OAI.Kadison.FreeMoments.Free.mono: intermediate lemma from OpenAI, Kadison's similarity theorem through uniform derivation estimates (result family 288)
ProvedOAI.Kadison.FreeMoments.Free.monoopenai-math
An intermediate lemma of the split of OpenAI's proof of OAI.Kadison.uniform_commutator, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Kadison's similarity theorem through uniform derivation estimates (result family 288), OpenAI Math Release, September 23, 2026, result family 288, github.com/openai/math, Apache License 2.0), ported to Lean 4.33.1 and Mathlib 0df444a.
The statement, as published (namespace OAI.Kadison.FreeMoments):
lemma Free.mono (hfree : Free T τ) (hle : ∀ i,S i≤T i) : Free S τ
Definitions. The statement is written with OAI.Kadison.FreeMoments.Free (definitions bundle OAIQ039C010 of this split).
Used by. The proof of OAI.Kadison.TracialSequence.freeCornerPosition_of_unitary_abstract of this split imports it.
Preamble
import Mathlib import Definitions.Def_OAIQ039C010
Formal statement
section
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false
namespace OAI
noncomputable section
section
noncomputable section
namespace Kadison.FreeMoments
variable {A ι : Type*} [Ring A] [Algebra ℂ A] [StarRing A] [StarModule ℂ A]
variable {S T : ι → StarSubalgebra ℂ A} {τ : A →ₗ[ℂ] ℂ}
lemma Free.mono (hfree : Free T τ) (hle : ∀ i,S i≤T i) : Free S τ := by
sorry
end Kadison.FreeMoments
end
end
end
end OAI
end
Source
OpenAI, Kadison's similarity theorem through uniform derivation estimates (result family 288), OpenAI Math Release, September 23, 2026, https://github.com/openai/math (result family 288; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Kadisons-similarity-theorem-through-uniform-derivation-estimates-September-23-2026/paper.pdf; Apache License 2.0)