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 |
|---|---|---|---|---|---|---|
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (kernel.ι g.unop).op = eqToHom (_ : Opposite.op B.unop = B) ≫ cokernel.π g ≫ (kernelUnopOp g).inv | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | simp | theorem kernel.ι_unop :
(kernel.ι g.unop).op = eqToHom (Opposite.op_unop _) ≫ cokernel.π g ≫ (kernelUnopOp g).inv := by
| Mathlib.CategoryTheory.Abelian.Opposite.124_0.3nBRs3fSYrCoEsT | theorem kernel.ι_unop :
(kernel.ι g.unop).op = eqToHom (Opposite.op_unop _) ≫ cokernel.π g ≫ (kernelUnopOp g).inv | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (cokernel.π g.unop).op = (cokernelUnopOp g).hom ≫ kernel.ι g ≫ eqToHom (_ : A = Opposite.op A.unop) | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | simp | theorem cokernel.π_unop :
(cokernel.π g.unop).op =
(cokernelUnopOp g).hom ≫ kernel.ι g ≫ eqToHom (Opposite.op_unop _).symm :=
by | Mathlib.CategoryTheory.Abelian.Opposite.129_0.3nBRs3fSYrCoEsT | theorem cokernel.π_unop :
(cokernel.π g.unop).op =
(cokernelUnopOp g).hom ≫ kernel.ι g ≫ eqToHom (Opposite.op_unop _).symm | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (image.ι g.unop).op ≫ (imageUnopOp g).hom = factorThruImage g | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | simp only [imageUnopOp, Iso.trans, Iso.symm, Iso.op, cokernelOpOp_inv, cokernelEpiComp_hom,
cokernelCompIsIso_hom, Abelian.coimageIsoImage'_hom, ← Category.assoc, ← op_comp] | theorem image_ι_op_comp_imageUnopOp_hom :
(image.ι g.unop).op ≫ (imageUnopOp g).hom = factorThruImage g := by
| Mathlib.CategoryTheory.Abelian.Opposite.164_0.3nBRs3fSYrCoEsT | theorem image_ι_op_comp_imageUnopOp_hom :
(image.ι g.unop).op ≫ (imageUnopOp g).hom = factorThruImage g | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (((((kernel.lift (cokernel.π g.unop) (cokernel.π (cokernel.π g.unop).op).unop
(_ : (cokernel.π (cokernel.π g.unop).op).unop ≫ cokernel.π g.unop = 0) ≫
(Abelian.imageIsoImage g.unop... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | simp only [Category.assoc, Abelian.imageIsoImage_hom_comp_image_ι, kernel.lift_ι,
Quiver.Hom.op_unop, cokernelIsoOfEq_hom_comp_desc_assoc, cokernel.π_desc_assoc,
cokernel.π_desc] | theorem image_ι_op_comp_imageUnopOp_hom :
(image.ι g.unop).op ≫ (imageUnopOp g).hom = factorThruImage g := by
simp only [imageUnopOp, Iso.trans, Iso.symm, Iso.op, cokernelOpOp_inv, cokernelEpiComp_hom,
cokernelCompIsIso_hom, Abelian.coimageIsoImage'_hom, ← Category.assoc, ← op_comp]
| Mathlib.CategoryTheory.Abelian.Opposite.164_0.3nBRs3fSYrCoEsT | theorem image_ι_op_comp_imageUnopOp_hom :
(image.ι g.unop).op ≫ (imageUnopOp g).hom = factorThruImage g | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ inv (eqToHom (_ : A = Opposite.op A.unop)) ≫ factorThruImage g = factorThruImage g | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | simp only [eqToHom_refl] | theorem image_ι_op_comp_imageUnopOp_hom :
(image.ι g.unop).op ≫ (imageUnopOp g).hom = factorThruImage g := by
simp only [imageUnopOp, Iso.trans, Iso.symm, Iso.op, cokernelOpOp_inv, cokernelEpiComp_hom,
cokernelCompIsIso_hom, Abelian.coimageIsoImage'_hom, ← Category.assoc, ← op_comp]
simp only [Category.asso... | Mathlib.CategoryTheory.Abelian.Opposite.164_0.3nBRs3fSYrCoEsT | theorem image_ι_op_comp_imageUnopOp_hom :
(image.ι g.unop).op ≫ (imageUnopOp g).hom = factorThruImage g | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ inv (𝟙 A) ≫ factorThruImage g = factorThruImage g | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | erw [IsIso.inv_id, Category.id_comp] | theorem image_ι_op_comp_imageUnopOp_hom :
(image.ι g.unop).op ≫ (imageUnopOp g).hom = factorThruImage g := by
simp only [imageUnopOp, Iso.trans, Iso.symm, Iso.op, cokernelOpOp_inv, cokernelEpiComp_hom,
cokernelCompIsIso_hom, Abelian.coimageIsoImage'_hom, ← Category.assoc, ← op_comp]
simp only [Category.asso... | Mathlib.CategoryTheory.Abelian.Opposite.164_0.3nBRs3fSYrCoEsT | theorem image_ι_op_comp_imageUnopOp_hom :
(image.ι g.unop).op ≫ (imageUnopOp g).hom = factorThruImage g | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (imageUnopOp g).hom ≫ image.ι g = (factorThruImage g.unop).op | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | simp only [← cancel_epi (image.ι g.unop).op, ← Category.assoc, image_ι_op_comp_imageUnopOp_hom,
← op_comp, image.fac, Quiver.Hom.op_unop] | theorem imageUnopOp_hom_comp_image_ι :
(imageUnopOp g).hom ≫ image.ι g = (factorThruImage g.unop).op := by
| Mathlib.CategoryTheory.Abelian.Opposite.175_0.3nBRs3fSYrCoEsT | theorem imageUnopOp_hom_comp_image_ι :
(imageUnopOp g).hom ≫ image.ι g = (factorThruImage g.unop).op | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ factorThruImage g ≫ (imageUnopOp g).inv = (image.ι g.unop).op | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | rw [Iso.comp_inv_eq, image_ι_op_comp_imageUnopOp_hom] | theorem factorThruImage_comp_imageUnopOp_inv :
factorThruImage g ≫ (imageUnopOp g).inv = (image.ι g.unop).op := by
| Mathlib.CategoryTheory.Abelian.Opposite.181_0.3nBRs3fSYrCoEsT | theorem factorThruImage_comp_imageUnopOp_inv :
factorThruImage g ≫ (imageUnopOp g).inv = (image.ι g.unop).op | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (imageUnopOp g).inv ≫ (factorThruImage g.unop).op = image.ι g | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | rw [Iso.inv_comp_eq, imageUnopOp_hom_comp_image_ι] | theorem imageUnopOp_inv_comp_op_factorThruImage :
(imageUnopOp g).inv ≫ (factorThruImage g.unop).op = image.ι g := by
| Mathlib.CategoryTheory.Abelian.Opposite.186_0.3nBRs3fSYrCoEsT | theorem imageUnopOp_inv_comp_op_factorThruImage :
(imageUnopOp g).inv ≫ (factorThruImage g.unop).op = image.ι g | Mathlib_CategoryTheory_Abelian_Opposite |
α : Type u_1
β : Type u_2
inst✝¹ : Monoid α
inst✝ : Monoid β
a b : ℕ
⊢ multiplicity ↑a ↑b = multiplicity a b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | apply Part.ext' | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by
| Mathlib.RingTheory.Multiplicity.66_0.uTHZeAJqYiw3Jx8 | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
case x
α : Type u_1
β : Type u_2
inst✝¹ : Monoid α
inst✝ : Monoid β
a b : ℕ
⊢ (multiplicity ↑a ↑b).Dom ↔ (multiplicity a b).Dom | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by
apply Part.ext'
· | Mathlib.RingTheory.Multiplicity.66_0.uTHZeAJqYiw3Jx8 | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
case x
α : Type u_1
β : Type u_2
inst✝¹ : Monoid α
inst✝ : Monoid β
a b : ℕ
⊢ (∃ n, ¬↑a ^ (n + 1) ∣ ↑b) ↔ ∃ n, ¬a ^ (n + 1) ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | norm_cast | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by
apply Part.ext'
· rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ]
| Mathlib.RingTheory.Multiplicity.66_0.uTHZeAJqYiw3Jx8 | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
case x
α : Type u_1
β : Type u_2
inst✝¹ : Monoid α
inst✝ : Monoid β
a b : ℕ
⊢ ∀ (h₁ : (multiplicity ↑a ↑b).Dom) (h₂ : (multiplicity a b).Dom),
Part.get (multiplicity ↑a ↑b) h₁ = Part.get (multiplicity a b) h₂ | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | intro h1 h2 | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by
apply Part.ext'
· rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ]
norm_cast
· | Mathlib.RingTheory.Multiplicity.66_0.uTHZeAJqYiw3Jx8 | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
case x
α : Type u_1
β : Type u_2
inst✝¹ : Monoid α
inst✝ : Monoid β
a b : ℕ
h1 : (multiplicity ↑a ↑b).Dom
h2 : (multiplicity a b).Dom
⊢ Part.get (multiplicity ↑a ↑b) h1 = Part.get (multiplicity a b) h2 | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | apply _root_.le_antisymm | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by
apply Part.ext'
· rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ]
norm_cast
· intro h1 h2
| Mathlib.RingTheory.Multiplicity.66_0.uTHZeAJqYiw3Jx8 | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
case x.a
α : Type u_1
β : Type u_2
inst✝¹ : Monoid α
inst✝ : Monoid β
a b : ℕ
h1 : (multiplicity ↑a ↑b).Dom
h2 : (multiplicity a b).Dom
⊢ Part.get (multiplicity ↑a ↑b) h1 ≤ Part.get (multiplicity a b) h2 | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | apply Nat.find_mono | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by
apply Part.ext'
· rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ]
norm_cast
· intro h1 h2
apply _root_.le_antisymm <;>
· | Mathlib.RingTheory.Multiplicity.66_0.uTHZeAJqYiw3Jx8 | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
case x.a.h
α : Type u_1
β : Type u_2
inst✝¹ : Monoid α
inst✝ : Monoid β
a b : ℕ
h1 : (multiplicity ↑a ↑b).Dom
h2 : (multiplicity a b).Dom
⊢ ∀ (n : ℕ), (fun n => ¬a ^ (n + 1) ∣ b) n → (fun n => ¬↑a ^ (n + 1) ∣ ↑b) n | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | norm_cast | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by
apply Part.ext'
· rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ]
norm_cast
· intro h1 h2
apply _root_.le_antisymm <;>
· apply Nat.find_mono
| Mathlib.RingTheory.Multiplicity.66_0.uTHZeAJqYiw3Jx8 | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
case x.a.h
α : Type u_1
β : Type u_2
inst✝¹ : Monoid α
inst✝ : Monoid β
a b : ℕ
h1 : (multiplicity ↑a ↑b).Dom
h2 : (multiplicity a b).Dom
⊢ ∀ (n : ℕ), (fun n => ¬a ^ (n + 1) ∣ b) n → (fun n => ¬a ^ (n + 1) ∣ b) n | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by
apply Part.ext'
· rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ]
norm_cast
· intro h1 h2
apply _root_.le_antisymm <;>
· apply Nat.find_mono
norm_cast
... | Mathlib.RingTheory.Multiplicity.66_0.uTHZeAJqYiw3Jx8 | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
case x.a
α : Type u_1
β : Type u_2
inst✝¹ : Monoid α
inst✝ : Monoid β
a b : ℕ
h1 : (multiplicity ↑a ↑b).Dom
h2 : (multiplicity a b).Dom
⊢ Part.get (multiplicity a b) h2 ≤ Part.get (multiplicity ↑a ↑b) h1 | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | apply Nat.find_mono | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by
apply Part.ext'
· rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ]
norm_cast
· intro h1 h2
apply _root_.le_antisymm <;>
· | Mathlib.RingTheory.Multiplicity.66_0.uTHZeAJqYiw3Jx8 | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
case x.a.h
α : Type u_1
β : Type u_2
inst✝¹ : Monoid α
inst✝ : Monoid β
a b : ℕ
h1 : (multiplicity ↑a ↑b).Dom
h2 : (multiplicity a b).Dom
⊢ ∀ (n : ℕ), (fun n => ¬↑a ^ (n + 1) ∣ ↑b) n → (fun n => ¬a ^ (n + 1) ∣ b) n | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | norm_cast | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by
apply Part.ext'
· rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ]
norm_cast
· intro h1 h2
apply _root_.le_antisymm <;>
· apply Nat.find_mono
| Mathlib.RingTheory.Multiplicity.66_0.uTHZeAJqYiw3Jx8 | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
case x.a.h
α : Type u_1
β : Type u_2
inst✝¹ : Monoid α
inst✝ : Monoid β
a b : ℕ
h1 : (multiplicity ↑a ↑b).Dom
h2 : (multiplicity a b).Dom
⊢ ∀ (n : ℕ), (fun n => ¬a ^ (n + 1) ∣ b) n → (fun n => ¬a ^ (n + 1) ∣ b) n | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by
apply Part.ext'
· rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ]
norm_cast
· intro h1 h2
apply _root_.le_antisymm <;>
· apply Nat.find_mono
norm_cast
... | Mathlib.RingTheory.Multiplicity.66_0.uTHZeAJqYiw3Jx8 | @[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Monoid α
inst✝ : Monoid β
a b : α
h : ¬Finite a b
n : ℕ
⊢ a ^ zero ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [_root_.pow_zero] | theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b :=
⟨fun h n =>
Nat.casesOn n
(by
| Mathlib.RingTheory.Multiplicity.78_0.uTHZeAJqYiw3Jx8 | theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Monoid α
inst✝ : Monoid β
a b : α
h : ¬Finite a b
n : ℕ
⊢ 1 ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact one_dvd _ | theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b :=
⟨fun h n =>
Nat.casesOn n
(by
rw [_root_.pow_zero]
| Mathlib.RingTheory.Multiplicity.78_0.uTHZeAJqYiw3Jx8 | theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Monoid α
inst✝ : Monoid β
a b : α
h : ¬Finite a b
n : ℕ
⊢ ∀ (n : ℕ), a ^ succ n ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simpa [Finite, Classical.not_not] using h | theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b :=
⟨fun h n =>
Nat.casesOn n
(by
rw [_root_.pow_zero]
exact one_dvd _)
(by | Mathlib.RingTheory.Multiplicity.78_0.uTHZeAJqYiw3Jx8 | theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Monoid α
inst✝ : Monoid β
a b : α
⊢ (∀ (n : ℕ), a ^ n ∣ b) → ¬Finite a b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp [Finite, multiplicity, Classical.not_not] | theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b :=
⟨fun h n =>
Nat.casesOn n
(by
rw [_root_.pow_zero]
exact one_dvd _)
(by simpa [Finite, Classical.not_not] using h),
by | Mathlib.RingTheory.Multiplicity.78_0.uTHZeAJqYiw3Jx8 | theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Monoid α
inst✝ : Monoid β
a b : α
⊢ (∀ (n : ℕ), a ^ n ∣ b) → ∀ (x : ℕ), a ^ (x + 1) ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | tauto | theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b :=
⟨fun h n =>
Nat.casesOn n
(by
rw [_root_.pow_zero]
exact one_dvd _)
(by simpa [Finite, Classical.not_not] using h),
by simp [Finite, multiplicity, Classical.not_not]; | Mathlib.RingTheory.Multiplicity.78_0.uTHZeAJqYiw3Jx8 | theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
k : ℕ
⊢ ↑k ≤ multiplicity a b → a ^ k ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [← PartENat.some_eq_natCast] | theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} :
(k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by
| Mathlib.RingTheory.Multiplicity.99_0.uTHZeAJqYiw3Jx8 | theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} :
(k : PartENat) ≤ multiplicity a b → a ^ k ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
k : ℕ
⊢ ↑k ≤ multiplicity a b → a ^ k ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact
Nat.casesOn k
(fun _ => by
rw [_root_.pow_zero]
exact one_dvd _)
fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk | theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} :
(k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by
rw [← PartENat.some_eq_natCast]
| Mathlib.RingTheory.Multiplicity.99_0.uTHZeAJqYiw3Jx8 | theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} :
(k : PartENat) ≤ multiplicity a b → a ^ k ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
k : ℕ
x✝ : ↑zero ≤ multiplicity a b
⊢ a ^ zero ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [_root_.pow_zero] | theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} :
(k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by
rw [← PartENat.some_eq_natCast]
exact
Nat.casesOn k
(fun _ => by
| Mathlib.RingTheory.Multiplicity.99_0.uTHZeAJqYiw3Jx8 | theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} :
(k : PartENat) ≤ multiplicity a b → a ^ k ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
k : ℕ
x✝ : ↑zero ≤ multiplicity a b
⊢ 1 ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact one_dvd _ | theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} :
(k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by
rw [← PartENat.some_eq_natCast]
exact
Nat.casesOn k
(fun _ => by
rw [_root_.pow_zero]
| Mathlib.RingTheory.Multiplicity.99_0.uTHZeAJqYiw3Jx8 | theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} :
(k : PartENat) ≤ multiplicity a b → a ^ k ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h : Finite a b
⊢ ↑(Part.get (multiplicity a b) h) ≤ multiplicity a b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [PartENat.natCast_get] | theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b :=
pow_dvd_of_le_multiplicity (by | Mathlib.RingTheory.Multiplicity.110_0.uTHZeAJqYiw3Jx8 | theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
m : ℕ
hm : multiplicity a b < ↑m
h : a ^ m ∣ b
⊢ False | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [PartENat.lt_coe_iff] at hm | theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by
| Mathlib.RingTheory.Multiplicity.114_0.uTHZeAJqYiw3Jx8 | theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
m : ℕ
hm : ∃ (h : (multiplicity a b).Dom), Part.get (multiplicity a b) h < m
h : a ^ m ∣ b
⊢ False | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) | theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by
rw [PartENat.lt_coe_iff] at hm; | Mathlib.RingTheory.Multiplicity.114_0.uTHZeAJqYiw3Jx8 | theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
m : ℕ
h : Finite a b
hm : Part.get (multiplicity a b) h < m
⊢ multiplicity a b < ↑m | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm | theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) :
¬a ^ m ∣ b :=
is_greatest (by | Mathlib.RingTheory.Multiplicity.118_0.uTHZeAJqYiw3Jx8 | theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) :
¬a ^ m ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
hfin : Finite a b
hdiv : a ∣ b
⊢ 0 < Part.get (multiplicity a b) hfin | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | refine' zero_lt_iff.2 fun h => _ | theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) :
0 < (multiplicity a b).get hfin := by
| Mathlib.RingTheory.Multiplicity.123_0.uTHZeAJqYiw3Jx8 | theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) :
0 < (multiplicity a b).get hfin | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
hfin : Finite a b
hdiv : a ∣ b
h : Part.get (multiplicity a b) hfin = 0
⊢ False | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) | theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) :
0 < (multiplicity a b).get hfin := by
refine' zero_lt_iff.2 fun h => _
| Mathlib.RingTheory.Multiplicity.123_0.uTHZeAJqYiw3Jx8 | theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) :
0 < (multiplicity a b).get hfin | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
k : ℕ
hk : a ^ k ∣ b
hsucc : ¬a ^ (k + 1) ∣ b
⊢ multiplicity a b ≤ ↑k | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | have : Finite a b := ⟨k, hsucc⟩ | theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
(k : PartENat) = multiplicity a b :=
le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by
| Mathlib.RingTheory.Multiplicity.129_0.uTHZeAJqYiw3Jx8 | theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
(k : PartENat) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
k : ℕ
hk : a ^ k ∣ b
hsucc : ¬a ^ (k + 1) ∣ b
this : Finite a b
⊢ multiplicity a b ≤ ↑k | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [PartENat.le_coe_iff] | theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
(k : PartENat) = multiplicity a b :=
le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by
have : Finite a b := ⟨k, hsucc⟩
| Mathlib.RingTheory.Multiplicity.129_0.uTHZeAJqYiw3Jx8 | theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
(k : PartENat) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
k : ℕ
hk : a ^ k ∣ b
hsucc : ¬a ^ (k + 1) ∣ b
this : Finite a b
⊢ ∃ (h : (multiplicity a b).Dom), Part.get (multiplicity a b) h ≤ k | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact ⟨this, Nat.find_min' _ hsucc⟩ | theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
(k : PartENat) = multiplicity a b :=
le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by
have : Finite a b := ⟨k, hsucc⟩
rw [PartENat.le_coe_iff]
| Mathlib.RingTheory.Multiplicity.129_0.uTHZeAJqYiw3Jx8 | theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
(k : PartENat) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
k : ℕ
hk : a ^ k ∣ b
hsucc : ¬a ^ (k + 1) ∣ b
⊢ k = Part.get (multiplicity a b) (_ : ∃ n, (fun n => ¬a ^ (n + 1) ∣ b) n) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] | theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
k = get (multiplicity a b) ⟨k, hsucc⟩ := by
| Mathlib.RingTheory.Multiplicity.137_0.uTHZeAJqYiw3Jx8 | theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
k = get (multiplicity a b) ⟨k, hsucc⟩ | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
k : ℕ
⊢ multiplicity a b < ↑k ↔ ¬a ^ k ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [pow_dvd_iff_le_multiplicity, not_le] | theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} :
multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by | Mathlib.RingTheory.Multiplicity.152_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} :
multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
n : ℕ
⊢ multiplicity a b = ↑n ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [← PartENat.some_eq_natCast] | theorem eq_coe_iff {a b : α} {n : ℕ} :
multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by
| Mathlib.RingTheory.Multiplicity.156_0.uTHZeAJqYiw3Jx8 | theorem eq_coe_iff {a b : α} {n : ℕ} :
multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
n : ℕ
⊢ multiplicity a b = ↑n ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact
⟨fun h =>
let ⟨h₁, h₂⟩ := eq_some_iff.1 h
h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by
rw [PartENat.lt_coe_iff]
exact ⟨h₁, lt_succ_self _⟩)⟩,
fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ | theorem eq_coe_iff {a b : α} {n : ℕ} :
multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by
rw [← PartENat.some_eq_natCast]
| Mathlib.RingTheory.Multiplicity.156_0.uTHZeAJqYiw3Jx8 | theorem eq_coe_iff {a b : α} {n : ℕ} :
multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
n : ℕ
h : multiplicity a b = ↑n
h₁ : (multiplicity a b).Dom
h₂ : Part.get (multiplicity a b) h₁ = n
⊢ multiplicity a b < ↑(Part.get (multiplicity a b) h₁ + 1) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [PartENat.lt_coe_iff] | theorem eq_coe_iff {a b : α} {n : ℕ} :
multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by
rw [← PartENat.some_eq_natCast]
exact
⟨fun h =>
let ⟨h₁, h₂⟩ := eq_some_iff.1 h
h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by
| Mathlib.RingTheory.Multiplicity.156_0.uTHZeAJqYiw3Jx8 | theorem eq_coe_iff {a b : α} {n : ℕ} :
multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
n : ℕ
h : multiplicity a b = ↑n
h₁ : (multiplicity a b).Dom
h₂ : Part.get (multiplicity a b) h₁ = n
⊢ ∃ (h : (multiplicity a b).Dom), Part.get (multiplicity a b) h <... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact ⟨h₁, lt_succ_self _⟩ | theorem eq_coe_iff {a b : α} {n : ℕ} :
multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by
rw [← PartENat.some_eq_natCast]
exact
⟨fun h =>
let ⟨h₁, h₂⟩ := eq_some_iff.1 h
h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by
rw [PartENat.lt_coe_iff]
| Mathlib.RingTheory.Multiplicity.156_0.uTHZeAJqYiw3Jx8 | theorem eq_coe_iff {a b : α} {n : ℕ} :
multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
⊢ (∀ (n : ℕ), ¬¬a ^ (n + 1) ∣ b) ↔ ∀ (n : ℕ), a ^ n ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp only [Classical.not_not] | theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b :=
(PartENat.find_eq_top_iff _).trans <| by
| Mathlib.RingTheory.Multiplicity.168_0.uTHZeAJqYiw3Jx8 | theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
⊢ (∀ (n : ℕ), a ^ (n + 1) ∣ b) ↔ ∀ (n : ℕ), a ^ n ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact
⟨fun h n =>
Nat.casesOn n
(by
rw [_root_.pow_zero]
exact one_dvd _)
fun n => h _,
fun h n => h _⟩ | theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b :=
(PartENat.find_eq_top_iff _).trans <| by
simp only [Classical.not_not]
| Mathlib.RingTheory.Multiplicity.168_0.uTHZeAJqYiw3Jx8 | theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h : ∀ (n : ℕ), a ^ (n + 1) ∣ b
n : ℕ
⊢ a ^ zero ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [_root_.pow_zero] | theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b :=
(PartENat.find_eq_top_iff _).trans <| by
simp only [Classical.not_not]
exact
⟨fun h n =>
Nat.casesOn n
(by
| Mathlib.RingTheory.Multiplicity.168_0.uTHZeAJqYiw3Jx8 | theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h : ∀ (n : ℕ), a ^ (n + 1) ∣ b
n : ℕ
⊢ 1 ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact one_dvd _ | theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b :=
(PartENat.find_eq_top_iff _).trans <| by
simp only [Classical.not_not]
exact
⟨fun h n =>
Nat.casesOn n
(by
rw [_root_.pow_zero]
| Mathlib.RingTheory.Multiplicity.168_0.uTHZeAJqYiw3Jx8 | theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : Finite a 1
⊢ Part.get (multiplicity a 1) ha = 0 | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] | @[simp]
theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by
| Mathlib.RingTheory.Multiplicity.191_0.uTHZeAJqYiw3Jx8 | @[simp]
theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : Finite a 1
⊢ 1 ∣ 1 ∧ ¬a ^ (0 + 1) ∣ 1 | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp [not_dvd_one_of_finite_one_right ha] | @[simp]
theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by
rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero]
| Mathlib.RingTheory.Multiplicity.191_0.uTHZeAJqYiw3Jx8 | @[simp]
theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
⊢ multiplicity a b = 0 ↔ ¬a ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [← Nat.cast_zero, eq_coe_iff] | theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by
| Mathlib.RingTheory.Multiplicity.202_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
⊢ a ^ 0 ∣ b ∧ ¬a ^ (0 + 1) ∣ b ↔ ¬a ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] | theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by
rw [← Nat.cast_zero, eq_coe_iff]
| Mathlib.RingTheory.Multiplicity.202_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
⊢ multiplicity a b ≠ ⊤ ↔ Finite a b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] | theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by
| Mathlib.RingTheory.Multiplicity.215_0.uTHZeAJqYiw3Jx8 | theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
⊢ multiplicity a b < ⊤ ↔ Finite a b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [lt_top_iff_ne_top, ne_top_iff_finite] | theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by
| Mathlib.RingTheory.Multiplicity.219_0.uTHZeAJqYiw3Jx8 | theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
hfin : Finite a b
⊢ ∃ c, b = a ^ Part.get (multiplicity a b) hfin * c ∧ ¬a ∣ c | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin | theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by
| Mathlib.RingTheory.Multiplicity.223_0.uTHZeAJqYiw3Jx8 | theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c | Mathlib_RingTheory_Multiplicity |
case intro
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
hfin : Finite a b
c : α
hc : b = a ^ Part.get (multiplicity a b) hfin * c
⊢ ∃ c, b = a ^ Part.get (multiplicity a b) hfin * c ∧ ¬a ∣ c | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | refine' ⟨c, hc, _⟩ | theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by
obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin
| Mathlib.RingTheory.Multiplicity.223_0.uTHZeAJqYiw3Jx8 | theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c | Mathlib_RingTheory_Multiplicity |
case intro
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
hfin : Finite a b
c : α
hc : b = a ^ Part.get (multiplicity a b) hfin * c
⊢ ¬a ∣ c | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rintro ⟨k, hk⟩ | theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by
obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin
refine' ⟨c, hc, _⟩
| Mathlib.RingTheory.Multiplicity.223_0.uTHZeAJqYiw3Jx8 | theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c | Mathlib_RingTheory_Multiplicity |
case intro.intro
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
hfin : Finite a b
c : α
hc : b = a ^ Part.get (multiplicity a b) hfin * c
k : α
hk : c = a * k
⊢ False | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc | theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by
obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin
refine' ⟨c, hc, _⟩
rintro ⟨k, hk⟩
| Mathlib.RingTheory.Multiplicity.223_0.uTHZeAJqYiw3Jx8 | theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c | Mathlib_RingTheory_Multiplicity |
case intro.intro
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
hfin : Finite a b
c k : α
hc : b = a ^ (Part.get (multiplicity a b) hfin + 1) * k
hk : c = a * k
⊢ False | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ | theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by
obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin
refine' ⟨c, hc, _⟩
rintro ⟨k, hk⟩
rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc
| Mathlib.RingTheory.Multiplicity.223_0.uTHZeAJqYiw3Jx8 | theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c | Mathlib_RingTheory_Multiplicity |
case intro.intro
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
hfin : Finite a b
c k : α
hc : b = a ^ (Part.get (multiplicity a b) hfin + 1) * k
hk : c = a * k
h₁ : a ^ (Part.get (multiplicity a b) hfin + 1) ∣ b... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ | theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by
obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin
refine' ⟨c, hc, _⟩
rintro ⟨k, hk⟩
rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc
have h₁ : a ^ ((multiplicity a b).... | Mathlib.RingTheory.Multiplicity.223_0.uTHZeAJqYiw3Jx8 | theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
hfin : Finite a b
c k : α
hc : b = a ^ (Part.get (multiplicity a b) hfin + 1) * k
hk : c = a * k
h₁ : a ^ (Part.get (multiplicity a b) hfin + 1) ∣ b
⊢ multiplicity a... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp | theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by
obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin
refine' ⟨c, hc, _⟩
rintro ⟨k, hk⟩
rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc
have h₁ : a ^ ((multiplicity a b).... | Mathlib.RingTheory.Multiplicity.223_0.uTHZeAJqYiw3Jx8 | theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
c d : β
h : ∀ (n : ℕ), a ^ n ∣ b → c ^ n ∣ d
this : Decidable (Finite a b) := Classical.dec (Finite a b)
hab : Finite a b
⊢ multiplicity a b ≤ multiplicity c d | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [← PartENat.natCast_get (finite_iff_dom.1 hab)] | theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d :=
⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h =>
letI := Classical.dec (Finite a b)
if hab : Finite a b then by
| Mathlib.RingTheory.Multiplicity.233_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
c d : β
h : ∀ (n : ℕ), a ^ n ∣ b → c ^ n ∣ d
this : Decidable (Finite a b) := Classical.dec (Finite a b)
hab : Finite a b
⊢ ↑(Part.get (multiplicity a b) (_ : (multi... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) | theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d :=
⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h =>
letI := Classical.dec (Finite a b)
if hab : Finite a b then by
rw ... | Mathlib.RingTheory.Multiplicity.233_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
c d : β
h : ∀ (n : ℕ), a ^ n ∣ b → c ^ n ∣ d
this : Decidable (Finite a b) := Classical.dec (Finite a b)
hab : ¬Finite a b
⊢ multiplicity a b ≤ multiplicity c d | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) | theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d :=
⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h =>
letI := Classical.dec (Finite a b)
if hab : Finite a b then by
rw ... | Mathlib.RingTheory.Multiplicity.233_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
c d : β
h : ∀ (n : ℕ), a ^ n ∣ b → c ^ n ∣ d
this✝ : Decidable (Finite a b) := Classical.dec (Finite a b)
hab : ¬Finite a b
this : ∀ (n : ℕ), c ^ n ∣ d
⊢ multiplicit... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)] | theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d :=
⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h =>
letI := Classical.dec (Finite a b)
if hab : Finite a b then by
rw ... | Mathlib.RingTheory.Multiplicity.233_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝⁴ : Monoid α
inst✝³ : Monoid β
inst✝² : DecidableRel fun x x_1 => x ∣ x_1
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
F : Type u_3
inst✝ : MonoidHomClass F α β
f : F
a b : α
n : ℕ
⊢ a ^ n ∣ b → f a ^ n ∣ f b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [← map_pow] | theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} :
multiplicity a b ≤ multiplicity (f a) (f b) :=
multiplicity_le_multiplicity_iff.mpr fun n ↦ by | Mathlib.RingTheory.Multiplicity.254_0.uTHZeAJqYiw3Jx8 | theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} :
multiplicity a b ≤ multiplicity (f a) (f b) | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝⁴ : Monoid α
inst✝³ : Monoid β
inst✝² : DecidableRel fun x x_1 => x ∣ x_1
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
F : Type u_3
inst✝ : MonoidHomClass F α β
f : F
a b : α
n : ℕ
⊢ a ^ n ∣ b → f (a ^ n) ∣ f b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact map_dvd f | theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} :
multiplicity a b ≤ multiplicity (f a) (f b) :=
multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; | Mathlib.RingTheory.Multiplicity.254_0.uTHZeAJqYiw3Jx8 | theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} :
multiplicity a b ≤ multiplicity (f a) (f b) | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝⁴ : Monoid α
inst✝³ : Monoid β
inst✝² : DecidableRel fun x x_1 => x ∣ x_1
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
F : Type u_3
inst✝ : MulEquivClass F α β
f : F
a b : α
n : ℕ
⊢ f a ^ n ∣ f b ↔ a ^ n ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [← map_pow] | theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} :
multiplicity (f a) (f b) = multiplicity a b :=
multiplicity_eq_multiplicity_iff.mpr fun n ↦ by | Mathlib.RingTheory.Multiplicity.258_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} :
multiplicity (f a) (f b) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝⁴ : Monoid α
inst✝³ : Monoid β
inst✝² : DecidableRel fun x x_1 => x ∣ x_1
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
F : Type u_3
inst✝ : MulEquivClass F α β
f : F
a b : α
n : ℕ
⊢ f (a ^ n) ∣ f b ↔ a ^ n ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact map_dvd_iff f | theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} :
multiplicity (f a) (f b) = multiplicity a b :=
multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; | Mathlib.RingTheory.Multiplicity.258_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} :
multiplicity (f a) (f b) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h : 0 < multiplicity a b
⊢ a ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [← pow_one a] | theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by
| Mathlib.RingTheory.Multiplicity.273_0.uTHZeAJqYiw3Jx8 | theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h : 0 < multiplicity a b
⊢ a ^ 1 ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | apply pow_dvd_of_le_multiplicity | theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by
rw [← pow_one a]
| Mathlib.RingTheory.Multiplicity.273_0.uTHZeAJqYiw3Jx8 | theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b | Mathlib_RingTheory_Multiplicity |
case a
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h : 0 < multiplicity a b
⊢ ↑1 ≤ multiplicity a b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h | theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by
rw [← pow_one a]
apply pow_dvd_of_le_multiplicity
| Mathlib.RingTheory.Multiplicity.273_0.uTHZeAJqYiw3Jx8 | theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
hdvd : a ∣ b
heq : 0 = multiplicity a b
⊢ multiplicity a b < 1 | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one | theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b :=
⟨dvd_of_multiplicity_pos, fun hdvd =>
lt_of_le_of_ne (zero_le _) fun heq =>
is_greatest
(show multiplicity a b < ↑1 by
| Mathlib.RingTheory.Multiplicity.279_0.uTHZeAJqYiw3Jx8 | theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
hdvd : a ∣ b
heq : 0 = multiplicity a b
⊢ a ^ 1 ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rwa [pow_one a] | theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b :=
⟨dvd_of_multiplicity_pos, fun hdvd =>
lt_of_le_of_ne (zero_le _) fun heq =>
is_greatest
(show multiplicity a b < ↑1 by
simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one)
... | Mathlib.RingTheory.Multiplicity.279_0.uTHZeAJqYiw3Jx8 | theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : ℕ
⊢ Finite a b ↔ a ≠ 1 ∧ 0 < b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt,
le_zero_iff] | theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by
| Mathlib.RingTheory.Multiplicity.288_0.uTHZeAJqYiw3Jx8 | theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : ℕ
⊢ (∀ (n : ℕ), a ^ n ∣ b) ↔ a = 1 ∨ b = 0 | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact
⟨fun h =>
or_iff_not_imp_right.2 fun hb =>
have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1
Classical.by_contradiction fun ha1 : a ≠ 1 =>
have ha_gt_one : 1 < a :=
lt_of_not_ge fun _ =>
match a with
| 0 => ... | theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by
rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt,
le_zero_iff]
| Mathlib.RingTheory.Multiplicity.288_0.uTHZeAJqYiw3Jx8 | theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : ℕ
h : ∀ (n : ℕ), a ^ n ∣ b
hb : ¬b = 0
ha : a = 0
⊢ 0 ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [ha] at h | theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by
rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt,
le_zero_iff]
exact
⟨fun h =>
or_iff_not_imp_right.2 fun hb =>
have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by | Mathlib.RingTheory.Multiplicity.288_0.uTHZeAJqYiw3Jx8 | theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : ℕ
h : ∀ (n : ℕ), 0 ^ n ∣ b
hb : ¬b = 0
ha : a = 0
⊢ 0 ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact h 1 | theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by
rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt,
le_zero_iff]
exact
⟨fun h =>
or_iff_not_imp_right.2 fun hb =>
have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; | Mathlib.RingTheory.Multiplicity.288_0.uTHZeAJqYiw3Jx8 | theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b✝ : ℕ
hb : ¬b✝ = 0
b : ℕ
h : ∀ (n : ℕ), (b + 2) ^ n ∣ b✝
ha : b + 2 ≠ 0
ha1 : b + 2 ≠ 1
x✝ : 1 ≥ b + 2
⊢ False | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | linarith | theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by
rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt,
le_zero_iff]
exact
⟨fun h =>
or_iff_not_imp_right.2 fun hb =>
have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exac... | Mathlib.RingTheory.Multiplicity.288_0.uTHZeAJqYiw3Jx8 | theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : ℕ
h : a = 1 ∨ b = 0
⊢ ∀ (n : ℕ), a ^ n ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | cases h | theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by
rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt,
le_zero_iff]
exact
⟨fun h =>
or_iff_not_imp_right.2 fun hb =>
have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exac... | Mathlib.RingTheory.Multiplicity.288_0.uTHZeAJqYiw3Jx8 | theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b | Mathlib_RingTheory_Multiplicity |
case inl
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : ℕ
h✝ : a = 1
⊢ ∀ (n : ℕ), a ^ n ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp [*] | theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by
rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt,
le_zero_iff]
exact
⟨fun h =>
or_iff_not_imp_right.2 fun hb =>
have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exac... | Mathlib.RingTheory.Multiplicity.288_0.uTHZeAJqYiw3Jx8 | theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b | Mathlib_RingTheory_Multiplicity |
case inr
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : ℕ
h✝ : b = 0
⊢ ∀ (n : ℕ), a ^ n ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp [*] | theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by
rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt,
le_zero_iff]
exact
⟨fun h =>
or_iff_not_imp_right.2 fun hb =>
have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exac... | Mathlib.RingTheory.Multiplicity.288_0.uTHZeAJqYiw3Jx8 | theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
a b c : α
⊢ Finite a (b * c) → Finite a c | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [mul_comm] | theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by
| Mathlib.RingTheory.Multiplicity.314_0.uTHZeAJqYiw3Jx8 | theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
a b c : α
⊢ Finite a (c * b) → Finite a c | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact finite_of_finite_mul_right | theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by
rw [mul_comm]; | Mathlib.RingTheory.Multiplicity.314_0.uTHZeAJqYiw3Jx8 | theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CommMonoid α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
ha : ¬IsUnit a
hb : IsUnit b
⊢ a ^ 0 ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp only [_root_.pow_zero, one_dvd] | theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 :=
eq_coe_iff.2
⟨show a ^ 0 ∣ b by | Mathlib.RingTheory.Multiplicity.320_0.uTHZeAJqYiw3Jx8 | theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CommMonoid α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
ha : ¬IsUnit a
hb : IsUnit b
⊢ ¬a ^ (0 + 1) ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [pow_one] | theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 :=
eq_coe_iff.2
⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by
| Mathlib.RingTheory.Multiplicity.320_0.uTHZeAJqYiw3Jx8 | theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CommMonoid α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
ha : ¬IsUnit a
hb : IsUnit b
⊢ ¬a ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact fun h => mt (isUnit_of_dvd_unit h) ha hb | theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 :=
eq_coe_iff.2
⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by
rw [pow_one]
| Mathlib.RingTheory.Multiplicity.320_0.uTHZeAJqYiw3Jx8 | theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝ : MonoidWithZero α
a b : α
h : Finite a b
n : ℕ
hn : ¬a ^ (n + 1) ∣ b
hb : b = 0
⊢ False | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp [hb] at hn | theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 :=
let ⟨n, hn⟩ := h
fun hb => by | Mathlib.RingTheory.Multiplicity.357_0.uTHZeAJqYiw3Jx8 | theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝² : CommMonoidWithZero α
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
⊢ multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | by_cases h : Finite a b | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by
| Mathlib.RingTheory.Multiplicity.382_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
case pos
α : Type u_1
β : Type u_2
inst✝² : CommMonoidWithZero α
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h : Finite a b
⊢ multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [← PartENat.natCast_get (finite_iff_dom.mp h)] | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by
by_cases h : Finite a b
· | Mathlib.RingTheory.Multiplicity.382_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
case pos
α : Type u_1
β : Type u_2
inst✝² : CommMonoidWithZero α
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h : Finite a b
⊢ multiplicity (Associates.mk a) (Associates.mk b) = ↑(Part.get (multiplicity a b) (_ : (multiplicity a b).Dom)) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | refine'
(multiplicity.unique
(show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by
by_cases h : Finite a b
· rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
| Mathlib.RingTheory.Multiplicity.382_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
case pos.refine'_1
α : Type u_1
β : Type u_2
inst✝² : CommMonoidWithZero α
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h : Finite a b
⊢ Associates.mk a ^ Part.get (multiplicity a b) h ∣ Associates.mk b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [← Associates.mk_pow, Associates.mk_dvd_mk] | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by
by_cases h : Finite a b
· rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
refine'
(multiplicity.unique
... | Mathlib.RingTheory.Multiplicity.382_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
case pos.refine'_2
α : Type u_1
β : Type u_2
inst✝² : CommMonoidWithZero α
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h : Finite a b
⊢ ¬Associates.mk a ^ (Part.get (multiplicity a b) h + 1) ∣ Associates.mk b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [← Associates.mk_pow, Associates.mk_dvd_mk] | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by
by_cases h : Finite a b
· rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
refine'
(multiplicity.unique
... | Mathlib.RingTheory.Multiplicity.382_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
case pos.refine'_1
α : Type u_1
β : Type u_2
inst✝² : CommMonoidWithZero α
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h : Finite a b
⊢ a ^ Part.get (multiplicity a b) h ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact pow_multiplicity_dvd h | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by
by_cases h : Finite a b
· rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
refine'
(multiplicity.unique
... | Mathlib.RingTheory.Multiplicity.382_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
case pos.refine'_2
α : Type u_1
β : Type u_2
inst✝² : CommMonoidWithZero α
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h : Finite a b
⊢ ¬a ^ (Part.get (multiplicity a b) h + 1) ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact is_greatest
((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by
by_cases h : Finite a b
· rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
refine'
(multiplicity.unique
... | Mathlib.RingTheory.Multiplicity.382_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
case neg
α : Type u_1
β : Type u_2
inst✝² : CommMonoidWithZero α
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h : ¬Finite a b
⊢ multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | suffices ¬Finite (Associates.mk a) (Associates.mk b) by
rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this
rw [h, this] | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by
by_cases h : Finite a b
· rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
refine'
(multiplicity.unique
... | Mathlib.RingTheory.Multiplicity.382_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝² : CommMonoidWithZero α
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h : ¬Finite a b
this : ¬Finite (Associates.mk a) (Associates.mk b)
⊢ multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by
by_cases h : Finite a b
· rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
refine'
(multiplicity.unique
... | Mathlib.RingTheory.Multiplicity.382_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝² : CommMonoidWithZero α
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h : multiplicity a b = ⊤
this : multiplicity (Associates.mk a) (Associates.mk b) = ⊤
⊢ multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [h, this] | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by
by_cases h : Finite a b
· rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
refine'
(multiplicity.unique
... | Mathlib.RingTheory.Multiplicity.382_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
case neg
α : Type u_1
β : Type u_2
inst✝² : CommMonoidWithZero α
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h : ¬Finite a b
⊢ ¬Finite (Associates.mk a) (Associates.mk b) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | refine'
not_finite_iff_forall.mpr fun n => by
rw [← Associates.mk_pow, Associates.mk_dvd_mk]
exact not_finite_iff_forall.mp h n | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by
by_cases h : Finite a b
· rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
refine'
(multiplicity.unique
... | Mathlib.RingTheory.Multiplicity.382_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝² : CommMonoidWithZero α
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h : ¬Finite a b
n : ℕ
⊢ Associates.mk a ^ n ∣ Associates.mk b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [← Associates.mk_pow, Associates.mk_dvd_mk] | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by
by_cases h : Finite a b
· rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
refine'
(multiplicity.unique
... | Mathlib.RingTheory.Multiplicity.382_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
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