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The concrete finite frame model and its sampled graphs

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

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    Natural Language Statement

    Lemma

    Assemble the actual moment coordinates, cut-profile space, descended testers, ordered mixer form, self-Gram matrix, frame tensors, and uniform raw law. This algebraic construction works for arbitrary mixers. It does not assert a connected-matching bound for those arbitrary choices.

    Concept map
    11 concepts
    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 exists_model proven

    2 graph_indepNum_le_two proven

    Lean source view on GitHub

    1import Lax342547.ConcreteCut
    2import Lax342547.RawLaw
    3import Lax342547.SampledGraph
    4
    5/-!
    6---
    7title: The concrete finite frame model and its sampled graphs
    8type: lemma
    9---
    10Assemble the actual moment coordinates, cut-profile space, descended
    11testers, ordered mixer form, self-Gram matrix, frame tensors, and uniform
    12raw law. This algebraic construction works for arbitrary mixers. It does
    13not assert a connected-matching bound for those arbitrary choices.
    14-/
    15
    16namespace Lax342547.ConcreteModel
    17
    18set_option backward.isDefEq.respectTransparency false
    19
    20open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.ConcreteGeometry
    21open Lax342547.CutProfiles Lax342547.ConcreteCut Lax342547.RawFrames
    22open Lax342547.HoleRelation Lax342547.SampledGraph
    23open scoped ENNReal
    24
    25abbrev RawVertex {k n b degree r : ℕ} (hr : 2 * r ≤ n) (hd : 1 ≤ degree)
    26 (M : Fin b → Matrix (Fin n) (Fin n) Binary) (h N : ℕ) :=
    27 Component (Tag k) → Frame (Coordinate k n b degree) (Fin h) (Fin N) (selfGram hr hd M)
    28
    29abbrev Ambient (k N : ℕ) := Component (Tag k) → Matrix (Fin N) (Fin N) Binary
    30
    31structure Model {k n b degree r J h N : ℕ} (hr : 2 * r ≤ n) (hd : 1 ≤ degree)
    32 (M : Fin b → Matrix (Fin n) (Fin n) Binary)
    33 (L R : Fin J → Component (Tag k) → Component (Tag k) → Moment k n b degree) where
    34 testers : Testers (k := k) (b := b) (degree := degree) hr
    35 frame : Frame (Coordinate k n b degree) (Fin h) (Fin N) (selfGram hr hd M)
    36 holes : HoleData (RawVertex (k := k) hr hd M h N) (Profile k n b degree) (Ambient k N)
    37 role_eq : holes.a = testers.role
    38 gradient_eq : holes.T = gradient testers L R
    39 embedding_eq : ∀ o, holes.U o = (profileMap o).comp (Profile k n b degree).subtype
    40 contraction_eq : ∀ o, holes.u o = profileContraction o
    41 hole_symmetric : ∀ {i j}, Hole holes i j → Hole holes j i
    42 law : PMF (RawVertex (k := k) hr hd M h N)
    43 law_uniform : ∀ o, law o = (Fintype.card (RawVertex (k := k) hr hd M h N) : ℝ≥0∞)⁻¹
    44
    45noncomputable def graph {k n b degree r J h N m : ℕ} {hr : 2 * r ≤ n} {hd : 1 ≤ degree}
    46 {M : Fin b → Matrix (Fin n) (Fin n) Binary}
    47 {L R : Fin J → Component (Tag k) → Component (Tag k) → Moment k n b degree}
    48 (D : Model (h := h) (N := N) hr hd M L R) (sample : Fin m → RawVertex (k := k) hr hd M h N) :
    49 SimpleGraph (Fin m) := sampleGraph (Hole D.holes) (fun {_ _} hij => D.hole_symmetric hij) sample
    50
    51axiom exists_model {k n b degree r J h N : ℕ} (hr : 2 * r ≤ n) (hd : 1 ≤ degree)
    52 (M : Fin b → Matrix (Fin n) (Fin n) Binary)
    53 (L R : Fin J → Component (Tag k) → Component (Tag k) → Moment k n b degree)
    54 (hN : 2 * (Fintype.card (Coordinate k n b degree) + h) ≤ N) :
    55 Nonempty (Model (h := h) (N := N) hr hd M L R)
    56
    57axiom graph_indepNum_le_two {k n b degree r J h N m : ℕ} {hr : 2 * r ≤ n} {hd : 1 ≤ degree}
    58 {M : Fin b → Matrix (Fin n) (Fin n) Binary}
    59 {L R : Fin J → Component (Tag k) → Component (Tag k) → Moment k n b degree}
    60 (D : Model (h := h) (N := N) hr hd M L R) (sample : Fin m → RawVertex (k := k) hr hd M h N) :
    61 (graph D sample).indepNum ≤ 2
    62
    63end Lax342547.ConcreteModel
    64
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