Ordered monoid structures on the order dual. #

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theorem OrderDual.contravariantClass_add_le.proof_1 {α : Type u_1} [LE α] [Add α] [c : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] :

ContravariantClass αᵒᵈ αᵒᵈ (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1

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instance OrderDual.contravariantClass_add_le {α : Type u} [LE α] [Add α] [c : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] :

ContravariantClass αᵒᵈ αᵒᵈ (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1

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@OrderDual.contravariantClass_add_le = @OrderDual.contravariantClass_add_le.proof_1

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instance OrderDual.contravariantClass_mul_le {α : Type u} [LE α] [Mul α] [c : ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] :

ContravariantClass αᵒᵈ αᵒᵈ (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1

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@OrderDual.contravariantClass_mul_le = @OrderDual.contravariantClass_mul_le.proof_1

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instance OrderDual.covariantClass_add_le {α : Type u} [LE α] [Add α] [c : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] :

CovariantClass αᵒᵈ αᵒᵈ (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1

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@OrderDual.covariantClass_add_le = @OrderDual.covariantClass_add_le.proof_1

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theorem OrderDual.covariantClass_add_le.proof_1 {α : Type u_1} [LE α] [Add α] [c : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] :

CovariantClass αᵒᵈ αᵒᵈ (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1

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instance OrderDual.covariantClass_mul_le {α : Type u} [LE α] [Mul α] [c : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] :

CovariantClass αᵒᵈ αᵒᵈ (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1

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@OrderDual.covariantClass_mul_le = @OrderDual.covariantClass_mul_le.proof_1

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instance OrderDual.contravariantClass_swap_add_le {α : Type u} [LE α] [Add α] [c : ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] :

ContravariantClass αᵒᵈ αᵒᵈ (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1

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@OrderDual.contravariantClass_swap_add_le = @OrderDual.contravariantClass_swap_add_le.proof_1

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theorem OrderDual.contravariantClass_swap_add_le.proof_1 {α : Type u_1} [LE α] [Add α] [c : ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] :

ContravariantClass αᵒᵈ αᵒᵈ (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1

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instance OrderDual.contravariantClass_swap_mul_le {α : Type u} [LE α] [Mul α] [c : ContravariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] :

ContravariantClass αᵒᵈ αᵒᵈ (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1

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@OrderDual.contravariantClass_swap_mul_le = @OrderDual.contravariantClass_swap_mul_le.proof_1

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instance OrderDual.covariantClass_swap_add_le {α : Type u} [LE α] [Add α] [c : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] :

CovariantClass αᵒᵈ αᵒᵈ (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1

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@OrderDual.covariantClass_swap_add_le = @OrderDual.covariantClass_swap_add_le.proof_1

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theorem OrderDual.covariantClass_swap_add_le.proof_1 {α : Type u_1} [LE α] [Add α] [c : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] :

CovariantClass αᵒᵈ αᵒᵈ (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1

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instance OrderDual.covariantClass_swap_mul_le {α : Type u} [LE α] [Mul α] [c : CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] :

CovariantClass αᵒᵈ αᵒᵈ (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1

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@OrderDual.covariantClass_swap_mul_le = @OrderDual.covariantClass_swap_mul_le.proof_1

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instance OrderDual.contravariantClass_add_lt {α : Type u} [LT α] [Add α] [c : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] :

ContravariantClass αᵒᵈ αᵒᵈ (fun x x_1 => x + x_1) fun x x_1 => x < x_1

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@OrderDual.contravariantClass_add_lt = @OrderDual.contravariantClass_add_lt.proof_1

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theorem OrderDual.contravariantClass_add_lt.proof_1 {α : Type u_1} [LT α] [Add α] [c : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] :

ContravariantClass αᵒᵈ αᵒᵈ (fun x x_1 => x + x_1) fun x x_1 => x < x_1

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instance OrderDual.contravariantClass_mul_lt {α : Type u} [LT α] [Mul α] [c : ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] :

ContravariantClass αᵒᵈ αᵒᵈ (fun x x_1 => x * x_1) fun x x_1 => x < x_1

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@OrderDual.contravariantClass_mul_lt = @OrderDual.contravariantClass_mul_lt.proof_1

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instance OrderDual.covariantClass_add_lt {α : Type u} [LT α] [Add α] [c : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] :

CovariantClass αᵒᵈ αᵒᵈ (fun x x_1 => x + x_1) fun x x_1 => x < x_1

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@OrderDual.covariantClass_add_lt = @OrderDual.covariantClass_add_lt.proof_1

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theorem OrderDual.covariantClass_add_lt.proof_1 {α : Type u_1} [LT α] [Add α] [c : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] :

CovariantClass αᵒᵈ αᵒᵈ (fun x x_1 => x + x_1) fun x x_1 => x < x_1

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instance OrderDual.covariantClass_mul_lt {α : Type u} [LT α] [Mul α] [c : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] :

CovariantClass αᵒᵈ αᵒᵈ (fun x x_1 => x * x_1) fun x x_1 => x < x_1

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@OrderDual.covariantClass_mul_lt = @OrderDual.covariantClass_mul_lt.proof_1

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instance OrderDual.contravariantClass_swap_add_lt {α : Type u} [LT α] [Add α] [c : ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] :

ContravariantClass αᵒᵈ αᵒᵈ (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1

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@OrderDual.contravariantClass_swap_add_lt = @OrderDual.contravariantClass_swap_add_lt.proof_1

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theorem OrderDual.contravariantClass_swap_add_lt.proof_1 {α : Type u_1} [LT α] [Add α] [c : ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] :

ContravariantClass αᵒᵈ αᵒᵈ (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1

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instance OrderDual.contravariantClass_swap_mul_lt {α : Type u} [LT α] [Mul α] [c : ContravariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] :

ContravariantClass αᵒᵈ αᵒᵈ (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1

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@OrderDual.contravariantClass_swap_mul_lt = @OrderDual.contravariantClass_swap_mul_lt.proof_1

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instance OrderDual.covariantClass_swap_add_lt {α : Type u} [LT α] [Add α] [c : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] :

CovariantClass αᵒᵈ αᵒᵈ (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1

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@OrderDual.covariantClass_swap_add_lt = @OrderDual.covariantClass_swap_add_lt.proof_1

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theorem OrderDual.covariantClass_swap_add_lt.proof_1 {α : Type u_1} [LT α] [Add α] [c : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] :

CovariantClass αᵒᵈ αᵒᵈ (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1

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instance OrderDual.covariantClass_swap_mul_lt {α : Type u} [LT α] [Mul α] [c : CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] :

CovariantClass αᵒᵈ αᵒᵈ (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1

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@OrderDual.covariantClass_swap_mul_lt = @OrderDual.covariantClass_swap_mul_lt.proof_1

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theorem OrderDual.orderedAddCommMonoid.proof_2 {α : Type u_1} [OrderedAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) :

a ≤ b → b ≤ a → a = b

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theorem OrderDual.orderedAddCommMonoid.proof_3 {α : Type u_1} [OrderedAddCommMonoid α] :

∀ (x x_1 : αᵒᵈ), x ≤ x_1 → ∀ (c : αᵒᵈ), c + x ≤ c + x_1

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theorem OrderDual.orderedAddCommMonoid.proof_1 {α : Type u_1} [OrderedAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) :

a + b = b + a

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instance OrderDual.orderedAddCommMonoid {α : Type u} [OrderedAddCommMonoid α] :

OrderedAddCommMonoid αᵒᵈ

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instance OrderDual.orderedCommMonoid {α : Type u} [OrderedCommMonoid α] :

OrderedCommMonoid αᵒᵈ

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OrderDual.orderedCommMonoid = let src := OrderDual.instPartialOrder α; let src_1 := OrderDual.instCommMonoid; OrderedCommMonoid.mk (_ : ∀ (x x_1 : αᵒᵈ), x ≤ x_1 → ∀ (c : αᵒᵈ), c * x ≤ c * x_1)

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theorem OrderDual.OrderedCancelAddCommMonoid.to_contravariantClass.proof_1 {α : Type u_1} [OrderedCancelAddCommMonoid α] :

ContravariantClass αᵒᵈ αᵒᵈ Add.add LE.le

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instance OrderDual.OrderedCancelAddCommMonoid.to_contravariantClass {α : Type u} [OrderedCancelAddCommMonoid α] :

ContravariantClass αᵒᵈ αᵒᵈ Add.add LE.le

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@OrderDual.OrderedCancelAddCommMonoid.to_contravariantClass = @OrderDual.OrderedCancelAddCommMonoid.to_contravariantClass.proof_1

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instance OrderDual.OrderedCancelCommMonoid.to_contravariantClass {α : Type u} [OrderedCancelCommMonoid α] :

ContravariantClass αᵒᵈ αᵒᵈ Mul.mul LE.le

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@OrderDual.OrderedCancelCommMonoid.to_contravariantClass = @OrderDual.OrderedCancelCommMonoid.to_contravariantClass.proof_1

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theorem OrderDual.orderedAddCancelCommMonoid.proof_2 {α : Type u_1} [OrderedCancelAddCommMonoid α] :

∀ (x x_1 x_2 : α), x + x_1 ≤ x + x_2 → x_1 ≤ x_2

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instance OrderDual.orderedAddCancelCommMonoid {α : Type u} [OrderedCancelAddCommMonoid α] :

OrderedCancelAddCommMonoid αᵒᵈ

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theorem OrderDual.orderedAddCancelCommMonoid.proof_1 {α : Type u_1} [OrderedCancelAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) :

a ≤ b → ∀ (c : αᵒᵈ), c + a ≤ c + b

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instance OrderDual.orderedCancelCommMonoid {α : Type u} [OrderedCancelCommMonoid α] :

OrderedCancelCommMonoid αᵒᵈ

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theorem OrderDual.linearOrderedAddCancelCommMonoid.proof_1 {α : Type u_1} [LinearOrderedCancelAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) :

a ≤ b → ∀ (c : αᵒᵈ), c + a ≤ c + b

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theorem OrderDual.linearOrderedAddCancelCommMonoid.proof_5 {α : Type u_1} [LinearOrderedCancelAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) :

max a b = if a ≤ b then b else a

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instance OrderDual.linearOrderedAddCancelCommMonoid {α : Type u} [LinearOrderedCancelAddCommMonoid α] :

LinearOrderedCancelAddCommMonoid αᵒᵈ

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theorem OrderDual.linearOrderedAddCancelCommMonoid.proof_6 {α : Type u_1} [LinearOrderedCancelAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) :

compare a b = compareOfLessAndEq a b

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theorem OrderDual.linearOrderedAddCancelCommMonoid.proof_3 {α : Type u_1} [LinearOrderedCancelAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) :

a ≤ b ∨ b ≤ a

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theorem OrderDual.linearOrderedAddCancelCommMonoid.proof_4 {α : Type u_1} [LinearOrderedCancelAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) :

min a b = if a ≤ b then a else b

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theorem OrderDual.linearOrderedAddCancelCommMonoid.proof_2 {α : Type u_1} [LinearOrderedCancelAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) (c : αᵒᵈ) :

a + b ≤ a + c → b ≤ c

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instance OrderDual.linearOrderedCancelCommMonoid {α : Type u} [LinearOrderedCancelCommMonoid α] :

LinearOrderedCancelCommMonoid αᵒᵈ

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theorem OrderDual.linearOrderedAddCommMonoid.proof_2 {α : Type u_1} [LinearOrderedAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) :

min a b = if a ≤ b then a else b

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theorem OrderDual.linearOrderedAddCommMonoid.proof_1 {α : Type u_1} [LinearOrderedAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) :

a ≤ b ∨ b ≤ a

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instance OrderDual.linearOrderedAddCommMonoid {α : Type u} [LinearOrderedAddCommMonoid α] :

LinearOrderedAddCommMonoid αᵒᵈ

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theorem OrderDual.linearOrderedAddCommMonoid.proof_5 {α : Type u_1} [LinearOrderedAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) :

a ≤ b → ∀ (c : αᵒᵈ), c + a ≤ c + b

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theorem OrderDual.linearOrderedAddCommMonoid.proof_4 {α : Type u_1} [LinearOrderedAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) :

compare a b = compareOfLessAndEq a b

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theorem OrderDual.linearOrderedAddCommMonoid.proof_3 {α : Type u_1} [LinearOrderedAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) :

max a b = if a ≤ b then b else a

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instance OrderDual.linearOrderedCommMonoid {α : Type u} [LinearOrderedCommMonoid α] :

LinearOrderedCommMonoid αᵒᵈ

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