Documentation

Mathlib.Algebra.Order.Ring.WithTop

Structures involving * and 0 on WithTop and WithBot #

The main results of this section are WithTop.canonicallyOrderedCommSemiring and WithBot.orderedCommSemiring.

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instance WithTop.instDecidableEqWithTop {α : Type u_1} [DecidableEq α] :
DecidableEq (WithTop α)
Equations
  • WithTop.instDecidableEqWithTop = instDecidableEqOption
source
instance WithTop.instMulZeroClassWithTop {α : Type u_1} [DecidableEq α] [Zero α] [Mul α] :
MulZeroClass (WithTop α)
Equations
  • One or more equations did not get rendered due to their size.
source
theorem WithTop.mul_def {α : Type u_1} [DecidableEq α] [Zero α] [Mul α] {a : WithTop α} {b : WithTop α} :
a * b = if a = 0 ∨ b = 0 then 0 else Option.map₂ (fun x x_1 => x * x_1) a b
source
theorem WithTop.top_mul_top {α : Type u_1} [DecidableEq α] [Zero α] [Mul α] :
⊤ * ⊤ = ⊤
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theorem WithTop.mul_top' {α : Type u_1} [DecidableEq α] [Zero α] [Mul α] (a : WithTop α) :
a * ⊤ = if a = 0 then 0 else ⊤
source
@[simp]
theorem WithTop.mul_top {α : Type u_1} [DecidableEq α] [Zero α] [Mul α] {a : WithTop α} (h : a ≠ 0) :
a * ⊤ = ⊤
source
theorem WithTop.top_mul' {α : Type u_1} [DecidableEq α] [Zero α] [Mul α] (a : WithTop α) :
⊤ * a = if a = 0 then 0 else ⊤
source
@[simp]
theorem WithTop.top_mul {α : Type u_1} [DecidableEq α] [Zero α] [Mul α] {a : WithTop α} (h : a ≠ 0) :
⊤ * a = ⊤
source
theorem WithTop.mul_eq_top_iff {α : Type u_1} [DecidableEq α] [Zero α] [Mul α] {a : WithTop α} {b : WithTop α} :
a * b = ⊤ ↔ a ≠ 0 ∧ b = ⊤ ∨ a = ⊤ ∧ b ≠ 0
source
theorem WithTop.mul_lt_top' {α : Type u_1} [DecidableEq α] [Zero α] [Mul α] [LT α] {a : WithTop α} {b : WithTop α} (ha : a < ⊤) (hb : b < ⊤) :
a * b < ⊤
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theorem WithTop.mul_lt_top {α : Type u_1} [DecidableEq α] [Zero α] [Mul α] [LT α] {a : WithTop α} {b : WithTop α} (ha : a ≠ ⊤) (hb : b ≠ ⊤) :
a * b < ⊤
source
instance WithTop.noZeroDivisors {α : Type u_1} [DecidableEq α] [Zero α] [Mul α] [NoZeroDivisors α] :
NoZeroDivisors (WithTop α)
Equations
source
@[simp]
theorem WithTop.coe_mul {α : Type u_1} [DecidableEq α] [MulZeroClass α] {a : α} {b : α} :
↑(a * b) = ↑a * ↑b
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theorem WithTop.mul_coe {α : Type u_1} [DecidableEq α] [MulZeroClass α] {b : α} (hb : b ≠ 0) {a : WithTop α} :
a * ↑b = Option.bind a fun a => some (a * b)
source
@[simp]
theorem WithTop.untop'_zero_mul {α : Type u_1} [DecidableEq α] [MulZeroClass α] (a : WithTop α) (b : WithTop α) :
WithTop.untop' 0 (a * b) = WithTop.untop' 0 a * WithTop.untop' 0 b
source
instance WithTop.instMulZeroOneClassWithTop {α : Type u_1} [DecidableEq α] [MulZeroOneClass α] [Nontrivial α] :
MulZeroOneClass (WithTop α)

Nontrivial α is needed here as otherwise we have 1 * ⊤ = ⊤ but also 0 * ⊤ = 0.

Equations
source
@[simp]
theorem MonoidWithZeroHom.withTopMap_apply {R : Type u_2} {S : Type u_3} [MulZeroOneClass R] [DecidableEq R] [Nontrivial R] [MulZeroOneClass S] [DecidableEq S] [Nontrivial S] (f : R →*₀ S) (hf : Function.Injective ↑f) :
↑(MonoidWithZeroHom.withTopMap f hf) = WithTop.map ↑f
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def MonoidWithZeroHom.withTopMap {R : Type u_2} {S : Type u_3} [MulZeroOneClass R] [DecidableEq R] [Nontrivial R] [MulZeroOneClass S] [DecidableEq S] [Nontrivial S] (f : R →*₀ S) (hf : Function.Injective ↑f) :
WithTop R →*₀ WithTop S

A version of WithTop.map for MonoidWithZeroHoms.

Equations
  • One or more equations did not get rendered due to their size.
Instances For
    source
    instance WithTop.instSemigroupWithZeroWithTop {α : Type u_1} [DecidableEq α] [SemigroupWithZero α] [NoZeroDivisors α] :
    SemigroupWithZero (WithTop α)
    Equations
    source
    instance WithTop.monoidWithZero {α : Type u_1} [DecidableEq α] [MonoidWithZero α] [NoZeroDivisors α] [Nontrivial α] :
    MonoidWithZero (WithTop α)
    Equations
    • One or more equations did not get rendered due to their size.
    source
    instance WithTop.commMonoidWithZero {α : Type u_1} [DecidableEq α] [CommMonoidWithZero α] [NoZeroDivisors α] [Nontrivial α] :
    CommMonoidWithZero (WithTop α)
    Equations
    source
    instance WithTop.commSemiring {α : Type u_1} [DecidableEq α] [CanonicallyOrderedCommSemiring α] [Nontrivial α] :
    CommSemiring (WithTop α)

    This instance requires CanonicallyOrderedCommSemiring as it is the smallest class that derives from both NonAssocNonUnitalSemiring and CanonicallyOrderedAddMonoid, both of which are required for distributivity.

    Equations
    • WithTop.commSemiring = let src := WithTop.addCommMonoidWithOne; let src_1 := WithTop.commMonoidWithZero; CommSemiring.mk (_ : ∀ (a b : WithTop α), a * b = b * a)
    source
    instance WithTop.instCanonicallyOrderedCommSemiringWithTop {α : Type u_1} [DecidableEq α] [CanonicallyOrderedCommSemiring α] [Nontrivial α] :
    CanonicallyOrderedCommSemiring (WithTop α)
    Equations
    • One or more equations did not get rendered due to their size.
    source
    @[simp]
    theorem RingHom.withTopMap_apply {R : Type u_2} {S : Type u_3} [CanonicallyOrderedCommSemiring R] [DecidableEq R] [Nontrivial R] [CanonicallyOrderedCommSemiring S] [DecidableEq S] [Nontrivial S] (f : R →+* S) (hf : Function.Injective ↑f) :
    ↑(RingHom.withTopMap f hf) = (↑(MonoidWithZeroHom.withTopMap (RingHom.toMonoidWithZeroHom f) hf)).toFun
    source
    def RingHom.withTopMap {R : Type u_2} {S : Type u_3} [CanonicallyOrderedCommSemiring R] [DecidableEq R] [Nontrivial R] [CanonicallyOrderedCommSemiring S] [DecidableEq S] [Nontrivial S] (f : R →+* S) (hf : Function.Injective ↑f) :
    WithTop R →+* WithTop S

    A version of WithTop.map for RingHoms.

    Equations
    • One or more equations did not get rendered due to their size.
    Instances For
      source
      instance WithBot.instDecidableEqWithBot {α : Type u_1} [DecidableEq α] :
      DecidableEq (WithBot α)
      Equations
      • WithBot.instDecidableEqWithBot = instDecidableEqOption
      source
      instance WithBot.instMulZeroClassWithBot {α : Type u_1} [DecidableEq α] [Zero α] [Mul α] :
      MulZeroClass (WithBot α)
      Equations
      • WithBot.instMulZeroClassWithBot = WithTop.instMulZeroClassWithTop
      source
      theorem WithBot.mul_def {α : Type u_1} [DecidableEq α] [Zero α] [Mul α] {a : WithBot α} {b : WithBot α} :
      a * b = if a = 0 ∨ b = 0 then 0 else Option.map₂ (fun x x_1 => x * x_1) a b
      source
      @[simp]
      theorem WithBot.mul_bot {α : Type u_1} [DecidableEq α] [Zero α] [Mul α] {a : WithBot α} (h : a ≠ 0) :
      a * ⊥ = ⊥
      source
      @[simp]
      theorem WithBot.bot_mul {α : Type u_1} [DecidableEq α] [Zero α] [Mul α] {a : WithBot α} (h : a ≠ 0) :
      ⊥ * a = ⊥
      source
      @[simp]
      theorem WithBot.bot_mul_bot {α : Type u_1} [DecidableEq α] [Zero α] [Mul α] :
      ⊥ * ⊥ = ⊥
      source
      theorem WithBot.mul_eq_bot_iff {α : Type u_1} [DecidableEq α] [Zero α] [Mul α] {a : WithBot α} {b : WithBot α} :
      a * b = ⊥ ↔ a ≠ 0 ∧ b = ⊥ ∨ a = ⊥ ∧ b ≠ 0
      source
      theorem WithBot.bot_lt_mul' {α : Type u_1} [DecidableEq α] [Zero α] [Mul α] [LT α] {a : WithBot α} {b : WithBot α} (ha : ⊥ < a) (hb : ⊥ < b) :
      ⊥ < a * b
      source
      theorem WithBot.bot_lt_mul {α : Type u_1} [DecidableEq α] [Zero α] [Mul α] [LT α] {a : WithBot α} {b : WithBot α} (ha : a ≠ ⊥) (hb : b ≠ ⊥) :
      ⊥ < a * b
      source
      @[simp]
      theorem WithBot.coe_mul {α : Type u_1} [DecidableEq α] [MulZeroClass α] {a : α} {b : α} :
      ↑(a * b) = ↑a * ↑b
      source
      theorem WithBot.mul_coe {α : Type u_1} [DecidableEq α] [MulZeroClass α] {b : α} (hb : b ≠ 0) {a : WithBot α} :
      a * ↑b = Option.bind a fun a => some (a * b)
      source
      instance WithBot.instMulZeroOneClassWithBot {α : Type u_1} [DecidableEq α] [MulZeroOneClass α] [Nontrivial α] :
      MulZeroOneClass (WithBot α)

      Nontrivial α is needed here as otherwise we have 1 * ⊥ = ⊥ but also = 0 * ⊥ = 0.

      Equations
      • WithBot.instMulZeroOneClassWithBot = WithTop.instMulZeroOneClassWithTop
      source
      instance WithBot.instNoZeroDivisorsWithBotToMulInstMulZeroClassWithBotToZeroZero {α : Type u_1} [DecidableEq α] [MulZeroClass α] [NoZeroDivisors α] :
      NoZeroDivisors (WithBot α)
      Equations
      source
      instance WithBot.instSemigroupWithZeroWithBot {α : Type u_1} [DecidableEq α] [SemigroupWithZero α] [NoZeroDivisors α] :
      SemigroupWithZero (WithBot α)
      Equations
      • WithBot.instSemigroupWithZeroWithBot = WithTop.instSemigroupWithZeroWithTop
      source
      instance WithBot.instMonoidWithZeroWithBot {α : Type u_1} [DecidableEq α] [MonoidWithZero α] [NoZeroDivisors α] [Nontrivial α] :
      MonoidWithZero (WithBot α)
      Equations
      • WithBot.instMonoidWithZeroWithBot = WithTop.monoidWithZero
      source
      instance WithBot.commMonoidWithZero {α : Type u_1} [DecidableEq α] [CommMonoidWithZero α] [NoZeroDivisors α] [Nontrivial α] :
      CommMonoidWithZero (WithBot α)
      Equations
      • WithBot.commMonoidWithZero = WithTop.commMonoidWithZero
      source
      instance WithBot.commSemiring {α : Type u_1} [DecidableEq α] [CanonicallyOrderedCommSemiring α] [Nontrivial α] :
      CommSemiring (WithBot α)
      Equations
      • WithBot.commSemiring = WithTop.commSemiring
      source
      instance WithBot.instPosMulMonoWithBotToMulInstMulZeroClassWithBotToZeroZeroPreorder {α : Type u_1} [DecidableEq α] [MulZeroClass α] [Preorder α] [PosMulMono α] :
      PosMulMono (WithBot α)
      Equations
      source
      instance WithBot.instMulPosMonoWithBotToMulInstMulZeroClassWithBotToZeroZeroPreorder {α : Type u_1} [DecidableEq α] [MulZeroClass α] [Preorder α] [MulPosMono α] :
      MulPosMono (WithBot α)
      Equations
      source
      instance WithBot.instPosMulStrictMonoWithBotToMulInstMulZeroClassWithBotToZeroZeroPreorder {α : Type u_1} [DecidableEq α] [MulZeroClass α] [Preorder α] [PosMulStrictMono α] :
      PosMulStrictMono (WithBot α)
      Equations
      source
      instance WithBot.instMulPosStrictMonoWithBotToMulInstMulZeroClassWithBotToZeroZeroPreorder {α : Type u_1} [DecidableEq α] [MulZeroClass α] [Preorder α] [MulPosStrictMono α] :
      MulPosStrictMono (WithBot α)
      Equations
      source
      instance WithBot.instPosMulReflectLTWithBotToMulInstMulZeroClassWithBotToZeroZeroPreorder {α : Type u_1} [DecidableEq α] [MulZeroClass α] [Preorder α] [PosMulReflectLT α] :
      PosMulReflectLT (WithBot α)
      Equations
      source
      instance WithBot.instMulPosReflectLTWithBotToMulInstMulZeroClassWithBotToZeroZeroPreorder {α : Type u_1} [DecidableEq α] [MulZeroClass α] [Preorder α] [MulPosReflectLT α] :
      MulPosReflectLT (WithBot α)
      Equations
      source
      instance WithBot.instPosMulMonoRevWithBotToMulInstMulZeroClassWithBotToZeroZeroPreorder {α : Type u_1} [DecidableEq α] [MulZeroClass α] [Preorder α] [PosMulMonoRev α] :
      PosMulMonoRev (WithBot α)
      Equations
      source
      instance WithBot.instMulPosMonoRevWithBotToMulInstMulZeroClassWithBotToZeroZeroPreorder {α : Type u_1} [DecidableEq α] [MulZeroClass α] [Preorder α] [MulPosMonoRev α] :
      MulPosMonoRev (WithBot α)
      Equations
      source
      instance WithBot.orderedCommSemiring {α : Type u_1} [DecidableEq α] [CanonicallyOrderedCommSemiring α] [Nontrivial α] :
      OrderedCommSemiring (WithBot α)
      Equations
      • One or more equations did not get rendered due to their size.