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 h₁.refine'_1
n : ℕ
n_big : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : ∏ x in S, f x = ∏ x in Finset.range (2 * n / 3 + 1), f x
... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | exact pow_factorization_choose_le (mul_pos two_pos n_pos) | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib_NumberTheory_Bertrand |
case h₁.refine'_2
n : ℕ
n_big : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : ∏ x in S, f x = ∏ x in Finset.range (2 * n / 3 + 1), f x
... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel] | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib_NumberTheory_Bertrand |
n : ℕ
n_big : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : ∏ x in S, f x = ∏ x in Finset.range (2 * n / 3 + 1), f x
⊢ Finset.card (Fin... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | rw [card_Icc, Nat.add_sub_cancel] | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib_NumberTheory_Bertrand |
case h₁.refine'_2
n : ℕ
n_big : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x
this✝ : ∏ x in S, f x = ∏ x in Finset.range (2 * n / 3 + 1), f x... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | rw [Finset.prod_const] | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib_NumberTheory_Bertrand |
case h₁.refine'_2
n : ℕ
n_big : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x
this✝ : ∏ x in S, f x = ∏ x in Finset.range (2 * n / 3 + 1), f x... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le) | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib_NumberTheory_Bertrand |
case h₁.refine'_2
n : ℕ
n_big : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x
this✝ : ∏ x in S, f x = ∏ x in Finset.range (2 * n / 3 + 1), f x... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | obtain ⟨h1, h2⟩ := Finset.mem_filter.1 hx | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib_NumberTheory_Bertrand |
case h₁.refine'_2.intro
n : ℕ
n_big : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x
this✝ : ∏ x in S, f x = ∏ x in Finset.range (2 * n / 3 + 1... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | exact Finset.mem_Icc.mpr ⟨(Finset.mem_filter.1 h1).2.one_lt.le, h2⟩ | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib_NumberTheory_Bertrand |
case h₂
n : ℕ
n_big : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : ∏ x in S, f x = ∏ x in Finset.range (2 * n / 3 + 1), f x
⊢ ∏ x in F... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | refine' le_trans _ (primorial_le_4_pow (2 * n / 3)) | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib_NumberTheory_Bertrand |
case h₂
n : ℕ
n_big : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : ∏ x in S, f x = ∏ x in Finset.range (2 * n / 3 + 1), f x
⊢ ∏ x in F... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | refine' (Finset.prod_le_prod' fun p hp => (_ : f p ≤ p)).trans _ | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib_NumberTheory_Bertrand |
case h₂.refine'_1
n : ℕ
n_big : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : ∏ x in S, f x = ∏ x in Finset.range (2 * n / 3 + 1), f x
... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | obtain ⟨h1, h2⟩ := Finset.mem_filter.1 hp | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib_NumberTheory_Bertrand |
case h₂.refine'_1.intro
n : ℕ
n_big : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : ∏ x in S, f x = ∏ x in Finset.range (2 * n / 3 + 1)... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib_NumberTheory_Bertrand |
case h₂.refine'_1.intro
n : ℕ
n_big : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : ∏ x in S, f x = ∏ x in Finset.range (2 * n / 3 + 1)... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2) | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib_NumberTheory_Bertrand |
case h₂.refine'_2
n : ℕ
n_big : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : ∏ x in S, f x = ∏ x in Finset.range (2 * n / 3 + 1), f x
... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _ | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib_NumberTheory_Bertrand |
case h₂.refine'_2
n : ℕ
n_big : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : ∏ x in S, f x = ∏ x in Finset.range (2 * n / 3 + 1), f x
... | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 *... | Mathlib_NumberTheory_Bertrand |
n : ℕ
n_big : 512 ≤ n
⊢ ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | by_contra no_prime | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : ℕ) (n_big : 512 ≤ n) :
∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n := by
-- Assume there is no prime in the range.
| Mathlib.NumberTheory.Bertrand.195_0.gJXoOT9Ce2wC0xc | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : ℕ) (n_big : 512 ≤ n) :
∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n | Mathlib_NumberTheory_Bertrand |
n : ℕ
n_big : 512 ≤ n
no_prime : ¬∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n
⊢ False | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := bertrand_main_inequality n_big | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : ℕ) (n_big : 512 ≤ n) :
∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential boun... | Mathlib.NumberTheory.Bertrand.195_0.gJXoOT9Ce2wC0xc | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : ℕ) (n_big : 512 ≤ n) :
∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n | Mathlib_NumberTheory_Bertrand |
n : ℕ
n_big : 512 ≤ n
no_prime : ¬∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n
H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n
⊢ False | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big) | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : ℕ) (n_big : 512 ≤ n) :
∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential boun... | Mathlib.NumberTheory.Bertrand.195_0.gJXoOT9Ce2wC0xc | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : ℕ) (n_big : 512 ≤ n) :
∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n | Mathlib_NumberTheory_Bertrand |
n : ℕ
n_big : 512 ≤ n
no_prime : ¬∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n
H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n
⊢ 4 ≤ 512 | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | norm_num1 | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : ℕ) (n_big : 512 ≤ n) :
∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential boun... | Mathlib.NumberTheory.Bertrand.195_0.gJXoOT9Ce2wC0xc | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : ℕ) (n_big : 512 ≤ n) :
∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n | Mathlib_NumberTheory_Bertrand |
n : ℕ
n_big : 512 ≤ n
no_prime : ¬∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n
H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n
H2 : 4 ^ n < n * centralBinom n
⊢ False | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | have H3 : n.centralBinom ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : ℕ) (n_big : 512 ≤ n) :
∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential boun... | Mathlib.NumberTheory.Bertrand.195_0.gJXoOT9Ce2wC0xc | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : ℕ) (n_big : 512 ≤ n) :
∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n | Mathlib_NumberTheory_Bertrand |
n : ℕ
n_big : 512 ≤ n
no_prime : ¬∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n
H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n
H2 : 4 ^ n < n * centralBinom n
⊢ 2 < 512 | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | norm_num1 | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : ℕ) (n_big : 512 ≤ n) :
∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential boun... | Mathlib.NumberTheory.Bertrand.195_0.gJXoOT9Ce2wC0xc | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : ℕ) (n_big : 512 ≤ n) :
∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n | Mathlib_NumberTheory_Bertrand |
n : ℕ
n_big : 512 ≤ n
no_prime : ¬∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n
H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n
H2 : 4 ^ n < n * centralBinom n
H3 : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)
⊢ False | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | rw [mul_assoc] at H1 | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : ℕ) (n_big : 512 ≤ n) :
∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential boun... | Mathlib.NumberTheory.Bertrand.195_0.gJXoOT9Ce2wC0xc | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : ℕ) (n_big : 512 ≤ n) :
∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n | Mathlib_NumberTheory_Bertrand |
n : ℕ
n_big : 512 ≤ n
no_prime : ¬∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n
H1 : n * ((2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)) ≤ 4 ^ n
H2 : 4 ^ n < n * centralBinom n
H3 : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)
⊢ False | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | exact not_le.2 H2 ((mul_le_mul_left' H3 n).trans H1) | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : ℕ) (n_big : 512 ≤ n) :
∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential boun... | Mathlib.NumberTheory.Bertrand.195_0.gJXoOT9Ce2wC0xc | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : ℕ) (n_big : 512 ≤ n) :
∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n | Mathlib_NumberTheory_Bertrand |
n q p : ℕ
prime_p : Prime p
covering : p ≤ 2 * q
H : n < q → ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n
hn : n < p
⊢ ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | by_cases h : p ≤ 2 * n | /-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number ≤ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : ℕ} (prime_p : Nat.Prime p)
(covering ... | Mathlib.NumberTheory.Bertrand.212_0.gJXoOT9Ce2wC0xc | /-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number ≤ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : ℕ} (prime_p : Nat.Prime p)
(covering ... | Mathlib_NumberTheory_Bertrand |
case pos
n q p : ℕ
prime_p : Prime p
covering : p ≤ 2 * q
H : n < q → ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n
hn : n < p
h : p ≤ 2 * n
⊢ ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | exact ⟨p, prime_p, hn, h⟩ | /-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number ≤ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : ℕ} (prime_p : Nat.Prime p)
(covering ... | Mathlib.NumberTheory.Bertrand.212_0.gJXoOT9Ce2wC0xc | /-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number ≤ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : ℕ} (prime_p : Nat.Prime p)
(covering ... | Mathlib_NumberTheory_Bertrand |
case neg
n q p : ℕ
prime_p : Prime p
covering : p ≤ 2 * q
H : n < q → ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n
hn : n < p
h : ¬p ≤ 2 * n
⊢ ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | exact H (lt_of_mul_lt_mul_left' (lt_of_lt_of_le (not_le.1 h) covering)) | /-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number ≤ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : ℕ} (prime_p : Nat.Prime p)
(covering ... | Mathlib.NumberTheory.Bertrand.212_0.gJXoOT9Ce2wC0xc | /-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number ≤ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : ℕ} (prime_p : Nat.Prime p)
(covering ... | Mathlib_NumberTheory_Bertrand |
n : ℕ
hn0 : n ≠ 0
⊢ ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | cases' lt_or_le 511 n with h h | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :
∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n := by
-- Split into cases whether `n` is large or small
| Mathlib.NumberTheory.Bertrand.223_0.gJXoOT9Ce2wC0xc | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :
∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n | Mathlib_NumberTheory_Bertrand |
case inl
n : ℕ
hn0 : n ≠ 0
h : 511 < n
⊢ ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | exact exists_prime_lt_and_le_two_mul_eventually n h | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :
∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_l... | Mathlib.NumberTheory.Bertrand.223_0.gJXoOT9Ce2wC0xc | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :
∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n | Mathlib_NumberTheory_Bertrand |
case inr
n : ℕ
hn0 : n ≠ 0
h : n ≤ 511
⊢ ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | replace h : n < 521 := h.trans_lt (by norm_num1) | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :
∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_l... | Mathlib.NumberTheory.Bertrand.223_0.gJXoOT9Ce2wC0xc | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :
∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n | Mathlib_NumberTheory_Bertrand |
n : ℕ
hn0 : n ≠ 0
h : n ≤ 511
⊢ 511 < 521 | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | norm_num1 | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :
∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_l... | Mathlib.NumberTheory.Bertrand.223_0.gJXoOT9Ce2wC0xc | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :
∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n | Mathlib_NumberTheory_Bertrand |
case inr
n : ℕ
hn0 : n ≠ 0
h : n < 521
⊢ ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | revert h | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :
∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_l... | Mathlib.NumberTheory.Bertrand.223_0.gJXoOT9Ce2wC0xc | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :
∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n | Mathlib_NumberTheory_Bertrand |
case inr
n : ℕ
hn0 : n ≠ 0
⊢ n < 521 → ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | open Lean Elab Tactic in
run_tac do
for i in [317, 163, 83, 43, 23, 13, 7, 5, 3, 2] do
let i : Term := quote i
evalTactic <| ←
`(tactic| refine' exists_prime_lt_and_le_two_mul_succ $i (by norm_num1) (by norm_num1) _)
exact fun h2 => ⟨2, prime_two, h2, Nat.mul_le_mul_left 2 (Nat.pos_of_ne_zer... | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :
∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_l... | Mathlib.NumberTheory.Bertrand.223_0.gJXoOT9Ce2wC0xc | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :
∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n | Mathlib_NumberTheory_Bertrand |
case inr
n : ℕ
hn0 : n ≠ 0
⊢ n < 521 → ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | run_tac do
for i in [317, 163, 83, 43, 23, 13, 7, 5, 3, 2] do
let i : Term := quote i
evalTactic <| ←
`(tactic| refine' exists_prime_lt_and_le_two_mul_succ $i (by norm_num1) (by norm_num1) _) | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :
∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_l... | Mathlib.NumberTheory.Bertrand.223_0.gJXoOT9Ce2wC0xc | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :
∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n | Mathlib_NumberTheory_Bertrand |
case inr
n : ℕ
hn0 : n ≠ 0
⊢ n < 521 → ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | run_tac do
for i in [317, 163, 83, 43, 23, 13, 7, 5, 3, 2] do
let i : Term := quote i
evalTactic <| ←
`(tactic| refine' exists_prime_lt_and_le_two_mul_succ $i (by norm_num1) (by norm_num1) _) | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :
∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_l... | Mathlib.NumberTheory.Bertrand.223_0.gJXoOT9Ce2wC0xc | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :
∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n | Mathlib_NumberTheory_Bertrand |
case inr
n : ℕ
hn0 : n ≠ 0
⊢ n < 2 → ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Conve... | exact fun h2 => ⟨2, prime_two, h2, Nat.mul_le_mul_left 2 (Nat.pos_of_ne_zero hn0)⟩ | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :
∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_l... | Mathlib.NumberTheory.Bertrand.223_0.gJXoOT9Ce2wC0xc | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :
∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n | Mathlib_NumberTheory_Bertrand |
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F F' : Filtration I M
i j : ℕ
⊢ I ^ i • N F j ≤ N F (i + 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... | induction' i with _ ih | theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by
| Mathlib.RingTheory.Filtration.64_0.wQ6WBws0g3n9213 | theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) | Mathlib_RingTheory_Filtration |
case 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
j : ℕ
⊢ I ^ Nat.zero • N F j ≤ N F (Nat.zero + 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... | simp | theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by
induction' i with _ ih
· | Mathlib.RingTheory.Filtration.64_0.wQ6WBws0g3n9213 | theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) | Mathlib_RingTheory_Filtration |
case 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
j n✝ : ℕ
ih : I ^ n✝ • N F j ≤ N F (n✝ + j)
⊢ I ^ Nat.succ n✝ • N F j ≤ N F (Nat.succ n✝ + 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 [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← add_assoc] | theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by
induction' i with _ ih
· simp
· | Mathlib.RingTheory.Filtration.64_0.wQ6WBws0g3n9213 | theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) | Mathlib_RingTheory_Filtration |
case 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
j n✝ : ℕ
ih : I ^ n✝ • N F j ≤ N F (n✝ + j)
⊢ I • I ^ n✝ • N F j ≤ N F (n✝ + j + 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 (Submodule.smul_mono_right ih).trans (F.smul_le _) | theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by
induction' i with _ ih
· simp
· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← add_assoc]
| Mathlib.RingTheory.Filtration.64_0.wQ6WBws0g3n9213 | theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) | 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
i j k : ℕ
⊢ I ^ (i + k) • N F j ≤ I ^ k • N F (i + 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 [add_comm, pow_add, mul_smul] | theorem pow_smul_le_pow_smul (i j k : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • F.N (i + j) := by
| Mathlib.RingTheory.Filtration.71_0.wQ6WBws0g3n9213 | theorem pow_smul_le_pow_smul (i j k : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • F.N (i + j) | 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
i j k : ℕ
⊢ I ^ k • I ^ i • N F j ≤ I ^ k • N F (i + 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 Submodule.smul_mono_right (F.pow_smul_le i j) | theorem pow_smul_le_pow_smul (i j k : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
| Mathlib.RingTheory.Filtration.71_0.wQ6WBws0g3n9213 | theorem pow_smul_le_pow_smul (i j k : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • F.N (i + j) | 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
S : Set (Filtration I M)
i : ℕ
⊢ sSup (N '' S) (i + 1) ≤ sSup (N '' S) 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 sSup_le_sSup_of_forall_exists_le _ | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
⟨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
| Mathlib.RingTheory.Filtration.94_0.wQ6WBws0g3n9213 | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) | 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
S : Set (Filtration I M)
i : ℕ
⊢ ∀ x ∈ Set.range fun f => ↑f (i + 1), ∃ y ∈ Set.range fun f => ↑f i, x ≤ y | /-
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 _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
⟨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
| Mathlib.RingTheory.Filtration.94_0.wQ6WBws0g3n9213 | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) | Mathlib_RingTheory_Filtration |
case intro.mk.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
S : Set (Filtration I M)
i : ℕ
F : Filtration I M
hF : F ∈ S
⊢ ∃ y ∈ Set.range fun f => ↑f i, (fun f => ↑f (i + 1)) { val := F.N, property := (_ : ∃ a ∈ S, a.N = F.N) } ≤ y | /-
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, hF, rfl⟩, rfl⟩, F.mono i⟩ | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
⟨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩
| Mathlib.RingTheory.Filtration.94_0.wQ6WBws0g3n9213 | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) | 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
S : Set (Filtration I M)
i : ℕ
⊢ I • sSup (N '' S) i ≤ sSup (N '' S) (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... | rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
⟨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩
exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, ... | Mathlib.RingTheory.Filtration.94_0.wQ6WBws0g3n9213 | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) | 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
S : Set (Filtration I M)
i : ℕ
⊢ ⨆ i_1, I • ↑i_1 i ≤ ⨆ i_1, ↑i_1 (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... | apply iSup_mono _ | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
⟨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩
exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, ... | Mathlib.RingTheory.Filtration.94_0.wQ6WBws0g3n9213 | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) | 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
S : Set (Filtration I M)
i : ℕ
⊢ ∀ (i_1 : ↑(N '' S)), I • ↑i_1 i ≤ ↑i_1 (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... | rintro ⟨_, F, hF, rfl⟩ | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
⟨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩
exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, ... | Mathlib.RingTheory.Filtration.94_0.wQ6WBws0g3n9213 | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) | Mathlib_RingTheory_Filtration |
case mk.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
S : Set (Filtration I M)
i : ℕ
F : Filtration I M
hF : F ∈ S
⊢ I • ↑{ val := F.N, property := (_ : ∃ a ∈ S, a.N = F.N) } i ≤
↑{ val := F.N, property := (_ : ∃ a ∈ S, a.N = F.N) } (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 F.smul_le i | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
⟨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩
exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, ... | Mathlib.RingTheory.Filtration.94_0.wQ6WBws0g3n9213 | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) | 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
S : Set (Filtration I M)
i : ℕ
⊢ sInf (N '' S) (i + 1) ≤ sInf (N '' S) 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 sInf_le_sInf_of_forall_exists_le _ | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
⟨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
| Mathlib.RingTheory.Filtration.114_0.wQ6WBws0g3n9213 | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) | 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
S : Set (Filtration I M)
i : ℕ
⊢ ∀ x ∈ Set.range fun f => ↑f i, ∃ y ∈ Set.range fun f => ↑f (i + 1), y ≤ 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 _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
⟨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
| Mathlib.RingTheory.Filtration.114_0.wQ6WBws0g3n9213 | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) | Mathlib_RingTheory_Filtration |
case intro.mk.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
S : Set (Filtration I M)
i : ℕ
F : Filtration I M
hF : F ∈ S
⊢ ∃ y ∈ Set.range fun f => ↑f (i + 1), y ≤ (fun f => ↑f i) { val := F.N, property := (_ : ∃ a ∈ S, a.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 ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
⟨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩
| Mathlib.RingTheory.Filtration.114_0.wQ6WBws0g3n9213 | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) | 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
S : Set (Filtration I M)
i : ℕ
⊢ I • sInf (N '' S) i ≤ sInf (N '' S) (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... | rw [sInf_eq_iInf', iInf_apply, iInf_apply] | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
⟨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩
exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, ... | Mathlib.RingTheory.Filtration.114_0.wQ6WBws0g3n9213 | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) | 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
S : Set (Filtration I M)
i : ℕ
⊢ I • ⨅ i_1, ↑i_1 i ≤ ⨅ i_1, ↑i_1 (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... | refine' Submodule.smul_iInf_le.trans _ | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
⟨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩
exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, ... | Mathlib.RingTheory.Filtration.114_0.wQ6WBws0g3n9213 | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) | 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
S : Set (Filtration I M)
i : ℕ
⊢ ⨅ i_1, I • ↑i_1 i ≤ ⨅ i_1, ↑i_1 (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... | apply iInf_mono _ | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
⟨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩
exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, ... | Mathlib.RingTheory.Filtration.114_0.wQ6WBws0g3n9213 | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) | 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
S : Set (Filtration I M)
i : ℕ
⊢ ∀ (i_1 : ↑(N '' S)), I • ↑i_1 i ≤ ↑i_1 (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... | rintro ⟨_, F, hF, rfl⟩ | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
⟨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩
exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, ... | Mathlib.RingTheory.Filtration.114_0.wQ6WBws0g3n9213 | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) | Mathlib_RingTheory_Filtration |
case mk.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
S : Set (Filtration I M)
i : ℕ
F : Filtration I M
hF : F ∈ S
⊢ I • ↑{ val := F.N, property := (_ : ∃ a ∈ S, a.N = F.N) } i ≤
↑{ val := F.N, property := (_ : ∃ a ∈ S, a.N = F.N) } (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 F.smul_le i | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
⟨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩
exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, ... | Mathlib.RingTheory.Filtration.114_0.wQ6WBws0g3n9213 | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) | 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
I : Ideal R
N : Submodule R M
i : ℕ
⊢ (fun i => I ^ i • N) (i + 1) ≤ (fun i => I ^ i • N) 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 | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i • N
mono i := by | Mathlib.RingTheory.Filtration.195_0.wQ6WBws0g3n9213 | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i | 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
I : Ideal R
N : Submodule R M
i : ℕ
⊢ I ^ (i + 1) • N ≤ I ^ 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... | rw [add_comm, pow_add, mul_smul] | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i • N
mono i := by dsimp only; | Mathlib.RingTheory.Filtration.195_0.wQ6WBws0g3n9213 | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i | 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
I : Ideal R
N : Submodule R M
i : ℕ
⊢ I ^ 1 • I ^ i • N ≤ I ^ 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... | exact Submodule.smul_le_right | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i • N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; | Mathlib.RingTheory.Filtration.195_0.wQ6WBws0g3n9213 | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i | 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
I : Ideal R
N : Submodule R M
i : ℕ
⊢ I • (fun i => I ^ i • N) i ≤ (fun i => I ^ i • N) (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... | dsimp only | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i • N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by | Mathlib.RingTheory.Filtration.195_0.wQ6WBws0g3n9213 | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i | 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
I : Ideal R
N : Submodule R M
i : ℕ
⊢ I • I ^ i • N ≤ I ^ (i + 1) • 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 [add_comm, pow_add, mul_smul, pow_one] | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i • N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; | Mathlib.RingTheory.Filtration.195_0.wQ6WBws0g3n9213 | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i | 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
I : Ideal R
N : Submodule R M
⊢ Stable (stableFiltration 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... | use 0 | theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
| Mathlib.RingTheory.Filtration.203_0.wQ6WBws0g3n9213 | theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable | 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
I : Ideal R
N : Submodule R M
⊢ ∀ n ≥ 0, I • Ideal.Filtration.N (stableFiltration I N) n = Ideal.Filtration.N (stableFiltration I N) (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 n _ | theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
| Mathlib.RingTheory.Filtration.203_0.wQ6WBws0g3n9213 | theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable | 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
I : Ideal R
N : Submodule R M
n : ℕ
a✝ : n ≥ 0
⊢ I • Ideal.Filtration.N (stableFiltration I N) n = Ideal.Filtration.N (stableFiltration I N) (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... | dsimp | theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
| Mathlib.RingTheory.Filtration.203_0.wQ6WBws0g3n9213 | theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable | 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
I : Ideal R
N : Submodule R M
n : ℕ
a✝ : n ≥ 0
⊢ I • I ^ n • N = I ^ (n + 1) • 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 [add_comm, pow_add, mul_smul, pow_one] | theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
| Mathlib.RingTheory.Filtration.203_0.wQ6WBws0g3n9213 | theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).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
⊢ ∃ n₀, ∀ (k : ℕ), N F (n₀ + k) = I ^ k • N F 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... | obtain ⟨n₀, hn⟩ := h | theorem Stable.exists_pow_smul_eq : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by
| Mathlib.RingTheory.Filtration.213_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ | Mathlib_RingTheory_Filtration |
case 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
n₀ : ℕ
hn : ∀ n ≥ n₀, I • N F n = N F (n + 1)
⊢ ∃ n₀, ∀ (k : ℕ), N F (n₀ + k) = I ^ k • N F 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... | use n₀ | theorem Stable.exists_pow_smul_eq : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by
obtain ⟨n₀, hn⟩ := h
| Mathlib.RingTheory.Filtration.213_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ | 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
n₀ : ℕ
hn : ∀ n ≥ n₀, I • N F n = N F (n + 1)
⊢ ∀ (k : ℕ), N F (n₀ + k) = I ^ k • N F 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... | intro k | theorem Stable.exists_pow_smul_eq : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by
obtain ⟨n₀, hn⟩ := h
use n₀
| Mathlib.RingTheory.Filtration.213_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ | 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
n₀ : ℕ
hn : ∀ n ≥ n₀, I • N F n = N F (n + 1)
k : ℕ
⊢ N F (n₀ + k) = I ^ k • N F 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... | induction' k with _ ih | theorem Stable.exists_pow_smul_eq : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by
obtain ⟨n₀, hn⟩ := h
use n₀
intro k
| Mathlib.RingTheory.Filtration.213_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ | Mathlib_RingTheory_Filtration |
case h.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
n₀ : ℕ
hn : ∀ n ≥ n₀, I • N F n = N F (n + 1)
⊢ N F (n₀ + Nat.zero) = I ^ Nat.zero • N F 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... | simp | theorem Stable.exists_pow_smul_eq : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by
obtain ⟨n₀, hn⟩ := h
use n₀
intro k
induction' k with _ ih
· | Mathlib.RingTheory.Filtration.213_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ | Mathlib_RingTheory_Filtration |
case h.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
n₀ : ℕ
hn : ∀ n ≥ n₀, I • N F n = N F (n + 1)
n✝ : ℕ
ih : N F (n₀ + n✝) = I ^ n✝ • N F n₀
⊢ N F (n₀ + Nat.succ n✝) = I ^ Nat.succ n✝ • N F 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 [Nat.succ_eq_add_one, ← add_assoc, ← hn, ih, add_comm, pow_add, mul_smul, pow_one] | theorem Stable.exists_pow_smul_eq : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by
obtain ⟨n₀, hn⟩ := h
use n₀
intro k
induction' k with _ ih
· simp
· | Mathlib.RingTheory.Filtration.213_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ | Mathlib_RingTheory_Filtration |
case h.succ.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
n₀ : ℕ
hn : ∀ n ≥ n₀, I • N F n = N F (n + 1)
n✝ : ℕ
ih : N F (n₀ + n✝) = I ^ n✝ • N F n₀
⊢ n₀ + 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... | linarith | theorem Stable.exists_pow_smul_eq : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by
obtain ⟨n₀, hn⟩ := h
use n₀
intro k
induction' k with _ ih
· simp
· rw [Nat.succ_eq_add_one, ← add_assoc, ← hn, ih, add_comm, pow_add, mul_smul, pow_one]
| Mathlib.RingTheory.Filtration.213_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ | 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 ≥ n₀, N F n = I ^ (n - n₀) • N F 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... | obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq | theorem Stable.exists_pow_smul_eq_of_ge : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by
| Mathlib.RingTheory.Filtration.223_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq_of_ge : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ | Mathlib_RingTheory_Filtration |
case 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
hn₀ : ∀ (k : ℕ), N F (n₀ + k) = I ^ k • N F n₀
⊢ ∃ n₀, ∀ n ≥ n₀, N F n = I ^ (n - n₀) • N F 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... | use n₀ | theorem Stable.exists_pow_smul_eq_of_ge : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by
obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq
| Mathlib.RingTheory.Filtration.223_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq_of_ge : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ | 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
n₀ : ℕ
hn₀ : ∀ (k : ℕ), N F (n₀ + k) = I ^ k • N F n₀
⊢ ∀ n ≥ n₀, N F n = I ^ (n - n₀) • N F 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... | intro n hn | theorem Stable.exists_pow_smul_eq_of_ge : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by
obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq
use n₀
| Mathlib.RingTheory.Filtration.223_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq_of_ge : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ | 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
n₀ : ℕ
hn₀ : ∀ (k : ℕ), N F (n₀ + k) = I ^ k • N F n₀
n : ℕ
hn : n ≥ n₀
⊢ N F n = I ^ (n - n₀) • N F 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... | convert hn₀ (n - n₀) | theorem Stable.exists_pow_smul_eq_of_ge : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by
obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq
use n₀
intro n hn
| Mathlib.RingTheory.Filtration.223_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq_of_ge : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ | Mathlib_RingTheory_Filtration |
case h.e'_2.h.e'_8
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
hn₀ : ∀ (k : ℕ), N F (n₀ + k) = I ^ k • N F n₀
n : ℕ
hn : n ≥ n₀
⊢ n = n₀ + (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, tsub_add_cancel_of_le hn] | theorem Stable.exists_pow_smul_eq_of_ge : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by
obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq
use n₀
intro n hn
convert hn₀ (n - n₀)
| Mathlib.RingTheory.Filtration.223_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq_of_ge : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ | 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
⊢ Stable F ↔ ∃ n₀, ∀ n ≥ n₀, N F n = I ^ (n - n₀) • N F 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... | refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ | theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by
| Mathlib.RingTheory.Filtration.231_0.wQ6WBws0g3n9213 | theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ | 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
h : ∃ n₀, ∀ n ≥ n₀, N F n = I ^ (n - n₀) • N F n₀
n : ℕ
hn : n ≥ Exists.choose h
⊢ 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 [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← pow_succ,
tsub_add_eq_add_tsub hn] | theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by
refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩
| Mathlib.RingTheory.Filtration.231_0.wQ6WBws0g3n9213 | theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ | 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
h : ∃ n₀, ∀ n ≥ n₀, N F n = I ^ (n - n₀) • N F n₀
n : ℕ
hn : n ≥ Exists.choose h
⊢ n + 1 ≥ Exists.choose h | /-
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 stable_iff_exists_pow_smul_eq_of_ge :
F.Stable ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by
refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by | Mathlib.RingTheory.Filtration.231_0.wQ6WBws0g3n9213 | theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ | 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✝ h : Stable F
e : N F 0 ≤ N F' 0
⊢ ∃ n₀, ∀ (n : ℕ), N F (n + n₀) ≤ 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... | obtain ⟨n₀, hF⟩ := h | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by
| Mathlib.RingTheory.Filtration.238_0.wQ6WBws0g3n9213 | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n | Mathlib_RingTheory_Filtration |
case 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
e : N F 0 ≤ N F' 0
n₀ : ℕ
hF : ∀ n ≥ n₀, I • N F n = N F (n + 1)
⊢ ∃ n₀, ∀ (n : ℕ), N F (n + n₀) ≤ 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... | use n₀ | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by
obtain ⟨n₀, hF⟩ := h
| Mathlib.RingTheory.Filtration.238_0.wQ6WBws0g3n9213 | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n | 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
e : N F 0 ≤ N F' 0
n₀ : ℕ
hF : ∀ n ≥ n₀, I • N F n = N F (n + 1)
⊢ ∀ (n : ℕ), N F (n + n₀) ≤ 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 n | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by
obtain ⟨n₀, hF⟩ := h
use n₀
| Mathlib.RingTheory.Filtration.238_0.wQ6WBws0g3n9213 | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n | 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
e : N F 0 ≤ N F' 0
n₀ : ℕ
hF : ∀ n ≥ n₀, I • N F n = N F (n + 1)
n : ℕ
⊢ N F (n + n₀) ≤ 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' n with n hn | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by
obtain ⟨n₀, hF⟩ := h
use n₀
intro n
| Mathlib.RingTheory.Filtration.238_0.wQ6WBws0g3n9213 | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n | Mathlib_RingTheory_Filtration |
case h.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
e : N F 0 ≤ N F' 0
n₀ : ℕ
hF : ∀ n ≥ n₀, I • N F n = N F (n + 1)
⊢ N F (Nat.zero + n₀) ≤ N F' Nat.zero | /-
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.antitone _).trans e | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by
obtain ⟨n₀, hF⟩ := h
use n₀
intro n
induction' n with n hn
· | Mathlib.RingTheory.Filtration.238_0.wQ6WBws0g3n9213 | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n | Mathlib_RingTheory_Filtration |
case h.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
e : N F 0 ≤ N F' 0
n₀ : ℕ
hF : ∀ n ≥ n₀, I • N F n = N F (n + 1)
⊢ 0 ≤ Nat.zero + 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... | simp | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by
obtain ⟨n₀, hF⟩ := h
use n₀
intro n
induction' n with n hn
· refine' (F.antitone _).trans e; | Mathlib.RingTheory.Filtration.238_0.wQ6WBws0g3n9213 | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n | Mathlib_RingTheory_Filtration |
case h.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
e : N F 0 ≤ N F' 0
n₀ : ℕ
hF : ∀ n ≥ n₀, I • N F n = N F (n + 1)
n : ℕ
hn : N F (n + n₀) ≤ N F' n
⊢ N F (Nat.succ n + n₀) ≤ N F' (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 [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← hF] | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by
obtain ⟨n₀, hF⟩ := h
use n₀
intro n
induction' n with n hn
· refine' (F.antitone _).trans e; simp
· | Mathlib.RingTheory.Filtration.238_0.wQ6WBws0g3n9213 | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n | Mathlib_RingTheory_Filtration |
case h.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
e : N F 0 ≤ N F' 0
n₀ : ℕ
hF : ∀ n ≥ n₀, I • N F n = N F (n + 1)
n : ℕ
hn : N F (n + n₀) ≤ N F' n
⊢ I • N F (n + n₀) ≤ N F' (n + 1)
case h.succ.a
R M : Type u
inst✝² : CommRing R
inst✝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 (Submodule.smul_mono_right hn).trans (F'.smul_le _) | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by
obtain ⟨n₀, hF⟩ := h
use n₀
intro n
induction' n with n hn
· refine' (F.antitone _).trans e; simp
· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← hF]
| Mathlib.RingTheory.Filtration.238_0.wQ6WBws0g3n9213 | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n | Mathlib_RingTheory_Filtration |
case h.succ.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
e : N F 0 ≤ N F' 0
n₀ : ℕ
hF : ∀ n ≥ n₀, I • N F n = N F (n + 1)
n : ℕ
hn : N F (n + n₀) ≤ N F' n
⊢ n + n₀ ≥ 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... | simp | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by
obtain ⟨n₀, hF⟩ := h
use n₀
intro n
induction' n with n hn
· refine' (F.antitone _).trans e; simp
· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← hF]
exact (Submodule.smul_mon... | Mathlib.RingTheory.Filtration.238_0.wQ6WBws0g3n9213 | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n | 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✝ h : Stable F
h' : Stable F'
e : N F 0 = N F' 0
⊢ ∃ n₀, ∀ (n : ℕ), N F (n + n₀) ≤ N F' n ∧ N F' (n + n₀) ≤ 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... | obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by
| Mathlib.RingTheory.Filtration.250_0.wQ6WBws0g3n9213 | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n | Mathlib_RingTheory_Filtration |
case 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✝ h : Stable F
h' : Stable F'
e : N F 0 = N F' 0
n₁ : ℕ
h₁ : ∀ (n : ℕ), N F (n + n₁) ≤ N F' n
⊢ ∃ n₀, ∀ (n : ℕ), N F (n + n₀) ≤ N F' n ∧ N F' (n + n₀) ≤ 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... | obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by
obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e)
| Mathlib.RingTheory.Filtration.250_0.wQ6WBws0g3n9213 | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n | Mathlib_RingTheory_Filtration |
case 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✝ h : Stable F
h' : Stable F'
e : N F 0 = N F' 0
n₁ : ℕ
h₁ : ∀ (n : ℕ), N F (n + n₁) ≤ N F' n
n₂ : ℕ
h₂ : ∀ (n : ℕ), N F' (n + n₂) ≤ N F n
⊢ ∃ n₀, ∀ (n : ℕ), N F (n + n₀) ≤ 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... | use max n₁ n₂ | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by
obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e)
obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm)
| Mathlib.RingTheory.Filtration.250_0.wQ6WBws0g3n9213 | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n | 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✝ h : Stable F
h' : Stable F'
e : N F 0 = N F' 0
n₁ : ℕ
h₁ : ∀ (n : ℕ), N F (n + n₁) ≤ N F' n
n₂ : ℕ
h₂ : ∀ (n : ℕ), N F' (n + n₂) ≤ N F n
⊢ ∀ (n : ℕ), N F (n + max n₁ n₂) ≤ N F' n ∧ N F' (n + max n₁ n₂... | /-
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 | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by
obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e)
obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm)
use max n₁ n₂
| Mathlib.RingTheory.Filtration.250_0.wQ6WBws0g3n9213 | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n | 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✝ h : Stable F
h' : Stable F'
e : N F 0 = N F' 0
n₁ : ℕ
h₁ : ∀ (n : ℕ), N F (n + n₁) ≤ N F' n
n₂ : ℕ
h₂ : ∀ (n : ℕ), N F' (n + n₂) ≤ N F n
n : ℕ
⊢ N F (n + max n₁ n₂) ≤ N F' n ∧ N F' (n + max n₁ n₂) ≤ 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' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by
obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e)
obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm)
use max n₁ n₂
intro n
| Mathlib.RingTheory.Filtration.250_0.wQ6WBws0g3n9213 | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n | Mathlib_RingTheory_Filtration |
case h.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✝ h : Stable F
h' : Stable F'
e : N F 0 = N F' 0
n₁ : ℕ
h₁ : ∀ (n : ℕ), N F (n + n₁) ≤ N F' n
n₂ : ℕ
h₂ : ∀ (n : ℕ), N F' (n + n₂) ≤ N F n
n : ℕ
⊢ n + n₁ ≤ n + max n₁ n₂ | /-
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 | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by
obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e)
obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm)
use max n₁ n₂
intro n
refine' ⟨(F.antitone _).... | Mathlib.RingTheory.Filtration.250_0.wQ6WBws0g3n9213 | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n | Mathlib_RingTheory_Filtration |
case h.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✝ h : Stable F
h' : Stable F'
e : N F 0 = N F' 0
n₁ : ℕ
h₁ : ∀ (n : ℕ), N F (n + n₁) ≤ N F' n
n₂ : ℕ
h₂ : ∀ (n : ℕ), N F' (n + n₂) ≤ N F n
n : ℕ
⊢ n + n₂ ≤ n + max n₁ n₂ | /-
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 | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by
obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e)
obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm)
use max n₁ n₂
intro n
refine' ⟨(F.antitone _).... | Mathlib.RingTheory.Filtration.250_0.wQ6WBws0g3n9213 | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n | 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
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... | rw [Subalgebra.smul_def, PolynomialModule.smul_apply] | /-- 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
| 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
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... | apply Submodule.sum_mem | /-- 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 |
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
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... | rintro ⟨j, k⟩ e | /-- 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 |
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... | rw [Finset.mem_antidiagonal] at e | /-- 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 |
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... | subst e | /-- 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 |
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