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All-rank tuple image caps from exact-pin leaf entropy

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

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    Natural Language Statement

    Lemma

    A tuple independent modulo the actual old pin defines an exact image pin of its full relative rank. The checked leaf cap therefore bounds any specified joint tuple image, uniformly over every tuple rank.

    Concept map
    4 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 image_pin_event proven

    2 image_pin_relative_rank proven

    3 leaf_tuple_image_cap proven

    Lean source view on GitHub

    1import Lax342547.ExactPins
    2import Lax342547.LeafExtraction
    3
    4/-!
    5---
    6title: All-rank tuple image caps from exact-pin leaf entropy
    7type: lemma
    8---
    9A tuple independent modulo the actual old pin defines an exact image pin of its full relative rank. The checked leaf cap therefore bounds any specified joint tuple image, uniformly over every tuple rank.
    10-/
    11
    12namespace Lax342547.LeafImages
    13
    14open Lax342547.MomentSpace Lax342547.ExactPins
    15open scoped ENNReal
    16
    17noncomputable def imagePin {Ω Axis I N : Type}
    18 (A : Ω → Axis → (I → Binary) →ₗ[Binary] (N → Binary))
    19 (S : Axis → Submodule Binary (I → Binary)) (o : Ω) : Pin Axis I N :=
    20 ⟨S, fun a => (A o a).comp (S a).subtype⟩
    21
    22axiom image_pin_event {Ω Axis I N : Type} {R : Axis → Type}
    23 [∀ a, AddCommGroup (R a)] [∀ a, Module Binary (R a)]
    24 (A : Ω → Axis → (I → Binary) →ₗ[Binary] (N → Binary))
    25 (f : ∀ a, R a →ₗ[Binary] (I → Binary)) (o : Ω) :
    26 (imagePin A (fun a => LinearMap.range (f a)) o).event A =
    27 {ω | ∀ a, (A ω a).comp (f a) = (A o a).comp (f a)}
    28
    29axiom image_pin_relative_rank {Ω Axis I N : Type} {R : Axis → Type}
    30 [Fintype Axis] [Fintype I]
    31 [∀ a, AddCommGroup (R a)] [∀ a, Module Binary (R a)] [∀ a, FiniteDimensional Binary (R a)]
    32 (A : Ω → Axis → (I → Binary) →ₗ[Binary] (N → Binary))
    33 (P : Pin Axis I N) (f : ∀ a, R a →ₗ[Binary] (I → Binary))
    34 (hf : ∀ a, Function.Injective ((P.space a).mkQ.comp (f a))) (o : Ω) :
    35 P.relativeRank (imagePin A (fun a => LinearMap.range (f a)) o) =
    36 ∑ a, Module.finrank Binary (R a)
    37
    38axiom leaf_tuple_image_cap {Ω Axis I N : Type} {R : Axis → Type}
    39 [Fintype Axis] [Fintype I]
    40 [∀ a, AddCommGroup (R a)] [∀ a, Module Binary (R a)] [∀ a, FiniteDimensional Binary (R a)]
    41 (p : PMF Ω) (A : Ω → Axis → (I → Binary) →ₗ[Binary] (N → Binary))
    42 (P : Pin Axis I N) (f : ∀ a, R a →ₗ[Binary] (I → Binary))
    43 (hf : ∀ a, Function.Injective ((P.space a).mkQ.comp (f a)))
    44 (y : ∀ a, R a →ₗ[Binary] (N → Binary)) (α : ℝ≥0∞)
    45 (hcap : ∀ Q : Pin Axis I N, 1 ≤ P.relativeRank Q →
    46 p.toOuterMeasure (Q.event A) ≤ α ^ (P.relativeRank Q))
    47 (hr : 1 ≤ ∑ a, Module.finrank Binary (R a)) :
    48 p.toOuterMeasure {ω | ∀ a, (A ω a).comp (f a) = y a} ≤
    49 α ^ (∑ a, Module.finrank Binary (R a))
    50
    51end Lax342547.LeafImages
    52
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