Lax Lean Archive

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

Right endpoint query-key caps via paired-list reindexing

Lax342547.RightQueryImages · concepts/Lax342547/RightQueryImages.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

    The flipped paired lists reindex the same incident positions and slots. Their actual left-key cap therefore gives the right-key cap with exactly the original exponent.

    Concept map
    92 concepts; 4 descendants hidden
    100%
    Exact-image bounds for independent affinecolumnsOrdered atom products in the actualgradient formPoint atoms and finite flavor distributionsSparse unselected primal vectors in barredspaces lie in the pinsBarred response spaces and the actualobstruction rank budgetCompression preserving allowed pointmoments and the actual cut domainThe exact space of binary base momentsRank control for the frozen baseline onprimal inputsFormal ordered product bits realizesymmetric correctionsBounded allowed derivatives preserving theactual linearized responseAllowed channel changes and the actual tableinjection testsRank-controlled factorization throughallowed orthogonal channelsA bounded-rank pure correction for theactual scalar recipeConcrete cut-space testers and the orderedmixer formConcrete coordinates, quadratic testers, andthe self-Gram formNumerical recipes prescribe the concretegradients on witness atomsThe affine minus-column law at a fixed plusframeCompression that fixes pin/key vectors andpreserves target matricesCut profiles and the constant kernelThe full linearized response on pairs ofactual cut profilesFull response obstructions are effectiveprofiles plus selected atomsExact image pins in nominal coefficientspacesFinite linear images and their uniform-lawdensity boundsAmbient symmetries and frame marginalsFresh key directions are independent modulotable spacesBaseline bilinear extensions retaining allfrozen rows and columnsBaseline contractions on the selected atomsThe symmetric binary gradient formGram-conditioned columns and theirrank-failure probabilityTwo-sided Gram normalization forindividually injective framesThe binary hole relationPaying the reference-image conditioning anddimension costsUniform injective frames and channeltranspose failureFresh point-ray spans are disjoint from thenominal table spacesLow-rank Boolean moments have boundedlabel supportTriangle relations separate into individuallabel blocksExtracting a leaf with all remainingexact-image capsAll-rank tuple image caps from exact-pin leafentropyBoolean point moments with restricted basecoordinatesThe 3K+28 baseline bound in the actualnominal coordinatesPrimal blocks of the nominal spaces andtheir bounded table partEndpoint projections of paired pure-responseannihilatorsActual paired-witness key spaces satisfy thebaseline hypothesesBinary prescriptions at all endpoints of apaired scalar recipeFull paired witness lists and scalar recipeequationsSparse pin exclusions with arbitrary basecoefficientsSparse residual contractions belong to theactual primal pinsRetractions with bounded rank on theprimal inputsExact images mixing independent injectiveframesJoint minus images after exposing severalplus framesPrimal projections and preservation ofeffective spacesPure obstructions on the actual pair of cutprofilesActual query keys independent modulo oldpinsReference keys and matched keys for actualpaired listsThe actual paired-list key slot budgetCompression on the actual barred nominalquotientsExtracting fresh label coefficients throughpin quotientsRank of a tensor killed in two quotientspacesRank-controlled pure forms on the actualbarred quotientsBounded baselines for both actual crossorientationsActual frame observations realize thenominal channel contractionsRaw matrix frames and their tensorrealizationThe finite uniform raw-vertex lawActual query-key image caps on raw leavesGradient residuals vanish on effective profilesand selected atomsCoupled scalar recipes give consistent atomgradientsJoint reference image caps across both signsand all drawsConstrained key reference spaces havepositive densityThe full reference cap for exact pin eventsRemoving selected atoms leaves onlyunselected component labelsThe bounded pure remainder of an actualscalar recipeMatrix representations and the boundedresidual rank ingredientsUnrestricted linearized solutions for actualscalar recipesRank loss under restriction of a bilinear formRight endpoint query-key caps via paired-listreindexingSelected tensor blocks of actual pureobstructionsA selected affine ray determines its momentblockSubtracting selected atoms preserves pureannihilationSelected label coefficients agree across thecut profileInterpolation of finitely many binary selectorlabelsNumerical cross tables, injection flags, andunary admissibilityUnselected sparse vectors cannot concealfresh key coefficientsA uniform label budget for all sparse pinvectorsLow-rank tester routing along tag starsConsistent symmetric binary prescriptions ontwo witness listsTable contractions on effective profiles andtheir full extensionsTable coordinates and private channelcomplementsMajority intersections in the cyclic taggeometryPure bilinear responses detect quotienttensorsQuotient extractors isolate individual tensorlabel blocksWell-defined channel contractions onprojected tensor spacesWitness atoms and their numerical testerrecords
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 5 statements. Each proof establishes one of them relative to its assumptions.

    1 actual_right_key_cap proven

    2 flipped_slot_count proven

    3 left_flip_right_tuple proven

    4 reindex_injective proven

    5 right_tuple_event proven

    Lean source view on GitHub

    1import Lax342547.RawQueryImages
    2import Lax342547.PairedRecipes
    3
    4/-!
    5---
    6title: Right endpoint query-key caps via paired-list reindexing
    7type: lemma
    8---
    9The flipped paired lists reindex the same incident positions and slots. Their actual left-key cap therefore gives the right-key cap with exactly the original exponent.
    10-/
    11
    12namespace Lax342547.RightQueryImages
    13
    14open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.ConcreteGeometry
    15open Lax342547.PairedWitnesses Lax342547.PairedRecipes Lax342547.QueryReference
    16open Lax342547.ReferenceKeys Lax342547.ReferencePins Lax342547.KeySpans
    17open scoped ENNReal
    18
    19noncomputable def flippedPosition {k n b degree r : ℕ} {hr : 2 * r ≤ n}
    20 (W : Lists k n b degree r hr) (e : Lax342547.CutProfiles.Component (Tag k)) :
    21 ComponentPosition (flip W) e ≃ ComponentPosition W e where
    22 toFun p := ⟨⟨p.val.2.1,p.val.1,p.val.2.2⟩, by
    23 have h := p.property
    24 change (W.right p.val.2.1 p.val.1 p.val.2.2).tag ∈ e.val at h
    25 rwa [W.same_tag]⟩
    26 invFun p := ⟨⟨p.val.2.1,p.val.1,p.val.2.2⟩, by
    27 change (W.right p.val.1 p.val.2.1 p.val.2.2).tag ∈ e.val
    28 rw [← W.same_tag]
    29 exact p.property⟩
    30 left_inv _ := rfl
    31 right_inv _ := rfl
    32
    33noncomputable def reindexTuple {k n b degree r : ℕ} {hr : 2 * r ≤ n} {N : Type}
    34 (W : Lists k n b degree r hr)
    35 (q : Tuples (Lax342547.CutProfiles.Component (Tag k)) (ComponentPosition W) N) :
    36 Tuples (Lax342547.CutProfiles.Component (Tag k)) (ComponentPosition (flip W)) N :=
    37 fun e => ⟨fun n p => (q e).1 n (flippedPosition W e p),
    38 fun n p => (q e).2 n (flippedPosition W e p)⟩
    39
    40axiom reindex_injective {k n b degree r : ℕ} {hr : 2 * r ≤ n} {N : Type}
    41 (W : Lists k n b degree r hr) : Function.Injective (reindexTuple (N := N) W)
    42
    43axiom left_flip_right_tuple {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type}
    44 [Fintype H] [Fintype N] {E : Moment k n b degree}
    45 (W : Lists k n b degree r hr) (o : Unit (H := H) (N := N) (E := E)) :
    46 leftTuple (flip W) o = reindexTuple W (rightTuple W o)
    47
    48axiom right_tuple_event {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type}
    49 [Fintype H] [Fintype N] {E : Moment k n b degree}
    50 (W : Lists k n b degree r hr) (o : Unit (H := H) (N := N) (E := E))
    51 (q : Tuples (Lax342547.CutProfiles.Component (Tag k)) (ComponentPosition W) N) :
    52 rightTuple W o = q ↔ leftTuple (flip W) o = reindexTuple W q
    53
    54axiom flipped_slot_count {k n b degree r : ℕ} {hr : 2 * r ≤ n}
    55 (W : Lists k n b degree r hr) : by
    56 classical
    57 exact Fintype.card (Lax342547.QuerySlots.Slot (flip W)) =
    58 Fintype.card (Lax342547.QuerySlots.Slot W)
    59
    60axiom actual_right_key_cap {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type}
    61 [Fintype H] [Fintype N] {E : Moment k n b degree}
    62 (W : Lists k n b degree r hr)
    63 (p : PMF (Unit (H := H) (N := N) (E := E)))
    64 (P : Lax342547.ExactPins.Pin (Lax342547.CutProfiles.Component (Tag k) × Bool)
    65 (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    66 (B : Fin 2 → Finset (Fin b → Binary)) (hB : Excludes P B)
    67 (hfresh : ∀ i z t, (W.right i z t).label ∉ B z)
    68 (q : Tuples (Lax342547.CutProfiles.Component (Tag k)) (ComponentPosition W) N)
    69 (α : ℝ≥0∞)
    70 (hcap : ∀ Q : Lax342547.ExactPins.Pin (Lax342547.CutProfiles.Component (Tag k) × Bool)
    71 (Fin 2 × (Coordinate k n b degree ⊕ H)) N,
    72 1 ≤ P.relativeRank Q → p.toOuterMeasure (Q.event observation) ≤ α ^ (P.relativeRank Q))
    73 (hrank : 1 ≤ Fintype.card (Lax342547.QuerySlots.Slot W)) :
    74 p.toOuterMeasure {o | rightTuple W o = q} ≤ α ^ Fintype.card (Lax342547.QuerySlots.Slot W)
    75
    76end Lax342547.RightQueryImages
    77
    Show ProofShow ProofShow 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…