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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.

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

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

Equations
@[simp]
theorem MonoidWithZeroHom.withTopMap_apply {R : Type u_1} {S : Type u_2} [inst : MulZeroOneClass R] [inst : DecidableEq R] [inst : Nontrivial R] [inst : MulZeroOneClass S] [inst : DecidableEq S] [inst : Nontrivial S] (f : R →*₀ S) (hf : Function.Injective f) :
def MonoidWithZeroHom.withTopMap {R : Type u_1} {S : Type u_2} [inst : MulZeroOneClass R] [inst : DecidableEq R] [inst : Nontrivial R] [inst : MulZeroOneClass S] [inst : DecidableEq S] [inst : Nontrivial S] (f : R →*₀ S) (hf : Function.Injective f) :

A version of WithTop.map for MonoidWithZeroHoms.

Equations
  • One or more equations did not get rendered due to their size.
Equations
instance WithTop.monoidWithZero {α : Type u_1} [inst : DecidableEq α] [inst : MonoidWithZero α] [inst : NoZeroDivisors α] [inst : Nontrivial α] :
Equations
  • One or more equations did not get rendered due to their size.
instance WithTop.commMonoidWithZero {α : Type u_1} [inst : DecidableEq α] [inst : CommMonoidWithZero α] [inst : NoZeroDivisors α] [inst : Nontrivial α] :
Equations
instance WithTop.commSemiring {α : Type u_1} [inst : DecidableEq α] [inst : CanonicallyOrderedCommSemiring α] [inst : Nontrivial 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)
Equations
  • One or more equations did not get rendered due to their size.
@[simp]
theorem RingHom.withTopMap_apply {R : Type u_1} {S : Type u_2} [inst : CanonicallyOrderedCommSemiring R] [inst : DecidableEq R] [inst : Nontrivial R] [inst : CanonicallyOrderedCommSemiring S] [inst : DecidableEq S] [inst : Nontrivial S] (f : R →+* S) (hf : Function.Injective f) :
def RingHom.withTopMap {R : Type u_1} {S : Type u_2} [inst : CanonicallyOrderedCommSemiring R] [inst : DecidableEq R] [inst : Nontrivial R] [inst : CanonicallyOrderedCommSemiring S] [inst : DecidableEq S] [inst : Nontrivial S] (f : R →+* S) (hf : Function.Injective f) :

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} [inst : DecidableEq α] :
Equations
  • WithBot.instDecidableEqWithBot = instDecidableEqOption
instance WithBot.instMulZeroClassWithBot {α : Type u_1} [inst : DecidableEq α] [inst : Zero α] [inst : Mul α] :
Equations
  • WithBot.instMulZeroClassWithBot = WithTop.instMulZeroClassWithTop
theorem WithBot.mul_def {α : Type u_1} [inst : DecidableEq α] [inst : Zero α] [inst : 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
@[simp]
theorem WithBot.mul_bot {α : Type u_1} [inst : DecidableEq α] [inst : Zero α] [inst : Mul α] {a : WithBot α} (h : a 0) :
@[simp]
theorem WithBot.bot_mul {α : Type u_1} [inst : DecidableEq α] [inst : Zero α] [inst : Mul α] {a : WithBot α} (h : a 0) :
@[simp]
theorem WithBot.bot_mul_bot {α : Type u_1} [inst : DecidableEq α] [inst : Zero α] [inst : Mul α] :
theorem WithBot.mul_eq_bot_iff {α : Type u_1} [inst : DecidableEq α] [inst : Zero α] [inst : Mul α] {a : WithBot α} {b : WithBot α} :
a * b = a 0 b = a = b 0
theorem WithBot.bot_lt_mul' {α : Type u_1} [inst : DecidableEq α] [inst : Zero α] [inst : Mul α] [inst : LT α] {a : WithBot α} {b : WithBot α} (ha : < a) (hb : < b) :
< a * b
theorem WithBot.bot_lt_mul {α : Type u_1} [inst : DecidableEq α] [inst : Zero α] [inst : Mul α] [inst : LT α] {a : WithBot α} {b : WithBot α} (ha : a ) (hb : b ) :
< a * b
@[simp]
theorem WithBot.coe_mul {α : Type u_1} [inst : DecidableEq α] [inst : MulZeroClass α] {a : α} {b : α} :
↑(a * b) = a * b
theorem WithBot.mul_coe {α : Type u_1} [inst : DecidableEq α] [inst : MulZeroClass α] {b : α} (hb : b 0) {a : WithBot α} :
a * b = Option.bind a fun a => some (a * b)
instance WithBot.instMulZeroOneClassWithBot {α : Type u_1} [inst : DecidableEq α] [inst : MulZeroOneClass α] [inst : Nontrivial α] :

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

Equations
  • WithBot.instMulZeroOneClassWithBot = WithTop.instMulZeroOneClassWithTop
Equations
  • WithBot.instSemigroupWithZeroWithBot = WithTop.instSemigroupWithZeroWithTop
instance WithBot.instMonoidWithZeroWithBot {α : Type u_1} [inst : DecidableEq α] [inst : MonoidWithZero α] [inst : NoZeroDivisors α] [inst : Nontrivial α] :
Equations
  • WithBot.instMonoidWithZeroWithBot = WithTop.monoidWithZero
instance WithBot.commMonoidWithZero {α : Type u_1} [inst : DecidableEq α] [inst : CommMonoidWithZero α] [inst : NoZeroDivisors α] [inst : Nontrivial α] :
Equations
  • WithBot.commMonoidWithZero = WithTop.commMonoidWithZero
instance WithBot.commSemiring {α : Type u_1} [inst : DecidableEq α] [inst : CanonicallyOrderedCommSemiring α] [inst : Nontrivial α] :
Equations
  • WithBot.commSemiring = WithTop.commSemiring
Equations
  • One or more equations did not get rendered due to their size.