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 |
|---|---|---|---|---|---|---|
case a.mk
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
r : ↥(reesAlgebra I)
f : PolynomialModule R M
hf :
f ∈
{
toAddSubsemigroup :=
{ carrier := {f | ∀ (i : ℕ), f i ∈ N F i},
add_mem' :=
... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k)) | /-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | ∀ i, f i ∈ F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw... | Mathlib.RingTheory.Filtration.263_0.wQ6WBws0g3n9213 | /-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
⊢ Filtration.submodule (F ⊓ F') = Filtration.submodule F ⊓ Filtration.submodule F' | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | ext | theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ F'.submodule := by
| Mathlib.RingTheory.Filtration.282_0.wQ6WBws0g3n9213 | theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ F'.submodule | Mathlib_RingTheory_Filtration |
case h
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
x✝ : PolynomialModule R M
⊢ x✝ ∈ Filtration.submodule (F ⊓ F') ↔ x✝ ∈ Filtration.submodule F ⊓ Filtration.submodule F' | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | exact forall_and | theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ F'.submodule := by
ext
| Mathlib.RingTheory.Filtration.282_0.wQ6WBws0g3n9213 | theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ F'.submodule | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
⊢ AddSubmonoid.closure (⋃ i, ⇑(single R i) '' ↑(N F i)) = (Filtration.submodule F).toAddSubmonoid | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | apply le_antisymm | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
| Mathlib.RingTheory.Filtration.298_0.wQ6WBws0g3n9213 | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid | Mathlib_RingTheory_Filtration |
case a
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
⊢ AddSubmonoid.closure (⋃ i, ⇑(single R i) '' ↑(N F i)) ≤ (Filtration.submodule F).toAddSubmonoid | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
· | Mathlib.RingTheory.Filtration.298_0.wQ6WBws0g3n9213 | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid | Mathlib_RingTheory_Filtration |
case a
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
⊢ ∀ (i : ℕ), ⇑(single R i) '' ↑(N F i) ⊆ ↑(Filtration.submodule F).toAddSubmonoid | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rintro i _ ⟨m, hm, rfl⟩ j | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
| Mathlib.RingTheory.Filtration.298_0.wQ6WBws0g3n9213 | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid | Mathlib_RingTheory_Filtration |
case a.intro.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
i : ℕ
m : M
hm : m ∈ ↑(N F i)
j : ℕ
⊢ ((single R i) m) j ∈ N F j | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rw [single_apply] | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ ⟨m, hm, rfl⟩ j
| Mathlib.RingTheory.Filtration.298_0.wQ6WBws0g3n9213 | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid | Mathlib_RingTheory_Filtration |
case a.intro.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
i : ℕ
m : M
hm : m ∈ ↑(N F i)
j : ℕ
⊢ (if i = j then m else 0) ∈ N F j | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | split_ifs with h | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ ⟨m, hm, rfl⟩ j
rw [single_apply]
| Mathlib.RingTheory.Filtration.298_0.wQ6WBws0g3n9213 | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid | Mathlib_RingTheory_Filtration |
case pos
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h✝ : Stable F
i : ℕ
m : M
hm : m ∈ ↑(N F i)
j : ℕ
h : i = j
⊢ m ∈ N F j | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rwa [← h] | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ ⟨m, hm, rfl⟩ j
rw [single_apply]
split_ifs with h
· | Mathlib.RingTheory.Filtration.298_0.wQ6WBws0g3n9213 | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid | Mathlib_RingTheory_Filtration |
case neg
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h✝ : Stable F
i : ℕ
m : M
hm : m ∈ ↑(N F i)
j : ℕ
h : ¬i = j
⊢ 0 ∈ N F j | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | exact (F.N j).zero_mem | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ ⟨m, hm, rfl⟩ j
rw [single_apply]
split_ifs with h
· rwa [← h]
· | Mathlib.RingTheory.Filtration.298_0.wQ6WBws0g3n9213 | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid | Mathlib_RingTheory_Filtration |
case a
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
⊢ (Filtration.submodule F).toAddSubmonoid ≤ AddSubmonoid.closure (⋃ i, ⇑(single R i) '' ↑(N F i)) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | intro f hf | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ ⟨m, hm, rfl⟩ j
rw [single_apply]
split_ifs with h
· rwa [← h]
· exact (F.N j).zer... | Mathlib.RingTheory.Filtration.298_0.wQ6WBws0g3n9213 | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid | Mathlib_RingTheory_Filtration |
case a
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
f : PolynomialModule R M
hf : f ∈ (Filtration.submodule F).toAddSubmonoid
⊢ f ∈ AddSubmonoid.closure (⋃ i, ⇑(single R i) '' ↑(N F i)) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rw [← f.sum_single] | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ ⟨m, hm, rfl⟩ j
rw [single_apply]
split_ifs with h
· rwa [← h]
· exact (F.N j).zer... | Mathlib.RingTheory.Filtration.298_0.wQ6WBws0g3n9213 | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid | Mathlib_RingTheory_Filtration |
case a
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
f : PolynomialModule R M
hf : f ∈ (Filtration.submodule F).toAddSubmonoid
⊢ Finsupp.sum f Finsupp.single ∈ AddSubmonoid.closure (⋃ i, ⇑(single R i) '' ↑(N F i)) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | apply AddSubmonoid.sum_mem _ _ | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ ⟨m, hm, rfl⟩ j
rw [single_apply]
split_ifs with h
· rwa [← h]
· exact (F.N j).zer... | Mathlib.RingTheory.Filtration.298_0.wQ6WBws0g3n9213 | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
f : PolynomialModule R M
hf : f ∈ (Filtration.submodule F).toAddSubmonoid
⊢ ∀ c ∈ f.support, (fun₀ | c => f c) ∈ AddSubmonoid.closure (⋃ i, ⇑(single R i) '' ↑(N F i)) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rintro c - | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ ⟨m, hm, rfl⟩ j
rw [single_apply]
split_ifs with h
· rwa [← h]
· exact (F.N j).zer... | Mathlib.RingTheory.Filtration.298_0.wQ6WBws0g3n9213 | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
f : PolynomialModule R M
hf : f ∈ (Filtration.submodule F).toAddSubmonoid
c : ℕ
⊢ (fun₀ | c => f c) ∈ AddSubmonoid.closure (⋃ i, ⇑(single R i) '' ↑(N F i)) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c)) | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ ⟨m, hm, rfl⟩ j
rw [single_apply]
split_ifs with h
· rwa [← h]
· exact (F.N j).zer... | Mathlib.RingTheory.Filtration.298_0.wQ6WBws0g3n9213 | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
⊢ Submodule.span (↥(reesAlgebra I)) (⋃ i, ⇑(single R i) '' ↑(N F i)) = Filtration.submodule F | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rw [← Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid] | theorem submodule_span_single :
Submodule.span (reesAlgebra I) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by
| Mathlib.RingTheory.Filtration.314_0.wQ6WBws0g3n9213 | theorem submodule_span_single :
Submodule.span (reesAlgebra I) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
⊢ Submodule.span ↥(reesAlgebra I) ↑(Filtration.submodule F) = Filtration.submodule F | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | exact Submodule.span_eq (Filtration.submodule F) | theorem submodule_span_single :
Submodule.span (reesAlgebra I) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [← Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
| Mathlib.RingTheory.Filtration.314_0.wQ6WBws0g3n9213 | theorem submodule_span_single :
Submodule.span (reesAlgebra I) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
⊢ Filtration.submodule F = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)) ↔
∀ n ≥ n₀, I • N F n = N F (n + 1) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
| Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
⊢ (∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆
↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))) ↔
∀ n ≥ n₀, I • N F n = N F (n + 1)
R M : T... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | swap | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
| Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
⊢ Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)) ≤
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⇑(single R i) '' ↑(N F i)) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
⊢ (∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆
↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))) ↔
∀ n ≥ n₀, I • N F n = N F (n + 1) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | constructor | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mp
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
⊢ (∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆
↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))) →
∀ n ≥ n₀, I • N F n = N F (n + 1) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | intro H n hn | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mp
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
H :
∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))
n : ℕ
hn : n ≥ n₀
⊢ I • N F n = N F (n + 1) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | refine' (F.smul_le n).antisymm _ | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mp
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
H :
∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))
n : ℕ
hn : n ≥ n₀
⊢ N F (n + 1) ≤ I • N F n | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | intro x hx | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mp
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
H :
∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))
n : ℕ
hn : n ≥ n₀
x : M
hx : x ∈ N F (n + 1)
⊢ ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mp.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
H :
∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))
n : ℕ
hn : n ≥ n₀
x : M
hx : x ∈ N F (n +... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mp.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
H :
∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))
n : ℕ
hn : n ≥ n₀
x : M
hx : x ∈ N F (n +... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | dsimp only at hl | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mp.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
H :
∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))
n : ℕ
hn : n ≥ n₀
x : M
hx : x ∈ N F (n +... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | erw [Finsupp.single_eq_same] at hl | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mp.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
H :
∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))
n : ℕ
hn : n ≥ n₀
x : M
hx : x ∈ N F (n +... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mp.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
H :
∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))
n : ℕ
hn : n ≥ n₀
x : M
hx : x ∈ N F (n +... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | apply Submodule.sum_mem _ _ | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
H :
∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))
n : ℕ
hn : n ≥ n₀
x : M
hx : x ∈ N F (n + 1)
l : ↑(⋃ i,... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mk.intro.intro.intro.intro.intro.intro.intro.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
H :
∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | dsimp only [Subtype.coe_mk] | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mk.intro.intro.intro.intro.intro.intro.intro.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
H :
∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
H :
∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))
n : ℕ
hn : n ≥ n₀
x : M
hx : x ∈ N F (n + 1)
l : ↑(⋃ i,... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | linarith | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mk.intro.intro.intro.intro.intro.intro.intro.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
H :
∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | have e : n' ≤ n := by linarith | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
H :
∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))
n : ℕ
hn : n ≥ n₀
x : M
hx : x ∈ N F (n + 1)
l : ↑(⋃ i,... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | linarith | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mk.intro.intro.intro.intro.intro.intro.intro.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
H :
∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | have := F.pow_smul_le_pow_smul (n - n') n' 1 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mk.intro.intro.intro.intro.intro.intro.intro.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
H :
∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, ← add_tsub_assoc_of_le e, add_comm] at this | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mk.intro.intro.intro.intro.intro.intro.intro.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
H :
∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)))... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mpr
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
n₀ : ℕ
⊢ (∀ n ≥ n₀, I • N F n = N F (n + 1)) →
∀ (i : ℕ),
⇑(single R i) '' ↑(N F i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mpr
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))
⊢ (∀ n ≥ n₀, I • N F n = N F (n + 1... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | intro hF i | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mpr
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))
hF : ∀ n ≥ n₀, I • N F n = N F (n +... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑(... | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))
hF : ∀ n ≥ n₀, I • N F n = N F (n + 1)
i : ℕ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | intro i hi | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))
hF : ∀ n ≥ n₀, I • N F n = N F (n + 1)
i✝ i ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | refine Set.Subset.trans ?_ Submodule.subset_span | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))
hF : ∀ n ≥ n₀, I • N F n = N F (n + 1)
i✝ i ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))
hF : ∀ n ≥ n₀, I • N F n = N F (n + 1)
i✝ i ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | exact hi | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mpr
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))
hF : ∀ n ≥ n₀, I • N F n = N F (n +... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | induction' i with j hj | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mpr.zero
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))
hF : ∀ n ≥ n₀, I • N F n = N F... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | exact this _ (zero_le _) | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case mpr.succ
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))
hF : ∀ n ≥ n₀, I • N F n = N F... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | by_cases hj' : j.succ ≤ n₀ | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case pos
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))
hF : ∀ n ≥ n₀, I • N F n = N F (n +... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | exact this _ hj' | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case neg
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))
hF : ∀ n ≥ n₀, I • N F n = N F (n +... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | simp only [not_le, Nat.lt_succ_iff] at hj' | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case neg
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))
hF : ∀ n ≥ n₀, I • N F n = N F (n +... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rw [Nat.succ_eq_add_one, ← hF _ hj'] | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case neg
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))
hF : ∀ n ≥ n₀, I • N F n = N F (n +... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rintro _ ⟨m, hm, rfl⟩ | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case neg.intro.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))
hF : ∀ n ≥ n₀, I • N F ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case neg.intro.intro.refine'_1
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))
hF : ∀ n ≥ n₀... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rw [add_comm, ← monomial_smul_single] | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case neg.intro.intro.refine'_1
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))
hF : ∀ n ≥ n₀... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | exact F'.smul_mem
⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))
hF : ∀ n ≥ n₀, I • N F n = N F (n + 1)
this ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rwa [pow_one] | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case neg.intro.intro.refine'_2
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))
hF : ∀ n ≥ n₀... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rw [map_add] | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
case neg.intro.intro.refine'_2
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))
hF : ∀ n ≥ n₀... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | exact F'.add_mem hx hy | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_... | Mathlib.RingTheory.Filtration.320_0.wQ6WBws0g3n9213 | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
⊢ Submodule.FG (Filtration.submodule F) ↔ Stable F | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_iff_stable_ge]
constructor
· rintro H
refine H.stablizes_of_iSup_eq
⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_
· intro n m e
rw [Submodule.span_le, Set.iUnion₂_subset_iff]
... | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
| Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
⊢ Submodule.FG (Filtration.submodule F) ↔ Stable F | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | delta Ideal.Filtration.Stable | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
| Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
⊢ Submodule.FG (Filtration.submodule F) ↔ ∃ n₀, ∀ n ≥ n₀, I • N F n = N F (n + 1) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | simp_rw [← F.submodule_eq_span_le_iff_stable_ge] | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
| Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
⊢ Submodule.FG (Filtration.submodule F) ↔
∃ n₀, Filtration.submodule F = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | constructor | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mp
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
⊢ Submodule.FG (Filtration.submodule F) →
∃ n₀, Filtration.submodule F = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rintro H | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mp
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
H : Submodule.FG (Filtration.submodule F)
⊢ ∃ n₀, Filtration.submodule F = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | refine H.stablizes_of_iSup_eq
⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mp.refine_1
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
H : Submodule.FG (Filtration.submodule F)
⊢ Monotone fun n₀ => Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | intro n m e | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mp.refine_1
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
H : Submodule.FG (Filtration.submodule F)
n m : ℕ
e : n ≤ m
⊢ (fun n₀ => Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i)... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rw [Submodule.span_le, Set.iUnion₂_subset_iff] | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mp.refine_1
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
H : Submodule.FG (Filtration.submodule F)
n m : ℕ
e : n ≤ m
⊢ ∀ i ≤ n,
⇑(single R i) '' ↑(N F i) ⊆
↑((fun n₀ => Submodule.span (↥(re... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | intro i hi | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mp.refine_1
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
H : Submodule.FG (Filtration.submodule F)
n m : ℕ
e : n ≤ m
i : ℕ
hi : i ≤ n
⊢ ⇑(single R i) '' ↑(N F i) ⊆
↑((fun n₀ => Submodule.span (↥(... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | refine Set.Subset.trans ?_ Submodule.subset_span | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mp.refine_1
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
H : Submodule.FG (Filtration.submodule F)
n m : ℕ
e : n ≤ m
i : ℕ
hi : i ≤ n
⊢ ⇑(single R i) '' ↑(N F i) ⊆ ⋃ i, ⋃ (_ : i ≤ m), ⇑(single R i) '... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mp.refine_1
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
H : Submodule.FG (Filtration.submodule F)
n m : ℕ
e : n ≤ m
i : ℕ
hi : i ≤ n
⊢ i ≤ m | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | exact hi.trans e | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mp.refine_2
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
H : Submodule.FG (Filtration.submodule F)
⊢ iSup
⇑{ toFun := fun n₀ => Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | dsimp | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mp.refine_2
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
H : Submodule.FG (Filtration.submodule F)
⊢ ⨆ n₀, Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i)) = Filtrat... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rw [← Submodule.span_iUnion, ← submodule_span_single] | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mp.refine_2
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
H : Submodule.FG (Filtration.submodule F)
⊢ Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ i_1, ⋃ (_ : i_1 ≤ i), ⇑(single R i_1) '' ↑(N F i_1)) =
... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | congr 1 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mp.refine_2.e_s
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
H : Submodule.FG (Filtration.submodule F)
⊢ ⋃ i, ⋃ i_1, ⋃ (_ : i_1 ≤ i), ⇑(single R i_1) '' ↑(N F i_1) = ⋃ i, ⇑(single R i) '' ↑(N F i) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | ext | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mp.refine_2.e_s.h
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
H : Submodule.FG (Filtration.submodule F)
x✝ : PolynomialModule R M
⊢ x✝ ∈ ⋃ i, ⋃ i_1, ⋃ (_ : i_1 ≤ i), ⇑(single R i_1) '' ↑(N F i_1) ↔ ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mp.refine_2.e_s.h
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
H : Submodule.FG (Filtration.submodule F)
x✝ : PolynomialModule R M
⊢ (∃ i, ∃ i_1 ≤ i, ∃ x ∈ N F i_1, (single R i_1) x = x✝) ↔ ∃ i, ∃ x ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | constructor | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mp.refine_2.e_s.h.mp
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
H : Submodule.FG (Filtration.submodule F)
x✝ : PolynomialModule R M
⊢ (∃ i, ∃ i_1 ≤ i, ∃ x ∈ N F i_1, (single R i_1) x = x✝) → ∃ i, ∃... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rintro ⟨-, i, -, e⟩ | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mp.refine_2.e_s.h.mp.intro.intro.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
H : Submodule.FG (Filtration.submodule F)
x✝ : PolynomialModule R M
i : ℕ
e : ∃ x ∈ N F i, (single R i) x = x✝
⊢ ∃ ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | exact ⟨i, e⟩ | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mp.refine_2.e_s.h.mpr
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
H : Submodule.FG (Filtration.submodule F)
x✝ : PolynomialModule R M
⊢ (∃ i, ∃ x ∈ N F i, (single R i) x = x✝) → ∃ i, ∃ i_1 ≤ i, ∃ x ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rintro ⟨i, e⟩ | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mp.refine_2.e_s.h.mpr.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
H : Submodule.FG (Filtration.submodule F)
x✝ : PolynomialModule R M
i : ℕ
e : ∃ x ∈ N F i, (single R i) x = x✝
⊢ ∃ i, ∃ i_1 ≤ ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | exact ⟨i, i, le_refl i, e⟩ | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mpr
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
⊢ (∃ n₀, Filtration.submodule F = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(N F i))) →
Submodule.FG (Filtration.s... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rintro ⟨n, hn⟩ | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mpr.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
n : ℕ
hn : Filtration.submodule F = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n), ⇑(single R i) '' ↑(N F i))
⊢ Submodule.FG (Filtration... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rw [hn] | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mpr.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
n : ℕ
hn : Filtration.submodule F = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n), ⇑(single R i) '' ↑(N F i))
⊢ Submodule.FG (Submodule.... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mpr.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
n : ℕ
hn : Filtration.submodule F = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n), ⇑(single R i) '' ↑(N F i))
⊢ Submodule.FG (⨆ x, Submo... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | apply Submodule.fg_iSup | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mpr.intro.h
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
n : ℕ
hn : Filtration.submodule F = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n), ⇑(single R i) '' ↑(N F i))
⊢ ∀ (i : { i // i ∈ Fins... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rintro ⟨i, hi⟩ | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mpr.intro.h.mk
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
n : ℕ
hn : Filtration.submodule F = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n), ⇑(single R i) '' ↑(N F i))
i : ℕ
hi : i ∈ Finset... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | obtain ⟨s, hs⟩ := hF' i | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mpr.intro.h.mk.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
n : ℕ
hn : Filtration.submodule F = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n), ⇑(single R i) '' ↑(N F i))
i : ℕ
hi : i ∈ ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
n : ℕ
hn : Filtration.submodule F = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n), ⇑(single R i) '' ↑(N F i))
i : ℕ
hi : i ∈ Finset.range (Nat.succ n)
... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rw [Finset.coe_image, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs] | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
n : ℕ
hn : Filtration.submodule F = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n), ⇑(single R i) '' ↑(N F i))
i : ℕ
hi : i ∈ Finset.range (Nat.succ n)
... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rfl | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mpr.intro.h.mk.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
n : ℕ
hn : Filtration.submodule F = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n), ⇑(single R i) '' ↑(N F i))
i : ℕ
hi : i ∈ ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rw [Subtype.coe_mk, ← this] | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
case mpr.intro.h.mk.intro
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
hF' : ∀ (i : ℕ), Submodule.FG (N F i)
n : ℕ
hn : Filtration.submodule F = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n), ⇑(single R i) '' ↑(N F i))
i : ℕ
hi : i ∈ ... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | exact ⟨_, rfl⟩ | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_... | Mathlib.RingTheory.Filtration.366_0.wQ6WBws0g3n9213 | /-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R M
hF : Stable F
F' : Filtration I M
hf : F' ≤ F
⊢ Stable F' | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | rw [← submodule_fg_iff_stable] at hF ⊢ | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' ≤ F) : F'.Stable := by
| Mathlib.RingTheory.Filtration.405_0.wQ6WBws0g3n9213 | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' ≤ F) : F'.Stable | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R M
hF : Submodule.FG (Filtration.submodule F)
F' : Filtration I M
hf : F' ≤ F
⊢ Submodule.FG (Filtration.submodule F')
case hF'
R M : Type u
ins... | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | any_goals intro i; exact IsNoetherian.noetherian _ | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' ≤ F) : F'.Stable := by
rw [← submodule_fg_iff_stable] at hF ⊢
| Mathlib.RingTheory.Filtration.405_0.wQ6WBws0g3n9213 | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' ≤ F) : F'.Stable | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R M
hF : Submodule.FG (Filtration.submodule F)
F' : Filtration I M
hf : F' ≤ F
⊢ Submodule.FG (Filtration.submodule F') | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | intro i | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' ≤ F) : F'.Stable := by
rw [← submodule_fg_iff_stable] at hF ⊢
any_goals | Mathlib.RingTheory.Filtration.405_0.wQ6WBws0g3n9213 | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' ≤ F) : F'.Stable | Mathlib_RingTheory_Filtration |
case hF'
R M : Type u
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R M
hF : Submodule.FG (Filtration.submodule F)
F' : Filtration I M
hf : F' ≤ F
⊢ ∀ (i : ℕ), Submodule.FG (N F' i) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | intro i | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' ≤ F) : F'.Stable := by
rw [← submodule_fg_iff_stable] at hF ⊢
any_goals | Mathlib.RingTheory.Filtration.405_0.wQ6WBws0g3n9213 | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' ≤ F) : F'.Stable | Mathlib_RingTheory_Filtration |
case hF'
R M : Type u
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R M
hF : Submodule.FG (Filtration.submodule F)
F' : Filtration I M
hf : F' ≤ F
i : ℕ
⊢ Submodule.FG (N F' i) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | exact IsNoetherian.noetherian _ | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' ≤ F) : F'.Stable := by
rw [← submodule_fg_iff_stable] at hF ⊢
any_goals intro i; | Mathlib.RingTheory.Filtration.405_0.wQ6WBws0g3n9213 | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' ≤ F) : F'.Stable | Mathlib_RingTheory_Filtration |
case hF'
R M : Type u
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R M
hF : Stable F
F' : Filtration I M
hf : F' ≤ F
⊢ ∀ (i : ℕ), Submodule.FG (N F i) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | intro i | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' ≤ F) : F'.Stable := by
rw [← submodule_fg_iff_stable] at hF ⊢
any_goals | Mathlib.RingTheory.Filtration.405_0.wQ6WBws0g3n9213 | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' ≤ F) : F'.Stable | Mathlib_RingTheory_Filtration |
case hF'
R M : Type u
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R M
hF : Stable F
F' : Filtration I M
hf : F' ≤ F
i : ℕ
⊢ Submodule.FG (N F i) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | exact IsNoetherian.noetherian _ | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' ≤ F) : F'.Stable := by
rw [← submodule_fg_iff_stable] at hF ⊢
any_goals intro i; | Mathlib.RingTheory.Filtration.405_0.wQ6WBws0g3n9213 | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' ≤ F) : F'.Stable | Mathlib_RingTheory_Filtration |
R M : Type u
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
I : Ideal R
F F'✝ : Filtration I M
h : Stable F
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R M
hF : Submodule.FG (Filtration.submodule F)
F' : Filtration I M
hf : F' ≤ F
⊢ Submodule.FG (Filtration.submodule F') | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Dat... | have := isNoetherian_of_fg_of_noetherian _ hF | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' ≤ F) : F'.Stable := by
rw [← submodule_fg_iff_stable] at hF ⊢
any_goals intro i; exact IsNoetherian.noetherian _
| Mathlib.RingTheory.Filtration.405_0.wQ6WBws0g3n9213 | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' ≤ F) : F'.Stable | Mathlib_RingTheory_Filtration |
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