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And in docstrings.
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Julian committed Oct 9, 2024
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10 changes: 6 additions & 4 deletions Mathlib/RingTheory/LaurentSeries.lean
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Expand Up @@ -17,10 +17,10 @@ import Mathlib.Topology.UniformSpace.Cauchy
# Laurent Series
## Main Definitions
* Defines `LaurentSeries` as an abbreviation for `HahnSeries ℤ`.
* Defines `hasseDeriv` of a Laurent series with coefficients in a module over a ring.
* Provides a coercion `PowerSeries R` into `LaurentSeries R` given by
`HahnSeries.ofPowerSeries`.
* Provides a coercion `from power series `R⟦X⟧` into `R⸨X⸩` given by `HahnSeries.ofPowerSeries`.
* Defines `LaurentSeries.powerSeriesPart`
* Defines the localization map `LaurentSeries.of_powerSeries_localization` which evaluates to
`HahnSeries.ofPowerSeries`.
Expand All @@ -34,6 +34,7 @@ element `LaurentSeries.Cauchy.coeff ℱ d` in `K` that serves as `d`th coefficie
series to which the filter `ℱ` converges.
## Main Results
* Basic properties of Hasse derivatives
### About the `X`-Adic valuation:
* The (integral) valuation of a power series is the order of the first non-zero coefficient, see
Expand All @@ -46,9 +47,10 @@ series to which the filter `ℱ` converges.
`instLaurentSeriesComplete`.
## Implementation details
* Since `LaurentSeries` is just an abbreviation of `HahnSeries ℤ _`, the definition of the
coefficients is given in terms of `HahnSeries.coeff` and this forces sometimes to go back-and-forth
from `X : LaurentSeries _` to `single 1 1 : HahnSeries ℤ _`.
from `X : _⸨X⸩` to `single 1 1 : HahnSeries ℤ _`.
-/
universe u
Expand Down Expand Up @@ -648,7 +650,7 @@ theorem uniformContinuous_coeff {uK : UniformSpace K} (d : ℤ) :
exact mem_uniformity_of_eq hS rfl

/-- Since extracting coefficients is uniformly continuous, every Cauchy filter in
`laurentSeries K` gives rise to a Cauchy filter in `K` for every `d : ℤ`, and such Cauchy filter
`K⸨X⸩` gives rise to a Cauchy filter in `K` for every `d : ℤ`, and such Cauchy filter
in `K` converges to a principal filter -/
def Cauchy.coeff {ℱ : Filter K⸨X⸩} (hℱ : Cauchy ℱ) : ℤ → K :=
let _ : UniformSpace K := ⊥
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