Lax Lean Archive

While this submission is a draft, it cannot be used by other submissions.

Exact leaf entropy bounds joint key and additional images

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

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Lemma

    The full key plus additional tuple has the sum of the two ranks. Applying the actual old-pin leaf cap before any cell conditioning bounds every specified joint image event.

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

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax342547.JointDirections
    2
    3/-!
    4---
    5title: Exact leaf entropy bounds joint key and additional images
    6type: lemma
    7---
    8The full key plus additional tuple has the sum of the two ranks. Applying the actual old-pin leaf cap before any cell conditioning bounds every specified joint image event.
    9-/
    10
    11namespace Lax342547.JointKeyCap
    12
    13open Lax342547.MomentSpace Lax342547.ExactPins
    14open scoped ENNReal
    15
    16axiom leaf_key_fresh_image_cap {Ω Axis I N : Type} {U V : Axis → Type}
    17 [Fintype Axis] [Fintype I]
    18 [∀ a, AddCommGroup (U a)] [∀ a, Module Binary (U a)] [∀ a, FiniteDimensional Binary (U a)]
    19 [∀ a, AddCommGroup (V a)] [∀ a, Module Binary (V a)] [∀ a, FiniteDimensional Binary (V a)]
    20 (p : PMF Ω) (A : Ω → Axis → (I → Binary) →ₗ[Binary] (N → Binary))
    21 (P : Pin Axis I N) (keys : ∀ a, U a →ₗ[Binary] (I → Binary))
    22 (fresh : ∀ a, V a →ₗ[Binary] (I → Binary))
    23 (hkeys : ∀ a, Function.Injective ((P.space a).mkQ.comp (keys a)))
    24 (hfresh : ∀ a, Function.Injective (((P.space a) ⊔ LinearMap.range (keys a)).mkQ.comp (fresh a)))
    25 (yk : ∀ a, U a →ₗ[Binary] (N → Binary)) (yf : ∀ a, V a →ₗ[Binary] (N → Binary))
    26 (α : ℝ≥0∞)
    27 (hcap : ∀ Q : Pin Axis I N, 1 ≤ P.relativeRank Q →
    28 p.toOuterMeasure (Q.event A) ≤ α ^ (P.relativeRank Q))
    29 (hr : 1 ≤ (∑ a, Module.finrank Binary (U a)) + ∑ a, Module.finrank Binary (V a)) :
    30 p.toOuterMeasure {ω | ∀ a, (A ω a).comp (keys a) = yk a ∧ (A ω a).comp (fresh a) = yf a} ≤
    31 α ^ ((∑ a, Module.finrank Binary (U a)) + ∑ a, Module.finrank Binary (V a))
    32
    33end Lax342547.JointKeyCap
    34
    Show Proof
    Builds on
    Used by
    From Mathlib

    none

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…