OAI.ElasticityBoundaryForceSupport.transferred_force_all: intermediate lemma from OpenAI, Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem

Proved

OAI.ElasticityBoundaryForceSupport.transferred_force_all

by dbenbenn · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.Elasticity.global_uniqueness, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem, OpenAI Math Release, September 24, 2026, result family 372, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.ElasticityBoundaryForceSupport.transferred_force_all of lean/OAI/MathematicalPhysics/Elasticity/BoundaryPackets.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.ElasticityBoundaryForceSupport, with open Set Filter, open scoped Topology BigOperators, open ElasticityBoundary ElasticityBoundaryPhysicalTransfer ElasticityBoundaryOperatorLocal):

theorem transferred_force_all (e : OpenPartialHomeomorph X X) {K : Set X}
    (hK : IsCompact K) (hs : K⊆e.source) {η : X → ℝ} (hηs : tsupport η⊆e.target)
    (hη : ∀ z∈e '' K, η z=1) (Q : X ≃ₗᵢ[ℝ] X) (u : I → Y → ℂ)
    (hus : ∀ j, Function.support (fun z : X => u j z.ofLp)⊆K)
    (hu : ∀ j, ContDiff ℝ (⊤ : ℕ∞) (u j))
    (Φ Ψ : X → X) (hΨ : ContDiff ℝ (⊤ : ℕ∞) Ψ)
    (hΦ : ∀ y∈K, Φ y=e y) (hΨe : ∀ z∈e '' K, Ψ =ᶠ[𝓝 z] e.symm)
    {l m : X → ℂ} (hl : ContDiff ℝ (⊤ : ℕ∞) l) (hm : ContDiff ℝ (⊤ : ℕ∞) m)
    (x : X) (hx : x∈e.source) (s : I) :
    (∑ i, frameMatrix Q i s*force l m (transferred e η Q u) i (e x))=
      strong (chartPrincipal Q (fun z => Φ (WithLp.toLp 2 z)) (fun z => (Ψ z).ofLp) l m)
        (chartLower Q (fun z => Φ (WithLp.toLp 2 z)) (fun z => (Ψ z).ofLp) l m) u s x.ofLp

Definitions. The statement is written with OAI.ElasticityBoundaryChartOperator.X, OAI.ElasticityBoundaryChartOperator.Y, OAI.ElasticityBoundaryChartOperator.chartPrincipal, OAI.ElasticityBoundaryForceSupport.I, OAI.ElasticityBoundaryForceSupport.X, OAI.ElasticityBoundaryForceSupport.Y, OAI.ElasticityBoundaryPhysicalTransfer.transferred, OAI.ElasticityBoundaryRotation.I, OAI.ElasticityBoundaryRotation.frameMatrix and OAI.ElasticityBoundaryStrongLayer.strong (definitions bundle OAIQ6El001 of this split); OAI.ElasticityBoundaryChartOperator.chartLower and OAI.ElasticityBoundaryPhysicalTransfer.force (definitions bundle OAIQ6El004 of this split).

Used by. The proof of OAI.ElasticityBoundaryPhysicalMass.transferred_mass_bound of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ6El004

Formal statement

section

attribute [-instance] _root_.OAI.ElasticityGeneralSlicing.cylinder_temperate _root_.OAI.ElasticityODE.instFactLeRealOfNat_solutions
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

section

noncomputable section
open Set Filter
open scoped Topology BigOperators
namespace ElasticityBoundaryForceSupport
open ElasticityBoundary ElasticityBoundaryPhysicalTransfer ElasticityBoundaryOperatorLocal
  ElasticityBoundaryLocalParametrix ElasticityBoundaryStrongLayer
  ElasticityBoundaryChartOperator ElasticityBoundaryFrameOperator
  ElasticityBoundaryFrameInverse ElasticityBoundaryRotation

theorem transferred_force_all (e : OpenPartialHomeomorph X X) {K : Set X}
    (hK : IsCompact K) (hs : K⊆e.source) {η : X → ℝ} (hηs : tsupport η⊆e.target)
    (hη : ∀ z∈e '' K, η z=1) (Q : X ≃ₗᵢ[ℝ] X) (u : I → Y → ℂ)
    (hus : ∀ j, Function.support (fun z : X => u j z.ofLp)⊆K)
    (hu : ∀ j, ContDiff ℝ (⊤ : ℕ∞) (u j))
    (Φ Ψ : X → X) (hΨ : ContDiff ℝ (⊤ : ℕ∞) Ψ)
    (hΦ : ∀ y∈K, Φ y=e y) (hΨe : ∀ z∈e '' K, Ψ =ᶠ[𝓝 z] e.symm)
    {l m : X → ℂ} (hl : ContDiff ℝ (⊤ : ℕ∞) l) (hm : ContDiff ℝ (⊤ : ℕ∞) m)
    (x : X) (hx : x∈e.source) (s : I) :
    (∑ i, frameMatrix Q i s*force l m (transferred e η Q u) i (e x))=
      strong (chartPrincipal Q (fun z => Φ (WithLp.toLp 2 z)) (fun z => (Ψ z).ofLp) l m)
        (chartLower Q (fun z => Φ (WithLp.toLp 2 z)) (fun z => (Ψ z).ofLp) l m) u s x.ofLp := by
  sorry

end ElasticityBoundaryForceSupport

end
end

end OAI
end

Source

OpenAI, Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem, OpenAI Math Release, September 24, 2026, https://github.com/openai/math (result family 372; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/MathematicalPhysics/Elasticity/BoundaryPackets.lean (declaration OAI.ElasticityBoundaryForceSupport.transferred_force_all)

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