Project-declaredLean 4.33.0-rc1 · mathlib@79d0395a1825

Rdist add rdist add cond Mutual eq

rdist_add_rdist_add_condMutual_eq

additive combinatorics entropy probability first estimate

Plain-language statement

A fibring identity in the $\tau$ -minimizer setup. Let $X_1',X_2'$ be independent copies of $X_1,X_2$ and put $k=d[X_1;X_2]$ . Then $d[X_1+X_2';X_2+X_1']+d[X_1\mid X_1+X_2';X_2\mid X_2+X_1']+I[X_1+X_2:X_1'+X_2\mid X_1+X_2+X_1'+X_2']=2k$ .

Exact Lean statement

lemma rdist_add_rdist_add_condMutual_eq [Module (ZMod 2) G] :
    d[X₁ + X₂' # X₂ + X₁'] + d[X₁ | X₁ + X₂' # X₂ | X₂ + X₁']
      + I[X₁ + X₂ : X₁' + X₂ | X₁ + X₂ + X₁' + X₂'] = 2 * k

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma rdist_add_rdist_add_condMutual_eq [Module (ZMod 2) G] :    d[X₁ + X₂' # X₂ + X₁'] + d[X₁ | X₁ + X₂' # X₂ | X₂ + X₁']      + I[X₁ + X₂ : X₁' + X₂ | X₁ + X₂ + X₁' + X₂'] = 2 * k := by  cases nonempty_fintype G  have h0 : ![X₁, X₂, X₂', X₁'] 0 = X₁ := rfl  have h1 : ![X₁, X₂, X₂', X₁'] 1 = X₂ := rfl  have h2 : ![X₁, X₂, X₂', X₁'] 2 = X₂' := rfl  have h3 : ![X₁, X₂, X₂', X₁'] 3 = X₁' := rfl  have h := sum_of_rdist_eq_char_2 ![X₁, X₂, X₂', X₁'] h_indep    (fun i => by fin_cases i <;> assumption)  rw [h0, h1, h2, h3] at h  have heq : d[X₂' # X₁'] = k := by    rw [rdist_symm]    apply h₁.symm.rdist_congr h₂.symm  rw [heq] at h  convert h.symm using 1  · congr 2 <;> abel  · ring

Project: Polynomial Freiman-Ruzsa project
License: Apache-2.0
Commit: a177b2e4abe4
Source: PFR/FirstEstimate.lean:59-76

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Plain-language statement

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Project documentation

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Person-level attribution pending.

View proof record