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Independent characters of actual mutual Gram entries

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

proven

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    Natural Language Statement

    Lemma

    The two reciprocal cross Gram blocks occupy disjoint coefficient entries. Their characters have no duplicate descriptions: every nonzero pattern gives a nonzero cross-batch matrix and therefore at least N bits of rank.

    Concept map
    11 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 coefficient_blocks proven

    2 mutual_gram_bits proven

    3 nonzero_coefficient proven

    Lean source view on GitHub

    1import Lax342547.CrossBatchMixing
    2
    3/-!
    4---
    5title: Independent characters of actual mutual Gram entries
    6type: lemma
    7---
    8The two reciprocal cross Gram blocks occupy disjoint coefficient entries.
    9Their characters have no duplicate descriptions: every nonzero pattern
    10gives a nonzero cross-batch matrix and therefore at least N bits of rank.
    11-/
    12
    13namespace Lax342547.CrossGramBasis
    14
    15open Lax342547.MomentSpace Lax342547.CrossBatchMixing
    16open scoped BigOperators
    17
    18variable {I J K L : Type} [Fintype I] [Fintype J] [Fintype K] [Fintype L]
    19 [DecidableEq I] [DecidableEq J] [DecidableEq K] [DecidableEq L]
    20
    21abbrev Slots (I J K L : Type) := (I × L) ⊕ (J × K)
    22
    23def tests {I J K L : Type} [DecidableEq I] [DecidableEq J] [DecidableEq K] [DecidableEq L] :
    24 Slots I J K L → Matrix (I ⊕ J) (K ⊕ L) Binary
    25 | Sum.inl (i,l) => fun a b => if a = Sum.inl i ∧ b = Sum.inr l then 1 else 0
    26 | Sum.inr (j,k) => fun a b => if a = Sum.inr j ∧ b = Sum.inl k then 1 else 0
    27
    28axiom coefficient_blocks (t : Slots I J K L → Binary) :
    29 coefficient tests t = Matrix.fromBlocks 0 (fun i l => t (Sum.inl (i,l)))
    30 (fun j k => t (Sum.inr (j,k))) 0
    31
    32axiom nonzero_coefficient (t : Slots I J K L → Binary) (ht : t ≠ 0) : coefficient tests t ≠ 0
    33
    34axiom mutual_gram_bits (N : ℕ)
    35 (xy : (Fin N × (I ⊕ J) → Binary) × (Fin N × (K ⊕ L) → Binary)) :
    36 (∀ i l,crossBits tests N xy (Sum.inl (i,l)) = ∑ a,xy.1 (a,Sum.inl i)*xy.2 (a,Sum.inr l)) ∧
    37 (∀ j k,crossBits tests N xy (Sum.inr (j,k)) = ∑ a,xy.1 (a,Sum.inr j)*xy.2 (a,Sum.inl k))
    38
    39end Lax342547.CrossGramBasis
    40
    Show ProofShow ProofShow Proof
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