RegularFunction.lean`

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6 declarations, 0 contain sorry, 0 declared as axioms

Regular functions between affine spaces

A regular function from $k^σ$ to $k^τ$ is a $τ$-indexed family of polynomials in the variables $σ$, i.e. a vector valued polynomial function. This file provides the basic API: the Jacobian matrix, composition, the identity, and evaluation at a point.

Imports (3)

Imported by 1 file

FormalConjecturesForMathlib.lean

abbrev MvPolynomial.RegularFunction The type of regular functions from $k^σ$ to $k^τ$ line 39 source
Source documentation
The type of regular functions from $k^σ$ to $k^τ$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing
Proof uses: MvPolynomial
1 instances, coercions and auxiliary constants left out.
```
abbrev RegularFunction := τ → MvPolynomial σ k
…
```
def MvPolynomial.RegularFunction.Jacobian The Jacobian of a vector valued polynomial function, viewed as a polynomial line 44 source
Source documentation
The Jacobian of a vector valued polynomial function, viewed as a polynomial.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, MvPolynomial.RegularFunction, Matrix, MvPolynomial
Proof uses: Equiv, Matrix.of, Derivation, Finsupp, Nat, MvPolynomial.pderiv
18 instances, coercions and auxiliary constants left out.
```
noncomputable def Jacobian (F : RegularFunction k σ τ) :
    Matrix σ τ (MvPolynomial σ k) :=
…
```
Mentions: MvPolynomial.RegularFunction
def MvPolynomial.RegularFunction.comp The composition of two vector valued polynomial functions line 49 source
Source documentation
The composition of two vector valued polynomial functions.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, MvPolynomial.RegularFunction
Proof uses: AlgHom, MvPolynomial, Finsupp, Nat, MvPolynomial.bind₁
11 instances, coercions and auxiliary constants left out.
```
noncomputable def comp
    (F : RegularFunction k σ τ) (G : RegularFunction k τ ι) :
    RegularFunction k σ ι :=
…
```
Mentions: MvPolynomial.RegularFunction
def MvPolynomial.RegularFunction.id line 55 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, MvPolynomial.RegularFunction
Proof uses: MvPolynomial.X
1 instances, coercions and auxiliary constants left out.
def MvPolynomial.RegularFunction.aeval The evaluation of a regular function f over k at some point a with coordinates in some algebra over k line 59 source
Source documentation
The evaluation of a regular function f over k at some point a with coordinates in some algebra over k

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, CommSemiring, Algebra, MvPolynomial.RegularFunction
Proof uses: AlgHom, MvPolynomial, Finsupp, Nat, MvPolynomial.aeval
13 instances, coercions and auxiliary constants left out.
```
noncomputable def aeval {σ τ : Type*} {S₁ : Type*} [CommSemiring S₁] [Algebra k S₁]
    (F : RegularFunction k σ τ) : (σ → S₁) → τ → S₁ :=
…
```
Mentions: MvPolynomial.RegularFunction
lemma MvPolynomial.RegularFunction.comp_aeval aeval is compatible with composition of regular functions line 64 source
Source documentation
aeval is compatible with composition of regular functions.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, MvPolynomial.RegularFunction, Eq, MvPolynomial.RegularFunction.aeval, MvPolynomial.RegularFunction.comp
Proof uses: funext, Eq.mpr, AlgHom, MvPolynomial, Finsupp, Nat, MvPolynomial.aeval, id, congrArg, MvPolynomial.RegularFunction.aeval.eq_1, MvPolynomial.bind₁, MvPolynomial.RegularFunction.comp.eq_1, MvPolynomial.aeval_bind₁, Eq.symm, Eq.refl
13 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
lemma comp_aeval
    {σ τ ι : Type*}
    (F : RegularFunction k σ τ) (G : RegularFunction k τ ι)
    (a : σ → k) : (F.comp G).aeval a = G.aeval (F.aeval a) := by
…
```
Mentions: MvPolynomial.RegularFunction, MvPolynomial.RegularFunction.aeval