# Ordered monoid structures on the order dual. #

instance OrderDual.contravariantClass_add_le {α : Type u} [LE α] [Add α] [c : ContravariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x x_1] :
ContravariantClass αᵒᵈ αᵒᵈ (fun (x x_1 : αᵒᵈ) => x + x_1) fun (x x_1 : αᵒᵈ) => x x_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 : α) => x * x_1) fun (x x_1 : α) => x x_1] :
ContravariantClass αᵒᵈ αᵒᵈ (fun (x x_1 : αᵒᵈ) => x * x_1) fun (x x_1 : αᵒᵈ) => x x_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 : α) => x + x_1) fun (x x_1 : α) => x x_1] :
CovariantClass αᵒᵈ αᵒᵈ (fun (x x_1 : αᵒᵈ) => x + x_1) fun (x x_1 : αᵒᵈ) => 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 : α) => x * x_1) fun (x x_1 : α) => x x_1] :
CovariantClass αᵒᵈ αᵒᵈ (fun (x x_1 : αᵒᵈ) => x * x_1) fun (x x_1 : αᵒᵈ) => x x_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 : α) => x + x_1) fun (x x_1 : α) => x x_1] :
ContravariantClass αᵒᵈ αᵒᵈ (Function.swap fun (x x_1 : αᵒᵈ) => x + x_1) fun (x x_1 : αᵒᵈ) => 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 : α) => x * x_1) fun (x x_1 : α) => x x_1] :
ContravariantClass αᵒᵈ αᵒᵈ (Function.swap fun (x x_1 : αᵒᵈ) => x * x_1) fun (x x_1 : αᵒᵈ) => x x_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 : α) => x + x_1) fun (x x_1 : α) => x x_1] :
CovariantClass αᵒᵈ αᵒᵈ (Function.swap fun (x x_1 : αᵒᵈ) => x + x_1) fun (x x_1 : αᵒᵈ) => 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 : α) => x * x_1) fun (x x_1 : α) => x x_1] :
CovariantClass αᵒᵈ αᵒᵈ (Function.swap fun (x x_1 : αᵒᵈ) => x * x_1) fun (x x_1 : αᵒᵈ) => x x_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 : α) => x + x_1) fun (x x_1 : α) => x < x_1] :
ContravariantClass αᵒᵈ αᵒᵈ (fun (x x_1 : αᵒᵈ) => x + x_1) fun (x x_1 : αᵒᵈ) => 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 : α) => x * x_1) fun (x x_1 : α) => x < x_1] :
ContravariantClass αᵒᵈ αᵒᵈ (fun (x x_1 : αᵒᵈ) => x * x_1) fun (x x_1 : αᵒᵈ) => x < x_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 : α) => x + x_1) fun (x x_1 : α) => x < x_1] :
CovariantClass αᵒᵈ αᵒᵈ (fun (x x_1 : αᵒᵈ) => x + x_1) fun (x x_1 : αᵒᵈ) => 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 : α) => x * x_1) fun (x x_1 : α) => x < x_1] :
CovariantClass αᵒᵈ αᵒᵈ (fun (x x_1 : αᵒᵈ) => x * x_1) fun (x x_1 : αᵒᵈ) => x < x_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 : α) => x + x_1) fun (x x_1 : α) => x < x_1] :
ContravariantClass αᵒᵈ αᵒᵈ (Function.swap fun (x x_1 : αᵒᵈ) => x + x_1) fun (x x_1 : αᵒᵈ) => 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 : α) => x * x_1) fun (x x_1 : α) => x < x_1] :
ContravariantClass αᵒᵈ αᵒᵈ (Function.swap fun (x x_1 : αᵒᵈ) => x * x_1) fun (x x_1 : αᵒᵈ) => x < x_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 : α) => x + x_1) fun (x x_1 : α) => x < x_1] :
CovariantClass αᵒᵈ αᵒᵈ (Function.swap fun (x x_1 : αᵒᵈ) => x + x_1) fun (x x_1 : αᵒᵈ) => 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 : α) => x * x_1) fun (x x_1 : α) => x < x_1] :
CovariantClass αᵒᵈ αᵒᵈ (Function.swap fun (x x_1 : αᵒᵈ) => x * x_1) fun (x x_1 : αᵒᵈ) => x < x_1
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theorem OrderDual.orderedAddCommMonoid.proof_1 {α : Type u_1} :
∀ (x x_1 : αᵒᵈ), x x_1∀ (c : αᵒᵈ), c + x c + x_1
instance OrderDual.orderedCommMonoid {α : Type u} :
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• OrderDual.orderedCommMonoid =
theorem OrderDual.orderedAddCancelCommMonoid.proof_1 {α : Type u_1} :
∀ (x x_1 x_2 : α), x + x_1 x + x_2x_1 x_2
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• OrderDual.orderedCancelCommMonoid =
theorem OrderDual.linearOrderedAddCancelCommMonoid.proof_5 {α : Type u_1} (a : αᵒᵈ) (b : αᵒᵈ) :
max a b = if a b then b else a
theorem OrderDual.linearOrderedAddCancelCommMonoid.proof_4 {α : Type u_1} (a : αᵒᵈ) (b : αᵒᵈ) :
min a b = if a b then a else b
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• One or more equations did not get rendered due to their size.
theorem OrderDual.linearOrderedAddCancelCommMonoid.proof_1 {α : Type u_1} (a : αᵒᵈ) (b : αᵒᵈ) :
a b∀ (c : αᵒᵈ), c + a c + b
theorem OrderDual.linearOrderedAddCancelCommMonoid.proof_2 {α : Type u_1} (a : αᵒᵈ) (b : αᵒᵈ) (c : αᵒᵈ) :
a + b a + cb c
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• One or more equations did not get rendered due to their size.
theorem OrderDual.linearOrderedAddCommMonoid.proof_4 {α : Type u_1} (a : αᵒᵈ) (b : αᵒᵈ) :
max a b = if a b then b else a
theorem OrderDual.linearOrderedAddCommMonoid.proof_1 {α : Type u_1} (a : αᵒᵈ) (b : αᵒᵈ) :
a b∀ (c : αᵒᵈ), c + a c + b
theorem OrderDual.linearOrderedAddCommMonoid.proof_2 {α : Type u_1} (a : αᵒᵈ) (b : αᵒᵈ) :
a b b a
theorem OrderDual.linearOrderedAddCommMonoid.proof_3 {α : Type u_1} (a : αᵒᵈ) (b : αᵒᵈ) :
min a b = if a b then a else b
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• One or more equations did not get rendered due to their size.
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• One or more equations did not get rendered due to their size.