Documentation

Mathlib.Algebra.Order.Group.Synonym

Group structure on the order type synonyms #

Transfer algebraic instances from α to αᵒᵈ and Lex α.

OrderDual #

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instance instOneOrderDual {α : Type u_1} [h : One α] :
One αᵒᵈ
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instance instZeroOrderDual {α : Type u_1} [h : Zero α] :
Zero αᵒᵈ
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instance instMulOrderDual {α : Type u_1} [h : Mul α] :
Mul αᵒᵈ
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instance instAddOrderDual {α : Type u_1} [h : Add α] :
Add αᵒᵈ
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instance instInvOrderDual {α : Type u_1} [h : Inv α] :
Inv αᵒᵈ
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instance instNegOrderDual {α : Type u_1} [h : Neg α] :
Neg αᵒᵈ
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instance instDivOrderDual {α : Type u_1} [h : Div α] :
Div αᵒᵈ
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instance instSubOrderDual {α : Type u_1} [h : Sub α] :
Sub αᵒᵈ
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instance OrderDual.instPow {α : Type u_1} {β : Type u_2} [h : Pow α β] :
Pow αᵒᵈ β
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instance OrderDual.instSMul {β : Type u_2} {α : Type u_1} [h : SMul β α] :
SMul β αᵒᵈ
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instance OrderDual.instVAdd {β : Type u_2} {α : Type u_1} [h : VAdd β α] :
VAdd β αᵒᵈ
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instance OrderDual.instPow' {α : Type u_1} {β : Type u_2} [h : Pow α β] :
Pow α βᵒᵈ
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instance OrderDual.instVAdd' {β : Type u_2} {α : Type u_1} [h : VAdd β α] :
VAdd βᵒᵈ α
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instance OrderDual.instSMul' {β : Type u_2} {α : Type u_1} [h : SMul β α] :
SMul βᵒᵈ α
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instance instSemigroupOrderDual {α : Type u_1} [h : Semigroup α] :
Semigroup αᵒᵈ
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instance instAddSemigroupOrderDual {α : Type u_1} [h : AddSemigroup α] :
AddSemigroup αᵒᵈ
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instance instCommSemigroupOrderDual {α : Type u_1} [h : CommSemigroup α] :
CommSemigroup αᵒᵈ
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instance instAddCommSemigroupOrderDual {α : Type u_1} [h : AddCommSemigroup α] :
AddCommSemigroup αᵒᵈ
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instance instIsLeftCancelMulOrderDual {α : Type u_1} [Mul α] [h : IsLeftCancelMul α] :
IsLeftCancelMul αᵒᵈ
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instance instIsLeftCancelAddOrderDual {α : Type u_1} [Add α] [h : IsLeftCancelAdd α] :
IsLeftCancelAdd αᵒᵈ
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instance instIsRightCancelMulOrderDual {α : Type u_1} [Mul α] [h : IsRightCancelMul α] :
IsRightCancelMul αᵒᵈ
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instance instIsRightCancelAddOrderDual {α : Type u_1} [Add α] [h : IsRightCancelAdd α] :
IsRightCancelAdd αᵒᵈ
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instance instIsCancelMulOrderDual {α : Type u_1} [Mul α] [h : IsCancelMul α] :
IsCancelMul αᵒᵈ
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instance instIsCancelAddOrderDual {α : Type u_1} [Add α] [h : IsCancelAdd α] :
IsCancelAdd αᵒᵈ
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instance instLeftCancelSemigroupOrderDual {α : Type u_1} [h : LeftCancelSemigroup α] :
LeftCancelSemigroup αᵒᵈ
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instance instAddLeftCancelSemigroupOrderDual {α : Type u_1} [h : AddLeftCancelSemigroup α] :
AddLeftCancelSemigroup αᵒᵈ
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instance instRightCancelSemigroupOrderDual {α : Type u_1} [h : RightCancelSemigroup α] :
RightCancelSemigroup αᵒᵈ
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instance instAddRightCancelSemigroupOrderDual {α : Type u_1} [h : AddRightCancelSemigroup α] :
AddRightCancelSemigroup αᵒᵈ
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instance instMulOneClassOrderDual {α : Type u_1} [h : MulOneClass α] :
MulOneClass αᵒᵈ
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instance instAddZeroClassOrderDual {α : Type u_1} [h : AddZeroClass α] :
AddZeroClass αᵒᵈ
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instance instMonoidOrderDual {α : Type u_1} [h : Monoid α] :
Monoid αᵒᵈ
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instance instAddMonoidOrderDual {α : Type u_1} [h : AddMonoid α] :
AddMonoid αᵒᵈ
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instance OrderDual.instCommMonoid {α : Type u_1} [h : CommMonoid α] :
CommMonoid αᵒᵈ
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instance OrderDual.instAddCommMonoid {α : Type u_1} [h : AddCommMonoid α] :
AddCommMonoid αᵒᵈ
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instance instLeftCancelMonoidOrderDual {α : Type u_1} [h : LeftCancelMonoid α] :
LeftCancelMonoid αᵒᵈ
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instance instAddLeftCancelMonoidOrderDual {α : Type u_1} [h : AddLeftCancelMonoid α] :
AddLeftCancelMonoid αᵒᵈ
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instance instRightCancelMonoidOrderDual {α : Type u_1} [h : RightCancelMonoid α] :
RightCancelMonoid αᵒᵈ
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instance instAddRightCancelMonoidOrderDual {α : Type u_1} [h : AddRightCancelMonoid α] :
AddRightCancelMonoid αᵒᵈ
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instance instCancelMonoidOrderDual {α : Type u_1} [h : CancelMonoid α] :
CancelMonoid αᵒᵈ
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instance instAddCancelMonoidOrderDual {α : Type u_1} [h : AddCancelMonoid α] :
AddCancelMonoid αᵒᵈ
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instance OrderDual.instCancelCommMonoid {α : Type u_1} [h : CancelCommMonoid α] :
CancelCommMonoid αᵒᵈ
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instance OrderDual.instAddCancelCommMonoid {α : Type u_1} [h : AddCancelCommMonoid α] :
AddCancelCommMonoid αᵒᵈ
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instance instInvolutiveInvOrderDual {α : Type u_1} [h : InvolutiveInv α] :
InvolutiveInv αᵒᵈ
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instance instInvolutiveNegOrderDual {α : Type u_1} [h : InvolutiveNeg α] :
InvolutiveNeg αᵒᵈ
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instance instDivInvMonoidOrderDual {α : Type u_1} [h : DivInvMonoid α] :
DivInvMonoid αᵒᵈ
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instance instSubNegAddMonoidOrderDual {α : Type u_1} [h : SubNegMonoid α] :
SubNegMonoid αᵒᵈ
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instance instDivisionMonoidOrderDual {α : Type u_1} [h : DivisionMonoid α] :
DivisionMonoid αᵒᵈ
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instance instSubtractionMonoidOrderDual {α : Type u_1} [h : SubtractionMonoid α] :
SubtractionMonoid αᵒᵈ
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instance instDivisionCommMonoidOrderDual {α : Type u_1} [h : DivisionCommMonoid α] :
DivisionCommMonoid αᵒᵈ
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instance instDivisionAddCommMonoidOrderDual {α : Type u_1} [h : SubtractionCommMonoid α] :
SubtractionCommMonoid αᵒᵈ
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instance OrderDual.instGroup {α : Type u_1} [h : Group α] :
Group αᵒᵈ
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instance OrderDual.instAddGroup {α : Type u_1} [h : AddGroup α] :
AddGroup αᵒᵈ
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instance instCommGroupOrderDual {α : Type u_1} [h : CommGroup α] :
CommGroup αᵒᵈ
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instance instAddCommGroupOrderDual {α : Type u_1} [h : AddCommGroup α] :
AddCommGroup αᵒᵈ
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@[simp]
theorem toDual_one {α : Type u_1} [One α] :
OrderDual.toDual 1 = 1
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@[simp]
theorem toDual_zero {α : Type u_1} [Zero α] :
OrderDual.toDual 0 = 0
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@[simp]
theorem ofDual_one {α : Type u_1} [One α] :
OrderDual.ofDual 1 = 1
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@[simp]
theorem ofDual_zero {α : Type u_1} [Zero α] :
OrderDual.ofDual 0 = 0
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@[simp]
theorem toDual_mul {α : Type u_1} [Mul α] (a b : α) :
OrderDual.toDual (a * b) = OrderDual.toDual a * OrderDual.toDual b
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@[simp]
theorem toDual_add {α : Type u_1} [Add α] (a b : α) :
OrderDual.toDual (a + b) = OrderDual.toDual a + OrderDual.toDual b
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@[simp]
theorem ofDual_mul {α : Type u_1} [Mul α] (a b : αᵒᵈ) :
OrderDual.ofDual (a * b) = OrderDual.ofDual a * OrderDual.ofDual b
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@[simp]
theorem ofDual_add {α : Type u_1} [Add α] (a b : αᵒᵈ) :
OrderDual.ofDual (a + b) = OrderDual.ofDual a + OrderDual.ofDual b
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@[simp]
theorem toDual_inv {α : Type u_1} [Inv α] (a : α) :
OrderDual.toDual a⁻¹ = (OrderDual.toDual a)⁻¹
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@[simp]
theorem toDual_neg {α : Type u_1} [Neg α] (a : α) :
OrderDual.toDual (-a) = -OrderDual.toDual a
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@[simp]
theorem ofDual_inv {α : Type u_1} [Inv α] (a : αᵒᵈ) :
OrderDual.ofDual a⁻¹ = (OrderDual.ofDual a)⁻¹
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@[simp]
theorem ofDual_neg {α : Type u_1} [Neg α] (a : αᵒᵈ) :
OrderDual.ofDual (-a) = -OrderDual.ofDual a
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@[simp]
theorem toDual_div {α : Type u_1} [Div α] (a b : α) :
OrderDual.toDual (a / b) = OrderDual.toDual a / OrderDual.toDual b
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@[simp]
theorem toDual_sub {α : Type u_1} [Sub α] (a b : α) :
OrderDual.toDual (a - b) = OrderDual.toDual a - OrderDual.toDual b
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@[simp]
theorem ofDual_div {α : Type u_1} [Div α] (a b : αᵒᵈ) :
OrderDual.ofDual (a / b) = OrderDual.ofDual a / OrderDual.ofDual b
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@[simp]
theorem ofDual_sub {α : Type u_1} [Sub α] (a b : αᵒᵈ) :
OrderDual.ofDual (a - b) = OrderDual.ofDual a - OrderDual.ofDual b
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@[simp]
theorem toDual_pow {α : Type u_1} {β : Type u_2} [Pow α β] (a : α) (b : β) :
OrderDual.toDual (a ^ b) = OrderDual.toDual a ^ b
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@[simp]
theorem toDual_smul {β : Type u_2} {α : Type u_1} [SMul β α] (b : β) (a : α) :
OrderDual.toDual (b • a) = b • OrderDual.toDual a
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@[simp]
theorem toDual_vadd {β : Type u_2} {α : Type u_1} [VAdd β α] (b : β) (a : α) :
OrderDual.toDual (b +ᵥ a) = b +ᵥ OrderDual.toDual a
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@[simp]
theorem ofDual_pow {α : Type u_1} {β : Type u_2} [Pow α β] (a : αᵒᵈ) (b : β) :
OrderDual.ofDual (a ^ b) = OrderDual.ofDual a ^ b
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@[simp]
theorem ofDual_vadd {β : Type u_2} {α : Type u_1} [VAdd β α] (b : β) (a : αᵒᵈ) :
OrderDual.ofDual (b +ᵥ a) = b +ᵥ OrderDual.ofDual a
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@[simp]
theorem ofDual_smul {β : Type u_2} {α : Type u_1} [SMul β α] (b : β) (a : αᵒᵈ) :
OrderDual.ofDual (b • a) = b • OrderDual.ofDual a
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@[simp]
theorem pow_toDual {α : Type u_1} {β : Type u_2} [Pow α β] (a : α) (b : β) :
a ^ OrderDual.toDual b = a ^ b
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@[simp]
theorem toDual_smul' {β : Type u_2} {α : Type u_1} [SMul β α] (b : β) (a : α) :
OrderDual.toDual b • a = b • a
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@[simp]
theorem toDual_vadd' {β : Type u_2} {α : Type u_1} [VAdd β α] (b : β) (a : α) :
OrderDual.toDual b +ᵥ a = b +ᵥ a
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@[simp]
theorem pow_ofDual {α : Type u_1} {β : Type u_2} [Pow α β] (a : α) (b : βᵒᵈ) :
a ^ OrderDual.ofDual b = a ^ b
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@[simp]
theorem ofDual_vadd' {β : Type u_2} {α : Type u_1} [VAdd β α] (b : βᵒᵈ) (a : α) :
OrderDual.ofDual b +ᵥ a = b +ᵥ a
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@[simp]
theorem ofDual_smul' {β : Type u_2} {α : Type u_1} [SMul β α] (b : βᵒᵈ) (a : α) :
OrderDual.ofDual b • a = b • a

Lexicographical order #

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instance instOneLex {α : Type u_1} [h : One α] :
One (Lex α)
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instance instZeroLex {α : Type u_1} [h : Zero α] :
Zero (Lex α)
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instance instMulLex {α : Type u_1} [h : Mul α] :
Mul (Lex α)
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instance instAddLex {α : Type u_1} [h : Add α] :
Add (Lex α)
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instance instInvLex {α : Type u_1} [h : Inv α] :
Inv (Lex α)
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instance instNegLex {α : Type u_1} [h : Neg α] :
Neg (Lex α)
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instance instDivLex {α : Type u_1} [h : Div α] :
Div (Lex α)
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instance instSubLex {α : Type u_1} [h : Sub α] :
Sub (Lex α)
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instance Lex.instPow {α : Type u_1} {β : Type u_2} [h : Pow α β] :
Pow (Lex α) β
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instance Lex.instSMul {β : Type u_2} {α : Type u_1} [h : SMul β α] :
SMul β (Lex α)
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instance Lex.instVAdd {β : Type u_2} {α : Type u_1} [h : VAdd β α] :
VAdd β (Lex α)
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instance Lex.instPow' {α : Type u_1} {β : Type u_2} [h : Pow α β] :
Pow α (Lex β)
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instance Lex.instSMul' {β : Type u_2} {α : Type u_1} [h : SMul β α] :
SMul (Lex β) α
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instance Lex.instVAdd' {β : Type u_2} {α : Type u_1} [h : VAdd β α] :
VAdd (Lex β) α
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instance instSemigroupLex {α : Type u_1} [h : Semigroup α] :
Semigroup (Lex α)
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instance instAddSemigroupLex {α : Type u_1} [h : AddSemigroup α] :
AddSemigroup (Lex α)
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instance instCommSemigroupLex {α : Type u_1} [h : CommSemigroup α] :
CommSemigroup (Lex α)
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instance instAddCommSemigroupLex {α : Type u_1} [h : AddCommSemigroup α] :
AddCommSemigroup (Lex α)
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instance instIsLeftCancelMulLex {α : Type u_1} [Mul α] [IsLeftCancelMul α] :
IsLeftCancelMul (Lex α)
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instance instIsLeftCancelAddLex {α : Type u_1} [Add α] [IsLeftCancelAdd α] :
IsLeftCancelAdd (Lex α)
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instance instIsRightCancelMulLex {α : Type u_1} [Mul α] [IsRightCancelMul α] :
IsRightCancelMul (Lex α)
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instance instIsRightCancelAddLex {α : Type u_1} [Add α] [IsRightCancelAdd α] :
IsRightCancelAdd (Lex α)
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instance instIsCancelMulLex {α : Type u_1} [Mul α] [IsCancelMul α] :
IsCancelMul (Lex α)
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instance instIsCancelAddLex {α : Type u_1} [Add α] [IsCancelAdd α] :
IsCancelAdd (Lex α)
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instance instLeftCancelSemigroupLex {α : Type u_1} [h : LeftCancelSemigroup α] :
LeftCancelSemigroup (Lex α)
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instance instAddLeftCancelSemigroupLex {α : Type u_1} [h : AddLeftCancelSemigroup α] :
AddLeftCancelSemigroup (Lex α)
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instance instRightCancelSemigroupLex {α : Type u_1} [h : RightCancelSemigroup α] :
RightCancelSemigroup (Lex α)
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instance instAddRightCancelSemigroupLex {α : Type u_1} [h : AddRightCancelSemigroup α] :
AddRightCancelSemigroup (Lex α)
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instance instMulOneClassLex {α : Type u_1} [h : MulOneClass α] :
MulOneClass (Lex α)
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instance instAddZeroClassLex {α : Type u_1} [h : AddZeroClass α] :
AddZeroClass (Lex α)
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instance instMonoidLex {α : Type u_1} [h : Monoid α] :
Monoid (Lex α)
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instance instAddMonoidLex {α : Type u_1} [h : AddMonoid α] :
AddMonoid (Lex α)
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instance instCommMonoidLex {α : Type u_1} [h : CommMonoid α] :
CommMonoid (Lex α)
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instance instAddCommMonoidLex {α : Type u_1} [h : AddCommMonoid α] :
AddCommMonoid (Lex α)
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instance instLeftCancelMonoidLex {α : Type u_1} [h : LeftCancelMonoid α] :
LeftCancelMonoid (Lex α)
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instance instAddLeftCancelMonoidLex {α : Type u_1} [h : AddLeftCancelMonoid α] :
AddLeftCancelMonoid (Lex α)
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instance instRightCancelMonoidLex {α : Type u_1} [h : RightCancelMonoid α] :
RightCancelMonoid (Lex α)
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instance instAddRightCancelMonoidLex {α : Type u_1} [h : AddRightCancelMonoid α] :
AddRightCancelMonoid (Lex α)
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instance instCancelMonoidLex {α : Type u_1} [h : CancelMonoid α] :
CancelMonoid (Lex α)
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instance instAddCancelMonoidLex {α : Type u_1} [h : AddCancelMonoid α] :
AddCancelMonoid (Lex α)
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instance instCancelCommMonoidLex {α : Type u_1} [h : CancelCommMonoid α] :
CancelCommMonoid (Lex α)
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instance instAddCancelCommMonoidLex {α : Type u_1} [h : AddCancelCommMonoid α] :
AddCancelCommMonoid (Lex α)
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instance instInvolutiveInvLex {α : Type u_1} [h : InvolutiveInv α] :
InvolutiveInv (Lex α)
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instance instInvolutiveNegLex {α : Type u_1} [h : InvolutiveNeg α] :
InvolutiveNeg (Lex α)
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instance instDivInvMonoidLex {α : Type u_1} [h : DivInvMonoid α] :
DivInvMonoid (Lex α)
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instance instSubNegAddMonoidLex {α : Type u_1} [h : SubNegMonoid α] :
SubNegMonoid (Lex α)
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instance instDivisionMonoidLex {α : Type u_1} [h : DivisionMonoid α] :
DivisionMonoid (Lex α)
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instance instSubtractionMonoidLex {α : Type u_1} [h : SubtractionMonoid α] :
SubtractionMonoid (Lex α)
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instance instDivisionCommMonoidLex {α : Type u_1} [h : DivisionCommMonoid α] :
DivisionCommMonoid (Lex α)
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instance instDivisionAddCommMonoidLex {α : Type u_1} [h : SubtractionCommMonoid α] :
SubtractionCommMonoid (Lex α)
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instance instGroupLex {α : Type u_1} [h : Group α] :
Group (Lex α)
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instance instAddGroupLex {α : Type u_1} [h : AddGroup α] :
AddGroup (Lex α)
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instance instCommGroupLex {α : Type u_1} [h : CommGroup α] :
CommGroup (Lex α)
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instance instAddCommGroupLex {α : Type u_1} [h : AddCommGroup α] :
AddCommGroup (Lex α)
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@[simp]
theorem toLex_one {α : Type u_1} [One α] :
toLex 1 = 1
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@[simp]
theorem toLex_zero {α : Type u_1} [Zero α] :
toLex 0 = 0
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@[simp]
theorem toLex_eq_one {α : Type u_1} [One α] {a : α} :
toLex a = 1 ↔ a = 1
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@[simp]
theorem toLex_eq_zero {α : Type u_1} [Zero α] {a : α} :
toLex a = 0 ↔ a = 0
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@[simp]
theorem ofLex_one {α : Type u_1} [One α] :
ofLex 1 = 1
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@[simp]
theorem ofLex_zero {α : Type u_1} [Zero α] :
ofLex 0 = 0
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@[simp]
theorem ofLex_eq_one {α : Type u_1} [One α] {a : Lex α} :
ofLex a = 1 ↔ a = 1
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@[simp]
theorem ofLex_eq_zero {α : Type u_1} [Zero α] {a : Lex α} :
ofLex a = 0 ↔ a = 0
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@[simp]
theorem toLex_mul {α : Type u_1} [Mul α] (a b : α) :
toLex (a * b) = toLex a * toLex b
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@[simp]
theorem toLex_add {α : Type u_1} [Add α] (a b : α) :
toLex (a + b) = toLex a + toLex b
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@[simp]
theorem ofLex_mul {α : Type u_1} [Mul α] (a b : Lex α) :
ofLex (a * b) = ofLex a * ofLex b
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@[simp]
theorem ofLex_add {α : Type u_1} [Add α] (a b : Lex α) :
ofLex (a + b) = ofLex a + ofLex b
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@[simp]
theorem toLex_inv {α : Type u_1} [Inv α] (a : α) :
toLex a⁻¹ = (toLex a)⁻¹
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@[simp]
theorem toLex_neg {α : Type u_1} [Neg α] (a : α) :
toLex (-a) = -toLex a
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@[simp]
theorem ofLex_inv {α : Type u_1} [Inv α] (a : Lex α) :
ofLex a⁻¹ = (ofLex a)⁻¹
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@[simp]
theorem ofLex_neg {α : Type u_1} [Neg α] (a : Lex α) :
ofLex (-a) = -ofLex a
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@[simp]
theorem toLex_div {α : Type u_1} [Div α] (a b : α) :
toLex (a / b) = toLex a / toLex b
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@[simp]
theorem toLex_sub {α : Type u_1} [Sub α] (a b : α) :
toLex (a - b) = toLex a - toLex b
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@[simp]
theorem ofLex_div {α : Type u_1} [Div α] (a b : Lex α) :
ofLex (a / b) = ofLex a / ofLex b
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@[simp]
theorem ofLex_sub {α : Type u_1} [Sub α] (a b : Lex α) :
ofLex (a - b) = ofLex a - ofLex b
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@[simp]
theorem toLex_pow {α : Type u_1} {β : Type u_2} [Pow α β] (a : α) (b : β) :
toLex (a ^ b) = toLex a ^ b
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@[simp]
theorem toLex_smul {β : Type u_2} {α : Type u_1} [SMul β α] (b : β) (a : α) :
toLex (b • a) = b • toLex a
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@[simp]
theorem toLex_vadd {β : Type u_2} {α : Type u_1} [VAdd β α] (b : β) (a : α) :
toLex (b +ᵥ a) = b +ᵥ toLex a
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@[simp]
theorem ofLex_pow {α : Type u_1} {β : Type u_2} [Pow α β] (a : Lex α) (b : β) :
ofLex (a ^ b) = ofLex a ^ b
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@[simp]
theorem ofLex_smul {β : Type u_2} {α : Type u_1} [SMul β α] (b : β) (a : Lex α) :
ofLex (b • a) = b • ofLex a
source
@[simp]
theorem ofLex_vadd {β : Type u_2} {α : Type u_1} [VAdd β α] (b : β) (a : Lex α) :
ofLex (b +ᵥ a) = b +ᵥ ofLex a
source
@[simp]
theorem pow_toLex {α : Type u_1} {β : Type u_2} [Pow α β] (a : α) (b : β) :
a ^ toLex b = a ^ b
source
@[simp]
theorem toLex_smul' {β : Type u_2} {α : Type u_1} [SMul β α] (b : β) (a : α) :
toLex b • a = b • a
source
@[simp]
theorem toLex_vadd' {β : Type u_2} {α : Type u_1} [VAdd β α] (b : β) (a : α) :
toLex b +ᵥ a = b +ᵥ a
source
@[simp]
theorem pow_ofLex {α : Type u_1} {β : Type u_2} [Pow α β] (a : α) (b : Lex β) :
a ^ ofLex b = a ^ b
source
@[simp]
theorem ofLex_vadd' {β : Type u_2} {α : Type u_1} [VAdd β α] (b : Lex β) (a : α) :
ofLex b +ᵥ a = b +ᵥ a
source
@[simp]
theorem ofLex_smul' {β : Type u_2} {α : Type u_1} [SMul β α] (b : Lex β) (a : α) :
ofLex b • a = b • a