{"name":"PFR_conjecture_pos_aux","declaration":"/-- Record positivity results that are useful in the proof of PFR. -/\ntheorem PFR_conjecture_pos_aux {G : Type u_1} [AddCommGroup G] {A : Set G} [Finite ↑A] {K : ℝ} (h₀A : Set.Nonempty A) (hA : ↑(Nat.card ↑(A - A)) ≤ K * ↑(Nat.card ↑A)) : 0 < ↑(Nat.card ↑A) ∧ 0 < ↑(Nat.card ↑(A - A)) ∧ 0 < K"}
{"name":"PFR_conjecture","declaration":"/-- The polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of an elementary abelian\n2-group of doubling constant at most $K$, then $A$ can be covered by at most $2K^{12}$ cosets of\na subgroup of cardinality at most $|A|$. -/\ntheorem PFR_conjecture {G : Type u_1} [AddCommGroup G] {A : Set G} [Finite ↑A] {K : ℝ} [Countable G] [ElementaryAddCommGroup G 2] [Fintype G] (h₀A : Set.Nonempty A) (hA : ↑(Nat.card ↑(A + A)) ≤ K * ↑(Nat.card ↑A)) : ∃ H c, ↑(Nat.card ↑c) < 2 * K ^ 12 ∧ Nat.card ↥H ≤ Nat.card ↑A ∧ A ⊆ c + ↑H"}
{"name":"PFR_conjecture_pos_aux'","declaration":"theorem PFR_conjecture_pos_aux' {G : Type u_1} [AddCommGroup G] {A : Set G} [Finite ↑A] {K : ℝ} (h₀A : Set.Nonempty A) (hA : ↑(Nat.card ↑(A + A)) ≤ K * ↑(Nat.card ↑A)) : 0 < ↑(Nat.card ↑A) ∧ 0 < ↑(Nat.card ↑(A + A)) ∧ 0 < K"}
{"name":"rdist_le_of_isUniform_of_card_add_le","declaration":"/-- A uniform distribution on a set with doubling constant `K` has self Rusza distance\nat most `log K`. -/\ntheorem rdist_le_of_isUniform_of_card_add_le {G : Type u_1} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G] {A : Set G} [Finite ↑A] {K : ℝ} [Countable G] (h₀A : Set.Nonempty A) (hA : ↑(Nat.card ↑(A - A)) ≤ K * ↑(Nat.card ↑A)) {Ω : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {U₀ : Ω → G} (U₀unif : ProbabilityTheory.IsUniform A U₀ MeasureTheory.volume) (U₀meas : Measurable U₀) : d[U₀ # U₀] ≤ Real.log K"}
{"name":"ProbabilityTheory.IsUniform.measureReal_preimage_sub","declaration":"/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V + x` with probability `# (A ∩ (B + x)) / #A ⬝ #B`. -/\ntheorem ProbabilityTheory.IsUniform.measureReal_preimage_sub {G : Type u_1} {Ω : Type u_2} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] {A : Finset G} {B : Finset G} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {U : Ω → G} {V : Ω → G} (Uunif : ProbabilityTheory.IsUniform (↑A) U MeasureTheory.volume) (Umeas : Measurable U) (Vunif : ProbabilityTheory.IsUniform (↑B) V MeasureTheory.volume) (Vmeas : Measurable V) (hindep : ProbabilityTheory.IndepFun U V MeasureTheory.volume) (x : G) : MeasureTheory.volume.real ((U - V) ⁻¹' {x}) =\n  ↑(Nat.card ↑(↑A ∩ (↑B + {x}))) / (↑(Nat.card { x // x ∈ A }) * ↑(Nat.card { x // x ∈ B }))"}
{"name":"sumset_eq_sub","declaration":"theorem sumset_eq_sub {G : Type u_1} [AddCommGroup G] {A : Set G} [ElementaryAddCommGroup G 2] : A + A = A - A"}
{"name":"ProbabilityTheory.IsUniform.measureReal_preimage_sub_zero","declaration":"/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V` with probability `# (A ∩ B) / #A ⬝ #B`. -/\ntheorem ProbabilityTheory.IsUniform.measureReal_preimage_sub_zero {G : Type u_1} {Ω : Type u_2} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] {A : Finset G} {B : Finset G} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {U : Ω → G} {V : Ω → G} (Uunif : ProbabilityTheory.IsUniform (↑A) U MeasureTheory.volume) (Umeas : Measurable U) (Vunif : ProbabilityTheory.IsUniform (↑B) V MeasureTheory.volume) (Vmeas : Measurable V) (hindep : ProbabilityTheory.IndepFun U V MeasureTheory.volume) : MeasureTheory.volume.real ((U - V) ⁻¹' {0}) =\n  ↑(Nat.card ↑(↑A ∩ ↑B)) / (↑(Nat.card { x // x ∈ A }) * ↑(Nat.card { x // x ∈ B }))"}
{"name":"PFR_conjecture_aux","declaration":"/-- Auxiliary statement towards the polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of\nan elementary abelian 2-group of doubling constant at most $K$, then there exists a subgroup $H$\nsuch that $A$ can be covered by at most $K^{13/2} |A|^{1/2} / |H|^{1/2}$ cosets of $H$, and $H$ has\nthe same cardinality as $A$ up to a multiplicative factor $K^11$. -/\ntheorem PFR_conjecture_aux {G : Type u_1} [AddCommGroup G] {A : Set G} [Finite ↑A] {K : ℝ} [Countable G] [ElementaryAddCommGroup G 2] [Fintype G] (h₀A : Set.Nonempty A) (hA : ↑(Nat.card ↑(A + A)) ≤ K * ↑(Nat.card ↑A)) : ∃ H c,\n  ↑(Nat.card ↑c) ≤ K ^ (13 / 2) * ↑(Nat.card ↑A) ^ (1 / 2) * ↑(Nat.card ↑↑H) ^ (-1 / 2) ∧\n    ↑(Nat.card ↥H) ≤ K ^ 11 * ↑(Nat.card ↑A) ∧ ↑(Nat.card ↑A) ≤ K ^ 11 * ↑(Nat.card ↥H) ∧ A ⊆ c + ↑H"}
{"name":"PFR_conjecture'","declaration":"/-- Corollary of `PFR_conjecture` in which the ambient group is not required to be finite (but) then\n$H$ and $c$ are finite. -/\ntheorem PFR_conjecture' {G : Type u_2} [AddCommGroup G] [ElementaryAddCommGroup G 2] {A : Set G} {K : ℝ} (h₀A : Set.Nonempty A) (Afin : Set.Finite A) (hA : ↑(Nat.card ↑(A + A)) ≤ K * ↑(Nat.card ↑A)) : ∃ H c, Set.Finite c ∧ Set.Finite ↑H ∧ ↑(Nat.card ↑c) < 2 * K ^ 12 ∧ Nat.card ↥H ≤ Nat.card ↑A ∧ A ⊆ c + ↑H"}