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Rank and trace of shared frame tensors

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

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    Natural Language Statement

    Lemma

    Injective primal frame maps preserve coefficient-matrix rank. A tensor shared by two endpoints has its column space in their primal-span intersection. Its trace is the entrywise pairing with the self-Gram matrix.

    Concept map
    8 concepts; 3 descendants hidden
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 4 statements. Each proof establishes one of them relative to its assumptions.

    1 sandwich_rank proven

    2 shared_rank proven

    3 shared_total_rank proven

    4 trace_sandwich proven

    Lean source view on GitHub

    1import Lax342547.RawFrames
    2import Lax342547.ConcreteGeometry
    3import Mathlib.LinearAlgebra.Matrix.Rank
    4
    5/-!
    6---
    7title: Rank and trace of shared frame tensors
    8type: lemma
    9---
    10Injective primal frame maps preserve coefficient-matrix rank. A tensor
    11shared by two endpoints has its column space in their primal-span
    12intersection. Its trace is the entrywise pairing with the self-Gram matrix.
    13-/
    14
    15namespace Lax342547.TensorIntersections
    16
    17open Lax342547.MomentSpace Lax342547.RawFrames Lax342547.ConcreteGeometry
    18
    19variable {B H N : Type} [Fintype B] [Fintype H] [Fintype N]
    20
    21def intersectionSpace (P Q : Matrix N B Binary) : Submodule Binary (N → Binary) :=
    22 LinearMap.range P.mulVecLin ⊓ LinearMap.range Q.mulVecLin
    23
    24axiom sandwich_rank {E : Matrix B B Binary} (F : Frame B H N E)
    25 (w : Matrix B B Binary) : (sandwich F.P F.Q w).rank = w.rank
    26
    27axiom shared_rank {E : Matrix B B Binary} (F G : Frame B H N E)
    28 (w v : Matrix B B Binary) (h : sandwich F.P F.Q w = sandwich G.P G.Q v) :
    29 w.rank = v.rank ∧ w.rank ≤ Module.finrank Binary (intersectionSpace F.P G.P)
    30
    31axiom shared_total_rank {Comp : Type} [Fintype Comp] {E : Matrix B B Binary}
    32 (F G : Comp → Frame B H N E) (w v : Comp → Matrix B B Binary)
    33 (h : profileMap F w = profileMap G v) :
    34 (∑ e, (w e).rank) = ∑ e, (v e).rank ∧
    35 (∑ e, (w e).rank) ≤ ∑ e, Module.finrank Binary (intersectionSpace (F e).P (G e).P)
    36
    37axiom trace_sandwich {E : Matrix B B Binary} (F : Frame B H N E)
    38 (w : Matrix B B Binary) : Matrix.trace (sandwich F.P F.Q w) = matrixPair E w
    39
    40end Lax342547.TensorIntersections
    41
    Show ProofShow ProofShow ProofShow Proof
    Builds on
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    From Mathlib

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