Algebraic structures on the set of positive numbers

In this file we define various instances (AddSemigroup, OrderedCommMonoid etc) on the type {x : R // 0 < x}. In each case we try to require the weakest possible typeclass assumptions on R but possibly, there is a room for improvements.

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instance Positive.instAddSubtypeLtToLTOfNatToOfNat0ToZero {M : Type u_1} [inst : AddMonoid M] [inst : Preorder M] [inst : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] : Add { x // 0 < x }

Equations

• Positive.instAddSubtypeLtToLTOfNatToOfNat0ToZero = { add := fun x y => { val := ↑x + ↑y, property := (_ : 0 < ↑x + ↑y) } }

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@[simp]

theorem Positive.coe_add {M : Type u_1} [inst : AddMonoid M] [inst : Preorder M] [inst : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (x : { x // 0 < x }) (y : { x // 0 < x }) : ↑(x + y) = ↑x + ↑y

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instance Positive.addSemigroup {M : Type u_1} [inst : AddMonoid M] [inst : Preorder M] [inst : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] : AddSemigroup { x // 0 < x }

Equations

• Positive.addSemigroup = Function.Injective.addSemigroup (fun a => ↑a) (_ : Function.Injective fun a => ↑a) (_ : ∀ (x y : { x // 0 < x }), ↑(x + y) = ↑x + ↑y)

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instance Positive.addCommSemigroup {M : Type u_1} [inst : AddCommMonoid M] [inst : Preorder M] [inst : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] : AddCommSemigroup { x // 0 < x }

Equations

• Positive.addCommSemigroup = Function.Injective.addCommSemigroup (fun a => ↑a) (_ : Function.Injective fun a => ↑a) (_ : ∀ (x y : { x // 0 < x }), ↑(x + y) = ↑x + ↑y)

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instance Positive.addLeftCancelSemigroup {M : Type u_1} [inst : AddLeftCancelMonoid M] [inst : Preorder M] [inst : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] : AddLeftCancelSemigroup { x // 0 < x }

Equations

• Positive.addLeftCancelSemigroup = Function.Injective.addLeftCancelSemigroup (fun a => ↑a) (_ : Function.Injective fun a => ↑a) (_ : ∀ (x y : { x // 0 < x }), ↑(x + y) = ↑x + ↑y)

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instance Positive.addRightCancelSemigroup {M : Type u_1} [inst : AddRightCancelMonoid M] [inst : Preorder M] [inst : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] : AddRightCancelSemigroup { x // 0 < x }

Equations

• Positive.addRightCancelSemigroup = Function.Injective.addRightCancelSemigroup (fun a => ↑a) (_ : Function.Injective fun a => ↑a) (_ : ∀ (x y : { x // 0 < x }), ↑(x + y) = ↑x + ↑y)

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instance Positive.covariantClass_add_lt {M : Type u_1} [inst : AddMonoid M] [inst : Preorder M] [inst : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] : CovariantClass { x // 0 < x } { x // 0 < x } (fun x x_1 => x + x_1) fun x x_1 => x < x_1

Equations

• @Positive.covariantClass_add_lt = @Positive.covariantClass_add_lt.proof_1

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instance Positive.covariantClass_swap_add_lt {M : Type u_1} [inst : AddMonoid M] [inst : Preorder M] [inst : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [inst : CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] : CovariantClass { x // 0 < x } { x // 0 < x } (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1

Equations

• @Positive.covariantClass_swap_add_lt = @Positive.covariantClass_swap_add_lt.proof_1

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instance Positive.contravariantClass_add_lt {M : Type u_1} [inst : AddMonoid M] [inst : Preorder M] [inst : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [inst : ContravariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] : ContravariantClass { x // 0 < x } { x // 0 < x } (fun x x_1 => x + x_1) fun x x_1 => x < x_1

Equations

• @Positive.contravariantClass_add_lt = @Positive.contravariantClass_add_lt.proof_1

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instance Positive.contravariantClass_swap_add_lt {M : Type u_1} [inst : AddMonoid M] [inst : Preorder M] [inst : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [inst : ContravariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] : ContravariantClass { x // 0 < x } { x // 0 < x } (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1

Equations

• @Positive.contravariantClass_swap_add_lt = @Positive.contravariantClass_swap_add_lt.proof_1

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instance Positive.contravariantClass_add_le {M : Type u_1} [inst : AddMonoid M] [inst : Preorder M] [inst : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [inst : ContravariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] : ContravariantClass { x // 0 < x } { x // 0 < x } (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1

Equations

• @Positive.contravariantClass_add_le = @Positive.contravariantClass_add_le.proof_1

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instance Positive.contravariantClass_swap_add_le {M : Type u_1} [inst : AddMonoid M] [inst : Preorder M] [inst : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [inst : ContravariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] : ContravariantClass { x // 0 < x } { x // 0 < x } (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1

Equations

• @Positive.contravariantClass_swap_add_le = @Positive.contravariantClass_swap_add_le.proof_1

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instance Positive.covariantClass_add_le {M : Type u_1} [inst : AddMonoid M] [inst : PartialOrder M] [inst : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] : CovariantClass { x // 0 < x } { x // 0 < x } (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1

Equations

• @Positive.covariantClass_add_le = @Positive.covariantClass_add_le.proof_1

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instance Positive.instMulSubtypeLtToLTToPreorderToPartialOrderOfNatToOfNat0ToZeroToMonoidWithZeroToSemiring {R : Type u_1} [inst : StrictOrderedSemiring R] : Mul { x // 0 < x }

Equations

• Positive.instMulSubtypeLtToLTToPreorderToPartialOrderOfNatToOfNat0ToZeroToMonoidWithZeroToSemiring = { mul := fun x y => { val := ↑x * ↑y, property := (_ : 0 < ↑x * ↑y) } }

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theorem Positive.val_mul {R : Type u_1} [inst : StrictOrderedSemiring R] (x : { x // 0 < x }) (y : { x // 0 < x }) : ↑(x * y) = ↑x * ↑y

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instance Positive.instPowSubtypeLtToLTToPreorderToPartialOrderOfNatToOfNat0ToZeroToMonoidWithZeroToSemiringNat {R : Type u_1} [inst : StrictOrderedSemiring R] : Pow { x // 0 < x } ℕ

Equations

• Positive.instPowSubtypeLtToLTToPreorderToPartialOrderOfNatToOfNat0ToZeroToMonoidWithZeroToSemiringNat = { pow := fun x n => { val := ↑x ^ n, property := (_ : 0 < ↑x ^ n) } }

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theorem Positive.val_pow {R : Type u_1} [inst : StrictOrderedSemiring R] (x : { x // 0 < x }) (n : ℕ) : ↑(x ^ n) = ↑x ^ n

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instance Positive.instSemigroupSubtypeLtToLTToPreorderToPartialOrderOfNatToOfNat0ToZeroToMonoidWithZeroToSemiring {R : Type u_1} [inst : StrictOrderedSemiring R] : Semigroup { x // 0 < x }

Equations

• One or more equations did not get rendered due to their size.

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instance Positive.instDistribSubtypeLtToLTToPreorderToPartialOrderOfNatToOfNat0ToZeroToMonoidWithZeroToSemiring {R : Type u_1} [inst : StrictOrderedSemiring R] : Distrib { x // 0 < x }

Equations

• One or more equations did not get rendered due to their size.

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instance Positive.instOneSubtypeLtToLTToPreorderToPartialOrderOfNatToOfNat0ToZeroToMonoidWithZeroToSemiring {R : Type u_1} [inst : StrictOrderedSemiring R] [inst : Nontrivial R] : One { x // 0 < x }

Equations

• Positive.instOneSubtypeLtToLTToPreorderToPartialOrderOfNatToOfNat0ToZeroToMonoidWithZeroToSemiring = { one := { val := 1, property := (_ : 0 < 1) } }

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theorem Positive.val_one {R : Type u_1} [inst : StrictOrderedSemiring R] [inst : Nontrivial R] : ↑1 = 1

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instance Positive.instMonoidSubtypeLtToLTToPreorderToPartialOrderOfNatToOfNat0ToZeroToMonoidWithZeroToSemiring {R : Type u_1} [inst : StrictOrderedSemiring R] [inst : Nontrivial R] : Monoid { x // 0 < x }

Equations

• One or more equations did not get rendered due to their size.

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instance Positive.orderedCommMonoid {R : Type u_1} [inst : StrictOrderedCommSemiring R] [inst : Nontrivial R] : OrderedCommMonoid { x // 0 < x }

Equations

• One or more equations did not get rendered due to their size.

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instance Positive.linearOrderedCancelCommMonoid {R : Type u_1} [inst : LinearOrderedCommSemiring R] : LinearOrderedCancelCommMonoid { x // 0 < x }

If R is a nontrivial linear ordered commutative semiring, then {x : R // 0 < x} is a linear ordered cancellative commutative monoid.

Equations

• One or more equations did not get rendered due to their size.