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Project-declaredLean 4.31.0 · mathlib@fabf563a7c95

Indicator eq 1 on pos

Polynomial.indicator_eq_1_on_pos

Plain-language statement

Indicator evaluated on an element of pos is equal to 1.

Exact Lean statement

lemma indicator_eq_1_on_pos {pos neg : Finset F} {x : F}
  (h_pos : x ∈ pos) :
  (indicator pos neg).eval x = 1

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
lemma indicator_eq_1_on_pos {pos neg : Finset F} {x : F}  (h_pos : x  pos) :  (indicator pos neg).eval x = 1 := by  unfold indicator  have {x} {y} (hy : y  (pos ∪ neg).erase x) :    (x - y)⁻¹ * (x - y) = 1 :=      inv_mul_cancel₀ (sub_ne_zero_of_ne (by aesop))  rw [Polynomial.eval]  aesop      (add simp        [Polynomial.eval_prod,          Polynomial.eval₂_finsetSum,          Lagrange.basis,          Finset.prod_eq_zero_iff,          Lagrange.basis,          Lagrange.basisDivisor,          Finset.prod_eq_one])      (add safe [(by rw [Finset.sum_eq_single x])])
Project
ArkLib
License
Apache-2.0
Commit
fad5cbf80877
Source
ArkLib/Data/Polynomial/Indicator.lean:98-115

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Project documentation

This lemma proves the final algebraic step in the DG25 Theorem 3.1 proof. It shows that if R > e + 1, then e * (R / (R - 1)) < e + 1. The intuition is that the fraction R / (R - 1) is always greater than 1, but as R gets larger, it gets closer to 1. The hypothesis R > e + 1 provides a strong enough bound to ensure the product e * (fraction) do...

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Project-declaredLean 4.31.0

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Plain-language statement

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cryptographyproof systemscoding theory

Source project: ArkLib

Person-level attribution pending.

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Project-declaredLean 4.31.0

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Plain-language statement

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cryptographyproof systemscoding theory

Source project: ArkLib

Person-level attribution pending.

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