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Independent free-direction spaces at distinct selector labels

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

proven

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    Natural Language Statement

    Lemma

    The linear variation of a point input when a single base block varies. For distinct labels the two direction spaces are independent, because the empty selector and a separating singleton selector recover both sets of input coefficients. This is the geometric input to the ordered binary mixer tests in Lemma 5.5.

    Concept map
    7 concepts; 8 descendants hidden
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 blockDirections_injective proven

    2 distinct_directions_injective proven

    3 point_varyBlock proven

    Lean source view on GitHub

    1import Lax342547.ConcreteGeometry
    2import Mathlib.Data.Matrix.ColumnRowPartitioned
    3
    4/-!
    5---
    6title: Independent free-direction spaces at distinct selector labels
    7type: lemma
    8---
    9The linear variation of a point input when a single base block varies.
    10For distinct labels the two direction spaces are independent, because
    11the empty selector and a separating singleton selector recover both
    12sets of input coefficients. This is the geometric input to the ordered
    13binary mixer tests in Lemma 5.5.
    14-/
    15
    16namespace Lax342547.AtomDirections
    17
    18open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.ConcreteGeometry
    19
    20noncomputable def blockDirections {k n b degree : ℕ} (s : Fin b → Binary)
    21 (c : Fin n → Base k n) : Matrix (Coordinate k n b degree) (Fin n) Binary := by
    22 classical
    23 exact fun i j => if i.2 = some (c j) then selectorEval s i.1 else 0
    24
    25noncomputable def varyBlock {k n : ℕ} (z : Base k n → Binary) (c : Fin n → Base k n)
    26 (v : Fin n → Binary) : Base k n → Binary := by
    27 classical
    28 exact fun i => z i + ∑ j, if c j = i then v j else 0
    29
    30axiom point_varyBlock {k n b degree : ℕ} (s : Fin b → Binary) (z : Base k n → Binary)
    31 (c : Fin n → Base k n) (v : Fin n → Binary) :
    32 point (selectorEval (degree := degree)) s (varyBlock z c v) =
    33 point selectorEval s z + (blockDirections s c).mulVec v
    34
    35axiom blockDirections_injective {k n b degree : ℕ} (s : Fin b → Binary)
    36 (c : Fin n → Base k n) (hc : Function.Injective c) :
    37 Function.Injective (blockDirections (degree := degree) s c).mulVec
    38
    39axiom distinct_directions_injective {k n b degree : ℕ} (hd : 1 ≤ degree)
    40 (s t : Fin b → Binary) (hst : s ≠ t) (c : Fin n → Base k n) (hc : Function.Injective c) :
    41 Function.Injective (Matrix.fromCols (blockDirections (degree := degree) s c)
    42 (blockDirections t c)).mulVec
    43
    44end Lax342547.AtomDirections
    45
    Show ProofShow ProofShow Proof
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