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Polynomial degree of actual tensor evaluations

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

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    Natural Language Statement

    Lemma

    Linear tensor evaluations retain the sum of coordinate degrees, including the exact twice-selector-degree bound for point tensors.

    Concept map
    17 concepts; 3 descendants hidden
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    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 linear_outer_product_degree_bound proven

    2 selector_tensor_degree proven

    Lean source view on GitHub

    1import Lax342547.QuadraticDegree
    2
    3/-!
    4---
    5title: Polynomial degree of actual tensor evaluations
    6type: lemma
    7---
    8Linear tensor evaluations retain the sum of coordinate degrees, including the exact twice-selector-degree bound for point tensors.
    9-/
    10
    11namespace Lax342547.PolynomialTensors
    12
    13open Lax342547.MomentSpace Lax342547.SelectorInterpolation
    14open scoped BigOperators
    15
    16axiom linear_outer_product_degree_bound {b d : ℕ} {I : Type} [Fintype I]
    17 (φ : Module.Dual Binary (Matrix I I Binary)) (z : I → (Fin b → Binary) → Binary)
    18 (hz : ∀ i, z i ∈ selectorSpan b d) :
    19 (fun x => φ (Matrix.of (fun i j => z i x*z j x))) ∈ selectorSpan b (d+d)
    20
    21axiom selector_tensor_degree {b degree : ℕ} {Base : Type} [Fintype Base]
    22 (φ : Module.Dual Binary (Matrix (SelectorCoordinates b degree × Option Base)
    23 (SelectorCoordinates b degree × Option Base) Binary)) (z : Base → Binary) :
    24 (fun s => φ (pointMoment selectorEval s z)) ∈ selectorSpan b (degree+degree)
    25
    26end Lax342547.PolynomialTensors
    27
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    From Mathlib

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