Monad operations on MvPolynomial

This file defines two monadic operations on MvPolynomial. Given p : MvPolynomial σ R,

• MvPolynomial.bind₁ and MvPolynomial.join₁ operate on the variable type σ.
• MvPolynomial.bind₂ and MvPolynomial.join₂ operate on the coefficient type R.

• MvPolynomial.bind₁ f φ with f : σ → MvPolynomial τ R and φ : MvPolynomial σ R, is the polynomial φ(f 1, ..., f i, ...) : MvPolynomial τ R.
• MvPolynomial.join₁ φ with φ : MvPolynomial (MvPolynomial σ R) R collapses φ to a MvPolynomial σ R, by evaluating φ under the map X f ↦ f for f : MvPolynomial σ R. In other words, if you have a polynomial φ in a set of variables indexed by a polynomial ring, you evaluate the polynomial in these indexing polynomials.
• MvPolynomial.bind₂ f φ with f : R →+* MvPolynomial σ S and φ : MvPolynomial σ R is the MvPolynomial σ S obtained from φ by mapping the coefficients of φ through f and considering the resulting polynomial as polynomial expression in MvPolynomial σ R.
• MvPolynomial.join₂ φ with φ : MvPolynomial σ (MvPolynomial σ R) collapses φ to a MvPolynomial σ R, by considering φ as polynomial expression in MvPolynomial σ R.

These operations themselves have algebraic structure: MvPolynomial.bind₁ and MvPolynomial.join₁ are algebra homs and MvPolynomial.bind₂ and MvPolynomial.join₂ are ring homs.

They interact in convenient ways with MvPolynomial.rename, MvPolynomial.map, MvPolynomial.vars, and other polynomial operations. Indeed, MvPolynomial.rename is the "map" operation for the (bind₁, join₁) pair, whereas MvPolynomial.map is the "map" operation for the other pair.

Implementation notes

We add a LawfulMonad instance for the (bind₁, join₁) pair. The second pair cannot be instantiated as a Monad, since it is not a monad in Type but in CommRingCat (or rather CommSemiRingCat).

def MvPolynomial.bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] (f : σ → MvPolynomial τ R) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R

bind₁ is the "left-hand side" bind operation on MvPolynomial, operating on the variable type. Given a polynomial p : MvPolynomial σ R and a map f : σ → MvPolynomial τ R taking variables in p to polynomials in the variable type τ, bind₁ f p replaces each variable in p with its value under f, producing a new polynomial in τ. The coefficient type remains the same. This operation is an algebra hom.

Equations

• MvPolynomial.bind₁ f = MvPolynomial.aeval f

def MvPolynomial.bind₂ {σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* MvPolynomial σ S) : MvPolynomial σ R →+* MvPolynomial σ S

bind₂ is the "right-hand side" bind operation on MvPolynomial, operating on the coefficient type. Given a polynomial p : MvPolynomial σ R and a map f : R → MvPolynomial σ S taking coefficients in p to polynomials over a new ring S, bind₂ f p replaces each coefficient in p with its value under f, producing a new polynomial over S. The variable type remains the same. This operation is a ring hom.

Equations

• MvPolynomial.bind₂ f = MvPolynomial.eval₂Hom f MvPolynomial.X

def MvPolynomial.join₁ {σ : Type u_1} {R : Type u_3} [CommSemiring R] : MvPolynomial (MvPolynomial σ R) R →ₐ[R] MvPolynomial σ R

join₁ is the monadic join operation corresponding to MvPolynomial.bind₁. Given a polynomial p with coefficients in R whose variables are polynomials in σ with coefficients in R, join₁ p collapses p to a polynomial with variables in σ and coefficients in R. This operation is an algebra hom.

Equations

• MvPolynomial.join₁ = MvPolynomial.aeval id

def MvPolynomial.join₂ {σ : Type u_1} {R : Type u_3} [CommSemiring R] : MvPolynomial σ (MvPolynomial σ R) →+* MvPolynomial σ R

join₂ is the monadic join operation corresponding to MvPolynomial.bind₂. Given a polynomial p with variables in σ whose coefficients are polynomials in σ with coefficients in R, join₂ p collapses p to a polynomial with variables in σ and coefficients in R. This operation is a ring hom.

Equations

• MvPolynomial.join₂ = MvPolynomial.eval₂Hom (RingHom.id (MvPolynomial σ R)) MvPolynomial.X

@[simp]

theorem MvPolynomial.aeval_eq_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] (f : σ → MvPolynomial τ R) : aeval f = bind₁ f

@[simp]

theorem MvPolynomial.eval₂Hom_C_eq_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] (f : σ → MvPolynomial τ R) : eval₂Hom C f = ↑(bind₁ f)

@[simp]

theorem MvPolynomial.eval₂Hom_eq_bind₂ {σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* MvPolynomial σ S) : eval₂Hom f X = bind₂ f

@[simp]

theorem MvPolynomial.aeval_id_eq_join₁ (σ : Type u_1) (R : Type u_3) [CommSemiring R] : aeval id = join₁

theorem MvPolynomial.eval₂Hom_C_id_eq_join₁ (σ : Type u_1) (R : Type u_3) [CommSemiring R] (φ : MvPolynomial (MvPolynomial σ R) R) : (eval₂Hom C id) φ = join₁ φ

@[simp]

theorem MvPolynomial.eval₂Hom_id_X_eq_join₂ (σ : Type u_1) (R : Type u_3) [CommSemiring R] : eval₂Hom (RingHom.id (MvPolynomial σ R)) X = join₂

@[simp]

theorem MvPolynomial.bind₁_X_right {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] (f : σ → MvPolynomial τ R) (i : σ) : (bind₁ f) (X i) = f i

@[simp]

theorem MvPolynomial.bind₂_X_right {σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* MvPolynomial σ S) (i : σ) : (bind₂ f) (X i) = X i

@[simp]

theorem MvPolynomial.bind₁_X_left {σ : Type u_1} {R : Type u_3} [CommSemiring R] : bind₁ X = AlgHom.id R (MvPolynomial σ R)

theorem MvPolynomial.bind₁_C_right {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] (f : σ → MvPolynomial τ R) (x : R) : (bind₁ f) (C x) = C x

@[simp]

theorem MvPolynomial.bind₂_C_right {σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* MvPolynomial σ S) (r : R) : (bind₂ f) (C r) = f r

@[simp]

theorem MvPolynomial.bind₂_C_left {σ : Type u_1} {R : Type u_3} [CommSemiring R] : bind₂ C = RingHom.id (MvPolynomial σ R)

@[simp]

theorem MvPolynomial.bind₂_comp_C {σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* MvPolynomial σ S) : (bind₂ f).comp C = f

@[simp]

theorem MvPolynomial.join₂_map {σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* MvPolynomial σ S) (φ : MvPolynomial σ R) : join₂ ((map f) φ) = (bind₂ f) φ

@[simp]

theorem MvPolynomial.join₂_comp_map {σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* MvPolynomial σ S) : join₂.comp (map f) = bind₂ f

theorem MvPolynomial.aeval_id_rename {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] (f : σ → MvPolynomial τ R) (p : MvPolynomial σ R) : (aeval id) ((rename f) p) = (aeval f) p

@[simp]

theorem MvPolynomial.join₁_rename {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] (f : σ → MvPolynomial τ R) (φ : MvPolynomial σ R) : join₁ ((rename f) φ) = (bind₁ f) φ

@[simp]

theorem MvPolynomial.bind₁_id {σ : Type u_1} {R : Type u_3} [CommSemiring R] : bind₁ id = join₁

@[simp]

theorem MvPolynomial.bind₂_id {σ : Type u_1} {R : Type u_3} [CommSemiring R] : bind₂ (RingHom.id (MvPolynomial σ R)) = join₂

theorem MvPolynomial.bind₁_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] {υ : Type u_6} (f : σ → MvPolynomial τ R) (g : τ → MvPolynomial υ R) (φ : MvPolynomial σ R) : (bind₁ g) ((bind₁ f) φ) = (bind₁ fun (i : σ) => (bind₁ g) (f i)) φ

theorem MvPolynomial.bind₁_comp_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] {υ : Type u_6} (f : σ → MvPolynomial τ R) (g : τ → MvPolynomial υ R) : (bind₁ g).comp (bind₁ f) = bind₁ fun (i : σ) => (bind₁ g) (f i)

theorem MvPolynomial.bind₂_comp_bind₂ {σ : Type u_1} {R : Type u_3} {S : Type u_4} {T : Type u_5} [CommSemiring R] [CommSemiring S] [CommSemiring T] (f : R →+* MvPolynomial σ S) (g : S →+* MvPolynomial σ T) : (bind₂ g).comp (bind₂ f) = bind₂ ((bind₂ g).comp f)

theorem MvPolynomial.bind₂_bind₂ {σ : Type u_1} {R : Type u_3} {S : Type u_4} {T : Type u_5} [CommSemiring R] [CommSemiring S] [CommSemiring T] (f : R →+* MvPolynomial σ S) (g : S →+* MvPolynomial σ T) (φ : MvPolynomial σ R) : (bind₂ g) ((bind₂ f) φ) = (bind₂ ((bind₂ g).comp f)) φ

theorem MvPolynomial.rename_comp_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] {υ : Type u_6} (f : σ → MvPolynomial τ R) (g : τ → υ) : (rename g).comp (bind₁ f) = bind₁ fun (i : σ) => (rename g) (f i)

theorem MvPolynomial.rename_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] {υ : Type u_6} (f : σ → MvPolynomial τ R) (g : τ → υ) (φ : MvPolynomial σ R) : (rename g) ((bind₁ f) φ) = (bind₁ fun (i : σ) => (rename g) (f i)) φ

theorem MvPolynomial.map_bind₂ {σ : Type u_1} {R : Type u_3} {S : Type u_4} {T : Type u_5} [CommSemiring R] [CommSemiring S] [CommSemiring T] (f : R →+* MvPolynomial σ S) (g : S →+* T) (φ : MvPolynomial σ R) : (map g) ((bind₂ f) φ) = (bind₂ ((map g).comp f)) φ

theorem MvPolynomial.bind₁_comp_rename {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] {υ : Type u_6} (f : τ → MvPolynomial υ R) (g : σ → τ) : (bind₁ f).comp (rename g) = bind₁ (f ∘ g)

theorem MvPolynomial.bind₁_rename {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] {υ : Type u_6} (f : τ → MvPolynomial υ R) (g : σ → τ) (φ : MvPolynomial σ R) : (bind₁ f) ((rename g) φ) = (bind₁ (f ∘ g)) φ

theorem MvPolynomial.bind₂_map {σ : Type u_1} {R : Type u_3} {S : Type u_4} {T : Type u_5} [CommSemiring R] [CommSemiring S] [CommSemiring T] (f : S →+* MvPolynomial σ T) (g : R →+* S) (φ : MvPolynomial σ R) : (bind₂ f) ((map g) φ) = (bind₂ (f.comp g)) φ

@[simp]

theorem MvPolynomial.map_comp_C {σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* S) : (map f).comp C = C.comp f

theorem MvPolynomial.hom_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : MvPolynomial τ R →+* S) (g : σ → MvPolynomial τ R) (φ : MvPolynomial σ R) : f ((bind₁ g) φ) = (eval₂Hom (f.comp C) fun (i : σ) => f (g i)) φ

theorem MvPolynomial.map_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* S) (g : σ → MvPolynomial τ R) (φ : MvPolynomial σ R) : (map f) ((bind₁ g) φ) = (bind₁ fun (i : σ) => (map f) (g i)) ((map f) φ)

@[simp]

theorem MvPolynomial.eval₂Hom_comp_C {σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* S) (g : σ → S) : (eval₂Hom f g).comp C = f

theorem MvPolynomial.eval₂Hom_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* S) (g : τ → S) (h : σ → MvPolynomial τ R) (φ : MvPolynomial σ R) : (eval₂Hom f g) ((bind₁ h) φ) = (eval₂Hom f fun (i : σ) => (eval₂Hom f g) (h i)) φ

theorem MvPolynomial.aeval_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] [Algebra R S] (f : τ → S) (g : σ → MvPolynomial τ R) (φ : MvPolynomial σ R) : (aeval f) ((bind₁ g) φ) = (aeval fun (i : σ) => (aeval f) (g i)) φ

theorem MvPolynomial.aeval_comp_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] [Algebra R S] (f : τ → S) (g : σ → MvPolynomial τ R) : (aeval f).comp (bind₁ g) = aeval fun (i : σ) => (aeval f) (g i)

theorem MvPolynomial.eval₂Hom_comp_bind₂ {σ : Type u_1} {R : Type u_3} {S : Type u_4} {T : Type u_5} [CommSemiring R] [CommSemiring S] [CommSemiring T] (f : S →+* T) (g : σ → T) (h : R →+* MvPolynomial σ S) : (eval₂Hom f g).comp (bind₂ h) = eval₂Hom ((eval₂Hom f g).comp h) g

theorem MvPolynomial.eval₂Hom_bind₂ {σ : Type u_1} {R : Type u_3} {S : Type u_4} {T : Type u_5} [CommSemiring R] [CommSemiring S] [CommSemiring T] (f : S →+* T) (g : σ → T) (h : R →+* MvPolynomial σ S) (φ : MvPolynomial σ R) : (eval₂Hom f g) ((bind₂ h) φ) = (eval₂Hom ((eval₂Hom f g).comp h) g) φ

theorem MvPolynomial.aeval_bind₂ {σ : Type u_1} {R : Type u_3} {S : Type u_4} {T : Type u_5} [CommSemiring R] [CommSemiring S] [CommSemiring T] [Algebra S T] (f : σ → T) (g : R →+* MvPolynomial σ S) (φ : MvPolynomial σ R) : (aeval f) ((bind₂ g) φ) = (eval₂Hom ((↑(aeval f)).comp g) f) φ

theorem MvPolynomial.eval₂Hom_C_left {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] (f : σ → MvPolynomial τ R) : eval₂Hom C f = ↑(bind₁ f)

Alias of MvPolynomial.eval₂Hom_C_eq_bind₁.

theorem MvPolynomial.bind₁_monomial {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] (f : σ → MvPolynomial τ R) (d : σ →₀ ℕ) (r : R) : (bind₁ f) ((monomial d) r) = C r * ∏ i ∈ d.support, f i ^ d i

theorem MvPolynomial.bind₂_monomial {σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* MvPolynomial σ S) (d : σ →₀ ℕ) (r : R) : (bind₂ f) ((monomial d) r) = f r * (monomial d) 1

@[simp]

theorem MvPolynomial.bind₂_monomial_one {σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* MvPolynomial σ S) (d : σ →₀ ℕ) : (bind₂ f) ((monomial d) 1) = (monomial d) 1

theorem MvPolynomial.vars_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] [DecidableEq τ] (f : σ → MvPolynomial τ R) (φ : MvPolynomial σ R) : ((bind₁ f) φ).vars ⊆ φ.vars.biUnion fun (i : σ) => (f i).vars

theorem MvPolynomial.mem_vars_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] (f : σ → MvPolynomial τ R) (φ : MvPolynomial σ R) {j : τ} (h : j ∈ ((bind₁ f) φ).vars) : ∃ i ∈ φ.vars, j ∈ (f i).vars

instance MvPolynomial.monad {R : Type u_3} [CommSemiring R] : Monad fun (σ : Type u_6) => MvPolynomial σ R

Equations

• One or more equations did not get rendered due to their size.

instance MvPolynomial.lawfulFunctor {R : Type u_3} [CommSemiring R] : LawfulFunctor fun (σ : Type u_6) => MvPolynomial σ R

instance MvPolynomial.lawfulMonad {R : Type u_3} [CommSemiring R] : LawfulMonad fun (σ : Type u_6) => MvPolynomial σ R