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Fix minor typos and textbook rendering issues in C06S03
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benjaminfjones authored and avigad committed Jan 20, 2025
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6 changes: 3 additions & 3 deletions MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean
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Expand Up @@ -241,7 +241,7 @@ function :math:`N : R \to \mathbb{N}` with the following two properties:
- For every :math:`a` and :math:`b \ne 0` in :math:`R`, there are
:math:`q` and :math:`r` in :math:`R` such that :math:`a = bq + r` and
either :math:`r = 0` or `N(r) < N(b)`.
either :math:`r = 0` or :math:`N(r) < N(b)`.
- For every :math:`a` and :math:`b \ne 0`, :math:`N(a) \le N(ab)`.
The ring of integers :math:`\Bbb{Z}` with :math:`N(a) = |a|` is an
Expand Down Expand Up @@ -288,7 +288,7 @@ implies that every irreducible element is prime.
The axioms for a Euclidean domain imply that one can write any nonzero element
as a finite product of irreducible elements. They also imply that one can use
the Euclidean algorithm to find a greatest common divisor of any two
nonzero elements ``a`` and ``b``, i.e.~an element that is divisible by any
nonzero elements ``a`` and ``b``, i.e. an element that is divisible by any
other common divisor. This, in turn, implies that factorization
into irreducible elements is unique up to multiplication by units.
Expand Down Expand Up @@ -348,7 +348,7 @@ See the file `GaussianInt.lean
Here we will instead carry out an argument that stays in the integers.
This illustrates a choice one commonly faces when formalizing mathematics.
Given an argument that requires concepts or machinery that is not already
in the library, one has two choices: either formalize the concepts or machinery
in the library, one has two choices: either formalize the concepts and machinery
needed, or adapt the argument to make use of concepts and machinery you
already have.
The first choice is generally a good investment of time when the results
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