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Well-defined cut functionals

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

proven

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    Natural Language Statement

    Theorem

    A tester vanishing on the common kernel gives a well-defined sum over all representative values. For an odd number of tags, summing any tester over all tags except a fixed one is also well-defined. These are precisely the two descent arguments for at,a,bta_t,a,b_t in (2.7).

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
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    1import Lax342547.CutProfiles
    2import Mathlib.Algebra.Ring.Parity
    3
    4/-!
    5---
    6title: Well-defined cut functionals
    7type: theorem
    8---
    9A tester vanishing on the common kernel gives a well-defined sum over
    10all representative values. For an odd number of tags, summing any
    11tester over all tags except a fixed one is also well-defined. These
    12are precisely the two descent arguments for at,a,bta_t,a,b_t in (2.7).
    13-/
    14
    15namespace Lax342547.CutFunctionals
    16
    17open Lax342547.MomentSpace Lax342547.CutProfiles
    18
    19axiom representative_sums_eq {Tag V : Type} [Fintype Tag] [DecidableEq Tag]
    20 [Nonempty Tag] [AddCommGroup V] [Module Binary V]
    21 (W : Tag → Submodule Binary V) (ηO ηS : V →ₗ[Binary] Binary)
    22 (hO : ∀ c ∈ ⨅ t, W t, ηO c = 0) (hodd : Odd (Fintype.card Tag))
    23 (w v : Tag → V) (hw : ∀ t, w t ∈ W t) (hv : ∀ t, v t ∈ W t)
    24 (heq : cutMap w = cutMap v) :
    25 (∑ t, ηO (w t)) = (∑ t, ηO (v t)) ∧
    26 ∀ t, (∑ l ∈ Finset.univ.erase t, ηS (w l)) =
    27 ∑ l ∈ Finset.univ.erase t, ηS (v l)
    28
    29end Lax342547.CutFunctionals
    30
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