Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion. • 103 items • Updated • 3
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values | symbolic_name stringlengths 1 115 | library stringclasses 88
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values | imports listlengths 0 89 | deps listlengths 0 64 | docstring stringlengths 0 1.64k | source_url stringclasses 1
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getCategoryStatsMarkdown : CoreM String | do
let stats ← getCategoryStats
let formalProofCount := (← getFormalProofTags).size
let githubSearchBaseUrl := "https://github.com/search?type=code&q=repo%3Agoogle-deepmind%2Fformal-conjectures+"
return s!"| Count | Category |
| ----- | ----------------- |
| {stats (Category.research ProblemStatus.open... | def | getCategoryStatsMarkdown | docbuild.scripts | docbuild/scripts/overwrite_index.lean | [
"MD4Lean",
"Lean",
"Batteries.Data.String.Matcher",
"FormalConjectures.Util.Attributes.Basic",
"Mathlib.Data.String.Defs"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
getSubjectStatsMarkdown : CoreM String | do
let tags ← getSubjectTags
let mut counts : Std.HashMap AMS Nat := {}
for tag in tags do
for subject in tag.subjects do
counts := counts.insert subject (counts.getD subject 0 + 1)
let sortedCounts := counts.toArray.qsort (lt := fun (_, c1) (_, c2) => c2 < c1)
let mut markdownTable := "| Count | ... | def | getSubjectStatsMarkdown | docbuild.scripts | docbuild/scripts/overwrite_index.lean | [
"MD4Lean",
"Lean",
"Batteries.Data.String.Matcher",
"FormalConjectures.Util.Attributes.Basic",
"Mathlib.Data.String.Defs"
] | [
"AMS"
] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
replaceTag (tag : String) (inputHtmlContent : String) (newContent : String) : IO String | do
let openTag := s!"<{tag}>"
let closeTag := s!"</{tag}>"
let prefixParts := (inputHtmlContent.toSlice.split openTag).toArray
if prefixParts.size < 2 then
throw <| IO.userError s!"Opening {openTag} tag not found in inputHtmlContent."
-- Split string by closeTag
let suffixParts := (inputHtmlContent.... | def | replaceTag | docbuild.scripts | docbuild/scripts/overwrite_index.lean | [
"MD4Lean",
"Lean",
"Batteries.Data.String.Matcher",
"FormalConjectures.Util.Attributes.Basic",
"Mathlib.Data.String.Defs"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
runWithImports {α : Type} (actionToRun : CoreM α) : IO α | do
-- This assumes a run of `lake exe mk_all; mv FormalConjectures.lean FormalConjectures/All.lean` took place before.
-- TODO(firsching): avoid this by instead using `Lake.Glob.forEachModuleIn` to generate a list of all modules instead.
-- Then it would be easily possible to sort out the statements from the Util... | def | runWithImports | docbuild.scripts | docbuild/scripts/overwrite_index.lean | [
"MD4Lean",
"Lean",
"Batteries.Data.String.Matcher",
"FormalConjectures.Util.Attributes.Basic",
"Mathlib.Data.String.Defs"
] | [] | Runs a `CoreM α` action in an environment where all FormalConjectures modules are imported.
This is useful for accessing declarations and attributes defined in the project. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
main (args : List String) : IO Unit | do
let .some (file : String) := args[0]?
| IO.println "Usage: stats <file>
overwrites the contents of the `main` tag of a html `file` with a welcome page including stats."
let inputHtmlContent ← IO.FS.readFile file
let .some (graphFileDark : String) := args[1]?
| IO.println "Repository growth graph not su... | def | main | docbuild.scripts | docbuild/scripts/overwrite_index.lean | [
"MD4Lean",
"Lean",
"Batteries.Data.String.Matcher",
"FormalConjectures.Util.Attributes.Basic",
"Mathlib.Data.String.Defs"
] | [
"getCategoryStatsMarkdown",
"getSubjectStatsMarkdown",
"replaceTag",
"runWithImports"
] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
testMain : IO Unit | do
let res1 ← replaceTag "main" "prefix<main>old</main>suffix" "new"
if res1 != "prefix<main>new</main>suffix" then
throw <| IO.userError s!"Test 1 failed: {res1}"
-- Test multiple closing tags
let res2 ← replaceTag "main" "prefix<main>old</main>suffix</main>extra" "new"
if res2 != "prefix<main>new</main... | def | testMain | docbuild.scripts | docbuild/scripts/test_overwrite_index.lean | [
"overwrite_index"
] | [
"replaceTag"
] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
arxiv.id0911_2077.conjecture6_3
(p : ℝ) (h_p : p ∈ Set.Ioo 0 (1 / 2)) (k : ℕ) (hk : 0 < k)
(σ : ℝ) (h_σ : σ = (p * (1 - p)).sqrt) :
letI hp' : (⟨p, le_of_lt h_p.1⟩ : ℝ≥0) ≤ 1 | by
have : p ≤ 1 := le_trans (le_of_lt (Set.mem_Ioo.mp h_p).right) (by linarith)
exact this
1 - Φ ((1 / 2 - p) * sqrt (2 * k : ℝ≥0) / σ)
+ (1 / 2) * ((2 * k).choose k) * σ ^ (2 * k)
≤ ((PMF.binomial (⟨p, le_of_lt h_p.1⟩) hp' (2 * k)).toMeasure
(Set.Ici ⟨k, by omega⟩)).toReal := by
... | theorem | Arxiv..arxiv.id0911_2077.conjecture6_3 | Arxiv.0911.2077 | FormalConjectures/Arxiv/0911.2077/Conjecture6_3.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Empirical evidence seems to suggest that Slud's bound does not hold for all $p$, and in fact, as $n\to\infty$,
the maximal permissible $p$ shrinks to $\frac{1}{2}$. Also, the following appears to be true:
When $p\in(0,1/2)$ and
$m = 2k$ is even, and $\sigma := \sqrt{p(1-p)}$,
$$
\mathbb{P}[B(p,m) \geq m/2] \geq 1 - ... | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
k (S : List ℤ) : ℕ | sSup {k : ℕ | ∃ X Y : List ℤ, Y ≠ [] ∧ S = X ++ (List.replicate k Y).flatten} | def | Arxiv..k | Arxiv.0912.2382 | FormalConjectures/Arxiv/0912.2382/CurlingNumberConjecture.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The curling number
Let $S$ be a finite nonempty sequence of integers. By grouping adjacent terms, it is always possible
to write it as $S = X Y Y . . . Y = X Y^k$, where $X$ and $Y$ are sequences of integers and $Y$ is nonempty
($X$ is allowed to be the empty sequence $∅$). There may be several ways to do this: choose... | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
S (S₀ : List ℤ) (n : ℕ) : List ℤ | match n with
| 0 => S₀
| n + 1 => (S S₀ n) ++ [Int.ofNat (k (S S₀ n))] | def | Arxiv..S | Arxiv.0912.2382 | FormalConjectures/Arxiv/0912.2382/CurlingNumberConjecture.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | One starts with any initial
sequence of integers $S₀$, and extends it by repeatedly appending the curling number of the current
sequence. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
curling_number_conjecture (S₀ : List ℤ) (h : S₀ ≠ []) : ∃ m, k (S S₀ m) = 1 | by
sorry | theorem | Arxiv..curling_number_conjecture | Arxiv.0912.2382 | FormalConjectures/Arxiv/0912.2382/CurlingNumberConjecture.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The sequence will eventually reach $1$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
Formula : Type
/-- `Atom n` is a propositional variable indexed by `n`. -/
| Atom : Nat → Formula
/-- `Falsum` is the constant ⊥. -/
| Falsum : Formula
/-- `Imp α β` is implication `(α → β)`. -/
| Imp : Formula → Formula → Formula
/-- `Nec α` is the necessity operator `□α`. -/
| Nec : Formula → Formula | inductive | Arxiv..Formula | Arxiv.1308.0994 | FormalConjectures/Arxiv/1308.0994/BoxdotConjecture.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | `Formula` is the inductive type of propositional modal formulas:
* `Atom n` is a propositional variable indexed by `n`.
* `Falsum` is the constant ⊥.
* `Imp α β` is implication `(α → β)`.
* `Nec α` is the necessity operator `□α`. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
Conj (α β : Formula) : Formula | ~(α ~> ~β) | def | Arxiv..Conj | Arxiv.1308.0994 | FormalConjectures/Arxiv/1308.0994/BoxdotConjecture.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | `Conj α β` is the conjunction `α ∧ β`. We define `α & β` as `~(α ~> ~β)` for simplicity. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
t (φ : Formula) : Formula | match φ with
| α ~> β => t α ~> t β
| □α => □t α & t α
| _ => φ | def | Arxiv..t | Arxiv.1308.0994 | FormalConjectures/Arxiv/1308.0994/BoxdotConjecture.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | `t φ` is the Boxdot translation of a formula `φ`. Roughly, t is the mapping `φ ↦ t φ`
from the language of monomodal logic into itself that preserves variables and the logical constant `⊥`,
commutes with the standard truth-functional operators, and is such that `t □a` = `□t a & t a`.
This implementation follows the def... | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
KProof : Set Formula → Formula → Prop
/-- Assumption rule: if `α ∈ Γ` then `α` is provable from `Γ`. -/
| ax {Γ} {α} (h : α ∈ Γ) : KProof Γ α
/-- Ax1: every instance of the schema `α → (β → α)` is a theorem. -/
| ax1 {Γ} {α β} : KProof Γ (α ~> β ~> α)
/-- Ax2: every instance of the schema `(α ~> β ~> γ) ~> (α ~> β) ~> ... | inductive | Arxiv..KProof | Arxiv.1308.0994 | FormalConjectures/Arxiv/1308.0994/BoxdotConjecture.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | `KProof Γ φ` is the usual Hilbert‐style proof relation for the minimal normal modal logic K,
with assumptions drawn from `Γ`. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
KTProof : Set Formula → Formula → Prop
/-- Embedding of K proofs into KT. -/
| lift_K {Γ} {α} (h : KProof Γ α) : KTProof Γ α
/-- T-axiom schema: every instance of `□α ~> α` is a theorem. -/
| axT {Γ} {α} : KTProof Γ (□ α ~> α)
/-- Modus Ponens: if `Γ ⊢ α ~> β` and `Γ ⊢ α`, then `Γ ⊢ β`. -/
| mp {Γ} {α β} (_ : KTProof Γ... | inductive | Arxiv..KTProof | Arxiv.1308.0994 | FormalConjectures/Arxiv/1308.0994/BoxdotConjecture.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | `KTProof Γ φ` denotes that `φ` is provable from the premises `Γ` in the normal modal logic KT
(also called T). KT extends system K by adding the instances of the T-axiom schema `□φ ~> φ` to K’s
usual axioms and rules of inference. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
KTExtendsK {Γ φ} (h : KProof Γ φ) : KTProof Γ φ | lift_K h | lemma | Arxiv..KTExtendsK | Arxiv.1308.0994 | FormalConjectures/Arxiv/1308.0994/BoxdotConjecture.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | If `KProof Γ φ`, then `KTProof Γ φ`. In other words, KT extends K. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
NormalModalLogic : Type where
/-- `thms` is the set of formulas proveable in the logic. -/
thms : Set Formula
/-- `extK` means that if `K ⊢ φ`, then `φ ∈ thms`. That is, the logic extends system K. -/
extK : ∀ {φ}, KProof ∅ φ → φ ∈ thms
/-- `mp` means that if `φ ∈ thms` and `(φ ~> ψ) ∈ thms`, then `ψ ∈ thms`.... | structure | Arxiv..NormalModalLogic | Arxiv.1308.0994 | FormalConjectures/Arxiv/1308.0994/BoxdotConjecture.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | A “normal modal logic” L is any `Set Formula` such that:
1. If `K ⊢ φ`, then `φ ∈ L` (L extends K)
2. If `φ ∈ L` and `(φ ~> ψ) ∈ L`, then `ψ ∈ L` (Closed under MP)
3. If `φ ∈ L`, then `□φ ∈ L` (Closed under Necessitation) | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
proves (L : NormalModalLogic) (φ : Formula) | φ ∈ L.thms | def | Arxiv..proves | Arxiv.1308.0994 | FormalConjectures/Arxiv/1308.0994/BoxdotConjecture.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
KT : NormalModalLogic | by
constructor
case thms =>
exact {φ | KTProof ∅ φ}
case extK =>
intro _ h
exact KTExtendsK h
case mp =>
intro φ ψ h₁ h₂
simp [Set.mem_setOf_eq] at *
exact KTProof.mp h₂ h₁
case nec =>
intro φ h
simp [Set.mem_setOf_eq] at *
exact KTProof.nec h | def | Arxiv..KT | Arxiv.1308.0994 | FormalConjectures/Arxiv/1308.0994/BoxdotConjecture.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | `KT` is the specific normal modal logic whose theorems are exactly those provable
in `KTProof` from the empty context.
This corresponds to `K ⊕ (□φ → φ)` as in both AJL (Steinsvold) and Jeřábek. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
BoxdotConjecture (L : NormalModalLogic) (H : ∀ φ, L ⊢ ■ φ ↔ KT ⊢ φ) : L ⊆ KT | by
sorry | theorem | Arxiv..BoxdotConjecture | Arxiv.1308.0994 | FormalConjectures/Arxiv/1308.0994/BoxdotConjecture.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Boxdot Conjecture: every normal modal logic that faithfully interprets KT
by the boxdot translation is included in KT. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
IsCrystalWithComponents (n a b : ℕ) : Prop | Odd n ∧ 1 < a ∧ 1 < b ∧ n = a * b ∧ 2 * (a + 1) * (b + 1) ∣ (a + b)^2 + (a * b + 1)^2 | def | Arxiv..IsCrystalWithComponents | Arxiv.1601.03081 | FormalConjectures/Arxiv/1601.03081/UniqueCrystalComponents.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | An odd number $n$ is called a crystal if $n = ab$, with $a, b > 1$
and $B(a, b) ∈ ℕ$, where $B(a, b) := ((a + b)^2 + (a b + 1)^2) / (2 (a + 1) (b + 1))$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
isCrystalWithComponents_35_5_7 : IsCrystalWithComponents 35 5 7 | by
norm_num [IsCrystalWithComponents] | theorem | Arxiv..isCrystalWithComponents_35_5_7 | Arxiv.1601.03081 | FormalConjectures/Arxiv/1601.03081/UniqueCrystalComponents.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
crystals_components_unique (n a b c d : ℕ)
(hab : IsCrystalWithComponents n a b) (hcd : IsCrystalWithComponents n c d) :
({a, b} : Finset ℕ) = {c, d} | by
sorry | theorem | Arxiv..crystals_components_unique | Arxiv.1601.03081 | FormalConjectures/Arxiv/1601.03081/UniqueCrystalComponents.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | If $n = ab$ is a crystal, then there are no other pairs of
positive integers $c, d > 1$, different from the couple $a, b$, such that $n = cd$ and
$B(c, d) ∈ ℕ$, i.e., the components of the crystals are unique. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
lt₂ {α : Type*} [LT α] (a b : Fin 3 → α) : Prop | ∃ (i j : Fin 3), i ≠ j ∧ a i < b i ∧ a j < b j | def | Arxiv..lt₂ | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Let $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$ be two triples of integers.
Say that $a$ is $2$-less than $b$, or $a <_2 b$, if $a_i < b_i$ for at least
two co-ordinates $i$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
not_lt₂ {α : Type*} [LinearOrder α] {a b : Fin 3 → α} :
¬a <₂ b ↔ ∀ i j, i ≠ j → a i < b i → b j ≤ a j | by simp [lt₂] | theorem | Arxiv..not_lt₂ | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
not_lt₂_of_forall_le {α : Type*} [LinearOrder α] {a b : Fin 3 → α}
(h : ∀ i, b i ≤ a i) : ¬a <₂ b | not_lt₂.2 fun _ _ _ _ => h _ | theorem | Arxiv..not_lt₂_of_forall_le | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
not_lt₂_of_exists {α : Type*} [LinearOrder α] {a b : Fin 3 → α}
(i j : Fin 3) (hij : i ≠ j) (hi : b i ≤ a i) (hj : b j ≤ a j) :
¬a <₂ b | by
refine not_lt₂.2 fun k l hkl h => ?_
have : k ≠ i := fun hk => not_lt.2 hi (hk ▸ h)
have : k ≠ j := fun hk => not_lt.2 hj (hk ▸ h)
have : l = i ∨ l = j := by omega
rcases this with (rfl | rfl); exact hi; exact hj | theorem | Arxiv..not_lt₂_of_exists | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
not_lt₂_self {α : Type*} [LinearOrder α] (a : Fin 3 → α) : ¬a <₂ a | by
simp | theorem | Arxiv..not_lt₂_self | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
lt₂_example_1 : ![3, 3, 9] <₂ ![5, 6, 1] | ⟨0, 1, zero_ne_one, by simp⟩ | theorem | Arxiv..lt₂_example_1 | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | For example, $(3, 3, 9) <_2 (5, 6, 1)$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
lt₂_example_2 : ![5, 6, 1] <₂ ![7, 7, 7] | ⟨0, 2, by simp, by simp⟩ | theorem | Arxiv..lt₂_example_2 | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | $(5, 6, 1) <_2 (7, 7, 7)$ | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
lt₂_example_3 : ![7, 7, 7] <₂ ![7, 8, 9] | ⟨1, 2, by simp, by simp⟩ | theorem | Arxiv..lt₂_example_3 | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | $(7, 7, 7) <_2 (7, 8, 9)$ | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
not_lt₂_example : ¬![1, 2, 3] <₂ ![1, 2, 4] | not_lt₂_of_exists 0 1 zero_ne_one (by simp) (by simp) | theorem | Arxiv..not_lt₂_example | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | but $(1, 2, 3)$ is not $2$-less than $(1, 2, 4). | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
not_trans_lt₂_nat : ∃ (a b c : Fin 3 → ℕ),
a <₂ b ∧ b <₂ c ∧ ¬a <₂ c | ⟨![1, 2, 3], ![2, 3, 1], ![3, 1, 2], ⟨0, 1, zero_ne_one, by simp⟩,
⟨0, 2, by simp, by simp⟩, not_lt₂_of_exists 1 2 (by simp) (by simp) (by simp)⟩ | theorem | Arxiv..not_trans_lt₂_nat | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The $2$-less relation is not transitive on the naturals. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
IsIncreasing₂ {α : Type*} [LT α] (s : List (Fin 3 → α)) : Prop | s.Pairwise lt₂ | def | Arxiv..IsIncreasing₂ | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Since the $2$-less relation is not transitive, we make a further definition to
specify transivity. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
isIncreasing₂_nil {α : Type*} [LT α] : IsIncreasing₂ (α := α) [] | by
simp [IsIncreasing₂] | theorem | Arxiv..isIncreasing₂_nil | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
isIncreasing₂_singleton {α : Type*} [LT α] (a : Fin 3 → α) : IsIncreasing₂ [a] | by
simp [IsIncreasing₂] | theorem | Arxiv..isIncreasing₂_singleton | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
isIncreasing₂_const_length {α : Type*} [LinearOrder α] {val : α} {s : List (Fin 3 → α)}
(h : IsIncreasing₂ s)
(h_const : ∀ a ∈ s, ∀ j, a j = val) : s.length < 2 | by
by_contra!
have h₀ : s[0] = fun _ => val := funext fun i => by simp [h_const s[0] (by simp)]
have h₁ : s[1] = fun _ => val := funext fun i => by simp [h_const s[1] (by simp)]
have := List.pairwise_iff_getElem.1 h 0 1 (by linarith) (by linarith) zero_lt_one
simp [h₀, h₁] at this
exact not_lt₂_self _ this | theorem | Arxiv..isIncreasing₂_const_length | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
maximalLength (n : ℕ) : ℕ | sSup { List.length s | (s) (_ : ∀ a ∈ s, Set.range a ⊆ Set.Icc 1 n) (_ : IsIncreasing₂ s) } | def | Arxiv..maximalLength | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Let $F(n)$ be the maximal length of a $2$-increasing sequence of triples with each coordinate
belong to $[n]$ ($= \{1, 2, ..., n\}$). | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
maximalLength_zero : maximalLength 0 = 0 | by
have (x : ℕ) (s : List (Fin 3 → ℕ)) :
IsIncreasing₂ s ∧ (∀ a, a ∉ s) ∧ s.length = x ↔ s = [] ∧ x = 0 := by
refine ⟨fun ⟨ha₁, ha₂, rfl⟩ => ?_, fun ⟨h₁, h₂⟩ => by simp [h₁, h₂]⟩
simp only [List.length_eq_zero_iff, and_self]
refine List.eq_nil_of_subset_nil fun ai hai => ?_
simpa using ha₂ ai ha... | theorem | Arxiv..maximalLength_zero | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
maximalLength_one : maximalLength 1 = 1 | by
classical
have (x : ℕ) (s : List (Fin 3 → ℕ)) :
IsIncreasing₂ s ∧ (∀ a ∈ s, ∀ i, a i = 1) ∧ s.length = x ↔
s = [fun _ => 1] ∧ x = 1 ∨ s = [] ∧ x = 0 := by
refine ⟨fun ⟨hs₁, hs₂, hx⟩ => ?_, fun h => by aesop⟩
have := hx ▸ isIncreasing₂_const_length hs₁ hs₂
interval_cases x; simp [List.le... | theorem | Arxiv..maximalLength_one | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
maximalLength_four : maximalLength 4 = 8 | by
sorry | theorem | Arxiv..maximalLength_four | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
exists_pair_of_mem_Icc {s : List (Fin 3 → ℕ)} {n : ℕ} (hn : 2 ≤ n)
(hs₁ : ∀ a ∈ s, Set.range a ⊆ Set.Icc 1 n) (hs₂ : s.length > n ^ 2) :
∃ (i j : Fin s.length), i ≠ j ∧ s[i] 0 = s[j] 0 ∧ s[i] 1 = s[j] 1 | by
sorry | lemma | Arxiv..exists_pair_of_mem_Icc | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | In a set of more than $n^2$ triples with coordinates from $\{1, ..., n\}$ we must
have two triples that are equal in their first two coordinates. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
maximalLength_le (n : ℕ) : F n ≤ n ^ 2 | by
sorry | theorem | Arxiv..maximalLength_le | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | For all $n$ we have $F(n) \leq n^2$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
maximalLength_ge_of_isSquare {n : ℕ} (h : IsSquare n) :
n.sqrt ^ 3 ≤ F n | by
sorry | theorem | Arxiv..maximalLength_ge_of_isSquare | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Moreover, whenever $n$ is a perfect square we have $F(n) \geq n^{3/2}$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
IsComparable₂ {α : Type*} [LT α] (t₁ t₂ : Fin 3 → α) : Prop | t₁ <₂ t₂ ∨ t₂ <₂ t₁ | def | Arxiv..IsComparable₂ | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Two triples $t_1$ and $t_2$ are $2$-comparable if one of them is $2$-less
than the other. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
IsComparableSet₂ {α : Type*} [LT α] (s : List (Fin 3 → α)) : Prop | ∃ t₁ t₂, t₁ ≠ t₂ ∧ t₁ ∈ s ∧ t₂ ∈ s ∧ IsComparable₂ t₁ t₂ | def | Arxiv..IsComparableSet₂ | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | A set of triples is $2$-comparable if any two of them are $2$-comparable. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
maximalLength_le_isBigO : ∃ Ω : ℕ → ℝ,
(fun (n : ℕ) => (Real.iteratedLog n : ℝ)) =O[atTop] Ω ∧
∀ n, F n ≤ n ^ 2 / Real.exp (Ω n) | by
sorry | theorem | Arxiv..maximalLength_le_isBigO | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [
"Real.iteratedLog"
] | $F(n) \leq n^2 / \exp(\Omega(\log^*(n)))$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
tripleProduct {α : Type*} (a b : Fin 3 → α) : Πₗ (_ : Fin 3), α × α | toLex (Pi.prod a b) | def | Arxiv..tripleProduct | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | We define the product of two triples $(a, b, c)$ and $(d, e, f)$ by
$((a, d), (b, e), (c, f))$, where the pairs are arranged in lexicographical order. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
tripleProduct_const {α : Type*} (a : α) :
tripleProduct (fun _ => a) (fun _ => a) = toLex (fun _ => (a, a)) | by
simpa [tripleProduct] using funext fun i => by simp | theorem | Arxiv..tripleProduct_const | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
tripleProduct_vecConst_const {α : Type*} (a : α) :
tripleProduct ![a, a, a] ![a, a, a] = toLex ![(a, a), (a, a), (a, a)] | by
simp [tripleProduct]
ext i <;> fin_cases i <;> simp | theorem | Arxiv..tripleProduct_vecConst_const | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
sequenceProduct {α : Type*} (s t : List (Fin 3 → α)) : Lex (List (Πₗ (_ : Fin 3), α × α)) | toLex (s.flatMap (fun a => List.map (tripleProduct a) t)) | def | Arxiv..sequenceProduct | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | We define the product $\otimes$ of two sequences $(a_i, b_i, c_i)$ and
$(d_i, e_i, f_i)$ by the sequence $((a_i, d_j), (b_i, e_j), (c_i, f_j))$, where
the indices $(i, j)$ are arranged lexicographically, and the pairs are also
ordered lexicographically. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
sequenceProduct_example : [![1, 1, 1]] ⊗₂ [![1, 1, 1]] = toLex [toLex ![(1, 1), (1, 1), (1, 1)]] | by
simp [sequenceProduct] | theorem | Arxiv..sequenceProduct_example | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
maximalLength_pow {n : ℕ} {e : ℝ} (hn : 1 < n) (h : F n = (n : ℝ) ^ e) :
∀ᶠ m : ℕ in Filter.atTop, (m : ℝ) ^ e ≤ F m | by
sorry | theorem | Arxiv..maximalLength_pow | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Suppose that for some $n$ we have $F(n) = n ^ {\alpha}$. Then there are arbitrarily
large $m$ such that $F(m) \geq m^{\alpha}$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
maximalLength_le_strong (n : ℕ) : F n ≤ Real.sqrt n ^ 3 | by
sorry | theorem | Arxiv..maximalLength_le_strong | Arxiv.1609.08688 | FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | $F(n) \leq n^{3/2}$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
independentDominationEven (hIso : 0 < G.minDegree) (hEven : Even G.maxDegree) :
let D | G.maxDegree
let i := G.indepDominationNumber
let n := Fintype.card V
(D + 2)^2 * i ≤ (D^2 + 4) * n := by
sorry | theorem | Arxiv..independentDominationEven | Arxiv.2107.00295 | FormalConjectures/Arxiv/2107.00295/IndependentDomination.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | **Conjecture 1.6 (Even case).**
For a nonempty isolate-free graph $G$ on $n$ vertices,
if $D$ is even, then $(D + 2)^2 \cdot i(G) \leq (D^2 + 4) \cdot n$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
independentDominationOdd (hIso : 0 < G.minDegree) (hOdd : Odd G.maxDegree) :
let D | G.maxDegree
let i := G.indepDominationNumber
let n := Fintype.card V
(D + 1) * (D + 3) * i ≤ (D^2 + 3) * n := by
sorry | theorem | Arxiv..independentDominationOdd | Arxiv.2107.00295 | FormalConjectures/Arxiv/2107.00295/IndependentDomination.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | **Conjecture 1.6 (Odd case).**
For a nonempty isolate-free graph $G$ on $n$ vertices,
if $D$ is odd, then $(D + 1)(D + 3) \cdot i(G) \leq (D^2 + 3) \cdot n$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
CollatzLike (n : ℕ) (hn : 8 < n) : 2 ∈ Nat.digits 3 (2^n) | by
sorry | theorem | Arxiv..CollatzLike | Arxiv.2107.12475 | FormalConjectures/Arxiv/2107.12475/CollatzLike.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | For $n > 8$, $2^n$ is not the the sum of distinct powers of $3$. Expressed here in terms of the base $3$ digits of $n$.
This conjecture is equivalent to the halting of a $15$-state $2$-symbol Turing Machine.
TODO(lezeau): Formalize the Turing Machine version of this problem.
Source: *Hardness of Busy Beaver Value BB... | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
two_not_in_digits_three_pow_eight : 2 ∉ Nat.digits 3 (2^8) | by norm_num | theorem | Arxiv..two_not_in_digits_three_pow_eight | Arxiv.2107.12475 | FormalConjectures/Arxiv/2107.12475/CollatzLike.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | For $n = 8$, $2$ is not contained in the base $3$ digits of $n$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
IsCancellative (k A : Type*) [Field k]
[CommRing A] [Algebra k A] [Algebra.FiniteType k A] : Prop | ∀ {B : Type*}
[CommRing B] [Algebra k B] [Algebra.FiniteType k B], Nonempty (A[X] ≃ₐ[k] B[X]) →
Nonempty (A ≃ₐ[k] B) | def | Arxiv..IsCancellative | Arxiv.2208.14736 | FormalConjectures/Arxiv/2208.14736/ZariskiCancellation.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | A finitely generated `k`-algebra `A` is cancellative if for all finitely generated `k` algebras `B` such that
`B[X] ≅ₖ A[X]` we have `B ≅ₖ A`. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
zariski_cancellation_problem {k : Type*} [Field k]
[CharZero k] {ι : Type*} [Fintype ι] : IsCancellative k (MvPolynomial ι k) | by
sorry | theorem | Arxiv..zariski_cancellation_problem | Arxiv.2208.14736 | FormalConjectures/Arxiv/2208.14736/ZariskiCancellation.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The **Zariski Cancellation Problem**: every polynomial ring over a field `k` of characteristic
`0` is cancellative. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
zariski_cancellation_problem.variants.dim_one
{k : Type*} [Field k] : IsCancellative k k[X] | by
sorry | theorem | Arxiv..zariski_cancellation_problem.variants.dim_one | Arxiv.2208.14736 | FormalConjectures/Arxiv/2208.14736/ZariskiCancellation.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The single variable polynomial ring `k[X]` is cancellative in any characteristic | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
zariski_cancellation_problem.variants.dim_two {k : Type*} [Field k] :
IsCancellative k (MvPolynomial (Fin 2) k) | by
sorry | theorem | Arxiv..zariski_cancellation_problem.variants.dim_two | Arxiv.2208.14736 | FormalConjectures/Arxiv/2208.14736/ZariskiCancellation.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The two variable polynomial ring `k[X]` is cancellative in any characteristic | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
zariski_cancellation_problem.variants.false_pos_card
(p : ℕ) [hp : Fact p.Prime] {ι : Type*} [Fintype ι] (hι : Fintype.card ι = 3) :
¬ IsCancellative (ZMod p) (MvPolynomial ι (ZMod p)) | by
sorry | theorem | Arxiv..zariski_cancellation_problem.variants.false_pos_card | Arxiv.2208.14736 | FormalConjectures/Arxiv/2208.14736/ZariskiCancellation.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The positive characteristic case of the Zariski Cancellation Problem is false in dimension `3` | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
MultiplicativelyIndependent (p q : ℕ) : Prop | Irrational (Real.log p / Real.log q) | def | Arxiv.id2303_01089.MultiplicativelyIndependent | Arxiv.2303.01089 | FormalConjectures/Arxiv/2303.01089/FurstenbergTimesPTimesQ.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Two integers $p, q \ge 2$ are multiplicatively independent if
$\log p / \log q$ is irrational. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
Tn (n : ℕ) (x : 𝕋) | n • x | def | Arxiv.id2303_01089.Tn | Arxiv.2303.01089 | FormalConjectures/Arxiv/2303.01089/FurstenbergTimesPTimesQ.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The map $T_n$ sends $x$ to $nx \bmod 1$ on the additive circle. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
Tn_continuous (n : ℕ) : Continuous (Tn n) | continuous_nsmul n | lemma | Arxiv.id2303_01089.Tn_continuous | Arxiv.2303.01089 | FormalConjectures/Arxiv/2303.01089/FurstenbergTimesPTimesQ.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
IsTnInvariant (n : ℕ) (F : Set 𝕋) : Prop | Tn n '' F ⊆ F | def | Arxiv.id2303_01089.IsTnInvariant | Arxiv.2303.01089 | FormalConjectures/Arxiv/2303.01089/FurstenbergTimesPTimesQ.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | A set $F$ is $T_n$-invariant if $T_n(F) \subseteq F$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
MeasureTheory.IsAtom {α : Type*} {m0 : MeasurableSpace α} (μ : Measure α) (A : Set α) : Prop | 0 < μ A ∧ ∀ B ⊆ A, MeasurableSet B → μ B = 0 ∨ μ B = μ A | def | Arxiv.id2303_01089.MeasureTheory.IsAtom | Arxiv.2303.01089 | FormalConjectures/Arxiv/2303.01089/FurstenbergTimesPTimesQ.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | A set $A$ is an atom if it has positive measure and for all $B \subseteq A$ measurable,
either $\mu(B) = 0$ or $\mu(B) = \mu(A)$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
MeasureTheory.IsAtomLess {α : Type*} {m0 : MeasurableSpace α} (μ : Measure α) : Prop where
NoAtoms : ∀ A, MeasurableSet A → ¬ MeasureTheory.IsAtom μ A | class | Arxiv.id2303_01089.MeasureTheory.IsAtomLess | Arxiv.2303.01089 | FormalConjectures/Arxiv/2303.01089/FurstenbergTimesPTimesQ.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | A measure is atomless if it has no atoms. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
UnitAddCircle.ProbabilityMeasure : ProbabilityMeasure 𝕋 | ⟨volume, IsProbabilityMeasure.mk UnitAddCircle.measure_univ⟩ | def | Arxiv.id2303_01089.UnitAddCircle.ProbabilityMeasure | Arxiv.2303.01089 | FormalConjectures/Arxiv/2303.01089/FurstenbergTimesPTimesQ.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
conjecture_1_3 {p q : ℕ} (hp : 2 <= p) (hq : 2 <= q) (hpq : MultiplicativelyIndependent p q)
{μ : Measure 𝕋} [IsProbabilityMeasure μ] [MeasureTheory.IsAtomLess μ]
(hmup : MeasurePreserving (Tn p) μ μ)
(hmuq : MeasurePreserving (Tn q) μ μ) :
μ = volume | by
sorry | theorem | Arxiv.id2303_01089.conjecture_1_3 | Arxiv.2303.01089 | FormalConjectures/Arxiv/2303.01089/FurstenbergTimesPTimesQ.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | **Conjecture 1.3** (the $\times p, \times q$ conjecture): the only atomless Borel probability
measure on $\mathbb{T}$ which is both $T_p$- and $T_q$-invariant is the Lebesgue measure. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
conjecture_1_4 :
answer(False) ↔
∀ p q : ℕ, 2 <= p → 2 <= q → MultiplicativelyIndependent p q →
∀ μ : ProbabilityMeasure 𝕋,
MeasureTheory.IsAtomLess μ.1 → MeasurePreserving (Tn p) μ μ →
Tendsto (fun n : ℕ => μ.map (Tn_continuous (q ^ n)).aemeasurable) atTop
(𝓝 UnitAdd... | by
sorry | theorem | Arxiv.id2303_01089.conjecture_1_4 | Arxiv.2303.01089 | FormalConjectures/Arxiv/2303.01089/FurstenbergTimesPTimesQ.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | **Conjecture 1.4**: if $\mu$ is an atomless $T_p$-invariant Borel probability measure on
$\mathbb{T}$, then $T_{q^n}\mu$ converges weak-star to Lebesgue measure.
This paper disproves the conjecture. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
S' (h k : ℕ) : ℤ | ∑ j ∈ Finset.Ico 1 k, (-1 : ℤ) ^ (j + 1 + ⌊(h * j : ℚ) / k⌋₊) | def | Arxiv..S' | Arxiv.2501.03234 | FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
S (k : ℕ) : ℤ | ∑ h ∈ Finset.Ico 1 k, S' h k | def | Arxiv..S | Arxiv.2501.03234 | FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Define the sum
$$S(k) := \sum_{h=1}^{k-1}S'(h, k)$$ | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
S_fst_10 : List.map S (List.range 10) = [0, 0, 1, 2, 5, 4, 7, 10, 11, 8] | by
unfold S
decide +kernel | theorem | Arxiv..S_fst_10 | Arxiv.2501.03234 | FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Note that in Table 1 in https://arxiv.org/abs/2501.03234v1, there seems to be an error:
11 appears twice. The first 10 values of $S$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
conjecture_1_1 (k : ℕ) (hprim : k.Prime) (hodd : Odd k) : 0 < S k | by
sorry | theorem | Arxiv..conjecture_1_1 | Arxiv.2501.03234 | FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | **Conjecture 1.1**: For any odd prime $k$, the sum associated with the classical theta function $θ_3$,
$S(k)$ is positive. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
conjecture_4_1 (k : ℕ) (hprim : k.Prime) (hodd : Odd k) (hgt : k > 5) : k < S k | by
sorry | theorem | Arxiv..conjecture_4_1 | Arxiv.2501.03234 | FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | **Conjecture 4.1**: For any prime $k$ larger than $5$, $S(k) > k$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
conjecture_4_2 (k : ℕ) (hprim : k.Prime) (hodd : Odd k) (hgt : k > 233) : 2 * k < S k | by
sorry | theorem | Arxiv..conjecture_4_2 | Arxiv.2501.03234 | FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | **Conjecture 4.2**: For any prime $k$ larger than $233$, $S(k) > 2k$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
conjecture_4_3 (k : ℕ) (hprim : k.Prime) (hodd : Odd k) (hgt : k > 3119) : 3 * k < S k | by
sorry | theorem | Arxiv..conjecture_4_3 | Arxiv.2501.03234 | FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | **Conjecture 4.3**: For any prime $k$ larger than $3119$, $S(k) > 3k$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
conjecture_4_4 (n : ℕ) : ∀ᶠ (k : ℕ) in Filter.atTop, k.Prime → Odd k → n * k < S k | by
sorry | theorem | Arxiv..conjecture_4_4 | Arxiv.2501.03234 | FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | **Conjecture 4.4**: Given a natural number $n ∈ ℕ$, for all large enough odd prime $k$ (depending on $n$),
$nk < S(k)$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
conjecture_4_4_def_0 (hc1_1: type_of% conjecture_1_1) : type_of% (conjecture_4_4 0) | by
simp only [Filter.Eventually, CharP.cast_eq_zero, zero_mul, Filter.mem_atTop_sets]
exact ⟨0, fun b sb bprim bodd ↦ hc1_1 b bprim bodd⟩ | theorem | Arxiv..conjecture_4_4_def_0 | Arxiv.2501.03234 | FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | **Conjecture 1.1 → Conjecture 4.4**: If conjecture 1.1 holds true, then this implies a special
case of conjecture 4.4 where $n = 0$. In this case the lower bound for the odd prime $k$
would be $0$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
conjecture_4_4_def_1 (hc4_1: type_of% conjecture_4_1) : type_of% (conjecture_4_4 1) | by
simp [Filter.Eventually, Filter.mem_atTop_sets]
exact ⟨5+1, fun b sb bprim bodd ↦ hc4_1 b bprim bodd (by linarith)⟩ | theorem | Arxiv..conjecture_4_4_def_1 | Arxiv.2501.03234 | FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | **Conjecture 4.1 → Conjecture 4.4**: If conjecture 4.1 holds true, then this implies a special
case of conjecture 4.4 where $n = 1$. In this case the lower bound would be $5$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
conjecture_4_4_def_2 (hc4_2: type_of% conjecture_4_2) : type_of% (conjecture_4_4 2) | by
simp only [Filter.Eventually, Filter.mem_atTop_sets]
exact ⟨233+1, fun b sb bprim bodd ↦ hc4_2 b bprim bodd (by linarith)⟩ | theorem | Arxiv..conjecture_4_4_def_2 | Arxiv.2501.03234 | FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | **Conjecture 4.2 → Conjecture 4.4**: If conjecture 4.2 holds true, then this implies a special
case of conjecture 4.4 for $n = 2$. For this scenario, the lower bound is now $233$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
conjecture_4_4_def_3 (hc4_3: type_of% conjecture_4_3) : type_of% (conjecture_4_4 3) | by
simp only [Filter.Eventually, Filter.mem_atTop_sets]
exact ⟨3119+1, fun b sb bprim bodd ↦ hc4_3 b bprim bodd (by linarith)⟩ | theorem | Arxiv..conjecture_4_4_def_3 | Arxiv.2501.03234 | FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | **Conjecture 4.3 → Conjecture 4.4**: If conjecture 4.3 holds true, then a special
case of conjecture 4.4 for $n = 3$ is obtained, and the lower bound is $3119$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
conjecture_1_1 {n : ℕ} (hn : 3 ≤ n)
(g : SL(n, ℝ) ⧸ Subgroup.map (map (Int.castRingHom ℝ)) ⊤)
(hg : IsCompact <| closure (MulAction.orbit (diagonalSubgroup (Fin n) ℝ) g)) :
IsClosed <| MulAction.orbit (diagonalSubgroup (Fin n) ℝ) g | by
sorry | theorem | Margulis.conjecture_1_1 | Arxiv.2504.17644 | FormalConjectures/Arxiv/2504.17644/Margulis.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Let `D` be the diagonal group of `SL_n(ℝ)` where n ≥ 3.
Then any relatively compact `D`-orbit in `SL_n(ℝ) / SL_n(ℤ)` is closed. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
polyToLaurent : F[X] →+* F⸨X⸩ | (HahnSeries.ofPowerSeries ℤ F).comp Polynomial.coeToPowerSeries.ringHom | def | Margulis.polyToLaurent | Arxiv.2504.17644 | FormalConjectures/Arxiv/2504.17644/Margulis.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The natural inclusion `F[t] →+* F((t⁻¹))`. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
huang_shi_theorem_1_2
(hchar : ringChar F ∈ ({3, 5, 7, 11} : Finset ℕ)) :
∃ z : SL(4, F⸨X⸩) ⧸ ( Matrix.SpecialLinearGroup.map
(polyToLaurent F)).range,
IsCompact (closure (MulAction.orbit (diagonalSubgroup (Fin 4) (F⸨X⸩)) z)) ∧
¬ IsClosed (MulAction.orbit (diagonalSubgroup (Fin 4) (F⸨X⸩)) z) | by
-- Placeholder: a Lean formalization would require a full development
-- of the Huang–Shi paper in mathlib.
sorry | theorem | Margulis.huang_shi_theorem_1_2 | Arxiv.2504.17644 | FormalConjectures/Arxiv/2504.17644/Margulis.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | **Huang–Shi, Theorem 1.2**
Let `F` be a finite field of characteristic `p ∈ {3, 5, 7, 11}`, and set
`K = F((t⁻¹))`, `A = F[t]`. Let
* `D` be the diagonal subgroup of `SL₄(K)`,
* `Γ = SL₄(A)` the lattice subgroup embedded into `SL₄(K)` via the natural inclusion `A →+* K`.
Then there exists `z : SL₄(K)/Γ` such that th... | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
finiteAdditiveConvolution (n : ℕ) (p q : F[X]) : F[X] | let c := fun k => ∑ ij ∈ antidiagonal (k : ℕ),
((n - ij.1)! * (n - ij.2)! : F) / (n ! * (n - k)! : F) *
p.coeff (n - ij.1) * q.coeff (n - ij.2)
∑ k ∈ range (n + 1), c k • X^(n - k) | def | Arxiv..finiteAdditiveConvolution | Arxiv.2602.05192 | FormalConjectures/Arxiv/2602.05192/FirstProof4.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Define $p \boxplus_n q(x)$ to be the polynomial
$$
(p \boxplus_n q)(x) = \sum_{k=0}^n c_k x^{n-k}
$$
where the coefficients $c_k$ are given by the formula:
$$
c_k = \sum_{i+j=k} \frac{(n-i)! (n-j)!}{n! (n-k)!} a_i b_j
$$
for $k = 0, 1, \dots, n$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
finiteAdditiveConvolution_comm (n : ℕ) (p q : F[X]) :
p (⊞_n) q = q (⊞_n) p | by
show ∑ a ∈_, _= ∑ a ∈_, _
exact sum_congr rfl fun m hm =>
(congr_arg₂ _) (sum_equiv (.prodComm _ _) (by simp [add_comm]) fun _ _ => by ring!) rfl | theorem | Arxiv..finiteAdditiveConvolution_comm | Arxiv.2602.05192 | FormalConjectures/Arxiv/2602.05192/FirstProof4.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
finiteAdditiveConvolution_degree (n : ℕ) (p q : ℝ[X])
(hp : p.degree = n) (hq : q.degree = n):
(p (⊞_n) q).degree = n | by
sorry | theorem | Arxiv..finiteAdditiveConvolution_degree | Arxiv.2602.05192 | FormalConjectures/Arxiv/2602.05192/FirstProof4.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
finiteAdditiveConvolution_monic' (n : ℕ) (p q : ℝ[X]) (hn : 0 < n)
(hp_deg : p.degree = n) (hq_deg : q.degree = n) (hp_monic : p.Monic) (hq_monic : q.Monic) :
(p (⊞_n) q).Monic | by
have hc0 : ∑ ij ∈ antidiagonal 0, ((n - ij.1)! * (n - ij.2)! : ℝ) / (n ! * (n - 0)! : ℝ) *
p.coeff (n - ij.1) * q.coeff (n - ij.2) = 1 := by
rw [antidiagonal_zero]
simp
have hp1 : p.coeff n = 1 := by
have : p.natDegree = n := natDegree_eq_of_degree_eq_some hp_deg
rw [← this]
exa... | theorem | Arxiv..finiteAdditiveConvolution_monic' | Arxiv.2602.05192 | FormalConjectures/Arxiv/2602.05192/FirstProof4.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
Φ (p : ℝ[X]) : ℝ≥0∞ | if p.roots.Nodup then
let roots := p.roots.toFinset
(∑ i ∈ roots, (∑ j ∈ roots.erase i, 1 / (i - j)) ^ 2).toNNReal
else
⊤ | def | Arxiv..Φ | Arxiv.2602.05192 | FormalConjectures/Arxiv/2602.05192/FirstProof4.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | For a monic polynomial $p(x)=\prod_{i\le n}(x- \lambda_i)$, define
$$\Phi_n(p):=\sum_{i\le n}(\sum_{j\neq i} \frac1{\lambda_i-\lambda_j})^2$$
and $\Phi_n(p):=\infty$ if $p$ has a multiple root. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
FourProp (p q : ℝ[X]) (n : ℕ) : Prop | p.degree = n → p.roots.card = n → q.degree = n → q.roots.card = n → p.Monic → q.Monic →
1 / Φ p + 1 / Φ q ≤ 1 / Φ (p (⊞_n) q) | def | Arxiv..FourProp | Arxiv.2602.05192 | FormalConjectures/Arxiv/2602.05192/FirstProof4.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | A predicate that holds if $p(x)$ and $q(x)$ are monic real-rooted polynomials of
degree $n$, then
$$\frac{1}{\Phi_n(p\boxplus_n q)} \ge \frac{1}{\Phi_n(p)}+\frac{1}{\Phi_n(q)}?$$ | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
four : answer(True) ↔ ∀ (p q : ℝ[X]) (n : ℕ), FourProp p q n | by
sorry | theorem | Arxiv..four | Arxiv.2602.05192 | FormalConjectures/Arxiv/2602.05192/FirstProof4.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Is it true that if $p(x)$ and $q(x)$ are monic real-rooted polynomials of
degree $n$, then
$$\frac{1}{\Phi_n(p\boxplus_n q)} \ge \frac{1}{\Phi_n(p)}+\frac{1}{\Phi_n(q)}?$$
[arxiv/2602.05192v2](https://arxiv.org/abs/2602.05192v2) contains a proof. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
four_2 : answer(True) ↔ ∀ (p q : ℝ[X]), FourProp p q 2 | by
sorry | theorem | Arxiv..four_2 | Arxiv.2602.05192 | FormalConjectures/Arxiv/2602.05192/FirstProof4.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Is it true that if $p(x)$ and $q(x)$ are monic real-rooted polynomials of
degree $2$, then
$$\frac{1}{\Phi_2(p\boxplus_n q)} \ge \frac{1}{\Phi_2(p)}+\frac{1}{\Phi_2(q)}?$$ | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
four_3 : answer(True ) ↔ ∀ (p q : ℝ[X]), FourProp p q 3 | by
sorry | theorem | Arxiv..four_3 | Arxiv.2602.05192 | FormalConjectures/Arxiv/2602.05192/FirstProof4.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Is it true that if $p(x)$ and $q(x)$ are monic real-rooted polynomials of
degree $3$, then
$$\frac{1}{\Phi_3(p\boxplus_n q)} \ge \frac{1}{\Phi_3(p)}+\frac{1}{\Phi_3(q)}?$$ | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
IsEpsilonLight (G : SimpleGraph V) (ε : ℝ) (S : Finset V) : Prop | letI G_S := G.induce S |>.spanningCoe
letI L := lapMatrix ℝ G
letI L_S := lapMatrix ℝ (G_S)
PosSemidef (ε • L - L_S) | def | Arxiv..IsEpsilonLight | Arxiv.2602.05192 | FormalConjectures/Arxiv/2602.05192/FirstProof6.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | For a graph $G = (V, E)$, let $G_S = (V, E(S,S))$ denote the graph with the same vertex set,
but only the edges between vertices in $S$.
Let $L$ be the Laplacian matrix of $G$ and let $L_S$ be the Laplacian of $G_S$.
I say that a set of vertices $S$ is $\epsilon$-light if the matrix $\epsilon L - L_S$ is
positive semi... | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
epsilon_light_subset_exists : answer(True) ↔
∃ (c : ℝ), c > 0 ∧ ∀ (n : ℕ) (G : SimpleGraph (Fin n)) (ε : ℝ),
0 < ε → ε < 1 →
∃ (S : Finset (Fin n)), IsEpsilonLight G ε S ∧ (S.card : ℝ) ≥ c * ε * n | by
sorry | theorem | Arxiv..epsilon_light_subset_exists | Arxiv.2602.05192 | FormalConjectures/Arxiv/2602.05192/FirstProof6.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Does there exist a constant $c > 0$ so that for every graph $G$ and every $\epsilon$ between
$0$ and $1$, $V$ contains an $\epsilon$-light subset $S$ of size at least $c \epsilon |V|$? | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
banach_mazur_rotation_problem : answer(sorry) ↔
∀ (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [SeparableSpace E]
[IsPretransitive (E ≃ₗᵢ[ℝ] E) (sphere (0 : E) 1)], ∃ (H : Type*) (_ : NormedAddCommGroup H)
(_ : InnerProductSpace ℝ H),
Nonempty (E ≃ₗᵢ[ℝ] H) | by
sorry | theorem | Arxiv..banach_mazur_rotation_problem | Arxiv.math.0110202 | FormalConjectures/Arxiv/math.0110202/BanachMazurRotation.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The Banach--Mazur rotation problem asks whether every separable Banach space whose group of linear
isometric equivalences acts transitively on the unit sphere is linearly isometric to a Hilbert
space. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
banach_mazur_rotation_problem.finite_dimensional {E : Type*} [NormedAddCommGroup E]
[NormedSpace ℝ E] [FiniteDimensional ℝ E] [IsPretransitive (E ≃ₗᵢ[ℝ] E) (sphere (0 : E) 1)] :
InnerProductSpaceable E | by
sorry | theorem | Arxiv..banach_mazur_rotation_problem.finite_dimensional | Arxiv.math.0110202 | FormalConjectures/Arxiv/math.0110202/BanachMazurRotation.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Every finite-dimensional real normed space whose isometry group acts transitively on the
unit sphere is Euclidean. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
End of preview. Expand in Data Studio
Lean4-FormalConjectures
Structured dataset from formal-conjectures — Google DeepMind formalized conjectures.
Source
- Repository: https://github.com/google-deepmind/formal-conjectures
- Commit:
b9b8aa0fd6170e482798f7c78c163acacd40e083 - Files: 896
- License: apache-2.0
Schema
| Column | Type | Description |
|---|---|---|
| statement | string | Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof |
| proof | string | Verbatim proof/body, empty if the declaration has none |
| type | string | Declaration keyword |
| symbolic_name | string | Declaration identifier |
| library | string | Sub-library |
| filename | string | Repository-relative source path |
| imports | list[string] | File-level Require/Import modules |
| deps | list[string] | Intra-corpus identifiers referenced |
| docstring | string | Preceding documentation comment, empty if absent |
| source_url | string | Upstream repository |
| commit | string | Upstream commit extracted |
Statistics
- Entries: 4,557
- With proof: 4,472 (98.1%)
- With docstring: 3,759 (82.5%)
- Libraries: 88
By type
| Type | Count |
|---|---|
| theorem | 2,859 |
| def | 1,143 |
| lemma | 364 |
| abbrev | 81 |
| structure | 53 |
| instance | 24 |
| inductive | 15 |
| class | 13 |
| elab | 3 |
| macro | 2 |
Example
S (S₀ : List ℤ) (n : ℕ) : List ℤ
match n with
| 0 => S₀
| n + 1 => (S S₀ n) ++ [Int.ofNat (k (S S₀ n))]
- type: def | symbolic_name:
Arxiv..S| FormalConjectures/Arxiv/0912.2382/CurlingNumberConjecture.lean
Use
Each declaration is split into a statement (signature/claim) and a proof (body) that are disjoint
and together form the complete declaration, for proof modeling, autoformalization, retrieval, and
dependency analysis via deps.
Citation
@misc{lean4_formalconjectures_dataset,
title = {Lean4-FormalConjectures},
author = {Norton, Charles},
year = {2026},
note = {Extracted from https://github.com/google-deepmind/formal-conjectures, commit b9b8aa0fd617},
url = {https://huggingface.co/datasets/phanerozoic/Lean4-FormalConjectures}
}
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