orderAddMonoidHom — Mathlib

English

In LocallyFiniteOrder, the order-preserving embedding addMonoidHom G agrees with the plain additive monoid hom; i.e., the two notion align on the underlying map.

Русский

В LocallyFiniteOrder вложение addMonoidHom G согласуется с простым моноидом сложения; т.е. две концепции совпадают по базовой карте.

LaTeX

$$$$ \operatorname{orderAddMonoidHom} G = \operatorname{addMonoidHom} G. $$$$

Lean4

/-- The canonical embedding (as an ordered monoid hom) from a linearly ordered cancellative
group into `ℤ`. This is either surjective or zero. -/
def orderAddMonoidHom : G →+o ℤ where
  __ := addMonoidHom G
  monotone' a b
    hab := by
    obtain ⟨b, rfl⟩ := add_left_surjective a b
    replace hab : 0 ≤ b := by simpa using hab
    suffices 0 ≤ addMonoidHom G b by simpa
    simp [addMonoidHom, hab]

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