Documentation

Mathlib.Algebra.Order.Monoid.Units

Units in ordered monoids #

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@[instance_reducible]
instance Units.instPreorder {α : Type u_1} [Monoid α] [Preorder α] :
Preorder αˣ
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@[instance_reducible]
instance AddUnits.instPreorder {α : Type u_1} [AddMonoid α] [Preorder α] :
Preorder (AddUnits α)
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@[simp]
theorem Units.val_le_val {α : Type u_1} [Monoid α] [Preorder α] {a b : αˣ} :
↑a ≤ ↑b ↔ a ≤ b
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@[simp]
theorem AddUnits.val_le_val {α : Type u_1} [AddMonoid α] [Preorder α] {a b : AddUnits α} :
↑a ≤ ↑b ↔ a ≤ b
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@[simp]
theorem Units.val_lt_val {α : Type u_1} [Monoid α] [Preorder α] {a b : αˣ} :
↑a < ↑b ↔ a < b
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@[simp]
theorem AddUnits.val_lt_val {α : Type u_1} [AddMonoid α] [Preorder α] {a b : AddUnits α} :
↑a < ↑b ↔ a < b
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@[instance_reducible]
instance Units.instPartialOrderUnits {α : Type u_1} [Monoid α] [PartialOrder α] :
PartialOrder αˣ
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@[instance_reducible]
instance AddUnits.instPartialOrderAddUnits {α : Type u_1} [AddMonoid α] [PartialOrder α] :
PartialOrder (AddUnits α)
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@[instance_reducible]
instance Units.instMaxOfLinearOrder {α : Type u_1} [Monoid α] [LinearOrder α] :
Max αˣ
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@[instance_reducible]
instance AddUnits.instMaxOfLinearOrder {α : Type u_1} [AddMonoid α] [LinearOrder α] :
Max (AddUnits α)
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@[instance_reducible]
instance Units.instMinOfLinearOrder {α : Type u_1} [Monoid α] [LinearOrder α] :
Min αˣ
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@[instance_reducible]
instance AddUnits.instMinOfLinearOrder {α : Type u_1} [AddMonoid α] [LinearOrder α] :
Min (AddUnits α)
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@[simp]
theorem Units.max_val {α : Type u_1} [Monoid α] [LinearOrder α] (a b : αˣ) :
↑(max a b) = max ↑a ↑b
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@[simp]
theorem AddUnits.max_val {α : Type u_1} [AddMonoid α] [LinearOrder α] (a b : AddUnits α) :
↑(max a b) = max ↑a ↑b
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@[simp]
theorem Units.min_val {α : Type u_1} [Monoid α] [LinearOrder α] (a b : αˣ) :
↑(min a b) = min ↑a ↑b
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@[simp]
theorem AddUnits.min_val {α : Type u_1} [AddMonoid α] [LinearOrder α] (a b : AddUnits α) :
↑(min a b) = min ↑a ↑b
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@[instance_reducible]
instance Units.instOrd {α : Type u_1} [Monoid α] [Ord α] :
Ord αˣ
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@[instance_reducible]
instance AddUnits.instOrd {α : Type u_1} [AddMonoid α] [Ord α] :
Ord (AddUnits α)
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theorem Units.compare_val {α : Type u_1} [Monoid α] [Ord α] (a b : αˣ) :
compare ↑a ↑b = compare a b
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theorem AddUnits.compare_val {α : Type u_1} [AddMonoid α] [Ord α] (a b : AddUnits α) :
compare ↑a ↑b = compare a b
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@[instance_reducible]
instance Units.instLinearOrder {α : Type u_1} [Monoid α] [LinearOrder α] :
LinearOrder αˣ
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@[instance_reducible]
instance AddUnits.instLinearOrder {α : Type u_1} [AddMonoid α] [LinearOrder α] :
LinearOrder (AddUnits α)
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def Units.orderEmbeddingVal {α : Type u_1} [Monoid α] [LinearOrder α] :
αˣ ↪o α

val : αˣ → α as an order embedding.

Instances For
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    def AddUnits.orderEmbeddingVal {α : Type u_1} [AddMonoid α] [LinearOrder α] :
    AddUnits α ↪o α

    val : add_units α → α as an order embedding.

    Instances For
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      @[simp]
      theorem AddUnits.orderEmbeddingVal_apply {α : Type u_1} [AddMonoid α] [LinearOrder α] :
      ⇑orderEmbeddingVal = val
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      @[simp]
      theorem Units.orderEmbeddingVal_apply {α : Type u_1} [Monoid α] [LinearOrder α] :
      ⇑orderEmbeddingVal = val