File: Mathlib/RingTheory/MvPolynomial/Basic.lean

full_name: MvPolynomial.instCharP
code: instance [CharP R p] : CharP (MvPolynomial σ R) p 
goal: CharP (MvPolynomial σ R) p

full_name: MvPolynomial.instCharZero
code: instance [CharZero R] : CharZero (MvPolynomial σ R) 
goal: CharZero (MvPolynomial σ R)

full_name: MvPolynomial.mapRange_eq_map
code: theorem mapRange_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] (p : MvPolynomial σ R)
    (f : R →+* S) : Finsupp.mapRange f f.map_zero p = map f p 
goal: Finsupp.mapRange ⇑f ⋯ p = (MvPolynomial.map f) p

full_name: MvPolynomial.restrictSupport
code: /-- The submodule of polynomials that are sum of monomials in the set `s`. -/
def restrictSupport (s : Set (σ →₀ ℕ)) : Submodule R (MvPolynomial σ R) 
goal: Submodule R (MvPolynomial σ R)

full_name: MvPolynomial.basisRestrictSupport
code: /-- `restrictSupport R s` has a canonical `R`-basis indexed by `s`. -/
def basisRestrictSupport (s : Set (σ →₀ ℕ)) : Basis s R (restrictSupport R s) 
goal: Basis (↑s) R ↥(MvPolynomial.restrictSupport R s)

full_name: MvPolynomial.restrictSupport_mono
code: theorem restrictSupport_mono {s t : Set (σ →₀ ℕ)} (h : s ⊆ t) :
    restrictSupport R s ≤ restrictSupport R t 
goal: MvPolynomial.restrictSupport R s ≤ MvPolynomial.restrictSupport R t

full_name: MvPolynomial.restrictTotalDegree
code: /-- The submodule of polynomials of total degree less than or equal to `m`. -/
def restrictTotalDegree (m : ℕ) : Submodule R (MvPolynomial σ R) 
goal: Submodule R (MvPolynomial σ R)

full_name: MvPolynomial.restrictDegree
code: /-- The submodule of polynomials such that the degree with respect to each individual variable is
less than or equal to `m`. -/
def restrictDegree (m : ℕ) : Submodule R (MvPolynomial σ R) 
goal: Submodule R (MvPolynomial σ R)

full_name: MvPolynomial.mem_restrictTotalDegree
code: theorem mem_restrictTotalDegree (p : MvPolynomial σ R) :
    p ∈ restrictTotalDegree σ R m ↔ p.totalDegree ≤ m 
goal: p ∈ MvPolynomial.restrictTotalDegree σ R m ↔ p.totalDegree ≤ m

full_name: MvPolynomial.mem_restrictDegree
code: theorem mem_restrictDegree (p : MvPolynomial σ R) (n : ℕ) :
    p ∈ restrictDegree σ R n ↔ ∀ s ∈ p.support, ∀ i, (s : σ →₀ ℕ) i ≤ n 
goal: p ∈ MvPolynomial.restrictDegree σ R n ↔ ∀ s ∈ p.support, ∀ (i : σ), s i ≤ n

full_name: MvPolynomial.mem_restrictDegree_iff_sup
code: theorem mem_restrictDegree_iff_sup [DecidableEq σ] (p : MvPolynomial σ R) (n : ℕ) :
    p ∈ restrictDegree σ R n ↔ ∀ i, p.degrees.count i ≤ n 
goal: p ∈ MvPolynomial.restrictDegree σ R n ↔ ∀ (i : σ), Multiset.count i p.degrees ≤ n

full_name: MvPolynomial.restrictTotalDegree_le_restrictDegree
code: theorem restrictTotalDegree_le_restrictDegree (m : ℕ) :
    restrictTotalDegree σ R m ≤ restrictDegree σ R m 
goal: MvPolynomial.restrictTotalDegree σ R m ≤ MvPolynomial.restrictDegree σ R m

full_name: MvPolynomial.basisMonomials
code: /-- The monomials form a basis on `MvPolynomial σ R`. -/
def basisMonomials : Basis (σ →₀ ℕ) R (MvPolynomial σ R) 
goal: Basis (σ →₀ ℕ) R (MvPolynomial σ R)

full_name: MvPolynomial.coe_basisMonomials
code: @[simp]
theorem coe_basisMonomials :
    (basisMonomials σ R : (σ →₀ ℕ) → MvPolynomial σ R) = fun s => monomial s 1 
goal: ⇑(MvPolynomial.basisMonomials σ R) = fun s => (MvPolynomial.monomial s) 1

full_name: MvPolynomial.instFree
code: /-- The `R`-module `MvPolynomial σ R` is free. -/
instance : Module.Free R (MvPolynomial σ R) 
goal: Module.Free R (MvPolynomial σ R)

full_name: MvPolynomial.linearIndependent_X
code: theorem linearIndependent_X : LinearIndependent R (X : σ → MvPolynomial σ R) 
goal: LinearIndependent R MvPolynomial.X

full_name: _private.Mathlib.RingTheory.MvPolynomial.Basic.0.MvPolynomial.finite_setOf_bounded
code: private lemma finite_setOf_bounded (α) [Finite α] (n : ℕ) : Finite {f : α →₀ ℕ | ∀ a, f a ≤ n} 
goal: Finite ↑{f | ∀ (a : α), f a ≤ n}

full_name: MvPolynomial.instFiniteSubtypeMemSubmoduleRestrictDegreeOfFinite
code: instance [Finite σ] (N : ℕ) : Module.Finite R (restrictDegree σ R N) 
goal: Module.Finite R ↥(MvPolynomial.restrictDegree σ R N)

full_name: MvPolynomial.instFiniteSubtypeMemSubmoduleRestrictTotalDegreeOfFinite
code: instance [Finite σ] (N : ℕ) : Module.Finite R (restrictTotalDegree σ R N) 
goal: Module.Finite R ↥(MvPolynomial.restrictTotalDegree σ R N)

full_name: MvPolynomial.algebraMvPolynomial
code: /--
If `S` is an `R`-algebra, then `MvPolynomial σ S` is a `MvPolynomial σ R` algebra.

Warning: This produces a diamond for
`Algebra (MvPolynomial σ R) (MvPolynomial σ (MvPolynomial σ S))`. That's why it is not a
global instance.
-/
noncomputable def algebraMvPolynomial : Algebra (MvPolynomial σ R) (MvPolynomial σ S) 
goal: Algebra (MvPolynomial σ R) (MvPolynomial σ S)

full_name: MvPolynomial.algebraMap_def
code: @[simp]
lemma algebraMap_def :
    algebraMap (MvPolynomial σ R) (MvPolynomial σ S) = MvPolynomial.map (algebraMap R S) 
goal: algebraMap (MvPolynomial σ R) (MvPolynomial σ S) = MvPolynomial.map (algebraMap R S)

full_name: MvPolynomial.instIsScalarTower
code: instance : IsScalarTower R (MvPolynomial σ R) (MvPolynomial σ S) 
goal: IsScalarTower R (MvPolynomial σ R) (MvPolynomial σ S)

full_name: Polynomial.basisMonomials
code: /-- The monomials form a basis on `R[X]`. -/
noncomputable def basisMonomials : Basis ℕ R R[X] 
goal: Basis ℕ R R[X]

full_name: Polynomial.coe_basisMonomials
code: @[simp]
theorem coe_basisMonomials : (basisMonomials R : ℕ → R[X]) = fun s => monomial s 1 
goal: ⇑(Polynomial.basisMonomials R) = fun s => (Polynomial.monomial s) 1