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Bounded tests under the actual two-batch Gram law

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

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    Natural Language Statement

    Lemma

    The union orbit is exactly the product of its separate orbit laws conditioned on reciprocal Gram entries and joint column injectivity. Removing only the joint-injectivity event costs its actual conditional mass; the remaining comparison uses the separately capped batch laws and independent Gram characters.

    Concept map
    21 concepts; 10 descendants hidden
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 union_conditional_integral proven

    2 union_test_comparison proven

    Lean source view on GitHub

    1import Lax342547.JoinedGramOrbits
    2import Lax342547.JointGramInjection
    3
    4/-!
    5---
    6title: Bounded tests under the actual two-batch Gram law
    7type: lemma
    8---
    9The union orbit is exactly the product of its separate orbit laws conditioned
    10on reciprocal Gram entries and joint column injectivity. Removing only the
    11joint-injectivity event costs its actual conditional mass; the remaining
    12comparison uses the separately capped batch laws and independent Gram characters.
    13-/
    14
    15namespace Lax342547.RawBatchComparison
    16noncomputable section
    17open Lax342547.MomentSpace Lax342547.FrameTuples Lax342547.RealCellLaws
    18open Lax342547.GramOrbitDensity Lax342547.RetainedImages
    19open scoped BigOperators
    20
    21variable {I J K L N : Type} [Fintype I] [Fintype J] [Fintype K] [Fintype L] [Fintype N]
    22 [DecidableEq I] [DecidableEq J] [DecidableEq K] [DecidableEq L] [DecidableEq N]
    23
    24def flat (G : Matrix I J Binary) (H : Matrix K L Binary)
    25 (z : Orbit N G × Orbit N H) :
    26 (N × (I ⊕ J) → Binary) × (N × (K ⊕ L) → Binary) :=
    27 (batchEquiv z.1.val,batchEquiv z.2.val)
    28def target (A : Matrix I L Binary) (B : Matrix K J Binary) :
    29 Lax342547.CrossGramBasis.Slots I J K L → Binary
    30 | Sum.inl (i,l) => A i l
    31 | Sum.inr (j,k) => B k j
    32
    33axiom union_conditional_integral (n : ℕ) (G : Matrix I J Binary) (H : Matrix K L Binary)
    34 (A : Matrix I L Binary) (B : Matrix K J Binary)
    35 [Nonempty (Orbit (Fin n) G)] [Nonempty (Orbit (Fin n) H)]
    36 [Nonempty (Orbit (Fin n) (Matrix.fromBlocks G A B H))]
    37 (test : ((Fin n × (I ⊕ J) → Binary) × (Fin n × (K ⊕ L) → Binary)) → ℝ) :
    38 (∑ z : Orbit (Fin n) (Matrix.fromBlocks G A B H),
    39 weights (PMF.uniformOfFintype _) z*test (Lax342547.JointGramInjection.ambientEquiv.symm z.val)) =
    40 ∑ x,conditionalLaw (fun x => batchLaw G x.1*batchLaw H x.2)
    41 (fun x => Lax342547.CrossBatchMixing.crossBits Lax342547.CrossGramBasis.tests n x = target A B ∧
    42 Lax342547.JointGramInjection.Good x) x*test x
    43
    44axiom union_test_comparison (n : ℕ) (G : Matrix I J Binary) (H : Matrix K L Binary)
    45 (A : Matrix I L Binary) (B : Matrix K J Binary)
    46 [Nonempty (Orbit (Fin n) G)] [Nonempty (Orbit (Fin n) H)]
    47 [Nonempty (Orbit (Fin n) (Matrix.fromBlocks G A B H))]
    48 (f : (Fin n × (I ⊕ J) → Binary) → ℝ) (g : (Fin n × (K ⊕ L) → Binary) → ℝ)
    49 (hN₁ : Fintype.card I+Fintype.card J+1 ≤ n)
    50 (hN₂ : Fintype.card K+Fintype.card L+1 ≤ n)
    51 (hf : ∀ x,|f x| ≤ 1) (hg : ∀ y,|g y| ≤ 1)
    52 (hsmall : ((2 : ℝ)^(Fintype.card I*Fintype.card J+2))*
    53 ((2 : ℝ)^(Fintype.card K*Fintype.card L+2))/Real.sqrt ((2 : ℝ)^n) ≤
    54 1/(2 : ℝ)^(Fintype.card (Lax342547.CrossGramBasis.Slots I J K L)+1)) :
    55 |(∑ z : Orbit (Fin n) (Matrix.fromBlocks G A B H),weights (PMF.uniformOfFintype _) z*
    56 (f (Lax342547.JointGramInjection.ambientEquiv.symm z.val).1*
    57 g (Lax342547.JointGramInjection.ambientEquiv.symm z.val).2))-
    58 (∑ x,batchLaw G x*f x)*(∑ y,batchLaw H y*g y)| ≤
    59 (2 : ℝ)^(Fintype.card (Lax342547.CrossGramBasis.Slots I J K L)+2)*
    60 ((((2 : ℝ)^(Fintype.card I*Fintype.card J+2))*
    61 ((2 : ℝ)^(Fintype.card K*Fintype.card L+2))/Real.sqrt ((2 : ℝ)^n))+
    62 (((2 : ℝ)^(Fintype.card I*Fintype.card J+2))*
    63 ((2 : ℝ)^(Fintype.card K*Fintype.card L+2))*
    64 (((2 : ℝ)^(Fintype.card I+Fintype.card K)+(2 : ℝ)^(Fintype.card J+Fintype.card L))/(2 : ℝ)^n)))
    65
    66end
    67end Lax342547.RawBatchComparison
    68
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