Pulling back linearly ordered fields along injective maps. #
def
Function.Injective.linearOrderedSemifield
{α : Type u_1}
{β : Type u_2}
[inst : LinearOrderedSemifield α]
[inst : Zero β]
[inst : One β]
[inst : Add β]
[inst : Mul β]
[inst : Pow β ℕ]
[inst : SMul ℕ β]
[inst : NatCast β]
[inst : Inv β]
[inst : Div β]
[inst : Pow β ℤ]
[inst : Sup β]
[inst : Inf β]
(f : β → α)
(hf : Function.Injective f)
(zero : f 0 = 0)
(one : f 1 = 1)
(add : ∀ (x y : β), f (x + y) = f x + f y)
(mul : ∀ (x y : β), f (x * y) = f x * f y)
(inv : ∀ (x : β), f x⁻¹ = (f x)⁻¹)
(div : ∀ (x y : β), f (x / y) = f x / f y)
(nsmul : ∀ (x : β) (n : ℕ), f (n • x) = n • f x)
(npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n)
(zpow : ∀ (x : β) (n : ℤ), f (x ^ n) = f x ^ n)
(nat_cast : ∀ (n : ℕ), f ↑n = ↑n)
(hsup : ∀ (x y : β), f (x ⊔ y) = max (f x) (f y))
(hinf : ∀ (x y : β), f (x ⊓ y) = min (f x) (f y))
:
Pullback a LinearOrderedSemifield under an injective map.
Equations
- One or more equations did not get rendered due to their size.
def
Function.Injective.linearOrderedField
{α : Type u_1}
{β : Type u_2}
[inst : LinearOrderedField α]
[inst : Zero β]
[inst : One β]
[inst : Add β]
[inst : Mul β]
[inst : Neg β]
[inst : Sub β]
[inst : Pow β ℕ]
[inst : SMul ℕ β]
[inst : SMul ℤ β]
[inst : SMul ℚ β]
[inst : NatCast β]
[inst : IntCast β]
[inst : RatCast β]
[inst : Inv β]
[inst : Div β]
[inst : Pow β ℤ]
[inst : Sup β]
[inst : Inf β]
(f : β → α)
(hf : Function.Injective f)
(zero : f 0 = 0)
(one : f 1 = 1)
(add : ∀ (x y : β), f (x + y) = f x + f y)
(mul : ∀ (x y : β), f (x * y) = f x * f y)
(neg : ∀ (x : β), f (-x) = -f x)
(sub : ∀ (x y : β), f (x - y) = f x - f y)
(inv : ∀ (x : β), f x⁻¹ = (f x)⁻¹)
(div : ∀ (x y : β), f (x / y) = f x / f y)
(nsmul : ∀ (x : β) (n : ℕ), f (n • x) = n • f x)
(zsmul : ∀ (x : β) (n : ℤ), f (n • x) = n • f x)
(qsmul : ∀ (x : β) (n : ℚ), f (n • x) = n • f x)
(npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n)
(zpow : ∀ (x : β) (n : ℤ), f (x ^ n) = f x ^ n)
(nat_cast : ∀ (n : ℕ), f ↑n = ↑n)
(int_cast : ∀ (n : ℤ), f ↑n = ↑n)
(rat_cast : ∀ (n : ℚ), f ↑n = ↑n)
(hsup : ∀ (x y : β), f (x ⊔ y) = max (f x) (f y))
(hinf : ∀ (x y : β), f (x ⊓ y) = min (f x) (f y))
:
Pullback a LinearOrderedField under an injective map.
Equations
- One or more equations did not get rendered due to their size.