Documentation

Mathlib.Algebra.Order.Pi

Pi instances for ordered groups and monoids #

This file defines instances for ordered group, monoid, and related structures on Pi types.

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instance Pi.orderedCommMonoid {ι : Type u_6} {Z : ι → Type u_7} [(i : ι) → OrderedCommMonoid (Z i)] :
OrderedCommMonoid ((i : ι) → Z i)

The product of a family of ordered commutative monoids is an ordered commutative monoid.

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instance Pi.orderedAddCommMonoid {ι : Type u_6} {Z : ι → Type u_7} [(i : ι) → OrderedAddCommMonoid (Z i)] :
OrderedAddCommMonoid ((i : ι) → Z i)

The product of a family of ordered additive commutative monoids is an ordered additive commutative monoid.

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instance Pi.existsMulOfLe {ι : Type u_6} {α : ι → Type u_7} [(i : ι) → LE (α i)] [(i : ι) → Mul (α i)] [∀ (i : ι), ExistsMulOfLE (α i)] :
ExistsMulOfLE ((i : ι) → α i)
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  • ⋯ = ⋯
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instance Pi.existsAddOfLe {ι : Type u_6} {α : ι → Type u_7} [(i : ι) → LE (α i)] [(i : ι) → Add (α i)] [∀ (i : ι), ExistsAddOfLE (α i)] :
ExistsAddOfLE ((i : ι) → α i)
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  • ⋯ = ⋯
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instance Pi.instCanonicallyOrderedCommMonoidForall {ι : Type u_6} {Z : ι → Type u_7} [(i : ι) → CanonicallyOrderedCommMonoid (Z i)] :
CanonicallyOrderedCommMonoid ((i : ι) → Z i)

The product of a family of canonically ordered monoids is a canonically ordered monoid.

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instance Pi.instCanonicallyOrderedAddCommMonoidForall {ι : Type u_6} {Z : ι → Type u_7} [(i : ι) → CanonicallyOrderedAddCommMonoid (Z i)] :
CanonicallyOrderedAddCommMonoid ((i : ι) → Z i)

The product of a family of canonically ordered additive monoids is a canonically ordered additive monoid.

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instance Pi.orderedCancelCommMonoid {I : Type u_1} {f : I → Type u_5} [(i : I) → OrderedCancelCommMonoid (f i)] :
OrderedCancelCommMonoid ((i : I) → f i)
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instance Pi.orderedAddCancelCommMonoid {I : Type u_1} {f : I → Type u_5} [(i : I) → OrderedCancelAddCommMonoid (f i)] :
OrderedCancelAddCommMonoid ((i : I) → f i)
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instance Pi.orderedCommGroup {I : Type u_1} {f : I → Type u_5} [(i : I) → OrderedCommGroup (f i)] :
OrderedCommGroup ((i : I) → f i)
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instance Pi.orderedAddCommGroup {I : Type u_1} {f : I → Type u_5} [(i : I) → OrderedAddCommGroup (f i)] :
OrderedAddCommGroup ((i : I) → f i)
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instance Pi.orderedSemiring {I : Type u_1} {f : I → Type u_5} [(i : I) → OrderedSemiring (f i)] :
OrderedSemiring ((i : I) → f i)
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instance Pi.orderedCommSemiring {I : Type u_1} {f : I → Type u_5} [(i : I) → OrderedCommSemiring (f i)] :
OrderedCommSemiring ((i : I) → f i)
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instance Pi.orderedRing {I : Type u_1} {f : I → Type u_5} [(i : I) → OrderedRing (f i)] :
OrderedRing ((i : I) → f i)
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instance Pi.orderedCommRing {I : Type u_1} {f : I → Type u_5} [(i : I) → OrderedCommRing (f i)] :
OrderedCommRing ((i : I) → f i)
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theorem Function.one_le_const_of_one_le {α : Type u_2} (β : Type u_3) [One α] [Preorder α] {a : α} (ha : 1 ≤ a) :
1 ≤ Function.const β a
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theorem Function.const_nonneg_of_nonneg {α : Type u_2} (β : Type u_3) [Zero α] [Preorder α] {a : α} (ha : 0 ≤ a) :
0 ≤ Function.const β a
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theorem Function.const_le_one_of_le_one {α : Type u_2} (β : Type u_3) [One α] [Preorder α] {a : α} (ha : a ≤ 1) :
Function.const β a ≤ 1
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theorem Function.const_nonpos_of_nonpos {α : Type u_2} (β : Type u_3) [Zero α] [Preorder α] {a : α} (ha : a ≤ 0) :
Function.const β a ≤ 0
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@[simp]
theorem Function.one_le_const {α : Type u_2} {β : Type u_3} [One α] [Preorder α] {a : α} [Nonempty β] :
1 ≤ Function.const β a ↔ 1 ≤ a
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@[simp]
theorem Function.const_nonneg {α : Type u_2} {β : Type u_3} [Zero α] [Preorder α] {a : α} [Nonempty β] :
0 ≤ Function.const β a ↔ 0 ≤ a
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@[simp]
theorem Function.one_lt_const {α : Type u_2} {β : Type u_3} [One α] [Preorder α] {a : α} [Nonempty β] :
1 < Function.const β a ↔ 1 < a
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@[simp]
theorem Function.const_pos {α : Type u_2} {β : Type u_3} [Zero α] [Preorder α] {a : α} [Nonempty β] :
0 < Function.const β a ↔ 0 < a
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@[simp]
theorem Function.const_le_one {α : Type u_2} {β : Type u_3} [One α] [Preorder α] {a : α} [Nonempty β] :
Function.const β a ≤ 1 ↔ a ≤ 1
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@[simp]
theorem Function.const_nonpos {α : Type u_2} {β : Type u_3} [Zero α] [Preorder α] {a : α} [Nonempty β] :
Function.const β a ≤ 0 ↔ a ≤ 0
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@[simp]
theorem Function.const_lt_one {α : Type u_2} {β : Type u_3} [One α] [Preorder α] {a : α} [Nonempty β] :
Function.const β a < 1 ↔ a < 1
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@[simp]
theorem Function.const_neg' {α : Type u_2} {β : Type u_3} [Zero α] [Preorder α] {a : α} [Nonempty β] :
Function.const β a < 0 ↔ a < 0
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theorem Function.one_le_extend {α : Type u_2} {β : Type u_3} {γ : Type u_4} [One γ] [LE γ] {f : α → β} {g : α → γ} {e : β → γ} (hg : 1 ≤ g) (he : 1 ≤ e) :
1 ≤ Function.extend f g e
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theorem Function.extend_nonneg {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Zero γ] [LE γ] {f : α → β} {g : α → γ} {e : β → γ} (hg : 0 ≤ g) (he : 0 ≤ e) :
0 ≤ Function.extend f g e
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theorem Function.extend_le_one {α : Type u_2} {β : Type u_3} {γ : Type u_4} [One γ] [LE γ] {f : α → β} {g : α → γ} {e : β → γ} (hg : g ≤ 1) (he : e ≤ 1) :
Function.extend f g e ≤ 1
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theorem Function.extend_nonpos {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Zero γ] [LE γ] {f : α → β} {g : α → γ} {e : β → γ} (hg : g ≤ 0) (he : e ≤ 0) :
Function.extend f g e ≤ 0
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@[simp]
theorem Pi.mulSingle_le_mulSingle {ι : Type u_6} {α : ι → Type u_7} [DecidableEq ι] [(i : ι) → One (α i)] [(i : ι) → Preorder (α i)] {i : ι} {a b : α i} :
Pi.mulSingle i a ≤ Pi.mulSingle i b ↔ a ≤ b
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@[simp]
theorem Pi.single_le_single {ι : Type u_6} {α : ι → Type u_7} [DecidableEq ι] [(i : ι) → Zero (α i)] [(i : ι) → Preorder (α i)] {i : ι} {a b : α i} :
Pi.single i a ≤ Pi.single i b ↔ a ≤ b
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theorem Pi.GCongr.mulSingle_mono {ι : Type u_6} {α : ι → Type u_7} [DecidableEq ι] [(i : ι) → One (α i)] [(i : ι) → Preorder (α i)] {i : ι} {a b : α i} :
a ≤ b → Pi.mulSingle i a ≤ Pi.mulSingle i b

Alias of the reverse direction of Pi.mulSingle_le_mulSingle.

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theorem Pi.GCongr.single_mono {ι : Type u_6} {α : ι → Type u_7} [DecidableEq ι] [(i : ι) → Zero (α i)] [(i : ι) → Preorder (α i)] {i : ι} {a b : α i} :
a ≤ b → Pi.single i a ≤ Pi.single i b
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@[simp]
theorem Pi.one_le_mulSingle {ι : Type u_6} {α : ι → Type u_7} [DecidableEq ι] [(i : ι) → One (α i)] [(i : ι) → Preorder (α i)] {i : ι} {a : α i} :
1 ≤ Pi.mulSingle i a ↔ 1 ≤ a
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@[simp]
theorem Pi.single_nonneg {ι : Type u_6} {α : ι → Type u_7} [DecidableEq ι] [(i : ι) → Zero (α i)] [(i : ι) → Preorder (α i)] {i : ι} {a : α i} :
0 ≤ Pi.single i a ↔ 0 ≤ a
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@[simp]
theorem Pi.mulSingle_le_one {ι : Type u_6} {α : ι → Type u_7} [DecidableEq ι] [(i : ι) → One (α i)] [(i : ι) → Preorder (α i)] {i : ι} {a : α i} :
Pi.mulSingle i a ≤ 1 ↔ a ≤ 1
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@[simp]
theorem Pi.single_nonpos {ι : Type u_6} {α : ι → Type u_7} [DecidableEq ι] [(i : ι) → Zero (α i)] [(i : ι) → Preorder (α i)] {i : ι} {a : α i} :
Pi.single i a ≤ 0 ↔ a ≤ 0