Documentation

Mathlib.Algebra.Order.Kleene

Kleene Algebras #

This file defines idempotent semirings and Kleene algebras, which are used extensively in the theory of computation.

An idempotent semiring is a semiring whose addition is idempotent. An idempotent semiring is naturally a semilattice by setting a ≤ b if a + b = b.

A Kleene algebra is an idempotent semiring equipped with an additional unary operator , the Kleene star.

Main declarations #

Notation #

a∗ is notation for kstar a in locale Computability.

References #

TODO #

Instances for AddOpposite, MulOpposite, ULift, Subsemiring, Subring, Subalgebra.

Tags #

kleene algebra, idempotent semiring

class IdemSemiring (α : Type u) extends Semiring α, SemilatticeSup α :

An idempotent semiring is a semiring with the additional property that addition is idempotent.

Instances
class IdemCommSemiring (α : Type u) extends CommSemiring α, IdemSemiring α :

An idempotent commutative semiring is a commutative semiring with the additional property that addition is idempotent.

Instances
class KStar (α : Type u_5) :
Type u_5

Notation typeclass for the Kleene star .

  • kstar : αα

    The Kleene star operator on a Kleene algebra

Instances

    The Kleene star operator on a Kleene algebra

    Equations
    class KleeneAlgebra (α : Type u_5) extends IdemSemiring α, KStar α :
    Type u_5

    A Kleene Algebra is an idempotent semiring with an additional unary operator kstar (for Kleene star) that satisfies the following properties:

    • 1 + a * a∗ ≤ a∗
    • 1 + a∗ * a ≤ a∗
    • If a * c + b ≤ c, then a∗ * b ≤ c
    • If c * a + b ≤ c, then b * a∗ ≤ c
    Instances
    @[instance 100]
    instance IdemSemiring.toOrderBot {α : Type u_1} [IdemSemiring α] :
    Equations
    @[reducible, inline]
    abbrev IdemSemiring.ofSemiring {α : Type u_1} [Semiring α] (h : ∀ (a : α), a + a = a) :

    Construct an idempotent semiring from an idempotent addition.

    Equations
    theorem add_eq_sup {α : Type u_1} [IdemSemiring α] (a b : α) :
    a + b = a b
    theorem add_idem {α : Type u_1} [IdemSemiring α] (a : α) :
    a + a = a
    theorem nsmul_eq_self {α : Type u_1} [IdemSemiring α] {n : } :
    n 0∀ (a : α), n a = a
    theorem add_eq_left_iff_le {α : Type u_1} [IdemSemiring α] {a b : α} :
    a + b = a b a
    theorem add_eq_right_iff_le {α : Type u_1} [IdemSemiring α] {a b : α} :
    a + b = b a b
    theorem LE.le.add_eq_left {α : Type u_1} [IdemSemiring α] {a b : α} :
    b aa + b = a

    Alias of the reverse direction of add_eq_left_iff_le.

    theorem LE.le.add_eq_right {α : Type u_1} [IdemSemiring α] {a b : α} :
    a ba + b = b

    Alias of the reverse direction of add_eq_right_iff_le.

    theorem add_le_iff {α : Type u_1} [IdemSemiring α] {a b c : α} :
    a + b c a c b c
    theorem add_le {α : Type u_1} [IdemSemiring α] {a b c : α} (ha : a c) (hb : b c) :
    a + b c
    @[instance 100]
    Equations
    @[instance 100]
    Equations
    • =
    @[instance 100]
    Equations
    • =
    @[simp]
    theorem one_le_kstar {α : Type u_1} [KleeneAlgebra α] {a : α} :
    theorem mul_kstar_le_kstar {α : Type u_1} [KleeneAlgebra α] {a : α} :
    theorem kstar_mul_le_kstar {α : Type u_1} [KleeneAlgebra α] {a : α} :
    theorem mul_kstar_le_self {α : Type u_1} [KleeneAlgebra α] {a b : α} :
    b * a bb * KStar.kstar a b
    theorem kstar_mul_le_self {α : Type u_1} [KleeneAlgebra α] {a b : α} :
    a * b bKStar.kstar a * b b
    theorem mul_kstar_le {α : Type u_1} [KleeneAlgebra α] {a b c : α} (hb : b c) (ha : c * a c) :
    theorem kstar_mul_le {α : Type u_1} [KleeneAlgebra α] {a b c : α} (hb : b c) (ha : a * c c) :
    theorem kstar_le_of_mul_le_left {α : Type u_1} [KleeneAlgebra α] {a b : α} (hb : 1 b) :
    b * a bKStar.kstar a b
    theorem kstar_le_of_mul_le_right {α : Type u_1} [KleeneAlgebra α] {a b : α} (hb : 1 b) :
    a * b bKStar.kstar a b
    @[simp]
    theorem le_kstar {α : Type u_1} [KleeneAlgebra α] {a : α} :
    theorem kstar_mono {α : Type u_1} [KleeneAlgebra α] :
    Monotone KStar.kstar
    @[simp]
    theorem kstar_eq_one {α : Type u_1} [KleeneAlgebra α] {a : α} :
    @[simp]
    theorem kstar_zero {α : Type u_1} [KleeneAlgebra α] :
    @[simp]
    theorem kstar_one {α : Type u_1} [KleeneAlgebra α] :
    @[simp]
    theorem kstar_mul_kstar {α : Type u_1} [KleeneAlgebra α] (a : α) :
    @[simp]
    theorem kstar_eq_self {α : Type u_1} [KleeneAlgebra α] {a : α} :
    KStar.kstar a = a a * a = a 1 a
    @[simp]
    theorem kstar_idem {α : Type u_1} [KleeneAlgebra α] (a : α) :
    @[simp]
    theorem pow_le_kstar {α : Type u_1} [KleeneAlgebra α] {a : α} {n : } :
    instance Prod.instIdemSemiring {α : Type u_1} {β : Type u_2} [IdemSemiring α] [IdemSemiring β] :
    Equations
    instance Prod.instIdemCommSemiring {α : Type u_1} {β : Type u_2} [IdemCommSemiring α] [IdemCommSemiring β] :
    Equations
    instance Prod.instKleeneAlgebra {α : Type u_1} {β : Type u_2} [KleeneAlgebra α] [KleeneAlgebra β] :
    Equations
    theorem Prod.kstar_def {α : Type u_1} {β : Type u_2} [KleeneAlgebra α] [KleeneAlgebra β] (a : α × β) :
    @[simp]
    theorem Prod.fst_kstar {α : Type u_1} {β : Type u_2} [KleeneAlgebra α] [KleeneAlgebra β] (a : α × β) :
    @[simp]
    theorem Prod.snd_kstar {α : Type u_1} {β : Type u_2} [KleeneAlgebra α] [KleeneAlgebra β] (a : α × β) :
    instance Pi.instIdemSemiring {ι : Type u_3} {π : ιType u_4} [(i : ι) → IdemSemiring (π i)] :
    IdemSemiring ((i : ι) → π i)
    Equations
    instance Pi.instIdemCommSemiringForall {ι : Type u_3} {π : ιType u_4} [(i : ι) → IdemCommSemiring (π i)] :
    IdemCommSemiring ((i : ι) → π i)
    Equations
    instance Pi.instKleeneAlgebraForall {ι : Type u_3} {π : ιType u_4} [(i : ι) → KleeneAlgebra (π i)] :
    KleeneAlgebra ((i : ι) → π i)
    Equations
    theorem Pi.kstar_def {ι : Type u_3} {π : ιType u_4} [(i : ι) → KleeneAlgebra (π i)] (a : (i : ι) → π i) :
    KStar.kstar a = fun (i : ι) => KStar.kstar (a i)
    @[simp]
    theorem Pi.kstar_apply {ι : Type u_3} {π : ιType u_4} [(i : ι) → KleeneAlgebra (π i)] (a : (i : ι) → π i) (i : ι) :
    @[reducible, inline]
    abbrev Function.Injective.idemSemiring {α : Type u_1} {β : Type u_2} [IdemSemiring α] [Zero β] [One β] [Add β] [Mul β] [Pow β ] [SMul β] [NatCast β] [Max β] [Bot β] (f : βα) (hf : Function.Injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (nsmul : ∀ (n : ) (x : β), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ), f n = n) (sup : ∀ (a b : β), f (a b) = f a f b) (bot : f = ) :

    Pullback an IdemSemiring instance along an injective function.

    Equations
    @[reducible, inline]
    abbrev Function.Injective.idemCommSemiring {α : Type u_1} {β : Type u_2} [IdemCommSemiring α] [Zero β] [One β] [Add β] [Mul β] [Pow β ] [SMul β] [NatCast β] [Max β] [Bot β] (f : βα) (hf : Function.Injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (nsmul : ∀ (n : ) (x : β), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ), f n = n) (sup : ∀ (a b : β), f (a b) = f a f b) (bot : f = ) :

    Pullback an IdemCommSemiring instance along an injective function.

    Equations
    @[reducible, inline]
    abbrev Function.Injective.kleeneAlgebra {α : Type u_1} {β : Type u_2} [KleeneAlgebra α] [Zero β] [One β] [Add β] [Mul β] [Pow β ] [SMul β] [NatCast β] [Max β] [Bot β] [KStar β] (f : βα) (hf : Function.Injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (nsmul : ∀ (n : ) (x : β), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ), f n = n) (sup : ∀ (a b : β), f (a b) = f a f b) (bot : f = ) (kstar : ∀ (a : β), f (KStar.kstar a) = KStar.kstar (f a)) :

    Pullback a KleeneAlgebra instance along an injective function.

    Equations