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[Merged by Bors] - chore(Algebra/Order/Module/PositiveLinearMap): generalize OrderHomClass.ofLinear #29626

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[Merged by Bors] - chore(Algebra/Order/Module/PositiveLinearMap): generalize OrderHomClass.ofLinear #29626

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chore: generalize
themathqueen Sep 13, 2025
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rename and golf
themathqueen Sep 13, 2025
fca370a
Merge branch 'leanprover-community:master' into chore_generalize_orde…
themathqueen Sep 15, 2025
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themathqueen Sep 15, 2025
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19 changes: 9 additions & 10 deletions Mathlib/Algebra/Order/Module/PositiveLinearMap.lean
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Expand Up @@ -50,17 +50,16 @@ def toPositiveLinearMap (f : F) : E₁ →ₚ[R] E₂ :=
instance instCoeToLinearMap : CoeHead F (E₁ →ₚ[R] E₂) where
coe f := toPositiveLinearMap f

/-- A linear map that maps nonnegative elements to nonnegative elements is an order
homomorphism. -/
lemma _root_.OrderHomClass.ofLinear {F' E₁' E₂' : Type*} [FunLike F' E₁' E₂'] [AddCommGroup E₁']
[PartialOrder E₁'] [AddCommGroup E₂'] [PartialOrder E₂'] [Module R E₁'] [Module R E₂']
[LinearMapClass F' R E₁' E₂'] [IsOrderedAddMonoid E₁'] [IsOrderedAddMonoid E₂']
/-- An additive group homomorphism that maps nonnegative elements to nonnegative elements
is an order homomorphism. -/
lemma _root_.OrderHomClass.ofAddMonoidHom {F' E₁' E₂' : Type*} [FunLike F' E₁' E₂'] [AddGroup E₁']
[LE E₁'] [AddRightMono E₁'] [AddGroup E₂'] [LE E₂'] [AddRightMono E₂']
[AddMonoidHomClass F' E₁' E₂']
(h : ∀ f : F', ∀ x, 0 ≤ x → 0 ≤ f x) : OrderHomClass F' E₁' E₂' where
map_rel := by
intro f a b hab
rw [← sub_nonneg] at hab ⊢
have : 0 ≤ f (b - a) := h f (b - a) hab
simpa using this
map_rel f a b hab := by simpa using h f (b - a) (sub_nonneg.mpr hab)

@[deprecated (since := "2025-09-13")] alias _root_.OrderHomClass.ofLinear :=
OrderHomClass.ofAddMonoidHom

end PositiveLinearMapClass

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2 changes: 1 addition & 1 deletion Mathlib/Analysis/CStarAlgebra/CompletelyPositiveMap.lean
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Expand Up @@ -91,7 +91,7 @@ open CStarMatrix in
/-- Linear maps which are completely positive are order homomorphisms (i.e., positive maps). -/
lemma _root_.OrderHomClass.of_map_cstarMatrix_nonneg
(h : ∀ (φ : F) (k : ℕ) (M : CStarMatrix (Fin k) (Fin k) A₁), 0 ≤ M → 0 ≤ M.map φ) :
OrderHomClass F A₁ A₂ := .ofLinear <| by
OrderHomClass F A₁ A₂ := .ofAddMonoidHom <| by
intro φ a ha
let Ma := toOneByOne (Fin 1) ℂ A₁ a
have h₁ : 0 ≤ Ma := map_nonneg (toOneByOne (Fin 1) ℂ A₁) ha
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