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

Proved

OAI.ElasticityAugmented.forced_equations_at

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.ElasticityAugmented.forced_equations_at of lean/OAI/MathematicalPhysics/Elasticity/ForcedTransport.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.ElasticityAugmented, with open scoped BigOperators):

lemma forced_equations_at (θ : Fin 3 → ℂ) (k r : Smooth)
    (hr : ∀ x,(r : X → ℂ) x≠0) (hk : ∀ x,(k : X → ℂ) x≠0)
    (hl : ∀ x,((k+r*r : Smooth) : X → ℂ) x≠0)
    (H : Fin 3 → Smooth) (K : Smooth) (a : Fin 3 → Smooth) (b : Smooth) (x : X)
    (hp : ∀ i,(direction (A := Smooth) coord θ (changeP r a i) : X → ℂ) x=
      θ i*(changeS k r hr a b : X → ℂ) x+(topSource r hr H i : X → ℂ) x)
    (hs : (direction (A := Smooth) coord θ (changeS k r hr a b) : X → ℂ) x=
      ((transportT θ k r hk hl*changeS k r hr a b+
        ∑ i,transportQ θ k r hr hk hl i*changeP r a i : Smooth) : X → ℂ) x+
          (scalarSource k r hr hl H K : X → ℂ) x) :
    (∀ i,((leadingR coord θ (k-r*r) (r*r) a b i : Smooth) : X → ℂ) x=(H i : X → ℂ) x) ∧
      ((leadingS coord θ (k-r*r) (r*r) a b : Smooth) : X → ℂ) x=(K : X → ℂ) x

Definitions. The statement is written with OAI.ElasticityAugmented.Smooth, OAI.ElasticityAugmented.X, OAI.ElasticityAugmented.changeP, OAI.ElasticityAugmented.changeS, OAI.ElasticityAugmented.coord, OAI.ElasticityAugmented.direction, OAI.ElasticityAugmented.leadingR and OAI.ElasticityAugmented.leadingS (definitions bundle OAIQ6El000 of this split); OAI.ElasticityAugmented.scalarSource, OAI.ElasticityAugmented.topSource, OAI.ElasticityAugmented.transportQ and OAI.ElasticityAugmented.transportT (definitions bundle OAIQ6El004 of this split).

Used by. The proof of OAI.ElasticityAugmented.exists_physical_forced_transport of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ6El004

Formal statement

section

attribute [-instance] _root_.OAI.ElasticityGeneralSlicing.cylinder_temperate
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 scoped BigOperators
namespace ElasticityAugmented

lemma forced_equations_at (θ : Fin 3 → ℂ) (k r : Smooth)
    (hr : ∀ x,(r : X → ℂ) x≠0) (hk : ∀ x,(k : X → ℂ) x≠0)
    (hl : ∀ x,((k+r*r : Smooth) : X → ℂ) x≠0)
    (H : Fin 3 → Smooth) (K : Smooth) (a : Fin 3 → Smooth) (b : Smooth) (x : X)
    (hp : ∀ i,(direction (A := Smooth) coord θ (changeP r a i) : X → ℂ) x=
      θ i*(changeS k r hr a b : X → ℂ) x+(topSource r hr H i : X → ℂ) x)
    (hs : (direction (A := Smooth) coord θ (changeS k r hr a b) : X → ℂ) x=
      ((transportT θ k r hk hl*changeS k r hr a b+
        ∑ i,transportQ θ k r hr hk hl i*changeP r a i : Smooth) : X → ℂ) x+
          (scalarSource k r hr hl H K : X → ℂ) x) :
    (∀ i,((leadingR coord θ (k-r*r) (r*r) a b i : Smooth) : X → ℂ) x=(H i : X → ℂ) x) ∧
      ((leadingS coord θ (k-r*r) (r*r) a b : Smooth) : X → ℂ) x=(K : X → ℂ) x := by
  sorry
end ElasticityAugmented

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/ForcedTransport.lean (declaration OAI.ElasticityAugmented.forced_equations_at)

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