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feat(data/power_series): Add multivariate power series and prove basic API #1244
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jcommelin
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Jul 20, 2019
ChrisHughes24
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Jul 22, 2019
anrddh
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May 15, 2020
…c API (leanprover-community#1244) * First start on power_series * Innocent changes * Almost a comm_semiring * Defined hom from mv_polynomial to mv_power_series; sorrys remain * Attempt that seem to go nowhere * Working on coeff_mul for polynomials * Small progress * Finish mv_polynomial.coeff_mul * Cleaner proof of mv_polynomial.coeff_mul * Fix build * WIP * Finish proof of mul_assoc * WIP * Golfing coeff_mul * WIP * Crazy wf is crazy * mv_power_series over local ring is local * WIP * Add empty line * wip * wip * WIP * WIP * WIP * Add header comments * WIP * WIP * Fix finsupp build * Fix build, hopefully * Fix build: ideals * More docs * Update src/data/power_series.lean Fix typo. * Fix build -- bump instance search depth * Make changes according to some of the review comments * Use 'formal' in the names * Use 'protected' in more places, remove '@simp's * Make 'inv_eq_zero' an iff * Generalize to non-commutative scalars * Move file * Undo name change, back to 'power_series' * spelling mistake * spelling mistake
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formal power series
This file defines (multivariate) formal power series
and develops the basic properties of these objects.
A formal power series is to a polynomial like an infinite sum is to a finite sum.
We provide the natural inclusion from polynomials to power series.
trunc n φ
truncates a power series to the polynomialthat has the same coefficients as φ, for all m ≤ n, and 0 otherwise.
If the constant coefficient of a formal power series is invertible,
then this formal power series is invertible.
Formal power series over a local ring form a local ring.
Implementation notes
In this file we define multivariate power series with coefficients in
α
asmv_power_series σ α := (σ →₀ ℕ) → α
Unfortunately there is not yet enough API to show that they are the completion
of the ring of multivariate polynomials. However, we provide most of the infrastructure
that is needed to do this. Once I-adic completion (topological or algebraic) is available
it should not be hard to fill in the details.
Formal power series in one variable are defined as
power_series α := mv_power_series unit α
This allows us to port a lot of proofs and properties
from the multivariate case to the single variable case.
However, it means that power series are indexed by (unit →₀ ℕ),
which is of course canonically isomorphic to ℕ.
We then build some glue to treat power series as if they are indexed by ℕ.
Occasionally this leads to proofs that are uglier than expected.