Documentation

Mathlib.Algebra.Star.Basic

Star monoids, rings, and modules #

We introduce the basic algebraic notions of star monoids, star rings, and star modules. A star algebra is simply a star ring that is also a star module.

These are implemented as "mixin" typeclasses, so to summon a star ring (for example) one needs to write (R : Type*) [Ring R] [StarRing R]. This avoids difficulties with diamond inheritance.

For now we simply do not introduce notations, as different users are expected to feel strongly about the relative merits of r^*, r†, rᘁ, and so on.

Our star rings are actually star non-unital, non-associative, semirings, but of course we can prove star_neg : star (-r) = - star r when the underlying semiring is a ring.

class Star (R : Type u) :

Notation typeclass (with no default notation!) for an algebraic structure with a star operation.

  • star : RR

    A star operation (e.g. complex conjugate).

Instances
    class StarMemClass (S : Type u_1) (R : Type u_2) [Star R] [SetLike S R] :

    StarMemClass S G states S is a type of subsets s ⊆ G closed under star.

    • star_mem : ∀ {s : S} {r : R}, r sstar r s

      Closure under star.

    Instances
      theorem StarMemClass.star_mem {S : Type u_1} {R : Type u_2} :
      ∀ {inst : Star R} {inst_1 : SetLike S R} [self : StarMemClass S R] {s : S} {r : R}, r sstar r s

      Closure under star.

      instance StarMemClass.instStar {R : Type u} {S : Type w} [Star R] [SetLike S R] [hS : StarMemClass S R] (s : S) :
      Star s
      Equations
      @[simp]
      theorem StarMemClass.coe_star {R : Type u} {S : Type w} [Star R] [SetLike S R] [hS : StarMemClass S R] (s : S) (x : s) :
      (star x) = star x
      class InvolutiveStar (R : Type u) extends Star :

      Typeclass for a star operation with is involutive.

      Instances

        Involutive condition.

        @[simp]
        theorem star_star {R : Type u} [InvolutiveStar R] (r : R) :
        star (star r) = r
        @[simp]
        theorem star_inj {R : Type u} [InvolutiveStar R] {x : R} {y : R} :
        star x = star y x = y

        star as an equivalence when it is involutive.

        Equations
        Instances For
          theorem eq_star_of_eq_star {R : Type u} [InvolutiveStar R] {r : R} {s : R} (h : r = star s) :
          s = star r
          theorem eq_star_iff_eq_star {R : Type u} [InvolutiveStar R] {r : R} {s : R} :
          r = star s s = star r
          theorem star_eq_iff_star_eq {R : Type u} [InvolutiveStar R] {r : R} {s : R} :
          star r = s star s = r
          class TrivialStar (R : Type u) [Star R] :

          Typeclass for a trivial star operation. This is mostly meant for .

          • star_trivial : ∀ (r : R), star r = r

            Condition that star is trivial

          Instances
            @[simp]
            theorem TrivialStar.star_trivial {R : Type u} :
            ∀ {inst : Star R} [self : TrivialStar R] (r : R), star r = r

            Condition that star is trivial

            class StarMul (R : Type u) [Mul R] extends InvolutiveStar :

            A *-magma is a magma R with an involutive operation star such that star (r * s) = star s * star r.

            Instances
              @[simp]
              theorem StarMul.star_mul {R : Type u} :
              ∀ {inst : Mul R} [self : StarMul R] (r s : R), star (r * s) = star s * star r

              star skew-distributes over multiplication.

              theorem star_star_mul {R : Type u} [Mul R] [StarMul R] (x : R) (y : R) :
              star (star x * y) = star y * x
              theorem star_mul_star {R : Type u} [Mul R] [StarMul R] (x : R) (y : R) :
              star (x * star y) = y * star x
              @[simp]
              theorem semiconjBy_star_star_star {R : Type u} [Mul R] [StarMul R] {x : R} {y : R} {z : R} :
              theorem SemiconjBy.star_star_star {R : Type u} [Mul R] [StarMul R] {x : R} {y : R} {z : R} :
              SemiconjBy x y zSemiconjBy (star x) (star z) (star y)

              Alias of the reverse direction of semiconjBy_star_star_star.

              @[simp]
              theorem commute_star_star {R : Type u} [Mul R] [StarMul R] {x : R} {y : R} :
              theorem Commute.star_star {R : Type u} [Mul R] [StarMul R] {x : R} {y : R} :
              Commute x yCommute (star x) (star y)

              Alias of the reverse direction of commute_star_star.

              theorem commute_star_comm {R : Type u} [Mul R] [StarMul R] {x : R} {y : R} :
              @[simp]
              theorem star_mul' {R : Type u} [CommSemigroup R] [StarMul R] (x : R) (y : R) :
              star (x * y) = star x * star y

              In a commutative ring, make simp prefer leaving the order unchanged.

              @[simp]
              theorem starMulEquiv_apply {R : Type u} [Mul R] [StarMul R] (x : R) :
              starMulEquiv x = MulOpposite.op (star x)
              def starMulEquiv {R : Type u} [Mul R] [StarMul R] :

              star as a MulEquiv from R to Rᵐᵒᵖ

              Equations
              • One or more equations did not get rendered due to their size.
              Instances For
                @[simp]
                theorem starMulAut_apply {R : Type u} [CommSemigroup R] [StarMul R] :
                ∀ (a : R), starMulAut a = star a
                def starMulAut {R : Type u} [CommSemigroup R] [StarMul R] :

                star as a MulAut for commutative R.

                Equations
                • starMulAut = { toFun := star, invFun := (Function.Involutive.toPerm star ).invFun, left_inv := , right_inv := , map_mul' := }
                Instances For
                  @[simp]
                  theorem star_one (R : Type u) [MulOneClass R] [StarMul R] :
                  star 1 = 1
                  @[simp]
                  theorem star_pow {R : Type u} [Monoid R] [StarMul R] (x : R) (n : ) :
                  star (x ^ n) = star x ^ n
                  @[simp]
                  theorem star_inv {R : Type u} [Group R] [StarMul R] (x : R) :
                  @[simp]
                  theorem star_zpow {R : Type u} [Group R] [StarMul R] (x : R) (z : ) :
                  star (x ^ z) = star x ^ z
                  @[simp]
                  theorem star_div {R : Type u} [CommGroup R] [StarMul R] (x : R) (y : R) :
                  star (x / y) = star x / star y

                  When multiplication is commutative, star preserves division.

                  @[reducible, inline]
                  abbrev starMulOfComm {R : Type u_1} [CommMonoid R] :

                  Any commutative monoid admits the trivial *-structure.

                  See note [reducible non-instances].

                  Equations
                  Instances For
                    theorem star_id_of_comm {R : Type u_1} [CommSemiring R] {x : R} :
                    star x = x

                    Note that since starMulOfComm is reducible, simp can already prove this.

                    class StarAddMonoid (R : Type u) [AddMonoid R] extends InvolutiveStar :

                    A *-additive monoid R is an additive monoid with an involutive star operation which preserves addition.

                    Instances
                      @[simp]
                      theorem StarAddMonoid.star_add {R : Type u} :
                      ∀ {inst : AddMonoid R} [self : StarAddMonoid R] (r s : R), star (r + s) = star r + star s

                      star commutes with addition

                      @[simp]
                      theorem starAddEquiv_apply {R : Type u} [AddMonoid R] [StarAddMonoid R] :
                      ∀ (a : R), starAddEquiv a = star a
                      def starAddEquiv {R : Type u} [AddMonoid R] [StarAddMonoid R] :
                      R ≃+ R

                      star as an AddEquiv

                      Equations
                      • starAddEquiv = { toFun := star, invFun := (Function.Involutive.toPerm star ).invFun, left_inv := , right_inv := , map_add' := }
                      Instances For
                        @[simp]
                        theorem star_zero (R : Type u) [AddMonoid R] [StarAddMonoid R] :
                        star 0 = 0
                        @[simp]
                        theorem star_eq_zero {R : Type u} [AddMonoid R] [StarAddMonoid R] {x : R} :
                        star x = 0 x = 0
                        theorem star_ne_zero {R : Type u} [AddMonoid R] [StarAddMonoid R] {x : R} :
                        star x 0 x 0
                        @[simp]
                        theorem star_neg {R : Type u} [AddGroup R] [StarAddMonoid R] (r : R) :
                        star (-r) = -star r
                        @[simp]
                        theorem star_sub {R : Type u} [AddGroup R] [StarAddMonoid R] (r : R) (s : R) :
                        star (r - s) = star r - star s
                        @[simp]
                        theorem star_nsmul {R : Type u} [AddMonoid R] [StarAddMonoid R] (n : ) (x : R) :
                        star (n x) = n star x
                        @[simp]
                        theorem star_zsmul {R : Type u} [AddGroup R] [StarAddMonoid R] (n : ) (x : R) :
                        star (n x) = n star x
                        class StarRing (R : Type u) [NonUnitalNonAssocSemiring R] extends StarMul :

                        A *-ring R is a non-unital, non-associative (semi)ring with an involutive star operation which is additive which makes R with its multiplicative structure into a *-multiplication (i.e. star (r * s) = star s * star r).

                        Instances
                          theorem StarRing.star_add {R : Type u} :
                          ∀ {inst : NonUnitalNonAssocSemiring R} [self : StarRing R] (r s : R), star (r + s) = star r + star s

                          star commutes with addition

                          @[instance 100]
                          Equations
                          @[simp]
                          theorem starRingEquiv_apply {R : Type u} [NonUnitalNonAssocSemiring R] [StarRing R] (x : R) :
                          starRingEquiv x = MulOpposite.op (star x)

                          star as a RingEquiv from R to Rᵐᵒᵖ

                          Equations
                          • starRingEquiv = { toFun := fun (x : R) => MulOpposite.op (star x), invFun := (starAddEquiv.trans MulOpposite.opAddEquiv).invFun, left_inv := , right_inv := , map_mul' := , map_add' := }
                          Instances For
                            @[simp]
                            theorem star_natCast {R : Type u} [NonAssocSemiring R] [StarRing R] (n : ) :
                            star n = n
                            @[simp]
                            theorem star_ofNat {R : Type u} [NonAssocSemiring R] [StarRing R] (n : ) [n.AtLeastTwo] :
                            @[simp]
                            theorem star_intCast {R : Type u} [Ring R] [StarRing R] (z : ) :
                            star z = z
                            @[simp]
                            theorem starRingAut_apply {R : Type u} [CommSemiring R] [StarRing R] :
                            ∀ (a : R), starRingAut a = star a

                            star as a ring automorphism, for commutative R.

                            Equations
                            • starRingAut = { toFun := star, invFun := starAddEquiv.invFun, left_inv := , right_inv := , map_mul' := , map_add' := }
                            Instances For
                              def starRingEnd (R : Type u) [CommSemiring R] [StarRing R] :
                              R →+* R

                              star as a ring endomorphism, for commutative R. This is used to denote complex conjugation, and is available under the notation conj in the locale ComplexConjugate.

                              Note that this is the preferred form (over starRingAut, available under the same hypotheses) because the notation E →ₗ⋆[R] F for an R-conjugate-linear map (short for E →ₛₗ[starRingEnd R] F) does not pretty-print if there is a coercion involved, as would be the case for (↑starRingAut : R →* R).

                              Equations
                              Instances For

                                star as a ring endomorphism, for commutative R. This is used to denote complex conjugation, and is available under the notation conj in the locale ComplexConjugate.

                                Note that this is the preferred form (over starRingAut, available under the same hypotheses) because the notation E →ₗ⋆[R] F for an R-conjugate-linear map (short for E →ₛₗ[starRingEnd R] F) does not pretty-print if there is a coercion involved, as would be the case for (↑starRingAut : R →* R).

                                Equations
                                Instances For
                                  theorem starRingEnd_apply {R : Type u} [CommSemiring R] [StarRing R] (x : R) :

                                  This is not a simp lemma, since we usually want simp to keep starRingEnd bundled. For example, for complex conjugation, we don't want simp to turn conj x into the bare function star x automatically since most lemmas are about conj x.

                                  theorem starRingEnd_self_apply {R : Type u} [CommSemiring R] [StarRing R] (x : R) :
                                  (starRingEnd R) ((starRingEnd R) x) = x
                                  Equations
                                  theorem RingHom.star_def {R : Type u} [CommSemiring R] [StarRing R] {S : Type u_1} [NonAssocSemiring S] (f : S →+* R) :
                                  star f = (starRingEnd R).comp f
                                  theorem RingHom.star_apply {R : Type u} [CommSemiring R] [StarRing R] {S : Type u_1} [NonAssocSemiring S] (f : S →+* R) (s : S) :
                                  (star f) s = star (f s)
                                  theorem Complex.conj_conj {R : Type u} [CommSemiring R] [StarRing R] (x : R) :
                                  (starRingEnd R) ((starRingEnd R) x) = x

                                  Alias of starRingEnd_self_apply.

                                  theorem RCLike.conj_conj {R : Type u} [CommSemiring R] [StarRing R] (x : R) :
                                  (starRingEnd R) ((starRingEnd R) x) = x

                                  Alias of starRingEnd_self_apply.

                                  @[simp]
                                  theorem conj_trivial {R : Type u} [CommSemiring R] [StarRing R] [TrivialStar R] (a : R) :
                                  (starRingEnd R) a = a
                                  @[simp]
                                  theorem star_inv' {R : Type u} [DivisionSemiring R] [StarRing R] (x : R) :
                                  @[simp]
                                  theorem star_zpow₀ {R : Type u} [DivisionSemiring R] [StarRing R] (x : R) (z : ) :
                                  star (x ^ z) = star x ^ z
                                  @[simp]
                                  theorem star_div' {R : Type u} [Semifield R] [StarRing R] (x : R) (y : R) :
                                  star (x / y) = star x / star y

                                  When multiplication is commutative, star preserves division.

                                  @[reducible, inline]
                                  abbrev starRingOfComm {R : Type u_1} [CommSemiring R] :

                                  Any commutative semiring admits the trivial *-structure.

                                  See note [reducible non-instances].

                                  Equations
                                  Instances For
                                    Equations
                                    Equations
                                    class StarModule (R : Type u) (A : Type v) [Star R] [Star A] [SMul R A] :

                                    A star module A over a star ring R is a module which is a star add monoid, and the two star structures are compatible in the sense star (r • a) = star r • star a.

                                    Note that it is up to the user of this typeclass to enforce [Semiring R] [StarRing R] [AddCommMonoid A] [StarAddMonoid A] [Module R A], and that the statement only requires [Star R] [Star A] [SMul R A].

                                    If used as [CommRing R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A], this represents a star algebra.

                                    Instances
                                      @[simp]
                                      theorem StarModule.star_smul {R : Type u} {A : Type v} :
                                      ∀ {inst : Star R} {inst_1 : Star A} {inst_2 : SMul R A} [self : StarModule R A] (r : R) (a : A), star (r a) = star r star a

                                      star commutes with scalar multiplication

                                      instance StarMul.toStarModule {R : Type u} [CommMonoid R] [StarMul R] :

                                      A commutative star monoid is a star module over itself via Monoid.toMulAction.

                                      Equations
                                      • =
                                      Equations
                                      • =
                                      Equations
                                      • =

                                      Instance needed to define star-linear maps over a commutative star ring (ex: conjugate-linear maps when R = ℂ).

                                      Equations
                                      • =
                                      class StarHomClass (F : Type u_1) (R : outParam (Type u_2)) (S : outParam (Type u_3)) [Star R] [Star S] [FunLike F R S] :

                                      StarHomClass F R S states that F is a type of star-preserving maps from R to S.

                                      • map_star : ∀ (f : F) (r : R), f (star r) = star (f r)

                                        the maps preserve star

                                      Instances
                                        theorem StarHomClass.map_star {F : Type u_1} {R : outParam (Type u_2)} {S : outParam (Type u_3)} :
                                        ∀ {inst : Star R} {inst_1 : Star S} {inst_2 : FunLike F R S} [self : StarHomClass F R S] (f : F) (r : R), f (star r) = star (f r)

                                        the maps preserve star

                                        Instances #

                                        instance Units.instStarMul {R : Type u} [Monoid R] [StarMul R] :
                                        Equations
                                        @[simp]
                                        theorem Units.coe_star {R : Type u} [Monoid R] [StarMul R] (u : Rˣ) :
                                        (star u) = star u
                                        @[simp]
                                        theorem Units.coe_star_inv {R : Type u} [Monoid R] [StarMul R] (u : Rˣ) :
                                        (star u)⁻¹ = star u⁻¹
                                        instance Units.instStarModule {R : Type u} [Monoid R] [StarMul R] {A : Type u_1} [Star A] [SMul R A] [StarModule R A] :
                                        Equations
                                        • =
                                        theorem IsUnit.star {R : Type u} [Monoid R] [StarMul R] {a : R} :
                                        IsUnit aIsUnit (star a)
                                        @[simp]
                                        theorem isUnit_star {R : Type u} [Monoid R] [StarMul R] {a : R} :
                                        instance Invertible.star {R : Type u_1} [MulOneClass R] [StarMul R] (r : R) [Invertible r] :
                                        Equations
                                        theorem star_invOf {R : Type u_1} [Monoid R] [StarMul R] (r : R) [Invertible r] [Invertible (star r)] :
                                        theorem IsLeftRegular.star {R : Type u} [Mul R] [StarMul R] {x : R} (hx : IsLeftRegular x) :
                                        theorem IsRightRegular.star {R : Type u} [Mul R] [StarMul R] {x : R} (hx : IsRightRegular x) :
                                        theorem IsRegular.star {R : Type u} [Mul R] [StarMul R] {x : R} (hx : IsRegular x) :
                                        @[simp]
                                        @[simp]
                                        @[simp]
                                        theorem isRegular_star_iff {R : Type u} [Mul R] [StarMul R] {x : R} :

                                        The opposite type carries the same star operation.

                                        Equations
                                        @[simp]
                                        theorem MulOpposite.op_star {R : Type u} [Star R] (r : R) :
                                        Equations
                                        Equations
                                        Equations
                                        Equations

                                        A commutative star monoid is a star module over its opposite via Monoid.toOppositeMulAction.

                                        Equations
                                        • =