Structures involving * and 0 on WithTop and WithBot

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

instance WithTop.instDecidableEqWithTop {α : Type u_1} [DecidableEq α] : DecidableEq (WithTop α)

Equations

• WithTop.instDecidableEqWithTop = instDecidableEqOption

instance WithTop.instMulZeroClass {α : Type u_1} [DecidableEq α] [MulZeroClass α] : MulZeroClass (WithTop α)

Equations

• WithTop.instMulZeroClass = MulZeroClass.mk (_ : ∀ (b : WithTop α), 0 * b = 0) (_ : ∀ (a : WithTop α), a * 0 = 0)

@[simp]

theorem WithTop.coe_mul {α : Type u_1} [DecidableEq α] [MulZeroClass α] (a : α) (b : α) : ↑(a * b) = ↑a * ↑b

theorem WithTop.mul_top' {α : Type u_1} [DecidableEq α] [MulZeroClass α] (a : WithTop α) : a * ⊤ = if a = 0 then 0 else ⊤

@[simp]

theorem WithTop.mul_top {α : Type u_1} [DecidableEq α] [MulZeroClass α] {a : WithTop α} (h : a ≠ 0) : a * ⊤ = ⊤

theorem WithTop.top_mul' {α : Type u_1} [DecidableEq α] [MulZeroClass α] (b : WithTop α) : ⊤ * b = if b = 0 then 0 else ⊤

@[simp]

theorem WithTop.top_mul {α : Type u_1} [DecidableEq α] [MulZeroClass α] {b : WithTop α} (hb : b ≠ 0) : ⊤ * b = ⊤

@[simp]

theorem WithTop.top_mul_top {α : Type u_1} [DecidableEq α] [MulZeroClass α] : ⊤ * ⊤ = ⊤

theorem WithTop.mul_def {α : Type u_1} [DecidableEq α] [MulZeroClass α] (a : WithTop α) (b : WithTop α) : a * b = if a = 0 ∨ b = 0 then 0 else WithTop.map₂ (fun (x x_1 : α) => x * x_1) a b

theorem WithTop.mul_eq_top_iff {α : Type u_1} [DecidableEq α] [MulZeroClass α] {a : WithTop α} {b : WithTop α} : a * b = ⊤ ↔ a ≠ 0 ∧ b = ⊤ ∨ a = ⊤ ∧ b ≠ 0

theorem WithTop.mul_coe_eq_bind {α : Type u_1} [DecidableEq α] [MulZeroClass α] {b : α} (hb : b ≠ 0) (a : WithTop α) : a * ↑b = Option.bind a fun (a : α) => some (a * b)

theorem WithTop.coe_mul_eq_bind {α : Type u_1} [DecidableEq α] [MulZeroClass α] {a : α} (ha : a ≠ 0) (b : WithTop α) : ↑a * b = Option.bind b fun (b : α) => some (a * b)

@[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

theorem WithTop.mul_lt_top' {α : Type u_1} [DecidableEq α] [MulZeroClass α] [LT α] {a : WithTop α} {b : WithTop α} (ha : a < ⊤) (hb : b < ⊤) : a * b < ⊤

theorem WithTop.mul_lt_top {α : Type u_1} [DecidableEq α] [MulZeroClass α] [LT α] {a : WithTop α} {b : WithTop α} (ha : a ≠ ⊤) (hb : b ≠ ⊤) : a * b < ⊤

instance WithTop.instNoZeroDivisors {α : Type u_1} [DecidableEq α] [MulZeroClass α] [NoZeroDivisors α] : NoZeroDivisors (WithTop α)

Equations

• (_ : NoZeroDivisors (WithTop α)) = (_ : NoZeroDivisors (WithTop α))

instance WithTop.instMulZeroOneClass {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

• WithTop.instMulZeroOneClass = let src := WithTop.instMulZeroClass; MulZeroOneClass.mk (_ : ∀ (a : WithTop α), 0 * a = 0) (_ : ∀ (a : WithTop α), a * 0 = 0)

@[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

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.

instance WithTop.instSemigroupWithZero {α : Type u_1} [DecidableEq α] [SemigroupWithZero α] [NoZeroDivisors α] : SemigroupWithZero (WithTop α)

Equations

• WithTop.instSemigroupWithZero = let src := WithTop.instMulZeroClass; SemigroupWithZero.mk (_ : ∀ (a : WithTop α), 0 * a = 0) (_ : ∀ (a : WithTop α), a * 0 = 0)

instance WithTop.instMonoidWithZero {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.

@[simp]

theorem WithTop.coe_pow {α : Type u_1} [DecidableEq α] [MonoidWithZero α] [NoZeroDivisors α] [Nontrivial α] (a : α) (n : ℕ) : ↑(a ^ n) = ↑a ^ n

instance WithTop.instCommMonoidWithZero {α : Type u_1} [DecidableEq α] [CommMonoidWithZero α] [NoZeroDivisors α] [Nontrivial α] : CommMonoidWithZero (WithTop α)

Equations

• WithTop.instCommMonoidWithZero = let src := WithTop.instMonoidWithZero; CommMonoidWithZero.mk (_ : ∀ (a : WithTop α), 0 * a = 0) (_ : ∀ (a : WithTop α), a * 0 = 0)

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 CanonicallyOrderedAddCommMonoid, both of which are required for distributivity.

Equations

• WithTop.commSemiring = let src := WithTop.addCommMonoidWithOne; let src_1 := WithTop.instCommMonoidWithZero; CommSemiring.mk (_ : ∀ (a b : WithTop α), a * b = b * a)

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.

@[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

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.

instance WithBot.instDecidableEqWithBot {α : Type u_1} [DecidableEq α] : DecidableEq (WithBot α)

Equations

• WithBot.instDecidableEqWithBot = instDecidableEqOption

instance WithBot.instMulZeroClassWithBot {α : Type u_1} [DecidableEq α] [MulZeroClass α] : MulZeroClass (WithBot α)

Equations

• WithBot.instMulZeroClassWithBot = WithTop.instMulZeroClass

@[simp]

theorem WithBot.coe_mul {α : Type u_1} [DecidableEq α] [MulZeroClass α] (a : α) (b : α) : ↑(a * b) = ↑a * ↑b

theorem WithBot.mul_bot' {α : Type u_1} [DecidableEq α] [MulZeroClass α] (a : WithBot α) : a * ⊥ = if a = 0 then 0 else ⊥

@[simp]

theorem WithBot.mul_bot {α : Type u_1} [DecidableEq α] [MulZeroClass α] {a : WithBot α} (h : a ≠ 0) : a * ⊥ = ⊥

theorem WithBot.bot_mul' {α : Type u_1} [DecidableEq α] [MulZeroClass α] (b : WithBot α) : ⊥ * b = if b = 0 then 0 else ⊥

@[simp]

theorem WithBot.bot_mul {α : Type u_1} [DecidableEq α] [MulZeroClass α] {b : WithBot α} (hb : b ≠ 0) : ⊥ * b = ⊥

@[simp]

theorem WithBot.bot_mul_bot {α : Type u_1} [DecidableEq α] [MulZeroClass α] : ⊥ * ⊥ = ⊥

theorem WithBot.mul_def {α : Type u_1} [DecidableEq α] [MulZeroClass α] (a : WithBot α) (b : WithBot α) : a * b = if a = 0 ∨ b = 0 then 0 else WithBot.map₂ (fun (x x_1 : α) => x * x_1) a b

theorem WithBot.mul_eq_bot_iff {α : Type u_1} [DecidableEq α] [MulZeroClass α] {a : WithBot α} {b : WithBot α} : a * b = ⊥ ↔ a ≠ 0 ∧ b = ⊥ ∨ a = ⊥ ∧ b ≠ 0

theorem WithBot.mul_coe_eq_bind {α : Type u_1} [DecidableEq α] [MulZeroClass α] {b : α} (hb : b ≠ 0) (a : WithBot α) : a * ↑b = Option.bind a fun (a : α) => some (a * b)

theorem WithBot.coe_mul_eq_bind {α : Type u_1} [DecidableEq α] [MulZeroClass α] {a : α} (ha : a ≠ 0) (b : WithBot α) : ↑a * b = Option.bind b fun (b : α) => some (a * b)

@[simp]

theorem WithBot.unbot'_zero_mul {α : Type u_1} [DecidableEq α] [MulZeroClass α] (a : WithBot α) (b : WithBot α) : WithBot.unbot' 0 (a * b) = WithBot.unbot' 0 a * WithBot.unbot' 0 b

theorem WithBot.bot_lt_mul' {α : Type u_1} [DecidableEq α] [MulZeroClass α] [LT α] {a : WithBot α} {b : WithBot α} (ha : ⊥ < a) (hb : ⊥ < b) : ⊥ < a * b

theorem WithBot.bot_lt_mul {α : Type u_1} [DecidableEq α] [MulZeroClass α] [LT α] {a : WithBot α} {b : WithBot α} (ha : a ≠ ⊥) (hb : b ≠ ⊥) : ⊥ < a * b

instance WithBot.instNoZeroDivisors {α : Type u_1} [DecidableEq α] [MulZeroClass α] [NoZeroDivisors α] : NoZeroDivisors (WithBot α)

Equations

• (_ : NoZeroDivisors (WithBot α)) = (_ : NoZeroDivisors (WithTop α))

instance WithBot.instMulZeroOneClass {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.instMulZeroOneClass = WithTop.instMulZeroOneClass

instance WithBot.instSemigroupWithZero {α : Type u_1} [DecidableEq α] [SemigroupWithZero α] [NoZeroDivisors α] : SemigroupWithZero (WithBot α)

Equations

• WithBot.instSemigroupWithZero = WithTop.instSemigroupWithZero

instance WithBot.instMonoidWithZero {α : Type u_1} [DecidableEq α] [MonoidWithZero α] [NoZeroDivisors α] [Nontrivial α] : MonoidWithZero (WithBot α)

Equations

• WithBot.instMonoidWithZero = WithTop.instMonoidWithZero

@[simp]

theorem WithBot.coe_pow {α : Type u_1} [DecidableEq α] [MonoidWithZero α] [NoZeroDivisors α] [Nontrivial α] (a : α) (n : ℕ) : ↑(a ^ n) = ↑a ^ n

instance WithBot.commMonoidWithZero {α : Type u_1} [DecidableEq α] [CommMonoidWithZero α] [NoZeroDivisors α] [Nontrivial α] : CommMonoidWithZero (WithBot α)

Equations

• WithBot.commMonoidWithZero = WithTop.instCommMonoidWithZero

instance WithBot.commSemiring {α : Type u_1} [DecidableEq α] [CanonicallyOrderedCommSemiring α] [Nontrivial α] : CommSemiring (WithBot α)

Equations

• WithBot.commSemiring = WithTop.commSemiring

instance WithBot.instPosMulMonoWithBotToMulInstMulZeroClassWithBotZeroToZeroPreorder {α : Type u_1} [DecidableEq α] [MulZeroClass α] [Preorder α] [PosMulMono α] : PosMulMono (WithBot α)

Equations

• One or more equations did not get rendered due to their size.

instance WithBot.instMulPosMonoWithBotToMulInstMulZeroClassWithBotZeroToZeroPreorder {α : Type u_1} [DecidableEq α] [MulZeroClass α] [Preorder α] [MulPosMono α] : MulPosMono (WithBot α)

Equations

• One or more equations did not get rendered due to their size.

instance WithBot.instPosMulStrictMonoWithBotToMulInstMulZeroClassWithBotZeroToZeroPreorder {α : Type u_1} [DecidableEq α] [MulZeroClass α] [Preorder α] [PosMulStrictMono α] : PosMulStrictMono (WithBot α)

Equations

• One or more equations did not get rendered due to their size.

instance WithBot.instMulPosStrictMonoWithBotToMulInstMulZeroClassWithBotZeroToZeroPreorder {α : Type u_1} [DecidableEq α] [MulZeroClass α] [Preorder α] [MulPosStrictMono α] : MulPosStrictMono (WithBot α)

Equations

• One or more equations did not get rendered due to their size.

instance WithBot.instPosMulReflectLTWithBotToMulInstMulZeroClassWithBotZeroToZeroPreorder {α : Type u_1} [DecidableEq α] [MulZeroClass α] [Preorder α] [PosMulReflectLT α] : PosMulReflectLT (WithBot α)

Equations

• One or more equations did not get rendered due to their size.

instance WithBot.instMulPosReflectLTWithBotToMulInstMulZeroClassWithBotZeroToZeroPreorder {α : Type u_1} [DecidableEq α] [MulZeroClass α] [Preorder α] [MulPosReflectLT α] : MulPosReflectLT (WithBot α)

Equations

• One or more equations did not get rendered due to their size.

instance WithBot.instPosMulReflectLEWithBotToMulInstMulZeroClassWithBotZeroToZeroPreorder {α : Type u_1} [DecidableEq α] [MulZeroClass α] [Preorder α] [PosMulReflectLE α] : PosMulReflectLE (WithBot α)

Equations

• One or more equations did not get rendered due to their size.

instance WithBot.instMulPosReflectLEWithBotToMulInstMulZeroClassWithBotZeroToZeroPreorder {α : Type u_1} [DecidableEq α] [MulZeroClass α] [Preorder α] [MulPosReflectLE α] : MulPosReflectLE (WithBot α)

Equations

• One or more equations did not get rendered due to their size.

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.