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Joint reference image caps across both signs and all draws

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

proven

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    Natural Language Statement

    Lemma

    Expose the plus frames, then integrate the uniform conditional minus estimate over the plus prescription. Nominal ranks are taken in the direct sum over all draws. The finite bound retains every conditioning factor and the complete common-annihilator dimension loss.

    Concept map
    13 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

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

    1 all_components_cap proven

    2 joint_cap proven

    3 joint_rank_cap proven

    Lean source view on GitHub

    1import Lax342547.ProductMinusImages
    2
    3/-!
    4---
    5title: Joint reference image caps across both signs and all draws
    6type: lemma
    7---
    8Expose the plus frames, then integrate the uniform conditional minus
    9estimate over the plus prescription. Nominal ranks are taken in the
    10direct sum over all draws. The finite bound retains every conditioning
    11factor and the complete common-annihilator dimension loss.
    12-/
    13
    14namespace Lax342547.ReferenceImages
    15
    16open Lax342547.MomentSpace Lax342547.RawFrames Lax342547.FrameSymmetry Lax342547.ProductImages
    17open scoped ENNReal
    18
    19variable {Draw B H T U N : Type} [Fintype Draw] [Fintype B] [Fintype H]
    20 [Fintype T] [Fintype U] [Fintype N]
    21
    22def plusImages {E : Matrix B B Binary} (C : Matrix (Draw × (B ⊕ H)) T Binary)
    23 (o : Draw → Frame B H N E) : Matrix N T Binary :=
    24 jointMatrix (fun i => (plus (o i)).val) * C
    25
    26def minusImages {E : Matrix B B Binary} (C : Matrix (Draw × (B ⊕ H)) U Binary)
    27 (o : Draw → Frame B H N E) : Matrix N U Binary :=
    28 jointMatrix (fun i => (minus (o i)).val) * C
    29
    30axiom joint_cap [DecidableEq Draw] [DecidableEq B] [DecidableEq H]
    31 [DecidableEq T] [DecidableEq U] [DecidableEq N]
    32 {E : Matrix B B Binary} [Nonempty (Frame B H N E)]
    33 (hN : 2 * (Fintype.card B + Fintype.card H) + 1 ≤ Fintype.card N)
    34 (C : Matrix (Draw × (B ⊕ H)) T Binary) (D : Matrix (Draw × (B ⊕ H)) U Binary)
    35 (hC : Function.Injective C.mulVec) (hD : Function.Injective D.mulVec)
    36 (y : Matrix N T Binary) (z : Matrix N U Binary) :
    37 (PMF.uniformOfFintype (Draw → Frame B H N E)).toOuterMeasure
    38 {o | plusImages C o = y ∧ minusImages D o = z} ≤
    39 (4 : ℝ≥0∞) ^ Fintype.card Draw /
    40 2 ^ (Fintype.card N * Fintype.card T + Fintype.card U *
    41 (Fintype.card N - Fintype.card Draw * (Fintype.card B + Fintype.card H)))
    42
    43axiom joint_rank_cap [DecidableEq Draw] [DecidableEq B] [DecidableEq H]
    44 [DecidableEq T] [DecidableEq U] [DecidableEq N]
    45 {E : Matrix B B Binary} [Nonempty (Frame B H N E)]
    46 (hN : 2 * (Fintype.card B + Fintype.card H) + 1 ≤ Fintype.card N)
    47 (C : Matrix (Draw × (B ⊕ H)) T Binary) (D : Matrix (Draw × (B ⊕ H)) U Binary)
    48 (y : Matrix N T Binary) (z : Matrix N U Binary) :
    49 (PMF.uniformOfFintype (Draw → Frame B H N E)).toOuterMeasure
    50 {o | plusImages C o = y ∧ minusImages D o = z} ≤
    51 (4 : ℝ≥0∞) ^ Fintype.card Draw /
    52 2 ^ (Fintype.card N * C.rank + D.rank *
    53 (Fintype.card N - Fintype.card Draw * (Fintype.card B + Fintype.card H)))
    54
    55axiom all_components_cap {Comp : Type} [Fintype Comp] [DecidableEq Comp]
    56 {K L : Comp → Type} [∀ e, Fintype (K e)] [∀ e, Fintype (L e)]
    57 [DecidableEq Draw] [DecidableEq B] [DecidableEq H] [DecidableEq N]
    58 {E : Matrix B B Binary} [Nonempty (Frame B H N E)]
    59 (hN : 2 * (Fintype.card B + Fintype.card H) + 1 ≤ Fintype.card N)
    60 (C : ∀ e, Matrix (Draw × (B ⊕ H)) (K e) Binary)
    61 (D : ∀ e, Matrix (Draw × (B ⊕ H)) (L e) Binary)
    62 (y : ∀ e, Matrix N (K e) Binary) (z : ∀ e, Matrix N (L e) Binary) :
    63 (PMF.uniformOfFintype (Draw → Comp → Frame B H N E)).toOuterMeasure
    64 {o | ∀ e, plusImages (C e) (fun i => o i e) = y e ∧ minusImages (D e) (fun i => o i e) = z e} ≤
    65 (4 : ℝ≥0∞) ^ (Fintype.card Draw * Fintype.card Comp) /
    66 2 ^ (Fintype.card N * ∑ e, (C e).rank +
    67 (Fintype.card N - Fintype.card Draw * (Fintype.card B + Fintype.card H)) * ∑ e, (D e).rank)
    68
    69end Lax342547.ReferenceImages
    70
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