English
Given a function f: M → α with monotonicity along archimedean classes, there exists an order-preserving lift MulArchimedeanClass M → α.
Русский
Заданной функции f: M → α, удовлетворяющей монотонности вдоль архимедеанских классов, существует ордоподобное возведение MulArchimedeanClass M → α.
LaTeX
$$$\\text{exists } \\text{OrderHom}(\\text{MulArchimedeanClass } M, \\alpha)$ under the given monotonicity condition$$
Lean4
/-- Lift a function `M → α` that's monotone along archimedean classes to a
monotone function `MulArchimedeanClass M →o α`. -/
@[to_additive /-- Lift a function `M → α` that's monotone along archimedean classes to a
monotone function `ArchimedeanClass M →o α`. -/
]
noncomputable def liftOrderHom (f : M → α) (h : ∀ a b, mk a ≤ mk b → f a ≤ f b) : MulArchimedeanClass M →o α
where
toFun := lift f fun a b heq ↦ le_antisymm (h a b heq.le) (h b a heq.ge)
monotone' A B
hle :=
by
induction A using ind with
| mk a
induction B using ind with
| mk b
simpa using h a b (mk_le_mk.mp hle)
