Lax Lean Archive

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

Ordered atom products in the actual gradient form

Lax342547.AtomProducts · concepts/Lax342547/AtomProducts.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

    Atom-pair mixing is exactly the product sum indexed by the two incident stars. Thus matching prescribed distinct-atom bits gives the prescribed gradient entries. No claim of occurrence or independence of those bits is part of this deterministic identity.

    Concept map
    12 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 gradient_atoms proven

    2 gradient_of_prescriptions proven

    3 mixing_atoms proven

    Lean source view on GitHub

    1import Lax342547.WitnessAtoms
    2
    3/-!
    4---
    5title: Ordered atom products in the actual gradient form
    6type: lemma
    7---
    8Atom-pair mixing is exactly the product sum indexed by the two incident
    9stars. Thus matching prescribed distinct-atom bits gives the prescribed
    10gradient entries. No claim of occurrence or independence of those bits
    11is part of this deterministic identity.
    12-/
    13
    14namespace Lax342547.AtomProducts
    15
    16open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.ConcreteGeometry
    17open Lax342547.ConcreteCut Lax342547.CutProfiles Lax342547.WitnessAtoms
    18open Lax342547.BinaryPrescriptions
    19
    20noncomputable def evaluatedBits {k n b degree J : ℕ}
    21 (M : Fin J → Component (Tag k) → Component (Tag k) → Moment k n b degree)
    22 (x y : PointAtom k n b degree) (t : ProductIndex k J) : Binary := by
    23 classical
    24 exact if x.tag ∈ t.2.1.val ∧ y.tag ∈ t.2.2.val then
    25 (M t.1 t.2.1 t.2.2).toBilin' x.vector y.vector else 0
    26
    27axiom mixing_atoms {k n b degree J : ℕ}
    28 (L R : Fin J → Component (Tag k) → Component (Tag k) → Moment k n b degree)
    29 (x y : PointAtom k n b degree) :
    30 mixingForm L R x.profile y.profile = productSum (evaluatedBits L) (evaluatedBits R) x y
    31
    32axiom gradient_atoms {k n b degree r J : ℕ} {hr : 2 * r ≤ n}
    33 (D : Testers (k := k) (b := b) (degree := degree) hr)
    34 (L R : Fin J → Component (Tag k) → Component (Tag k) → Moment k n b degree)
    35 (x y : PointAtom k n b degree) :
    36 gradient D L R x.profile y.profile = testerEntry (x.numerical hr) (y.numerical hr) +
    37 productSum (evaluatedBits L) (evaluatedBits R) x y +
    38 productSum (evaluatedBits L) (evaluatedBits R) y x
    39
    40axiom gradient_of_prescriptions {I : Type} {k n b degree r J : ℕ} {hr : 2 * r ≤ n}
    41 (D : Testers (k := k) (b := b) (degree := degree) hr)
    42 (L R : Fin J → Component (Tag k) → Component (Tag k) → Moment k n b degree)
    43 (atoms : I → PointAtom k n b degree) (lbits rbits : I → I → ProductIndex k J → Binary)
    44 (hzero : ∀ x y t, ¬ allowedIndex ((atoms x).numerical hr) ((atoms y).numerical hr) t →
    45 lbits x y t = 0 ∧ rbits x y t = 0)
    46 (hbits : ∀ x y, x ≠ y → ∀ t,
    47 allowedIndex ((atoms x).numerical hr) ((atoms y).numerical hr) t →
    48 (L t.1 t.2.1 t.2.2).toBilin' (atoms x).vector (atoms y).vector = lbits x y t ∧
    49 (R t.1 t.2.1 t.2.2).toBilin' (atoms x).vector (atoms y).vector = rbits x y t) :
    50 ∀ x y, gradient D L R (atoms x).profile (atoms y).profile =
    51 testerEntry ((atoms x).numerical hr) ((atoms y).numerical hr) +
    52 productSum lbits rbits x y + productSum lbits rbits y x
    53
    54end Lax342547.AtomProducts
    55
    Show ProofShow ProofShow 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…