remainder — Mathlib

English

Further consequences about addVal for products and powers follow from the valuation axioms.

Русский

Последствия для произведений и степеней описываются аксиомами оценки.

LaTeX

$$$\mathrm{addVal}(ab) = \mathrm{addVal}(a) + \mathrm{addVal}(b)$$$

Lean4

/-- A noncomputable remainder to define the Euclidean domain structure. The GCD algorithm only takes
two steps to terminate. Given `GCD(x,y)`, if `x ∣ y` then `y%x = 0` so we're done in one step;
otherwise `y%x = y` and then `GCD(x,y) = GCD(y,x)` which brings us back to the first case. -/
def remainder (x y : R) : R :=
  open Classical in if y ∣ x then 0 else x

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