebraicODE n f g)
    (N : ℕ) (hN : IsCoeffIntegralAdjointInvNat f N) :
    IsAlgebraic (Polynomial ℚ) f := by
…
```
Mentions: MvRatFunc, LamLitt.IsSolutionOfAlgebraicODE, LamLitt.IsCoeffIntegralAdjointInvNat