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Explicit pruning and scalar agreement exponent margins

Lax342547.CollisionExponents · concepts/Lax342547/CollisionExponents.lean · lax-342547

proven

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    Natural Language Statement

    Lemma

    The numerical bounds in the collision argument have explicit scale conditions. Original-cell pruning is exponentially smaller than overlap, and the actual scalar tests cost at most .02N when their bit count is at most .01N.

    Concept map
    5 concepts; 4 descendants hidden
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 6 statements. Each proof establishes one of them relative to its assumptions.

    1 collision_lower_from_overlap proven

    2 paper_scalar_cost proven

    3 paper_scalar_smallness proven

    4 pruning_overlap_bound proven

    5 pruning_power_identity proven

    6 scalar_agreement_power proven

    Lean source view on GitHub

    1import Lax342547.RetainedImages
    2
    3/-!
    4---
    5title: Explicit pruning and scalar agreement exponent margins
    6type: lemma
    7---
    8The numerical bounds in the collision argument have explicit scale conditions. Original-cell pruning is exponentially smaller than overlap, and the actual scalar tests cost at most .02N when their bit count is at most .01N.
    9-/
    10
    11namespace Lax342547.CollisionExponents
    12
    13
    14
    15axiom pruning_power_identity (k Cn N : ℝ) :
    16 2 * (2 : ℝ)^(k*N) * (2 : ℝ)^Cn * (2 : ℝ)^(-(k+1/100)*N) =
    17 (2 : ℝ)^(1+Cn-N/100)
    18
    19axiom pruning_overlap_bound (keyCount cellCount k Cn N : ℝ)
    20 (_hkey0 : 0 ≤ keyCount) (hcell0 : 0 ≤ cellCount)
    21 (hkey : keyCount ≤ (2 : ℝ)^(k*N)) (hcell : cellCount ≤ (2 : ℝ)^Cn)
    22 (hscale : 1+Cn ≤ N/200) :
    23 2 * keyCount * cellCount * (2 : ℝ)^(-(k+1/100)*N) ≤ (2 : ℝ)^(-N/200)
    24
    25axiom scalar_agreement_power (s : ℕ) :
    26 1 / (2 : ℝ)^(s+1) = (2 : ℝ)^(-((s : ℝ)+1))
    27
    28axiom paper_scalar_smallness (s : ℕ) (N : ℝ) (hs : (s : ℝ) ≤ N/100) (hN : 100 ≤ N) :
    29 (2 : ℝ)^(-(45/100 : ℝ)*N) ≤ 1/(2 : ℝ)^(s+1)
    30
    31axiom paper_scalar_cost (s : ℕ) (N : ℝ) (hs : (s : ℝ) ≤ N/100) (hN : 100 ≤ N) :
    32 (2 : ℝ)^(-N/50) ≤ 1/(2 : ℝ)^(s+1)
    33
    34axiom collision_lower_from_overlap (overlap retained error keys agreement holes : ℝ)
    35 (hkeys : 0 < keys) (hagreement : 0 ≤ agreement)
    36 (hpruning : overlap - keys * retained ≤ error)
    37 (hholes : agreement * retained ≤ holes) :
    38 agreement * (overlap-error) / keys ≤ holes
    39
    40end Lax342547.CollisionExponents
    41
    Show ProofShow ProofShow ProofShow ProofShow ProofShow Proof
    Builds on
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    From Mathlib

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