state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
α β γ : Type u
t : Type u → Type u
inst✝¹ : Traversable t
inst✝ : LawfulTraversable t
head✝ : α
tail✝ : List α
ih : FreeMonoid.toList (List.traverse (Const.mk' ∘ FreeMonoid.of) tail✝) = tail✝
| head✝ :: tail✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | rw [← ih]; rfl | @[simp]
theorem toList_eq_self {xs : List α} : toList xs = xs := by
simp only [toList_spec, foldMap, traverse]
induction xs
case nil => rfl
case cons _ _ ih => conv_rhs => | Mathlib.Control.Fold.382_0.ilkJEkQU7vZZ6HB | @[simp]
theorem toList_eq_self {xs : List α} : toList xs = xs | Mathlib_Control_Fold |
α β γ : Type u
t : Type u → Type u
inst✝¹ : Traversable t
inst✝ : LawfulTraversable t
head✝ : α
tail✝ : List α
ih : FreeMonoid.toList (List.traverse (Const.mk' ∘ FreeMonoid.of) tail✝) = tail✝
| head✝ :: tail✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | rw [← ih] | @[simp]
theorem toList_eq_self {xs : List α} : toList xs = xs := by
simp only [toList_spec, foldMap, traverse]
induction xs
case nil => rfl
case cons _ _ ih => conv_rhs => | Mathlib.Control.Fold.382_0.ilkJEkQU7vZZ6HB | @[simp]
theorem toList_eq_self {xs : List α} : toList xs = xs | Mathlib_Control_Fold |
α β γ : Type u
t : Type u → Type u
inst✝¹ : Traversable t
inst✝ : LawfulTraversable t
head✝ : α
tail✝ : List α
ih : FreeMonoid.toList (List.traverse (Const.mk' ∘ FreeMonoid.of) tail✝) = tail✝
| head✝ :: FreeMonoid.toList (List.traverse (Const.mk' ∘ FreeMonoid.of) tail✝) | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | rfl | @[simp]
theorem toList_eq_self {xs : List α} : toList xs = xs := by
simp only [toList_spec, foldMap, traverse]
induction xs
case nil => rfl
case cons _ _ ih => conv_rhs => rw [← ih]; | Mathlib.Control.Fold.382_0.ilkJEkQU7vZZ6HB | @[simp]
theorem toList_eq_self {xs : List α} : toList xs = xs | Mathlib_Control_Fold |
α β γ : Type u
t : Type u → Type u
inst✝¹ : Traversable t
inst✝ : LawfulTraversable t
xs : t α
⊢ length xs = List.length (toList xs) | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | unfold length | theorem length_toList {xs : t α} : length xs = List.length (toList xs) := by
| Mathlib.Control.Fold.390_0.ilkJEkQU7vZZ6HB | theorem length_toList {xs : t α} : length xs = List.length (toList xs) | Mathlib_Control_Fold |
α β γ : Type u
t : Type u → Type u
inst✝¹ : Traversable t
inst✝ : LawfulTraversable t
xs : t α
⊢ (foldl (fun l x => { down := l.down + 1 }) { down := 0 } xs).down = List.length (toList xs) | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | rw [foldl_toList] | theorem length_toList {xs : t α} : length xs = List.length (toList xs) := by
unfold length
| Mathlib.Control.Fold.390_0.ilkJEkQU7vZZ6HB | theorem length_toList {xs : t α} : length xs = List.length (toList xs) | Mathlib_Control_Fold |
α β γ : Type u
t : Type u → Type u
inst✝¹ : Traversable t
inst✝ : LawfulTraversable t
xs : t α
⊢ (List.foldl (fun l x => { down := l.down + 1 }) { down := 0 } (toList xs)).down = List.length (toList xs) | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | generalize toList xs = ys | theorem length_toList {xs : t α} : length xs = List.length (toList xs) := by
unfold length
rw [foldl_toList]
| Mathlib.Control.Fold.390_0.ilkJEkQU7vZZ6HB | theorem length_toList {xs : t α} : length xs = List.length (toList xs) | Mathlib_Control_Fold |
α β γ : Type u
t : Type u → Type u
inst✝¹ : Traversable t
inst✝ : LawfulTraversable t
xs : t α
ys : List α
⊢ (List.foldl (fun l x => { down := l.down + 1 }) { down := 0 } ys).down = List.length ys | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | rw [← Nat.add_zero ys.length] | theorem length_toList {xs : t α} : length xs = List.length (toList xs) := by
unfold length
rw [foldl_toList]
generalize toList xs = ys
| Mathlib.Control.Fold.390_0.ilkJEkQU7vZZ6HB | theorem length_toList {xs : t α} : length xs = List.length (toList xs) | Mathlib_Control_Fold |
α β γ : Type u
t : Type u → Type u
inst✝¹ : Traversable t
inst✝ : LawfulTraversable t
xs : t α
ys : List α
⊢ (List.foldl (fun l x => { down := l.down + 1 }) { down := 0 } ys).down = List.length ys + 0 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | generalize 0 = n | theorem length_toList {xs : t α} : length xs = List.length (toList xs) := by
unfold length
rw [foldl_toList]
generalize toList xs = ys
rw [← Nat.add_zero ys.length]
| Mathlib.Control.Fold.390_0.ilkJEkQU7vZZ6HB | theorem length_toList {xs : t α} : length xs = List.length (toList xs) | Mathlib_Control_Fold |
α β γ : Type u
t : Type u → Type u
inst✝¹ : Traversable t
inst✝ : LawfulTraversable t
xs : t α
ys : List α
n : ℕ
⊢ (List.foldl (fun l x => { down := l.down + 1 }) { down := n } ys).down = List.length ys + n | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | induction' ys with _ _ ih generalizing n | theorem length_toList {xs : t α} : length xs = List.length (toList xs) := by
unfold length
rw [foldl_toList]
generalize toList xs = ys
rw [← Nat.add_zero ys.length]
generalize 0 = n
| Mathlib.Control.Fold.390_0.ilkJEkQU7vZZ6HB | theorem length_toList {xs : t α} : length xs = List.length (toList xs) | Mathlib_Control_Fold |
case nil
α β γ : Type u
t : Type u → Type u
inst✝¹ : Traversable t
inst✝ : LawfulTraversable t
xs : t α
n : ℕ
⊢ (List.foldl (fun l x => { down := l.down + 1 }) { down := n } []).down = List.length [] + n | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | simp | theorem length_toList {xs : t α} : length xs = List.length (toList xs) := by
unfold length
rw [foldl_toList]
generalize toList xs = ys
rw [← Nat.add_zero ys.length]
generalize 0 = n
induction' ys with _ _ ih generalizing n
· | Mathlib.Control.Fold.390_0.ilkJEkQU7vZZ6HB | theorem length_toList {xs : t α} : length xs = List.length (toList xs) | Mathlib_Control_Fold |
case cons
α β γ : Type u
t : Type u → Type u
inst✝¹ : Traversable t
inst✝ : LawfulTraversable t
xs : t α
head✝ : α
tail✝ : List α
ih : ∀ (n : ℕ), (List.foldl (fun l x => { down := l.down + 1 }) { down := n } tail✝).down = List.length tail✝ + n
n : ℕ
⊢ (List.foldl (fun l x => { down := l.down + 1 }) { down := n } (head✝... | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | simp_arith [ih] | theorem length_toList {xs : t α} : length xs = List.length (toList xs) := by
unfold length
rw [foldl_toList]
generalize toList xs = ys
rw [← Nat.add_zero ys.length]
generalize 0 = n
induction' ys with _ _ ih generalizing n
· simp
· | Mathlib.Control.Fold.390_0.ilkJEkQU7vZZ6HB | theorem length_toList {xs : t α} : length xs = List.length (toList xs) | Mathlib_Control_Fold |
α β γ : Type u
t : Type u → Type u
inst✝³ : Traversable t
inst✝² : LawfulTraversable t
m : Type u → Type u
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → β → m α
x : α
xs : t β
⊢ foldlm f x xs = unop ((foldlM.ofFreeMonoid f) (FreeMonoid.ofList (toList xs))) x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | simp only [foldlm, toList_spec, foldMap_hom_free (foldlM.ofFreeMonoid f),
foldlm.ofFreeMonoid_comp_of, foldlM.get, FreeMonoid.ofList_toList] | theorem foldlm_toList {f : α → β → m α} {x : α} {xs : t β} :
foldlm f x xs = List.foldlM f x (toList xs) :=
calc
foldlm f x xs = unop (foldlM.ofFreeMonoid f (FreeMonoid.ofList <| toList xs)) x :=
by | Mathlib.Control.Fold.403_0.ilkJEkQU7vZZ6HB | theorem foldlm_toList {f : α → β → m α} {x : α} {xs : t β} :
foldlm f x xs = List.foldlM f x (toList xs) | Mathlib_Control_Fold |
α β γ : Type u
t : Type u → Type u
inst✝³ : Traversable t
inst✝² : LawfulTraversable t
m : Type u → Type u
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → β → m α
x : α
xs : t β
⊢ unop ((foldlM.ofFreeMonoid f) (FreeMonoid.ofList (toList xs))) x = List.foldlM f x (toList xs) | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | simp [foldlM.ofFreeMonoid, unop_op, flip] | theorem foldlm_toList {f : α → β → m α} {x : α} {xs : t β} :
foldlm f x xs = List.foldlM f x (toList xs) :=
calc
foldlm f x xs = unop (foldlM.ofFreeMonoid f (FreeMonoid.ofList <| toList xs)) x :=
by simp only [foldlm, toList_spec, foldMap_hom_free (foldlM.ofFreeMonoid f),
foldlm.ofFreeMonoid_comp_... | Mathlib.Control.Fold.403_0.ilkJEkQU7vZZ6HB | theorem foldlm_toList {f : α → β → m α} {x : α} {xs : t β} :
foldlm f x xs = List.foldlM f x (toList xs) | Mathlib_Control_Fold |
α β γ : Type u
t : Type u → Type u
inst✝³ : Traversable t
inst✝² : LawfulTraversable t
m : Type u → Type u
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → β → m β
x : β
xs : t α
⊢ foldrm f x xs = List.foldrM f x (toList xs) | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | change _ = foldrM.ofFreeMonoid f (FreeMonoid.ofList <| toList xs) x | theorem foldrm_toList (f : α → β → m β) (x : β) (xs : t α) :
foldrm f x xs = List.foldrM f x (toList xs) := by
| Mathlib.Control.Fold.412_0.ilkJEkQU7vZZ6HB | theorem foldrm_toList (f : α → β → m β) (x : β) (xs : t α) :
foldrm f x xs = List.foldrM f x (toList xs) | Mathlib_Control_Fold |
α β γ : Type u
t : Type u → Type u
inst✝³ : Traversable t
inst✝² : LawfulTraversable t
m : Type u → Type u
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → β → m β
x : β
xs : t α
⊢ foldrm f x xs = (foldrM.ofFreeMonoid f) (FreeMonoid.ofList (toList xs)) x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | simp only [foldrm, toList_spec, foldMap_hom_free (foldrM.ofFreeMonoid f),
foldrm.ofFreeMonoid_comp_of, foldrM.get, FreeMonoid.ofList_toList] | theorem foldrm_toList (f : α → β → m β) (x : β) (xs : t α) :
foldrm f x xs = List.foldrM f x (toList xs) := by
change _ = foldrM.ofFreeMonoid f (FreeMonoid.ofList <| toList xs) x
| Mathlib.Control.Fold.412_0.ilkJEkQU7vZZ6HB | theorem foldrm_toList (f : α → β → m β) (x : β) (xs : t α) :
foldrm f x xs = List.foldrM f x (toList xs) | Mathlib_Control_Fold |
α β γ : Type u
t : Type u → Type u
inst✝³ : Traversable t
inst✝² : LawfulTraversable t
m : Type u → Type u
inst✝¹ : Monad m
inst✝ : LawfulMonad m
g : β → γ
f : α → γ → m α
a : α
l : t β
⊢ foldlm f a (g <$> l) = foldlm (fun x y => f x (g y)) a l | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | simp only [foldlm, foldMap_map, (· ∘ ·), flip] | @[simp]
theorem foldlm_map (g : β → γ) (f : α → γ → m α) (a : α) (l : t β) :
foldlm f a (g <$> l) = foldlm (fun x y => f x (g y)) a l := by
| Mathlib.Control.Fold.419_0.ilkJEkQU7vZZ6HB | @[simp]
theorem foldlm_map (g : β → γ) (f : α → γ → m α) (a : α) (l : t β) :
foldlm f a (g <$> l) = foldlm (fun x y => f x (g y)) a l | Mathlib_Control_Fold |
α β γ : Type u
t : Type u → Type u
inst✝³ : Traversable t
inst✝² : LawfulTraversable t
m : Type u → Type u
inst✝¹ : Monad m
inst✝ : LawfulMonad m
g : β → γ
f : γ → α → m α
a : α
l : t β
⊢ foldrm f a (g <$> l) = foldrm (f ∘ g) a l | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | simp only [foldrm, foldMap_map, (· ∘ ·), flip] | @[simp]
theorem foldrm_map (g : β → γ) (f : γ → α → m α) (a : α) (l : t β) :
foldrm f a (g <$> l) = foldrm (f ∘ g) a l := by | Mathlib.Control.Fold.425_0.ilkJEkQU7vZZ6HB | @[simp]
theorem foldrm_map (g : β → γ) (f : γ → α → m α) (a : α) (l : t β) :
foldrm f a (g <$> l) = foldrm (f ∘ g) a l | Mathlib_Control_Fold |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
⊢ LeftInverse (Polynomial.eval₂ C (X PUnit.unit)) (eval₂ Polynomial.C fun x => Polynomial.X) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | let f : R[X] →+* MvPolynomial PUnit R := Polynomial.eval₂RingHom MvPolynomial.C (X PUnit.unit) | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun := eval₂ Polynomial.C fun _ => Polynomial.X
invFun := Polynomial.eval₂ MvPolynomial.C (X PUnit.unit)
left_inv := by
| Mathlib.Data.MvPolynomial.Equiv.61_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
f : R[X] →+* MvPolynomial PUnit.{?u.1943 + 1} R := eval₂RingHom C (X PUnit.unit)
⊢ LeftInverse (Polynomial.eval₂ C (X PUnit.unit)) (eval₂ Polynomial.C fun x => Polynomial.X) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | let g : MvPolynomial PUnit R →+* R[X] := eval₂Hom Polynomial.C fun _ => Polynomial.X | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun := eval₂ Polynomial.C fun _ => Polynomial.X
invFun := Polynomial.eval₂ MvPolynomial.C (X PUnit.unit)
left_inv := by
... | Mathlib.Data.MvPolynomial.Equiv.61_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
f : R[X] →+* MvPolynomial PUnit.{?u.1943 + 1} R := eval₂RingHom C (X PUnit.unit)
g : MvPolynomial PUnit.{?u.2335 + 1} R →+* R[X] := eval₂Hom Polynomial.C fun x => Polynomial.X
⊢ LeftInverse (Polynomial.eva... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | show ∀ p, f.comp g p = p | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun := eval₂ Polynomial.C fun _ => Polynomial.X
invFun := Polynomial.eval₂ MvPolynomial.C (X PUnit.unit)
left_inv := by
... | Mathlib.Data.MvPolynomial.Equiv.61_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
f : R[X] →+* MvPolynomial PUnit.{?u.2335 + 1} R := eval₂RingHom C (X PUnit.unit)
g : MvPolynomial PUnit.{?u.2335 + 1} R →+* R[X] := eval₂Hom Polynomial.C fun x => Polynomial.X
⊢ ∀ (p : MvPolynomial PUnit.{... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | apply is_id | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun := eval₂ Polynomial.C fun _ => Polynomial.X
invFun := Polynomial.eval₂ MvPolynomial.C (X PUnit.unit)
left_inv := by
... | Mathlib.Data.MvPolynomial.Equiv.61_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun | Mathlib_Data_MvPolynomial_Equiv |
case hC
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
f : R[X] →+* MvPolynomial PUnit.{?u.2335 + 1} R := eval₂RingHom C (X PUnit.unit)
g : MvPolynomial PUnit.{?u.2335 + 1} R →+* R[X] := eval₂Hom Polynomial.C fun x => Polynomial.X
⊢ RingHom.comp (RingH... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | ext a | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun := eval₂ Polynomial.C fun _ => Polynomial.X
invFun := Polynomial.eval₂ MvPolynomial.C (X PUnit.unit)
left_inv := by
... | Mathlib.Data.MvPolynomial.Equiv.61_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun | Mathlib_Data_MvPolynomial_Equiv |
case hC.a.a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a✝ a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
f : R[X] →+* MvPolynomial PUnit.{?u.2335 + 1} R := eval₂RingHom C (X PUnit.unit)
g : MvPolynomial PUnit.{?u.2335 + 1} R →+* R[X] := eval₂Hom Polynomial.C fun x => Polynomial.X
a : R
m✝ : PUnit... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | dsimp | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun := eval₂ Polynomial.C fun _ => Polynomial.X
invFun := Polynomial.eval₂ MvPolynomial.C (X PUnit.unit)
left_inv := by
... | Mathlib.Data.MvPolynomial.Equiv.61_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun | Mathlib_Data_MvPolynomial_Equiv |
case hC.a.a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a✝ a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
f : R[X] →+* MvPolynomial PUnit.{?u.2335 + 1} R := eval₂RingHom C (X PUnit.unit)
g : MvPolynomial PUnit.{?u.2335 + 1} R →+* R[X] := eval₂Hom Polynomial.C fun x => Polynomial.X
a : R
m✝ : PUnit... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [eval₂_C, Polynomial.eval₂_C] | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun := eval₂ Polynomial.C fun _ => Polynomial.X
invFun := Polynomial.eval₂ MvPolynomial.C (X PUnit.unit)
left_inv := by
... | Mathlib.Data.MvPolynomial.Equiv.61_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun | Mathlib_Data_MvPolynomial_Equiv |
case hX
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
f : R[X] →+* MvPolynomial PUnit.{?u.2335 + 1} R := eval₂RingHom C (X PUnit.unit)
g : MvPolynomial PUnit.{?u.2335 + 1} R →+* R[X] := eval₂Hom Polynomial.C fun x => Polynomial.X
⊢ ∀ (n : PUnit.{?u.23... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rintro ⟨⟩ | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun := eval₂ Polynomial.C fun _ => Polynomial.X
invFun := Polynomial.eval₂ MvPolynomial.C (X PUnit.unit)
left_inv := by
... | Mathlib.Data.MvPolynomial.Equiv.61_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun | Mathlib_Data_MvPolynomial_Equiv |
case hX.unit
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
f : R[X] →+* MvPolynomial PUnit.{?u.2335 + 1} R := eval₂RingHom C (X PUnit.unit)
g : MvPolynomial PUnit.{?u.2335 + 1} R →+* R[X] := eval₂Hom Polynomial.C fun x => Polynomial.X
⊢ (RingHom.comp ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | dsimp | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun := eval₂ Polynomial.C fun _ => Polynomial.X
invFun := Polynomial.eval₂ MvPolynomial.C (X PUnit.unit)
left_inv := by
... | Mathlib.Data.MvPolynomial.Equiv.61_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun | Mathlib_Data_MvPolynomial_Equiv |
case hX.unit
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
f : R[X] →+* MvPolynomial PUnit.{?u.2335 + 1} R := eval₂RingHom C (X PUnit.unit)
g : MvPolynomial PUnit.{?u.2335 + 1} R →+* R[X] := eval₂Hom Polynomial.C fun x => Polynomial.X
⊢ Polynomial.eva... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [eval₂_X, Polynomial.eval₂_X] | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun := eval₂ Polynomial.C fun _ => Polynomial.X
invFun := Polynomial.eval₂ MvPolynomial.C (X PUnit.unit)
left_inv := by
... | Mathlib.Data.MvPolynomial.Equiv.61_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a✝ a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
p : R[X]
a : R
⊢ eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) (Polynomial.C a)) = Polynomial.C a | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [Polynomial.eval₂_C, MvPolynomial.eval₂_C] | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun := eval₂ Polynomial.C fun _ => Polynomial.X
invFun := Polynomial.eval₂ MvPolynomial.C (X PUnit.unit)
left_inv := by
... | Mathlib.Data.MvPolynomial.Equiv.61_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
p✝ p q : R[X]
hp : eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) p) = p
hq : eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) q) = q
⊢ eval₂ Pol... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [Polynomial.eval₂_add, MvPolynomial.eval₂_add, hp, hq] | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun := eval₂ Polynomial.C fun _ => Polynomial.X
invFun := Polynomial.eval₂ MvPolynomial.C (X PUnit.unit)
left_inv := by
... | Mathlib.Data.MvPolynomial.Equiv.61_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
p✝ : R[X]
p : ℕ
n : R
x✝ :
eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) (Polynomial.C n * Polynomial.X ^ p)) =
Polynomial.C n * Polynomial.X ^ p
⊢ eval₂ Polynomial.C ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [Polynomial.eval₂_mul, Polynomial.eval₂_pow, Polynomial.eval₂_X, Polynomial.eval₂_C,
eval₂_mul, eval₂_C, eval₂_pow, eval₂_X] | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun := eval₂ Polynomial.C fun _ => Polynomial.X
invFun := Polynomial.eval₂ MvPolynomial.C (X PUnit.unit)
left_inv := by
... | Mathlib.Data.MvPolynomial.Equiv.61_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def pUnitAlgEquiv : MvPolynomial PUnit R ≃ₐ[R] R[X] where
toFun | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e✝ : ℕ
s : σ →₀ ℕ
inst✝³ : CommSemiring R
inst✝² : CommSemiring S₁
inst✝¹ : CommSemiring S₂
inst✝ : CommSemiring S₃
e : S₁ ≃+* S₂
f : S₂ ≃+* S₃
p : MvPolynomial σ S₁
⊢ (RingEquiv.trans (mapEquiv σ e) (mapEquiv σ f)) p = (mapEquiv σ (RingEquiv.tr... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [RingEquiv.coe_trans, comp_apply, mapEquiv_apply, RingEquiv.coe_ringHom_trans,
map_map] | @[simp]
theorem mapEquiv_trans [CommSemiring S₁] [CommSemiring S₂] [CommSemiring S₃] (e : S₁ ≃+* S₂)
(f : S₂ ≃+* S₃) : (mapEquiv σ e).trans (mapEquiv σ f) = mapEquiv σ (e.trans f) :=
RingEquiv.ext fun p => by
| Mathlib.Data.MvPolynomial.Equiv.115_0.88gPfxLltQQTcHM | @[simp]
theorem mapEquiv_trans [CommSemiring S₁] [CommSemiring S₂] [CommSemiring S₃] (e : S₁ ≃+* S₂)
(f : S₂ ≃+* S₃) : (mapEquiv σ e).trans (mapEquiv σ f) = mapEquiv σ (e.trans f) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e✝ : ℕ
s : σ →₀ ℕ
inst✝⁶ : CommSemiring R
A₁ : Type u_2
A₂ : Type u_3
A₃ : Type u_4
inst✝⁵ : CommSemiring A₁
inst✝⁴ : CommSemiring A₂
inst✝³ : CommSemiring A₃
inst✝² : Algebra R A₁
inst✝¹ : Algebra R A₂
inst✝ : Algebra R A₃
e : A₁ ≃ₐ[R] A₂
f : A... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | ext | @[simp]
theorem mapAlgEquiv_trans (e : A₁ ≃ₐ[R] A₂) (f : A₂ ≃ₐ[R] A₃) :
(mapAlgEquiv σ e).trans (mapAlgEquiv σ f) = mapAlgEquiv σ (e.trans f) := by
| Mathlib.Data.MvPolynomial.Equiv.143_0.88gPfxLltQQTcHM | @[simp]
theorem mapAlgEquiv_trans (e : A₁ ≃ₐ[R] A₂) (f : A₂ ≃ₐ[R] A₃) :
(mapAlgEquiv σ e).trans (mapAlgEquiv σ f) = mapAlgEquiv σ (e.trans f) | Mathlib_Data_MvPolynomial_Equiv |
case h.a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e✝ : ℕ
s : σ →₀ ℕ
inst✝⁶ : CommSemiring R
A₁ : Type u_2
A₂ : Type u_3
A₃ : Type u_4
inst✝⁵ : CommSemiring A₁
inst✝⁴ : CommSemiring A₂
inst✝³ : CommSemiring A₃
inst✝² : Algebra R A₁
inst✝¹ : Algebra R A₂
inst✝ : Algebra R A₃
e : A₁ ≃ₐ[R]... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [AlgEquiv.trans_apply, mapAlgEquiv_apply, map_map] | @[simp]
theorem mapAlgEquiv_trans (e : A₁ ≃ₐ[R] A₂) (f : A₂ ≃ₐ[R] A₃) :
(mapAlgEquiv σ e).trans (mapAlgEquiv σ f) = mapAlgEquiv σ (e.trans f) := by
ext
| Mathlib.Data.MvPolynomial.Equiv.143_0.88gPfxLltQQTcHM | @[simp]
theorem mapAlgEquiv_trans (e : A₁ ≃ₐ[R] A₂) (f : A₂ ≃ₐ[R] A₃) :
(mapAlgEquiv σ e).trans (mapAlgEquiv σ f) = mapAlgEquiv σ (e.trans f) | Mathlib_Data_MvPolynomial_Equiv |
case h.a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e✝ : ℕ
s : σ →₀ ℕ
inst✝⁶ : CommSemiring R
A₁ : Type u_2
A₂ : Type u_3
A₃ : Type u_4
inst✝⁵ : CommSemiring A₁
inst✝⁴ : CommSemiring A₂
inst✝³ : CommSemiring A₃
inst✝² : Algebra R A₁
inst✝¹ : Algebra R A₂
inst✝ : Algebra R A₃
e : A₁ ≃ₐ[R]... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rfl | @[simp]
theorem mapAlgEquiv_trans (e : A₁ ≃ₐ[R] A₂) (f : A₂ ≃ₐ[R] A₃) :
(mapAlgEquiv σ e).trans (mapAlgEquiv σ f) = mapAlgEquiv σ (e.trans f) := by
ext
simp only [AlgEquiv.trans_apply, mapAlgEquiv_apply, map_map]
| Mathlib.Data.MvPolynomial.Equiv.143_0.88gPfxLltQQTcHM | @[simp]
theorem mapAlgEquiv_trans (e : A₁ ≃ₐ[R] A₂) (f : A₂ ≃ₐ[R] A₃) :
(mapAlgEquiv σ e).trans (mapAlgEquiv σ f) = mapAlgEquiv σ (e.trans f) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
he : IsEmpty σ
⊢ AlgHom.comp (aeval fun a => IsEmpty.elim he a) (Algebra.ofId R (MvPolynomial σ R)) = AlgHom.id R R | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | ext | /-- The algebra isomorphism between multivariable polynomials in no variables
and the ground ring. -/
@[simps!]
def isEmptyAlgEquiv [he : IsEmpty σ] : MvPolynomial σ R ≃ₐ[R] R :=
AlgEquiv.ofAlgHom (aeval (IsEmpty.elim he)) (Algebra.ofId _ _)
(by | Mathlib.Data.MvPolynomial.Equiv.212_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in no variables
and the ground ring. -/
@[simps!]
def isEmptyAlgEquiv [he : IsEmpty σ] : MvPolynomial σ R ≃ₐ[R] R | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
he : IsEmpty σ
⊢ AlgHom.comp (Algebra.ofId R (MvPolynomial σ R)) (aeval fun a => IsEmpty.elim he a) = AlgHom.id R (MvPolynomial σ R) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | ext i m | /-- The algebra isomorphism between multivariable polynomials in no variables
and the ground ring. -/
@[simps!]
def isEmptyAlgEquiv [he : IsEmpty σ] : MvPolynomial σ R ≃ₐ[R] R :=
AlgEquiv.ofAlgHom (aeval (IsEmpty.elim he)) (Algebra.ofId _ _)
(by ext)
(by
| Mathlib.Data.MvPolynomial.Equiv.212_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in no variables
and the ground ring. -/
@[simps!]
def isEmptyAlgEquiv [he : IsEmpty σ] : MvPolynomial σ R ≃ₐ[R] R | Mathlib_Data_MvPolynomial_Equiv |
case hf.a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
he : IsEmpty σ
i : σ
m : σ →₀ ℕ
⊢ coeff m ((AlgHom.comp (Algebra.ofId R (MvPolynomial σ R)) (aeval fun a => IsEmpty.elim he a)) (X i)) =
coeff m ((AlgHom.id R (MvPolynomial σ R)) (X i)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | exact IsEmpty.elim' he i | /-- The algebra isomorphism between multivariable polynomials in no variables
and the ground ring. -/
@[simps!]
def isEmptyAlgEquiv [he : IsEmpty σ] : MvPolynomial σ R ≃ₐ[R] R :=
AlgEquiv.ofAlgHom (aeval (IsEmpty.elim he)) (Algebra.ofId _ _)
(by ext)
(by
ext i m
| Mathlib.Data.MvPolynomial.Equiv.212_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in no variables
and the ground ring. -/
@[simps!]
def isEmptyAlgEquiv [he : IsEmpty σ] : MvPolynomial σ R ≃ₐ[R] R | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
⊢ MvPolynomial (S₁ ⊕ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | apply mvPolynomialEquivMvPolynomial R (Sum S₁ S₂) _ _ (sumToIter R S₁ S₂) (iterToSum R S₁ S₂) | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := by
| Mathlib.Data.MvPolynomial.Equiv.247_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
case hfgC
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
⊢ RingHom.comp (RingHom.comp (sumToIter R S₁ S₂) (iterToSum R S₁ S₂)) C = C | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | refine RingHom.ext (hom_eq_hom _ _ ?hC ?hX) | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := by
apply mvPolynomialEquivM... | Mathlib.Data.MvPolynomial.Equiv.247_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
case hC
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
⊢ RingHom.comp (RingHom.comp (RingHom.comp (sumToIter R S₁ S₂) (iterToSum R S₁ S₂)) C) C = RingHom.comp C C
case hX
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | case hC => ext1; simp only [RingHom.comp_apply, iterToSum_C_C, sumToIter_C] | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := by
apply mvPolynomialEquivM... | Mathlib.Data.MvPolynomial.Equiv.247_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
⊢ RingHom.comp (RingHom.comp (RingHom.comp (sumToIter R S₁ S₂) (iterToSum R S₁ S₂)) C) C = RingHom.comp C C | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | case hC => ext1; simp only [RingHom.comp_apply, iterToSum_C_C, sumToIter_C] | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := by
apply mvPolynomialEquivM... | Mathlib.Data.MvPolynomial.Equiv.247_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
⊢ RingHom.comp (RingHom.comp (RingHom.comp (sumToIter R S₁ S₂) (iterToSum R S₁ S₂)) C) C = RingHom.comp C C | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | ext1 | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := by
apply mvPolynomialEquivM... | Mathlib.Data.MvPolynomial.Equiv.247_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
case a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
x✝ : R
⊢ (RingHom.comp (RingHom.comp (RingHom.comp (sumToIter R S₁ S₂) (iterToSum R S₁ S₂)) C) C) x✝ = (RingHom.comp C C) x✝ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [RingHom.comp_apply, iterToSum_C_C, sumToIter_C] | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := by
apply mvPolynomialEquivM... | Mathlib.Data.MvPolynomial.Equiv.247_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
case hX
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
⊢ ∀ (n : S₂), (RingHom.comp (RingHom.comp (sumToIter R S₁ S₂) (iterToSum R S₁ S₂)) C) (X n) = C (X n) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | case hX => intro; simp only [RingHom.comp_apply, iterToSum_C_X, sumToIter_Xr] | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := by
apply mvPolynomialEquivM... | Mathlib.Data.MvPolynomial.Equiv.247_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
⊢ ∀ (n : S₂), (RingHom.comp (RingHom.comp (sumToIter R S₁ S₂) (iterToSum R S₁ S₂)) C) (X n) = C (X n) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | case hX => intro; simp only [RingHom.comp_apply, iterToSum_C_X, sumToIter_Xr] | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := by
apply mvPolynomialEquivM... | Mathlib.Data.MvPolynomial.Equiv.247_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
⊢ ∀ (n : S₂), (RingHom.comp (RingHom.comp (sumToIter R S₁ S₂) (iterToSum R S₁ S₂)) C) (X n) = C (X n) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | intro | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := by
apply mvPolynomialEquivM... | Mathlib.Data.MvPolynomial.Equiv.247_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n✝ : S₂
⊢ (RingHom.comp (RingHom.comp (sumToIter R S₁ S₂) (iterToSum R S₁ S₂)) C) (X n✝) = C (X n✝) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [RingHom.comp_apply, iterToSum_C_X, sumToIter_Xr] | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := by
apply mvPolynomialEquivM... | Mathlib.Data.MvPolynomial.Equiv.247_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
case hfgX
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
⊢ ∀ (n : S₁), (sumToIter R S₁ S₂) ((iterToSum R S₁ S₂) (X n)) = X n | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp [iterToSum_X, sumToIter_Xl] | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := by
apply mvPolynomialEquivM... | Mathlib.Data.MvPolynomial.Equiv.247_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
case hgfC
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
⊢ RingHom.comp (RingHom.comp (iterToSum R S₁ S₂) (sumToIter R S₁ S₂)) C = C | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | ext1 | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := by
apply mvPolynomialEquivM... | Mathlib.Data.MvPolynomial.Equiv.247_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
case hgfC.a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
x✝ : R
⊢ (RingHom.comp (RingHom.comp (iterToSum R S₁ S₂) (sumToIter R S₁ S₂)) C) x✝ = C x✝ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [RingHom.comp_apply, sumToIter_C, iterToSum_C_C] | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := by
apply mvPolynomialEquivM... | Mathlib.Data.MvPolynomial.Equiv.247_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
case hgfX
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
⊢ ∀ (n : S₁ ⊕ S₂), (iterToSum R S₁ S₂) ((sumToIter R S₁ S₂) (X n)) = X n | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rintro ⟨⟩ | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := by
apply mvPolynomialEquivM... | Mathlib.Data.MvPolynomial.Equiv.247_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
case hgfX.inl
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
val✝ : S₁
⊢ (iterToSum R S₁ S₂) ((sumToIter R S₁ S₂) (X (Sum.inl val✝))) = X (Sum.inl val✝) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [sumToIter_Xl, iterToSum_X, sumToIter_Xr, iterToSum_C_X] | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := by
apply mvPolynomialEquivM... | Mathlib.Data.MvPolynomial.Equiv.247_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
case hgfX.inr
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
val✝ : S₂
⊢ (iterToSum R S₁ S₂) ((sumToIter R S₁ S₂) (X (Sum.inr val✝))) = X (Sum.inr val✝) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [sumToIter_Xl, iterToSum_X, sumToIter_Xr, iterToSum_C_X] | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := by
apply mvPolynomialEquivM... | Mathlib.Data.MvPolynomial.Equiv.247_0.88gPfxLltQQTcHM | /-- The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumRingEquiv : MvPolynomial (Sum S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
src✝ : MvPolynomial (S₁ ⊕ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := sumRingEquiv R S₁ S₂
⊢ ∀ (r : R),
Equiv.toFun src✝.toEquiv ((algebraMap R (MvPolynomial (S₁ ⊕ S₂) R)) r) =
(algebraMap R... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | intro r | /-- The algebra isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumAlgEquiv : MvPolynomial (Sum S₁ S₂) R ≃ₐ[R] MvPolynomial S₁ (MvPolynomial S₂ R) :=
{ sumRingEquiv R S₁ S₂ ... | Mathlib.Data.MvPolynomial.Equiv.261_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumAlgEquiv : MvPolynomial (Sum S₁ S₂) R ≃ₐ[R] MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
src✝ : MvPolynomial (S₁ ⊕ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := sumRingEquiv R S₁ S₂
r : R
⊢ Equiv.toFun src✝.toEquiv ((algebraMap R (MvPolynomial (S₁ ⊕ S₂) R)) r) =
(algebraMap R (MvPolynom... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | have A : algebraMap R (MvPolynomial S₁ (MvPolynomial S₂ R)) r = (C (C r) : _) := rfl | /-- The algebra isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumAlgEquiv : MvPolynomial (Sum S₁ S₂) R ≃ₐ[R] MvPolynomial S₁ (MvPolynomial S₂ R) :=
{ sumRingEquiv R S₁ S₂ ... | Mathlib.Data.MvPolynomial.Equiv.261_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumAlgEquiv : MvPolynomial (Sum S₁ S₂) R ≃ₐ[R] MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
src✝ : MvPolynomial (S₁ ⊕ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := sumRingEquiv R S₁ S₂
r : R
A : (algebraMap R (MvPolynomial S₁ (MvPolynomial S₂ R))) r = C (C r)
⊢ Equiv.toFun src✝.toEquiv ((algeb... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | have B : algebraMap R (MvPolynomial (Sum S₁ S₂) R) r = C r := rfl | /-- The algebra isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumAlgEquiv : MvPolynomial (Sum S₁ S₂) R ≃ₐ[R] MvPolynomial S₁ (MvPolynomial S₂ R) :=
{ sumRingEquiv R S₁ S₂ ... | Mathlib.Data.MvPolynomial.Equiv.261_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumAlgEquiv : MvPolynomial (Sum S₁ S₂) R ≃ₐ[R] MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
src✝ : MvPolynomial (S₁ ⊕ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R) := sumRingEquiv R S₁ S₂
r : R
A : (algebraMap R (MvPolynomial S₁ (MvPolynomial S₂ R))) r = C (C r)
B : (algebraMap R (MvPolynomial (S... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [sumRingEquiv, mvPolynomialEquivMvPolynomial, Equiv.toFun_as_coe,
Equiv.coe_fn_mk, B, sumToIter_C, A] | /-- The algebra isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumAlgEquiv : MvPolynomial (Sum S₁ S₂) R ≃ₐ[R] MvPolynomial S₁ (MvPolynomial S₂ R) :=
{ sumRingEquiv R S₁ S₂ ... | Mathlib.Data.MvPolynomial.Equiv.261_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficients in multivariable polynomials in the other type.
-/
def sumAlgEquiv : MvPolynomial (Sum S₁ S₂) R ≃ₐ[R] MvPolynomial S₁ (MvPolynomial S₂ R) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
⊢ AlgHom.comp (aeval fun o => Option.elim o Polynomial.X fun s => Polynomial.C (X s))
(Polynomial.aevalTower (rename Option.some) (X none)) =
AlgHom.id R (MvPolynomial S₁ R)[X] | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | ext : 2 | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
polynomials with coefficients in `MvPolynomial S₁ R`.
-/
@[simps!]
def optionEquivLeft : MvPolynomial (Option S₁) R ≃ₐ[R] Polynomial (MvPolynomial S₁ R) :=
AlgEquiv.ofAlgHom (MvPolynomial.aeval fun o => o.elim Polynomial.X fun s => Polyn... | Mathlib.Data.MvPolynomial.Equiv.280_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
polynomials with coefficients in `MvPolynomial S₁ R`.
-/
@[simps!]
def optionEquivLeft : MvPolynomial (Option S₁) R ≃ₐ[R] Polynomial (MvPolynomial S₁ R) | Mathlib_Data_MvPolynomial_Equiv |
case hC.hf
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
i✝ : S₁
⊢ (AlgHom.comp
(AlgHom.comp (aeval fun o => Option.elim o Polynomial.X fun s => Polynomial.C (X s))
(Polynomial.aevalTower (rename Option.some) (X none)))
CAlgH... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp [← Polynomial.C_eq_algebraMap] | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
polynomials with coefficients in `MvPolynomial S₁ R`.
-/
@[simps!]
def optionEquivLeft : MvPolynomial (Option S₁) R ≃ₐ[R] Polynomial (MvPolynomial S₁ R) :=
AlgEquiv.ofAlgHom (MvPolynomial.aeval fun o => o.elim Polynomial.X fun s => Polyn... | Mathlib.Data.MvPolynomial.Equiv.280_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
polynomials with coefficients in `MvPolynomial S₁ R`.
-/
@[simps!]
def optionEquivLeft : MvPolynomial (Option S₁) R ≃ₐ[R] Polynomial (MvPolynomial S₁ R) | Mathlib_Data_MvPolynomial_Equiv |
case hX.a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n✝ : ℕ
⊢ Polynomial.coeff
((AlgHom.comp (aeval fun o => Option.elim o Polynomial.X fun s => Polynomial.C (X s))
(Polynomial.aevalTower (rename Option.some) (X none)))
Poly... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp [← Polynomial.C_eq_algebraMap] | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
polynomials with coefficients in `MvPolynomial S₁ R`.
-/
@[simps!]
def optionEquivLeft : MvPolynomial (Option S₁) R ≃ₐ[R] Polynomial (MvPolynomial S₁ R) :=
AlgEquiv.ofAlgHom (MvPolynomial.aeval fun o => o.elim Polynomial.X fun s => Polyn... | Mathlib.Data.MvPolynomial.Equiv.280_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
polynomials with coefficients in `MvPolynomial S₁ R`.
-/
@[simps!]
def optionEquivLeft : MvPolynomial (Option S₁) R ≃ₐ[R] Polynomial (MvPolynomial S₁ R) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
⊢ AlgHom.comp (Polynomial.aevalTower (rename Option.some) (X none))
(aeval fun o => Option.elim o Polynomial.X fun s => Polynomial.C (X s)) =
AlgHom.id R (MvPolynomial (Option S₁) R) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | ext i : 2 | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
polynomials with coefficients in `MvPolynomial S₁ R`.
-/
@[simps!]
def optionEquivLeft : MvPolynomial (Option S₁) R ≃ₐ[R] Polynomial (MvPolynomial S₁ R) :=
AlgEquiv.ofAlgHom (MvPolynomial.aeval fun o => o.elim Polynomial.X fun s => Polyn... | Mathlib.Data.MvPolynomial.Equiv.280_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
polynomials with coefficients in `MvPolynomial S₁ R`.
-/
@[simps!]
def optionEquivLeft : MvPolynomial (Option S₁) R ≃ₐ[R] Polynomial (MvPolynomial S₁ R) | Mathlib_Data_MvPolynomial_Equiv |
case hf.a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
i : Option S₁
m✝ : Option S₁ →₀ ℕ
⊢ coeff m✝
((AlgHom.comp (Polynomial.aevalTower (rename Option.some) (X none))
(aeval fun o => Option.elim o Polynomial.X fun s => Polynomial.C (... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | cases i | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
polynomials with coefficients in `MvPolynomial S₁ R`.
-/
@[simps!]
def optionEquivLeft : MvPolynomial (Option S₁) R ≃ₐ[R] Polynomial (MvPolynomial S₁ R) :=
AlgEquiv.ofAlgHom (MvPolynomial.aeval fun o => o.elim Polynomial.X fun s => Polyn... | Mathlib.Data.MvPolynomial.Equiv.280_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
polynomials with coefficients in `MvPolynomial S₁ R`.
-/
@[simps!]
def optionEquivLeft : MvPolynomial (Option S₁) R ≃ₐ[R] Polynomial (MvPolynomial S₁ R) | Mathlib_Data_MvPolynomial_Equiv |
case hf.a.none
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
m✝ : Option S₁ →₀ ℕ
⊢ coeff m✝
((AlgHom.comp (Polynomial.aevalTower (rename Option.some) (X none))
(aeval fun o => Option.elim o Polynomial.X fun s => Polynomial.C (X s)))
... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
polynomials with coefficients in `MvPolynomial S₁ R`.
-/
@[simps!]
def optionEquivLeft : MvPolynomial (Option S₁) R ≃ₐ[R] Polynomial (MvPolynomial S₁ R) :=
AlgEquiv.ofAlgHom (MvPolynomial.aeval fun o => o.elim Polynomial.X fun s => Polyn... | Mathlib.Data.MvPolynomial.Equiv.280_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
polynomials with coefficients in `MvPolynomial S₁ R`.
-/
@[simps!]
def optionEquivLeft : MvPolynomial (Option S₁) R ≃ₐ[R] Polynomial (MvPolynomial S₁ R) | Mathlib_Data_MvPolynomial_Equiv |
case hf.a.some
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
m✝ : Option S₁ →₀ ℕ
val✝ : S₁
⊢ coeff m✝
((AlgHom.comp (Polynomial.aevalTower (rename Option.some) (X none))
(aeval fun o => Option.elim o Polynomial.X fun s => Polynomial.C ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
polynomials with coefficients in `MvPolynomial S₁ R`.
-/
@[simps!]
def optionEquivLeft : MvPolynomial (Option S₁) R ≃ₐ[R] Polynomial (MvPolynomial S₁ R) :=
AlgEquiv.ofAlgHom (MvPolynomial.aeval fun o => o.elim Polynomial.X fun s => Polyn... | Mathlib.Data.MvPolynomial.Equiv.280_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
polynomials with coefficients in `MvPolynomial S₁ R`.
-/
@[simps!]
def optionEquivLeft : MvPolynomial (Option S₁) R ≃ₐ[R] Polynomial (MvPolynomial S₁ R) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
⊢ AlgHom.comp (aeval fun o => Option.elim o (C Polynomial.X) X)
(aevalTower (Polynomial.aeval (X none)) fun i => X (Option.some i)) =
AlgHom.id R (MvPolynomial S₁ R[X]) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | ext : 2 | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
multivariable polynomials with coefficients in polynomials.
-/
def optionEquivRight : MvPolynomial (Option S₁) R ≃ₐ[R] MvPolynomial S₁ R[X] :=
AlgEquiv.ofAlgHom (MvPolynomial.aeval fun o => o.elim (C Polynomial.X) X)
(MvPolynomial.ae... | Mathlib.Data.MvPolynomial.Equiv.292_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
multivariable polynomials with coefficients in polynomials.
-/
def optionEquivRight : MvPolynomial (Option S₁) R ≃ₐ[R] MvPolynomial S₁ R[X] | Mathlib_Data_MvPolynomial_Equiv |
case h₁.hX
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
⊢ (AlgHom.comp
(AlgHom.comp (aeval fun o => Option.elim o (C Polynomial.X) X)
(aevalTower (Polynomial.aeval (X none)) fun i => X (Option.some i)))
(IsScalarTower.toAlgH... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [MvPolynomial.algebraMap_eq, Option.elim, AlgHom.coe_comp, AlgHom.id_comp,
IsScalarTower.coe_toAlgHom', comp_apply, aevalTower_C, Polynomial.aeval_X, aeval_X,
Option.elim', aevalTower_X, AlgHom.coe_id, id.def, eq_self_iff_true, imp_true_iff] | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
multivariable polynomials with coefficients in polynomials.
-/
def optionEquivRight : MvPolynomial (Option S₁) R ≃ₐ[R] MvPolynomial S₁ R[X] :=
AlgEquiv.ofAlgHom (MvPolynomial.aeval fun o => o.elim (C Polynomial.X) X)
(MvPolynomial.ae... | Mathlib.Data.MvPolynomial.Equiv.292_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
multivariable polynomials with coefficients in polynomials.
-/
def optionEquivRight : MvPolynomial (Option S₁) R ≃ₐ[R] MvPolynomial S₁ R[X] | Mathlib_Data_MvPolynomial_Equiv |
case h₂.a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
i✝ : S₁
m✝ : S₁ →₀ ℕ
⊢ coeff m✝
((AlgHom.comp (aeval fun o => Option.elim o (C Polynomial.X) X)
(aevalTower (Polynomial.aeval (X none)) fun i => X (Option.some i)))
(X i✝)... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [MvPolynomial.algebraMap_eq, Option.elim, AlgHom.coe_comp, AlgHom.id_comp,
IsScalarTower.coe_toAlgHom', comp_apply, aevalTower_C, Polynomial.aeval_X, aeval_X,
Option.elim', aevalTower_X, AlgHom.coe_id, id.def, eq_self_iff_true, imp_true_iff] | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
multivariable polynomials with coefficients in polynomials.
-/
def optionEquivRight : MvPolynomial (Option S₁) R ≃ₐ[R] MvPolynomial S₁ R[X] :=
AlgEquiv.ofAlgHom (MvPolynomial.aeval fun o => o.elim (C Polynomial.X) X)
(MvPolynomial.ae... | Mathlib.Data.MvPolynomial.Equiv.292_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
multivariable polynomials with coefficients in polynomials.
-/
def optionEquivRight : MvPolynomial (Option S₁) R ≃ₐ[R] MvPolynomial S₁ R[X] | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
⊢ AlgHom.comp (aevalTower (Polynomial.aeval (X none)) fun i => X (Option.some i))
(aeval fun o => Option.elim o (C Polynomial.X) X) =
AlgHom.id R (MvPolynomial (Option S₁) R) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | ext ⟨i⟩ : 2 | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
multivariable polynomials with coefficients in polynomials.
-/
def optionEquivRight : MvPolynomial (Option S₁) R ≃ₐ[R] MvPolynomial S₁ R[X] :=
AlgEquiv.ofAlgHom (MvPolynomial.aeval fun o => o.elim (C Polynomial.X) X)
(MvPolynomial.ae... | Mathlib.Data.MvPolynomial.Equiv.292_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
multivariable polynomials with coefficients in polynomials.
-/
def optionEquivRight : MvPolynomial (Option S₁) R ≃ₐ[R] MvPolynomial S₁ R[X] | Mathlib_Data_MvPolynomial_Equiv |
case hf.none.a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
m✝ : Option S₁ →₀ ℕ
⊢ coeff m✝
((AlgHom.comp (aevalTower (Polynomial.aeval (X none)) fun i => X (Option.some i))
(aeval fun o => Option.elim o (C Polynomial.X) X))
(X... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [Option.elim, AlgHom.coe_comp, comp_apply, aeval_X, aevalTower_C,
Polynomial.aeval_X, AlgHom.coe_id, id.def, aevalTower_X] | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
multivariable polynomials with coefficients in polynomials.
-/
def optionEquivRight : MvPolynomial (Option S₁) R ≃ₐ[R] MvPolynomial S₁ R[X] :=
AlgEquiv.ofAlgHom (MvPolynomial.aeval fun o => o.elim (C Polynomial.X) X)
(MvPolynomial.ae... | Mathlib.Data.MvPolynomial.Equiv.292_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
multivariable polynomials with coefficients in polynomials.
-/
def optionEquivRight : MvPolynomial (Option S₁) R ≃ₐ[R] MvPolynomial S₁ R[X] | Mathlib_Data_MvPolynomial_Equiv |
case hf.some.a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
val✝ : S₁
m✝ : Option S₁ →₀ ℕ
⊢ coeff m✝
((AlgHom.comp (aevalTower (Polynomial.aeval (X none)) fun i => X (Option.some i))
(aeval fun o => Option.elim o (C Polynomial.X) X))
... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [Option.elim, AlgHom.coe_comp, comp_apply, aeval_X, aevalTower_C,
Polynomial.aeval_X, AlgHom.coe_id, id.def, aevalTower_X] | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
multivariable polynomials with coefficients in polynomials.
-/
def optionEquivRight : MvPolynomial (Option S₁) R ≃ₐ[R] MvPolynomial S₁ R[X] :=
AlgEquiv.ofAlgHom (MvPolynomial.aeval fun o => o.elim (C Polynomial.X) X)
(MvPolynomial.ae... | Mathlib.Data.MvPolynomial.Equiv.292_0.88gPfxLltQQTcHM | /-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
multivariable polynomials with coefficients in polynomials.
-/
def optionEquivRight : MvPolynomial (Option S₁) R ≃ₐ[R] MvPolynomial S₁ R[X] | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
⊢ ↑(finSuccEquiv R n) =
eval₂Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | ext i : 2 | theorem finSuccEquiv_eq :
(finSuccEquiv R n : MvPolynomial (Fin (n + 1)) R →+* Polynomial (MvPolynomial (Fin n) R)) =
eval₂Hom (Polynomial.C.comp (C : R →+* MvPolynomial (Fin n) R)) fun i : Fin (n + 1) =>
Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i := by
| Mathlib.Data.MvPolynomial.Equiv.318_0.88gPfxLltQQTcHM | theorem finSuccEquiv_eq :
(finSuccEquiv R n : MvPolynomial (Fin (n + 1)) R →+* Polynomial (MvPolynomial (Fin n) R)) =
eval₂Hom (Polynomial.C.comp (C : R →+* MvPolynomial (Fin n) R)) fun i : Fin (n + 1) =>
Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i | Mathlib_Data_MvPolynomial_Equiv |
case hC.a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
i : R
⊢ (RingHom.comp (↑(finSuccEquiv R n)) C) i =
(RingHom.comp
(eval₂Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i) C)
... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [finSuccEquiv, optionEquivLeft_apply, aeval_C, AlgEquiv.coe_trans, RingHom.coe_coe,
coe_eval₂Hom, comp_apply, renameEquiv_apply, eval₂_C, RingHom.coe_comp, rename_C] | theorem finSuccEquiv_eq :
(finSuccEquiv R n : MvPolynomial (Fin (n + 1)) R →+* Polynomial (MvPolynomial (Fin n) R)) =
eval₂Hom (Polynomial.C.comp (C : R →+* MvPolynomial (Fin n) R)) fun i : Fin (n + 1) =>
Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i := by
ext i : 2
· | Mathlib.Data.MvPolynomial.Equiv.318_0.88gPfxLltQQTcHM | theorem finSuccEquiv_eq :
(finSuccEquiv R n : MvPolynomial (Fin (n + 1)) R →+* Polynomial (MvPolynomial (Fin n) R)) =
eval₂Hom (Polynomial.C.comp (C : R →+* MvPolynomial (Fin n) R)) fun i : Fin (n + 1) =>
Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i | Mathlib_Data_MvPolynomial_Equiv |
case hC.a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
i : R
⊢ (algebraMap R (MvPolynomial (Fin n) R)[X]) i = Polynomial.C (C i) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rfl | theorem finSuccEquiv_eq :
(finSuccEquiv R n : MvPolynomial (Fin (n + 1)) R →+* Polynomial (MvPolynomial (Fin n) R)) =
eval₂Hom (Polynomial.C.comp (C : R →+* MvPolynomial (Fin n) R)) fun i : Fin (n + 1) =>
Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i := by
ext i : 2
· simp only [finSuccEq... | Mathlib.Data.MvPolynomial.Equiv.318_0.88gPfxLltQQTcHM | theorem finSuccEquiv_eq :
(finSuccEquiv R n : MvPolynomial (Fin (n + 1)) R →+* Polynomial (MvPolynomial (Fin n) R)) =
eval₂Hom (Polynomial.C.comp (C : R →+* MvPolynomial (Fin n) R)) fun i : Fin (n + 1) =>
Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i | Mathlib_Data_MvPolynomial_Equiv |
case hX.a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
i : Fin (n + 1)
n✝ : ℕ
⊢ Polynomial.coeff (↑(finSuccEquiv R n) (X i)) n✝ =
Polynomial.coeff
((eval₂Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => Poly... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | refine' Fin.cases _ _ i | theorem finSuccEquiv_eq :
(finSuccEquiv R n : MvPolynomial (Fin (n + 1)) R →+* Polynomial (MvPolynomial (Fin n) R)) =
eval₂Hom (Polynomial.C.comp (C : R →+* MvPolynomial (Fin n) R)) fun i : Fin (n + 1) =>
Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i := by
ext i : 2
· simp only [finSuccEq... | Mathlib.Data.MvPolynomial.Equiv.318_0.88gPfxLltQQTcHM | theorem finSuccEquiv_eq :
(finSuccEquiv R n : MvPolynomial (Fin (n + 1)) R →+* Polynomial (MvPolynomial (Fin n) R)) =
eval₂Hom (Polynomial.C.comp (C : R →+* MvPolynomial (Fin n) R)) fun i : Fin (n + 1) =>
Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i | Mathlib_Data_MvPolynomial_Equiv |
case hX.a.refine'_1
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
i : Fin (n + 1)
n✝ : ℕ
⊢ Polynomial.coeff (↑(finSuccEquiv R n) (X 0)) n✝ =
Polynomial.coeff
((eval₂Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp [finSuccEquiv] | theorem finSuccEquiv_eq :
(finSuccEquiv R n : MvPolynomial (Fin (n + 1)) R →+* Polynomial (MvPolynomial (Fin n) R)) =
eval₂Hom (Polynomial.C.comp (C : R →+* MvPolynomial (Fin n) R)) fun i : Fin (n + 1) =>
Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i := by
ext i : 2
· simp only [finSuccEq... | Mathlib.Data.MvPolynomial.Equiv.318_0.88gPfxLltQQTcHM | theorem finSuccEquiv_eq :
(finSuccEquiv R n : MvPolynomial (Fin (n + 1)) R →+* Polynomial (MvPolynomial (Fin n) R)) =
eval₂Hom (Polynomial.C.comp (C : R →+* MvPolynomial (Fin n) R)) fun i : Fin (n + 1) =>
Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i | Mathlib_Data_MvPolynomial_Equiv |
case hX.a.refine'_2
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
i : Fin (n + 1)
n✝ : ℕ
⊢ ∀ (i : Fin n),
Polynomial.coeff (↑(finSuccEquiv R n) (X (Fin.succ i))) n✝ =
Polynomial.coeff
((eval₂Hom (RingHom.comp Polynomial.C C) fu... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp [finSuccEquiv] | theorem finSuccEquiv_eq :
(finSuccEquiv R n : MvPolynomial (Fin (n + 1)) R →+* Polynomial (MvPolynomial (Fin n) R)) =
eval₂Hom (Polynomial.C.comp (C : R →+* MvPolynomial (Fin n) R)) fun i : Fin (n + 1) =>
Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i := by
ext i : 2
· simp only [finSuccEq... | Mathlib.Data.MvPolynomial.Equiv.318_0.88gPfxLltQQTcHM | theorem finSuccEquiv_eq :
(finSuccEquiv R n : MvPolynomial (Fin (n + 1)) R →+* Polynomial (MvPolynomial (Fin n) R)) =
eval₂Hom (Polynomial.C.comp (C : R →+* MvPolynomial (Fin n) R)) fun i : Fin (n + 1) =>
Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
⊢ (finSuccEquiv R n) p =
(eval₂Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i) p | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [← finSuccEquiv_eq, RingHom.coe_coe] | @[simp]
theorem finSuccEquiv_apply (p : MvPolynomial (Fin (n + 1)) R) :
finSuccEquiv R n p =
eval₂Hom (Polynomial.C.comp (C : R →+* MvPolynomial (Fin n) R))
(fun i : Fin (n + 1) => Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i) p := by
| Mathlib.Data.MvPolynomial.Equiv.329_0.88gPfxLltQQTcHM | @[simp]
theorem finSuccEquiv_apply (p : MvPolynomial (Fin (n + 1)) R) :
finSuccEquiv R n p =
eval₂Hom (Polynomial.C.comp (C : R →+* MvPolynomial (Fin n) R))
(fun i : Fin (n + 1) => Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i) p | Mathlib_Data_MvPolynomial_Equiv |
R✝ : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R✝
e : ℕ
s : σ →₀ ℕ
inst✝¹ : CommSemiring R✝
n✝ : ℕ
R : Type u
inst✝ : CommSemiring R
n : ℕ
⊢ RingHom.comp (↑(AlgEquiv.symm (finSuccEquiv R n))) (RingHom.comp Polynomial.C C) = C | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | refine' RingHom.ext fun x => _ | theorem finSuccEquiv_comp_C_eq_C {R : Type u} [CommSemiring R] (n : ℕ) :
(↑(MvPolynomial.finSuccEquiv R n).symm : Polynomial (MvPolynomial (Fin n) R) →+* _).comp
(Polynomial.C.comp MvPolynomial.C) =
(MvPolynomial.C : R →+* MvPolynomial (Fin n.succ) R) := by
| Mathlib.Data.MvPolynomial.Equiv.337_0.88gPfxLltQQTcHM | theorem finSuccEquiv_comp_C_eq_C {R : Type u} [CommSemiring R] (n : ℕ) :
(↑(MvPolynomial.finSuccEquiv R n).symm : Polynomial (MvPolynomial (Fin n) R) →+* _).comp
(Polynomial.C.comp MvPolynomial.C) =
(MvPolynomial.C : R →+* MvPolynomial (Fin n.succ) R) | Mathlib_Data_MvPolynomial_Equiv |
R✝ : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R✝
e : ℕ
s : σ →₀ ℕ
inst✝¹ : CommSemiring R✝
n✝ : ℕ
R : Type u
inst✝ : CommSemiring R
n : ℕ
x : R
⊢ (RingHom.comp (↑(AlgEquiv.symm (finSuccEquiv R n))) (RingHom.comp Polynomial.C C)) x = C x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [RingHom.comp_apply] | theorem finSuccEquiv_comp_C_eq_C {R : Type u} [CommSemiring R] (n : ℕ) :
(↑(MvPolynomial.finSuccEquiv R n).symm : Polynomial (MvPolynomial (Fin n) R) →+* _).comp
(Polynomial.C.comp MvPolynomial.C) =
(MvPolynomial.C : R →+* MvPolynomial (Fin n.succ) R) := by
refine' RingHom.ext fun x => _
| Mathlib.Data.MvPolynomial.Equiv.337_0.88gPfxLltQQTcHM | theorem finSuccEquiv_comp_C_eq_C {R : Type u} [CommSemiring R] (n : ℕ) :
(↑(MvPolynomial.finSuccEquiv R n).symm : Polynomial (MvPolynomial (Fin n) R) →+* _).comp
(Polynomial.C.comp MvPolynomial.C) =
(MvPolynomial.C : R →+* MvPolynomial (Fin n.succ) R) | Mathlib_Data_MvPolynomial_Equiv |
R✝ : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R✝
e : ℕ
s : σ →₀ ℕ
inst✝¹ : CommSemiring R✝
n✝ : ℕ
R : Type u
inst✝ : CommSemiring R
n : ℕ
x : R
⊢ ↑(AlgEquiv.symm (finSuccEquiv R n)) ((RingHom.comp Polynomial.C C) x) = C x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | refine'
(MvPolynomial.finSuccEquiv R n).injective
(Trans.trans ((MvPolynomial.finSuccEquiv R n).apply_symm_apply _) _) | theorem finSuccEquiv_comp_C_eq_C {R : Type u} [CommSemiring R] (n : ℕ) :
(↑(MvPolynomial.finSuccEquiv R n).symm : Polynomial (MvPolynomial (Fin n) R) →+* _).comp
(Polynomial.C.comp MvPolynomial.C) =
(MvPolynomial.C : R →+* MvPolynomial (Fin n.succ) R) := by
refine' RingHom.ext fun x => _
rw [RingH... | Mathlib.Data.MvPolynomial.Equiv.337_0.88gPfxLltQQTcHM | theorem finSuccEquiv_comp_C_eq_C {R : Type u} [CommSemiring R] (n : ℕ) :
(↑(MvPolynomial.finSuccEquiv R n).symm : Polynomial (MvPolynomial (Fin n) R) →+* _).comp
(Polynomial.C.comp MvPolynomial.C) =
(MvPolynomial.C : R →+* MvPolynomial (Fin n.succ) R) | Mathlib_Data_MvPolynomial_Equiv |
R✝ : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R✝
e : ℕ
s : σ →₀ ℕ
inst✝¹ : CommSemiring R✝
n✝ : ℕ
R : Type u
inst✝ : CommSemiring R
n : ℕ
x : R
⊢ (RingHom.comp Polynomial.C C) x = (finSuccEquiv R n) (C x) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [MvPolynomial.finSuccEquiv_apply, MvPolynomial.eval₂Hom_C] | theorem finSuccEquiv_comp_C_eq_C {R : Type u} [CommSemiring R] (n : ℕ) :
(↑(MvPolynomial.finSuccEquiv R n).symm : Polynomial (MvPolynomial (Fin n) R) →+* _).comp
(Polynomial.C.comp MvPolynomial.C) =
(MvPolynomial.C : R →+* MvPolynomial (Fin n.succ) R) := by
refine' RingHom.ext fun x => _
rw [RingH... | Mathlib.Data.MvPolynomial.Equiv.337_0.88gPfxLltQQTcHM | theorem finSuccEquiv_comp_C_eq_C {R : Type u} [CommSemiring R] (n : ℕ) :
(↑(MvPolynomial.finSuccEquiv R n).symm : Polynomial (MvPolynomial (Fin n) R) →+* _).comp
(Polynomial.C.comp MvPolynomial.C) =
(MvPolynomial.C : R →+* MvPolynomial (Fin n.succ) R) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
⊢ (finSuccEquiv R n) (X 0) = Polynomial.X | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp | theorem finSuccEquiv_X_zero : finSuccEquiv R n (X 0) = Polynomial.X := by | Mathlib.Data.MvPolynomial.Equiv.352_0.88gPfxLltQQTcHM | theorem finSuccEquiv_X_zero : finSuccEquiv R n (X 0) = Polynomial.X | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
j : Fin n
⊢ (finSuccEquiv R n) (X (Fin.succ j)) = Polynomial.C (X j) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp | theorem finSuccEquiv_X_succ {j : Fin n} : finSuccEquiv R n (X j.succ) = Polynomial.C (X j) := by
| Mathlib.Data.MvPolynomial.Equiv.356_0.88gPfxLltQQTcHM | theorem finSuccEquiv_X_succ {j : Fin n} : finSuccEquiv R n (X j.succ) = Polynomial.C (X j) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
m : Fin n →₀ ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
⊢ coeff m (Polynomial.coeff ((finSuccEquiv R n) f) i) = coeff (cons i m) f | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | induction' f using MvPolynomial.induction_on' with j r p q hp hq generalizing i m | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f := by
| Mathlib.Data.MvPolynomial.Equiv.361_0.88gPfxLltQQTcHM | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f | Mathlib_Data_MvPolynomial_Equiv |
case h1
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
j : Fin (n + 1) →₀ ℕ
r : R
m : Fin n →₀ ℕ
i : ℕ
⊢ coeff m (Polynomial.coeff ((finSuccEquiv R n) ((monomial j) r)) i) = coeff (cons i m) ((monomial j) r)
case h2
R : Type u
S₁ : Type v
S₂ : Ty... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | swap | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f := by
induc... | Mathlib.Data.MvPolynomial.Equiv.361_0.88gPfxLltQQTcHM | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f | Mathlib_Data_MvPolynomial_Equiv |
case h2
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
p q : MvPolynomial (Fin (n + 1)) R
hp : ∀ (m : Fin n →₀ ℕ) (i : ℕ), coeff m (Polynomial.coeff ((finSuccEquiv R n) p) i) = coeff (cons i m) p
hq : ∀ (m : Fin n →₀ ℕ) (i : ℕ), coeff m (Polynomi... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [(finSuccEquiv R n).map_add, Polynomial.coeff_add, coeff_add, hp, hq] | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f := by
induc... | Mathlib.Data.MvPolynomial.Equiv.361_0.88gPfxLltQQTcHM | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f | Mathlib_Data_MvPolynomial_Equiv |
case h1
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
j : Fin (n + 1) →₀ ℕ
r : R
m : Fin n →₀ ℕ
i : ℕ
⊢ coeff m (Polynomial.coeff ((finSuccEquiv R n) ((monomial j) r)) i) = coeff (cons i m) ((monomial j) r) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [finSuccEquiv_apply, coe_eval₂Hom, eval₂_monomial, RingHom.coe_comp, prod_pow,
Polynomial.coeff_C_mul, coeff_C_mul, coeff_monomial, Fin.prod_univ_succ, Fin.cases_zero,
Fin.cases_succ, ← map_prod, ← RingHom.map_pow, Function.comp_apply] | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f := by
induc... | Mathlib.Data.MvPolynomial.Equiv.361_0.88gPfxLltQQTcHM | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f | Mathlib_Data_MvPolynomial_Equiv |
case h1
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
j : Fin (n + 1) →₀ ℕ
r : R
m : Fin n →₀ ℕ
i : ℕ
⊢ r * coeff m (Polynomial.coeff (Polynomial.X ^ j 0 * Polynomial.C (∏ x : Fin n, X x ^ j (Fin.succ x))) i) =
if j = cons i m then r else 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [← mul_boole, mul_comm (Polynomial.X ^ j 0), Polynomial.coeff_C_mul_X_pow] | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f := by
induc... | Mathlib.Data.MvPolynomial.Equiv.361_0.88gPfxLltQQTcHM | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f | Mathlib_Data_MvPolynomial_Equiv |
case h1
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
j : Fin (n + 1) →₀ ℕ
r : R
m : Fin n →₀ ℕ
i : ℕ
⊢ r * coeff m (if i = j 0 then ∏ x : Fin n, X x ^ j (Fin.succ x) else 0) = r * if j = cons i m then 1 else 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | congr 1 | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f := by
induc... | Mathlib.Data.MvPolynomial.Equiv.361_0.88gPfxLltQQTcHM | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f | Mathlib_Data_MvPolynomial_Equiv |
case h1.e_a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
j : Fin (n + 1) →₀ ℕ
r : R
m : Fin n →₀ ℕ
i : ℕ
⊢ coeff m (if i = j 0 then ∏ x : Fin n, X x ^ j (Fin.succ x) else 0) = if j = cons i m then 1 else 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | obtain rfl | hjmi := eq_or_ne j (m.cons i) | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f := by
induc... | Mathlib.Data.MvPolynomial.Equiv.361_0.88gPfxLltQQTcHM | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f | Mathlib_Data_MvPolynomial_Equiv |
case h1.e_a.inl
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
r : R
m : Fin n →₀ ℕ
i : ℕ
⊢ coeff m (if i = (cons i m) 0 then ∏ x : Fin n, X x ^ (cons i m) (Fin.succ x) else 0) =
if cons i m = cons i m then 1 else 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simpa only [cons_zero, cons_succ, if_pos rfl, monomial_eq, C_1, one_mul, prod_pow] using
coeff_monomial m m (1 : R) | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f := by
induc... | Mathlib.Data.MvPolynomial.Equiv.361_0.88gPfxLltQQTcHM | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f | Mathlib_Data_MvPolynomial_Equiv |
case h1.e_a.inr
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
j : Fin (n + 1) →₀ ℕ
r : R
m : Fin n →₀ ℕ
i : ℕ
hjmi : j ≠ cons i m
⊢ coeff m (if i = j 0 then ∏ x : Fin n, X x ^ j (Fin.succ x) else 0) = if j = cons i m then 1 else 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [hjmi, if_false] | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f := by
induc... | Mathlib.Data.MvPolynomial.Equiv.361_0.88gPfxLltQQTcHM | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f | Mathlib_Data_MvPolynomial_Equiv |
case h1.e_a.inr
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
j : Fin (n + 1) →₀ ℕ
r : R
m : Fin n →₀ ℕ
i : ℕ
hjmi : j ≠ cons i m
⊢ coeff m (if i = j 0 then ∏ x : Fin n, X x ^ j (Fin.succ x) else 0) = 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | obtain hij | rfl := ne_or_eq i (j 0) | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f := by
induc... | Mathlib.Data.MvPolynomial.Equiv.361_0.88gPfxLltQQTcHM | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f | Mathlib_Data_MvPolynomial_Equiv |
case h1.e_a.inr.inl
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
j : Fin (n + 1) →₀ ℕ
r : R
m : Fin n →₀ ℕ
i : ℕ
hjmi : j ≠ cons i m
hij : i ≠ j 0
⊢ coeff m (if i = j 0 then ∏ x : Fin n, X x ^ j (Fin.succ x) else 0) = 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [hij, if_false, coeff_zero] | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f := by
induc... | Mathlib.Data.MvPolynomial.Equiv.361_0.88gPfxLltQQTcHM | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f | Mathlib_Data_MvPolynomial_Equiv |
case h1.e_a.inr.inr
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
j : Fin (n + 1) →₀ ℕ
r : R
m : Fin n →₀ ℕ
hjmi : j ≠ cons (j 0) m
⊢ coeff m (if j 0 = j 0 then ∏ x : Fin n, X x ^ j (Fin.succ x) else 0) = 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [eq_self_iff_true, if_true] | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f := by
induc... | Mathlib.Data.MvPolynomial.Equiv.361_0.88gPfxLltQQTcHM | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f | Mathlib_Data_MvPolynomial_Equiv |
case h1.e_a.inr.inr
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
j : Fin (n + 1) →₀ ℕ
r : R
m : Fin n →₀ ℕ
hjmi : j ≠ cons (j 0) m
⊢ coeff m (∏ x : Fin n, X x ^ j (Fin.succ x)) = 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | have hmj : m ≠ j.tail := by
rintro rfl
rw [cons_tail] at hjmi
contradiction | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f := by
induc... | Mathlib.Data.MvPolynomial.Equiv.361_0.88gPfxLltQQTcHM | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
j : Fin (n + 1) →₀ ℕ
r : R
m : Fin n →₀ ℕ
hjmi : j ≠ cons (j 0) m
⊢ m ≠ tail j | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rintro rfl | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f := by
induc... | Mathlib.Data.MvPolynomial.Equiv.361_0.88gPfxLltQQTcHM | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
j : Fin (n + 1) →₀ ℕ
r : R
hjmi : j ≠ cons (j 0) (tail j)
⊢ False | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [cons_tail] at hjmi | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f := by
induc... | Mathlib.Data.MvPolynomial.Equiv.361_0.88gPfxLltQQTcHM | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
j : Fin (n + 1) →₀ ℕ
r : R
hjmi : j ≠ j
⊢ False | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | contradiction | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f := by
induc... | Mathlib.Data.MvPolynomial.Equiv.361_0.88gPfxLltQQTcHM | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f | Mathlib_Data_MvPolynomial_Equiv |
case h1.e_a.inr.inr
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
j : Fin (n + 1) →₀ ℕ
r : R
m : Fin n →₀ ℕ
hjmi : j ≠ cons (j 0) m
hmj : m ≠ tail j
⊢ coeff m (∏ x : Fin n, X x ^ j (Fin.succ x)) = 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simpa only [monomial_eq, C_1, one_mul, prod_pow, Finsupp.tail_apply, if_neg hmj.symm] using
coeff_monomial m j.tail (1 : R) | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f := by
induc... | Mathlib.Data.MvPolynomial.Equiv.361_0.88gPfxLltQQTcHM | /-- The coefficient of `m` in the `i`-th coefficient of `finSuccEquiv R n f` equals the
coefficient of `Finsupp.cons i m` in `f`. -/
theorem finSuccEquiv_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :
coeff m (Polynomial.coeff (finSuccEquiv R n f) i) = coeff (m.cons i) f | Mathlib_Data_MvPolynomial_Equiv |
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