Star on product types

We put a Star structure on product types that operates elementwise.

instance Prod.instStarProd {R : Type u} {S : Type v} [Star R] [Star S] : Star (R × S)

Equations

• Prod.instStarProd = { star := fun (x : R × S) => (star x.1, star x.2) }

@[simp]

theorem Prod.fst_star {R : Type u} {S : Type v} [Star R] [Star S] (x : R × S) : (star x).1 = star x.1

@[simp]

theorem Prod.snd_star {R : Type u} {S : Type v} [Star R] [Star S] (x : R × S) : (star x).2 = star x.2

theorem Prod.star_def {R : Type u} {S : Type v} [Star R] [Star S] (x : R × S) : star x = (star x.1, star x.2)

instance Prod.instTrivialStarProdInstStarProd {R : Type u} {S : Type v} [Star R] [Star S] [TrivialStar R] [TrivialStar S] : TrivialStar (R × S)

Equations

• ⋯ = ⋯

instance Prod.instInvolutiveStarProd {R : Type u} {S : Type v} [InvolutiveStar R] [InvolutiveStar S] : InvolutiveStar (R × S)

Equations

• Prod.instInvolutiveStarProd = InvolutiveStar.mk ⋯

instance Prod.instStarMulProdInstMul {R : Type u} {S : Type v} [Mul R] [Mul S] [StarMul R] [StarMul S] : StarMul (R × S)

Equations

• Prod.instStarMulProdInstMul = StarMul.mk ⋯

instance Prod.instStarAddMonoidProdInstAddMonoid {R : Type u} {S : Type v} [AddMonoid R] [AddMonoid S] [StarAddMonoid R] [StarAddMonoid S] : StarAddMonoid (R × S)

Equations

• Prod.instStarAddMonoidProdInstAddMonoid = StarAddMonoid.mk ⋯

instance Prod.instStarRingProdInstNonUnitalNonAssocSemiring {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [StarRing R] [StarRing S] : StarRing (R × S)

Equations

• Prod.instStarRingProdInstNonUnitalNonAssocSemiring = let __src := inferInstanceAs (StarAddMonoid (R × S)); let __src_1 := inferInstanceAs (StarMul (R × S)); StarRing.mk ⋯

instance Prod.instStarModuleProdInstStarProdSmul {R : Type u} {S : Type v} {α : Type w} [SMul α R] [SMul α S] [Star α] [Star R] [Star S] [StarModule α R] [StarModule α S] : StarModule α (R × S)

Equations

• ⋯ = ⋯

theorem Units.embed_product_star {R : Type u} [Monoid R] [StarMul R] (u : Rˣ) : (Units.embedProduct R) (star u) = star ((Units.embedProduct R) u)