Documentation

Mathlib.Algebra.Star.Pi

star on pi types #

We put a Star structure on pi types that operates elementwise, such that it describes the complex conjugation of vectors.

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instance Pi.instStarForAll {I : Type u} {f : I → Type v} [(i : I) → Star (f i)] :
Star ((i : I) → f i)
Equations
  • Pi.instStarForAll = { star := fun (x : (i : I) → f i) (i : I) => star (x i) }
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@[simp]
theorem Pi.star_apply {I : Type u} {f : I → Type v} [(i : I) → Star (f i)] (x : (i : I) → f i) (i : I) :
star x i = star (x i)
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theorem Pi.star_def {I : Type u} {f : I → Type v} [(i : I) → Star (f i)] (x : (i : I) → f i) :
star x = fun (i : I) => star (x i)
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instance Pi.instTrivialStarForAllInstStarForAll {I : Type u} {f : I → Type v} [(i : I) → Star (f i)] [∀ (i : I), TrivialStar (f i)] :
TrivialStar ((i : I) → f i)
Equations
  • ⋯ = ⋯
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instance Pi.instInvolutiveStarForAll {I : Type u} {f : I → Type v} [(i : I) → InvolutiveStar (f i)] :
InvolutiveStar ((i : I) → f i)
Equations
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instance Pi.instStarMulForAllInstMul {I : Type u} {f : I → Type v} [(i : I) → Mul (f i)] [(i : I) → StarMul (f i)] :
StarMul ((i : I) → f i)
Equations
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instance Pi.instStarAddMonoidForAllAddMonoid {I : Type u} {f : I → Type v} [(i : I) → AddMonoid (f i)] [(i : I) → StarAddMonoid (f i)] :
StarAddMonoid ((i : I) → f i)
Equations
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instance Pi.instStarRingForAllNonUnitalNonAssocSemiringToNonUnitalNonAssocSemiring {I : Type u} {f : I → Type v} [(i : I) → NonUnitalSemiring (f i)] [(i : I) → StarRing (f i)] :
StarRing ((i : I) → f i)
Equations
  • Pi.instStarRingForAllNonUnitalNonAssocSemiringToNonUnitalNonAssocSemiring = StarRing.mk ⋯
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instance Pi.instStarModuleForAllInstStarForAllInstSMul {I : Type u} {f : I → Type v} {R : Type w} [(i : I) → SMul R (f i)] [Star R] [(i : I) → Star (f i)] [∀ (i : I), StarModule R (f i)] :
StarModule R ((i : I) → f i)
Equations
  • ⋯ = ⋯
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theorem Pi.single_star {I : Type u} {f : I → Type v} [(i : I) → AddMonoid (f i)] [(i : I) → StarAddMonoid (f i)] [DecidableEq I] (i : I) (a : f i) :
Pi.single i (star a) = star (Pi.single i a)
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@[simp]
theorem Pi.conj_apply {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → CommSemiring (α i)] [(i : ι) → StarRing (α i)] (f : (i : ι) → α i) (i : ι) :
(starRingEnd ((i : ι) → α i)) f i = (starRingEnd (α i)) (f i)
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theorem Function.update_star {I : Type u} {f : I → Type v} [(i : I) → Star (f i)] [DecidableEq I] (h : (i : I) → f i) (i : I) (a : f i) :
Function.update (star h) i (star a) = star (Function.update h i a)
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theorem Function.star_sum_elim {I : Type u_1} {J : Type u_2} {α : Type u_3} (x : I → α) (y : J → α) [Star α] :
star (Sum.elim x y) = Sum.elim (star x) (star y)